Citation
A mathematical model of singers' vibrato based on waveform analysis

Material Information

Title:
A mathematical model of singers' vibrato based on waveform analysis
Creator:
Diaz, Jose Antonio, 1967-
Publication Date:
Language:
English
Physical Description:
xxi, 234 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Acoustic spectra ( jstor )
Amplitude ( jstor )
Boxes ( jstor )
Parametric models ( jstor )
Signals ( jstor )
Singers ( jstor )
Software ( jstor )
Spectral methods ( jstor )
Vibrational spectra ( jstor )
Vibrato ( jstor )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 230-233).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jose A. Diaz.

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Source Institution:
University of Florida
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The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. §107) for non-profit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
Resource Identifier:
40074031 ( OCLC )
ocm40074031

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Full Text










A MATHEMATICAL MODEL OF SINGERS' VIBRATO
BASED ON WAVEFORM ANALYSIS
















By

JOSE ANTONIO DIAZ













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998




























Copyright 1998 by

Jose Antonio Diaz






























I want to dedicate this dissertation to my father, my mother and my sister. They encouraged me at all times and provided their unconditional support during the most difficult times. Their love and affection kept me going through the entire Ph.D. program.
I also desire to dedicate it to my relatives and friends who always keep me in their minds and prayers, and have kept in touch with me even though the distance between us is large.














ACKNOWLEDGMENTS



First, I want to acknowledge the importance of Dr. Antonio Arroyo's involvement in the success of my Ph.D. and this dissertation. He led me during the entire Ph.D. program, from the moment I entered into it until the end. He gave me the right advice and encouragement while I was taking classes, in preparing for the written and oral qualifying exams, and during my dissertation experimental phase. Dr. Arroyo provided me with the guidance and technical support needed to accomplish this research.

I also need to recognize the contribution of Dr. Howard Rothman to this project. He provided his expertise and knowledge of the musical field which were vital part in the success of this dissertation. His ideas and feedback played an important role in the development of the algorithms.

Finally, I give thanks to all the professors in my committee, the department of electrical and computer engineering, the department of communication sciences and disorders, all the people who were directly or indirectly involved in this dissertation during all its stages, and especially to God, the creator of all, and Jesus Christ, the way, the truth and the life.










iv















TABLE OF CONTENTS
page


ACKN OW LED GM EN TS ........................................................ ........................ iv

L IST O F T A B L E S ................................................................. ................................ xi

LIST OF FIGU RE S ............................................................... .. ...................... xii

A B STR A C T ....................................................... xx

CHAPTERS

1 INTRODUCTION ............................................... .... ................ 1
Importance of the Study of Vibrato ...... .... ................................................. 1
Organization of the Dissertation ........................................................ .................2

2 LITERATURE REVIEW AND OVERVIEW ................................... 4
The Singers' V ibrato .............................................................. 4
Effect of Vibrato on Pitch Perception ........... ........... .......................... 6
The Existence of Vibrato.............................................7
The Desirability of the Vibrato ................................. ................
Universality of use ................. ......................... 10
A utom atic nature....................................................................... .............. 10
Advantage over precision ....................................... 10
Im portance in tone quality ...................... ........... .. .................. ......... 10
Subjective Factors that Affect the Appreciation of the Vibrato ..................................... 11
The vibrato ear..................... ............................. 11
The emotionality of the individual .................. ................. ................... 11
Attitude and training.................. .................... ................ 11
The listener's disposition..................................... ... ....................... 11
Objective Factors that Determine the Quality of the Vibrato ............................. 12
E xtrem e extent ........................................................ 12
Vibrato in Different Types of Singing ...................... ............. ...... 13
Vibrato in Musical Instruments.......................... ........ ................ 13
Vibrato in the violin.............................................. 13
Vibrato in wind instruments ........................ .......... .......... 14
The Singers' Formant .................. .................................... ......... 14
R elated W o rk .......................................................................................... 15
V ibrato ........................................ 15


V









Measurement of the Vibrato Rate of Ten Singers ........... .......... 15
Frequency Modulation Characteristics of Sustained /a/ Sung in Vocal
Vibrato................................. ................ ......... 16
An Investigation of Vocal Vibrato for Synthesis ................. ................ 18
Acoustic and Psychoacoustic Aspects of Vocal Vibrato ................. ................. 20
Synthesis of Sung Vowels Using a Time-Domain Approach ..............................21
The FCSV Software ...................................................................... .... ......... .................. 22
Fundam ental C oncepts ............................................................................................. 25
The Discrete Fourier Transform ... ............. ...................... 25
Linear Prediction Coding ...................................... ...... .......... ................ 26
D efinitions ............................ .... ............................ ... 26
Relationship between the model parameters and the autocorrelation
function ............................................. 28
The autocorrelation method ...................................... 30
The covariance m ethod................................................... 30
M ultiple Signal Classification M ethod ...................................... .....................31

3 PROPOSED RESEARCH .......................................................... 33
V ibrato M odel ....................................................... 33
Frequency and Amplitude Vibrato Analysis .......................................................35
The Singers' Formant ................................. ........................... .... ..... 36

4 ALGORITHM DEVELOPMENT...................... ................ 37
Conversion of the FCSV Software (Frequency Characterization of Singers'
Vibrato) from Matlab 4.2 to 5.1 ............................................ ........... 37
Objective of the Functions Implemented up to this Point ......................... .......... 38
F ile, O pen W ave F ile ..........................................................................................38
File, Save W ave File .................. ................................... 38
File, Save Parameters ................................. ........................... ...... 39
File, Print Figure ................................................................. .............. 39
F ile E x it ................................................................. ................................... 3 9
Edit, Edit W ave File ........................................................... ................ 39
E dit, Play W ave File ....................................................... 39
Edit, Zoom in and out ............................................................... ................. 39
Edit, Options, Spectrogram ..................................... 39
Edit, O ptions, Elliptical Filter ............................................................................. 40
Edit, Options, Full Length Model ................. .............. 41
V iew F ilter R esponse ............................................................... ................. 42
A nalysis, Spectrogram ........................................................................ 43
A nalysis, G et V ibrato ................................................................ ................. 44
Analysis, Get Parameters .................................................................. ..... ... 44
F ilter, E lliptical ....................................................... 45
Filter, M edian ............................................................ 45
M odel, Full Length M odel .................................................. ............... ... 46
H e lp ......................................................................................... 4 8



vi









Relationship Between the Model Parameters in the Time-Domain and the ZD om ain .................... ..................................................... 48
Calculation of the Instantaneous Frequency and Amplitude of the Frequency
Vibrato W ave.................................... .... ........................ 52
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave ................. 60
Software Development ............................................ ........ 63
Edit, Options, Short Length Model .......... ......................... 66
V iew W indow ............................ ......................................... 66
Filter, Elliptical, Frequency Vibrato, Instantaneous Frequency and
Instantaneous Am plitude .................................. ........... .................. 67
Filter, Get Frequency, Power Spectrum and Instantaneous Power Spectrum....... 68 Model, Short Length, Frequency Vibrato, Instantaneous Frequency ................... 68
Model, Short Length, Frequency Vibrato, Instantaneous Amplitude ................... 68
M odel, Short Length, Amplitude Vibrato ........................................................... 69
Softw are Im provem ents ................................. ........................... .. ................ 69
V alid atio n ................................ .......................................................................... 7 0
Signal with no Modulation. .................................. .. ... ........ ....70
Signal with Frequency and Amplitude Modulation .......................................... 74

5 DATA ANALYSIS AND RESULTS ........................................................... 80
Sample Selection.......................... ............................... 80
Data Analysis ................... .............. .... ................... 81
Jussi Bjorling......................... ......... .......... ........... 84
File bjor5.w av ............................. .... ...................... 84
File bjorll.w av. ........................................... ................ 86
File bjor25.w av. .......................................................... 88
Enrico Carusso ......................... .......... ......... 90
File ec04 .w av ............................... .. ............................ .................. 90
F ile ec08.w av ................................................... ...........................92
File ecl6.w av .................. .................................. ........ 94
Luciano Pavarotti ...... ............................... ......... 96
File pav02.w av ....................... .................... .................................. 96
File pav03a.wav ............................................. 98
File pavl4.wav ................................ ........................... ... ...... 100
Placido Domingo.................................................. 102
File pldom01.wav ......................................................... 102
File pldom02.wav ................................................... ............... 104
File pldom24.wav ............................................... ............... 106
Kathleen Battle ............ .................... .......... 108
File kbat0l.wav .. ...... ............... .................. 108
File kbat20.w av .................................................................... 110
File kbat22 .w av ................................................................... 112
Monserrat Caballe ...... .................... ...... ........... 114
File moncab30.wav ............................................... 114
File moncab31.wav ............................ ................................ ...... 116



vii









File moncab33.wav.................... ................. 118
V ictoria D eLosAngeles ............................................................ 120
File delosa0l.wav .................................... ......... 120
File delosa07.w av ........................................................ .......... ........ 122
File delosa09.wav ......................................................... 124

6 VIBRATO MODEL .......... ................................. 126
Frequency Vibrato M odel .......................... ......................... ............... 126
Amplitude Vibrato M odel ........................................ 134

7 STATISTICAL ANALYSIS.................................. 139
M odel Param eters ....................................................... 139
Parameter Statistics....................... ................ 140

8 THE SINGERS' FORM ANT .......................................................... 147
Calculation of the Singers' Formant Parameters .................................. ...... 147
D ata A nalysis ................9..............................................
Jussi B jorling ...................................................... 149
Enrico Caruso ................................. ............................... ....... 150
L uciano Pavarotti ........................................................ 150
Placido D omingo.......................... ................... 151
M onserrat Caballe ................................... ....... .. .. ............................... 151
V ictoria D eL osA ngeles .................................................................................... 152
Singers' Formant Variation Among Six Different Singers ................................ 152
Singers' Formant Parameters for all the Samples ............................................... 153
Amplitude Study .......... ................................. 155
Frequency Study ................................ .... ....... .......... .... ....... 156
Singers' Formant Variation Within Samples of the Same Singer ............................. 157
Ju ssi B jo rling ................................................................ ................ 15 8
Enrico C aruso .................. ................................... 159
Luciano Pavarotti ......................... ............ ............................. 160
Placido Domingo................................................. 161
Monserrat Caballe ................................. ......... 162
Victoria DeLosAngeles ................................................. 163
Summary of Results ......... ............................... 164

9 CONCLUSIONS AND FUTURE WORK ...................................... 165
C o n clu sio n s ............................................ ...................... .. .............. 16 5
Frequency and Amplitude Vibrato Analysis ................................. ....... 165
Vibrato Model ....... ............ ............... .. ................ 166
Singers' Formant .......... ................................. 169
Future W ork ................................................................................................ ........ 170
A lgorithm s ................................................................................. .............. 170
Vibrato Model............ ................................. 171
Singers' Formant ........................................... 171



VIii









APPENDICES

A USER'S MANUAL........... ................................. 173
Software Installation ................ .................... .................. 173
O pening a V ibrato Sam ple ....................................................... 174
Analysis of the Frequency and Amplitude Vibrato Waves ............... ......... .. 178
Full L ength M odel ................................ ....... .. ................................................ 184
Instantaneous Frequency and Amplitude of the Frequency Vibrato Wave ............... 187
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave ............... 195
Saving the Vibrato Parameters ....................................................... 197
Editing the Vibrato Sample File........................................ ..... 198

B SOFTW ARE LISTIN G S ................................................................ 201
M-file mmsv.m .................................................................... ............. ................... 201
M-file ampfrl.m. ............... ................................ 201
M -file am pl4.m ..................................... .............. ....... ... ...... ................ 203
M -file auto .m ........................................................................................... 2 0 5
M -file covar.m .................................... ............................ ........ 205
M -file curvam l.m .................................................................... 206
M-file curve2.m ................ ......... ................. 208
M -file curvfrl .m .......................................... .... ............. ........ 209
M-file delfig.m .................. ......... .................... 210
M -file edit3.m ................................ ............................... .......... 210
M -file ellipam l.m ...................................................................... ................ ......... 211
M-file ellipfrl.m ............. ......... ................. 212
M-file ellipinstaml.m ................ ....... .................. 212
M-file ellipinstfrl.m ................................................................ ...... ........................... 213
M -file freqfrl.m ....................................................................... 214
M-file freqn4.m ................................. ....................... 215
M-file fullaml.m ............................ ........ ...................... 218
M -file fullfrl.m ................... ............................. .................................. 2 19
M -file getvall.m .................. .................................. 220
M -file instam l.m .................................................... ........................ 221
M -file instfr4.m ............................................................ ............... 222
M -file loadw av1.m .................. .................................... 223
M -file medaml.m ................................................................... ............................ 223
M -file medfrl.m ...... .................. ............................... 224
M -file savedat5.m ...................................................................... 225
M -file savew avl.m .......... ........... ........................................................ .......... 226
M -file spect3.m ....................................................... 226
M-file varinit.m ............... .................................. 227
M -file view polinst.m ................................................................... .......... 227
M -file view pol.m .................. ................................... 228
M -file view res.m ........................................................ 228
M-file zoominl.m ......................................................... 228
M -file zoom out l.m .................................................................................... 229


ix









LIST OF REFERENCES ............................................................... ......... 230

BIOGRAPHICAL SKETCH ................................................ ............... 234















LIST OF TABLES


Table page 4.1. Variable names and descriptions....................................71

5.1. Singers' names and file names. ............. .. ........................... 80

5.2. Filter frequency for the instantaneous amplitude waves. ......................................... 83

7.1. Description of the model parameters. ........................................ ................ 139

7.2. Model parameters and statistics .......................................... ........... 141

8.1. Singers' formant parameters. ............................................... 153

8.2. Relationship between groups and singers' names .............................................. 156

8.3. Singers' formant variation within samples of the same singer. ............................. 164





























xi















LIST OF FIGURES


Figure page 2.1. Exam ple of vibrato w ave ............................................................................... ...... 5

2.2. The singers' form ant. ................................ ... .......................... ................. 14

4.1. M ain screen. ............................... .............................. .......... 38

4.2. V ibrato sam ple ........................................................ 38

4.3. Spectrogram options .......................................................... ................ 40

4.4. Elliptical filter options ................................................................... .... ........ ................41

4.5. Full length model options ........................................................ 42

4.6. Filter frequency response. ............................................................. ................. 43

4.7. Exam ple of spectrogram ....................................................... 43

4.8. Example of frequency and amplitude vibrato waves ........................ .................. 44

4.9. Example of frequency and amplitude vibrato power spectra .................................. 45

4.10. LPC model of the frequency vibrato .............................................. ............ 46

4.11. Pole location of the frequency vibrato model ................................................. 47

4.12. Aspect of the computer's screen................................... ......... 48

4.13. Frequency and amplitude vibrato waves. .................................. 50

4.14. Pole location for the covariance method ................................... 51

4.15. Power spectrum for the covariance model ...................................51

4.16. Pole location for the autocorrelation method. ................. .................. 52

4.17. Power spectrum for the autocorrelation model. ................................. 52

4.18. Spectrogram of the frequency vibrato wave .............................. ................. 53




xii









4.19. Pole location for the spectrogram ...................................... 54

4.20. Instantaneous frequency vector. ................................... ..................... 55

4.21. Instantaneous frequency vector. ................................................ 55

4.22. Power spectrum of instantaneous frequency vector. ........................................ 56

4.23. Instantaneous frequency and amplitude vectors. ............... .... ................. 57

4.24. Spectrogram using the autocorrelation method..................................... 59

4.25. Instantaneous frequency and amplitude of the frequency vibrato wave ................... 59

4.26. Power spectra of the instantaneous waves. ............................................................ 60

4.27. Example of amplitude vibrato wave ............................................................... 61

4.28. Power spectrum of amplitude vibrato wave. .................................................. 61

4.29. Spectrogram of an amplitude vibrato wave ............................... ................ 62

4.30. Pole location for figure 4.29 ........................................................ .... 63

4.31. Options for the short length model. ................ ................. 67

4.32. V iew w indow option ...................................................... 67

4.33. Spectrogram of signal validl.wav .... ..............................................70

4.34. Frequency and amplitude vibrato waves. ......................................................72

4.35. Frequency and amplitude vibrato power spectra. .............. .. .................. 73

4.36. Spectrogram of signal valid3.wav. .................................................................. 74

4.37. Frequency and amplitude vibrato waves. ...................................................... 75

4.38. Frequency and amplitude vibrato power spectra .........................................76

4.39. Instantaneous frequency and amplitude of the frequency vibrato wave ................... 78

4.40. Power spectra of the instantaneous waves ................................. ..................... 78

5.1. Frequency and amplitude vibrato waves for bjor5.wav. ......................................... 84

5.2. Vibrato power spectra for bjor5.wav ................................. ............................ .... 84

5.3. Instantaneous waves for bjor5.wav ......................................... ............. 85

5.4. Power spectra of the instantaneous waves in bjor5.wav......................................... 85


XIll









5.5. Frequency and amplitude vibrato waves for bjorl 1.wav................... ..................... 86

5.6. Vibrato power spectra for bjorl 1.wav ........ ............................................ 86

5.7. Instantaneous waves for bjorl 1.wav ......................... ....................... 87

5.8. Power spectra of the instantaneous waves in bjorl 1 .wav .................................. 87

5.9. Frequency and amplitude vibrato waves for bjor25.wav..... ............... 88

5.10. Vibrato power spectra for bjor25.wav ................................... 88

5.11. Instantaneous waves for bjor25.wav ...................................... 89

5.12. Power spectra of the instantaneous waves in bjor25.wav. ................................. 89

5.13. Frequency and amplitude vibrato waves for ecO4.wav. ............... ..... .... 90

5.14. Vibrato power spectra for ec04.wav ................... ........................ .............. .. 90

5.15. Instantaneous w aves for ec04.w av. ................................................................. 91

5.16. Power spectra of the instantaneous waves in ec04.wav ..................... ........... 91

5.17. Frequency and amplitude vibrato waves for ec08.wav. ................................92

5.18. Vibrato power spectra for ec08.wav ................................ ................................ 92

5.19. Instantaneous w aves for ec08.w av. ....................................................................... 93

5.20. Power spectra of the instantaneous waves in ec08.wav...................................... 93

5.21. Frequency and amplitude vibrato waves for ecl 6.wav. ......................................... 94

5.22. Vibrato power spectra for ecl 6.wav ................................ .............................. .. 94

5.23. Instantaneous waves for ecl6.wav ................................... 95

5.24. Power spectra of the instantaneous waves in ecl 6.wav .................................. 95

5.25. Frequency and amplitude vibrato waves for pav02.wav. .................. ............... 96

5.26. Vibrato power spectra for pav02.wav ................... ........................................... 96

5.27. Instantaneous waves for pav02.wav. ............................................97

5.28. Power spectra of the instantaneous waves in pav02.wav .................................. 97

5.29. Frequency and amplitude vibrato waves for pav03a.wav ...... .................... 98

5.30. Vibrato power spectra for pav03a.wav .........................................................98


xiv









5.31. Instantaneous waves for pav03a.wav ................. ..........................................99

5.32. Power spectra of the instantaneous waves in pav03a.wav ................................... 99

5.33. Frequency and amplitude vibrato waves for pavl4.wav. ..................................... 100

5.34. Vibrato power spectra for pavl4.wav ................... .......................................... 100

5.35. Instantaneous waves for pavl4.w av. ........................................ 101

5.36. Power spectra of the instantaneous waves in pavl4.wav ..................................... 101

5.37. Frequency and amplitude vibrato waves for pldom01l.wav ................................ 102

5.38. Vibrato power spectra for pldom01l.wav. ............... ................ ................. 102

5.39. Instantaneous waves for pldom01.wav .................0................ ..... 103

5.40. Power spectra of the instantaneous waves in pldom01l.wav................ 103

5.41. Frequency and amplitude vibrato waves for pldom02.wav. ................................ 104

5.42. Vibrato power spectra for pldom02.wav. .................................................. 104

5.43. Instantaneous waves for pldom02.wav. .................................. 105

5.44. Power spectra of the instantaneous waves in pldom02.wav ................................. 105

5.45. Frequency and amplitude vibrato waves for pldom24.wav .................................. 106

5.46. Vibrato power spectra for pldom24.wav. ..................... ................. 106

5.47. Instantaneous waves for pldom24.wav. ........................................ 107

5.48. Power spectra of the instantaneous waves in pldom24.wav ............................... 107

5.49. Frequency and amplitude vibrato waves for kbat0l.wav ................................... 108

5.50. Vibrato power spectra for kbat0l.wav. .......................................... ................ 108

5.51. Instantaneous waves for kbat01.wav. ........................................ 109

5.52. Power spectra of the instantaneous waves in kbat01.wav .................. ................ 109

5.53. Frequency and amplitude vibrato waves for kbat20.wav ................................... 110

5.54. Vibrato power spectra for kbat20.wav. ........................................ ............. 110

5.55. Instantaneous waves for kbat20.wav. ................................ ...................... 111

5.56. Power spectra of the instantaneous waves in kbat20.wav .................. ................. 111


xv









5.57. Frequency and amplitude vibrato waves for kbat22.wav. ..................................... 112

5.58. Vibrato power spectra for kbat22.wav. .................................. 112

5.59. Instantaneous waves for kbat22.wav. ........................................ 113

5.60. Power spectra of the instantaneous waves in kbat22.wav .................................. 113

5.61. Frequency and amplitude vibrato waves for moncab30.wav .............................................. 114

5.62. Vibrato power spectra for moncab30.wav. ............................................... .. 114

5.63. Instantaneous waves for moncab30.wav ............................................................. 115

5.64. Power spectra of the instantaneous waves in moncab30.wav. ........................... 115

5.65. Frequency and amplitude vibrato waves for moncab31 .wav ................................. 116

5.66. Vibrato power spectra for moncab31 .wav. ............................................... .. 116

5.67. Instantaneous waves for moncab31.wav ................... .................... ............ .... 117

5.68. Power spectra of the instantaneous waves in moncab31 .wav. ........................... 117

5.69. Frequency and amplitude vibrato waves for moncab33.wav ................................. 118

5.70. Vibrato power spectra for moncab33.wav. ..................................................... 118

5.71. Instantaneous waves for moncab33.w av ............................................................ 119

5.73. Frequency and amplitude vibrato waves for delosa0l.wav ................................ 120

5.74. Vibrato power spectra for delosa01.wav. ................................................... 120

5.75. Instantaneous waves for delosa01.wav ............................................................... 121

5.76. Power spectra of the instantaneous waves in delosa0l.wav. ............................. 121

5.77. Frequency and amplitude vibrato waves for delosa07.wav ................................ 122

5.78. Vibrato power spectra for delosa07.wav. .................................. 122

5.79. Instantaneous waves for delosa07.wav. ......................................................... 123

5.80. Power spectra of the instantaneous waves in delosa07.wav. .............................. 123

5.81. Frequency and amplitude vibrato waves for delosa09.wav ................................ 124

5.82. Vibrato power spectra for delosa09.wav. .................................... 124

5.83. Instantaneous waves for delosa09.wav. ......................................................... 125


xvi









5.84. Power spectra of the instantaneous waves in delosa09.wav. ................................ 125

6.1. Pure sinusoidal synthesis ....................................... ............... 129

6.2. Error for pure sinusoidal m odel .......................................................... .......... 130

6.3. Synthesized wave using the proposed model. ............... ................... 130

6.4. Error for proposed model. ...................................................... 131

6.5. Pure sinusoidal synthesis. ......... .............................. 132

6.6. Error for pure sinusoidal model ............................................ 132

6.7. Synthesized wave using the proposed model. .................. ..................... 133

6.8. Error for proposed model. ..................................... 133

6.9. Synthesized amplitude vibrato wave ................................... 136

6.10. Error for amplitude model .............................. ................... .... ....... 137

6.11. Synthesized amplitude vibrato wave ................................. ................ 138

6.12. Error for amplitude model .......................................... 138

8.1. FFT and LPC power spectra ..................................................... 148

8.2. Singers' formant in sample bjor05.wav. .................................. 149

8.3. Singers' formant in sample ec04.wav ................................... 150

8.4. Singers' formant in sample pav02.wav. .................................. 150

8.5. Singers' formant in sample pldom01.wav ....................................................... 151

8.6. Singers' formant in sample moncab30.wav..................................... 151

8.7. Singers' formant in sample delosa01.wav ....................................................... 152

8.8. Analysis of variance table ................................................................ .............. 155

8.9 B ox plots. ................................................... 155

8.10. Analysis of variance table ................................................. 156

8.11. Box plots. ............................................... ................ 157

8.12. Box plots for the amplitude analysis of Bjorling ........................... 158

8.13. Box plots for the frequency analysis of Bjorling ............................................... 158


xvii









8.14. Box plots for the amplitude analysis of Caruso ................ ........................ 159

8.15. Box plots for the frequency analysis of Caruso ............................. 159

8.16. Box plots for the amplitude analysis of Pavarotti ................................... 160

8.17. Box plots for the frequency analysis of Pavarotti ................................... 160

8.18. Box plots for the amplitude analysis of Domingo .................. ................. 161

8.19. Box plots for the frequency analysis of Domingo ................................... 161

8.20. Box plots for the amplitude analysis of Caballe ................................................. 162

8.21. Box plots for the frequency analysis of Caballe .................................................... 162

8.22. Box plots for the amplitude analysis of DeLosAngeles ......................... .. 163

8.23. Box plots for the frequency analysis of DeLosAngeles.................................... 163

A 1. M M SV m ain screen. ................................. ..................... ............. .......... 174

A.2. Open wave file window ......................................................... 177

A .3. V ibrato sam ple ........................................................ 177

A.4. Example of a spectrogram ........................................ 178

A. 5. Input required window. ........................................ ........ .......... 179

A.6. Window for harmonic number input. ................................... 179

A.7. Spectrogram showing frequency vibrato wave .................................................. 180

A.8. Input required window. .............................. .................................... 180

A.9. Frequency and amplitude vibrato waves. ...................................... 182

A. 10. Frequency and amplitude vibrato power spectra. ................. ................. 182

A. 11. Aspect of the computer's screen. ................................. 183

A. 12. Matlab command window. ..................................................... 183

A 13. Full length m odel options. ............................................................ ..... 184

A. 14. Parametric model of the frequency vibrato wave ................................................ 185

A. 15. Pole location for full length model................. ............................................... 186

A. 16. Parametric model of the amplitude vibrato wave ................................................ 186


xviii









A 17. Elliptical filter options. ....................................................... 188

A. 18. Instantaneous power spectrum of the frequency vibrato ................................ 189

A. 19. Input required window ...................................... ................... .......... 189

A.20. Instantaneous frequency wave. ...................................... 190

A.21. Instantaneous frequency wave. ................................... ........... 190

A.22. Instantaneous frequency power spectrum .... ................. .................. 191

A.23. Pole location for the instantaneous power spectrum .................. ................... 191

A.24. Instantaneous power spectrum of the frequency vibrato ................................... 192

A.25. Instantaneous frequency wave. ................................. 193

A.26. Instantaneous amplitude wave. .................................. 193

A.27. Instantaneous frequency power spectrum ...................... ................. 194

A.28. Filtered instantaneous amplitude wave .................. ......................................... 195

A.29. Power spectrum of the filtered wave ........................... .................. 195

A .30. Short length model options ................................................ ......................... 196

A.3 1. Instantaneous power spectrum of the amplitude vibrato ............................... 197

A.32. Save parameters window ............................ ...................................... 198

A.33. Input required window ........................................................ 199

A.34. Edit wave window .................... .............. ................... 199

A .35. Save w ave file w indow ........................................................................ 200

















xix














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

A MATHEMATICAL MODEL OF SINGERS' VIBRATO BASED ON WAVEFORM ANALYSIS By

Jose Antonio Diaz

August 1998


Chairman: A. Antonio Arroyo Major Department: Electrical and Computer Engineering

The main objective of this dissertation is the development of a mathematical model of the singers' vibrato. This model describes the frequency and amplitude characteristics of good vibrato samples found in the singing voice.

A software tool was developed in Matlab 5.1 to analyze and extract the parameters of the singers' vibrato. The software creates a spectrogram of the sample set, from which the user can select different harmonics. The frequency and amplitude vibrato waves are extracted from the harmonic selected, and their power spectra and parameters are calculated. Then, the instantaneous frequency and amplitude of the frequency vibrato wave are generated, and their power spectra and parameters are calculated.

Sample sets were digitized and analyzed using the software developed. The results obtained show that the instantaneous frequency and amplitude of the frequency vibrato wave do not stay constant. The variations in each wave are shown as the sum of three


xx








sinusoidal components. The variations in the amplitude vibrato wave were caused by three factors: frequency variations, formants, and the amplitude of the glottal pulse. The variations due to the amplitude of the glottal pulse are shown as a single sine wave.

A new model was developed for the frequency and amplitude vibrato. The

frequency vibrato model is based on the sinusoidal model, but with the frequency and amplitude of the wave varying over time. These variations are in turn sinusoidal, having three sinusoidal components each. The amplitude vibrato model describes the variations due to frequency variations, formants and the amplitude of the glottal pulse. The variations due to the amplitude of the glottal pulse are modeled as the sum of two sine waves.

A study was made of the singers' formant. A software tool was developed to

measure the singers' formant frequency and amplitude. Sample sets were analyzed using software and the parameters calculated. A statistical analysis of the parameters showed that the frequency varied among the different singers. A comparison made among samples of the same singer showed that the amplitude varied in 4 out of 6 singers, and the frequency varied in 3 out of 6.




















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CHAPTER 1
INTRODUCTION


Importance of the Study of Vibrato


The vibrato (the frequency variation/modulation of tone patterns) is one of the elements that determines voice quality during singing. A tone produced with vibrato is more rich and beautiful than a constant tone. This is evidenced by the fact that all great classically trained singers, as well as popular ones, use vibrato a high percentage of the time. Although it may be unnoticed by individual listeners, the importance and effect in voice quality is unquestioned.

In spite of the significance of the vibrato in voice quality during singing, little effort has been made to investigate and analyze this phenomenon throughout the years. The most complete study of vibrato was completed in the 1930s by Seashore at the University of Iowa (Seashore, 1932, p. 1; Seashore, 1936, p. 1). His studies set the basic rules and principles for the study of vibrato which are still in use today. Few and isolated studies have been accomplished since.

Since the time of Seashore's studies a great number of improvements have been made in the equipment and techniques available for signal analysis. These techniques for the analysis of digital signals may reveal new information about vibrato.







1






2


Organization of the Dissertation


This dissertation has been divided into 9 chapters titled introduction, literature

review and overview, proposed research, algorithm development, data analysis and results, vibrato model, statistical analysis, the singers' formant, and conclusions and future work. There are also two appendices: user's manual, and software listings.

The literature review and overview chapter covers four topics that are of vital importance for the understanding of this dissertation: the singers' vibrato, related work, the FCSV (Frequency Characterization of the Singers' Vibrato) software, and fundamental concepts.

The objective of this dissertation is described in detail in the proposed research

chapter. It divides the present study into three main sections: vibrato model, frequency and amplitude vibrato analysis, and the singers' formant.

Chapter four describes the steps followed to develop and implement the

mathematical algorithms and the MMSV (Mathematical Model of the Singers' Vibrato) software. It contains a description of each of the software modules, and a section on the validation of the software results.

The criteria used to select the samples and the results obtained with the MMSV software are shown in the data analysis and results chapter. The results are presented in graphic form, and are then analyzed.

The process of developing the vibrato model is described in chapter six. Two models are presented: one for the frequency vibrato wave and one for the amplitude vibrato wave. The models are compared against the original waves and against a pure






3


sinusoidal model to verify their advantage over the pure sinusoidal model. An error measure is made to quantify the model's accuracy.

The statistical analysis chapter shows the parameters obtained with the MMSV

software for the samples selected, and their statistics. All the parameters used in the model are presented and analyzed.

The singers' formant chapter contains a complete study of the singers' formant

based on 6 singers. A technique for the measurement of the singers' formant is described, and used to extract the singers' formant parameters. Two studies are made: comparison of the singers' formant among different singers, and comparison of the singers' formant parameters among samples of the same singer.

The last chapter presents a summary of the contributions in this dissertation and

several applications, and also provides future directions and guidelines for the continuation of this research.

Appendix A contains the MMSV users' manual. The objective of this appendix is to give the user a step by step guide of the MMSV software to obtain the vibrato sample waves and parameters.

The Matlab code developed during the present research is presented in appendix B. The software, some vibrato samples, and the data files obtained are provided on a floppy disk.














CHAPTER 2
LITERATURE REVIEW AND OVERVIEW The Singers' Vibrato


The discussion that follows deals primarily with Western operatic vibrato.

Most professional opera singers acquire vibrato almost involuntarily without

pursuing or seeking it actively. It has been conjectured that vibrato develops by itself when the singer's training progresses satisfactorily.

This phenomenon is produced by a nearly sinusoidal variation of the vibration

frequency of the vocal folds. The degree in which a singer possesses a periodic vibrato is considered a measure of his voice quality. The singer's voice is generally considered better if he has a more periodic vibrato (Sundberg, 1987, p. 163).

Since the vibrato is a variation in the frequency of oscillation of the vocal folds, it has been characterized mainly by two parameters: the rate (frequency) and the extent (amplitude) of the variations (see figure 2.1). The vibrato rate specifies the number of cycles per second of the frequency oscillations. The extent specifies how wide the frequency changes from peak to peak during a cycle.

The vibrato rate is considered to be a constant for most singers. They are generally unable to alter it (Sundberg, 1987, p. 165). However, some singers can adapt their vibrato frequency at will. The range of typical vibrato frequencies is between 5.5 Hz and 7.5 Hz.





4






5


Vibrato rates below the lower limit are considered slow, and rates above the upper limit convey nervousness to the listener.



Frequency vibrato
3000 .


2950 2900 2850


2800 2750


2700
0 0.5 1 1.5 2 Time(s)


Figure 2.1. Example of vibrato wave.



The amplitude of the vibrato varies with the voice loudness. Its extent is +1 or +2 semitones in average. Vibrato extents smaller than 0.5 are not common for singers, and extents greater than 2 semitones are judged to be bad.

There are two other types of frequency modulation in singing which are similar to vibrato: tremolo and trillo. Tremolo can be distinguished by a more irregular and rapid oscillation. Its frequency is of 7 Hz or more. The trillo is characterized by a higher amplitude modulation, which is of +2 semitones or more.





6


As a result of the variation in the frequency of oscillation of the vocal folds, all the upper harmonics vary accordingly and in phase with the fundamental frequency. The frequency variation of all the harmonics is accompanied by an amplitude variation.

It can be inferred with relatively high confidence that the production of vibrato will produce a change in the overall amplitude of the sung tone. An increase in the fundamental frequency causes a shift in the strongest harmonic closer or farther from the first formant. Therefore, the amplitude of that harmonic will increase or decrease, and correspondingly change the overall amplitude.

Alternatively, the mean frequency of the strongest harmonic may occur at the same frequency as the formant. Consequently, the harmonic varies above and below the frequency of the first formant. In this case, the rate of the amplitude vibrato will be twice the rate of the frequency vibrato.

The main effect of the vibrato on the listener is caused by the frequency vibrato (Sundberg, 1987, p. 165). If the extent of the amplitude vibrato is small, which happens when the strongest harmonic is far from the formant, or if it is large, which happens when the strongest harmonic is closer to the formant, the effect on the listener is approximately the same.

Effect of Vibrato on Pitch Perception

Vibrato corresponds to a periodic variation in the fundamental frequency, while the fundamental frequency determines the pitch we hear. Therefore, this variation affects pitch perception. One might hypothesize that the pitch from a vibrato tone is not as accurately perceived as the pitch from a constant note. However, if this were true, the reason for





7


producing vibrato would be an increase in the accuracy in the singer's fundamental frequency, which is not the case.

In a study conducted by Sundberg (Sundberg, 1978, p.51) some singers were

asked to adjust the fundamental frequency of a constant tone to match the mean frequency of a vibrato tone. The results of this study revealed that the singers adjusted the frequency of the tone at the value of the linear average of the oscillating fundamental frequency with a small margin of error. It was also found that we really perceive the logarithmic average and not the linear average. In order to interpret the previous results correctly it is important to note that they are only valid when someone listens to a single isolated tone.

The vibrato is also detected in the EMG (electromyography) signal from the

laryngeal muscles. From this we may deduce that the presence, absence, or type of vibrato may provide an aural indication of the physical conditions of the singer's larynx. Research by Sundberg and Askenfelt (Sundberg and Askenfelt, 1983, p.307) indicates that there is a relationship between the absence of vibrato and phonatory problems. In order for the singer to produce an artistically satisfactory result it is necessary to perform difficult passages without apparent difficulty. Therefore, the singer is likely showing his healthy voice to the audience when he is able to produce vibrato in high or difficult notes. The Existence of Vibrato

Vibrato has been controversial from Mozart's time. There is confusion in the musical field about vibrato which can be attributed to the following reasons (Seashore, 1936, p.47):

1. Lack of knowledge about what vibrato really is.






8


2. Many people cannot hear the vibrato. This happens to individuals with
unmusical ears, and to musicians with skillful ears but with lack of training in
tone analysis.

3. The vibrato is generally perceived to a much smaller degree than its true
extent. A vibrato extent of 1 tone is always heard as a fifth of a tone or less.

4. The assumption that the elimination of the exaggerated types of vibrato
eliminate all vibrato. Most of the perceptions of a bad vibrato fail to distinguish
between tremolo and wobble.

5. Many musicians are able to discern tone quality but fail to judge vibrato.

Based on laboratory experiments, Seashore reached the following conclusions

regarding the existence of the frequency vibrato:

1. The vibrato is present in 95 percent of all the tones sung by professional
singers. This percentage includes both good and bad vibrato.

2. All types of tones, such as sustained tones, short tones, attacks, releases, and
other types of changes in pitch contain vibrato. These types of vibrato are hard
to hear, and can only be detected by a highly musical ear. From this, it can be concluded that the vibrato is not an enrichment forced on a portion of a note,
but under typical conditions it is a vibrant characteristic of the voice organ
throughout the entire performance.

3. Vibrato is present in all successful voice students, well trained amateurs and in
recognized artists. Primitive people display vibrato when singing with authentic
feeling.

4. The vibrato may start to develop at an early age at the time a child begins to
sing naturally and with true feeling. The percentage of cases of vibrato is
greater as the individuals grow older and get musical training.

5. People who confess to not using vibrato and who are opposed to its use, such
as great singers, teachers of voice, and voice students, employ it in their best
singing.

6. A person with the appropriate singing skills who has no vibrato may develop it
to a good level in a few lessons. It has been found in the untrained voice as an
irregular pulsation which by appropriate training and direction can be
converted into a good vibrato.






9


7. It can be difficult for well trained singers to produce songs or even a tone
without the use of vibrato. In this aspect a wide range of individual differences
can exist.

8. Vibrato is commonly found in emotional speech. There is almost no vibrato in
common speech, but for emotional conversation when sustained vowels
appear, vibrato tends to appear.

9. String players tend to use vibrato on all sustained tones. It is likely that all
professional violinists use vibrato in most of the sustained tones, except for
some specific effects.

10. Band or orchestral instruments can produce vibrato, but its use is usually
discouraged for woodwind or brass instruments in most cases. Probably, the
main cause for the low percentage of use is difficulty in controlling it properly
and easily, and not because it is not desirable.

11. Vibrato occurs in the sincere laughter of the adult and in the energetic crying
of a child. It is likely to be a characteristic of the neuromuscular organism
which appears under emotional situations.

In short, it can be said that a variation of tone in the form of a periodic increase and decrease in pitch is nearly universal in good singing, is commonly found in instruments, especially in string instruments, and is often encountered in emotional speech. The Desirability of the Vibrato

The fact that the vibrato is almost always present in good singing and is universally accepted in string instruments does not demonstrate that it is desirable. Moreover, not all types of vibrato are desirable.

The desirability of the vibrato is confirmed by the universality of its use, its involuntary nature, its presence in instruments, its preference over precision, and its importance in tone quality.






10


Universality of use

A very important fact is that all excellent singers and violinists, recognized as the best in musical accomplishment, display vibrato. Vibrato may change with time, but it is very likely to stay as a fundamental parameter of musical productions. Automatic nature

The fact that, in voice, the vibrato appears naturally whenever a person sings with authentic feeling indicates that the vibrato is a form of esthetic expression. This statement is reinforced by the fact that the vibrato can be found in primitive people who are completely untrained musically and are not conscious of the phenomenon, and by the fact that children begin to develop it at the time they start singing in propitious conditions for self-expression. Moreover, vibrato production is seldom taught to singing students. Most of the teaching deals with its modification or attenuation. Many of the best singers do not know how they develop vibrato.

Advantage over precision

One more proof of the exceptional musical value of the vibrato is its preference to accuracy and smoothness of tone in the playing of violinists, that is, the vibrato has a higher importance than the precision of the tone in the violin. Importance in tone quality

When appreciating the vibrato quality, it is not thought of as a periodic variation in pitch, but rather as a flexible, rich, and tender factor in tone quality, all of which are considered desirable.






11


Subjective Factors that Affect the Appreciation of the Vibrato The vibrato ear

There are ears that can detect differences in vibrato quality in the same way that there are ears for pitch, intensity, and time (Seashore, 1936, p. 147). The sharp vibrato ear detects in a higher degree the elements of elegance and beauty that are not appreciated by the less sensitive vibrato ear.

The emotionality of the individual

At the same level of importance as the ear is the natural and temperamental

response or emotionality of the individual. A person with a sharp vibrato ear may detect every aspect of the vibrato with the highest distinction and still be unable to experience the feeling that the artist communicates into it. The main reason for this is the inability of the listener to perceive emotions in music. This inability varies widely among different people. Attitude and training

The influence of attitude and training for perception and feeling in vibrato has been corroborated in a clear way by considering the widely varying attitudes toward vibrato. The listener's disposition

We are considering the vibrato esthetic, and esthetics is systematic and precise.

Consequently, when we consider norms of beauty for vibrato we must restrict ourselves to judgments made by listeners through a critical attitude and not through a musical attitude.






12


Objective Factors that Determine the Quality of the Vibrato Extreme extent

The most commonly found deficiency in vibrato is an extreme extent of frequency and amplitude (Seashore, 1936, p. 149). The excess may be found in frequency or amplitude, or both. An excessive extent produces a negative response in a refined musical ear. The following rules may be used safely for judging the vibrato in general (Seashore, 1936, p. 150). However, we should keep in mind that appreciation is a relative subject, and what is extreme for one person may not be for another, and when deciding what is excessive we must consider the particular desired purpose.

1. The best average extent of pitch and intensity is that which causes the desired
tone quality, but does not lead to the perception of changes in frequency.

2. Regularity in extent is remarkably important for good vibrato. A large and
irregular extent of the vibrato converts it to an unpleasant flutter. The extent of
the vibrato may be large or small, may increase or decrease, change within a single tone or in a sequence of tones, but the change must be progressive and
smooth.

3. A relatively small extent of the vibrato fails to contribute to the enhancement of
tone quality in an amount proportional to the smallness.

4. A slow rate makes the pitch change more evident. High rate produces a new
effect known as chatter.

5. In voice vibrato the variations in pitch should be dominant over the variations
in extent.

6. Artistic performance requires variation in extent and rate throughout a
presentation. A uniform vibrato becomes monotonous and fails to expose the
feelings of the performer.

7. In vocal and instrumental performances the artist has more freedom to utilize
the vibrato than he has when playing in an ensemble.





13


These rules are used in judging different types of vibrato (musical criticism), objective training, and esthetic theory.

In short, a good vibrato probably contains smooth transitions in rate and extent,

has a nearly sinusoidal shape, is adjusted by the singer to solo and ensemble performances, and is present in most tones and transitions except where it is not used for specific purposes.

Vibrato in Different Types of Singing

A study performed by Easley (Easley, 1932, p. 1) shows that in most cases the

frequency and amplitude of the frequency vibrato wave are higher in opera songs than in concert songs. Higher frequencies and amplitudes are also measured in opera songs compared to concert songs even when they are sung by the same singers.

When comparing historical and contemporary opera singers with historical and contemporary Jewish cantors, Rothman (Rothman, Diaz and Vincent, 1998, p.5) found most of the significant differences between eras, and not between groups within the same time period. The mean frequency of the frequency vibrato wave is 7 Hz for the historical group while the frequency of the contemporary group is slower. The amplitude variation of the frequency vibrato wave is higher for the contemporary group than in the historical group.

Vibrato in Musical Instruments

Vibrato in the violin

A study made by Rothman (Rothman and Arroyo, 1988, p.7) about the acoustic parameters in violin vibrato among pre and post World War II performers shows that the






14


frequency of the frequency vibrato wave increased slightly after World War II. The mean value post World war II is 6.74 Hz, and 6.41 Hz pre World War II. The amplitude variation of the frequency vibrato wave around the mean value is 0.98 semi-tones for the more recent group and 0.63 semi-tones for the older group. Vibrato in wind instruments

According to Hattwick (Hattwick, 1932, p. 1) a recording of 742 sec. of several

performances with different types of wind instruments shows that vibrato is present during 131 sec., that is, 18% of the time. The average amplitude of the frequency vibrato wave is

0.4 of a tone and the average frequency is 7 Hz. Hattwick noticed a tendency for the use of vibrato in brass instruments rather than in reed instruments. Vibrato was also found to a greater extent in jazz music than in the classics.

The Singers' Formant


A contrast of the spectrum of a vowel as it is spoken and sung shows that the amplitude is much higher between 2,500 and 3,500 Hz (see figure 2.2). This peak is typical of all vowels sung by professional male singers, and has been named the "singers' formant."


0

-20

-70
withotext' f


-90
Frequency (kHz)

Figure 2.2. The singers' formant (Sundberg, 1987, p. 118).






15


Related Work


Vibrato

The book Vibrato by DeJonckere, Minoru, and Sundberg (DeJonckere, Minoru, and Sundberg, 1995, p. 1), contains a collection of papers dealing with vibrato. These papers refer to all the different aspects of vibrato. They present vibrato from the physiological, physical, and psychological points of view.

The more relevant papers for the development of my dissertation are in chapters 2 and 7. Chapter 2 presents a paper by J. Sundberg called "Acoustic and Psychoacoustic Aspects of Vocal Vibrato." This paper is discussed later in this document. In chapter 7 we find a paper from E. Prame called "Measurement of the Vibrato Rate of Ten Singers," which is described below.

We find several studies about the frequency and extent of the frequency vibrato for different artists but none of them analyze the curve shape, nor are signal processing techniques like FFT, LPC, Burg, MUSIC, etc. applied to the curves for analysis. The amplitude vibrato is studied very briefly. No models for vibrato are presented. Measurement of the Vibrato Rate of Ten Singers

The main objective of this study by E. Prame (Prame, 1994, p. 1979) was the analysis of the vibrato rate. Three aspects of the vibrato rate were analyzed:

1. Intratone variability, which shows the variation of the vibrato rate within a
tone.

2. Intertone variability, with the objective of comparing vibrato rates among
different tones within a song.






16


3. Interartist variability, to compare the average vibrato rate among different
artists.

The material used in this study came from recordings made in a real musical

context. Ten different singers were used and 25 tones per singer were chosen for analysis.

The frequency measurements were obtained from spectrograms by measuring the time from one wave trough to the next.

There is a tendency in most of the tones analyzed to have a higher vibrato rate at the end, this is, during the last 1 to 5 periods. The rate increase averages about 15%. The intertone variation between the maximum and minimum for an artist was +8% of the artist's average. Each tone was calculated as a 3 cycle frequency average. The average vibrato frequency for all artists was 6.0 Hz, with a maximum of 6.4 Hz and a minimum of

5.5 Hz.

Frequency Modulation Characteristics of Sustained /a/ Sung in Vocal Vibrato

This is a paper by Yoshiyuki Horii (Horii, 1989, p. 1). Horii noticed that there are differences in the values reported in the literature for the amplitude modulations in singing vibrato. The values found vary from nonmeasurable to several decibels.

It is understood by most of the researchers that the curve shape for frequency vibrato is almost sinusoidal (Horii, 1989, p. 1).

Winckel (Winckel, 1953, p.252) considered nonsinusoidal vibrato curves as

characterizing poor vibrato. Programs used to synthesize the singing voice use sinusoidal patterns for singing vibrato because there are no good mathematical models for vibrato.

The main objective of this study, therefore, was to investigate modulation rates, extents, and rates of the fundamental frequency (Fo) increase and decrease for each





17


individual cycle at low, mid and high tone vibrato samples. The curve shapes and the cycle-to-cycle variations of the rate and extent were also analyzed.

In this study eight professional male singers were used as subjects. Each of them was recorded sustaining an /a/ with vibrato for at least three seconds at low, mid and high pitch levels. The analysis program was based on a peak picking method to detect the fundamental frequency.

For each of the modulation cycles the following variables were calculated:

modulation period, modulation extent, rate of Fo increase, rate of Fo decrease. Also, the following two sequencial measures were obtained: modulation jitter, and modulation shimmer.

The entire set of samples exhibited three different vibrato curve shapes: sinusoidal, triangular, and trapezoidal. From the whole set of samples only two of them exhibited a sinusoidal shape, five of them showed triangular shape, and ten of them trapezoidal shape. Asymmetry was found in the frequency vibrato, which was perceived in the form of a higher slope during the increase of the vibrato curve and a smaller slope during the decrease.

Singers' performance in the conditions of the present study in which they are not performing in front of a crowd and with an orchestra, and do not need to express emotions, may be different from normal singing conditions.

Research done in the past has found no relation between pitch level and vibrato rates, although, loud vibrato samples have been reported to produce higher vibrato rates than soft vibrato samples. However, the data collected in this study shows that there is less variation in vibrato rate than in vibrato extent.





18


Particular patterns shown by the vibrato samples were not associated with specific singers as suggested by Winckel (Winckel, 1953, p.252).

The triangular and trapezoidal curve shapes found in this research may be used in synthesis in order to produce realistic vibrato. Also, the different patterns and asymmetry in vibrato may have an effect on perception which has not been studied as yet. An Investigation of Vocal Vibrato for Synthesis

In their paper, Maher and Beauchamp (Maher and Beauchamp, 1990, p.219), state that vibrato plays an important role in voice quality, and therefore in singing synthesis methods. If a natural sounding effect is desired in synthesized singing, vibrato must be studied carefully.

In this study, spectrograms and a peak identification and tracking technique were used to calculate the harmonics. The synthesis method used decomposes the signal as a sum of sinusoids.

Most research indicates that the shape of the vocal tract remains fixed in vibrato

while the excitation from the vocal folds changes frequency in an almost sinusoidal manner (Bennet, 1981, p.34; Rossing, Sundberg and Ternstrom, 1987, p.830; Sundberg, 1977, p.257). However, some studies have emphasized the importance of random variations in vibrato (Bennet, 1981, p.34; Chowning, 1980, p.4).

Four singers were used as subjects in this study. Each of them was recorded singing the vowel la/ at a comfortable level in a range of pitches. All the subjects had formal singing training but none of them were professional singers.






19


The original samples were analyzed and several modifications were made to

remove frequency and/or amplitude variations. Then, the original and modified signals were judged and comparisons were made (Maher and Beauchamp, 1990, p.219).

According to the listeners judgment, the best quality was found in signals with frequency and amplitude modulations. Signals with only frequency variations were of secondary quality, followed by signals with only amplitude variations. Signals with neither amplitude nor frequency variations were found to have the poorest quality (Maher and Beauchamp, 1990, p.219).

It was noted that amplitude variations affect perception, and are important for good synthesis quality (Maher and Beauchamp, 1990, p.219).

Maher and Beauchamp reported that the frequency vibrato was nearly sinusoidal with some drift. Its amplitude, rate and fundamental frequency appear to contain random components. The following model was used to describe frequency vibrato (Maher and Beauchamp, 1990, p.219):

f,(t)= fo +d,(t) (2-1) d(t) = ar,(t) (2-2)

d, (t) = d(t) + Af (t) sin(O, (t) + q0) = frequency deviation (2-3) (t) = 2r fo [I + azr3(t)It = phase of fundamental (2-4) Af(t) = f [ + a2r2 (t)] = vibrato depth in Hz (2-5)


0,(t) = 27cf f,(t)dt (2-6)





20


The listeners did not show a preference toward vibrato with random components versus pure sinusoidal vibrato.

Further study of a descriptive model and of the effect of random variations in vibrato are suggested (Maher and Beauchamp, 1990, p.219). Also, there is a need for developing synthesis models with a more natural sound quality. Acoustic and Psychoacoustic Aspects of Vocal Vibrato

In this paper, Sundberg (Sundberg, 1994, p.45), reported that the vibrato frequency is not as constant for a particular singer as has been usually assumed. An investigation made by Prame (Prame, 1994, p. 1979) found that vibrato rates change within a single tone and also in individual tones in a performance.

The extent of vibrato is usually less than +1 semitones which correspond to a frequency change of about 6%.

The waveform of the frequency vibrato is similar to a sine wave. However, substantial deviations from this wave shape have been found as reported by SchultzCoulon (Schultz-Coulon, 1976, p.335; Schultz-Coulon, 1978, p. 142; Schultz-Coulon & Battmer, 1981, p.1).

There are three possible causes for the amplitude variation during vibrato. The first is a frequency variation, which causes the harmonics to move closer and farther from the formants. Another source of variation is the characteristics of the voice source in the voice organ, which may change during vibrato. A final source of variation are changes in the vocal tract shape, which determines the formants.

The voice source amplitude and the formant frequencies are not always constant during vibrato. In particular types of vibrato the voice source amplitude varies. In a






21


different type of vibrato the formant frequencies vary in synchrony with vibrato. This can be caused by movements of the tongue or the pharynx sidewalls.

The effect of amplitude vibrato on listeners has often been overestimated, but no formal investigation has been made. From results of experiments with synthesized singing, frequency variations and not amplitude variations cause the main perceptual effect on listeners.

A possible reason for the use of vibrato is to avoid beats (the addition/cancellation of sounds with slightly different frequencies). Beats are avoided if the notes in a consonant interval are sung with vibrato. Also, vibrato may help make the singer's voice more distinguishable when singing with a loud orchestra. A final incentive for the use of vibrato could be that it is used to show the audience that the singer is capable of solving difficult tasks without struggle (Sundberg, 1994, p.45). Synthesis of Sung Vowels Using a Time-Domain Approach

Titze (Titze, 1983, p.90) has used a time domain approach to synthesize sung vowels. In this study, the glottal volume velocity waveform was specified in the time domain by

g(t) = At" sin(7t / yT) for t < T (2-7) g(t) = 0 for t > T (2-8)

Titze modeled the effect of the vocal tract and the lip radiation using the articulatory model.

The amplitude and frequency of the glottal pulse were modulated according to

A= A(1+0.01a, sin(2yft)+O.O1Nr, r(t)) (2-9)





22


f = F( + 0.01a, sin(2ft) + O.O1Nr2(t)) (2-10)

Here, Ao and Fo represent the pulse fundamental amplitude and frequency, at and av correspond to percent amplitude and frequency modulation (called tremolo and vibrato in this study), ft and fv define the tremolo and vibrato frequency, Ns and Nj are the shimmer factor and the jitter factor, and ri(t) and r2(t) are pseudo-random functions.

From this model we can see that amplitude and frequency vibrato are modeled as a sinusoidal variation of the amplitude and frequency of the glottal pulse waveform. Jitter and shimmer are modeled as random variations in frequency and amplitude.

The inclusion of vibrato in singing samples has the important effect of transforming a poor synthetic speech quality into a fairly good singing quality.

For the same percentage of frequency and amplitude modulations, frequency modulations are perceived up to 10 times more effectively than amplitude variations.

The FCSV Software


The FCSV (frequency characterization of singers' vibrato) software was developed for use with Matlab (Diaz, 1995, p.56) for the extraction and analysis of singers' vibrato. It provides a fast and precise measure of the vibrato and its parameters, and gives us a signal representation of the vibrato wave.

The main menu consists of four options which naturally subdivide into four main functions:

1. Spec (Spectrogram).

2. Curv (Curve).

3. Calc (Calculations).





23


4. Per (Periodicity).

Spec reads the vibrato sample file, graphs the power spectrum vs. time

(Spectrogram), and stores all the variables in memory, which is used later when building a database. Curv gives the user the option of analyzing a (user specified) single harmonic. Calc extracts frequency and amplitude parameters over the user-specified harmonic chosen earlier in Curv. In addition, Calc displays the data on the screen and optionally saves the data in a text file. Per allows the examination of the most important components of the frequency and amplitude power spectra, and displays and plays back the original wave for comparison purposes.

Each of these four modules (composed of several functions) performs one of the main tasks in FCSV, which have been modularized in an effort to simplify the use of the software and reduce operator errors.

The Spec module is composed of two main functions. One generates the

Spectrogram while the other provides sound wave editing and play back. The module starts by generating the Spectrogram, then plays back the sound wave and asks the user to select the segment to be analyzed. In this way, the user is forced to check that the segment he wants to analyze contains vibrato. After editing (if necessary), the Spectrogram is regenerated displaying the selected portion while generating values for the variables.

The main function of the Curv module is the determination and display of the

vibrato pulse. First, Curv generates a vector containing the vibrato pulse of the harmonic selected, and then displays it on a Spectrogram plot, with the objective of comparing the calculated and the actual vibrato. If they match, the user can proceed to the next module.






24


If not, the user can retry the Curv option to recalculate the vibrato wave with another window size.

The software may fail to identify vibrato because of an inappropriate window size or a difficult vibrato wave. A difficult vibrato wave is one that has one or more of the following characteristics:

1. It is noisy (the recording is not good because of equipment noise, or instrument
or voice interference).

2. Small inter-harmonic distances, common with male voices (the distances
between harmonics are smaller than in the female voice) may mask the
harmonics.

3. High extent of the frequency vibrato, which occurs when the frequency change
of the harmonics is of such magnitude that the high frequencies of one
harmonic get close to the low frequencies of the next upper harmonic, masking
the effect.

The most important functions accomplished by the Calc module are the

calculation of the parameters of the frequency vibrato, the calculation of the parameters of the amplitude vibrato, the determination of the most important components of the frequency and amplitude power spectrum, and the option of saving the resulting data to the hard drive.

The last module is the periodicity analysis. It groups two different primary

functions: displaying of the most important frequency components of the frequency and amplitude vibrato, and plotting of the original wave.






25


Fundamental Concepts


The Discrete Fourier Transform

As in the case of continuous time periodic signals, if we have a sequence x(n) which is periodic with period N so that x(n) = x(n+rN) for any value of r where r is an integer, this signal can be represented by a Fourier series composed of a sum of complex exponentials at integer multiple frequencies of the fundamental frequency (2xt/N) of the signal x(n). These complex exponentials have the form: ek (n) = ej(2r/)kn" ek (n + rN) (2-11) where k is an integer. Thus, the Fourier series representation becomes (Oppenheim and Schafer, 1989, p.515)

x(n)= 1 X(k)e (2r/N)kn (2-12) Nk

The Fourier series representation for a discrete time signal with period N requires only N complex exponentials at integer multiple frequencies, while the Fourier series representation of a continuous time periodic signal usually requires infinite complex exponentials at integer multiple frequencies. This can be seen in equation 2-11 since the complex exponentials ek(n) are the same for values of k separated by N. This is ek (n)= ek+1 (n) (2-13)

where 1 is an integer. The N periodic complex exponentials eo(n), el(n),..., eNl1(n) constitute all the periodic complex exponentials that are integer multiples of 2xt/N. Therefore, the Fourier series representation of the periodic signal x(n) requires only N of these complex exponentials and has the form:






26


1 N-I
x(n)= X(k)e j(2,/N)kn (2-14) N k=0

The Fourier series coefficients X(K) can be obtained from x(n) by the relation:

N-I
X(k) = Z x(n)e j(2"N)k" (2-15)
n=O

The sequence X(k) in equation 2-15 is periodic with period N, that is, X(k) = X(N+k) for any integer k.

The Fourier series coefficients can be interpreted in two ways: as a sequence of finite length given by equation 2-15 for k = 0, 1,...., N-1, and zero otherwise, or as a periodic sequence valid for all N as given by equation 2-15. Both interpretations are valid since equation 2-14 only makes use of the values of X(k) for 0 < k < N-1.

Equations 2-14 and 2-15 constitute an analysis/synthesis pair and are usually referred to as the discrete Fourier series of a periodic sequence. Linear Prediction Coding

Definitions

Many discrete time random processes can be properly approximated by a time series or rational transfer function model. In this model, the input sequence u(n) and the output sequence x(n) being modeled are related by the following equation:

P q
x(n)= --a(k)x(n- k)+ _b(k)u(n-k) (2-16)
k=- k=O

This general lineal model is called an ARMA (auto regressive moving average) model. The driving noise u(n) in the ARMA model is not the observation noise usually






27


present in signal processing applications. Any observed noised should be modeled by the ARMA process by changing its parameters.

The transfer function H(z) between the input and output of the ARMA model are given by

B(z)
H(z) = (2-17) A(z)

where

q
B(z) = Ib(k)z-k = Z-transform of MA branch (2-18) k=O


A(z) =~ a(k)z-k =Z-transform of AR branch (2-19)
k=O

It is assumed that all the zeros of A(z) are inside the unit circle of the Z-plane. In this case H(z) is guaranteed to be a stable, causal filter. The parameters a(k) are called the autoregressive coefficients, and b(k) the moving average coefficients.

The Z-transform of the autocorrelation function at the output of a linear filter (Pxx(z)) is related to the autocorrelation at the input (P0u(z)) by the following equation:


P, (z)= H(z)H* (1/ z)P,,(z) z)B*(z* P,, (z) (2-20) A(z)A'(1/z')

When equation 2-20 is evaluated along the unit circle it becomes the power

spectral density (PSD) of P(f). Frequently u(n) is assumed to be a white noise process of zero mean and variance 02. The PSD of this type of noise is C2. Therefore, the PSD given by equation 2-20 becomes: PARA (f)= P(f) = B(f 2 (2-21)
(f)= 2 )f (





28


The specification of a(k), b(k) and 02 is equivalent to specifying the PSD of the signal x(n). It is assumed that a(O) = 1, and b(O) = 1, since the filter gain can be incorporated into a2.

If all the b(k) coefficients are assumed to be zero except b(O) = 1, equation 2-16 becomes


x(n)= -f a(k)x(n k) + u(n) (2-22)
k=1

which is an AR model of order p. This process is called an autoregressive process because the signal x(n) is a linear regression on itself For this kind of process the PSD is

2
PAR (f) = )2 (2-23) A(f)

This model is also called an all-pole model, and is frequently denoted as an AR(p) process.

Relationship between the model parameters and the autocorrelation function

A relationship between the ARMA model parameters and the autocorrelation

function of the signal x(n) can be found by taking the inverse Z-transform of equation 220, using the causality property of H(z), and doing some manipulation. After going through these steps we obtain the following results (Kay, 1988, p. 115):

p q-k
r,(k) a()r(k -l)+Z 2 h*(l)b(l+k) for k= 0, 1,...,q
l=1 1=0


r,(k)= a()r,(k -1) for k q+1 (2-24)
l=1

For the particular case of an AR process b(l) = 5(1). Applying this to equations 224, they become





29

p
r, (k)= a()r, (k -1) + 2h* (-k) (2-25)
1=1

Also,

h*(-k) = 0 for k > 0 (2-26) and


h*(0) = lim H(z) = 1 (2-27) Applying equations 2-26 and 2-27 to 2-25 we obtain

p
r. (k) =- a(l)r,(k- 1) fork> 1
l=1

P
r.(k) =- a()r,.(-1)+2 fork= 0 (2-28)
/=1

The set of equations 2-28 are called the Yule-Walker equations. They provide a

relationship between the AR model parameters and the autocorrelation function of x(n). If the autocorrelation values ofx(n) are known we can determine the AR coefficients by solving the following set of linear equations (Kay, 1988, p. 116):

r, (0) r (- 1) ... r,[-(p-1)] a(l) r(1)
r, (1) r, (0) .. r,[-(p-2)]j a(2) r, (2) (2-29)

r (p 1) r, (p 2) -. r, (O) La(p) r, ( p)

Equations 2-29 can be arranged to incorporate the second equation in 2-28. This results in (Kay, 1988, p.116)

r, (0) r.(-1) ... r,(-p) i 1
r. (1) r, (0) ... r,[-(p-1)] a(1) 0
(2-30)

r. (p) r (P) r (0) a(p) _0






30


The autocorrelation method

In the autocorrelation method the AR parameters are determined by minimizing an estimate of the prediction error power:

2

1k=1

It is assumed that x(O), x(1),...,x(N-1) are known. The samples of x(n) outside the 0, 1,...,N- 1 range are equal to 0 in 2-31. After minimizing and manipulating 2-31 we get the following set of equations in matrix form (Kay, 1988, p.221):

i. (0) (-1) [-(p-1)] i(l) -?(1)
(1) i (0) - i [-(p-2)] c(2) i (2) (2-32)


(,(p-1) i (p-2) (0) La(p)- L,(p) where

1 N-1-k
F,(k) = N x*(n)x(n + k) for k = 0, 1,...,p
N n=O

r,(k) = F (-k) for k = -(p-1), -(p-2),...,-1 (2-33)

Equations 2-33 are known as the biased autocorrelation function estimator. The estimate of the white noise variance can be obtained from the estimate of pmin by using

p
2 = ,(0) + Z (k) (-k) (2-34)
k=1

The covariance method

The parameters for the covariance method can be found by minimizing the prediction error power:






31


1 N-1 P 2
= x(n) + Ia(k)x(n- k) (2-35) N n=p k=1

The prediction error power for the covariance method and the autocorrelation method are identical except for the range of summation, which is from p to N- for the covariance method. In this method all the data points needed to calculate the estimate of pi are known.

After minimizing and manipulating 2-35, we get the following set of equations:

c (1,1) c, (1,2) --- c,(l,p) l ) (1) c,(1,0) c, (2,1) c, (2,2) c, (2, p) a(2) c, (2,0)
= (2-36) c, (p,1) c, (p,2) -.. c,(p,p) La(p) c, (p, p) where

1 N-1
c,(j,k)= x*(n-j)x(n-k) (2-37) N- p ,=p

The estimate of the white noise variance can be found by using

P
S= = c (0,0) + a(k)c, (0,k) (2-38)
k=1

The matrix in equation 2-36 will be singular if the data consist of less than p complex sinusoids, however, any observation noise will make the matrix nonsingular. Multiple Signal Classification Method

The multiple signal classification (MUSIC) method is based on the orthogonality property (Kay, 1988, p.424):

M
eIH "aiv, = 0 i=, 2,....p (2-39) j=p+l






32


where {el, e2,....,ep}are the noise vectors and {vi, v2,....,vp}are the signal vectors.

The signal vectors are orthogonal to all the vectors in the subspace formed by the noise vectors. The frequencies of the sinusoidal components in the signal are calculated as the peaks of the spectral estimator:

1
MUSIC w E (2-40) P c rE, *r (2-40)
Noise fNoise

where ENoise is the matrix of noise subspace eigenvectors obtained from the signal autocorrelation matrix.

1
eJwl
= ejw2

-jw(N1)


M = Number of complex sinusoids in the signal

N > M+1

In theory, when w = wi (the ith sinusoidal frequency) MUSIC -->

Due to estimation errors the frequencies of the peaks given by the MUSIC estimator will be at or near the true frequency values.














CHAPTER 3
PROPOSED RESEARCH


Vibrato Model


Several models have been proposed and used for synthesizing vibrato (Maher, and Beauchamp, 1990, p. 219; Titze, 1983, p.90), however, they all have been developed by applying the analysis by synthesis method. Synthesized voice samples are created with different vibrato characteristics, the synthesized data is judged, and the results analyzed to select the best representation for vibrato. No formal analysis has been made on the effects of the vibrato waveform on perception. Previous investigations have analyzed the vibrato in a simple way, assuming that the waveform is sinusoidal and the parameters to be measured are frequency and amplitude in every cycle. Also, none of the more recent high resolution techniques like LPC, Burg, MUSIC, etc. have been applied to the vibrato signal (see chapter 2 for the most relevant papers published lately in this area).

There is also controversy regarding the effect of the random patterns in vibrato. Maher (Maher, and Beauchamp, 1990, p.219) found that there was no preference toward vibrato with random components versus pure sinusoidal vibrato, while Titze (Titze, 1983, p.90) includes random patterns in his vibrato model.

The results obtained from FCSV in conjunction with available data will be used to produce a mathematical model for good vibrato. This new work will make use of my master's thesis work and will build upon it.



33






34


There are several groups of data available for analysis, the largest being 12-bit samples. The total number of 12-bit samples is 574, from which 163 were used for analysis in my master's thesis since they contained good vibrato samples (Diaz, 1995, p.78). There is also a substantial amount of 16-bit data in .wav format which has been collected in the department of Communication Sciences and Disorders at the University of Florida.

The characteristics of a good vibrato can be obtained by applying FCSV to the data, using perceptual judgments, and correlating the perceptual judgments with the curves and parameters. There is evidence (Horii, 1989, p.1) that a good vibrato is produced by a symmetric frequency vibrato curve, but there are no clear results about the effects of the amplitude vibrato on perception, perhaps because it has not been studied in depth. The perceptual judgments will be provided by Dr. Rothman, a professor in communication sciences, who has a lot of experience in this area and is involved in this Ph.D. thesis as a committee member.

A mathematical model for good vibrato will be developed using the software, data and perceptual judgments. The software will provide parameters for the vibrato samples. The good vibrato samples will be identified using the perceptual judgments and a model that fits them will be developed. This model will describe the frequency and amplitude characteristics of good vibrato. The process of finding the right model will involve the testing of several models. Due to the sinusoidal shape of the frequency vibrato curve, an all poles model should fit properly. The amplitude vibrato curve is more complicated and will require a more detailed analysis of the model to be used. The errors of each model will





35


be calculated to make sure that a low error model is found, which describes good vibrato properly.

Frequency and Amplitude Vibrato Analysis


The frequency and amplitude curves obtained with FCSV are analyzed by the

software using FFT's. It was suggested in my master's thesis that more information can be obtained from these curves by using other methods, e.g., Burg or LPC. These methods can be used to improve the frequency resolution, and increase the precision of the results. Arroyo & Rothman did some preliminary work in this area in 1987-89.

The frequency vibrato curve possesses a sinusoidal-like shape. This curve can be properly represented by an LPC model. Also, for poor vibrato the frequency and amplitude curves change from cycle to cycle. These changes can be quantified by analyzing a small segment of the frequency vibrato using LPC and sliding the window along the time axis until the entire sample has been analyzed. This will show the instantaneous frequency and extent characteristics of the frequency vibrato with high resolution. This technique has not been used in the past and constitutes a new way to analyze the vibrato curves. The same technique can be applied to the amplitude vibrato to produce the instantaneous frequency and extent characteristics, but the model used should be of a higher order to represent the curve properly since the amplitude vibrato curve is more complex. The results obtained will help in the development of the mathematical model for good vibrato.






36


The Singers' Formant


All the samples analyzed during my master's thesis showed the so-called singers' formant. However, this may not always be the case. Some samples used in different studies at the Communication Processes and Disorders at the University Florida show that the singers' formant is not always present in singing samples.

The singers' formant can be seen in FCSV as an increase in the amplitude of the harmonics at about 2500-3500 Hz. The amplitude of the singers' formant in the samples analyzed in my masters' thesis was approximately the same as that of the first formant.

A study can be conducted with the data already available and the use of FCSV or any other adequate software to determine if the singer's formant varies within different samples of the same singer and among different singers.

The variables to be measured will be the frequency and amplitude of the singer's formant for each sample. The study will be based on at least 25 singers and 4 samples per singer. Statistical methods will be applied to determine significant differences between samples and singers.














CHAPTER 4
ALGORITHM DEVELOPMENT


Conversion of the FCSV Software (Frequency Characterization of Singers' Vibrato) from Matlab 4.2 to 5.1


This study began by converting each of the m-files in the FCSV software from Matlab 4.2 to 5.1 since FCSV calculates the frequency and amplitude vibrato waves that are going to be used for the analysis and determination of the vibrato model.

At the same time that each m-file was being converted, the user interface was

being changed to make it more flexible, and provide more information simultaneously on the screen. The added flexibility was provided by separating some of the functions that were performed by the activation of a single button, into two or more options.

The software was modified from button-driven to menu-driven, which improves the screen utilization by allowing more space for figures. Also, parameters that were fixed in the past, can now be changed through sliders and pop-up menus.

Figure 4.1 shows the window that appears when the software is activated. This

window contains axes for the display of the original vibrato wave, and a menu bar with all the choices available through the software. All the functions are selected and controlled through the use of this menu.







37






38



File Edit View Analysis Filter Model Help

1


0
0 0.2 0.4 0.6 0.8 1 Figure 4. 1. Main screen.




Objective of the Functions Implemented up to this Point File, Open Wave File

The objective of this function is to load the original vibrato wave and show it on

the screen. It also initializes the variables. Figure 4.2 shows how the sample is presented in this window.





File Edit View Analysis Filter Model x 104 VIbrato sample
5



0 0.5 1 1.5 File: Pldomi .wav Time(s) Figure 4.2. Vibrato sample.



File, Save Wave File

This function allows the vibrato sample to be saved. It is useful for saving the .wav file after changes have been made using the edit option.






39


File, Save Parameters

This function is used to save the vibrato parameters on disk after the analysis has been performed.

File, Print Figure

This option allows the user to print the original vibrato wave. The other figures can be printed by using the print option on each figure's menu. File, Exit

By selecting this option all figures are closed and the program finishes execution. Edit, Edit Wave File

This function allows the vibrato sample to be edited to remove non vibrato

segments. The user indicates the segment to be removed on the spectrogram figure, since this figure provides the best visual information about the vibrato sample. Edit, Play Wave File

This function allows the user to listen to the vibrato sample. Edit, Zoom in and out

These two options allow the user to zoom in and out of the vibrato sample figure to look at the details of the original wave. Edit, Options, Spectrogram

By choosing this option the user can adjust several parameters to be used for

calculating the spectrogram. Figure 4.3 shows the window that appears when this option is selected.






40



File Edit Window Help

0 a 10000 Min Freq J
1000 4000 11100 Max Freq i J


Window Hanning 1 Acce cancelI

Figure 4.3. Spectrogram options.



The available options are

1. Minimum frequency. This represents the minimum frequency displayed in the
spectrogram. The valid values are from 0 to 10,000 Hz. The callback function checks the value of the maximum frequency and increases it if it is lower than
the minimum frequency or higher, but less than 1000 Hz. This avoids user
errors.

2. Maximum frequency. This represents the maximum frequency displayed in the
spectrogram. The valid values are from 1000 to 11,000 Hz. The callback
function checks the value of the minimum frequency and decreases it if it is
higher than the maximum frequency or lower, but less than 1000 Hz. This
avoids user errors.

3. Window. This option allows the user to change the window used to calculate
the spectrogram. The choices available are: Blackman, Chebyshev, Hamming,
Hanning, Kaiser, and rectangular. Edit, Options, Elliptical Filter

The user can change some parameters of the elliptical filter by using this option. Figure 4.4 shows the window displayed when this option is selected.






41



File Edit Window Help

0 20 40 0 20 100



Filter te Low pa ss Accept Cancel

Figure 4.4. Elliptical filter options.



The parameters that can be adjusted are

1. Cutoff frequency. This is the frequency of transition for the filter. The range of
values is from 0 to 40 Hz.

2. Attenuation. This parameter represents the filter attenuation in the rejection
band. The range of values is from 0 to 100 db.

3. Type. The user can select between low pass and high pass filters using this
pop-up menu.

Edit, Options, Full Length Model

This option allows the selection of the parametric model to be used in the full

length model, and the adjusting of some of its parameters. Figure 4.5 shows the window by which the changes can be made.






42



File Edit Window Help

1 1 41
Dec factor

1 3 21

Poles -j I

Mehod Covce

Accept Cancel

Figure 4.5. Full length model options.



The parameters that can be adjusted are

1. Decimation factor. The decimation factor allows a reduction of the sampling
rate of the vibrato sample. This option was added to the software since in
experiments previously performed with synthetic vibrato waves applying the
LPC model by covariance method, it was noticed that the method had
difficulties when modelling frequency components which are close to DC
value. The only way to move the poles away from the unit circle is by decimation. After decimation, the LPC method detects more easily the
frequency components that were initially close to DC. The Matlab algorithm
for decimation performs filtering before decimation to avoid aliasing. The
range of valid values is from 1 to 41.

2. Poles. This function allows the user to adjust the number of poles used in the
full length model. The range of values is from 1 to 21 poles.

3. Method. The user can select the analysis method for the full length model. The
options are LPC by autocorrelation, LPC by covariance, and MUSIC (Multiple
Signal Classification).

View, Filter Response

The objective of this function is to display the elliptical filter frequency response. In this way the user can see the frequency response of the filter and make changes if needed. Figure 4.6 shows the frequency response of the default filter.






43






File Edit Window Help
Filter frequency response


10




10-4
0 10 20 30 40 Close

Figure 4.6. Filter frequency response.



Analysis, Spectrogram

This function calculates and displays the vibrato sample spectrogram. A color bar was added to show the amplitude scale. An example is shown in figure 4.7.





File Edit Window Help Spectrogram 4000 120

3000 60


40
LL 1000

0

0 0.5 1 1.5 Time(s)

Figure 4.7. Example of spectrogram.






44


Analysis, Get Vibrato

This option allows the user to select one of the harmonics to be analyzed. The

algorithm calculates the frequency and amplitude vibrato waves, and displays the vibrato wave on the spectrogram for comparison purposes. Analysis, Get Parameters

The user can display the frequency and amplitude vibrato waves with their

respective power spectra by choosing this option. It also calculates the vibrato parameters. The parameters are printed in the command window, the frequency and amplitude vibrato waves are displayed in a new window as is shown in figure 4.8, and the power spectra are displayed as shown in figure 4.9.





File Edit Window Help Frequency and amplitude vibrato 200 4



._ 100 I -2
E 2
-200 -4
0 0.5 1 1.5 Blue=Freq; Red=Amp Time(s)

Figure 4.8. Example of frequency and amplitude vibrato waves.






45



File Edit Window Help
Freq and amp vibrato power spectrum 8000 80 6000 60 4000 40 E 2000 20
0 0 0 10 20 30 40 Blue=Freq; Red=Amp Frequenc(Hz) Figure 4.9. Example of frequency and amplitude vibrato power spectra.



Filter, Elliptical

This function allows the frequency and/or amplitude vibrato waves to be filtered with an elliptical filter. This is a new option in the software, which was included for the following reasons:

1. The discrete nature of the FFT introduces some sharp discontinuities which are
not present in the real wave. These discontinuities introduce high frequency
components that can be removed using a low pass filter.

2. It is known that for the frequency vibrato wave there are no frequency
components above 10 Hz, and for the amplitude vibrato there no components above 20 Hz. Therefore, a low pass filter can be used to reduce the noise level
and improve the performance of the LPC method.

3. Unwanted frequency components can be removed before applying the full
length model.

Filter, Median

The user can apply a median filter to the frequency and/or amplitude vibrato by

using this option. This gives the user the choice of a different type of filter to remove rapid variations not present in the real wave.






46


Model, Full Length Model

This option allows the application of a parametric model to the frequency and/or amplitude vibrato waves. The frequency response of the model is plotted in the same figure of the FFT power spectrum to compare how well the model matches the FFT power spectrum (see figure 4.10). Different colors are used to facilitate the reading. A green curve represents the frequency vibrato model and a pink curve the amplitude vibrato model. The frequency and amplitude of the highest peak and the error in the frequency measure are calculated, and the pole locations are shown in a separate window (see figure

4.11). The error in the frequency measure is 0.7 percent, or lower when decimation is performed, compared to 5 percent with the FFT power spectrum. This represents a reduction of almost 90 percent.





File Edit Window Help
Freq and amp vibrato power spectrum 6000 80 400 160

-40
S2000 20 00
0 10 20 30 40 Blue=Freq; Red=Amp Frequency(Hz)

Figure 4.10. LPC model of the frequency vibrato.






47



File Edit Window Help
Pole-zero plot





-0.5
-- -- -- -- - - - - -
E

-2 -1 0 1 2 Real part Cose


Figure 4.11. Pole location of the frequency vibrato model.



The covariance method was implemented first since in an experiment this investigator performed with a synthetic vibrato wave, the covariance method outperformed the autocorrelation and MUSIC methods in calculating the frequency of the wave.

It was noticed from the experiments conducted that the LPC method works better after the wave has been filtered with the elliptical filter. This agrees with the literature since it has been reported that the performance of the LPC method is better for low noise level signals. Also, the LPC method has some difficulties with signals of relative low frequency, that is, signals whose poles are close to DC. This effect can be counteracted by shifting the poles away from the DC value, which can be accomplished by decimating the signal with the option provided for it.

The location and sizes of the windows were changed so that 4 four windows can be presented at the same time on the screen. This allows an easy comparison of the different representations of the vibrato wave (see figure 4.12).






48






File Edit View Anays: Filer Model File Edit window Help S104 Vibrato sample Frequency and amplitude vibrato
15 200 4 SP 10 Iio 2
5 0
0 05 1 15
File: Pldom way Time(s) -100 -2

a -20 -4 File EdA W dow Help 0 0,5 1 1.5 Spectrogram Blue=Freq, Red=Amp Time(s)
4000 120 X.x 0Fe Edit Window Help
q30100 Freq and amp vibrato power spectrum 80 6000 80 S2000 60 40 60 S4000
2 -40
1000ooo 20 2000 E20
0 < 0
0 0.5 1 15 0 10 20 30 40
Times) Blue=Freq; Red=Amp Frequency(Hz) !Start y Micros MATL Figuie f Figure. R Figuee [ Figw. $K a 1PM Figure 4.12. Aspect of the computer's screen.



Help


This function displays a windows that tells the user where to find information

about this software, and who developed it.


Relationship Between the Model Parameters in the Time-Domain and the Z-Domain


Although the model in the Z-domain provides a unique representation of the

frequency or amplitude vibrato, the parameters in the Z-domain do not provide much

information to people not familiar with the Z-transform. Therefore, it was decided to





49


provide parameters in the time domain which would be obtained from the Z-domain parameters.

As the frequency vibrato is almost sinusoidal, it can be properly represented by a sine-wave. The frequency of the sine wave can be easily obtained from the frequency of the poles in the Z-domain, but the amplitude calculation is not trivial and no information was found on how to determine it. Therefore, research was undertaken as part of this study to find the relationship between the Z-domain model and the amplitude of the sine wave in the time domain.

First, It was noticed that there was an error in the calculation of-the gain by the LPC function in Matlab. After contacting Matlab, they provided the fix. However, the model did not match the FFT in all cases. For poles located near the unit circle, the model and the FFT did not match. In all other cases they did match.

The model and the FFT matched in all cases when the number of points used to calculate the power spectrum using the model was equal to the number of samples in the signal. Therefore, the gain of the signal had to be adjusted to compensate for this. If the minimum square error is multiplied by the number of samples used to calculate the power spectrum, instead of the length of the signal, both spectra match in all cases.

In order to calculate the amplitude of the sine wave in the time domain, the area of the peak in the power spectrum has to be calculated. This can be done by adding the vectors used to obtain the power spectrum. This summation has to be corrected by two factors in order to obtain the amplitude. It has to be multiplied by two in order to compensate for the other half of the spectrum in the frequency domain, and divided by the






50


number of samples used to calculate the power spectrum as if we were doing an inverse Fourier transform.

The following is an example to show how the method works. Figure 4.13 shows a frequency vibrato wave from Placido Domingo. The frequency vibrato wave is shown in blue. The wave was filtered with a 10 Hz low-pass filter to eliminate noise and improve the results. Figure 4.14 shows the pole location given by the Covariance method, and figure 4.15 shows the corresponding power spectra. The FFT power spectrum is shown in blue and the LPC model spectrum in green. By applying the method described, an amplitude of 114.049 Hz was obtained from the LPC model spectrum, which agrees with the signal shown in figure 4.13.





Fe dit Window Help
Frequency and amplitude vibrato 200 4






-200 -4 0 0


-200 -4
0 05 1 1.5
Blue=Freq; Red=Amp Time(s) Figure 4.13. Frequency and amplitude vibrato waves.






51



File Edit Window Help Pole-zero plot
1


(U 0 - --- -- -- -. -0.5
-1
-2 1 0 1 2 Real part Close

Figure 4.14. Pole location for the covariance method.




File Edit Window Help Freq and amp vibrato power spectra
6000 fl 60
-4000
40

E 20

0 10 20 30 40 Blue=Freq, Red=Amp Frequency(Hz) Figure 4.15. Power spectrum for the covariance model.



Figure 4.16 shows the pole location calculated by the autocorrelation method, and figure 4.17 shows the corresponding power spectra. The FFT power spectrum is shown in blue and the LPC spectrum in green. Using the method described an amplitude of 88.380 Hz was obtained from the LPC model spectrum, which also agrees with the signal in figure 4.13.






52



File Edit Window Help
Pole-zero plot
~a + O.


(U 0 -------------------D-----5 -0.5
-1
-2 -1 0 1 2 Real part Close


Figure 4.16. Pole location for the autocorrelation method.




File Edit Window Help Freq and amp vibrato power spectra 6000 80

60
40
a 2000
0 0

0 10 20 30 40 Blue=Freq; Red=Amp FrequencV(HZ) Figure 4.17. Power spectrum for the autocorrelation model.



Calculation of the Instantaneous Frequency and Amplitude of the Frequency Vibrato Wave


An easy to way to determine the symmetry of the vibrato wave is by calculating its instantaneous frequency and amplitude. If the signal is symmetric, the frequency and amplitude should remain constant.






53


A program was developed to calculate the instantaneous frequency and amplitude (similar to a spectrogram) of the frequency vibrato wave. This "spectrogram" is created by taking a small segment or window of the signal and applying the LPC by covariance method, then the window is shifted to the right and the new segment is analyzed. This process is repeated until the whole signal has been analyzed. The length of the segment was chosen to be equal to one cycle of the vibrato wave since a longer segment will show average values and a shorter segment would not contain enough data for a reliable result. The covariance method was chosen since it outperforms the other two methods being used (autocorrelation and MUSIC). Figure 4.18 shows the spectrogram of the curve shown in figure 4.13, which belongs to Placido Domingo. The red color represents high amplitude and the blue color low amplitude. We can see that the frequency varies between 6 and 7 Hz as expected for frequency vibrato. The software also calculates the pole location for each segment. Figure 4.19 shows the pole location for the spectrogram in figure 4.18.





File Edit Window Help
Inst freq and amp of freq vitb 40
80
30
I60 2 0 40 .20
LL10

00 0
0 0.5 1 1.5 Time(s)
Figure 4.18. Spectrogram of the frequency vibrato wave.






54


Fie Edt Endow help




-0.5






-1 -0.5 0 5 1 15
Real Dart
Figure 4.19. Pole location for the spectrogram.



The properties and advantages of the Wigner distribution and Wavelets for the calculation of the instantaneous frequency were investigated. The results provided by these two methods show a resolution comparable to that of the spectrogram, and with cross terms. Since these methods do not make assumptions regarding the signal properties, the parametric methods will provide higher resolution and more accurate results.

In order to determine that the frequency variations are real and not an artifact of the covariance method, a 6 Hz sine wave was generated, and the method described above was applied to it. The spectrogram showed a straight line at 5.97 Hz.

Another Matlab module was developed to calculate the frequency vector and

display it on the spectrogram for comparison purposes. Figure 4.20 shows the frequency vector of the sample being analyzed as a blue line.






55



File Edit Window Help Inst freq and amp of freq vib 40
80
30
60

C20 .40
-r"
20

10

0 0.5 1 1.5


Figure 4.20. Instantaneous frequency vector.



Another module displays the frequency vector in a new window and calculates its power spectrum. Figures 4.21 and 4.22 display the frequency vector and power spectrum corresponding to figure 4.20.





File Edit Window Help Inst freq and amp of freq vib 10 1

S0.5
5 0

--0.5

0 -1
0 0.5 1 1.5 Blue=Freq,; Red=Amp Time(s) Figure 4.21. Instantaneous frequency vector.






56



File Edit Window Help
Power spectra of inst freq and amp of freq viL 30 1

t 20
0.5



0 10 20 30 40 Blue=Freq; Red=Amp Frequency(Hz)

Figure 4.22. Power spectrum of instantaneous frequency vector.



Another module calculates the instantaneous amplitude in each segment using the method described previously, and displays the amplitude in the window of the frequency vector. Figure 4.23 shows the amplitude vector corresponding to the analyzed sample in red. The most important feature of this curve is that the variations in amplitude are much larger than the ones in the original wave. The maximum amplitude in figure 4.23 is 50.425 and the minimum is 2.192. This indicates that the relationship between the maximum and minimum amplitude is approximately 25 to 1. This is obviously wrong since the frequency vibrato wave not does change in amplitude that much.

Where was the problem? Was the method being used incorrect? Or was there a limitation regarding the sample length? If the frequencies were being calculated properly, why were the amplitudes incorrect?






57



Ede Edt Window Help
Inst freq and amp of freq vib 10 60

t--4__ -.- 40




0 0
0 0.5 1 1.5 Blue=Freq; Red=Amp Time(s) Figure 4.23. Instantaneous frequency and amplitude vectors.



Answering these questions is important in order to determine if there was an error in the methodology. Therefore, it was decided to perform experiments to find out where the problem was.

Many experiments were performed. Only the most important ones will be described here:

1. The number of vectors added to calculate each instantaneous amplitude was
changed to several different values. This did not significantly change the
results. Even though this smoothed the curve somewhat, it did not produce a
significant change.

2. The mean value used to remove the DC component of the frequency vibrato
wave was changed from the mean value of the whole signal to the mean value
of the analyzed segment. This did not produce any change.

3. The number of poles in the model was changed to higher and lower values.
Higher values produced spurious peaks and lower values did not improve the
results.

4. The decimation factor was increased to different values. This did not produce
any effect on the results.





58


5. The number of samples used to calculate the model power spectrum was
reduced. This increased the difference between the highest and lowest
amplitudes in the instantaneous amplitude curve.

6. The window length was increased from one cycle to almost the whole length.
The results approached the expected results as the window length approached
the whole signal. A window length close to whole signal does not allow the
calculation of the instantaneous amplitude and therefore cannot be used.

7. The algorithms were applied to a non filtered signal. This reduced the distance
between the highest and lowest peaks in the instantaneous amplitude curve,
especially when the number of poles was reduced to two. The number of peaks was reduced and the mean valued tended to approach the true value, but there
were still spurious peaks.

8. The window overlap in the spectrogram used to get the frequency vibrato
curve was increased, so that instead of 14 samples per cycle, I had 250 or 1000
samples. This did not improve the results, needed an enormous amount of
RAM, and slowed down the software considerably.

9. The autocorrelation and MUSIC methods were used to calculate the
spectrogram. This showed that the pole location obtained using these methods
had significant errors.

After these experiments it was concluded that the method being used for the

calculation of the amplitude was correct but the amplitude given by the covariance method was not precise for that particular sample length. Therefore, it was decided to calculate the amplitude using the autocorrelation method, even though the frequency given by this method had some error.

Figure 4.24 shows the spectrogram created using the autocorrelation method. Using the same method described before to extract the instantaneous frequency and amplitude from the spectrogram, the curves shown in figure 4.25 were obtained. The instantaneous frequency is shown in blue and the instantaneous amplitude in red. There is some ripple in both curves.






59





File Edit Window H elp Inst freq and amp offreq vib


60
30

a) 50 c 20

u 40



0 0.5 1 1.5 Time(s)

Figure 4.24. Spectrogram using the autocorrelation method.




File Edit Windo Help inst treq and amp of freq vib 10
rh' I', N .. 100


5 "50



0 0.5 1 t1.5
Blue=Freq; Red=Amp Time(s) Figure 4.25. Instantaneous frequency and amplitude of the frequency vibrato wave.



The amplitude curve in figure 4.25 matches the instantaneous amplitude of the signal shown in figure 4.13. The amplitude curve in figure 4.25 has a maximum value of 108.163, a minimum of 70.812, and a mean of 87.508. The peaks at 0.1 and 0.9 sec., and the valley at 0.25 sec., match the characteristics of the wave in figure 4.13. The ripple






60


shown is due to deviations of the frequency vibrato wave from the sinusoidal pattern. This ripple can be reduced by filtering the signal with a low pass filter.

Figure 4.26 shows the power spectrum of the instantaneous frequency and

amplitude waves shown in figure 4.25. The high peaks at 12 Hz are due to the ripple in the curves.




File Edit Window Help
Power spectra of inst treq and amp of treq vib 30 300

-20 200 a 10 10 Ii


0 10 20 30 40 Blue=Freq, Red=Amp FrequencV(Hz) Figure 4.26. Power spectra of the instantaneous waves.



From all these experiments it was concluded that for very short signals (only one cycle of the signal in the sample being analyzed) the frequencies calculated by the covariance method are very precise but not the amplitudes, and the autocorrelation method provides better amplitude estimates than the covariance method.

Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave


Figure 4.27 shows an example of an amplitude vibrato wave. The amplitude

vibrato wave is shown in red. This wave is taken from sample pldom01l.wav which belong






61


to Placido Domingo. The harmonic selected is number 6. Figure 4.28 shows the power spectrum of the amplitude vibrato wave in red.





File Edit Window Help
Frequency and amplitude vibrato



2I0 1.5 2




Blue=Freq; Red=Amp Time(s)

Figure 4.27. Example of amplitude vibrato wave.




File Edit Window Help
Freq and amp vibrato power spectra 8000 150 -10000







i 0 E 2000


E20 -' U
0 10 0 1.5 2 Blue=Freq; Red=Amp Freimuenc(Hs)









Figure 4.28. Power spectrum of amplitude vibrato wave.



As can be seen in figure 4.27, the amplitude vibrato wave is noisier than the


frequency vibrato wave and its frequencies and amplitudes change more rapidly than in the frequency vibrato wave. Therefore, the instantaneous frequency and amplitude techniques used for the frequency vibrato should not be used to obtain instantaneous frequency and 8000 -:- -- 150




















used for the frequency vibrato should not be used to obtain instantaneous frequency and






62


amplitude curves (see Kay's book p. 198 for a comparison between LPC and FFT based spectral estimators). However, a menu option was created to show the instantaneous frequencies and amplitudes of the wave as they are shown in a spectrogram, but the values are not quantified. This has the objective of showing how periodic the wave is. The method used to calculate this "spectrogram" is LPC by autocorrelation, since it produces relatively good frequency and amplitude values. Figure 4.29 shows the instantaneous frequency and amplitude plot of the amplitude wave in figure 4.27. We can see that the frequency of oscillation of the amplitude vibrato wave changes, which indicates that the formants shift. The number of poles used to create this plot was 7, and the segment length is 174 msec. This segment length guarantees that at least one cycle of the slowest frequency present (6 Hz) will be contained in the segment. Figure 4.30 shows the pole location for figure 4.29.





File Edit Window Help
Inst freq and amp of amp vib

30
30 25 20






0 0.5 1 1.5
Time(s)
Figure 4.29. Spectrogram of an amplitude vibrato wave.






63



File Edt Window Help
Pole-zero plot




S0.5



-2 -1 0 1 2 Real part c Figure 4.30. Pole location for figure 4.29.



Software Development


The number of individual programs and screens increased considerably with the addition of the modules for the calculation of the instantaneous frequency and amplitude of the frequency and amplitude vibrato.

The total number of windows increased from 4 to 9. It would be difficult to read more than 4 windows simultaneously on a 15 inch screen. Therefore, it was decided to display the windows for the instantaneous power spectra, instantaneous frequency and amplitude of the frequency vibrato, and power spectrum of the instantaneous curves on top of other windows. The user can bring the hidden windows to the front by clicking on the buttons of the windows 95 task bar.

The inclusion of each of the programs developed in the menu of options would

increase the number of options to the point where it would be difficult to use the program






64


and would lead to user errors. Therefore, software modules were grouped to reduce the number of options and then added to the menu of options.

After changes and additions the final version of the menu of options looked like this:

1. File

1.1. Open wave file.

1.2. Save wave file

1.3. Save parameters

1.4. Print figure

1.5. Exit

2. Edit

2.1. Edit wave file 2.2. Play wave file

2.3. Zoom

2.3.1 In

2.3.2. Out

2.4. Options

2.4.1. Spectrogram

2.4.2. Elliptical filter

2.4.3. Full length model

2.4.4. Short length model

3. View

3.1. Window






65


3..2. Filter response

4. Analysis

4.1. Spectrogram

4.2. Get vibrato

4.3. Get parameters

5. Filter

5.1. Elliptical

5.1.1. Frequency vibrato

5.1.1.1. Wave

5.1.1.2. Instantaneous frequency 5.1.1.3. Instantaneous amplitude

5.1.2. Amplitude vibrato

5.1.2.1. Wave

5.2. Median

5.2.1. Frequency vibrato wave 5.2.2. Amplitude vibrato wave

5.3. Get frequency

5.3.1. Power spectrum

5.3.2. Instantaneous power spectrum

6. Model

6.1. Full length

6.1.1. Frequency vibrato 6.1.2. Amplitude vibrato





66


6.2. Short length

6.2.1. Frequency vibrato

6.2.1.1. Instantaneous frequency 6.2.1.2. Instantaneous amplitude

6.2.2. Amplitude vibrato

7. Help

7.1. Where to find help

7.2. About

The following section will describe the tasks performed by each of the most recently added menu options. Edit, Options, Short Length Model

This option allows the user to change the number of poles used for the short length model. The default value is 3 poles since the frequency vibrato wave is almost sinusoidal. The range of valid values is between 1 and 11 poles. Figure 4.31 shows the window displayed when this option is selected. Changing the number of poles for the short length model does not change the number of poles used for the full length model. View, Window

This option allows the user to observe the shape of the window currently in use. Figure 4.32 shows the window displayed when the option is selected.






67



File Edt Window Help





1 3 11 Potes l




Accept Cancel

Figure 4.31. Options for the short length model.




File Edit Window Help Current window




01




0 200 400 600 800 1000
Close

Figure 4.32. View window option.



Filter, Elliptical, Frequency Vibrato, Instantaneous Frequency and Instantaneous Amplitude

By using this option the user can apply an elliptical filter to the instantaneous frequency or instantaneous amplitude waves. The filter parameters can be changed by selecting the Edit, Options, Elliptical filter option.






68


Filter, Get Frequency, Power Spectrum and Instantaneous Power Spectrum

This option is used to obtain a frequency value from the power spectrum or the instantaneous power spectrum. This is useful to determine the cutoff frequency for the elliptical filter. When this option is activated a message appears telling the user that an input is required. After clicking the OK button the power spectrum or instantaneous power spectrum window is activated and the cursor becomes a crosshair. The desired value is displayed in the Matlab command window after clicking on the desired frequency. Model, Short Length, Frequency Vibrato, Instantaneous Frequency

This option allows the user to obtain the instantaneous power spectrum of the

frequency vibrato wave, its instantaneous frequency wave, and the power spectrum of the instantaneous frequency wave. This option is composed of the following modules:

1. The first module calculates the instantaneous power spectrum of the frequency
vibrato wave using the LPC by covariance method.

2. The function of the second module is to calculate the instantaneous frequency
and amplitude waves, and display the instantaneous frequency wave on top of the spectrogram. This is done for verification purposes. The user should select
the range of frequencies where the wave is located.

3. The third module displays the instantaneous frequency wave, calculates and
displays its average, maximum, and minimum values, calculates its power
spectrum and displays it.

4. Finally, the pole location of all the segments analyzed is displayed. Model, Short Length, Frequency Vibrato, Instantaneous Amplitude

This option is very similar to the option described in the previous section, but

instead it uses the LPC by autocorrelation method, displays the instantaneous amplitude






69


wave, and calculates its power spectrum. This option is composed of the following modules:

1. The first module calculates the instantaneous power spectrum of the frequency
vibrato wave using the LPC by autocorrelation method.

2. The function of the second module is to calculate the instantaneous frequency
and amplitude waves, and display the instantaneous frequency wave on top of the spectrogram. This is done for verification purposes. The user should select
the range of frequencies where the wave is located. The instantaneous
frequency and amplitude waves will contain some ripple, but the instantaneous
frequency wave calculated with this option does not replace the wave
calculated using the Instantaneous frequency option, since the latter is more
precise. Also, no calculations are made on this instantaneous frequency wave.

3. The third module displays the instantaneous amplitude wave, calculates and
displays its average, maximum, and minimum values, calculates its power
spectrum and displays it.

4. Finally, the pole location of all the segments analyzed is displayed. Model, Short Length. Amplitude Vibrato

This option allows the user to calculate and display the instantaneous power

spectrum of the amplitude vibrato. This option uses the LPC by autocorelation method.

Software Improvements


After all the programs developed were included in the menu of options, additional work was done on the software for some minor improvements:

1. All variables were checked for consistency in their names, and renamed as
needed.

2. Temporary variables were cleared to keep the amount of memory used to a
minimum. This also helps in debugging the software since the workspace
became smaller.






70


3. The Save Parameters module was modified to also save the mean, maximum
and minimum values of the instantaneous waves and the ten more dominant
components of the power spectra of the instantaneous waves. The file format was changed so that it saves the variables in a column vector. Also, I decided
to store information of one harmonic per data file. Table 4.1 describes the
contents of the data files.


Validation


Two test signals were created in order to verify the precision/accuracy of the

results given by the software developed. The first signal has no frequency nor amplitude modulation and the other has both frequency and amplitude modulation. Signal with no Modulation.

The signal with no amplitude nor frequency modulation is called valid .wav. This signal is a pure tone of 3000 Hz. This will emulate a harmonic at 3000 Hz with no frequency nor amplitude vibrato. Figure 4.33 shows the spectrogram of signal valid .wav. As expected, it is a straight line.




File Edit Window Help Spectrogram
4000
100
I 3000 "woo...m u.mm mmmmm. .
S50
5 2000

1 000
-50
0
0 0.5 1 1.5 Time(s)

Figure 4.33. Spectrogram of signal valid .wav.






71


Table 4.1. Variable names and descriptions.

Variable name Description pathname Path to the wave file fname File name of the wave file harm Harmonic analyzed fmean Mean frequency of the frequency vibrato fmax Maximum frequency of the frequency vibrato fmin Minimum frequency of the frequency vibrato fvarhz Mean frequency variation in Hz of the frequency vibrato fvarpc Mean frequency variation in percentage of the frequency vibrato fvarst Maximum frequency variation in semi-tones of the frequency vibrato
fvarab Mean frequency variation above the mean in Hz of the frequency vibrato
fvarbl Mean frequency variation below the mean in Hz of the frequency vibrato
amean Mean amplitude of the amplitude vibrato amax Maximum amplitude of the amplitude vibrato amin Minimum amplitude of the amplitude vibrato avardb Mean amplitude variation in db of the amplitude vibrato avarpc Mean amplitude variation in percentage of the amplitude vibrato avarab Mean amplitude variation above the mean in db of the amplitude vibrato
avarbl Mean amplitude variation below the mean in db of the amplitude vibrato
real(Yford(j)) Real and imaginary parts of the ten most dominant components of the imag(Yford(j)) frequency vibrato power spectrum j= 1:10
fy(If(j)) Frequencies of the ten most dominant components of the frequency j = 1:10 vibrato power spectrum Nf Number of samples of the frequency vibrato real(Yaord(j)) Real and imaginary parts of the ten most dominant components of the imag(Yaord(j)) amplitude vibrato power spectrum j= 1:10
fy(Ia(j)) Frequencies of the ten most dominant components of the amplitude j = 1:10 vibrato power spectrum Na Number of samples of the amplitude vibrato real(Yfordinst(j)) Real and imaginary parts of the ten most dominant components of the imag(Yfordinst(j)) instantaneous frequency power spectrum j = 1:10
fy(Ifinst(j)) Frequencies of the ten most dominant components of the instantaneous j = 1:10 frequency powerspectrum real(Yaordinst(j)) Real and imaginary parts of the ten most dominant components of the imag(Yaordinst(j)) instantaneous amplitude power spectrum j= 1:10
fy(Iainst(j)) Frequencies of the ten most dominant components of the instantaneous j = 1:10 amplitude power spectrum






72


Figure 4.34 shows the frequency and amplitude vibrato waves obtained from the spectrogram for signal validl.wav. The frequency vibrato wave is shown in blue and the amplitude vibrato wave in red. The frequency vibrato wave is a perfectly straight line, as it should be for a constant tone. The amplitude vibrato wave has some variations of very small magnitude, which reach a maximum of 2x10-6 db. These variations are small compared to the mean amplitude value of 133.616 db; therefore, it can be said that there is no error in the amplitude calculation either.




Fie Edit Window Help
Frequency and amplitude vibrato x 10.6
1 4

10.5
0 0



S 0 .5 1 1.5
Blue=Freq; Red=Amp Time(s)

Figure 4.34. Frequency and amplitude vibrato waves.



Figure 4.35 shows the power spectra of the waves in figure 4.34. The frequency vibrato power spectrum is shown in blue and the amplitude vibrato spectrum in red. The frequency vibrato spectrum is flat, indicating no oscillations in the frequency vibrato wave. The amplitude vibrato spectrum has a peak at 9 Hz of 3.7x105 units in amplitude, which indicates that the oscillations in the amplitude vibrato wave are very small.






73



File Edit Window Help Freq and amp vibrato power spectr_ 106
1 4


0o 2 E -05 i

-1 0 0 10 20 30 40 Blue=Freq, Red=Amp Frequency(Hz) Figure 4.35. Frequency and amplitude vibrato power spectra.



The parameters calculated by the software for the signals in figure 4.34 are shown below.

Frequency vibrato:

Mean frequency = 2993.115

Maximum frequency = 2993.115 Minimum frequency = 2993.115

Mean frequency variation in Hz = 0.000

Mean frequency variation in percentage = 0.000

Maximum frequency variation in semi-tones = 0.000

Amplitude vibrato:

Mean amplitude = 133.616

Maximum amplitude = 133.616 Minimum amplitude = 133.616

Mean amplitude variation in db = 0.000

Mean amplitude variation in percentage = 0.000






74


The mean frequency given by the software is 2993.115 Hz, compared to a real value of 3000 Hz, this represents an error of 0.23%. The software reports no frequency variations at all. The mean amplitude calculated is 133.616 db with no amplitude variations. In summary, the frequency and amplitude calculations made using signal valid .wav are very accurate. Signal with Frequency and Amplitude Modulation

The signal with frequency and amplitude modulations is called valid3.wav. This file contains a tone at 3000 Hz with sinusoidal frequency and amplitude modulations. The frequency is modulated 100 Hz at a rate of 6 Hz, and the amplitude is modulated 4000 SPL (Sound Pressure Level) units at a rate of 1 Hz. This will emulate a harmonic at 3000 Hz with vibrato. Figure 4.36 shows the spectrogram of signal valid3.wav.





Eile Edit Window Help
Spectrogram
4000

S3000

U 50 2000

u- 1000


0
0 0.5 1 1.5 Time(s)

Figure 4.36. Spectrogram of signal valid3.wav.






75


Figure 4.37 shows the frequency and amplitude vibrato waves obtained from figure

4.36. The frequency vibrato wave is shown in blue and the amplitude vibrato wave in red. The amplitude vibrato wave was filtered with a 5 Hz low pass filter to eliminate the higher frequency components. Both waves have sinusoidal shape but are not completely symmetrical. There are several possible causes for this:

1. The modulations in signal valid3.wav are not perfectly symmetrical.

2. The resolution of the FFT.

3. The algorithm used to calculate the amplitude.





File Edit Window Help
Frequency and amplitude vibrato 200 2





100 -2 O 0 5 1 1.5 Blue=Freq, Red=Amp Time(s) Figure 4.37. Frequency and amplitude vibrato waves.



Figure 4.38 shows the power spectrum of the waves in figure 4.37. These power spectra show that the waves in figure 4.37 have sinusoidal shape.






76



File Edit Wndow Help
Freq and amp vibrato power spectra 8000 150

6000
D100
4000
E- 50
E 2000

0
0 10 20 30 40 Blue=Freq; Red=Amp Frequency(Hz)

Figure 4.38. Frequency and amplitude vibrato power spectra.



The parameters calculated by the software developed are shown below. Frequency vibrato:

Mean frequency = 3000.611 Maximum frequency = 3100.781 Minimum frequency = 2906.982 Mean frequency variation in Hz = 64.013 Mean frequency variation in percentage = 2.133 Maximum frequency variation in semi-tones = 0.559 Mean frequency variation above the mean in Hz = 64.834 Mean frequency variation below the mean in Hz = 63.213 Maximum frequency variation above the mean in semi-tones = 0.569 Maximum frequency variation below the mean in semi-tones = 0.549 Maximum amplitude relative to the mean = 7020.281 Frequency of the maximum amplitude in Hz = 6.112 Maximum frequency error in percentage = 4.167






77


Amplitude vibrato:

Mean amplitude = 132.173

Maximum amplitude = 133.874 Minimum amplitude = 130.347

Mean amplitude variation in db = 1.093

Mean amplitude variation in percentage = 0.827

Mean amplitude variation above the mean in db = 1.079 Mean amplitude variation below the mean in db = 1.107

Maximum amplitude relative to the mean = 130.821 Frequency of the maximum amplitude in Hz = 1.019

Maximum frequency error in percentage = 25.000

The mean frequency of the frequency vibrato is 3000.611 Hz, compared with an expected value of 3000 Hz, and the maximum and minimum values are 3100.781 Hz and 2906.982 Hz, compared to expected values of 3100 Hz and 2900 Hz respectively. The frequency vibrato is calculated at 6.112 Hz compared to an expected value of 6 Hz. From these values we can see that the errors between the expected and the calculated values are very small.

The mean value of the amplitude vibrato is 133.874 db compared to 133.616 db

for signal valid .wav. The frequency of oscillation of the amplitude vibrato is calculated at

1.019 Hz compared to an expected value of 1 Hz. Again, the errors are very small.

Figure 4.39 shows the instantaneous frequency and amplitude of the frequency vibrato wave in figure 4.37. The instantaneous frequency is shown in blue and the instantaneous amplitude in red. The instantaneous frequency wave shows that there are






78


frequency variations in the signal. The instantaneous amplitude wave shows the amplitude variations in the wave. The algorithm used to calculate the instantaneous amplitude introduces some error. Figure 4.40 shows the power spectra of the waves in figure 4.39.




File Edit Window Help Inst freq and amp of freq vib 10
E10 100


S5 50


0 0
0 0.5 1 1.5 Blue=Freq; Red=Amp Time(s) Figure 4.39. Instantaneous frequency and amplitude of the frequency vibrato wave.




File Edit Window Help Power spectra of inst freq and amp
6 150 4 Ilk 100


2 r 50
0 0!
0 5 10 15 20 Blue=Freq; Red=Amp Frequency(Hz)

Figure 4.40. Power spectra of the instantaneous waves.



The software developed calculated the following parameters for the waves in figure 4.39.






79


Instantaneous frequency: Mean frequency = 6.038

Maximum frequency = 6.366 Minimum frequency = 5.737

Instantaneous amplitude: Mean amplitude = 98.271

Maximum amplitude = 106.844

Minimum amplitude = 87.800

The mean frequency value is 6.038 Hz compared to an expected value of 6 Hz.

The maximum and minimum values of 6.366 Hz and 5.737 Hz indicate the real maximum and minimum frequencies of the frequency vibrato wave.

The mean amplitude of 98.271 Hz is close to the expected value of 100 Hz. The maximum and minimum values of 106.844 and 87.800 Hz indicate the variations in the signal amplitude. There is an error in the measure of about 3 percent introduced by the algorithm.




Full Text
5.5. Frequency and amplitude vibrato waves for bjorl l.wav 86
5.6. Vibrato power spectra for bjorl 1 .wav 86
5.7. Instantaneous waves for bjorl 1 .wav 87
5.8. Power spectra of the instantaneous waves in bjorl 1 .wav 87
5.9. Frequency and amplitude vibrato waves for bjor25.wav 88
5.10. Vibrato power spectra for bjor25.wav 88
5.11. Instantaneous waves for bjor25.wav 89
5.12. Power spectra of the instantaneous waves in bjor25.wav 89
5.13. Frequency and amplitude vibrato waves for ec04.wav 90
5.14. Vibrato power spectra for ec04.wav 90
5.15. Instantaneous waves for ec04.wav 91
5.16. Power spectra of the instantaneous waves in ec04.wav 91
5.17. Frequency and amplitude vibrato waves for ec08.wav 92
5.18. Vibrato power spectra for ec08.wav 92
5.19. Instantaneous waves for ec08.wav 93
5.20. Power spectra of the instantaneous waves in ec08 .wav 93
5.21. Frequency and amplitude vibrato waves for ecl6.wav 94
5.22. Vibrato power spectra for ecl6.wav 94
5.23. Instantaneous waves for ecl6.wav 95
5.24. Power spectra of the instantaneous waves in ecl6.wav 95
5.25. Frequency and amplitude vibrato waves for pav02.wav 96
5.26. Vibrato power spectra for pav02.wav 96
5.27. Instantaneous waves for pav02.wav 97
5.28. Power spectra of the instantaneous waves in pav02.wav 97
5.29. Frequency and amplitude vibrato waves for pav03a.wav 98
5.30. Vibrato power spectra for pav03a.wav 98
xiv


164
Summary of Results
Table 8.3 shows the results of the analysis performed on each singer. Table 8.3
shows that the amplitude varies for Bj or ling, Caruso, Caballe, and DeLosAngeles, and the
frequency varies for Bjorling, Caruso, and DeLosAngeles. This means that the singers
formant is not constant for singers Bjorling, Caruso, Caballe, and DeLosAngeles. It is very
interesting to note that no significant results were found for Pavarotti and Domingo, who
are the most renowned opera singers of our age.
Table 8.3. Singers' formant variation within samples of the same singer.
Singer
Parameter
P value
a level
Conclusion
Significant
Bjorling
Amplitude
0.0001
0.05
Reject
Yes
Bjorling
Frequency
0.0117
0.05
Reject
Yes
Caruso
Amplitude
0.0004
0.05
Reject
Yes
Caruso
Frequency
0.0153
0.05
Reject
Yes
Pavarotti
Amplitude
0.5297
0.05
Do not reject
No
Pavarotti
Frequency
0.1581
0.05
Do not reject
No
Domingo
Amplitude
0.0662
0.05
Do not reject
No
Domingo
Frequency
0.3072
0.05
Do not reject
No
Caballe
Amplitude
0.0134
0.05
Reject
Yes
Caballe
Frequency
0.3221
0.05
Do not reject
No
DeLosAngeles
Amplitude
0.0121
0.05
Reject
Yes
DeLosAngeles
Frequency
0.0017
0.05
Reject
Yes


210
% Plots the instantaneous frequency wave on the spectrogram
hold on;
plot((t(1:length(t)14))Fmaxinst,'b-');
hold off;
% Clears temporary variables
clear handle j k 1 iinf isup width Amaxinsttemp;
M-file delfig.m
function delfig(figtag);
% Deletes figure if it exists
handle= findobj('Tag',figtag);
if -isempty(handle)
close(handle);
end;
M-file edit3 m
% M-file Edit3.m
% Edits a wave file
% Deletes figures 2, 3, 4, figinst, figinstfrcur,
% and instfrspec if they exist
delfig('Fig2');
delfig('Fig31);
delfig('Fig4');
delfig('Figinst');
delfig('FiglnstFrCur1);
delfig('FiglnstFrSpec1);
% Creates figure 2 without the colorbar
delfig('Fig2');
fig2;
imagesc(t',f',X);
axis('xy');
axis([0 max(t) liml lim2]);
colormap(jet);
title('Spectrogram');
xlabel('Time(s)') ;
ylabel('Frequency(Hz)');
% Displays message
handle= msgbox(1 Select range to analyzeInput required');
waitfor(handle);
drawnow;
% Gets input from mouse
[1,k]= ginput(2);
linf= round(1(1).*(length(t)-1)./max(t) + 1);
lsup= round(1(2).*(length(t)-1)./max(t) + 1);
if linf < 1
linf= 1;
end


144
Table 7.2continued.
File name
abs(Y aordinst(3))
angle(Yaordinst(3))
Frainst(l)
Frainst(2)
Frainst(3)
Bjor05.wav
210.1646774
-1.202137142
0.536
1.609
3.755
Bjorll.wav
292.3503504
-0.946578073
2.235
1.118
3.912
Bjor25.wav
328.941343
0.99341352
4.471
2.794
0.559
Ec04.wav
206.723911
-1.230666794
1.599
2.132
3.198
Ec08.wav
298.5207903
-0.269983041
0.513
1.025
2.05
Ecl6.wav
267.820549
-1.092420324
1.49
3.477
2.484
Pav02.wav
512.3911853
-1.318345437
1.588
0.529
1.059
Pav03a.wav
398.8213155
-1.464256542
2.647
1.059
0.529
Pavl4.wav
315.501213
0.519425634
0.536
1.073
3.219
Pldom01.wav
292.5294274
-2.021513141
0.523
1.568
2.613
Pldom02.wav
261.6376187
-0.996518678
2.077
2.596
4.673
Pldom24.wav
626.3397759
0.65479937
1.066
1.599
0.533
kbat01.wav
337.9308508
-1.979066261
0.485
2.424
2.909
Kbat20.wav
512.8935086
-0.680251121
0.529
1.059
1.588
Kbat22.wav
284.3988597
2.011376646
0.523
1.568
1.045
Moncab30.wav
548.3046941
-0.002485847
1.631
0.544
2.175
Moncab31 .wav
392.8751982
1.324270004
0.563
2.814
3.377
Moncab33.wav
338.8294541
-0.067362082
0.54
1.62
3.781
delosa01.wav
522.4234653
-2.873605537
2.19
0.547
1.095
delosa07.wav
1278.076778
-1.501360519
1.095
0.547
2.19
delosa09.wav
528.9989224
1.738538725
0.54
1.62
2.16
Mean
416.9749471
1.303667
1.586762
2.328762
Maximum
1278.076778
4.471
3.477
4.673
Minimum
206.723911
0.485
0.529
0.529
Std. dev.
225.9204717
0.988155
0.824669
1.201388
Cl Maximum
868.8158906
3.279977
3.2361
4.731538
Cl Minimum
-34.86599643
-0.67264
-0.06258
-0.07401


42
Figure 4.5. Full length model options.
The parameters that can be adjusted are
1. Decimation factor. The decimation factor allows a reduction of the sampling
rate of the vibrato sample. This option was added to the software since in
experiments previously performed with synthetic vibrato waves applying the
LPC model by covariance method, it was noticed that the method had
difficulties when modelling frequency components which are close to DC
value. The only way to move the poles away from the unit circle is by
decimation. After decimation, the LPC method detects more easily the
frequency components that were initially close to DC. The Matlab algorithm
for decimation performs filtering before decimation to avoid aliasing. The
range of valid values is from 1 to 41.
2. Poles. This function allows the user to adjust the number of poles used in the
full length model. The range of values is from 1 to 21 poles.
3. Method. The user can select the analysis method for the full length model. The
options are LPC by autocorrelation, LPC by covariance, and MUSIC (Multiple
Signal Classification).
View. Filter Response
The objective of this function is to display the elliptical filter frequency response.
In this way the user can see the frequency response of the filter and make changes if
needed. Figure 4.6 shows the frequency response of the default filter.


3
sinusoidal model to verify their advantage over the pure sinusoidal model. An error
measure is made to quantify the model's accuracy.
The statistical analysis chapter shows the parameters obtained with the MMSV
software for the samples selected, and their statistics. All the parameters used in the model
are presented and analyzed.
The singers' formant chapter contains a complete study of the singers' formant
based on 6 singers. A technique for the measurement of the singers' formant is described,
and used to extract the singers' formant parameters. Two studies are made: comparison of
the singers' formant among different singers, and comparison of the singers' formant
parameters among samples of the same singer.
The last chapter presents a summary of the contributions in this dissertation and
several applications, and also provides future directions and guidelines for the continuation
of this research.
Appendix A contains the MMSV users' manual. The objective of this appendix is
to give the user a step by step guide of the MMSV software to obtain the vibrato sample
waves and parameters.
The Matlab code developed during the present research is presented in appendix
B. The software, some vibrato samples, and the data files obtained are provided on a
floppy disk.


57
File Edit Window Help
Inst freq and amp of freq vlb
10
60
S2
40
20
0
0
0 0.5 1
Blue=Freq; Red=Amp
1.5
Time(s)
Figure 4.23. Instantaneous frequency and amplitude vectors.
Answering these questions is important in order to determine if there was an error
in the methodology. Therefore, it was decided to perform experiments to find out where
the problem was.
Many experiments were performed. Only the most important ones will be
described here:
1. The number of vectors added to calculate each instantaneous amplitude was
changed to several different values. This did not significantly change the
results. Even though this smoothed the curve somewhat, it did not produce a
significant change.
2. The mean value used to remove the DC component of the frequency vibrato
wave was changed from the mean value of the whole signal to the mean value
of the analyzed segment. This did not produce any change.
3. The number of poles in the model was changed to higher and lower values.
Higher values produced spurious peaks and lower values did not improve the
results.
4. The decimation factor was increased to different values. This did not produce
any effect on the results.


128
power spectra. The instantaneous amplitude and frequency waves can be synthesized using
the following formulas:
AAf (J) = 2(| Ya(\) | cos(2^ (l)/(y) + ZYa(l)) +
| Ya(2) j cos(2nfa (2)t(J) + Ya(2)) + (6-6)
| Ya(3) | cos(2nfa(3)t(j) + ZYa(3)))/(N-\4),j = l :N-l4
where:
AAf(j) = Frequency vibrato instantaneous amplitude variation respect to the mean
Ya(l), Ya(2), Ya(3) = Three most dominant components of the instantaneous
amplitude spectrum (complex numbers)
fa(l), fa(2), fa(3) = Frequencies of Ya(l), Ya(2), Ya(3)
t(j) = Time at instant j
N = Total number of samples of the frequency vibrato wave
4ft U) = 2(1 Yf( 1) I C0S(2^> mj) + Yf{ 1)) +
| 7/(2) | cos(2nff (2)t(j) + ZF/(2)) + (6-7)
17/(3) | cos(2^>(3>C/) + ZYf(3)))/(N -14), j = V.N-\4
where:
Aff(j) = Frequency vibrato instantaneous frequency variation respect to the mean
Yf(l), Yf(2), Yf(3) = Three most dominant components of the instantaneous
frequency spectrum (complex numbers)
ff(l), ff(2), ff(3) = Frequencies of Yf(l), Yf(2), Y{(3)
t(j) = Time at instant j
N = Total number of samples of the frequency vibrato wave


4.19. Pole location for the spectrogram 54
4.20. Instantaneous frequency vector 55
4.21. Instantaneous frequency vector 55
4.22. Power spectrum of instantaneous frequency vector 56
4.23. Instantaneous frequency and amplitude vectors 57
4.24. Spectrogram using the autocorrelation method 59
4.25. Instantaneous frequency and amplitude of the frequency vibrato wave 59
4.26. Power spectra of the instantaneous waves 60
4.27. Example of amplitude vibrato wave 61
4.28. Power spectrum of amplitude vibrato wave 61
4.29. Spectrogram of an amplitude vibrato wave 62
4.30. Pole location for figure 4.29 63
4.31. Options for the short length model 67
4.32. View window option 67
4.33. Spectrogram of signal validl.wav 70
4.34. Frequency and amplitude vibrato waves 72
4.35. Frequency and amplitude vibrato power spectra 73
4.36. Spectrogram of signal valid3.wav 74
4.37. Frequency and amplitude vibrato waves 75
4.38. Frequency and amplitude vibrato power spectra 76
4.39. Instantaneous frequency and amplitude of the frequency vibrato wave 78
4.40. Power spectra of the instantaneous waves 78
5.1. Frequency and amplitude vibrato waves for bjor5.wav 84
5.2. Vibrato power spectra for bjor5.wav 84
5.3. Instantaneous waves for bjor5.wav 85
5.4. Power spectra of the instantaneous waves in bjor5.wav 85
xiii


113
U3
h
File Edit Window Help
15
Power spectra of instfreq and amp
1st and 2nd peak
1st, 2nd, 3rd and 4th peak
o
0 5 10
Blue=Freq; Red=Amp
800
600
400
200
0
15 20
Frequency! Hz)
Figure 5.60. Power spectra of the instantaneous waves in kbat22.wav.


125
Figure 5.83. Instantaneous waves for delosa09.wav.
Figure 5.84. Power spectra of the instantaneous waves in delosa09.wav.


172
may be conducted to determine in which cases the singers' formant is present and in which
cases is not.
The singers' formant amplitude was measured as the absolute value of the formant
peak. This method depends on the sample recording level and may affect the results. A
better method would use a relative measure which would cancel out the effect of the
recording level.
In the comparison made among singers it was found that the singers' formant does
not change in the contemporary, male singers. This may be caused by better or different
training techniques. A study can be conducted with more singers' to corroborate or reject
this finding.


LIST OF REFERENCES
Bennet, G., Singing Synthesis in Electronic Music. Proceedings of the Stockholm Music
Acoustic Conference, Vol. 33, p. 34-50, 1981.
Boudreaux-Bartels, and G. F., Parks, T. W., Time-Varying Filtering and Signal Estimation
Using Wigner Distribution Synthesis Techniques. IEEE Transactions on
Acoustics, Speech and Signal processing, Vol. ASSP-34 No. 3, p. 442-451, June,
1986.
Breen, A., Speech Synthesis Models: A Review. Electronics & Communication
Engineering Journal, p. 19-31, February, 1992.
Brigham, E. 0., The Fast Fourier Transform. Prentice Hall, Englewoods Cliffs, New
Jersey, 1974.
Chan, Y. T., Lavoie, J. M. M., and Plan, J. B., A Parameter Estimation Approach to
Estimation of Frequencies of Sinusoids. IEEE Transactions on Acoustics, Speech
and Signal processing, Vol. ASSP-29 No. 2, p. 214-219, April, 1981.
Childers, D. G., Modem Spectrum Analysis. IEEE Press, New York, New York, 1978.
Chowning, J., Computer Sinthesis of the Singing Voice. Proceedings of the Stockholm
Music Acoustic Conference, Vol. 33, p. 4-13, 1980.
Culver, C. A., Musical Acoustics. The Maple Press Company, York, Pennsylvania, 1956.
DeJonckere, P. H., Minoru, H., and Sundberg, J., Vibrato. Singular Publishing Group,
Inc., San Diego, California, 1995.
Deller, J. R., Proakis, J. G., and Hansen, J. H. L., Discrete-Time Processing of Speech
Signals. Prentice Hall, Upper Saddle River, New Jersey, 1987.
Diaz, J. A., Frequency Characterization of Singers Vibrato. University of Florida,
Gainesville, Florida, December 1995.
Easley, E. B., A Comparison of the Vibrato in Concert and Opera Singing. University of
Iowa Studies in the Psychology of Music, p. 1-7, 1932.
230


151
Placido Domingo
Figure 8.5 shows the singers' formant of segment 36 in sample pldom01.wav.
Figure 8.5. Singers' formant in sample pldom01.wav.
Monserrat Caballe
Figure 8.6 shows the singers' formant of segment 32 in sample moncab30.wav.
Figure 8.6. Singers' formant in sample moncab30.wav.


117
Figure 5.67. Instantaneous waves for moncab31.wav.
Figure 5.68. Power spectra of the instantaneous waves in moncab31.wav.


130
maximum error is 97.72 Hz, the minimum is -86.13 Hz, and the average (£|error|/N) is
16.01 Hz.
1 figuie No. 5 HI £3
Fie £dt Window Help
100
80
60
40
20
0
-20
-40
|i| -60
-80
-100
0 0 5 1 1.5 2 2.5
Figure 6.2. Error for pure sinusoidal model.
Figure 6.3 shows the original frequency vibrato wave in blue and a synthesized
wave in red which was generated using the proposed model. We can see that the
synthesized wave follows the original wave more closely than a pure sinusoidal wave, both
in amplitude and frequency.
Figure 6.3. Synthesized wave using the proposed model.


5
Vibrato rates below the lower limit are considered slow, and rates above the upper limit
convey nervousness to the listener.
Frequency vibrato
Figure 2 .1. Example of vibrato wave.
The amplitude of the vibrato varies with the voice loudness. Its extent is 1 or 2
semitones in average. Vibrato extents smaller than 0.5 are not common for singers, and
extents greater than 2 semitones are judged to be bad.
There are two other types of frequency modulation in singing which are similar to
vibrato: tremolo and trillo. Tremolo can be distinguished by a more irregular and rapid
oscillation. Its frequency is of 7 Hz or more. The trillo is characterized by a higher
amplitude modulation, which is of 2 semitones or more.


62
amplitude curves (see Kay's book p. 198 for a comparison between LPC and FFT based
spectral estimators). However, a menu option was created to show the instantaneous
frequencies and amplitudes of the wave as they are shown in a spectrogram, but the values
are not quantified. This has the objective of showing how periodic the wave is. The
method used to calculate this spectrogram is LPC by autocorrelation, since it produces
relatively good frequency and amplitude values. Figure 4.29 shows the instantaneous
frequency and amplitude plot of the amplitude wave in figure 4.27. We can see that the
frequency of oscillation of the amplitude vibrato wave changes, which indicates that the
formants shift. The number of poles used to create this plot was 7, and the segment length
is 174 msec. This segment length guarantees that at least one cycle of the slowest
frequency present (6 FIz) will be contained in the segment. Figure 4.30 shows the pole
location for figure 4.29.
MFigure No. 5
File Edit Window Help
Inst freq and amp of amp vib
SE
Time(s)
Figure 4.29. Spectrogram of an amplitude vibrato wave.


232
Rioul, O., and Vetterli, M., Wavelets and Signal Processing. IEEE SP Magazine, p. 14-
38, October, 1991.
Root, A. R., Pitch Patterns and Tonal Movement in Speech. Psychol. Monog., No. 1, p.
109-159, 1930.
Rossing, T., Sundberg, J., and Ternstrom, S., Acoustic Comparison of Soprano Solo and
Choir Singing. Journal of the Acoustical Society of America, p. 830-836,
September 1987
Rothman, H. B., and Arroyo, A. A., Acoustic Parameters of Violin Vibrato. Unpublisehd
Paper, p. 1-30, 1998.
Rothman, H. B., Diaz, J. A., and Vincent, K. E., Comparing Historical and Contemporary
Opera Singers with Historical and Contemporary Jewish Cantors, Paper Presented
at the 27th Symposium: Care of the Professional Voice, p. 1-8, June, 1998.
Rothman, H. B., and Timberlake, C., Perceptual Evaluation of Singers Vibrato
Transcripts of the Thirteenth Symposium: Care of the Professional Voice, p. 111-
115, June, 1984
Rothman, H. B., Vibrato: What is it?. The Nats Journal, Vol. 43 No. 4, p. 16-19, March
1987.
Sataloff, R. T., The Science and Art of Clinical Care. Raven Press, New York, New York,
1991.
Schoen, M., The Pitch Factor in Artistic Singing. Psychol. Monog., No. 1, p. 230-259,
1922.
Schultz-Coulon, H., The Neuromuscular Phonatorv Control System and Vocal Function.
Acta Octolaryngol., p. 142-153, 1978.
Schultz-Coulon, H., Zur Bedeutung der Kinastetisch-Reflektorishen Phonations-Controlle
fur die Genauigkeit der Stimme. Folia Phoniatrica. p. 335-348, 1976.
Schultz-Coulon, H., and Battmer, R., Die Quantitative Bewertung des Sangervibratos.
Folia Phoniatrica, p. 1-14, 1981.
Schutte, H. K., Miller, D. G, and Svec, J. G., Measurements of Formant Frequencies and
Bandwidths in Singing. Journal of Voice, Vol. 9 No. 3, p. 290-296, 1995.
Seashore, C., The Vibrato. University of Iowa, Iowa City, Iowa, 1932.


169
The model developed can be used to create synthesized vibrato samples. By
applying the model developed and using the model parameters, the synthesized vibrato
wave will have the characteristics of a real vibrato wave.
The software and the mathematical model developed can be used to analyze
vibrato samples from music students or singers with vibrato problems who wish to
improve their vibrato. By using the software and the mathematical model the problem or
deficiency can be found out, and suggestions to correct it can be made.
The mathematical model and digital signal processing algorithms developed in this
dissertation can be applied to other fields. The spectral analysis algorithms are not limited
to vibrato signals and can be directly applied to other fields to determine the signal
parameters and characteristics.
Singers' Formant
The LPC power spectrum is more sensitive to the number of poles when
measuring the singing voice than when measuring normal speech. This is due to the larger
separation between harmonics in the singing voice.
The singers' formant was not found in one of the female singers' (Katherine Battle)
even though all of the females are opera singers with similar voice types (sopranos).
The frequency values of the singers' formant in the samples analyzed ranged from
2368 to 3273 Hz. These values are slightly lower than the values reported in the literature
(2500 to 3500 Hz).
It was found that the singers' formant frequency varies among the six singers under
study. This result contradicts the literature in which the singer's formant is considered
constant. No significant result was found in the amplitude comparison among singers.


233
Seashore, C., Studies in the Psychology of Music. University of Iowa, Iowa City, Iowa,
1936.
Shipp, T., Sunberg, J., and Haglund, S., A Model of Frequency Vibrato. Transcripts of
the Thirteenth Symposium: Care of the Professional Voice, p. 116-117, June,
1984.
Simon, C., The Variability of Consecutive Wave Lenghts in Vocal and Instrumental
Sound. Psychol. Monog., No. 1, p. 41-83, 1926.
Sundberg, J., Acoustic and Psvchoacoustic Aspects of Vocal Vibrato. KTH Speech
Transmission Laboratory Quarterly Progress and Status Report, p. 45-48,
October 15, 1994.
Sundberg, J., Effects of the Vibrato and the Singing Formant on Pitch. Musicologica
Slovaca, p. 51-69, 1978.
Sundberg, J., The Acoustics of the Singing Voice. Scientific American, p. 82-91, March
1977.
Sundberg, J., The Science of the Singing Voice. Northern Illinois University Press,
Dekalb, Illinois, 1987.
Sundberg, J., Vibrato and Vowel Identification. Arch. Acoustics, p. 257-266, 1977.
Sundberg, J., and Askenfelt, A., Larynx Height and Voice Source: A Relationship?. Voice
Physiology, p. 307-316, 1983.
Travis, L. E., Studies in Stuttering. Arch. Neurol, andPsychiatr., p. 298, 1927.
Titze, I. R., Synthesis of Sung Vowels Using a Time-Domain Approach. Transcripts of
the Eleventh Symposium: Care of the Professional Voice, p. 90-98, 1983.
William, T. C, Cooley, J. W., Favin, D. L., Helms, H. D., Kaenel, R. A., Lang, W. W.,
Maling, G. C., Nelson, D. E., Rader, C. M., and Welch, P. D., What is the Fourier
Transform?. IEEE Trans. Audio Electroacoustics, Vol. AU-15, p. 45-55, June
1967.
Winckel, F., Physikalische Kriterien fur Obiektive Stimmbeurteilung. Folia Phoniatrica, p.
232, 1953.
Yannis, S., Decomposition of Speech Signals into a Periodic and Non-periodic Part Based
on Sinusoidal Models. Proceedings of the IEEE International Conference on
Electronics, Circuits, and Systems, Vol. 1, p. 514-517, 1996.


28
The specification of a(k), b(k) and a2 is equivalent to specifying the PSD of the
signal x(n). It is assumed that a(0) = 1, and b(0) = 1, since the filter gain can be
incorporated into a2.
If all the b(k) coefficients are assumed to be zero except b(0) = 1, equation 2-16
becomes
p
x(n) = a(k)x{n k) + u(n) (2-22)
k=\
which is an AR model of order p. This process is called an autoregressive process because
the signal x(n) is a linear regression on itself. For this kind of process the PSD is
(2-23)
This model is also called an all-pole model, and is frequently denoted as an AR(p)
process.
Relationship between the model parameters and the autocorrelation function
A relationship between the ARMA model parameters and the autocorrelation
function of the signal x(n) can be found by taking the inverse Z-transform of equation 2-
20, using the causality property of H(z), and doing some manipulation. After going
through these steps we obtain the following results (Kay, 1988, p. 115):
= + + fork = 0, 1,. ,q
Z=1 1=0
r(*) = -I(Or0k ~ 0 for k > q+1 (2-24)
Z=1
For the particular case of an AR process b(l) = 8(1). Applying this to equations 2-
24, they become


CHAPTER 2
LITERATURE REVIEW AND OVERVIEW
The Singers' Vibrato
The discussion that follows deals primarily with Western operatic vibrato.
Most professional opera singers acquire vibrato almost involuntarily without
pursuing or seeking it actively. It has been conjectured that vibrato develops by itself when
the singer's training progresses satisfactorily.
This phenomenon is produced by a nearly sinusoidal variation of the vibration
frequency of the vocal folds. The degree in which a singer possesses a periodic vibrato is
considered a measure of his voice quality. The singer's voice is generally considered better
if he has a more periodic vibrato (Sundberg, 1987,p.163).
Since the vibrato is a variation in the frequency of oscillation of the vocal folds, it
has been characterized mainly by two parameters: the rate (frequency) and the extent
(amplitude) of the variations (see figure 2.1). The vibrato rate specifies the number of
cycles per second of the frequency oscillations. The extent specifies how wide the
frequency changes from peak to peak during a cycle.
The vibrato rate is considered to be a constant for most singers. They are generally
unable to alter it (Sundberg, 1987, p.165). However, some singers can adapt their vibrato
frequency at will. The range of typical vibrato frequencies is between 5.5 Hz and 7.5 Hz.
4


129
The objective of these formulas is to provide the best model for the instantaneous
frequency and amplitude waves with the smallest number of parameters; therefore, they do
not entirely match the original waves. The model would not be useful if the number of
parameters is too high. Also, this would violate one of the objectives of the model, which
is to reproduce the original wave with a relatively small number of parameters.
1 tested the model with two samples, one from a male singer and one from a female
singer. First, I used sample pldom01.wav which belongs to Placido Domingo. Figure 6.1
shows in blue the original vibrato wave for sample pldom01.wav, and a synthesized
vibrato wave in red which is purely sinusoidal. We can see that both waves start very
closely, but their amplitude and frequency do not match very well throughout the two
second time period.
Figure 6.1. Pure sinusoidal synthesis.
Figure 6.2 shows the error of the pure sinusoidal wave of figure 6.1. The error is
calculated by subtracting the values of the original and the synthesized wave. The


183
File Edit View Analysis Filter Model Help
Vibrato sample
0.5 1
File: PldomOl wav
1.5 2
Time(s)
File Edit Window Help
Spectrogram
4000
3000
VVV V yV V V/ w v V \yn
CD
=3
ur
(D
2000
1000
120
100
B0
60
40
20
0.5 1 1.5
Time(s)
File Edit Window Help
200
- 100
Frequency and amplitude vibrato
cu n
T3 U
f-100
£
< -200
lili
\l\l\j
\i\
AAA/
Ww
L]
l uy y
I/
f¡
0 0.5 1 1.5 2
Blue=Freq; Red=Amp Time(s)
10
5
0
-5
Q Figure No. 4
File Edit Window Help
Freq and amp vibrato power spectra
8000 150
S 6000
cu
T3
4000
E 2000
<
Q

0 10 20
Blue=Freq; Red=Amp
100
50
0
30 40
FrequencyfHz)
safesfesiSs
Start WjA icros... MATL I 5S Figure... S Figure... I S Figure ...II BE Figur...
I 4:51 F'M
Figure A. 11. Aspect of the computer's screen.
*} MAT LAB Command Window
File Edit Window Help
Maxinun amplitude relative to the nean = 7248.785
Frequency of the naxinun amplitude in Hz = 5.748
Maximum frequency error in percentage = 4.167
Amplitude uibrato:
Mean amplitude = 114.942
Maximum amplitude = 120.437
Minimum amplitude = 111.597
Mean amplitude uariation in db = 1.310
Mean amplitude variation in percentage = 1.139
Mean amplitude variation above the mean in db = 1.487
Mean amplitude variation belou the mean in db = 1.170
Maximum amplitude relative to the mean = 41.740
Frequency of the maximum amplitude in Hz = 11.496
Maximum frequency error in percentage = 2.083
.d
J
Figure A. 12. Matlab command window.


99
File Edit Window Help
Instfreq and amp of freq vib
100
50
Figure 5.31. Instantaneous waves for pav03a.wav.
File Edit Window Help
Power spectra of instfreq and amp
Blue=Freq, Red=Amp FrequencvCHz)
Figure 5.32. Power spectra of the instantaneous waves in pav03a.wav.


ACKNOWLEDGMENTS
First, I want to acknowledge the importance of Dr. Antonio Arroyo's involvement
in the success of my Ph.D. and this dissertation. He led me during the entire Ph D.
program, from the moment I entered into it until the end. He gave me the right advice and
encouragement while I was taking classes, in preparing for the written and oral qualifying
exams, and during my dissertation experimental phase. Dr. Arroyo provided me with the
guidance and technical support needed to accomplish this research.
I also need to recognize the contribution of Dr. Howard Rothman to this project.
He provided his expertise and knowledge of the musical field which were vital part in the
success of this dissertation. His ideas and feedback played an important role in the
development of the algorithms.
Finally, I give thanks to all the professors in my committee, the department of
electrical and computer engineering, the department of communication sciences and
disorders, all the people who were directly or indirectly involved in this dissertation during
all its stages, and especially to God, the creator of all, and Jesus Christ, the way, the truth
and the life.
IV


39
File. Save Parameters
This function is used to save the vibrato parameters on disk after the analysis has
been performed.
File. Print Figure
This option allows the user to print the original vibrato wave. The other figures
can be printed by using the print option on each figure's menu.
File. Exit
By selecting this option all figures are closed and the program finishes execution.
Edit. Edit Wave File
This function allows the vibrato sample to be edited to remove non vibrato
segments. The user indicates the segment to be removed on the spectrogram figure, since
this figure provides the best visual information about the vibrato sample.
Edit. Play Wave File
This function allows the user to listen to the vibrato sample.
Edit. Zoom in and out
These two options allow the user to zoom in and out of the vibrato sample figure
to look at the details of the original wave.
Edit. Options. Spectrogram
By choosing this option the user can adjust several parameters to be used for
calculating the spectrogram. Figure 4.3 shows the window that appears when this option
is selected.


208
M-file curve2.m
% M-file curve2.m
% Calculates and plots the frequency vibrato wave
% Deletes figures 3, 4, figinst, figinstfrcur,
% and instfrspec if they exist
delfig('Fig3');
delfig('Fig4');
delfig('Figinst');
delfig('FiglnstFrCur');
delfig('FiglnstFrSpec');
% Displays message
handle= msgbox('Select harmonic to analyzeInput required')
waitfor(handle);
drawnow;
handle= findobj('Tag', 'Fig2');
figure(handle);
% Gets input from mouse
[1,k]= ginput(2) ;
isup= round(k(1).*(length(f)-1)./max(f) +1);
iinf= round(k(2).*(length(f)-1)./max(f) + 1);
% Calculates the frequency vibrato wave
width= isup-iinf;
for j = l:size(X,2)
if iinf < 1
iinf = 1;
end
if isup > size(X,l)
isup = size(X,l);
end
[Max (j ) imax (j ) ] = max (X (iinf: isup, j )) ;
imax(j)= imax(j) + iinf 1;
iinf= imax(j) round(width./2);
isup= iinf + width;
end
Fmax= f(imax);
% Gets the harmonic number
harm= inputdlg('Harmonic number: Input required');
harm= char(harm);
% Deletes figure 2 if it exists
delfig('Fig2');
% Plots the spectrogram
fig2 ;
imagesc(t',f',X);
axis('xy');
axis([0 max(t) liml lim2]);
colormap(jet);
title('Spectrogram');
xlabel('Time(s)');
ylabel('Frequency(Hz)');
colorbar;


199
should be left outside. The areas outside the range 0 < t < tmax can be clicked on if the user
does not want to remove the beginning or ending segments of the signal.
Figure A.33. Input required window.
Figure A.34. Edit wave window.
The non vibrato parts of the signal will be removed, and the vibrato sample
window and the spectrogram window will be updated with the edited signal. It is
important to note that this changes are made on the computer's memory and do not affect
the .wav file on the hard disk. If the user wants to record the changes in the wav file on
the hard disk, he/she should select the File, Save wave file option. The window shown in
figure A.3 5 will appear on the screen. A directory/file name should be selected by clicking


197
Figure A.31. Instantaneous power spectrum of the amplitude vibrato.
Saving the Vibrato Parameters
The vibrato parameters can be saved into disk for later analysis. It is necessary to
perform the analysis described above in the Frequency and Amplitude Vibrato Analysis
and in the Instantaneous Frequency and Amplitude of the Frequency Vibrato Wave
sections, before saving the parameters.
The File, Save parameters option should be chosen. A window similar to figure
A. 3 2 will be shown. The user should select the directory and file name where the file will
be saved by clicking and/or typing on the desired directory/file name, and click the Save
button to save the parameters onto disk (see table 4.1 in chapter 4 for a list of the
variables saved onto disk). The extension of the data files is .dat and does not have to be
typed, as the software will add it automatically.


110
File kbat20.wav
Blue=Freq; Red=Amp Time(s)
Figure 5.53. Frequency and amplitude vibrato waves for kbat20.wav.
Figure 5.54. Vibrato power spectra for kbat20.wav.


221
% Determines figure to be used
if fig == 'full'
handle= findobj('Tag','Fig4');
else
handle= findobj('Tag','FiglnstFrSpec');
end
figure(handle);
% Gets value and prints it
[1,k]= ginput(1);
fprintf(1\n');
fprintf('Frequency = %5.3f\n',l)
% Clears temporary variables
clear handle 1 k;
M-file instaml.m
% M-file instaml.m
% Calculates and displays the "spectrogram" of the
% amplitude vibrato wave
% Initializes variables
hf= zeros(64,(length(Amax)-14));
af= zeros(polinst+1,(length(Amax)-14));
gf= zeros(1,(length(Amax)-14));
% Calculates the "spectrogram"
for j= 1:((length(Amax)-14));
ameanseg= mean(Amax(j:j +14));
[atemp,gf(j)]= auto((Amax(j:j+14)-ameanseg),polinst,128);
af(:,j)= atemp';
[hf(:,j),wf] = freqz(gf(j),af(:,j),64,l./tsy);
end
% Deletes figinst if it exists
handle= findobj('Tag','Figinst1);
if -isempty(handle)
close(handle);
end;
% Plots the "spectrogram"
figinst;
imagesc((t(1:length(t)-14))',wf',20.*logl0(abs(hf)));
axis('xy');
colormap(jet);
title('Inst freq and amp of amp vib');
xlabel('Time(s)1);
ylabel('Frequency(Hz)');
colorbar;
% Calculates and displays the poles
afpoles= zeros(polinst,length(Amax)-14);
for j= 1:(length(Amax)-14);
afpoles(:,j)= roots(af(:,j));
end


161
Placido Domingo
Figure 8.18 shows the box plots for the amplitude analysis and figure 8.19 shows
the box plots for the frequency analysis.
Figure 8.18. Box plots for the amplitude analysis of Domingo.
Figure 8 .19. Box plots for the frequency analysis of Domingo.



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145
Table 7.2continued.
File name
abs(Yford(l))
angle(Yford(l))
Nf
fmeaninst
ameaninst
fmean
Bjor05.wav
5909.619037
-1.313701378
164
6.324
75.27
2538.917
Bjorll.wav
4340.802087
0.885794937
158
6.376
74.742
2362.466
Bjor25.wav
4478.087709
2.62729487
158
5.904
84.002
2867.674
Ec04.wav
3912.021882
-0.090229752
165
6.204
54.038
2230.399
Ec08.wav
4582.729198
-1.645030337
171
6.32
64.326
2479.949
Ecl6.wav
3109.659544
0.312664547
176
6.15
56.192
2090.381
Pav02.wav
8641.810631
0.074050301
166
5.933
115.447
2794.74
Pav03a.wav
7280.894423
2.709355857
166
5.942
94.951
3223.633
Pavl4.wav
6987.02517
-2.794540998
164
6.029
96.589
2675.084
Pldom01.wav
6797.723235
-0.946049596
168
5.815
84.702
2847.463
Pldom02.wav
6617.304801
2.548543931
169
5.872
87.938
2588.01
Pldom24.wav
8716.904393
0.390141146
165
5.451
104.463
2864.254
kbat01.wav
6306.290785
-1.863819079
180
6.29
72.365
1871.066
Kbat20.wav
4126.232581
-0.552340964
166
6.216
53.672
1323.985
Kbat22.wav
4858.244362
2.411032507
168
6.231
58.524
1382.104
Moncab30.wav
10000.17742
1.605615644
162
5.412
125.575
2988.32
Moncab3 l.wav
7046.839842
-1.270963368
157
5.484
122.878
2814.454
Moncab33.wav
9410.474773
-2.644078226
163
5.476
115.67
2956.343
delosa01.wav
9330.580834
0.672759432
161
5.84
138.791
3110.775
delosa07.wav
11829.28335
-0.633842622
161
6.213
189.599
3931.437
delosa09.wav
8356.391194
-0.180678377
163
6.405
110.957
3173.353
Mean
6792.337964
5.994619
94.31862
Maximum
11829.28335
6.405
189.599
Minimum
3109.659544
5.412
53.672
Std. dev.
2276.922838
0.315313
32.74224
Cl Maximum
11346.18364
6.625245
159.8031
Cl Minimum
2238.492287
5.363993
28.83414


190
Figure A.20. Instantaneous frequency wave
Also, the instantaneous frequency wave and its corresponding power spectrum will
be displayed in new windows on top of the frequency vibrato wave and the frequency
vibrato power spectrum (see figures A. 21 and A. 22).
Figure A.21. Instantaneous frequency wave.


Copyright 1998
by
Jose Antonio Diaz


22
/ = F0(l + 0.0lav sin(2^vi) + 0.01ACr2(/)) (2-10)
Here, A<, and F0 represent the pulse fundamental amplitude and frequency, at and av
correspond to percent amplitude and frequency modulation (called tremolo and vibrato in
this study), ft and fv define the tremolo and vibrato frequency, Ns and Nj are the shimmer
factor and the jitter factor, and ri(t) and r2(t) are pseudo-random functions.
From this model we can see that amplitude and frequency vibrato are modeled as a
sinusoidal variation of the amplitude and frequency of the glottal pulse waveform. Jitter
and shimmer are modeled as random variations in frequency and amplitude.
The inclusion of vibrato in singing samples has the important effect of transforming
a poor synthetic speech quality into a fairly good singing quality.
For the same percentage of frequency and amplitude modulations, frequency
modulations are perceived up to 10 times more effectively than amplitude variations.
The FCSV Software
The FCSV (frequency characterization of singers vibrato) software was developed
for use with Matlab (Diaz, 1995, p.56) for the extraction and analysis of singers vibrato.
It provides a fast and precise measure of the vibrato and its parameters, and gives us a
signal representation of the vibrato wave.
The main menu consists of four options which naturally subdivide into four main
functions:
1. Spec (Spectrogram).
2. Curv (Curve).
3.Calc (Calculations).


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
94 j -g-
A. Antonio Arroyo, Chairman
Associate Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
iQf^del "if'
Donald G. Childers
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy^"! /
A/ HA-'(rS
Howard W. Beck
Associate Professor of Agricultural and
Biological Engineering
1 certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosofihy
Herman Lam
Associate Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Howard B. Rothman
Professor of Communication
Sciences and Disorders


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August 1998
Winfred M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School


88
File bior25.wav
Figure 5.9. Frequency and amplitude vibrato waves for bjor25.wav.
Figure 5.10. Vibrato power spectra for bjor25.wav.


180
File Edit Window Help
Spectrogram
4000, .
FT! 120
r¡- 3000
N
X
>.
g 2000
=3
O'
CD
LL 1000
WV w'yV y y yy wj
o
SAAAAAAAA/WW
MkkMiemHSbh4**MmI
0 0.5 1 1.5
Time(s)
100
80
60
40
20
Figure A.7. Spectrogram showing frequency vibrato wave.
If the calculated frequency vibrato wave does not match the real frequency vibrato
wave, the Get vibrato option can be selected again. This can be done as many times as
needed until the calculated frequency vibrato wave matches the real one. The software
may fail to extract noisy, too wide, or low level frequency vibrato waves.
By selecting the Get parameters option, the software will display the frequency
vibrato wave and its corresponding power spectrum. The window in figure A. 8 will be
displayed to inform the user that an input is required.
Figure A. 8. Input required window.


81
All these singers are or have been renowned professional opera singers who have
been judged to have good vibrato. The samples were selected based on vibrato quality,
low noise level, absence of musical instruments, and length.
Data Analysis
All the samples selected were analyzed using the software developed. Since the
number of graphs created is relatively high only the most important graphs were saved. All
parameters were saved for each sample. The graphs saved for each sample were the
following:
1. Frequency and amplitude vibrato waves.
2. Frequency and amplitude vibrato power spectra.
3. Instantaneous frequency and amplitude of the frequency vibrato.
4. Power spectra of the instantaneous waves.
Figures 5.1 to 5.84 show the four graphs saved for each sample. Figures 5.1, 5.5,
5.9, 5.13, 5.17, 5.21, 5.25, 5.29, 5.33, 5.37, 5.41, 5.45, 5.49, 5.53, 5.57, 5.61, 5.65, 5.69,
5.73, 5.77, and 5.81 show the frequency vibrato wave in blue and the amplitude vibrato
wave in red. All the frequency vibrato waves were filtered with a 10 Hz low pass filter to
reduce noise. Figures 5.2, 5.6, 5.10, 5.14, 5.18, 5.22, 5.26, 5.30, 5.34, 5.38, 5.42, 5.46,
5.50, 5.54, 5.58, 5.62, 5.66, 5.70, 5.74, 5.78, and 5.82 show the power spectra of the
frequency and amplitude vibrato waves in the selected samples. The power spectrum of
the frequency vibrato is shown in blue and the power spectrum of the amplitude vibrato in
red. The prominent characteristic of the frequency vibrato power spectrum is its high peak
at about 6 Hz, which is similar for all samples. This indicates that the wave is almost


19
The original samples were analyzed and several modifications were made to
remove frequency and/or amplitude variations. Then, the original and modified signals
were judged and comparisons were made (Maher and Beauchamp, 1990, p.219).
According to the listeners judgment, the best quality was found in signals with
frequency and amplitude modulations. Signals with only frequency variations were of
secondary quality, followed by signals with only amplitude variations. Signals with neither
amplitude nor frequency variations were found to have the poorest quality (Maher and
Beauchamp, 1990, p.219).
It was noted that amplitude variations affect perception, and are important for
good synthesis quality (Maher and Beauchamp, 1990, p.219).
Maher and Beauchamp reported that the frequency vibrato was nearly sinusoidal
with some drift. Its amplitude, rate and fundamental frequency appear to contain random
components. The following model was used to describe frequency vibrato (Maher and
Beauchamp, 1990, p.219):
/,(/) = fo +dXt)
d{t) = a,r,(t)
(2-1)
(2-2)
dr{f) = d{t) +/Sf{t)n(6v{f) +(/)^ = frequency deviation (2-3)
0V (/) = 2k jo /v0 [l + a3r3 {t)\dt = phase of fundamental (2-4)
Â¥(f) = A[l + t*2r2(t)] = vibrato depth in Hz (2-5)
W =
(2-5)
(2-6)


90
Enrico Carusso
File ec04.wav
Figure 5.13. Frequency and amplitude vibrato waves for ec04.wav.
File Edit Window Help
Freq and amp vibrato power spectra
Figure 5.14. Vibrato power spectra for ec04.wav.


158
Jussi Bjorling
Figure 8.12 shows the box plots for the amplitude analysis and figure 8.13 shows
the box plots for the frequency analysis.
Figure 8.12. Box plots for the amplitude analysis of Bjorling.
Figure 8 .13. Box plots for the frequency analysis of Bjorling.


71
Table 4.1. Variable names and descriptions.
Variable name
Description
pathname
Path to the wave file
fname
File name of the wave file
harm
Harmonic analyzed
fmean
Mean frequency of the frequency vibrato
ftnax
Maximum frequency of the frequency vibrato
fmin
Minimum frequency of the frequency vibrato
fvarhz
Mean frequency variation in Hz of the frequency vibrato
fvarpc
Mean frequency variation in percentage of the frequency vibrato
fvarst
Maximum frequency variation in semi-tones of the frequency vibrato
fvarab
Mean frequency variation above the mean in Hz of the frequency
vibrato
fvarbl
Mean frequency variation below the mean in Hz of the frequency
vibrato
amean
Mean amplitude of the amplitude vibrato
amax
Maximum amplitude of the amplitude vibrato
amin
Minimum amplitude of the amplitude vibrato
avardb
Mean amplitude variation in db of the amplitude vibrato
avarpc
Mean amplitude variation in percentage of the amplitude vibrato
avarab
Mean amplitude variation above the mean in db of the amplitude
vibrato
avarbl
Mean amplitude variation below the mean in db of the amplitude
vibrato
real(Yford(j))
imag(Yford(j))
i = 1:10
Real and imaginary parts of the ten most dominant components of the
frequency vibrato power spectrum
fy(If(j))
j = 1:10
Frequencies of the ten most dominant components of the frequency
vibrato power spectrum
Nf
Number of samples of the frequency vibrato
real(Y aord(j))
imag(Y aord(j))
1 = 1:10
Real and imaginary parts of the ten most dominant components of the
amplitude vibrato power spectrum
fy(IaCi))
j = 1:10
Frequencies of the ten most dominant components of the amplitude
vibrato power spectrum
Na
Number of samples of the amplitude vibrato
real(Yfordinst(j))
imag(Yfordinst(j))
1 = 1:10
Real and imaginary parts of the ten most dominant components of the
instantaneous frequency power spectrum
fy(Ifinst(j)>
1 = 1:10
Frequencies of the ten most dominant components of the instantaneous
frequency power spectrum
real(Y aordinst(j))
imag(Y aordinst(j))
j = 1:10
Real and imaginary parts of the ten most dominant components of the
instantaneous amplitude power spectrum
fy(Iainst(j))
1 = 1:10
Frequencies of the ten most dominant components of the instantaneous
amplitude power spectrum


181
The user should click on the OK button, move the cursor to the frequency vibrato
power spectrum and select the range of frequencies to be analyzed. The lower frequency
should be input first and then the higher frequency. The software will calculate the
frequency of the highest peak in the specified frequency range and the peak amplitude, and
will display the results on the Matlab command window.
The amplitude vibrato wave will be displayed in red on the same window of the
frequency vibrato wave, and the amplitude vibrato power spectrum in red on the same
window of the frequency vibrato power spectrum. The software will display the window
shown in figure A. 8 to indicate that another input is required. The user should click on the
range of frequencies to be analyzed on the amplitude vibrato power spectrum, first the
lower frequency and then the higher frequency. The software will calculate the frequency
of the highest peak in the specified frequency range and the peak amplitude, and will
display the results on the Matlab command window.
The vertical scales for the amplitude vibrato wave and the amplitude vibrato power
spectrum are shown on the right side of the figure. Figures A. 9 and A. 10 show the final
results of the Get parameters option. The mean values of the waves in figure A.9 have
been extracted, consequently, the waves oscillate around 0.


LIST OF TABLES
Table page
4.1. Variable names and descriptions 71
5.1. Singers names and file names 80
5.2. Filter frequency for the instantaneous amplitude waves 83
7.1. Description of the model parameters 139
7.2. Model parameters and statistics 141
8.1. Singers' formant parameters 153
8.2. Relationship between groups and singers names 156
8.3. Singers' formant variation within samples of the same singer 164
xi


118
File moncab33.wav
Figure 5.69. Frequency and amplitude vibrato waves for moncab33.wav.
File Edit Window Help
Freq and amp vibrato power spectra
Figure 5.70. Vibrato power spectra for moncab33.wav.


108
Kathleen Battle
File kbat01.wav
Blue=Freq; Red=Amp Time(s)
Figure 5.49. Frequency and amplitude vibrato waves for kbat01.wav.
File Edit Window Help
Freq and amp vibrato power spectra
0 10 20
Blue=Freq, Red=Amp
200
150
100
50
0
30 40
Frequencv(Hz)
Figure 5.50. Vibrato power spectra for kbat01.wav.


167
result we can see that the autocorrelation method has one advantage against the
covariance method, and is comparable to it in its usefulness, contrary to what is reported
in the literature.
The Instantaneous frequency and Instantaneous amplitude options allow the
observation of the variations in the frequency and amplitude of the frequency vibrato wave
with an accuracy not seen before. These waves show that these two parameters do not
stay constant during the vibrato segment.
The variations in the frequency and amplitude of the frequency vibrato wave were
found to be not random, they are rather sinusoidal containing several frequency
components. The number of sinusoidal components was 2 or 3 in almost all cases.
A slow frequency variation (below 5 Hz) was found in virtually all the amplitude
vibrato waves. This variation was seen as a single peak in 20 of the 21 samples, which
indicates that there is a variation in the amplitude of the glottal pulse during vibrato.
A model was developed to describe the frequency vibrato wave found in the
singing voice. This new model properly describes the patterns observed in all the vibrato
samples in this research. It is based on the sinusoidal model but the amplitude and
frequency of the sine wave vary with time.
A model was also proposed for the amplitude vibrato wave of the singing voice.
This model describes the amplitude variations due to the effect of the frequency variations,
the formants, and the glottal pulse amplitude. The amplitude variations due to the glottal
pulse amplitude were modeled as a sum of two sine waves.
The frequency vibrato model was compared against the pure sinusoidal model
based on an error measure which consisted of the difference between the original and the


220
% Plots the frequency vibrato power spectrum
handle= findobj('Tag', FreqVibCovarAxes');
axes(handle);
cl a;
hold on;
axis auto;
plot(wf,abs(hf) 'g');
limits= axis;
axis([0 40 limits(3) limits(4)]);
hold off;
handle= msgbox('Select range to analyzeInput required');
waitfor(handle);
drawnow;
handle= findobj('Tag','Fig4');
figure(handle);
% Gets input from mouse
[1,k]= ginput(2);
ninf= roundfl(1).*(length(wf)-1)./max(wf) + 1);
nsup= round(l(2).*(length(wf)-1)./max(wf) + 1);
if ninf < 2
ninf =2;
end
if nsup > length(wf)
nsup = length(wf);
end
% Maximum amplitude relative to the mean, frequency,
% and percent of frequency error
[hmax,nmax]= max(abs(hf(ninf:nsup)));
wmax= wf(nmax+ninf-1);
errorpc= ((wf(2)/2) / wmax) 100;
fprintf('\n');
disp('Frequency vibrato model:');
fprintf('Maximum amplitude = %5.3f\n',hmax)
fprintf('Frequency of the maximum amplitude in Hz = %5.3f\n',wmax)
fprintf('Maximum frequency error in percentage = %5.3f\n',errorpc)
% Plots the poles
if ((method == 'aut') | (method == 'cov'))
viewpol(af);
end
% Clears temporary variables
clear Fmaxdec fmeandec handle limits k 1 ninf nsup hmax nmax wmax
errorpc;
M-file getvall.m
function getvall(fig)
% Gets frequency from figure
% Displays message
handle= msgbox('Select frequency to obtain','Input required');
waitfor(handle);
drawnow;


124
File delosa09.wav
-ElWfflii
File Edit Window Help
HEsiO
0 0.5 1 1.5 2
Blue=Freq; Red=Amp Time(s)
Figure 5.81. Frequency and amplitude vibrato waves for delosa09.wav.
Figure 5.82. Vibrato power spectra for delosa09.wav.


54
Figure 4.19. Pole location for the spectrogram.
The properties and advantages of the Wigner distribution and Wavelets for the
calculation of the instantaneous frequency were investigated. The results provided by
these two methods show a resolution comparable to that of the spectrogram, and with
cross terms. Since these methods do not make assumptions regarding the signal
properties, the parametric methods will provide higher resolution and more accurate
results.
In order to determine that the frequency variations are real and not an artifact of
the covariance method, a6Hz sine wave was generated, and the method described above
was applied to it. The spectrogram showed a straight line at 5.97 Hz.
Another Matlab module was developed to calculate the frequency vector and
display it on the spectrogram for comparison purposes. Figure 4.20 shows the frequency
vector of the sample being analyzed as a blue line.


48
File Edil View Analysis Filter Model
-A Vibrato sample
Q. r
of
-D n
X 10
£ -3
< 0
0.5 1
File: Pldoml .wav
1.5
Tlme(s)
File Edit Window Help
Spectrogram
4000
pj' 3000
fe/itV W- \? vZW Wvzvz
>.
S 2000
rr
CD
^ 1000
0
120
100
80
60
40
20
0
0.5 1
Time(s)
1.5
File Edit Window Help
Frequency and amplitude vibrato
5.-100
E
<
-200
0 0.5 1 1.5
Blue=Freq; Red=Amp Tlme(s)
Figure No. 4
File Edit Window Help
HII3
Hj§§§|
Freq and amp vibrato power spectrum
6000
0 10 20
Blue=Freq; Red=Amp
80
60
40
20
0
30 40
Frequency(Hz)
/Vjj
Start
^Micros...
matl..
E Figure...
E Figure...
E Figure...
E Figur...
3:11 PM
Figure 4.12. Aspect of the computers screen.
Help
This function displays a windows that tells the user where to find information
about this software, and who developed it.
Relationship Between the Model Parameters in the Time-Domain and the Z-Domain
Although the model in the Z-domain provides a unique representation of the
frequency or amplitude vibrato, the parameters in the Z-domain do not provide much
information to people not familiar with the Z-transform. Therefore, it was decided to


12
Objective Factors that Determine the Quality of the Vibrato
Extreme extent
The most commonly found deficiency in vibrato is an extreme extent of frequency
and amplitude (Seashore, 1936, p. 149). The excess may be found in frequency or
amplitude, or both. An excessive extent produces a negative response in a refined musical
ear. The following rules may be used safely for judging the vibrato in general (Seashore,
1936, p. 150). However, we should keep in mind that appreciation is a relative subject, and
what is extreme for one person may not be for another, and when deciding what is
excessive we must consider the particular desired purpose.
1. The best average extent of pitch and intensity is that which causes the desired
tone quality, but does not lead to the perception of changes in frequency.
2. Regularity in extent is remarkably important for good vibrato. A large and
irregular extent of the vibrato converts it to an unpleasant flutter. The extent of
the vibrato may be large or small, may increase or decrease, change within a
single tone or in a sequence of tones, but the change must be progressive and
smooth.
3. A relatively small extent of the vibrato fails to contribute to the enhancement of
tone quality in an amount proportional to the smallness.
4. A slow rate makes the pitch change more evident. High rate produces a new
effect known as chatter.
5. In voice vibrato the variations in pitch should be dominant over the variations
in extent.
6. Artistic performance requires variation in extent and rate throughout a
presentation. A uniform vibrato becomes monotonous and fails to expose the
feelings of the performer.
7. In vocal and instrumental performances the artist has more freedom to utilize
the vibrato than he has when playing in an ensemble.


212
% Deletes ampl vibrato power spectrum and mean value line
% from the AmplVibSpecAxes
handle= findobj('Tag','Fig4');
handlel= findobj(handle,'Tag','AmplVibSpecAxes');
axes(handlel);
cl a ;
% Makes the calculations on the filtered signal and plots it
ampl4;
% Clears temporary variables
clear handle;
M-file ellipfrl.m
% M-file ellipfrl.m
% Applies an elliptical filter to the frequency vibrato wave
% Filters the frequency vibrato wave
Fmax= filtfilt(b,a,Fmax);
% Deletes the freq vibrato wave and mean value line (0)
% from the FreqVibCurAxes
handle= findobj('Tag','Fig3');
handle1= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
cl a;
% Deletes the freq vibrato power spectrum and mean value line
% from the FreqVibSpecAxes
handle= findobj('Tag','Fig4');
handlel= findobj(handle,'Tag','FreqVibSpecAxes');
axes(handlel);
cla;
% Makes the calculations on the filtered signal and plots it
freqn4;
% Clears temporary variables
clear handle handlel;
M-file ellipinstaml.m
% M-file ellipinstaml.m
% Applies an elliptical filter to the instantaneous
% amplitude wave
% Filters the instantaneous aplitude wave
Amaxinst= filtfilt(b,a,Amaxinst);


56
Figure 4.22. Power spectrum of instantaneous frequency vector.
Another module calculates the instantaneous amplitude in each segment using the
method described previously, and displays the amplitude in the window of the frequency
vector. Figure 4.23 shows the amplitude vector corresponding to the analyzed sample in
red. The most important feature of this curve is that the variations in amplitude are much
larger than the ones in the original wave. The maximum amplitude in figure 4.23 is 50.425
and the minimum is 2.192. This indicates that the relationship between the maximum and
minimum amplitude is approximately 25 to 1. This is obviously wrong since the frequency
vibrato wave not does change in amplitude that much.
Where was the problem? Was the method being used incorrect? Or was there a
limitation regarding the sample length? If the frequencies were being calculated properly,
why were the amplitudes incorrect?


A. 17. Elliptical filter options 188
A. 18. Instantaneous power spectrum of the frequency vibrato 189
A. 19. Input required window 189
A.20. Instantaneous frequency wave 190
A.21. Instantaneous frequency wave 190
A.22. Instantaneous frequency power spectrum 191
A. 23. Pole location for the instantaneous power spectrum 191
A. 24. Instantaneous power spectrum of the frequency vibrato 192
A.25. Instantaneous frequency wave 193
A.26. Instantaneous amplitude wave 193
A.27. Instantaneous frequency power spectrum 194
A.28. Filtered instantaneous amplitude wave 195
A. 29. Power spectrum of the filtered wave 195
A.30. Short length model options 196
A.31. Instantaneous power spectrum of the amplitude vibrato 197
A.32. Save parameters window 198
A.33. Input required window 199
A.34. Edit wave window 199
A.35. Save wave file window 200
xix


50
number of samples used to calculate the power spectrum as if we were doing an inverse
Fourier transform.
The following is an example to show how the method works. Figure 4.13 shows a
frequency vibrato wave from Placido Domingo. The frequency vibrato wave is shown in
blue. The wave was filtered with a 10 Hz low-pass filter to eliminate noise and improve
the results. Figure 4.14 shows the pole location given by the Covariance method, and
figure 4.15 shows the corresponding power spectra. The FFT power spectrum is shown in
blue and the LPC model spectrum in green. By applying the method described, an
amplitude of 114.049 Hz was obtained from the LPC model spectrum, which agrees with
the signal shown in figure 4.13.
Frequency and amplitude vibrato
Figure 4.13. Frequency and amplitude vibrato waves.


133
follows the original wave more closely than a pure sinusoidal wave, both in amplitude and
frequency.
Figure 6.7. Synthesized wave using the proposed model.
Figure 6.8 shows the error curve for the model in figure 6.7. The errors have been
reduced. The maximum error is 33.63 Hz, the minimum is -14.20 Hz, and the average is
5.91 Hz
Figure 6.8. Error for proposed model.


13
These rules are used in judging different types of vibrato (musical criticism),
objective training, and esthetic theory.
In short, a good vibrato probably contains smooth transitions in rate and extent,
has a nearly sinusoidal shape, is adjusted by the singer to solo and ensemble performances,
and is present in most tones and transitions except where it is not used for specific
purposes.
Vibrato in Different Types of Singing
A study performed by Easley (Easley, 1932, p. 1) shows that in most cases the
frequency and amplitude of the frequency vibrato wave are higher in opera songs than in
concert songs. Higher frequencies and amplitudes are also measured in opera songs
compared to concert songs even when they are sung by the same singers.
When comparing historical and contemporary opera singers with historical and
contemporary Jewish cantors, Rothman (Rothman, Diaz and Vincent, 1998, p.5) found
most of the significant differences between eras, and not between groups within the same
time period. The mean frequency of the frequency vibrato wave is 7 Hz for the historical
group while the frequency of the contemporary group is slower. The amplitude variation
of the frequency vibrato wave is higher for the contemporary group than in the historical
group.
Vibrato in Musical Instruments
Vibrato in the violin
A study made by Rothman (Rothman and Arroyo, 1988, p.7) about the acoustic
parameters in violin vibrato among pre and post World War II performers shows that the


141
Table 7.2. Model parameters and statistics.
File name
abs(Yfordinst(l))
angle(Yfordinst( 1))
abs(Yfordinst(2))
angle(Y fordinst(2))
Bjor05.wav
16.63550291
-0.30534487
9.130444951
1.53322086
Bjorll.wav
26.1680386
0.846559536
15.71893743
0.995303879
Bjor25.wav
11.06213768
-2.685424186
9.169079561
0.935104151
Ec04.wav
14.80996131
2.390673464
10.11211575
2.831882174
Ec08.wav
14.04505696
0.567957886
12.88104359
0.495802703
Ecl6.wav
14.81928001
-0.818709622
14.08774971
0.821645465
Pav02.wav
18.18044774
1.012301959
11.7520531
0.915245968
Pav03a.wav
14.73434943
-0.623492104
9.042248227
0.446187967
Pavl4.wav
9.196974394
-1.258810953
7.936250059
-1.409314203
Pldom01.wav
18.8440584
0.742494929
12.50905672
1.45139915
Pldom02.wav
13.11743306
0.953294087
8.327400615
0.532119825
Pldom24.wav
13.02653446
0.751954197
12.21040155
0.598065186
kbat01.wav
15.45786648
-1.409008409
9.984129807
3.092294482
Kbat20.wav
19.96453157
0.206835822
18.48602123
-2.03437136
Kbat22.wav
10.61398309
-2.713008033
6.573953225
0.496238842
Moncab30.wav
12.54787572
0.982152726
10.21093257
1.130713968
Moncab31.wav
18.80036417
0.771067755
18.49931631
1.996529078
Moncab33.wav
22.24263496
2.011541037
17.32084828
-1.696583523
delosa01.wav
7.581071164
1.315835912
5.527726838
2.75033617
delosa07.wav
10.91641058
1.288394617
10.56296384
0.697792849
delosa09.wav
13.17872839
0.692979784
13.06970409
-0.337357273
Mean
15.04491624
11.57677988
Maximum
26.1680386
18.49931631
Minimum
7.581071164
5.527726838
Std. dev.
4.375034222
3.576104254
Cl Maximum
23.79498468
18.72898839
Cl Minimum
6.294847797
4.424571371


CHAPTER 4
ALGORITHM DEVELOPMENT
Conversion of the FCSV Software (Frequency Characterization of Singers' Vibrato) from
Matlab 4.2 to 5.1
This study began by converting each of the m-files in the FCSV software from
Matlab 4.2 to 5.1 since FCSV calculates the frequency and amplitude vibrato waves that
are going to be used for the analysis and determination of the vibrato model.
At the same time that each m-file was being converted, the user interface was
being changed to make it more flexible, and provide more information simultaneously on
the screen. The added flexibility was provided by separating some of the functions that
were performed by the activation of a single button, into two or more options.
The software was modified from button-driven to menu-driven, which improves
the screen utilization by allowing more space for figures. Also, parameters that were fixed
in the past, can now be changed through sliders and pop-up menus.
Figure 4.1 shows the window that appears when the software is activated. This
window contains axes for the display of the original vibrato wave, and a menu bar with all
the choices available through the software. All the functions are selected and controlled
through the use of this menu.
37


102
Placido Domingo
File pldom01.wav
Figure 5.37. Frequency and amplitude vibrato waves for pldom01.wav.
Freq and amp vibrato power spectra
Figure 5.38. Vibrato power spectra for pldom01.wav.


67
Figure 4.31. Options for the short length model.
Figure 4.32. View window option.
Filter. Elliptical. Frequency Vibrato. Instantaneous Frequency and Instantaneous
Amplitude
By using this option the user can apply an elliptical filter to the instantaneous
frequency or instantaneous amplitude waves. The filter parameters can be changed by
selecting the Edit, Options, Elliptical filter option.


72
Figure 4.34 shows the frequency and amplitude vibrato waves obtained from the
spectrogram for signal valid 1 .wav. The frequency vibrato wave is shown in blue and the
amplitude vibrato wave in red. The frequency vibrato wave is a perfectly straight line, as it
should be for a constant tone. The amplitude vibrato wave has some variations of very
small magnitude, which reach a maximum of 2x1 O'6 db. These variations are small
compared to the mean amplitude value of 133.616 db; therefore, it can be said that there is
no error in the amplitude calculation either.
Figure 4.34. Frequency and amplitude vibrato waves.
Figure 4.35 shows the power spectra of the waves in figure 4.34. The frequency
vibrato power spectrum is shown in blue and the amplitude vibrato spectrum in red. The
frequency vibrato spectrum is flat, indicating no oscillations in the frequency vibrato wave
The amplitude vibrato spectrum has a peak at 9 Hz of 3.7x1 O'5 units in amplitude, which
indicates that the oscillations in the amplitude vibrato wave are very small.


150
Enrico Caruso
Figure 8.3 shows the singers' formant of segment 25 in sample ec04.wav.
Figure 8.3. Singers' formant in sample ec04.wav.
Luciano Pavarotti
Figure 8.4 shows the singers' formant of segment 38 in sample pav02.wav.
Figure 8.4. Singers' formant in sample pav02.wav.


BIOGRAPHICAL SKETCH
Jose Antonio Diaz was born in Caracas, Venezuela, in March 1967. He started his
studies in elementary school in 1972 in Valencia, Venezuela. In 1977, he finished
elementary school and started middle school in the same city. In 1982, he finished high
school.
He started his undergraduate studies at the University of Carabobo in Venezuela in
the year 1983, to pursue the electrical engineering degree, which he received in 1990. In
1991, he started his professional career as a full-time engineer at El Palito refinery, and
part-time professor at the University of Carabobo, where he taught the electronics circuits
I laboratory. By the year 1994, he moved to the United States to pursue his master's
degree in electrical engineering at the University of Florida, which he received in
December 1995.
In 1996, he entered the Ph D. program in Electrical Engineering at the University
of Florida, which he expects to finish in August 1998. His main concentration areas are
digital signal processing, computers, and communications. He is a member of Tau Beta Pi
and Eta Kappa Nu.
234


89
Figure 5.11. Instantaneous waves for bjor25.wav.
Figure 5.12. Power spectra of the instantaneous waves in bjor25.wav.


30
The autocorrelation method
In the autocorrelation method the AR parameters are determined by minimizing an
estimate of the prediction error power:
P =
x(ri) + ^a(k)x(n -k)
k=1
2
(2-31)
It is assumed that x(0), x(l),...,x(N-l) are known. The samples of x(n) outside the
0, range are equal to 0 in 2-31. After minimizing and manipulating 2-31 we get
the following set of equations in matrix form (Kay, 1988, p.221):
4(0) 4c ("O " 4r[-(^-l)]
'(!)'
40)"
40) 4(0) 4[-0>-2)]
a{2)
- _
4(2)
_4Cp-i) L(p~2) 4(0)
4 (p)_
where
4(*) = T7 Z **()*( + *) for k = 0, l,...,p
n=0
4(*) = r\-k) for k = -(p-1), -(p-2),...,-l (2-33)
Equations 2-33 are known as the biased autocorrelation function estimator. The
estimate of the white noise variance can be obtained from the estimate of pm¡ by using
?=L(0) + a(k)f(-k) (2-34)
k=\
The covariance method
The parameters for the covariance method can be found by minimizing the
prediction error power:


188
Figure A. 17. Elliptical filter options.
The input required window shown in figure A. 8 will appear to indicate to the user
the need for an input. The user should click on the OK button, move the mouse pointer to
the frequency vibrato power spectrum, and select the range of frequencies to analyze, first
on the lower frequency and then the higher frequency. The software will calculate the
parameters for the filtered frequency vibrato wave and will show them on the Matlab
command window.
The instantaneous frequency wave of the frequency vibrato can be obtained by
choosing the Model, Short length, Frequency vibrato, Instantaneous frequency option.
The software will calculate an instantaneous power spectrum of the frequency vibrato
wave and will display it in a new window on top of the sample spectrogram (see figure
A. 18).


101
Figure 5.35. Instantaneous waves for pavl4.wav.
Figure 5.36. Power spectra of the instantaneous waves in pavl4.wav.


209
% Plots the frequency vibrato wave on the spectrogram
hold on;
plot(t',Fmax, 'b-') ;
hold off;
% Clears temporary variables
clear handle j k 1 iinf isup width;
M-file curvfrl.m
% M-file curvfrl.m
% Calculates and plots the instantaneous frequency wave
% Displays message
handle= msgbox('Select range of frequenciesInput required')
waitfor(handle);
drawnow;
handle= findobj('Tag','Figlnst');
figure(handle);
% Gets input from mouse
[1, k]= ginput(2);
isup= round(k(1).*(length(wf)-1)./max(wf) + 1);
iinf= round(k(2).*(length(wf)-1)./max(wf) + 1);
% Calculates the instantaneous frequency wave
width= isup-iinf;
for j = l:size(hf,2)
if iinf < 1
iinf = 1;
end
if isup > size(hf,l)
isup = size(hf,l);
end
[Amaxinsttemp(j),imaxinst(j)] = max(hf(iinf:isup,j));
imaxinst(j)= imaxinst(j) + iinf 1;
iinf= imaxinst(j) round(width./2);
isup= iinf + width;
end
Fmaxinst= wf(imaxinst)';
% Deletes figinst
handle= findobj(1 Tag', 'Figlnst');
close(handle);
% Plots the spectrogram
figinst;
imagesc((t(1:length(t)-14)) ',wf',20.*logl0(abs(hf))) ;
axis('xy');
colormap(jet);
title('Inst freq and amp of freq vib');
xlabel('Time(s)');
ylabel('Frequency(Hz)');
colorbar;


223
% Clears temporary variables
clear fmeanseg atemp anal handle j;
M-file loadwavl m
% M-file loadwavl.m
% Loads the vibrato sample into memory
clear;
% Deletes figures 2, 3, 4, figinst, figinstfrcur,
% and instfrspec if they exist
delfig('Fig2');
delfig('Fig3');
delfig('Fig4');
delfig('Figinst');
delfig(1FiglnstFrCur');
delfig('FiglnstFrSpec');
% Reads the wave file and displays it
[fname pathname]^ uigetfile('*.wav','Load wave file (.wav)');
if fname ~= 0
[x,fs,format]= loadl6([pathname fname]);
ts= 1/fs;
tmax= ts (length(x)-1);
t= 0:ts:tmax;
handle= findobj('Tag','Figl');
figure(handle);
plot(t,x);
% This fixes a Matlab 5 bug with the plot function
% set(handle,'BackingStore','off');
set(handle,'Renderer','zbuffer');
% Improves the window aspect
limits= axis;
axis([0 tmax limits(3) limits(4)]);
title('Vibrato sample1);
xlabel(['File: fname Time(s)']);
ylabel('Amplitude(spl)');
% Variable initialization
varinit;
% Clears temporary variables
clear limits handle;
end
M-file medaml m
% M-file medaml.m
% Applies a median filter to the amplitude vibrato wave
% Filters the amplitude vibrato with a median filter
Amax= medfiltl(Amax,3);


73
Figure 4.35. Frequency and amplitude vibrato power spectra.
The parameters calculated by the software for the signals in figure 4.34 are shown
below.
Frequency vibrato:
Mean frequency = 2993.115
Maximum frequency = 2993.115
Minimum frequency = 2993.115
Mean frequency variation in Hz = 0.000
Mean frequency variation in percentage = 0.000
Maximum frequency variation in semi-tones = 0.000
Amplitude vibrato:
Mean amplitude = 133.616
Maximum amplitude = 133.616
Minimum amplitude = 133.616
Mean amplitude variation in db = 0.000
Mean amplitude variation in percentage = 0.000


CHAPTER 7
STATISTICAL ANALYSIS
Model Parameters
Table 7.1 shows the parameters that were used in the proposed mathematical
model and a description of each one.
Table 7 .1. Description of the model parameters.
Parameter
Description
abs(Yfordinst(j)), j=T:3
Absolute value of the three most dominant components of the
instantaneous frequency power spectrum
angle(Yfordinst(j)), j=l :3
Phase angle of the three most dominant components of the instantaneous
frequency power spectrum
Frfinst(j), j=l:3
Frequency of the three most dominant components of the instantaneous
frequency power spectrum
abs(Y aordinst(j)), j= 1:3
Absolute value of the three most dominant components of the
instantaneous amplitude power spectrum
angle(Y aordinst(j)). j= 1:3
Phase angle of the three most dominant components of the instantaneous
amplitude power spectrum
Frainst(j), j=l:3
Frequency of the three most dominant components of the instantaneous
amplitude power spectrum
abs(Yford(l))
Absolute value of the most dominant component of the frequency vibrato
power spectrum
angle(Yford(l))
Phase angle of the most dominant component of the frequency vibrato
power spectrum
Nf
Length in samples of the frequency vibrato wave
ftneaninst
Average value of the instantaneous frequency wave
ameaninst
Average value of the instantaneous amplitude wave
fmean
Average value of the frequency vibrato wave
abs(Yaord(j)), j=1:2
Absolute value of the two most dominant components of the amplitude
vibrato power spectrum
angle(Y aord(j)), j=1:2
Phase angle of the two most dominant components of the amplitude
vibrato power spectrum
fra(j),j=l:2
Frequency of the two most dominant components of the amplitude vibrato
power spectrum
Na
Length in samples of the amplitude vibrato wave
139


52
Figure 4.16. Pole location for the autocorrelation method.
Figure 4.17. Power spectrum for the autocorrelation model.
Calculation of the Instantaneous Frequency and Amplitude of the Frequency Vibrato
Wave
An easy to way to determine the symmetry of the vibrato wave is by calculating its
instantaneous frequency and amplitude. If the signal is symmetric, the frequency and
amplitude should remain constant.


6
As a result of the variation in the frequency of oscillation of the vocal folds, all the
upper harmonics vary accordingly and in phase with the fundamental frequency. The
frequency variation of all the harmonics is accompanied by an amplitude variation.
It can be inferred with relatively high confidence that the production of vibrato will
produce a change in the overall amplitude of the sung tone. An increase in the fundamental
frequency causes a shift in the strongest harmonic closer or farther from the first formant.
Therefore, the amplitude of that harmonic will increase or decrease, and correspondingly
change the overall amplitude.
Alternatively, the mean frequency of the strongest harmonic may occur at the same
frequency as the formant. Consequently, the harmonic varies above and below the
frequency of the first formant. In this case, the rate of the amplitude vibrato will be twice
the rate of the frequency vibrato.
The main effect of the vibrato on the listener is caused by the frequency vibrato
(Sundberg, 1987, p. 165). If the extent of the amplitude vibrato is small, which happens
when the strongest harmonic is far from the formant, or if it is large, which happens when
the strongest harmonic is closer to the formant, the effect on the listener is approximately
the same.
Effect of Vibrato on Pitch Perception
Vibrato corresponds to a periodic variation in the fundamental frequency, while the
fundamental frequency determines the pitch we hear. Therefore, this variation affects pitch
perception. One might hypothesize that the pitch from a vibrato tone is not as accurately
perceived as the pitch from a constant note. However, if this were true, the reason for


152
Victoria DeLosAngeles
Figure 8.7 shows the singers' formant of segment 17 in sample delosa01.wav.
Figure 8.7. Singers' formant in sample delosa01.wav.
Singers Formant Variation Among Six Different Singers
This section describes how the singers' formant varies among the six different
singers. The variations in amplitude and frequency were analyzed. The frequency and
amplitude of each .wav file were calculated as the average of the three measures made on
each of them. These values are shown in table 8.1.
The comparisons were made using the analysis of variance test. The confidence
level used was 95 percent. Therefore, p values smaller than 0.05 will cause the rejection of
the hypothesis that the mean values of each group are the same. Each of the singers
represents a group and each group had 3 samples.


123
Figure 5.79. Instantaneous waves for delosa07.wav.
Figure 5.80. Power spectra of the instantaneous waves in delosa07.wav.


107
Figure 5.47. Instantaneous waves for pldom24.wav.
Figure 5.48. Power spectra of the instantaneous waves in pldom24.wav.


27
present in signal processing applications. Any observed noised should be modeled by the
ARMA process by changing its parameters.
The transfer function H(z) between the input and output of the ARMA model are
given by
H(z) =
m
A(z)
(2-17)
where
B(z) = ^b(k)z~k = Z-transform of MA branch (2-18)
k=0
A(z) = ^a(k)z k = Z-transform of AR branch (2-19)
k=0
It is assumed that all the zeros of A(z) are inside the unit circle of the Z-plane. In
this case H(z) is guaranteed to be a stable, causal filter. The parameters a(k) are called the
autoregressive coefficients, and b(k) the moving average coefficients.
The Z-transform of the autocorrelation function at the output of a linear filter
(P^z)) is related to the autocorrelation at the input (Puu(z)) by the following equation:
P (2) = H(z)H (1 / z' )/> (z) = P (*) (2-20)
A(z)A (1/z )
When equation 2-20 is evaluated along the unit circle it becomes the power
spectral density (PSD) of Pxx(f). Frequently u(n) is assumed to be a white noise process of
zero mean and variance a2. The PSD of this type of noise is a2. Therefore, the PSD given
by equation 2-20 becomes:
B(f)
A(f)
2
(2-21)


148
Figure 8.1. FFT and LPC power spectra.
The software was tested with different numbers of poles ranging from 5 to 14. For
a low number of poles (5) the LPC spectrum was too smooth and with 14 poles the LPC
spectrum was starting to fit the FFT spectrum closely instead of giving the envelope. The
best results were obtained with 11 poles.
The software divides each sample into segments of 1024 samples and randomly
selects three segments per sample. In this way three sets of parameters per one wav file
were obtained. The file name, path, number of poles, segment number, and the frequency
and amplitude of the singers' formant are saved in ASCII format.


9
7. It can be difficult for well trained singers to produce songs or even a tone
without the use of vibrato. In this aspect a wide range of individual differences
can exist.
8. Vibrato is commonly found in emotional speech. There is almost no vibrato in
common speech, but for emotional conversation when sustained vowels
appear, vibrato tends to appear.
9. String players tend to use vibrato on all sustained tones. It is likely that all
professional violinists use vibrato in most of the sustained tones, except for
some specific effects.
10. Band or orchestral instruments can produce vibrato, but its use is usually
discouraged for woodwind or brass instruments in most cases. Probably, the
main cause for the low percentage of use is difficulty in controlling it properly
and easily, and not because it is not desirable.
11. Vibrato occurs in the sincere laughter of the adult and in the energetic crying
of a child. It is likely to be a characteristic of the neuromuscular organism
which appears under emotional situations.
In short, it can be said that a variation of tone in the form of a periodic increase
and decrease in pitch is nearly universal in good singing, is commonly found in
instruments, especially in string instruments, and is often encountered in emotional speech.
The Desirability of the Vibrato
The fact that the vibrato is almost always present in good singing and is universally
accepted in string instruments does not demonstrate that it is desirable. Moreover, not all
types of vibrato are desirable.
The desirability of the vibrato is confirmed by the universality of its use, its
involuntary nature, its presence in instruments, its preference over precision, and its
importance in tone quality.


119
Figure 5.71. Instantaneous waves for moncab33.wav.
File Edit Window Help
Power spectra of inst freq and amp
Blue=Freq; R e d = Am p Frequency(Hz)
Figure 5.72. Power spectra of the instantaneous waves in moncab33.wav.


207
% Calculates the instantaneous frequency wave
width= isup-iinf;
for j = l:size(hf,2)
if iinf < 1
iinf = 1;
end
if isup > size(hf,l)
isup = size(hf,l);
end
[Amaxinst(j),imaxinst(j)] = max(hf(iinf:isup,j));
imaxinst(j)= imaxinst(j) + iinf 1;
iinf= imaxinst(j) round(width./2);
isup= iinf + width;
end
% Calculates the instantaneous amplitude wave
for j = l:size(hf,2)
Amaxinst(j)= hf(imaxinst(j),j);
for k=l:l
if ((imaxinst(j)-k)>=1) & ((imaxinst(j)+k)<=64)
Amaxinst(j)= Amaxinst(j) + hf((imaxinst(j)-
k),j)+hf((imaxinst(j)+k),j);
else
disp('Warning: Error calculating Instantaneous amplitude
vector');
end
end
Amaxinst(j)= 2.*Amaxinst(j)./128;
end
% Deletes figinst
handle= findobj('Tag','Figinst');
close(handle);
% Plots the spectrogram
figinst;
imagesc((t(1:length(t)-14))',wf',20.*logl0(abs(hf)));
axis('xy');
colormap(jet);
title('Inst freq and amp of freq vib');
xlabel('Time(s)');
ylabel('Frequency(Hz)');
colorbar;
% Plots the instantaneous frequency wave on the spectrogram
hold on;
plot((t(1:length(t)-14))',wf(imaxinst),1b-');
hold off;
% Clears temporary variables
clear handle j k 1 iinf isup width;


CHAPTER 9
CONCLUSIONS AND FUTURE WORK
Conclusions
Frequency and Amplitude Vibrato Analysis
The MMSV software simultaneously provides ease of use and flexibility. The
software was made user friendly by the use of pull down menus, sliders, buttons,
windows, etc. This was made possible by the use of the Matlab Graphics User Interface
Development Environment tool (GUIDE), which reduced significantly the development
time.
The software was made flexible by providing several options for the editing,
processing, and analysis of the vibrato sample. The user can choose between a full length
model and a short length model. There are four possible methods for the full length model:
periodogram, autocorrelation, covariance, and MUSIC (Multiple Signal Classification).
Also, options are provided for changing the spectrogram, filter, full length model, and
short length model parameters. However, not all the parameters in the software can be
changed. Parameters that can negatively affect the results cannot be changed. For
example, the segment length in the spectrogram, and ripple for the elliptical filter.
The default Matlab window size was changed to fit several windows on the screen
at the same time. This allows comparisons among the different views of the vibrato signal.
Also, colors were used extensively to improve the graphs readability.
165


A MATHEMATICAL MODEL OF SINGERS' VIBRATO
BASED ON WAVEFORM ANALYSIS
By
JOSE ANTONIO DIAZ
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998


196
poles can be varied from 1 to 11 poles. The default value is 3. In most cases 7 poles will
produce a good result. The user should click on the Accept button to make the change
effective.
Figure A.30. Short length model options.
After choosing the Model, Short length, Amplitude vibrato option, the software
will display the instantaneous power spectrum on top of the vibrato sample spectrogram as
shown in figure A.31. No curves nor parameters are extracted from this figure. Its main
purpose is to show the amplitude vibrato variations in a qualitative way.
The pole location for the instantaneous power spectrum will be shown in a window
similar to figure A.23. The user should click on the OK button to close this window when
done with it to avoid window proliferation.


sinusoidal components. The variations in the amplitude vibrato wave were caused by three
factors: frequency variations, formants, and the amplitude of the glottal pulse. The
variations due to the amplitude of the glottal pulse are shown as a single sine wave.
A new model was developed for the frequency and amplitude vibrato. The
frequency vibrato model is based on the sinusoidal model, but with the frequency and
amplitude of the wave varying over time. These variations are in turn sinusoidal, having
three sinusoidal components each. The amplitude vibrato model describes the variations
due to frequency variations, formants and the amplitude of the glottal pulse. The variations
due to the amplitude of the glottal pulse are modeled as the sum of two sine waves.
A study was made of the singers' formant. A software tool was developed to
measure the singers' formant frequency and amplitude. Sample sets were analyzed using
software and the parameters calculated. A statistical analysis of the parameters showed
that the frequency varied among the different singers. A comparison made among samples
of the same singer showed that the amplitude varied in 4 out of 6 singers, and the
frequency varied in 3 out of 6.
xxi


86
File bjorl l.wav
Figure 5.5. Frequency and amplitude vibrato waves for bjorl 1 .wav.
Figure 5.6. Vibrato power spectra for bjorl l.wav.


146
Table 7.2continued.
File name
abs(Yaord(l))
angle(Yaord(l))
abs(Yaord(2))
angle(Y aord(2))
fra(l)
fta(2)
Na
Bjor05.wav
110.1495569
-1.175426972
58.84092951
2.070095111
0.491
0.981
164
Bjorll.wav
316.9036059
1.632883467
187.2870911
1.638342988
0.509
1.019
158
Bjor25.wav
82.99109387
1.476066246
45.72203046
-0.814678237
1.019
3.056
158
Ec04.wav
67.84109014
-2.147899489
37.66677046
-1.996239541
0.488
0.975
165
Ec08.wav
23.57872391
-2.181702963
21.50706865
1.877872327
3.294
1.412
171
Ecl6.wav
90.56036467
-0.39762901
49.02121445
-2.130353552
0.457
1.372
176
Pav02.wav
99.98171261
-1.951321277
49.55114209
-0.478801657
0.485
2.424
166
Pav03a.wav
92.52291187
0.561392438
65.27183039
-0.836032793
0.485
3.878
166
Pavl4.wav
44.51632339
-1.519601727
0
0
0.981
0
164
Pldom01.wav
130.0681718
-1.021012501
63.56640906
-1.530575567
0.479
0.958
168
Pldom02.wav
46.78183665
2.494910818
39.27091037
2.481646061
1.429
2.857
169
Pldom24.wav
91.5618424
-2.041076835
43.96561975
-0.552075184
0.488
0.975
165
kbat01.wav
177.1607963
1.968862617
97.93108885
2.014267261
0.447
0.894
180
Kbat20.wav
223.928329
-2.439205653
108.4897281
-2.14067005
0.485
0.97
166
Kbat22.wav
92.90212864
-2.530878153
44.05325416
2.858600301
1.437
0.479
168
Moncab30.wav
329.3090003
1.854629913
153.7872518
1.699058185
0.497
0.994
162
Moncab31.wav
277.6432748
1.366421872
78.15296314
2.572896781
0.513
1.025
157
Moncab33.wav
111.8025057
1.617422192
0
0
0.494
0
163
delosa01.wav
143.1976603
-0.855406722
52.87502677
3.072810334
0.5
2.499
161
delosa07.wav
175.3615807
-2.026813553
105.5745527
-1.080174848
0.5
1
161
delosa09.wav
171.7470522
1.982746348
85.48473062
2.15470768
0.494
1.481
163
Mean
138.1195029
73.05366381
0.761
1.539
Maximum
329.3090003
187.2870911
3.294
3.878
Minimum
23.57872391
21.50706865
0.447
0.479
Std. dev.
84.27966133
41.0739495
0.641
0.904
Cl Maximum
306.6788256
155.2015628
2.043
3.348
Cl Minimum
-30.4398197
-9.09423518
-0.52
-0.27


34
There are several groups of data available for analysis, the largest being 12-bit
samples. The total number of 12-bit samples is 574, from which 163 were used for
analysis in my masters thesis since they contained good vibrato samples (Diaz, 1995,
p.78). There is also a substantial amount of 16-bit data in wav format which has been
collected in the department of Communication Sciences and Disorders at the University of
Florida.
The characteristics of a good vibrato can be obtained by applying FCSV to the
data, using perceptual judgments, and correlating the perceptual judgments with the
curves and parameters. There is evidence (Horii, 1989, p. 1) that a good vibrato is
produced by a symmetric frequency vibrato curve, but there are no clear results about the
effects of the amplitude vibrato on perception, perhaps because it has not been studied in
depth. The perceptual judgments will be provided by Dr. Rothman, a professor in
communication sciences, who has a lot of experience in this area and is involved in this
Ph D. thesis as a committee member.
A mathematical model for good vibrato will be developed using the software, data
and perceptual judgments. The software will provide parameters for the vibrato samples.
The good vibrato samples will be identified using the perceptual judgments and a model
that fits them will be developed. This model will describe the frequency and amplitude
characteristics of good vibrato. The process of finding the right model will involve the
testing of several models. Due to the sinusoidal shape of the frequency vibrato curve, an
all poles model should fit properly. The amplitude vibrato curve is more complicated and
will require a more detailed analysis of the model to be used. The errors of each model will


168
synthesized wave. Two test cases were selected: one sample from a male singer and one
from a female singer. In the first case the pure sinusoidal model had an average error of
16.01 Hz, while the proposed frequency vibrato model had an error of 6.68 Hz. In the
second case the pure sinusoidal model had an error of 20.30 Hz, and the proposed model
an error of 5.91 Hz.
The amplitude vibrato model was tested with the same samples used to test the
frequency vibrato model. The synthesized waves were plotted against the original waves
and the error between the two was measured. The errors obtained were 0.344 db and
0.523 db, which are satisfactory values.
The analysis of the parameter statistics shows that there is high variance in the
parameter values. This can be attributed to several causes:
1. The parameter values vary randomly from one vibrato sample to another one.
2. The parameter values are different for each singer.
3. There are different types of vibrato, that is, the parameters may vary depending
on the vibrato sample.
4. The parameter values are dependent on the time the singer lived, that is,
contemporary singers parameters may be different from those of historic
singers.
In an informal analysis it was noticed that the collected data tends to support the
fourth hypothesis described above. There seems to be a difference in the vibrato model
parameters between contemporary and historic singers.
The amplitude vibrato variations at 3 times the frequency vibrato are caused by the
presence of a formant and a valley in the range of frequencies swept by the frequency
vibrato wave.


179
Figure A. 5. Input required window.
The user should click on the OK button and move the mouse pointer to the
spectrogram window. The mouse pointer will appear like a "cross hair. The user should
select a harmonic by clicking above and below the starting point of the harmonic, this is,
first above and then below the frequency value of the harmonic at t = 0. The window
shown in figure A6 will appear prompting the user for the harmonic number. The user
should click on the white rectangular area to place the cursor in it, type the harmonic
number and click the OK button. The software will extract the frequency and amplitude
vibrato waves and display the frequency vibrato wave in blue on top of the spectrogram.
Figure A. 7 shows the results for harmonic number 6.
Figure A.6. Window for harmonic number input.


43
Figure 4.6. Filter frequency response.
Analysis, Spectrogram
This function calculates and displays the vibrato sample spectrogram. A color bar
was added to show the amplitude scale. An example is shown in figure 4.7.
Figure 4.7. Example of spectrogram.


96
Luciano Pavarotti
File pav02.wav
Figure 5.25. Frequency and amplitude vibrato waves for pav02.wav.
Figure 5.26. Vibrato power spectra for pav02.wav.


26
,j(2tr/N)kn
(2-14)
The Fourier series coefficients X(K) can be obtained from x(n) by the relation:
X(k) = £x(y(2*W)te
(2-15)
The sequence X(k) in equation 2-15 is periodic with period N, that is, X(k) =
X(N+k) for any integer k.
The Fourier series coefficients can be interpreted in two ways: as a sequence of
finite length given by equation 2-15 for k = 0, 1,...N-l, and zero otherwise, or as a
periodic sequence valid for all N as given by equation 2-15. Both interpretations are valid
since equation 2-14 only makes use of the values of X(k) for 0 < k < N-l.
Equations 2-14 and 2-15 constitute an analysis/synthesis pair and are usually
referred to as the discrete Fourier series of a periodic sequence.
Linear Prediction Coding
Definitions
Many discrete time random processes can be properly approximated by a time
series or rational transfer function model. In this model, the input sequence u(n) and the
output sequence x(n) being modeled are related by the following equation:
p
x(ri) = -~y a(k)x(n k) + yb{k)u{n k) (2-16)
This general lineal model is called an ARMA (auto regressive moving average)
model. The driving noise u(n) in the ARMA model is not the observation noise usually


51
Figure 4.14. Pole location for the covariance method.
Figure 4.15. Power spectrum for the covariance model.
Figure 4.16 shows the pole location calculated by the autocorrelation method, and
figure 4.17 shows the corresponding power spectra. The FFT power spectrum is shown in
blue and the LPC spectrum in green. Using the method described an amplitude of 88.380
Hz was obtained from the LPC model spectrum, which also agrees with the signal in
figure 4.13.


46
Model. Full Length Model
This option allows the application of a parametric model to the frequency and/or
amplitude vibrato waves. The frequency response of the model is plotted in the same
figure of the FFT power spectrum to compare how well the model matches the FFT
power spectrum (see figure 4.10). Different colors are used to facilitate the reading. A
green curve represents the frequency vibrato model and a pink curve the amplitude vibrato
model. The frequency and amplitude of the highest peak and the error in the frequency
measure are calculated, and the pole locations are shown in a separate window (see figure
4.11). The error in the frequency measure is 0.7 percent, or lower when decimation is
performed, compared to 5 percent with the FFT power spectrum. This represents a
reduction of almost 90 percent.
Figure 4 .10. LPC model of the frequency vibrato.


116
File moncab31 .wav
Figure 5.65. Frequency and amplitude vibrato waves for moncab31 .wav.
Figure 5.66. Vibrato power spectra for moncab31.wav.


231
Hakes, J., Shipp, T., and Doherty, T., Acoustic Characteristics of Vocal Oscillations:
Vibrato. Exaggerated Vibrato. Trill, and Trillo. Journal of Voice, Vol.l No. 4, p.
326-331, 1988.
Hakes, J., Shipp, T., and Doherty, T., Acoustic Properties of Straight Tone. Vibrato. Trill,
and Trillo. Journal of Voice, Vol.l No. 2, p. 148-156, 1987.
Hattwick, M., The Vibrato in Wind Instruments. University of Iowa Studies in the
Psychology of Music, p. 1-5, 1932.
Horii, Y., Frequency Modulation Characteristics of Sustained /aJ sung in Vocal Vibrato.
Journal of Speech and Hearing Research, p. 1-8, December, 1989.
Kay, S. M., Modem Spectral Estimation: Theory and Application. Prentice Hall,
Englewood Cliffs, New Jersey, 1988.
Kwalwasser, J., The Vibrato. Psychol. Monog., No. 1, p. 84-108, 1926.
Large, J., An Air Flow Study of Vocal Vibrato. Transcripts of the Eighth Symposium:
Care of the Professional Voice, p. 39-45, June, 1979.
Maher, R., and Beauchamp, J., An Investigation of Vocal Vibrato for Synthesis. Applied
Acoustics, p. 219-245, 1990.
Mathews, M., and Pierce, J., Current Directions in Computer Music Research. The MIT
press, Cambridge, Massachusetts, 1989.
Matlab User's Guide. The Math Works, Natick, Masachussetts, 1992.
Metfessel, M., Sonance as a Form of Tonal Fusion. Psychol. Rev., p. 459-466, 1926.
Metfessel, M., Technique for Objective Studies of the Vocal Art. Psychol. Monog., No. 1,
p. 1-40, 1926.
Miller, R., The Structure of Singing. Schirmer Books, New York, New York, 1986.
Oppenheim, V. A., and Schafer, R. W., Discrete-time Signal Processing. Prentice Hall,
Englewood Cliffs, New Jersey, 1989.
Prame, E., Measurement of the Vibrato Rate of Ten Singers. Journal of the Acoustical
Society of America, p. 1979-1984, October 1994.
Prame, E., Vibrato Extent and Intonation in German Lied Singing. KTH Speech, Music
and Hearing Quarterly Progress and Status Report, p. 1-8, April, 1996.


142
Table 7.2continued.
File name
abs(Yfordinst(3))
angle(Yfordinst(3))
Frfinst(l)
Frfinst(2)
Frfmst(3)
Bjor05.wav
8.616500972
1.839196108
2.146
0.536
3.219
Bjorll.wav
14.12880858
0.886062792
0.559
1.118
2.235
Bjor25.wav
9.124209829
0.026525955
3.353
0.559
1.118
Ec04.wav
7.876012506
2.588565211
2.665
3.731
1.066
Ec08.wav
12.39444109
2.229546798
0.513
1.538
3.075
Ecl6.wav
11.68864338
1.944947934
2.484
1.49
0.497
Pav02.wav
8.293662641
-0.273231631
0.529
1.588
3.177
Pav03a.wav
8.346366874
1.333256792
1.059
2.118
2.647
Pavl4.wav
7.603803785
1.520010432
1.609
3.755
0.536
Pldom01.wav
11.01854913
1.549285484
0.523
1.045
3.135
Pldom02.wav
5.781440997
0.859093257
0.519
4.154
1.038
Pldom24.wav
11.17277155
1.148228071
1.066
0.533
3.731
kbat01.wav
7.583506445
0.441295407
2.909
2.424
3.394
Kbat20.wav
17.80233639
-0.892888451
3.177
0.529
1.059
Kbat22.wav
6.284570391
-2.112443825
2.09
1.568
4.18
Moncab30.wav
9.506931576
1.23886716
4.35
1.631
0.544
Moncab31.wav
15.44704839
0.637830092
1.126
3.377
0.563
Moncab33.wav
16.882484
-0.703801289
1.08
2.7
1.62
delosa01.wav
4.124225139
0.15777418
3.832
2.737
1.642
delosa07.wav
10.02368725
1.762009222
1.095
0.547
1.642
delosa09.wav
12.28671225
-0.712589986
0.54
1.62
2.16
Mean
10.28508158
1.77257
1.87133
2.01324
Maximum
17.80233639
4.35
4.154
4.18
Minimum
4.124225139
0.513
0.529
0.497
Std. dev.
3.52049266
1.19201
1.13261
1.15371
Cl Maximum
17.3260669
4.15659
4.13655
4.32066
Cl Minimum
3.244096259
-0.61145
-0.39388
-0.29418


192
The instantaneous amplitude wave of the frequency vibrato can be obtained by
choosing the Model, Short length, Frequency vibrato, Instantaneous amplitude option.
The software will calculate an instantaneous power spectrum of the frequency vibrato
wave and will display it in a window on top of the sample spectrogram (see figure A. 24).
Figure A.24. Instantaneous power spectrum of the frequency vibrato.
The window shown in figure A. 19 will appear on the screen to inform the user of
the need for an input. The user should click on the OK button, move the mouse pointer to
the frequency vibrato instantaneous power spectrum and click above and below the initial
frequency of the instantaneous frequency wave, that is, the frequency at t = 0. The
software will extract the instantaneous frequency wave and will display it in blue on top of
the frequency vibrato instantaneous power spectrum for comparison purposes (see figure
A.25).


Relationship Between the Model Parameters in the Time-Domain and the Z-
Domain 48
Calculation of the Instantaneous Frequency and Amplitude of the Frequency
Vibrato Wave 52
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave 60
Software Development 63
Edit, Options, Short Length Model 66
View, Window 66
Filter, Elliptical, Frequency Vibrato, Instantaneous Frequency and
Instantaneous Amplitude 67
Filter, Get Frequency, Power Spectrum and Instantaneous Power Spectrum 68
Model, Short Length, Frequency Vibrato, Instantaneous Frequency 68
Model, Short Length, Frequency Vibrato, Instantaneous Amplitude 68
Model, Short Length, Amplitude Vibrato 69
Software Improvements 69
Validation 70
Signal with no Modulation 70
Signal with Frequency and Amplitude Modulation 74
5 DATA ANALYSIS AND RESULTS 80
Sample Selection 80
Data Analysis 81
Jussi Bjorling 84
File bjor5.wav 84
File bjorl 1 .wav 86
File bjor25.wav 88
Enrico Carusso 90
File ec04.wav 90
File ec08.wav 92
File ecl6.wav 94
Luciano Pavarotti 96
File pav02.wav 96
File pav03a.wav 98
File pavl4.wav 100
Placido Domingo 102
File pldom01.wav 102
File pldom02.wav 104
File pldom24.wav 106
Kathleen Battle 108
File kbat01.wav 108
File kbat20.wav 110
File kbat22.wav 112
Monserrat Caballe 114
File moncab30.wav 114
File moncab31 .wav 116
vii


170
A statistical analysis showed that the singers' formant amplitude varies for Bjorling,
Caruso, Caballe, and DeLosAngeles, and the frequency varies for Bjorling, Caruso, and
DeLosAngeles. This means that the singers formant is not constant for singers Bjorling,
Caruso, Caballe, and DeLosAngeles. However, the amplitude variations may be influenced
by different recording levels in the samples. It is very interesting to note that no significant
results were found for Pavarotti and Domingo, who are the most renowned opera singers
of our age.
Future Work
Algorithms
The algorithm for the calculation of the instantaneous frequency wave is more
precise than the algorithm for the calculation of the amplitude vibrato. Although the error
introduced by the latter is not high (approximately 3%), it might be reduced. This will help
obtain more precise model parameters.
There is no known cause of the ripple in the instantaneous amplitude wave.
Although it is evident that this is caused by a dependence of the method on the signal
phase angle, the theory says that the model does not depend on it. A reduction in the
ripple would decrease the error in the instantaneous amplitude wave.
No formal analysis was made of the effect of the resolution of the instantaneous
power spectrum in the calculation of the instantaneous amplitude wave. The number of
points was chosen to be 64 since it provided a good combination of speed and resolution,
but better results might be obtained with different values.


7
producing vibrato would be an increase in the accuracy in the singer's fundamental
frequency, which is not the case.
In a study conducted by Sundberg (Sundberg, 1978, p.51) some singers were
asked to adjust the fundamental frequency of a constant tone to match the mean frequency
of a vibrato tone. The results of this study revealed that the singers adjusted the frequency
of the tone at the value of the linear average of the oscillating fundamental frequency with
a small margin of error. It was also found that we really perceive the logarithmic average
and not the linear average. In order to interpret the previous results correctly it is
important to note that they are only valid when someone listens to a single isolated tone.
The vibrato is also detected in the EMG (electromyography) signal from the
laryngeal muscles. From this we may deduce that the presence, absence, or type of vibrato
may provide an aural indication of the physical conditions of the singer's larynx. Research
by Sundberg and Askenfelt (Sundberg and Askenfelt, 1983, p.307) indicates that there is a
relationship between the absence of vibrato and phonatory problems. In order for the
singer to produce an artistically satisfactory result it is necessary to perform difficult
passages without apparent difficulty. Therefore, the singer is likely showing his healthy
voice to the audience when he is able to produce vibrato in high or difficult notes.
The Existence of Vibrato
Vibrato has been controversial from Mozart's time. There is confusion in the
musical field about vibrato which can be attributed to the following reasons (Seashore,
1936, p.47):
1. Lack of knowledge about what vibrato really is.


File moncab33.wav 118
Victoria DeLosAngeles 120
File delosa01.wav 120
File delosa07.wav 122
File delosa09.wav 124
6 VIBRATO MODEL 126
Frequency Vibrato Model 126
Amplitude Vibrato Model 134
7 STATISTICAL ANALYSIS 139
Model Parameters 139
Parameter Statistics 140
8 THE SINGERS' FORMANT 147
Calculation of the Singers Formant Parameters 147
Data Analysis 149
Jussi Bjorling 149
Enrico Caruso 150
Luciano Pavarotti 150
Placido Domingo 151
Monserrat Caballe 151
Victoria DeLosAngeles 152
Singers Formant Variation Among Six Different Singers 152
Singers' Formant Parameters for all the Samples 153
Amplitude Study 155
Frequency Study 156
Singers Formant Variation Within Samples of the Same Singer 157
Jussi Bjorling 158
Enrico Caruso 159
Luciano Pavarotti 160
Placido Domingo 161
Monserrat Caballe 162
Victoria DeLosAngeles 163
Summary of Results 164
9 CONCLUSIONS AND FUTURE WORK 165
Conclusions 165
Frequency and Amplitude Vibrato Analysis 165
Vibrato Model 166
Singers' Formant 169
Future Work 170
Algorithms 170
Vibrato Model 171
Singers' Formant 171
viii


Measurement of the Vibrato Rate of Ten Singers 15
Frequency Modulation Characteristics of Sustained /a/ Sung in Vocal
Vibrato 16
An Investigation of Vocal Vibrato for Synthesis 18
Acoustic and Psychoacoustic Aspects of Vocal Vibrato 20
Synthesis of Sung Vowels Using a Time-Domain Approach 21
The FCSV Software 22
Fundamental Concepts 25
The Discrete Fourier Transform 25
Linear Prediction Coding 26
Definitions 26
Relationship between the model parameters and the autocorrelation
function 28
The autocorrelation method 30
The covariance method 30
Multiple Signal Classification Method 31
3 PROPOSED RESEARCH 33
Vibrato Model 33
Frequency and Amplitude Vibrato Analysis 35
The Singers Formant 36
4 ALGORIT11M DEVELOPMENT 37
Conversion of the FCSV Software (Frequency Characterization of Singers'
Vibrato) from Matlab 4.2 to 5.1 37
Objective of the Functions Implemented up to this Point 38
File, Open Wave File 38
File, Save Wave File 38
File, Save Parameters 39
File, Print Figure 39
File, Exit 39
Edit, Edit Wave File 39
Edit, Play Wave File 39
Edit, Zoom in and out 39
Edit, Options, Spectrogram 39
Edit, Options, Elliptical Filter 40
Edit, Options, Full Length Model 41
View, Filter Response 42
Analysis, Spectrogram 43
Analysis, Get Vibrato 44
Analysis, Get Parameters 44
Filter, Elliptical 45
Filter, Median 45
Model, Full Length Model 46
Help 48
vi


LIST OF REFERENCES 230
BIOGRAPHICAL SKETCH 234
x


38
Figure 4.1. Main screen.
Objective of the Functions Implemented up to this Point
File. Open Wave File
The objective of this function is to load the original vibrato wave and show it on
the screen. It also initializes the variables. Figure 4.2 shows how the sample is presented in
this window.
Figure 4.2. Vibrato sample.
File. Save Wave File
This function allows the vibrato sample to be saved. It is useful for saving the .wav
file after changes have been made using the edit option.


177
Figure A.2. Open wave file window.
By clicking on the displayed directories/samples, or by typing a directory/file name,
the sample will be loaded into memory. It is important to note that the sample must be in
wav format at a sample rate of 22050 Hz and 16 bit resolution. Other file formats and/or
parameter values are not supported. The sample and its file name will be displayed as
shown in figure A. 3. The sample displayed is pldomOl wav.
Figure A.3. Vibrato sample.


18
Particular patterns shown by the vibrato samples were not associated with specific
singers as suggested by Winckel (Winckel, 1953, p.252).
The triangular and trapezoidal curve shapes found in this research may be used in
synthesis in order to produce realistic vibrato. Also, the different patterns and asymmetry
in vibrato may have an effect on perception which has not been studied as yet.
An Investigation of Vocal Vibrato for Synthesis
In their paper, Maher and Beauchamp (Maher and Beauchamp, 1990, p.219), state
that vibrato plays an important role in voice quality, and therefore in singing synthesis
methods. If a natural sounding effect is desired in synthesized singing, vibrato must be
studied carefully.
In this study, spectrograms and a peak identification and tracking technique were
used to calculate the harmonics. The synthesis method used decomposes the signal as a
sum of sinusoids.
Most research indicates that the shape of the vocal tract remains fixed in vibrato
while the excitation from the vocal folds changes frequency in an almost sinusoidal manner
(Bennet, 1981, p.34; Rossing, Sundberg and Ternstrom, 1987, p.830; Sundberg, 1977,
p.257). However, some studies have emphasized the importance of random variations in
vibrato (Bennet, 1981, p.34, Chowning, 1980, p.4).
Four singers were used as subjects in this study. Each of them was recorded
singing the vowel /a/ at a comfortable level in a range of pitches. All the subjects had
formal singing training but none of them were professional singers.


68
Filter. Get Frequency, Power Spectrum and Instantaneous Power Spectrum
This option is used to obtain a frequency value from the power spectrum or the
instantaneous power spectrum. This is useful to determine the cutoff frequency for the
elliptical filter. When this option is activated a message appears telling the user that an
input is required. After clicking the OK button the power spectrum or instantaneous
power spectrum window is activated and the cursor becomes a crosshair. The desired
value is displayed in the Matlab command window after clicking on the desired frequency.
Model. Short Length. Frequency Vibrato. Instantaneous Frequency
This option allows the user to obtain the instantaneous power spectrum of the
frequency vibrato wave, its instantaneous frequency wave, and the power spectrum of the
instantaneous frequency wave. This option is composed of the following modules:
1. The first module calculates the instantaneous power spectrum of the frequency
vibrato wave using the LPC by covariance method.
2. The function of the second module is to calculate the instantaneous frequency
and amplitude waves, and display the instantaneous frequency wave on top of
the spectrogram. This is done for verification purposes. The user should select
the range of frequencies where the wave is located.
3. The third module displays the instantaneous frequency wave, calculates and
displays its average, maximum, and minimum values, calculates its power
spectrum and displays it.
4. Finally, the pole location of all the segments analyzed is displayed.
Model. Short Length. Frequency Vibrato. Instantaneous Amplitude
This option is very similar to the option described in the previous section, but
instead it uses the LPC by autocorrelation method, displays the instantaneous amplitude


98
File pav03a.wav
File Edit Window Help
Frequency and amplitude vibrato
10
0
-10
-20
B!ue=Freq; Red=Amp
Time(s)
Figure 5.29. Frequency and amplitude vibrato waves for pav03a.wav.
Fite Edit Window Help
Freq and amp vibrato power spectra
8000 600
£ 6000
CD
TJ
4000
2000
0 10 20
Blue=Freq; Red=Amp
30 40
Frequency(Hz)
Figure 5.30. Vibrato power spectra for pav03a.wav.


195
Figure A.28. Filtered instantaneous amplitude wave.
Figure A.29. Power spectrum of the filtered wave.
The software will display one button for each figure on the screen in the taskbar.
The user can bring figures to the front by clicking on the taskbar buttons.
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave
The software can create an instantaneous power spectrum of the amplitude vibrato
wave. First, the pole number should be adjusted by selecting the Edit, Options, Short
length model option. The software will show the window in figure A.30. The number of


63
Figure 4.30. Pole location for figure 4.29.
Software Development
The number of individual programs and screens increased considerably with the
addition of the modules for the calculation of the instantaneous frequency and amplitude
of the frequency and amplitude vibrato.
The total number of windows increased from 4 to 9. It would be difficult to read
more than 4 windows simultaneously on a 15 inch screen. Therefore, it was decided to
display the windows for the instantaneous power spectra, instantaneous frequency and
amplitude of the frequency vibrato, and power spectrum of the instantaneous curves on
top of other windows. The user can bring the hidden windows to the front by clicking on
the buttons of the windows 95 task bar.
The inclusion of each of the programs developed in the menu of options would
increase the number of options to the point where it would be difficult to use the program


186
Figure A. 15. Pole location for full length model.
The Model, Full length, Amplitude vibrato option will display the parametric
model of the amplitude vibrato in purple in the frequency and amplitude vibrato power
spectrum window as shown in figure A. 16. The parameters used to obtain figure A. 16 are
1. Decimation factor: 1.
2. Poles: 7.
3. Method. Covariance.
Figure A. 16. Parametric model of the amplitude vibrato wave.


121
Figure 5.75. Instantaneous waves for delosaOl wav.
Figure 5.76. Power spectra of the instantaneous waves in delosaOl wav.


70
3. The Save Parameters module was modified to also save the mean, maximum
and minimum values of the instantaneous waves and the ten more dominant
components of the power spectra of the instantaneous waves. The file format
was changed so that it saves the variables in a column vector. Also, I decided
to store information of one harmonic per data file. Table 4.1 describes the
contents of the data files.
Validation
Two test signals were created in order to verily the precision/accuracy of the
results given by the software developed. The first signal has no frequency nor amplitude
modulation and the other has both frequency and amplitude modulation.
Signal with no Modulation.
The signal with no amplitude nor frequency modulation is called validl.wav. This
signal is a pure tone of 3000 Hz. This will emulate a harmonic at 3000 Hz with no
frequency nor amplitude vibrato. Figure 4.33 shows the spectrogram of signal validl .wav.
As expected, it is a straight line.
Figure 4.33. Spectrogram of signal validl.wav.


205
M-file auto.m
function [a,g]= auto(x,p,n)
% LPC by autocorrelation method
% Variable initialization
N= length(x);
A= zeros(p);
r= xcorr(x,'biased');
% Calculates the A matrix and the b vector
for k=l:p,
for 1=1:p;
A(k,l)= r(N+k-1);
end
end
for k=l:p,
b(k)= r (N+k) ;
end
% Obtains the a coefficients
a= inv(A)*(-b)';
% Calculates the c vector
for k=l:p,
c(k)= r(N-k);
end
% Calculates the gain
pmin= r(N) + (a')*(c');
a= [1 a'];
g= sqrt(pmin.*n) ;
M-file covar m
function [a,g]= covar(x,p)
% LPC by covariance method
% Variable initialization
N= length(x);
A= zeros(p);
% Calculates the A matrix and the b vector
for k=l:p,
for 1=1:p;
for m=p:N-l;
A(k,1)= A(k,1) + x(m+l-k).*x(m+1-1);
end
A(k,1)= A(k,1)./(N-p);
end
end


97
Figure 5.27. Instantaneous waves for pav02.wav.
Figure 5.28. Power spectra of the instantaneous waves in pav02.wav.


83
Table 5.2. Filter frequency for the instantaneous amplitude waves.
File name
Filter frequency (Hz)
Bjor5.wav
5
Bjorll.wav
7
Bjor25.wav
6
Ec04.wav
7
Ec08.wav
6
Ecl6.wav
6
Pav02.wav
6
Pav03a.wav
6
Pavl4.wav
6
Pldom01.wav
6
Pldom02.wav
7
Pldom24.wav
5
Kbat01.wav
6
Kbat20.wav
6
Kbat22.wav
6
Moncab30.wav
5
Moncab31 .wav
7
Moncab33.wav
5
Delosa01.wav
7
Delosa07.wav
6
Delosa09.wav
8


163
Victoria DeLosAngeles
Figure 8.22 shows the box plots for the amplitude analysis and figure 8.23 shows
the box plots for the frequency analysis.
Figure 8.22. Box plots for the amplitude analysis of DeLosAngeles.
Figure 8.23. Box plots for the frequency analysis of DeLosAngeles.


94
File eci6.wav
Figure 5.21. Frequency and amplitude vibrato waves for ecl6.wav.
File Edit Window Help
Freq and amp vibrato power spectra
150
100
50
0
30 40
Frequency(Hz)
0 10 20
Blue=Freq; Red=Amp
Figure 5.22. Vibrato power spectra for ecl6.wav.


219
% Gets input from mouse
[1,k]= ginput (2) ;
ninf= round(1(1).*(length(wa)-1)./max(wa) + 1) ;
nsup= round(1(2).*(length(wa)-1)./max(wa) +1);
if ninf < 2
ninf = 2;
end
if nsup > length(wa)
nsup = length(wa);
end
% Maximum amplitude relative to the mean, frequency,
% and percent of frequency error
[hmax,nmax]= max(abs(ha(ninf:nsup)));
wmax= wa(nmax+ninf-1);
errorpc= ((wa(2)/2) / wmax) 100;
fprintf(1\n');
disp('Amplitude vibrato model:');
fprintf('Maximum amplitude = %5.3f\n',hmax)
fprintf('Frequency of the maximum amplitude in Hz = %5.3f\n',wmax)
fprintf('Maximum frequency error in percentage = %5.3f\n',errorpc)
% Plots the poles
if ((method == 'aut') | (method == 'cov'))
viewpol(aa);
end
% Clears temporary variables
clear Amaxdec ameandec handle limits k 1 ninf nsup hmax nmax wmax
errorpc;
M-file fiillfrl.m
% M-file fullfrl.m
% Calculates full length model for the frequency vibrato
% Applies decimation factor if needed
if decfac > 1
Fmaxdec= decimate(Fmax,decfac);
else
Fmaxdec= Fmax;
end
% Calculates model
fmeandec= mean(Fmaxdec);
if ((method == 'aut') | (method == 'cov'))
if method == 'cov'
[af,gf] = covar((Fmaxdec-fmeandec),poles);
else
[af,gf] = auto((Fmaxdec-fmeandec),poles,1024);
end
[hf,wf] = freqz(gf,af,512,1./(decfac.*tsy));
else
[hf,wf] = pmusic((Fmaxdec-fmeandec),poles,1024,1./(decfac.*tsy));
end


120
Victoria DeLosAngeles
File delosa01.wav
File Edit Window Help
Frequency and amplitude vibrato
Blue=Freq; Red=Amp Time(s)
Figure 5.73. Frequency and amplitude vibrato waves for delosa01.wav.
Figure 5.74. Vibrato power spectra for delosa01.wav.


135
Aa(t) = Amplitude variation function due to the amplitude variation of the voice
source
Ai(t) = Amplitude of the frequency component at the vibrato frequency
A2(t) = Amplitude of the frequency component at twice the vibrato frequency
A3(t) = Amplitude of the frequency component at three times the vibrato frequency
fi(t)= Frequency function
t = Time
i, 2, 3 = Phase angles at t = 0
It is only necessary to model Aa(t) at this point since the frequency variations have
already been modeled and the effect of the formants will be achieved when the voice
source is passed through filters to create the desired vowel.
Since in 20 of the 21 samples there is a single peak below 5 Hz (see chapter 5), it
can be modeled from the amplitude vibrato power spectrum using the following model:
A(J) = Ao+ 2(| Ya{ 1) | cos(2^fl (1 + ZYa( 1)) +
Fa(2) | cos(2^(2)/0 + ZYa(2)))/N,j = 1: N (6-9)
where:
A(j) = Synthesized amplitude vibrato wave
Ao = Average value of the amplitude vibrato wave
Ya(l), Ya(2) = Two most dominant components below 5 Hz of the amplitude
vibrato spectrum (complex numbers)
fa(l), fa(2) = Frequencies of Ya(l), and Ya(2)
t(j) = Time at instant j
N = Total number of samples of the amplitude vibrato wave


112
File kbat22.wav
Figure 5.57. Frequency and amplitude vibrato waves for kbat22.wav.
Figure 5.58. Vibrato power spectra for kbat22.wav.


APPENDIX A
USER'S MANUAL
Software Installation
The MMSV (mathematical model of singers' vibrato) software was developed
using Matlab 5.1. It is not compatible with 4.x or lower versions. It may run properly on
versions 5.0 or 5.2, but has not been tested on them, and therefore, it is recommended
that it be run on version 5.1. The digital signal processing toolbox is required for proper
operation.
The software has been loaded onto a floppy disk which contains a directory called
mmsv, and the subdirectories samples and data under it. The mmsv directory contains all
the files necessary to run the MMSV software. The samples directory contains some
vibrato samples that can be used to test the software, and the data directory contains the
data files for the samples used in this dissertation.
The contents of the floppy disk should be loaded into the matlabVfiles directory. If
this directory does not exist, it should be created. Also, the following two lines should be
added to the startup.m file:
addpath c:\matlab\files\mmsv;
cd c:\matlab\files;
If the startup, m does not exist, it should be created and placed in the
\matlab\toolbox\local directory. At this point the software will be ready to run.
173


APPENDIX B
SOFTWARE LISTINGS
M-file mmsv.m
% M-file mmsv.m
% Mathematical model of singers' vibrato
% Complete vibrato analysis software
close all;
figl;
M-file ampfrl m
% M-file ampfrl.m
% Plots the instantaneous amplitude wave and calculates
% its power spectrum
% Creates figure 3 if it does not exist
handle= findobj('Tag','FiglnstFrCur');
if isempty(handle)
figinstfrcur;
handle= gcf;
handlel= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
axis([0 max(t(l:length(t)-14)) 0 1]);
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
axis([0 max(t(1:length(t)-14)) 0 1]);
end;
% Creates figure 4 if it does not exist
handle= findobj('Tag','FiglnstFrSpec');
if isempty(handle)
figinstfrspec;
handle= gcf;
handlel= findobj(handle,'Tag','FreqVibSpecAxes');
axes(handlel);
axis([0 20 0 1]) ;
handlel= findobj(handle,'Tag','AmplVibSpecAxes');
axes(handlel);
axis([02001]);
end;
201


15
Related Work
Vibrato
The book Vibrato by DeJonckere, Minoru, and Sundberg (DeJonckere, Minoru,
and Sundberg, 1995, p.l), contains a collection of papers dealing with vibrato. These
papers refer to all the different aspects of vibrato. They present vibrato from the
physiological, physical, and psychological points of view.
The more relevant papers for the development of my dissertation are in chapters 2
and 7. Chapter 2 presents a paper by J. Sundberg called Acoustic and Psychoacoustic
Aspects of Vocal Vibrato. This paper is discussed later in this document. In chapter 7 we
find a paper from E. Prame called Measurement of the Vibrato Rate of Ten Singers,
which is described below.
We find several studies about the frequency and extent of the frequency vibrato for
different artists but none of them analyze the curve shape, nor are signal processing
techniques like FFT, LPC, Burg, MUSIC, etc. applied to the curves for analysis. The
amplitude vibrato is studied very briefly. No models for vibrato are presented.
Measurement of the Vibrato Rate of Ten Singers
The main objective of this study by E. Prame (Prame, 1994, p 1979) was the
analysis of the vibrato rate. Three aspects of the vibrato rate were analyzed:
1. Intratone variability, which shows the variation of the vibrato rate within a
tone.
2. Intertone variability, with the objective of comparing vibrato rates among
different tones within a song.


226
for j=l:10
fprintf(fid,'%10.3f\n',fyinst(Ifinst(j)));
end
for j=l:10
fprintf(fid,'%10.3f\n',real(Yaordinst(j)) ) ;
fprintf(fid,1%10.3f\n1,imag(Yaordinst(j)));
end
for j=l:10
fprintf(fid,'%10.3f\n',fyinst(Iainst(j)));
end
fclose(fid);
end
% Clears temporary variables
clear j pathnamel fnamel fid mes;
M-file savewavl.m
% M-file savewavl.m
% Saves wave file
% Gets the file name and saves it
[fname pathname]= uiputfile('*.wav','Save wave file (.wav)')
if fname ~= 0
savel6([pathname fname], x,fs);
end
M-file spect3.m
% M-file spect3.m
% Calculates and plots the spectrogram
% Assumes mono sound
% Deletes figures 2, 3, 4, figinst, figinstfrcur,
% and instfrspec if they exist
delfig('Fig2');
delfig('Fig3');
delfig('Fig4');
delfig('Figinst');
delfig('FiglnstFrCur');
delfig('FiglnstFrSpec');
% Generates the spectrogram
X= specgram(x,N,fs,win,ovlap);
X= 20.*logl0(abs(X));
f= fs .* (0:(N/2-1))/N;
% Calculates the number of fit's (columns of X)
col= size(X,2);
% Generates the time vector
t= 0:tsy:(tsy.*(col-1));


114
Monserrat Caballe
File moncab30.wav
File Edit Window Help
Blue=Freq; Red=Amp Time(s)
Figure 5.61. Frequency and amplitude vibrato waves for moncab30.wav.
Figure 5.62. Vibrato power spectra for moncab30.wav.


136
This model includes two sine waves in case the peak is wide or located between
two frequency values in the power spectrum.
The model was tested with the same two samples used for the frequency vibrato
model. First, sample pldom01.wav which belongs to Placido Domingo was used. Figure
6.9 shows the original vibrato wave for sample pldom01.wav in blue, and the synthesized
amplitude vibrato wave in red. The original wave was filtered with a 4 Hz low pass filter
to eliminate the effects caused by the frequency variations and the formants. We can see
that the synthesized wave follows the shape of the original very well during almost the
entire time period, with some error at both ends.
Figure 6.9. Synthesized amplitude vibrato wave.
Figure 6.10 shows the error of the synthesized wave in figure 6.9. The error is
calculated by subtracting the values of the original and the synthesized wave. The
maximum error is 1.82 db, the minimum is -1.86 db, and the average (I|error|/N) is 0.344


217
fprintf('Maximum frequency variation below the mean in semi-tones
%5.3f\n',fvarblst)
% Plots the frequency vibrato wave
for j=l:length(t)
linel(j)= 0;
end
handle= findobj('Tag',1FreqVibCurAxes');
axes(handle);
hold on;
axis auto;
handle= plot(t',(Fmax-fmean),'b-',t',linel,'b-');
limits= axis;
axis([0 max(t) limits(3) limits(4)]);
hold off;
drawnow;
% Generates the frequency vibrato power spectrum
% The length of Max, imax, and t are equal
Nf= length(t);
Yf= fft((Fmax-fmean),Nf);
fsy= l./tsy;
fy= fsy.* (0:(Nf/2-1))/Nf;
Ameanf= mean(abs(Yf));
for j=l:length(fy)
line2(j)= Ameanf;
end
% Plots the frequency vibrato power spectrum
handle= findobj('Tag','FreqVibSpecAxes');
axes(handle);
hold on;
axis auto;
handle= plot(fy,abs(Yf(1:length(fy))),'b-1,fy,line2,'b-');
limits= axis;
axis([0 40 limits(3) limits(4)]);
hold off;
handle= msgbox('Select range to analyzeInput required');
waitfor(handle);
drawnow;
handle= findobj('Tag','Fig4');
figure(handle);
% Gets input from mouse
[1,k]= ginput(2);
ninf= round(1(1).*(length(fy)-1)./max(fy) +1);
nsup= round(1(2).*(length(fy)-1)./max(fy) +1);
if ninf < 2
ninf = 2;
end
if nsup > length(fy)
nsup = length(fy);
end
% Maximum amplitude relative to the mean, frequency,
% and percent of frequency error
[Ymaxf,nmax]= max(abs(Yf(ninf:nsup))) ;
Af= Ymaxf Ameanf;
ff= fy(nmax+ninf-1) ;
errorpc= ((fy(2)/2) / ff) 100;


166
The algorithms used in the program are efficient. The software does not require a
special computer to run on nor does it take much time during analysis. The most
computationally demanding options are the Instantaneous frequency option, and the
Amplitude vibrato option. When tested on a Pentium 133 MHz with 16 MB of RAM, it
only took 12 and 8 sec. respectively to run these options with a 2 seconds long sample.
The number of poles used for testing the Instantaneous frequency option was 3, and 7 for
the Amplitude vibrato option.
The use of the parametric models reduced the error on the frequency vibrato
calculation to very low levels. A typical error for a 2 sec sample is 0.7% compared to a
5% error using the traditional FFT method. This represents a reduction of 86%.
Vibrato Model
A relationship was found between the model parameters in the Z-domain and the
time domain. This was motivated by the fact that not many people are familiar with the Z-
transform. This method helped in the development of the frequency vibrato model, and
can be applied to any digital signal.
Different parametric methods like autocorrelation, covariance and Burg, and
subspace methods like principal components, MUSIC (Multiple Signal Classification) and
ESPRIT (Estimation of Signal Parameters via Rotational Invariance Technique), were
tested. From these methods, the autocorrelation and covariance methods produced the
best results for this particular application in terms of resolution and reliability, and
therefore, were used extensively.
The covariance method provided better results on the frequency calculation and
the autocorrelation method provided better results on the amplitude calculation. From this


105
File Edit Window Help
Instfreq and amp of freq vib
100
0 0.5 1 1.5
Blue=Freq; Red=Amp Time(s)
Figure 5.43. Instantaneous waves for pldom02.wav.
f~j Figure No. 7
File Edit Window Help
Power spectra of inst freq and amp
Figure 5 .44. Power spectra of the instantaneous waves in pldom02.wav.


137
db. The addition of another sinusoidal term to the model did not improve the error
significantly. These error values are satisfactory.
Figure 6.10. Error for amplitude model.
The sample chosen from the female singers was kbat22.wav, which belongs to
Kathleen Battle. Figure 6.11 shows the original amplitude wave for sample kbat22.wav in-
blue, and the synthesized amplitude vibrato wave in red. The original wave was filtered
with a 3 Hz low pass filter to eliminate the effects caused by the frequency variations and
the formats. We can see that the synthesized wave follows the shape of the original wave
during the entire time period.
Figure 6.12 shows the error of the synthesized wave in figure 6.11. The maximum
error is 1.16 db, the minimum is -1.28 db, and the average is 0.523 db. Again, the error
values are satisfactory.


85
Figure 5.3. Instantaneous waves for bjor5.wav.
Figure 5.4. Power spectra of the instantaneous waves in bjor5.wav.


CHAPTER 8
THE SINGERS' FORMANT
Calculation of the Singers Formant Parameters
First, it was necessary to develop software to calculate the frequency and
amplitude of the singers' formant for each sample. This software is completely separate
from the vibrato analysis software.
A decision had to be made regarding the length of the segment to be used for
analysis. It was decided to analyze a short segment (1024 samples) in case the singers'
formant parameters changed during the two second period of a sample. This allowed
several measures to be made on each sample, and consequently, to statistically compare
samples of the same singer.
During the software testing period it became apparent that it was not easy to
measure the singers' formant in a precise and consistent way from the power spectrum
given by the FFT. Therefore, an LPC was applied and the LPC power spectrum was
plotted on the same window of the FFT power spectrum. The singer's formant is shown
clearly in this plot. Figure 8.1 shows an example of the FFT and LPC power spectrum.
The FFT spectrum is shown in blue and the LPC power spectrum in red.
147


49
provide parameters in the time domain which would be obtained from the Z-domain
parameters.
As the frequency vibrato is almost sinusoidal, it can be properly represented by a
sine-wave. The frequency of the sine wave can be easily obtained from the frequency of
the poles in the Z-domain, but the amplitude calculation is not trivial and no information
was found on how to determine it. Therefore, research was undertaken as part of this
study to find the relationship between the Z-domain model and the amplitude of the sine
wave in the time domain.
First, It was noticed that there was an error in the calculation of the gain by the
LPC function in Matlab. After contacting Matlab, they provided the fix. However, the
model did not match the FFT in all cases. For poles located near the unit circle, the model
and the FFT did not match. In all other cases they did match.
The model and the FFT matched in all cases when the number of points used to
calculate the power spectrum using the model was equal to the number of samples in the
signal. Therefore, the gain of the signal had to be adjusted to compensate for this. If the
minimum square error is multiplied by the number of samples used to calculate the power
spectrum, instead of the length of the signal, both spectra match in all cases.
In order to calculate the amplitude of the sine wave in the time domain, the area of
the peak in the power spectrum has to be calculated. This can be done by adding the
vectors used to obtain the power spectrum. This summation has to be corrected by two
factors in order to obtain the amplitude. It has to be multiplied by two in order to
compensate for the other half of the spectrum in the frequency domain, and divided by the


198
_j data
I samples
File name: |SEEC
Save as type: pdat 3 Cancel
Figure A.32. Save parameters window.
The variables are saved in ASCII format, so that they can be read into a
spreadsheet like Excel and processed as desired. The parameters of only one harmonic can
be saved in each ,dat file.
Editing the Vibrato Sample File
The vibrato sample file (.wav file) can be edited to remove non vibrato parts. A
spectrogram of the sample must be made before the sample can be edited since the signal
is going to be edited based on the spectrogram plot.
The Edit, Edit wave file option should be selected. The window shown in figure
A. 3 3 will appear to inform the user of the need for an input. The user should click on the
OK button and move the mouse pointer to the spectrogram window. The spectrogram
window will appear as shown in figure A.34, with no color bar on the right side, and the
mouse pointer will look like a "cross hair. The user should click on the beginning and
ending time of the segment containing the desired vibrato part, and the non desired part


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
LIST OF TABLES xi
LIST OF FIGURES xii
ABSTRACT xx
CHAPTERS
1 INTRODUCTION 1
Importance of the Study of Vibrato 1
Organization of the Dissertation 2
2 LITERATURE REVIEW AND OVERVIEW 4
The Singers' Vibrato 4
Effect of Vibrato on Pitch Perception 6
The Existence of Vibrato 7
The Desirability of the Vibrato 9
Universality of use 10
Automatic nature 10
Advantage over precision 10
Importance in tone quality 10
Subjective Factors that Affect the Appreciation of the Vibrato 11
The vibrato ear 11
The emotionality of the individual 11
Attitude and training 11
The listener's disposition 11
Objective Factors that Determine the Quality of the Vibrato 12
Extreme extent 12
Vibrato in Different Types of Singing 13
Vibrato in Musical Instruments 13
Vibrato in the violin 13
Vibrato in wind instruments 14
The Singers' Formant 14
Related Work 15
Vibrato 15
v


229
% Clears temporary variables
clear handle k 1 ninf nsup limits;
M-file zoomoutl.m
% M-file zoomoutl.m
% Zooms out vibrato sample
% Activates the figure
handle= findobj('Tag','Figl1);
figure(handle);
% Changes the axis limits
limits= axis;
axis([0 tmax limits(3) limits(4)]);
% Clears temporary variables
clear handle limits;


23
4. Per (Periodicity).
Spec reads the vibrato sample file, graphs the power spectrum vs. time
(Spectrogram), and stores all the variables in memory, which is used later when building a
database. Curv gives the user the option of analyzing a (user specified) single harmonic.
Calc extracts frequency and amplitude parameters over the user-specified harmonic chosen
earlier in Curv. In addition, Calc displays the data on the screen and optionally saves the
data in a text file. Per allows the examination of the most important components of the
frequency and amplitude power spectra, and displays and plays back the original wave for
comparison purposes.
Each of these four modules (composed of several functions) performs one of the
main tasks in FCSV, which have been modularized in an effort to simplify the use of the
software and reduce operator errors.
The Spec module is composed of two main functions. One generates the
Spectrogram while the other provides sound wave editing and play back. The module
starts by generating the Spectrogram, then plays back the sound wave and asks the user to
select the segment to be analyzed. In this way, the user is forced to check that the segment
he wants to analyze contains vibrato. After editing (if necessary), the Spectrogram is
regenerated displaying the selected portion while generating values for the variables.
The main function of the Curv module is the determination and display of the
vibrato pulse. First, Curv generates a vector containing the vibrato pulse of the harmonic
selected, and then displays it on a Spectrogram plot, with the objective of comparing the
calculated and the actual vibrato. If they match, the user can proceed to the next module.


100
File pavl4.wav
Figure 5.33. Frequency and amplitude vibrato waves for pavl4.wav.
Figure No. 4
File Edit Window Help
nnsixi
Freq and amp vibrato power spectra
8000 i 1200
B 6000
(D
-a
4000
2000
0 10 20
Blue=Freq; Red=Amp
30 40
Frequencv(Hz)
Figure 5.34. Vibrato power spectra for pavl4.wav.


215
% Generates the power spectrum of the instantaneous
% frequency wave
% The length of Amax, imax, and t are equal
Nfinst= length(Fmaxinst);
Yfinst= fft((Fmaxinst-fmeaninst),Nfinst);
fsy= l./tsy;
fyinst= fsy.* (0:(Nfinst/2-1))/Nfinst;
% Plots the power spectrum of the instantaneous frequency
% wave in the same window of the power spectrum of the
% instantaneous amplitude wave
handle= findobj('Tag','FiglnstFrSpec');
handlel= findobj(handle,'Tag','FreqVibSpecAxes');
axes(handlel);
hold on;
axis auto;
handle= plot(fyinst,abs(Yfinst(1:length(fyinst))),'b');
limits^ axis;
axis([0 20 limits(3) limits(4)]);
hold off;
drawnow;
% Orders the components of the Inst frequency wave
[Yfordinst Ifinst]= sort(Yfinst(1:length(fyinst)));
Yfordinst= fliplr(Yfordinst);
Ifinst= fliplr(Ifinst);
% Clears temporary variables
clear handle handlel Nfinst limits;
M-file freqn4.m
% M-file freqn4.m
% Plots the frequency vibrato wave an calculates
% its power spectrum
% Deletes figures figinst, figinstfrcur,
% and instfrspec if they exist
delfig(1Figinst');
delfig(1FiglnstFrCur');
delfig('FiglnstFrSpec');
% Creates figure 3 if it does not exist
handle= findobj('Tag','Fig3');
if isempty(handle)
fig3;
handle= findobj('Tag',1FreqVibCurAxes');
axes(handle);
axis([0 max(t) -1 1]) ;
handle= findobj('Tag','AmplVibCurAxes');
axes(handle);
axis([0 max(t) -1 1]) ;
end;


CHAPTER 1
INTRODUCTION
Importance of the Study of Vibrato
The vibrato (the frequency variation/modulation of tone patterns) is one of the
elements that determines voice quality during singing. A tone produced with vibrato is
more rich and beautiful than a constant tone. This is evidenced by the fact that all great
classically trained singers, as well as popular ones, use vibrato a high percentage of the
time. Although it may be unnoticed by individual listeners, the importance and effect in
voice quality is unquestioned.
In spite of the significance of the vibrato in voice quality during singing, little effort
has been made to investigate and analyze this phenomenon throughout the years. The most
complete study of vibrato was completed in the 1930s by Seashore at the University of
Iowa (Seashore, 1932, p.l; Seashore, 1936, p.l). His studies set the basic rules and
principles for the study of vibrato which are still in use today. Few and isolated studies
have been accomplished since.
Since the time of Seashore's studies a great number of improvements have been
made in the equipment and techniques available for signal analysis. These techniques for
the analysis of digital signals may reveal new information about vibrato.
1


203
M-file ampl4 m
% M-file ampl4.m
% Plots the amplitude vibrato wave and calculates
% its power spectrum
% Mean, maximum, and minimum amplitude for the ampliude vibrato
fprintf('\n');
disp('Amplitude vibrato:');
amean= mean(Amax);
fprintf('Mean amplitude = %5.3f\n',amean)
amax= max(Amax);
amin= min(Amax);
fprintf('Maximum amplitude = %5.3f\n',amax)
fprintf('Minimum amplitude = %5.3f\n',amin)
% Amplitude variation in db and percentage
avardb= mean(abs(Amax-amean));
avarpc= (avardb / amean) 100;
fprintf('Mean amplitude variation in db = %5.3f\n',avardb)
fprintf('Mean amplitude variation in percentage = %5.3f\n',avarpc)
% Amplitude variation above and below the mean in db
amaxab= 0 ;
elemab= 0;
amaxbl= 0;
elembl= 0;
for j=l:length(t)
if Amax(j) > amean
amaxab= amaxab + Amax(j);
elemab= elemab + 1;
end
if Amax(j) < amean
amaxbl= amaxbl + Amax(j);
elembl= elembl +1;
end
end
avarab= amaxab/elemab amean;
avarbl= amean amaxbl/elembl;
fprintf('Mean amplitude variation above the mean in db =
%5.3f\n',avarab)
fprintf('Mean amplitude variation below the mean in db =
%5.3f\n',avarbl)
% Plots the amplitude vibrato in the same window
% of the frequency vibrato
for j=l:length(t)
linel(j)= 0;
end
handle= findobj('Tag','AmplVibCurAxes');
axes(handle);
hold on;
axis auto;
handle= plot(t',(Amax-amean),'r-',t',linel,'r-');
limits^ axis;
axis([0 max(t) limits(3) limits(4)]);
hold off;
drawnow;


176
5.2. Median
5.2.1. Frequency vibrato wave
5.2.2. Amplitude vibrato wave
5.3. Get frequency
5.3.1. Power spectrum
5.3.2. Instantaneous power spectrum
6. Model
6.1. Full length
6.1.1. Frequency vibrato
6.1.2. Amplitude vibrato
6.2. Short length
6.2.1. Frequency vibrato
6.2.11. Instantaneous frequency
6.2.1.2.Instantaneous amplitude
6.2.2. Amplitude vibrato
7. Help
7.1. Where to find help
7.2. About
To open a vibrato sample select File, Open wave file. A window similar to figure
A.2 will be displayed.


64
and would lead to user errors. Therefore, software modules were grouped to reduce the
number of options and then added to the menu of options.
After changes and additions the final version of the menu of options looked like
this:
1. File
1.1. Open wave file.
1.2. Save wave file
1.3. Save parameters
1.4. Print figure
1.5. Exit
2. Edit
2.1. Edit wave file
2.2. Play wave file
2.3. Zoom
2.3.1 In
2.3.2.Out
2.4. Options
2.4.1. Spectrogram
2.4.2. Elliptical filter
2.4.3. Full length model
2.4.4. Short length model
3. View
3.1.
Window


APPENDICES
A USER'S MANUAL 173
Software Installation 173
Opening a Vibrato Sample 174
Analysis of the Frequency and Amplitude Vibrato Waves 178
Full Length Model 184
Instantaneous Frequency and Amplitude of the Frequency Vibrato Wave 187
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave 195
Saving the Vibrato Parameters 197
Editing the Vibrato Sample File 198
B SOFTWARE LISTINGS 201
M-file mmsv.m 201
M-file ampffl.m 201
M-file ampl4.m 203
M-file auto.m 205
M-file covar.m 205
M-file curvaml .m 206
M-file curve2.m 208
M-file curvff 1 .m 209
M-file delfig.m 210
M-file edit3.m 210
M-file ellipaml .m 211
M-file ellipffl.m 212
M-file ellipinstaml.m 212
M-file ellipinstff 1 .m 213
M-file ffeqfiT.m 214
M-file freqn4.m 215
M-file fullaml.m 218
M-file fullfiT.m 219
M-file getvall m 220
M-file instaml .m 221
M-file instfH.m 222
M-file loadwavl .m 223
M-file medaml ,m 223
M-file medfir 1 m 224
M-file savedat5.m 225
M-file savewavl ,m 226
M-file spect3.m 226
M-file varinit.m 227
M-file viewpolinst.m 227
M-file viewpol.m 228
M-file viewres.m 228
M-file zoominl.m 228
M-file zoomoutl .m 229
IX


76
Figure 4.38. Frequency and amplitude vibrato power spectra.
The parameters calculated by the software developed are shown below.
Frequency vibrato:
Mean frequency = 3000.611
Maximum frequency = 3100.781
Minimum frequency = 2906.982
Mean frequency variation in Hz = 64.013
Mean frequency variation in percentage = 2.133
Maximum frequency variation in semi-tones = 0.559
Mean frequency variation above the mean in Flz = 64.834
Mean frequency variation below the mean in Flz = 63.213
Maximum frequency variation above the mean in semi-tones = 0.569
Maximum frequency variation below the mean in semi-tones = 0.549
Maximum amplitude relative to the mean = 7020.281
Frequency of the maximum amplitude in Hz = 6 .112
Maximum frequency error in percentage = 4.167


178
Analysis of the Frequency and Amplitude Vibrato Waves
The analysis of the frequency and amplitude vibrato waves is performed by
applying the following three options in the Analysis menu to a wave file:
1. Spectrogram.
2. Get vibrato.
3. Get parameters.
The Spectrogram option creates a spectrogram of the vibrato sample shown on the
MMSV main screen. Figure A.4 shows the spectrogram for sample pldom01.wav.
File Edil Window Help
Spectrogram
4000
3000
N
X
> r-
£ 2000
3
cr

t 1000
0
0 0.5 1 1.5 2
Time(s)
Figure A.4. Example of a spectrogram.
The Get vibrato option extracts the frequency and amplitude vibrato waves. After
choosing this option the software will indicate the need for data input by showing the
window displayed in figure A. 5.


24
If not, the user can retry the Curv option to recalculate the vibrato wave with another
window size.
The software may fail to identify vibrato because of an inappropriate window size
or a difficult vibrato wave. A difficult vibrato wave is one that has one or more of the
following characteristics:
1. It is noisy (the recording is not good because of equipment noise, or instrument
or voice interference).
2. Small inter-harmonic distances, common with male voices (the distances
between harmonics are smaller than in the female voice) may mask the
harmonics.
3. High extent of the frequency vibrato, which occurs when the frequency change
of the harmonics is of such magnitude that the high frequencies of one
harmonic get close to the low frequencies of the next upper harmonic, masking
the effect.
The most important functions accomplished by the Calc module are the
calculation of the parameters of the frequency vibrato, the calculation of the parameters of
the amplitude vibrato, the determination of the most important components of the
frequency and amplitude power spectrum, and the option of saving the resulting data to
the hard drive.
The last module is the periodicity analysis. It groups two different primary
functions: displaying of the most important frequency components of the frequency and
amplitude vibrato, and plotting of the original wave.


211
if lsup > col
lsup= col;
end
% Shortens and saves the sound wave
linf=(linf-1) points + 1;
lsup=(lsup-1) points + N + 1;
x= [x(linf:lsup)];
save temp pathname fname x fs format
% Clears all variables in memory
clear;
% Loads all the saved variables
load temp
delete('temp.mat1);
% Plots the edited wave
ts= 1/fs;
tmax= ts .* (length(x)-1);
t= 0:ts:tmax;
handle= findobj('Tag','Figl');
figure(handle);
plot(t,x);
limits= axis;
axis([0 tmax limits (3) limits(4)]);
title('Vibrato sample');
xlabel(['File: fname Time(s)']);
ylabel('Amplitude(spl)');
% Variable initialization
varinit;
% Clears temporary variables
clear limits handle;
% Creates the spectrogram
spect3;
M-file ellipaml.m
% M-file ellipaml.m
% Applies an elliptical filter to the amplitude vibrato wave
% Filters the amplitude vibrato wave
Amax= filtfilt(b,a,Amax);
% Deletes the amp vibrato wave and mean value line (0)
% from the AmplVibCurAxes
handle= findobj('Tag','Fig3');
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
cla;


156
Table 8.2. Relationship between groups and singers names.
Group
Singer's name
1
Jussi Bjorling
2
Enrico Caruso
3
Luciano Pavarotti
4
Placido Domingo
5
Monserrat Caballe
6
Victoria DeLosAngeles
Frequency Study
In order to study the variations in the singers' formant frequency, the frequency
values were input into the analysis of variance function in Matlab. Figure 8.10 shows the
analysis of variance table and figure 8.11 the box plots given by Matlab. The p value for
these data is 0.0258. Therefore, the null hypothesis is rejected. The mean values are
different. The box plots show the location of the frequency values for each singer. The
relationship between the group number and the singer's name is shown in table 8.2.
ANOVA Table
Source SS df MS F
Columns 3.539e+0Q5 5 7.078e+0043.854
Error 2.204e+005 12 1.836e+Q04
Total 5.743e+005 17
Figure 8.10. Analysis of variance table.


95
File Edit Window Help
Inst freq and amp of freq vib
Figure 5.23. Instantaneous waves for ecl6.wav.
Figure 5.24. Power spectra of the instantaneous waves in ecl6.wav.


134
In summary, the model proposed approximates a real vibrato wave. Those two
examples show that it can model a real vibrato wave. The model can also be used to create
vibrato waves with specific characteristics by giving the parameters the desired values.
Amplitude Vibrato Model
As explained by Sundberg (Sundberg, 1994, p.45), there are three possible causes
for the amplitude vibrato variations:
1. Frequency variations in the voice source.
2. Vocal tract shape variations.
3. Amplitude variations in the voice source.
The effect of the frequency variations and the vocal tract shape (formants) can be
seen as peaks in the amplitude vibrato spectrum at integer multiples of the frequency of
the frequency vibrato (see chapter 5). Therefore, I propose this model to describe the
amplitude vibrato:
A(t) A0 + Aa(t) +
Al (t) cos (2nf¡ (t)t + ]) +
A2 (/) cos(2^2/, (t)t + 2) + (6-8)
A3 (t) cos(2^3/, (t)t + 2)
where:
A(t) = synthesized amplitude vibrato wave
Av = Amplitude vibrato average value


155
Amplitude Study
In order to study the singers' formant amplitude, the amplitude values were input
into the analysis of variance function in Matlab. Figure 8.8 shows the analysis of variance
table and figure 8.9 shows the box plots given by Matlab. The p value for these data is
0.1568; therefore, the null hypothesis that all the mean amplitudes are the same is not
rejected. The box plots show the location of the amplitude values for each singer. Table
8.2 shows the relationship between the group number and the singer's name.
ANOVA Table
Source
ss
df
MS
F
Columns
276.6
5
55.31
1.964
Error
337.9
12
28 16
Total
614.5
17
Figure 8.8. Analysis of variance table.
Figure 8.9. Box plots.


200
and/or typing the desired directory/file name. The extension of the file will be wav and
does not have to be typed.
The edited signal can be saved with the original file name or a different one. If the
original name is used the edited wave file will replace the original file, which will be lost. If
a different name is used, there will be two copies of the file: the original and the edited file.
After clicking on the Save button the operation will be completed.
Figure A.35. Save wave file window.


21
different type of vibrato the formant frequencies vary in synchrony with vibrato. This can
be caused by movements of the tongue or the pharynx sidewalls.
The effect of amplitude vibrato on listeners has often been overestimated, but no
formal investigation has been made. From results of experiments with synthesized singing,
frequency variations and not amplitude variations cause the main perceptual effect on
listeners.
A possible reason for the use of vibrato is to avoid beats (the addition/cancellation
of sounds with slightly different frequencies). Beats are avoided if the notes in a consonant
interval are sung with vibrato. Also, vibrato may help make the singers voice more
distinguishable when singing with a loud orchestra. A final incentive for the use of vibrato
could be that it is used to show the audience that the singer is capable of solving difficult
tasks without struggle (Sundberg, 1994, p.45).
Synthesis of Sung Vowels Using a Time-Domain Approach
Titze (Titze, 1983, p.90) has used a time domain approach to synthesize sung
vowels. In this study, the glottal volume velocity waveform was specified in the time
domain by
g(t) Ata sin(;rf / yf)
for t < T
(2-7)
o
ll
'So
for t > T
(2-8)
Titze modeled the effect of the vocal tract and the lip radiation using the
articulatory model.
The amplitude and frequency of the glottal pulse were modulated according to
A = ^(l + O.Ola, n{l7rftt) + 0.01jVJ/\(/))
(2-9)


82
sinusoidal as reported in the literature. The common, prominent characteristics of the
amplitude vibrato waves are a peak at low frequency (below 5 Hz), and peaks at one, two
and three times the vibrato rate. In 20 of the 21 samples there is one peak below 5 Hz,
only sample ec08.wav from Enrico Caruso (see figure 5 .18) has two peaks of almost the
same height. Frequencies between 0 and 5 Hz were analyzed in order to investigate the
variations due to changes in the amplitude of the voice source.
Figures 5.3, 5.7, 5.11, 5.15, 5.19, 5.23, 5.27, 5.31, 5.35, 5.39, 5.43, 5.47, 5.51,
5.55, 5.59, 5.63, 5.67, 5.71, 5.75, 5.79, and 5.83 show the instantaneous frequency and
amplitude of the frequency vibrato in the selected samples. The instantaneous frequency
wave is shown in blue and the instantaneous amplitude wave in red. The instantaneous
amplitude waves were filtered with a low pass filter at the frequencies indicated by table
5.2. Figures 5.4, 5.8, 5.12, 5.16, 5.20, 5.24, 5.28, 5.32, 5.36, 5.40, 5.44, 5.48, 5.52, 5.56,
5.60, 5.64, 5.68, 5.72, 5.76, 5.80, and 5.84 show the power spectra of the instantaneous
waves. The power spectrum of the instantaneous frequency wave is shown in blue and the
power spectrum of the instantaneous amplitude in red. The prominent characteristics of
the spectrum of the instantaneous frequency and the instantaneous amplitude waves are
the high peaks located between 0 and 5 Hz. In 19 of the 21 figures the power spectrum of
the instantaneous frequency waves shows 2 or 3 peaks, except in figures 5.24 and 5.64
where it shows 4 peaks. In 20 of the 21 figures the power spectrum of the instantaneous
amplitude waves shows 2 or 3 peaks, except in figure 5.60 where it shows 4 peaks.


CHAPTER 6
VIBRATO MODEL
Frequency Vibrato Model
It has been reported in the literature over the years that the vibrato is almost
sinusoidal, and by looking at the waveform and power spectrum of the selected samples it
is evident that the frequency vibrato wave is almost sinusoidal. However, a sinusoidal
model does not provide a convincing, real effect to the human ear. Therefore, the
following model was proposed as a starting point:
fit) = Fo+Af(t) cos (2jrff (t)t + ) (6-1)
where:
f(t) = Synthesized frequency vibrato wave
F0 = Frequency vibrato average value
Af(t) = Amplitude function of the frequency vibrato
ff(t) = Frequency function of the frequency vibrato
t = Time
<|> Phase angle at t = 0
In this equation the frequency vibrato has a sinusoidal shape but the amplitude and
frequency of it are functions of time. These functions can be deterministic or random.
Several models which could make use of the information of instantaneous
frequency and amplitude provided by the software were tested, and this resulted in the
126


2
Organization of the Dissertation
This dissertation has been divided into 9 chapters titled introduction, literature
review and overview, proposed research, algorithm development, data analysis and results,
vibrato model, statistical analysis, the singers' formant, and conclusions and future work.
There are also two appendices: user's manual, and software listings.
The literature review and overview chapter covers four topics that are of vital
importance for the understanding of this dissertation: the singers' vibrato, related work,
the FCSV (Frequency Characterization of the Singers' Vibrato) software, and fundamental
concepts.
The objective of this dissertation is described in detail in the proposed research
chapter. It divides the present study into three main sections: vibrato model, frequency and
amplitude vibrato analysis, and the singers' formant.
Chapter four describes the steps followed to develop and implement the
mathematical algorithms and the MMS V (Mathematical Model of the Singers' Vibrato)
software. It contains a description of each of the software modules, and a section on the
validation of the software results.
The criteria used to select the samples and the results obtained with the MMSV
software are shown in the data analysis and results chapter. The results are presented in
graphic form, and are then analyzed.
The process of developing the vibrato model is described in chapter six. Two
models are presented: one for the frequency vibrato wave and one for the amplitude
vibrato wave. The models are compared against the original waves and against a pure


20
The listeners did not show a preference toward vibrato with random components
versus pure sinusoidal vibrato.
Further study of a descriptive model and of the effect of random variations in
vibrato are suggested (Maher and Beauchamp, 1990, p.219). Also, there is a need for
developing synthesis models with a more natural sound quality.
Acoustic and Psvchoacoustic Aspects of Vocal Vibrato
In this paper, Sundberg (Sundberg, 1994, p.45), reported that the vibrato
frequency is not as constant for a particular singer as has been usually assumed. An
investigation made by Prame (Prame, 1994, p. 1979) found that vibrato rates change within
a single tone and also in individual tones in a performance.
The extent of vibrato is usually less than 1 semitones which correspond to a
frequency change of about 6%.
The waveform of the frequency vibrato is similar to a sine wave. However,
substantial deviations from this wave shape have been found as reported by Schultz-
Coulon (Schultz-Coulon, 1976, p.335; Schultz-Coulon, 1978, p.142; Schultz-Coulon &
Battmer, 1981, p.l).
There are three possible causes for the amplitude variation during vibrato. The first
is a frequency variation, which causes the harmonics to move closer and farther from the
formants. Another source of variation is the characteristics of the voice source in the voice
organ, which may change during vibrato. A final source of variation are changes in the
vocal tract shape, which determines the formants.
The voice source amplitude and the formant frequencies are not always constant
during vibrato. In particular types of vibrato the voice source amplitude varies. In a


78
frequency variations in the signal. The instantaneous amplitude wave shows the amplitude
variations in the wave. The algorithm used to calculate the instantaneous amplitude
introduces some error. Figure 4.40 shows the power spectra of the waves in figure 4.39.
Figure 4.39. Instantaneous frequency and amplitude of the frequency vibrato wave.
Figure 4.40. Power spectra of the instantaneous waves.
The software developed calculated the following parameters for the waves in
figure 4.39.


69
wave, and calculates its power spectrum. This option is composed of the following
modules:
1. The first module calculates the instantaneous power spectrum of the frequency
vibrato wave using the LPC by autocorrelation method.
2. The function of the second module is to calculate the instantaneous frequency
and amplitude waves, and display the instantaneous frequency wave on top of
the spectrogram. This is done for verification purposes. The user should select
the range of frequencies where the wave is located. The instantaneous
frequency and amplitude waves will contain some ripple, but the instantaneous
frequency wave calculated with this option does not replace the wave
calculated using the Instantaneous frequency option, since the latter is more
precise. Also, no calculations are made on this instantaneous frequency wave.
3. The third module displays the instantaneous amplitude wave, calculates and
displays its average, maximum, and minimum values, calculates its power
spectrum and displays it.
4. Finally, the pole location of all the segments analyzed is displayed.
Model Short Length. Amplitude Vibrato
This option allows the user to calculate and display the instantaneous power
spectrum of the amplitude vibrato. This option uses the LPC by autocorelation method.
Software Improvements
After all the programs developed were included in the menu of options, additional
work was done on the software for some minor improvements.
1. All variables were checked for consistency in their names, and renamed as
needed.
2. Temporary variables were cleared to keep the amount of memory used to a
minimum. This also helps in debugging the software since the workspace
became smaller.


185
Choosing Model, Full length, Frequency vibrato will display the parametric model
of the frequency vibrato in green in the window of the frequency and amplitude vibrato
power spectrum as shown in figure A. 14. The parameters used to obtain figure A. 14 are
1. Decimation factor: 1.
2. Poles. 3.
3. Method: Covariance.
Figure A. 14. Parametric model of the frequency vibrato wave
The software will display figure A. 8 to indicate that an input is required. The user
should click on the OK button, move the mouse pointer to the model spectrum window,
and select the range of frequencies to analyze, first the lower frequency and then the
higher frequency. The software will calculate the frequency and height of the peak in the
selected range, and will display them in the Matlab command window. The pole location
will be shown in a window as shown in figure A. 15. The user should click on the close
button after finishing to avoid window proliferation.


75
Figure 4.37 shows the frequency and amplitude vibrato waves obtained from figure
4.36. The frequency vibrato wave is shown in blue and the amplitude vibrato wave in red.
The amplitude vibrato wave was filtered with a 5 Hz low pass filter to eliminate the higher
frequency components. Both waves have sinusoidal shape but are not completely
symmetrical. There are several possible causes for this:
1. The modulations in signal valid3.wav are not perfectly symmetrical.
2. The resolution of the FFT.
3. The algorithm used to calculate the amplitude.
Figure 4.37. Frequency and amplitude vibrato waves.
Figure 4.38 shows the power spectrum of the waves in figure 4.37. These power
spectra show that the waves in figure 4.37 have sinusoidal shape.


53
A program was developed to calculate the instantaneous frequency and amplitude
(similar to a spectrogram) of the frequency vibrato wave. This spectrogram is created by
taking a small segment or window of the signal and applying the LPC by covariance
method, then the window is shifted to the right and the new segment is analyzed. This
process is repeated until the whole signal has been analyzed. The length of the segment
was chosen to be equal to one cycle of the vibrato wave since a longer segment will show
average values and a shorter segment would not contain enough data for a reliable result.
The covariance method was chosen since it outperforms the other two methods being used
(autocorrelation and MUSIC). Figure 4 .18 shows the spectrogram of the curve shown in
figure 4.13, which belongs to Placido Domingo. The red color represents high amplitude
and the blue color low amplitude. We can see that the frequency varies between 6 and 7
Hz as expected for frequency vibrato. The software also calculates the pole location for
each segment. Figure 4.19 shows the pole location for the spectrogram in figure 4.18.
Figure 4.18. Spectrogram of the frequency vibrato wave.


213
% Deletes the instantaneous amplitude wave
% from the AmplVibCurAxes
handle= findobj('Tag','FiglnstFrCur') ;
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
cla;
% Deletes the power spectrum of the instantaneous amplitude
% wave from the AmplVibSpecAxes
handle= findobj('Tag','FiglnstFrSpec');
handlel= findobj(handle,'Tag','AmplVibSpecAxes');
axes(handlel);
cla;
% Makes the calculations on the filtered signal and plots it
ampfrl;
% Clears temporary variables
clear handle handlel;
M-file ellipinstfrl m
% M-file ellipinstfrl.m
% Applies an elliptical filter to the instantaneous
% frequency wave
% Filters the instantaneous frequency wave
Fmaxinst= filtfilt(b,a,Fmaxinst);
% Deletes the instantaneous frequency wave
% from the FreqVibCurAxes
handle= findobj('Tag','FiglnstFrCur');
handlel= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
cla;
% Deletes the power spectrum of the instantaneous frequency
% wave from the FreqVibSpecAxes
handle= findobj('Tag','FiglnstFrSpec');
handlel= findobj(handle,'Tag','FreqVibSpecAxes');
axes(handlel);
cla;
% Makes the calculations on the filtered signal and plots it
freqfrl;
% Clears temporary variables
clear handle handlel;
% M-file exitl.m
% Exits the mmsv software
% Closes all open windows
close all;


149
Data Analysis
It was agreed to use the same samples in this study that were previously used as
examples of good vibrato. Six singers out of seven were used. Kathleen Battle was not
used since her samples do not posses the singers' formant. Therefore, a total of 18 samples
(3 per singer) were analyzed. Three measures were made on each sample. Each measure
was randomly selected after dividing the sample into segments of 1024 samples.
In the following section I will only show one example of the singers' formant per
singer due to space limitations.
Jussi Biorling
Figure 8.2 shows the singers' formant of segment 24 in sample bjor05 .wav.
Figure 8.2. Singers' formant in sample bjor05.wav.


5.57. Frequency and amplitude vibrato waves for kbat22.wav 112
5.58. Vibrato power spectra for kbat22.wav 112
5.59. Instantaneous waves for kbat22.wav 113
5.60. Power spectra of the instantaneous waves in kbat22.wav 113
5.61. Frequency and amplitude vibrato waves for moncab30.wav 114
5.62. Vibrato power spectra for moncab30.wav 114
5.63. Instantaneous waves for moncab30.wav 115
5.64. Power spectra of the instantaneous waves in moncab30.wav 115
5.65. Frequency and amplitude vibrato waves for moncab31 wav 116
5.66. Vibrato power spectra for moncab31.wav 116
5.67. Instantaneous waves for moncab31.wav 117
5.68. Power spectra of the instantaneous waves in moncab31 .wav 117
5.69. Frequency and amplitude vibrato waves for moncab33.wav 118
5.70. Vibrato power spectra for moncab33.wav 118
5.71. Instantaneous waves for moncab33.wav 119
5.73. Frequency and amplitude vibrato waves for delosa01.wav 120
5.74. Vibrato power spectra for delosa01.wav 120
5.75. Instantaneous waves for delosa01.wav 121
5.76. Power spectra of the instantaneous waves in delosa01.wav 121
5.77. Frequency and amplitude vibrato waves for delosa07.wav 122
5.78. Vibrato power spectra for delosa07.wav 122
5.79. Instantaneous waves for delosa07.wav 123
5.80. Power spectra of the instantaneous waves in delosa07 .wav 123
5.81. Frequency and amplitude vibrato waves for delosa09.wav 124
5.82. Vibrato power spectra for delosa09.wav 124
5.83. Instantaneous waves for delosa09.wav 125
xvi


35
be calculated to make sure that a low error model is found, which describes good vibrato
properly.
Frequency and Amplitude Vibrato Analysis
The frequency and amplitude curves obtained with FCSV are analyzed by the
software using FFTs. It was suggested in my masters thesis that more information can be
obtained from these curves by using other methods, e.g., Burg or LPC. These methods
can be used to improve the frequency resolution, and increase the precision of the results.
Arroyo & Rothman did some preliminary work in this area in 1987-89.
The frequency vibrato curve possesses a sinusoidal-like shape. This curve can be
properly represented by an LPC model. Also, for poor vibrato the frequency and
amplitude curves change from cycle to cycle. These changes can be quantified by
analyzing a small segment of the frequency vibrato using LPC and sliding the window
along the time axis until the entire sample has been analyzed. This will show the
instantaneous frequency and extent characteristics of the frequency vibrato with high
resolution. This technique has not been used in the past and constitutes a new way to
analyze the vibrato curves. The same technique can be applied to the amplitude vibrato to
produce the instantaneous frequency and extent characteristics, but the model used should
be of a higher order to represent the curve properly since the amplitude vibrato curve is
more complex. The results obtained will help in the development of the mathematical
model for good vibrato.


47
Figure 4.11. Pole location of the frequency vibrato model.
The covariance method was implemented first since in an experiment this
investigator performed with a synthetic vibrato wave, the covariance method
outperformed the autocorrelation and MUSIC methods in calculating the frequency of the
wave.
It was noticed from the experiments conducted that the LPC method works better
after the wave has been filtered with the elliptical filter. This agrees with the literature
since it has been reported that the performance of the LPC method is better for low noise
level signals. Also, the LPC method has some difficulties with signals of relative low
frequency, that is, signals whose poles are close to DC. This effect can be counteracted by
shifting the poles away from the DC value, which can be accomplished by decimating the
signal with the option provided for it.
The location and sizes of the windows were changed so that 4 four windows can
be presented at the same time on the screen. This allows an easy comparison of the
different representations of the vibrato wave (see figure 4 .12).


174
Opening a Vibrato Sample
In order to start the MMSV software the user should type mmsv at the Matlab
command prompt. This will bring up the MMSV main screen, which is shown in Figure
A. 1.
Figure A.l. MMSV main screen.
The main figure contains axes to display the vibrato sample and a menu which
controls all the software functions. The functions available are
File
1.1.
Open wave file.
1.2.
Save wave file
1.3.
Save parameters
1.4.
Print figure
1.5.
Exit
Edit
2.1.
Edit wave file
2.2.
Play wave file


159
Enrico Caruso
Figure 8.14 shows the box plots for the amplitude analysis and figure 8.15 shows
the box plots for the frequency analysis.
Figure 8 .14. Box plots for the amplitude analysis of Caruso.
Figure 8.15. Box plots for the frequency analysis of Caruso.


93
File Edit Window Help
Instfreq and amp offreq vIP
Figure 5.19. Instantaneous waves for ec08.wav.
Figure 5.20. Power spectra of the instantaneous waves in ec08.wav.


103
Figure 5.39. Instantaneous waves for pldom01.wav.
Figure 5.40. Power spectra of the instantaneous waves in pldomOl wav.


138
Figure 6.11. Synthesized amplitude vibrato wave.
Figure 6.12. Error for amplitude model.


66
6.2.Short length
6.2.1. Frequency vibrato
6.2.11. Instantaneous frequency
6.2.1.2.Instantaneous amplitude
6.2.2. Amplitude vibrato
7. Help
7.1. Where to find help
7.2. About
The following section will describe the tasks performed by each of the most
recently added menu options.
Edit. Options. Short Length Model
This option allows the user to change the number of poles used for the short length
model. The default value is 3 poles since the frequency vibrato wave is almost sinusoidal.
The range of valid values is between 1 and 11 poles. Figure 4.31 shows the window
displayed when this option is selected. Changing the number of poles for the short length
model does not change the number of poles used for the full length model.
View. Window
This option allows the user to observe the shape of the window currently in use.
Figure 4.32 shows the window displayed when the option is selected.


25
Fundamental Concepts
The Discrete Fourier Transform
As in the case of continuous time periodic signals, if we have a sequence x(n)
which is periodic with period N so that x(n) = x(n+rN) for any value of r where r is an
integer, this signal can be represented by a Fourier series composed of a sum of complex
exponentials at integer multiple frequencies of the fundamental frequency (27t/N) of the
signal x(n). These complex exponentials have the form:
ek in) = ej(2*'N)kn =ek(n + rN) (2-11)
where k is an integer. Thus, the Fourier series representation becomes (Oppenheim and
Schafer, 1989, p.515)
x (n) = ^Yjxik)ej(2n,N)kn (2-12)
The Fourier series representation for a discrete time signal with period N requires
only N complex exponentials at integer multiple frequencies, while the Fourier series
representation of a continuous time periodic signal usually requires infinite complex
exponentials at integer multiple frequencies. This can be seen in equation 2-11 since the
complex exponentials ek(n) are the same for values of k separated by N. This is
ek(n) = ek+iN(n) (2-13)
where 1 is an integer. The N periodic complex exponentials eo(n), ei(n),..., e>j-i(n)
constitute all the periodic complex exponentials that are integer multiples of 2it/N.
Therefore, the Fourier series representation of the periodic signal x(n) requires only N of
these complex exponentials and has the form:


8
2. Many people cannot hear the vibrato. This happens to individuals with
unmusical ears, and to musicians with skillful ears but with lack of training in
tone analysis.
3. The vibrato is generally perceived to a much smaller degree than its true
extent. A vibrato extent of 1 tone is always heard as a fifth of a tone or less.
4. The assumption that the elimination of the exaggerated types of vibrato
eliminate all vibrato. Most of the perceptions of a bad vibrato fail to distinguish
between tremolo and wobble.
5. Many musicians are able to discern tone quality but fail to judge vibrato.
Based on laboratory experiments, Seashore reached the following conclusions
regarding the existence of the frequency vibrato:
1. The vibrato is present in 95 percent of all the tones sung by professional
singers. This percentage includes both good and bad vibrato.
2. All types of tones, such as sustained tones, short tones, attacks, releases, and
other types of changes in pitch contain vibrato. These types of vibrato are hard
to hear, and can only be detected by a highly musical ear. From this, it can be
concluded that the vibrato is not an enrichment forced on a portion of a note,
but under typical conditions it is a vibrant characteristic of the voice organ
throughout the entire performance.
3. Vibrato is present in all successful voice students, well trained amateurs and in
recognized artists. Primitive people display vibrato when singing with authentic
feeling.
4. The vibrato may start to develop at an early age at the time a child begins to
sing naturally and with true feeling. The percentage of cases of vibrato is
greater as the individuals grow older and get musical training.
5. People who confess to not using vibrato and who are opposed to its use, such
as great singers, teachers of voice, and voice students, employ it in their best
singing.
6. A person with the appropriate singing skills who has no vibrato may develop it
to a good level in a few lessons. It has been found in the untrained voice as an
irregular pulsation which by appropriate training and direction can be
converted into a good vibrato.


I want to dedicate this dissertation to my father, my mother and my sister. They
encouraged me at all times and provided their unconditional support during the most
difficult times. Their love and affection kept me going through the entire Ph.D. program.
I also desire to dedicate it to my relatives and friends who always keep me in their
minds and prayers, and have kept in touch with me even though the distance between us is
large.


182
Figure A.9. Frequency and amplitude vibrato waves.
File Edit Window Help
Freq and amp vibrato power spectra
150
100
50
0
30 40
Frequencv(Hz)
0 10 20
Blue=Freq; Red=Amp
Figure A. 10. Frequency and amplitude vibrato power spectra.
The Get parameters option prints the parameters of the frequency and amplitude
vibrato waves on the Matlab command window as it calculates them. Figure A ll shows
the aspect of the computer's screen at this point. By selecting the third button from left to
right on the taskbar, the Matlab command window will be brought to the front to display
the parameters (see figure A. 12).


109
Blue=Freq; Red=Amp Time(s)
Figure 5.51. Instantaneous waves for kbatOl .wav.
Figure 5.52. Power spectra of the instantaneous waves in kbat01.wav.


104
File pldom02.wav
Figure 5.41. Frequency and amplitude vibrato waves for pldom02.wav.
Figure 5.42. Vibrato power spectra for pldom02.wav.


32
where {ei, e2,....,ep}are the noise vectors and {vi, v2,....,vp}are the signal vectors.
The signal vectors are orthogonal to all the vectors in the subspace formed by the
noise vectors. The frequencies of the sinusoidal components in the signal are calculated as
the peaks of the spectral estimator:
w E E- w
w Noise Noise n
(2-40)
where Noise is the matrix of noise subspace eigenvectors obtained from the signal
autocorrelation matrix.
w
1
yj* 1
jw 2
jw(N-l)
M = Number of complex sinusoids in the signal
N > M+l
In theory, when w = w¡ (the ith sinusoidal frequency)
PMUSIC 00
Due to estimation errors the frequencies of the peaks given by the MUSIC
estimator will be at or near the true frequency values.


171
Vibrato Model
A lot of data in dat file format was obtained from the samples analyzed which has
not been thoroughly analyzed. This data can be used to make comparisons among the 7
singers to determine if there are differences among them. An informal analysis suggests
that parameters of contemporary and historic singers are different.
The samples used in this research belong to singers who posses good vibrato. The
software can be used to analyze samples of poor vibrato. The parameters of both types of
samples can be compared to determine what are the parameters that determine a good
vibrato. According to the literature a good vibrato should be more controlled, that is, it
should have smaller frequency and amplitude variations, which can be easily seen in the
instantaneous frequency and amplitude waves.
A listening test can be conducted to study the listeners' opinion on the model
developed. This study could include three types of vibrato: real vibrato, proposed model,
and pure sinusoidal model. Comparisons could be made among the three vibrato types.
The realistic effect of the proposed model could be judged.
The effect of the formants was not analyzed in this research. The formants are
considered to be constant during vibrato but may actually change during it. Some singers
who posses poor vibrato move their jaw while performing vibrato. This may alter the
vibrato wave and may have an effect on the listener.
Singers' Formant
It was expected to find the singers' formant in Katherine Battles samples as it was
in the other two female singers. This casts some doubt about the purpose of it. A study


218
fprintf('Maximum amplitude relative to the mean = %5.3f\n',Af)
fprintf('Frequency of the maximum amplitude in Hz = %5.3f\n',ff)
fprintf('Maximum frequency error in percentage = %5.3f\n',errorpc)
% Orders the components of the frequency vibrato
[Yford If]= sort(Yf(1:length(fy)));
Yford= fliplr(Yford);
If= fliplr(If);
% Clears temporary variables
clear handle elemab elembl j k 1 ninf nsup linel line2 limits;
clear Ameanf Ymaxf nmax Af ff errorpc;
M-file fullaml m
% M-file fullaml.m
% Calculates full length model for the amplitude vibrato
% Applies decimation factor if needed
if decfac > 1
Amaxdec= decimate(Amax,decfac);
else
Amaxdec= Amax;
end
% Calculates model
ameandec= mean(Amaxdec);
if ((method == 'aut') | (method == 'cov'))
if method == 'cov'
[aa,ga] = covar((Amaxdec-ameandec),poles);
else
[aa,ga] = auto((Amaxdec-ameandec),poles,1024);
end
[ha,wa] = freqz(ga,aa,512,1./(decfac.*tsy));
else
[ha,wa] = pmusic((Amaxdec-ameandec),poles,1024,1./(decfac.*tsy))
end
% Plots the amplitude vibrato power spectrum
handle= findobj('Tag','AmplVibCovarAxes');
axes(handle);
cl a;
hold on;
axis auto;
plot(wa,abs(ha),'m');
limits= axis;
axis([0 40 limits(3) limits(4)]);
hold off;
handle= msgbox('Select range to analyze','Input required');
waitfor(handle);
drawnow;
handle= findobj('Tag','Fig4');
figure(handle);


41
Figure 4.4. Elliptical filter options.
The parameters that can be adjusted are
1. Cutoff frequency. This is the frequency of transition for the filter. The range of
values is from 0 to 40 Hz.
2. Attenuation. This parameter represents the filter attenuation in the rejection
band. The range of values is from 0 to 100 db.
3. Type. The user can select between low pass and high pass filters using this
pop-up menu.
Edit. Options. Full Length Model
This option allows the selection of the parametric model to be used in the full
length model, and the adjusting of some of its parameters. Figure 4.5 shows the window
by which the changes can be made.


74
The mean frequency given by the software is 2993.115 Hz, compared to a real
value of 3000 Hz, this represents an error of 0.23%. The software reports no frequency
variations at all. The mean amplitude calculated is 133.616 db with no amplitude
variations. In summary, the frequency and amplitude calculations made using signal
valid 1 .wav are very accurate.
Signal with Frequency and Amplitude Modulation
The signal with frequency and amplitude modulations is called valid3.wav. This file
contains a tone at 3000 Hz with sinusoidal frequency and amplitude modulations. The
frequency is modulated 100 Hz at a rate of 6 Hz, and the amplitude is modulated 4000
SPL (Sound Pressure Level) units at a rate of 1 Hz. This will emulate a harmonic at 3000
Hz with vibrato. Figure 4.36 shows the spectrogram of signal valid3.wav.
Figure 4.36. Spectrogram of signal valid3 .wav.


214
M-file freqfrl.m
% M-file freqfrl.m
% Plots the instantaneous frequency wave and
% calculates its power spectrum
% Creates figure 3 if it does not exist
handle= findobj('Tag','FiglnstFrCur');
if isempty(handle)
figinstfrcur;
handle= gcf;
handlel= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
axis([0 max(t(lrlength(t)-14)) 0 1]);
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
axis([0 max(t(1:length(t)-14)) 0 1]);
end;
% Creates figure 4 if it does not exist
handle= findobj('Tag','FiglnstFrSpec');
if isempty(handle)
figinstfrspec;
handle= gcf;
handlel= findobj(handle,'Tag','FreqVibSpecAxes')
axes(handlel);
axis([0 20 0 1]);
handlel= findobj(handle,'Tag','AmplVibSpecAxes')
axes(handlel);
axis([0 20 0 1]);
end;
% Mean, maximum, and minimum frquency for the
% instantaneous frequency wave
fprintf('\n');
disp('Frequency vibrato:');
fmeaninst= mean(Fmaxinst);
fprintf('Mean frequency = %5.3f\n',fmeaninst)
fmaxinst= max(Fmaxinst);
fmininst= min(Fmaxinst);
fprintf('Maximum frequency = %5.3f\n,fmaxinst)
fprintf(Minimum frequency = %5.3f\n1,fmininst)
% Plots the instantaneous frequency wave
handle= findobj('Tag','FiglnstFrCur');
handlel= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
hold on;
axis auto;
handle= plot((t(1:length(t)-14))',Fmaxinst,'b-');
axis([0 max(t(1:length(t)-14)) 0 10]);
hold off;
drawnow;


132
Figure 6.5. Pure sinusoidal synthesis.
Figure 6.6 shows the error of the pure sinusoidal wave of figure 6.5. The
maximum error is 66.21 Hz, the minimum is -58.43 Hz, and the average is 20.30 Hz
Figure 6.6. Error for pure sinusoidal model.
Figure 6.7 shows the original frequency vibrato wave in blue and a synthesized
wave in red which was generated using the proposed model. Again, the synthesized wave


77
Amplitude vibrato:
Mean amplitude = 132.173
Maximum amplitude = 133.874
Minimum amplitude = 130.347
Mean amplitude variation in db = 1.093
Mean amplitude variation in percentage = 0.827
Mean amplitude variation above the mean in db = 1.079
Mean amplitude variation below the mean in db = 1.107
Maximum amplitude relative to the mean = 130.821
Frequency of the maximum amplitude in Hz = 1.019
Maximum frequency error in percentage = 25.000
The mean frequency of the frequency vibrato is 3000.611 Hz, compared with an
expected value of 3000 Hz, and the maximum and minimum values are 3100.781 Hz and
2906.982 Hz, compared to expected values of 3100 Hz and 2900 Hz respectively. The
frequency vibrato is calculated at 6.112 Hz compared to an expected value of 6 Hz. From
these values we can see that the errors between the expected and the calculated values are
very small.
The mean value of the amplitude vibrato is 133.874 db compared to 133.616 db
for signal validl.wav. The frequency of oscillation of the amplitude vibrato is calculated at
1.019 Hz compared to an expected value of 1 Hz. Again, the errors are very small.
Figure 4.39 shows the instantaneous frequency and amplitude of the frequency
vibrato wave in figure 4.37. The instantaneous frequency is shown in blue and the
instantaneous amplitude in red. The instantaneous frequency wave shows that there are


225
M-file savedat5 m
% M-file savedat5.m
% Saves the contents of the variables in memory
% Saves parameters from one harmonic
[fnamel pathnamel]= uiputfile(1 *.dat','Save data file (.dat)')
if fnamel ~= 0
[fid,mes]= fopen([pathnamel fnamel],'wt');
fprintf(fid,'%s\n',pathname);
fprintf(fid,'%s\n',fname);
fprintf(fid,'%3s\n',harm);
fprintf(fid,'%10.3f\n',fmean);
fprintf(fid,'%10.3f\n1,fmax);
fprintf(fid,'%10.3f\n',fmin);
fprintf(fid,'%10.3f\n',fvarhz);
fprintf(fid,'%10.3f\n',fvarpc);
fprintf(fid,'%10.3f\n',fvarst);
fprintf(fid,'%10.3f\n',fvarab);
fprintf(fid,'%10.3f\n',fvarbl);
fprintf(fid,'%10.3f\n',amean);
fprintf(fid,'%10.3f\n',amax);
fprintf(fid,'%10.3f\n',amin);
fprintf(fid,'%10.3f\n',avardb);
fprintf(fid,'%10.3f\n',avarpc);
fprintf(fid,1%10.3f\n1,avarab);
fprintf(fid,'%10.3f\n',avarbl);
for j=l:10
fprintf(fid,'%10.3f\n',real(Yford(j)));
fprintf(fid,'%10.3f\n',imag(Yford(j ) ) ) ;
end
for j=l:10
fprintf(fid,'%10.3f\n',fy(If(j)));
end
fprintf(fid,'%10.3f\n',Nf);
for j=l:10
fprintf(fid,'%10.3f\n',real(Yaord(j)));
fprintf(fid,'%10.3f\n',imag(Yaord(j)));
end
for j=l:10
fprintf(fid,'%10.3f\n',fy(Ia(j)));
end
fprintf(fid,'%10.3f\n',Na);
fprintf(fid,1%10.3f\n',fmeaninst);
fprintf(fid,'%10.3f\n',fmaxinst);
fprintf(fid,'%10.3f\n',fmininst);
fprintf(fid,'%10.3f\n1,ameaninst);
fprintf(fid,'%10.3f\n',amaxinst);
fprintf(fid,'%10.3f\n',amininst);
for j=l:10
fprintf(fid,'%10.3f\n',real(Yfordinst(j)));
fprintf(fid,'%10.3f\n',imag(Yfordinst(j)));
end


11
Subjective Factors that Affect the Appreciation of the Vibrato
The vibrato ear
There are ears that can detect differences in vibrato quality in the same way that
there are ears for pitch, intensity, and time (Seashore, 1936, p.147). The sharp vibrato ear
detects in a higher degree the elements of elegance and beauty that are not appreciated by
the less sensitive vibrato ear.
The emotionality of the individual
At the same level of importance as the ear is the natural and temperamental
response or emotionality of the individual. A person with a sharp vibrato ear may detect
every aspect of the vibrato with the highest distinction and still be unable to experience the
feeling that the artist communicates into it. The main reason for this is the inability of the
listener to perceive emotions in music. This inability varies widely among different people.
Attitude and training
The influence of attitude and training for perception and feeling in vibrato has been
corroborated in a clear way by considering the widely varying attitudes toward vibrato.
The listener's disposition
We are considering the vibrato esthetic, and esthetics is systematic and precise.
Consequently, when we consider norms of beauty for vibrato we must restrict ourselves to
judgments made by listeners through a critical attitude and not through a musical attitude.


204
% Generates the amplitude vibrato power spectrum
% The length of Max, imax, and t are equal
Na= length(t);
Ya= fft((Amax-amean),Na);
fsy= l./tsy;
fy= fsy.* (0:(Na/2-1))/Na;
Ameana= mean(abs(Ya));
for j=l:length(fy)
line2(j)= Ameana;
end
% Plots the amplitude vibrato power spectrum in the
% same window of the frequency vibrato power spectrum
handle= findobj('Tag','AmplVibSpecAxes');
axes(handle);
hold on;
axis auto;
handle= plot(fy,abs(Ya(1:length(fy))),'r-',fy, line2, 'r-') ;
limits= axis;
axis([0 40 limits(3) limits(4)]);
hold off;
handle= msgbox('Select range to analyzeInput required');
waitfor(handle);
drawnow;
handle= findobj('Tag','Fig4');
figure(handle);
% Gets input from mouse
[1, k]= ginput(2) ;
ninf= round(1(1).*(length(fy)-1)./max(fy) +1);
nsup= round(1(2).*(length(fy)-1)./max(fy) +1);
if ninf < 2
ninf = 2;
end
if nsup > length(fy)
nsup = length(fy);
end
% Maximum amplitude relative to the mean, frequency,
% and percent of frequency error
[Ymaxa,nmax]= max(abs(Ya(ninf:nsup)));
Aa= Ymaxa Ameana;
fa= fy(nmax+ninf-1) ;
errorpc= ((fy(2)./2) / fa) 100;
fprintf('Maximum amplitude relative to the mean = %5.3f\n',Aa)
fprintf('Frequency of the maximum amplitude in Hz = %5.3f\n',fa)
fprintf('Maximum frequency error in percentage = %5.3f\n',errorpc)
% Orders the components of the amplitude vibrato power spectrum
[Yaord Ia]= sort(Ya(1:length(fy)));
Yaord= fliplr(Yaord);
Ia= fliplr(la);
% Clears temporary variables
clear handle elemab elembl j k 1 ninf nsup linel line2 limits;
clear Ameana Ymaxa nmax Aa fa errorpc;


Table 7.2continued.
File name
abs(Yaordinst(l))
angle( Y aordinst( 1))
abs(Yaordinst(2))
angle(Y aordinst(2))
Bjor05.wav
428.6493648
-1.923949858
351.7142015
-0.547978468
Bjorll.wav
472.0123706
-2.221710869
350.5805972
-0.423298467
Bjor25.wav
403.4658652
1.487262165
342.9190815
-1.556492173
Ec04.wav
419.8408637
-1.01842387
221.2708611
-2.581262051
Ec08.wav
467.5129849
0.288751596
374.0404176
-1.526650395
Ecl6.wav
518.4697985
0.72683439
317.7147648
-0.528919066
Pav02.wav
611.7352461
-0.202764505
582.8538577
2.433107068
Pav03a.wav
633.6934994
1.298649058
433.3412993
2.132573381
Pavl4.wav
581.2453752
1.56328996
480.501224
0.341331552
Pldom01.wav
448.1944166
1.493149311
374.1665247
0.107888509
Pldom02.wav
280.7343454
-1.450277244
268.7163355
2.817862705
Pldom24.wav
717.6319202
1.81314831
653.412521
1.07944091
kbat01.wav
769.8594711
2.174353813
358.770321
2.733458554
Kbat20.wav
771.1820335
-0.91846969
555.0829229
-1.973449139
Kbat22.wav
613.8471095
1.249079258
360.6672215
0.698007122
Moncab30.wav
724.0891971
1.338317754
553.3504429
2.56807831
Moncab31.wav
1095.882152
0.583007719
559.1500202
-2.899005733
Moncab33.wav
716.5894113
1.33081096
499.8508554
-1.12022057
delosa01.wav
979.4838282
-2.986148109
747.9957197
1.871982054
delosa07.wav
1577.735819
-2.707784861
1377.721878
-2.098912677
delosa09.wav
1932.719508
2.489770635
621.2338802
0.045355961
Mean
722.122599
494.5264261
Maximum
1932.719508
1377.721878
Minimum
280.7343454
221.2708611
Std. dev.
388.9928377
238.5499356
Cl Maximum
1500.108274
971.6262973
Cl Minimum
-55.86307628
17.4265548


40
Figure 4.3. Spectrogram options.
The available options are
1. Minimum frequency. This represents the minimum frequency displayed in the
spectrogram. The valid values are from 0 to 10,000 Hz. The callback function
checks the value of the maximum frequency and increases it if it is lower than
the minimum frequency or higher, but less than 1000 Hz. This avoids user
errors.
2. Maximum frequency. This represents the maximum frequency displayed in the
spectrogram. The valid values are from 1000 to 11,000 Hz. The callback
function checks the value of the minimum frequency and decreases it if it is
higher than the maximum frequency or lower, but less than 1000 Flz. This
avoids user errors.
3. Window. This option allows the user to change the window used to calculate
the spectrogram. The choices available are: Blackman, Chebyshev, Hamming,
Hanning, Kaiser, and rectangular.
Edit, Options, Elliptical Filter
The user can change some parameters of the elliptical filter by using this option.
Figure 4.4 shows the window displayed when this option is selected.


14
frequency of the frequency vibrato wave increased slightly after World War II. The mean
value post World war II is 6.74 Hz, and 6.41 Hz pre World War II. The amplitude
variation of the frequency vibrato wave around the mean value is 0.98 semi-tones for the
more recent group and 0.63 semi-tones for the older group.
Vibrato in wind instruments
According to Hattwick (Hattwick, 1932, p.l) a recording of 742 sec. of several
performances with different types of wind instruments shows that vibrato is present during
131 sec., that is, 18% of the time. The average amplitude of the frequency vibrato wave is
0.4 of a tone and the average frequency is 7 Hz. Hattwick noticed a tendency for the use
of vibrato in brass instruments rather than in reed instruments. Vibrato was also found to a
greater extent in jazz music than in the classics.
The Singers' Formant
A contrast of the spectrum of a vowel as it is spoken and sung shows that the
amplitude is much higher between 2,500 and 3,500 Hz (see figure 2.2). This peak is
typical of all vowels sung by professional male singers, and has been named the "singers'
formant."
Figure 2.2. The singers formant (Sundberg, 1987, p.l 18).


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EF7P6CZG8_WD3G58 INGEST_TIME 2014-12-05T22:54:52Z PACKAGE AA00024485_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


45
Figure 4.9. Example of frequency and amplitude vibrato power spectra.
Filter. Elliptical
This function allows the frequency and/or amplitude vibrato waves to be filtered
with an elliptical filter. This is a new option in the software, which was included for the
following reasons:
1. The discrete nature of the FFT introduces some sharp discontinuities which are
not present in the real wave. These discontinuities introduce high frequency
components that can be removed using a low pass filter.
2. It is known that for the frequency vibrato wave there are no frequency
components above 10 Hz, and for the amplitude vibrato there no components
above 20 Hz. Therefore, a low pass filter can be used to reduce the noise level
and improve the performance of the LPC method.
3. Unwanted frequency components can be removed before applying the full
length model.
Filter. Median
The user can apply a median filter to the frequency and/or amplitude vibrato by
using this option. This gives the user the choice of a different type of filter to remove rapid
variations not present in the real wave.


87
File Edit Window Help
Instfreq and amp of Treq vib
Figure 5.7. Instantaneous waves for bjorl 1.wav.
Figure 5.8. Power spectra of the instantaneous waves in bjorl 1 .wav.


84
Jussi Biorling
File bjor5.wav
Blue=Freq, Red=Amp Time(s)
Figure 5.1. Frequency and amplitude vibrato waves for bjor5.wav.
ES
File Edit Window Help
Blue=Freq; Red=Amp
Frequency(Hz)
Figure 5.2. Vibrato power spectra for bjor5.wav.


122
File delosa07.wav
Figure 5.77. Frequency and amplitude vibrato waves for delosa07.wav.
Figure 5.78. Vibrato power spectra for delosa07.wav.


127
following model, which can reproduce the vibrato wave based on the instantaneous
frequency and amplitude information and their average values:
ft 1) = to (6-2)
/(l) = Fa+ Ao cos(^0 ) (6-3)
ftj) = ftj -1) + 2*CZf, (i)/(N -14) + Aff(j))At,j = 2:N (6-4)
i=l
f(j) = K+(ZA,(.OI(.N-\4) + AArU))oos(M)J =2:N (6-5)
1=1
where:
<(>(1) = Frequency vibrato starting phase angle
f(l) = Frequency vibrato starting value
ff(i) = Frequency vibrato instantaneous frequency
Afi(j) = Frequency variation respect to the average value
At = Time between two frequency vibrato samples
f(j) = Synthesized frequency vibrato wave
F0 = Average value of the frequency vibrato wave
Af(i) = Frequency vibrato instantaneous amplitude
AAf(j) = Amplitude variation respect to the average value
N = Total number of samples of the frequency vibrato wave
Since the number of dominant peaks located below 5 Hz was 2 and 3 in most cases
(see chapter 5), the instantaneous frequency and amplitude waves can be modeled with
reasonable accuracy by using the three most dominant components of the instantaneous


17
individual cycle at low, mid and high tone vibrato samples. The curve shapes and the
cycle-to-cycle variations of the rate and extent were also analyzed.
In this study eight professional male singers were used as subjects. Each of them
was recorded sustaining an /a/ with vibrato for at least three seconds at low, mid and high
pitch levels. The analysis program was based on a peak picking method to detect the
fundamental frequency.
For each of the modulation cycles the following variables were calculated:
modulation period, modulation extent, rate of Fo increase, rate of Fo decrease. Also, the
following two sequencial measures were obtained: modulation jitter, and modulation
shimmer.
The entire set of samples exhibited three different vibrato curve shapes: sinusoidal,
triangular, and trapezoidal. From the whole set of samples only two of them exhibited a
sinusoidal shape, five of them showed triangular shape, and ten of them trapezoidal shape.
Asymmetry was found in the frequency vibrato, which was perceived in the form of a
higher slope during the increase of the vibrato curve and a smaller slope during the
decrease.
Singers performance in the conditions of the present study in which they are not
performing in front of a crowd and with an orchestra, and do not need to express
emotions, may be different from normal singing conditions.
Research done in the past has found no relation between pitch level and vibrato
rates, although, loud vibrato samples have been reported to produce higher vibrato rates
than soft vibrato samples. However, the data collected in this study shows that there is less
variation in vibrato rate than in vibrato extent.


206
for k=l:p,
b(k) = 0;
for m=p:N-l;
b(k)= b(k) + x(m+l-k).*x(m+1);
end
b(k)= b(k)./(N-p);
end
% Obtains the a coefficients
a= inv(A)*(b)';
% Calculates the c vector
for k=l:p,
c(k) = 0;
for m=p:N-l;
c(k)= c(k) + x(m+1).*x(m+l-k);
end
c (k) = c (k) / (N-p) ;
end
% Calculates the coo coefficient
coo = 0;
for m=p:N-l;
coo= coo + x(m+1).*x(m+1);
end
coo= coo./(N-p);
% Calculates the gain
pmin= coo + (a')*(c');
a= [1 a1];
g= sqrt(pmin.*1024);
M-file curvaml m
% M-file curvaml.m
% Calculates and plots the instantaneous amplitude wave
% Displays message
handle= msgbox('Select range of frequenciesInput required')
waitfor(handle);
drawnow;
handle= findobj('Tag',1Figlnst');
figure(handle);
% Gets input from mouse
[1,k]= ginput (2);
isup= round(k(1).*(length(wf)-1)./max(wf) + 1);
iinf= round(k(2).*(length(wf)-1)./max(wf) +1);


10
Universality of use
A very important fact is that all excellent singers and violinists, recognized as the
best in musical accomplishment, display vibrato. Vibrato may change with time, but it is
very likely to stay as a fundamental parameter of musical productions.
Automatic nature
The fact that, in voice, the vibrato appears naturally whenever a person sings with
authentic feeling indicates that the vibrato is a form of esthetic expression. This statement
is reinforced by the fact that the vibrato can be found in primitive people who are
completely untrained musically and are not conscious of the phenomenon, and by the fact
that children begin to develop it at the time they start singing in propitious conditions for
self-expression. Moreover, vibrato production is seldom taught to singing students. Most
of the teaching deals with its modification or attenuation. Many of the best singers do not
know how they develop vibrato.
Advantage over precision
One more proof of the exceptional musical value of the vibrato is its preference to
accuracy and smoothness of tone in the playing of violinists, that is, the vibrato has a
higher importance than the precision of the tone in the violin.
Importance in tone quality
When appreciating the vibrato quality, it is not thought of as a periodic variation in
pitch, but rather as a flexible, rich, and tender factor in tone quality, all of which are
considered desirable.


36
The Singers Formant
All the samples analyzed during my masters thesis showed the so-called singers
formant. However, this may not always be the case. Some samples used in different
studies at the Communication Processes and Disorders at the University Florida show that
the singers formant is not always present in singing samples.
The singers formant can be seen in FCSV as an increase in the amplitude of the
harmonics at about 2500-3500 Hz. The amplitude of the singers formant in the samples
analyzed in my masters thesis was approximately the same as that of the first formant.
A study can be conducted with the data already available and the use of FCSV or
any other adequate software to determine if the singers formant varies within different
samples of the same singer and among different singers.
The variables to be measured will be the frequency and amplitude of the singer's
formant for each sample. The study will be based on at least 25 singers and 4 samples per
singer. Statistical methods will be applied to determine significant differences between
samples and singers.


184
Full Length Model
The Full length model option calculates a parametric model of the frequency or
amplitude vibrato waves. The following model parameters can be adjusted:
1. Decimation factor. The decimation factor reduces the sample rate of the
frequency or amplitude vibrato by the value selected. A decimation factor of 2
reduces the sample rate from 80 Hz to 40 Hz. This can be used when the
frequencies in the signal are low to improve the model performance. The
minimum value is 1 and the maximum is 41. The default value is 1.
2. Poles. The number of poles determines the maximum number of peaks in the
power spectrum, and therefore, the number of sine waves that can be modeled.
Two poles are requires per sine wave. The range of values is from 1 to 21. The
default value is 3.
3. Model. Three different models can be chosen: autocorrelation, covariance, and
MUSIC (multiple signal classification). The default value is Covariance.
These parameters can be adjusted by selecting Edit, Options, Full length model.
The software will display the window shown in figure A. 13. After selecting the desired
values, clicking on the Accept button will make the changes effective.
Figure A. 13. Full length model options.


65
4.
5.
6.
3..2. Filter response
Analysis
4.1. Spectrogram
4.2. Get vibrato
4.3. Get parameters
Filter
5.1. Elliptical
5.1.1. Frequency vibrato
5.1.1.1. Wave
5.1.12. Instantaneous frequency
5.1.13. Instantaneous amplitude
5.1.2. Amplitude vibrato
5.1.2.1. Wave
5.2. Median
5.2.1. Frequency vibrato wave
5.2.2. Amplitude vibrato wave
5.3. Get frequency
5.3.1. Power spectrum
5.3.2. Instantaneous power spectrum
Model
6.1.Full length
6.1.1.Frequency vibrato
6.1.2.Amplitude vibrato


189
Time(s)
Figure A. 18. Instantaneous power spectrum of the frequency vibrato.
The window shown in figure A. 19 will appear on the screen to inform the user of
the need for an input. The user should click on the OK button, move the mouse pointer to
the frequency vibrato instantaneous power spectrum and click above and below the initial
frequency of the instantaneous wave, that is, the frequency at t = 0. The software will
extract the instantaneous frequency wave and will display it in blue on top of the
frequency vibrato instantaneous power spectrum for comparison purposes (see figure
A.20).
Figure A. 19. Input required window


187
The software will display figure A.8 to indicate that an input is required. The user
should click on the OK button, move the mouse pointer to the model spectrum window
and select the range of frequencies to analyze, first the lower frequency and then the
higher frequency. The software will calculate the frequency and height of the highest peak
in the selected range, and will display them in the Matlab command window. The pole
location will be shown in a window similar to figure A. 15.
Instantaneous Frequency and Amplitude of the Frequency Vibrato Wave
In order to obtain clean instantaneous frequency and amplitude waves, filtering the
frequency vibrato wave with a low pass filter is recommended. To adjust the low pass
filter parameters click on Edit, Options, Elliptical filter. The window shown in figure A. 17
will appear on the screen. The parameters that can be adjusted are
1. Cutoff frequency. This is the filter transition frequency. It can be adjusted from
0 to 40 Hz. The default value is 20 Hz.
2. Attenuation. It represents the filter attenuation in decibels. The range of values
is from 0 to 100 db. The default value is 20 db.
3. Filter type. The filter type can be low pass or high pass. The default value is
low pass.
After the parameters have been changed to the desired values the user should click
on the OK button to make the changes effective. A cutoff frequency of 10 Hz should work
in most cases.
To filter the frequency vibrato wave the user, should click on Filter, Elliptical,
Frequency vibrato, Wave. The filtered frequency vibrato wave and its corresponding
power spectrum will replace the original ones.


59
Figure 4.24. Spectrogram using the autocorrelation method.
FHe Edit Window Help
Inst freq and amp of freq vIP
n
-a
N
n ->
13
"MAA/WlA/vvVvu-r'
Q-
E
<
0
100
50
0 0.5 1 1.5
Blue=Freq; Red=Amp Time(s)
Figure 4.25. Instantaneous frequency and amplitude of the frequency vibrato wave.
The amplitude curve in figure 4.25 matches the instantaneous amplitude of the
signal shown in figure 4.13. The amplitude curve in figure 4.25 has a maximum value of
108.163, a minimum of 70.812, and a mean of 87.508. The peaks at 0.1 and 0.9 sec., and
the valley at 0.25 sec., match the characteristics of the wave in figure 4.13. The ripple


31
P
A
N-\
z
n=p
x(n) + ^a(k)x(n k)
*=1
(2-35)
The prediction error power for the covariance method and the autocorrelation
method are identical except for the range of summation, which is from p to N-l for the
covariance method. In this method all the data points needed to calculate the estimate of
Pmin are known.
After minimizing and manipulating 2-35, we get the following set of equations:
^(1,1)
^(2,D
*( W)
cxx (2,2)
,P)
c(2 ,p)
a(l)'
a(2)
"cao)"
cxx (2,0)
2)
cAp,p)_
1
£>
W
1
S*P,P)_
where
(7,*) = t¡tjlx\n-j)x(n-k) (2-37)
N-pt'p
The estimate of the white noise variance can be found by using
2 =Pm, =c(0,0) + (*)c(CU) (2-38)
k=1
The matrix in equation 2-36 will be singular if the data consist of less than p
complex sinusoids, however, any observation noise will make the matrix nonsingular.
Multiple Signal Classification Method
The multiple signal classification (MUSIC) method is based on the orthogonality
property (Kay, 1988, p.424):
M
Tajvj = i=l, 2,....p (2-39)
j=p+\


194
Figure A.27. Instantaneous frequency power spectrum
The pole location for the instantaneous power spectrum will be shown in a figure
similar to figure A.23. The user should click on the OK button to close this window when
he is finished with it to avoid window proliferation. The parameters for the instantaneous
amplitude wave will be shown on the Matlab command window.
The ripple shown on the amplitude vibrato wave in figure A. 26 is an artifact of the
algorithm and should be eliminated by filtering the wave with a low pass filter. First, the
filter frequency should be adjusted by selecting the Edit, Options, Elliptical filter option,
changing the filter frequency, and clicking on the OK button (a filter frequency of 5 Hz
will work in most cases). And second, the filter should be applied by choosing the Filter,
Elliptical, Frequency vibrato, Instantaneous amplitude option.
The filtered instantaneous amplitude wave and its corresponding power spectrum
will replace the non filtered versions as shown in figures A.28 and A.29. The parameters
of the filtered wave will be shown in the Matlab command window.


Ill
Figure 5.55. Instantaneous waves for kbat20.wav.
File Edit Window Help
Power spectra of instfreq and amp
Blue=Freq; Red=Amp
Frequency!Hz')
Figure 5.56. Power spectra of the instantaneous waves in kbat20.wav.


106
File pldom24.wav
Figure 5.45. Frequency and amplitude vibrato waves for pldom24.wav.
Figure 5.46. Vibrato power spectra for pldom24.wav.


LIST OF FIGURES
Figure page
2.1. Example of vibrato wave 5
2.2. The singers formant 14
4.1. Main screen 38
4.2. Vibrato sample 38
4.3. Spectrogram options 40
4.4. Elliptical filter options 41
4.5. Full length model options 42
4.6. Filter frequency response 43
4.7. Example of spectrogram 43
4.8. Example of frequency and amplitude vibrato waves 44
4.9. Example of frequency and amplitude vibrato power spectra 45
4.10. LPC model of the frequency vibrato 46
4.11. Pole location of the frequency vibrato model 47
4.12. Aspect of the computers screen 48
4.13. Frequency and amplitude vibrato waves 50
4.14. Pole location for the covariance method 51
4.15. Power spectrum for the covariance model 51
4.16. Pole location for the autocorrelation method 52
4.17. Power spectrum for the autocorrelation model 52
4.18. Spectrogram of the frequency vibrato wave 53
xii


160
Luciano Pavarotti
Figure 8.16 shows the box plots for the amplitude analysis and figure 8.17 shows
the box plots for the frequency analysis.
Figure 8 .16. Box plots for the amplitude analysis of Pavarotti.
Figure 8.17. Box plots for the frequency analysis of Pavarotti.


CHAPTER 3
PROPOSED RESEARCH
Vibrato Model
Several models have been proposed and used for synthesizing vibrato (Maher, and
Beauchamp, 1990, p. 219; Titze, 1983, p.90), however, they all have been developed by
applying the analysis by synthesis method. Synthesized voice samples are created with
different vibrato characteristics, the synthesized data is judged, and the results analyzed to
select the best representation for vibrato. No formal analysis has been made on the effects
of the vibrato waveform on perception. Previous investigations have analyzed the vibrato
in a simple way, assuming that the waveform is sinusoidal and the parameters to be
measured are frequency and amplitude in every cycle. Also, none of the more recent high
resolution techniques like LPC, Burg, MUSIC, etc. have been applied to the vibrato signal
(see chapter 2 for the most relevant papers published lately in this area).
There is also controversy regarding the effect of the random patterns in vibrato.
Maher (Maher, and Beauchamp, 1990, p.219) found that there was no preference toward
vibrato with random components versus pure sinusoidal vibrato, while Titze (Titze, 1983,
p.90) includes random patterns in his vibrato model.
The results obtained from FCSV in conjunction with available data will be used to
produce a mathematical model for good vibrato. This new work will make use of my
master's thesis work and will build upon it.
33


227
% Plots the spectrogram
fig2;
imagesc(t',f',X);
axis('xy');
axis([0 max(t) liml lim2]);
colormap(jet);
title('Spectrogram');
xlabel('Time(s)');
ylabel('Frequency(Hz)');
colorbar;
M-file varinit.m
% M-file varinit.m
% Initializes variables
% Variable initialization
N= 1024; % window = 46.44 msec
ovlap= 750;
% Calculates the number of points (samples) between two fft's
points= N ovlap;
% Calculates the time between two fft's
tsy = ts.*points;
% Variable initialization
win= hanning(N);
liml= 0;
lim2= 4000;
ftype='lo';
order= 9;
cutoff= 20;
atten= 20;
[b,a]= ellip(order,0.5,atten,2.*cutoff.*tsy) ;
decfac= 1;
poles= 3;
method= 'cov';
polinst= 3;
M-file viewpolinst.m
function viewpolinst(afpoles);
% Displays the poles of the short length model
% Creates figure and displays the poles
figpoll;
zplane(1,afpoles');
title('Pole-zero plot');


5.84. Power spectra of the instantaneous waves in delosa09.wav 125
6.1. Pure sinusoidal synthesis 129
6.2. Error for pure sinusoidal model 130
6.3. Synthesized wave using the proposed model 130
6.4. Error for proposed model 131
6.5. Pure sinusoidal synthesis 132
6.6. Error for pure sinusoidal model 132
6.7. Synthesized wave using the proposed model 133
6.8. Error for proposed model 133
6.9. Synthesized amplitude vibrato wave 136
6.10. Error for amplitude model 137
6.11. Synthesized amplitude vibrato wave 138
6.12. Error for amplitude model 138
8.1. FFT and LPC power spectra 148
8.2. Singers' formant in sample bjor05.wav 149
8.3. Singers' formant in sample ec04.wav 150
8.4. Singers' formant in sample pav02.wav 150
8.5. Singers' formant in sample pldom01.wav 151
8.6. Singers' formant in sample moncab30.wav 151
8.7. Singers' formant in sample delosa01.wav 152
8.8. Analysis of variance table 155
8.9. Box plots 155
8.10. Analysis of variance table 156
8.11. Box plots 157
8.12. Box plots for the amplitude analysis of Bjorling 158
8.13. Box plots for the frequency analysis of Bjorling 158
xvii


153
Singers' Formant Parameters for all the Samples
Table 8.1 shows the results obtained for all the samples analyzed.
Table 8.1. Singers' formant parameters.
Singer
Path
File name
Poles
Segment
Amplitude
Frequency
Bjorling
C:\Jose\Samples\
Bjor05.wav
11
24
110.961
2648.584
Bjorling
C:\Jose\Samples\
Bjor05.wav
11
31
112.829
2734.717
Bjorling
C:\Jose\Samples\
Bjor05.wav
11
8
113.388
2777.783
Bjorling
C:\Jose\Samples\
Bjorl 1 .wav
11
11
119.391
2842.383
Bjorling
C:\Jose\Samples\
Bjorl 1 .wav
11
41
118.585
2627.051
Bjorling
C:\Jose\Samples\
Bjorl 1 .wav
11
6
114.513
2756.25
Bjorling
C:\Jose\Samples\
Bjor25.wav
11
5
103.164
2928.516
Bjorling
C:\Jose\Samples\
Bjor25.wav
11
37
102.763
3014.648
Bjorling
C:\Jose\Samples\
Bjor25.wav
11
33
102.443
2993.115
Caruso
C:\Jose\Samples\
Ec04.wav
11
25
120.754
2368.652
Caruso
C:\Jose\Samples\
Ec04.wav
11
32
119.74
2390.186
Caruso
C:\Jose\Samples\
Ec04.wav
11
33
119.165
2540.918
Caruso
C:\Jose\Samples\
Ec08.wav
11
46
111.797
2734.717
Caruso
C:\Jose\Samples\
Ec08.wav
11
34
110.645
2799.316
Caruso
C:\Jose\Samples\
Ec08.wav
11
19
107.787
2756.25
Caruso
C:\Jose\Samples\
Ecl6.wav
11
20
116.409
2777.783
Caruso
C:\Jose\Samples\
Ecl6.wav
11
32
117.046
2519.385
Caruso
C:\Jose\Samples\
Ecl6.wav
11
22
115.346
2734.717
Pavarotti
C:\Jose\Samples\
Pav02.wav
11
38
113.948
2863.916
Pavarotti
C:\Jose\Samples\
Pav02.wav
11
33
111.891
2906.982
Pavarotti
C:\Jose\Samples\
Pav02.wav
11
24
112.789
2906.982
Pavarotti
C:\Jose\Samples\
Pav03a.wav
11
18
117.751
2777.783
Pavarotti
C:\Jose\Samples\
Pav03a.wav
11
34
114.281
3122.314
Pavarotti
C:\Jose\Samples\
Pav03a.wav
11
12
111.443
3014.648
Pavarotti
C:\Jose\Samples\
Pavl4.wav
11
3
115.643
2820.85
Pavarotti
C:\Jose\Samples\
Pavl4.wav
11
29
112.909
2734.717
Pavarotti
C:\Jose\Samples\
Pavl4.wav
11
7
115.922
2777.783


115
Figure 5.63. Instantaneous waves for moncab30.wav.
Figure 5.64. Power spectra of the instantaneous waves in moncab30.wav.


29
r (k) = (k -1) + a2h* (-k) (2-25)
/=i
Also,
h\-k) = 0 for k > 0 (2-26)
and
h*(0) = (lim//(z)) =1 (2-27)
\z-CO /
Applying equations 2-26 and 2-27 to 2-25 we obtain
>** (*) = a(7)r- (* z) for k > 1
/=i
>(*) = -Z + 0-2 for k = 0 (2-28)
/=i
The set of equations 2-28 are called the Yule-Walker equations. They provide a
relationship between the AR model parameters and the autocorrelation function of x(n). If
the autocorrelation values of x(n) are known we can determine the AR coefficients by
solving the following set of linear equations (Kay, 1988, p. 116):
^(0) r(-l) " r[-{p-1)]
a(l)
r0)"
'O) ra(0) " r[-(p-2)]
a(2)
=
r(2)
JxxP-1) rxx(P ~ 2) ^(0)
ci(p)
1
' A
Equations 2-29 can be arranged to incorporate the second equation in 2-28. This
results in (Kay, 1988, p. 116)
>*(0)
i
i
a2
r-0)
*( o)
o]
a(l)
=
0
/**(/>) r(p-i)
i
(/>)_
0
(2-30)


159
8 .14. Box plots for the amplitude analysis of Caruso
8.15. Box plots for the frequency analysis of Caruso 159
8.16. Box plots for the amplitude analysis of Pavarotti 160
8 .17. Box plots for the frequency analysis of Pavarotti 160
8.18. Box plots for the amplitude analysis of Domingo 161
8.19. Box plots for the frequency analysis of Domingo 161
8.20. Box plots for the amplitude analysis of Caballe 162
8.21. Box plots for the frequency analysis of Caballe 162
8.22. Box plots for the amplitude analysis of DeLosAngeles 163
8.23. Box plots for the frequency analysis of DeLosAngeles 163
A.l. MMSV main screen 174
A.2. Open wave file window 177
A. 3. Vibrato sample 177
A.4. Example of a spectrogram 178
A. 5. Input required window 179
A.6. Window for harmonic number input 179
A. 7. Spectrogram showing frequency vibrato wave 180
A. 8. Input required window 180
A.9. Frequency and amplitude vibrato waves 182
A. 10. Frequency and amplitude vibrato power spectra 182
All. Aspect of the computer's screen 183
A. 12. Matlab command window 183
A. 13. Full length model options 184
A. 14. Parametric model of the frequency vibrato wave 185
A. 15. Pole location for lull length model 186
A. 16. Parametric model of the amplitude vibrato wave 186
xviii


61
to Placido Domingo. The harmonic selected is number 6. Figure 4.28 shows the power
spectrum of the amplitude vibrato wave in red.
Figure 4.27. Example of amplitude vibrato wave.
rum
File Edit Window Help
Freq and amp vibrato power spectra
0
10
20
Blue=Freq; Red=Amp
150
100
50
0
30 40
Frequency(Hz)
Figure 4.28. Power spectrum of amplitude vibrato wave.
As can be seen in figure 4.27, the amplitude vibrato wave is noisier than the
frequency vibrato wave and its frequencies and amplitudes change more rapidly than in the
frequency vibrato wave. Therefore, the instantaneous frequency and amplitude techniques
used for the frequency vibrato should not be used to obtain instantaneous frequency and


CHAPTER 5
DATA ANALYSIS AND RESULTS
Sample Selection
Dr. Rothmans experience with sung vibrato samples was used for the selection of
the good vibrato samples. We agreed to use 3 samples per singer. Since the total number
of samples chosen had been 20, this resulted in a total of 7 singers. Four male and three
female singers were selected. Table 5.1 shows the singers names and file names.
Table 5.1. Singers names and file names.
Singers name
File name
Jussi Bjorling
Bjor5.wav
Bjorl l.wav
Bjor25.wav
Enrico Caruso
Ec04.wav
Ec08.wav
Ecl6.wav
Luciano Pavarotti
Pav02.wav
Pav03a.wav
Pavl4.wav
Placido Domingo
Pldom01.wav
Pldom02.wav
Pldom24.wav
Kathleen Battle
Kbat01.wav
Kbat20.wav
Kbat22.wav
Monserrat Caballe
Moncab30.wav
Moncab31 .wav
Moncab33.wav
Victoria DeLosAngeles
Delosa01.wav
Delosa07.wav
Delosa09.wav
80


% Clears temporary variables
clear ameanseg atemp handle j;
M-file instfH.m
% M-file instfr4.m
% Calculates and displays the "spectrogram" of the
% frequency vibrato wave
% Initializes variables
if anal==,freq'
hf= zeros(512,(length(Fmax)-14));
else
hf= zeros(64,(length(Fmax)-14));
end
af= zeros(polinst+1,(length(Fmax)-14));
gf= zeros(1,(length(Fmax)-14));
% Determines the method and calculates the "spectrogram"
if anal=='freq'
for j= 1:((length(Fmax)-14));
[atemp,gf(j)]= covar((Fmax(j:j+14)-fmean),polinst);
af(:,j)= atemp';
[hf(:,j),wf] = freqz(gf(j),af(:,j),512,l./tsy);
end
else
for j= 1:((length(Fmax)-14));
fmeanseg= mean(Fmax(j:j+14));
[atemp,gf(j)]= auto((Fmax(j:j+14)-fmeanseg),polinst,128)
af(:,j)= atemp';
[hf(:,j),wf] = freqz(gf(j),af(:,j),64,l./tsy);
end
end
% Deletes figinst if it exists
handle= findobj('Tag','Figinst');
if -isempty(handle)
close(handle);
end;
% Plots the spectrogram
figinst;
imagesc((t(1:length(t)-14))',wf',20.*logl0(abs(hf)));
axis('xy');
colormap(jet) ;
title('Inst freq and amp of freq vib');
xlabel('Time(s)');
ylabel('Frequency(Hz)1);
colorbar;
% Calculates and displays the poles
afpoles= zeros(polinst,length(Fmax)-14);
for j= 1:(length(Fmax)-14);
afpoles(:,j)= roots(af(:,j));
end


228
M-file viewpol.m
function viewpol(a);
% Displays the poles of the full length model
% Creates figure and displays the poles
figpoll;
zplane(1,a);
title('Pole-zero plot');
M-file viewres.m
function viewres(b,a,tsy)
% Displays filter frequency response
% Creates figure
figresl;
% Calculates response and displays it
[h,w]= freqz(b,a,512,1./tsy) ;
plot(w,20.*logl0(abs(h) ) ) ;
limits= axis;
axis([0 40 limits (3) limits(4)]);
title('Filter frequency response');
M-file zoominl m
% M-file zoominl.m
% Zooms in vibrato sample
% Displays message and activates figure
handle= msgbox('Select range to displayInput required')
waitfor(handle);
drawnow;
handle= findobj('Tag','Figl');
figure(handle);
% Gets input from mouse
[1,k]= ginput(2);
ninf= round(1(1).*(length(x)-1)./tmax +1);
nsup= round(1(2).*(length(x)-1)./tmax + 1);
if ninf < 1
ninf= 1;
end
if nsup > length(x)
nsup= length(x);
end
% Changes the axis limits
limits= axis;
axis([(ninf-1).*ts (nsup-l).*ts limits(3) limits(4)]);


131
Figure 6.4 shows the error curve for the model in figure 6.3. The errors have been
reduced. The maximum error is 18.97 Hz, the minimum -26.64 Hz, and the average is
6.68 Hz.
Figure 6.4. Error for proposed model.
The sample chosen from the female singers was kbat22.wav. This sample belongs
to Kathleen Battle. Figure 6.5 shows the original vibrato wave for sample kbat22.wav in
blue, and a synthesized vibrato wave in red which is purely sinusoidal. We can see that
both waves match in frequency at the beginning, but not during the whole two second
span, and the amplitude of the synthesized wave does not follow the original wave.


60
shown is due to deviations of the frequency vibrato wave from the sinusoidal pattern. This
ripple can be reduced by filtering the signal with a low pass filter.
Figure 4.26 shows the power spectrum of the instantaneous frequency and
amplitude waves shown in figure 4.25. The high peaks at 12 Hz are due to the ripple in the
curves.
Figure 4.26. Power spectra of the instantaneous waves.
From all these experiments it was concluded that for very short signals (only one
cycle of the signal in the sample being analyzed) the frequencies calculated by the
covariance method are very precise but not the amplitudes, and the autocorrelation
method provides better amplitude estimates than the covariance method.
Instantaneous Frequency and Amplitude of the Amplitude Vibrato Wave
Figure 4.27 shows an example of an amplitude vibrato wave. The amplitude
vibrato wave is shown in red. This wave is taken from sample pldomOl .wav which belong


162
Monserrat Caballe
Figure 8.20 shows the box plots for the amplitude analysis and figure 8.21 shows
the box plots for the frequency analysis.
Figure 8.20. Box plots for the amplitude analysis of Caballe.
Figure 8.21. Box plots for the frequency analysis of Caballe.


79
Instantaneous frequency:
Mean frequency = 6.038
Maximum frequency = 6.366
Minimum frequency = 5.737
Instantaneous amplitude:
Mean amplitude = 98.271
Maximum amplitude = 106.844
Minimum amplitude = 87.800
The mean frequency value is 6.038 Hz compared to an expected value of 6 Hz.
The maximum and minimum values of 6.366 Hz and 5.737 Hz indicate the real maximum
and minimum frequencies of the frequency vibrato wave.
The mean amplitude of 98.271 Hz is close to the expected value of 100 Hz. The
maximum and minimum values of 106.844 and 87.800 Hz indicate the variations in the
signal amplitude. There is an error in the measure of about 3 percent introduced by the
algorithm.


157
Figure 8.11. Box plots.
Singers Formant Variation Within Samples of the Same Singer
The objective of this study is to find out if the singers' formant varies within
samples of the same singer. In order to determine this the analysis of variance test was
used. Each singer was studied separately. Both the amplitude and frequency of the singers'
formant were analyzed.
Each analysis of variance test had three groups and three samples per group. Each
.wav file represented a group and each sample segment represented a group sample. Due
to space limitations, only the box plots and p values for each of the singers will be shown.
The following sections show the results of this study.


154
Table 8.1continued.
Singer
Path
File name
Poles
Segment
Amplitude
Frequency
Domingo
C:\Jose\Samples\
Pldom01.wav
11
36
112.171
2734.717
Domingo
C:\Jose\Samples\
Pldom01.wav
11
8
116.172
2820.85
Domingo
C:\Jose\Samples\
Pldom01.wav
11
28
113.322
2540.918
Domingo
C:\Jose\Samples\
Pldom02.wav
11
34
117.652
2648.584
Domingo
C:\Jose\Samples\
Pldom02.wav
11
5
111.183
2734.717
Domingo
C:\Jose\Samples\
Pldom02.wav
11
26
119.684
2713.184
Domingo
C:\Jose\Samples\
Pldom24.wav
11
43
107.228
2734.717
Domingo
C:\Jose\Samples\
Pldom24.wav
11
35
105.161
2842.383
Domingo
C:\Jose\Samples\
Pldom24.wav
11
10
111.823
2863.916
Caballe
C:\Jose\Samples\
Moncab30.wav
11
32
115.069
3079.248
Caballe
C:\Jose\Samples\
Moncab30.wav
11
6
106.42
2906.982
Caballe
C:\Jose\Samples\
Moncab30.wav
11
33
113.086
2950.049
Caballe
C:\Jose\Samples\
Moncab31 wav
11
12
109.643
2885.449
Caballe
C:\Jose\Samples\
Moncab31 wav
11
14
106.021
2993.115
Caballe
C:\Jose\Samples\
Moncab31 .wav
11
9
107.261
2928.516
Caballe
C:\Jose\Samples\
Moncab33.wav
11
24
96.259
2777.783
Caballe
C:\Jose\Samples\
Moncab33.wav
11
15
103.369
2993.115
Caballe
C:\Jose\Samples\
Moncab33.wav
11
20
95.802
2799.316
DeLosAngeles
C:\Jose\Samples\
delosa01.wav
11
17
107.198
3014.648
DeLosAngeles
C:\Jose\Samples\
delosa01.wav
11
19
105.605
3208.447
DeLosAngeles
C:\Jose\Samples\
delosa01.wav
11
4
112.799
3186.914
DeLosAngeles
C:\Jose\Samples\
delosa07.wav
11
19
101.501
2885.449
DeLosAngeles
C:\Jose\Samples\
delosa07.wav
11
44
97.97
2777.783
DeLosAngeles
C:\Jose\Samples\
delosa07.wav
11
12
98.549
2842.383
DeLosAngeles
C:\Jose\Samples\
delosa09.wav
11
11
104.938
3251.514
DeLosAngeles
C:\Jose\Samples\
delosa09.wav
11
29
108.733
3165.381
DeLosAngeles
C:\Jose\Samples\
delosa09.wav
11
28
108.583
3273.047


175
2.3. Zoom
2.3.1 In
2.3.2.Out
2.4. Options
2.4.1. Spectrogram
2.4.2. Elliptical filter
2.4.3. Full length model
2.4.4. Short length model
View
3.1.
Window
3..2.
Filter response
Analysis
4.1.
Spectrogram
4.2.
Get vibrato
4.3.
Get parameters
Filter
5.1.
Elliptical
5.1.1. Frequency vibrato
5.1.1.1.Wave
5.1.12. Instantaneous frequency
5.1.13. Instantaneous amplitude
5.1.2. Amplitude vibrato
5.1.2.1.Wave


224
% Deletes ampl vibrato wave and mean value line (0)
% from the FreqVibCurAxes
handle= findobj('Tag','Fig3');
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
cla;
% Deletes ampl vibrato power spectrum and mean value line
% from the FreqVibSpecAxes
handle= findobj('Tag','Fig4');
handlel= findobj(handle,'Tag','AmplVibSpecAxes');
axes(handlel);
cla;
% Makes the calculations on the filtered signal and plots it
ampl4;
% Clears temporary variables
clear handle handlel;
M-file medfrl m
% M-file medfrl.m
% Applies median filter to the frequency vibrato wave
% Filters the frequency vibrato wave with a median filter
Fmax= medfiltl(Fmax, 3);
% Deletes freq vibrato wave and mean value line (0)
% from the FreqVibCurAxes
handle= findobj('Tag','Fig3');
handlel= findobj(handle,'Tag','FreqVibCurAxes');
axes(handlel);
cla;
% Deletes freq vibrato power spectrum and mean value line
% from the FreqVibSpecAxes
handle= findobj('Tag','Fig4');
handlel= findobj(handle,'Tag','FreqVibSpecAxes');
axes(handlel);
cla;
% Makes the calculations on the filtered signal and plots it
freqn4;
% Clears temporary variables
clear handle handlel;


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A MATHEMATICAL MODEL OF SINGERS' VIBRATO
BASED ON WAVEFORM ANALYSIS
By
Jose Antonio Diaz
August 1998
Chairman: A. Antonio Arroyo
Major Department: Electrical and Computer Engineering
The main objective of this dissertation is the development of a mathematical model
of the singers' vibrato. This model describes the frequency and amplitude characteristics of
good vibrato samples found in the singing voice.
A software tool was developed in Matlab 5.1 to analyze and extract the parameters
of the singers vibrato. The software creates a spectrogram of the sample set, from which
the user can select different harmonics. The frequency and amplitude vibrato waves are
extracted from the harmonic selected, and their power spectra and parameters are
calculated. Then, the instantaneous frequency and amplitude of the frequency vibrato
wave are generated, and their power spectra and parameters are calculated.
Sample sets were digitized and analyzed using the software developed. The results
obtained show that the instantaneous frequency and amplitude of the frequency vibrato
wave do not stay constant. The variations in each wave are shown as the sum of three
xx


44
Analysis. Get Vibrato
This option allows the user to select one of the harmonics to be analyzed. The
algorithm calculates the frequency and amplitude vibrato waves, and displays the vibrato
wave on the spectrogram for comparison purposes.
Analysis. Get Parameters
The user can display the frequency and amplitude vibrato waves with their
respective power spectra by choosing this option. It also calculates the vibrato parameters.
The parameters are printed in the command window, the frequency and amplitude vibrato
waves are displayed in a new window as is shown in figure 4.8, and the power spectra are
displayed as shown in figure 4.9.
Figure 4.8. Example of frequency and amplitude vibrato waves.


202
% Mean, maximum, and minimum amplitude for the
% instantaneous amplitude wave
fprintf('\n');
disp('Amplitude vibrato:');
ameaninst= mean(abs(Amaxinst));
fprintf('Mean amplitude = %5.3f\n',ameaninst)
amaxinst= max(abs(Amaxinst));
amininst= min(abs(Amaxinst));
fprintf('Maximum amplitude = %5.3f\n',amaxinst)
fprintf('Minimum amplitude = %5.3f\n',amininst)
% Plots the instantaneous amplitude wave in the same window
% of the instantaneous frequency wave
handle= findobj('Tag','FiglnstFrCur');
handlel= findobj(handle,'Tag','AmplVibCurAxes');
axes(handlel);
hold on;
axis auto;
handle= plot((t(1:length(t)-14))',abs(Amaxinst),'r- ) ;
limits= axis;
axis([0 max(t(1:length(t)-14)) 0 limits(4)]);
hold off;
drawnow;
% Generates the power spectrum of the instantaneous
% amplitude wave
% The length of Amax, imax, and t are equal
Nainst= length(Amaxinst);
Yainst= fft((abs(Amaxinst)-ameaninst),Nainst);
fsy= l./tsy;
fyinst= fsy.* (0:(Nainst/2-1))/Nainst;
% Plots the power spectrum of the instantaneous amplitude
% wave in the same window of the power spectrum of the
% instantaneous frequency wave
handle= findobj('Tag','FiglnstFrSpec');
handlel= findobj(handle,'Tag','AmplVibSpecAxes');
axes(handlel);
hold on;
axis auto;
handle= plot(fyinst,abs(Yainst(1:length(fyinst))),'r-');
limits= axis;
axis([0 20 limits(3) limits(4)]);
hold off;
drawnow;
% Orders the components of the instantaneous amplitude wave
[Yaordinst Iainst]= sort(Yainst(1:length(fyinst)));
Yaordinst= fliplr(Yaordinst);
Iainst= fliplr(Iainst);
% Clears temporary variables
clear handle handlel Nainst limits;


216
% Creates figure 4 if it does not exist
handle= findobj('Tag','Fig4');
if isempty(handle)
fig4 ;
handle= findobj('Tag','FreqVibSpecAxes');
axes(handle);
axis([0 40 0 1]);
handle= findobj('Tag','AmplVibSpecAxes');
axes(handle);
axis([0 40 0 1]);
end;
% Mean, maximum, and minimum frquency for the frequency
% vibrato wave
fprintf('\n');
disp('Frequency vibrato:');
fmean= mean(Fmax);
fprintf('Mean frequency = %5.3f\n',fmean)
fmax= max(Fmax);
fmin= min(Fmax);
fprintf('Maximum frequency = %5.3f\n',fmax)
fprintf('Minimum frequency = %5.3f\n',fmin)
% Frequency variation in Hz, percentage, and semi-tones
fvarhz= mean(abs(Fmax-fmean));
fvarpc= (fvarhz / fmean) 100;
fvarst= (39.863137 loglO(fmax/16.35159) 39.863137 *
loglO(fmin/16.35159))/2;
fprintf('Mean frequency variation in Hz = %5.3f\n',fvarhz)
fprintf('Mean frequency variation in percentage = %5.3f\n',fvarpc)
fprintf('Maximum frequency variation in semi-tones = %5.3f\n',fvarst)
% Frequency variation above and below the mean in Hz and semi-tones
fmaxab= 0;
elemab= 0;
fmaxbl= 0;
elembl= 0;
for j=l:length(t)
if Fmax(j) > fmean
fmaxab= fmaxab + Fmax(j);
elemab= elemab + 1;
end
i f Fmax (j ) < fmean
fmaxbl= fmaxbl + Fmax(j);
elembl= elembl + 1;
end
end
fvarab= fmaxab/elemab fmean;
fvarbl= fmean fmaxbl/elembl;
fvarabst= 39.863137 loglO(fmax/16.35159) 39.863137 *
loglO(fmean/16.35159);
fvarblst= 39.863137 loglO(fmean/16.35159) 39.863137 *
loglO(fmin/16.35159);
fprintf('Mean frequency variation above the mean in Hz =
%5.3f\n',fvarab)
fprintf('Mean frequency variation below the mean in Hz =
%5.3f\n',fvarbl)
fprintf('Maximum frequency variation above the mean in semi-tones =
%5.3f\n',fvarabst)


91
Figure 5.15. Instantaneous waves for ec04.wav.
Figure 5.16. Power spectra of the instantaneous waves in ec04.wav.


58
5. The number of samples used to calculate the model power spectrum was
reduced. This increased the difference between the highest and lowest
amplitudes in the instantaneous amplitude curve.
6. The window length was increased from one cycle to almost the whole length.
The results approached the expected results as the window length approached
the whole signal. A window length close to whole signal does not allow the
calculation of the instantaneous amplitude and therefore cannot be used.
7. The algorithms were applied to a non filtered signal. This reduced the distance
between the highest and lowest peaks in the instantaneous amplitude curve,
especially when the number of poles was reduced to two. The number of peaks
was reduced and the mean valued tended to approach the true value, but there
were still spurious peaks.
8. The window overlap in the spectrogram used to get the frequency vibrato
curve was increased, so that instead of 14 samples per cycle, I had 250 or 1000
samples. This did not improve the results, needed an enormous amount of
RAM, and slowed down the software considerably.
9. The autocorrelation and MUSIC methods were used to calculate the
spectrogram. This showed that the pole location obtained using these methods
had significant errors.
After these experiments it was concluded that the method being used for the
calculation of the amplitude was correct but the amplitude given by the covariance method
was not precise for that particular sample length. Therefore, it was decided to calculate the
amplitude using the autocorrelation method, even though the frequency given by this
method had some error.
Figure 4.24 shows the spectrogram created using the autocorrelation method.
Using the same method described before to extract the instantaneous frequency and
amplitude from the spectrogram, the curves shown in figure 4.25 were obtained. The
instantaneous frequency is shown in blue and the instantaneous amplitude in red. There is
some ripple in both curves.


191
Figure A.22. Instantaneous frequency power spectrum
The pole location for the instantaneous power spectrum will be shown as in figure
A.23. The user should click on the OK button to close this window when he is finished
with it to avoid window proliferation. The parameters for the instantaneous frequency
wave will be shown in the Matlab command window.
Figure A.23. Pole location for the instantaneous power spectrum.


92
File ec08.wav
Figure 5.17. Frequency and amplitude vibrato waves for ec08.wav.
Freq and amp vibrato power spectra
Figure 5.18. Vibrato power spectra for ec08.wav.


5.31. Instantaneous waves for pav03a.wav 99
5.32. Power spectra of the instantaneous waves in pav03a.wav 99
5.33. Frequency and amplitude vibrato waves for pavl4.wav 100
5.34. Vibrato power spectra for pavl4.wav 100
5.35. Instantaneous waves for pavl4.wav 101
5.36. Power spectra of the instantaneous waves in pavl4.wav 101
5.37. Frequency and amplitude vibrato waves for pldom01.wav 102
5.38. Vibrato power spectra for pldom01.wav 102
5.39. Instantaneous waves for pldom01.wav 103
5.40. Power spectra of the instantaneous waves in pldom01.wav 103
5.41. Frequency and amplitude vibrato waves for pldom02.wav 104
5.42. Vibrato power spectra for pldom02.wav 104
5.43. Instantaneous waves for pldom02.wav 105
5.44. Power spectra of the instantaneous waves in pldom02.wav 105
5.45. Frequency and amplitude vibrato waves for pldom24.wav 106
5.46. Vibrato power spectra for pldom24.wav 106
5.47. Instantaneous waves for pldom24.wav 107
5 .48. Power spectra of the instantaneous waves in pldom24.wav 107
5.49. Frequency and amplitude vibrato waves for kbat01.wav 108
5.50. Vibrato power spectra for kbat01.wav 108
5.51. Instantaneous waves for kbat01.wav 109
5.52. Power spectra of the instantaneous waves in kbat01.wav 109
5.53. Frequency and amplitude vibrato waves for kbat20.wav 110
5.54. Vibrato power spectra for kbat20.wav 110
5.55. Instantaneous waves for kbat20.wav Ill
5.56. Power spectra of the instantaneous waves in kbat20.wav Ill
xv


193
Figure A.25. Instantaneous frequency wave
Also, the instantaneous amplitude wave and its corresponding power spectrum will
be displayed in windows on top of the amplitude vibrato wave and the amplitude vibrato
power spectrum (see figures A.26 and A.27).
Figure A.26. Instantaneous amplitude wave.


16
3. Interartist variability, to compare the average vibrato rate among different
artists.
The material used in this study came from recordings made in a real musical
context. Ten different singers were used and 25 tones per singer were chosen for analysis.
The frequency measurements were obtained from spectrograms by measuring the
time from one wave trough to the next.
There is a tendency in most of the tones analyzed to have a higher vibrato rate at
the end, this is, during the last 1 to 5 periods. The rate increase averages about 15%. The
intertone variation between the maximum and minimum for an artist was 8% of the
artists average. Each tone was calculated as a 3 cycle frequency average. The average
vibrato frequency for all artists was 6.0 Hz, with a maximum of 6.4 Hz and a minimum of
5.5 Hz.
Frequency Modulation Characteristics of Sustained /a/ Sung in Vocal Vibrato
This is a paper by Yoshiyuki Horii (Horii, 1989, p. 1). Horii noticed that there are
differences in the values reported in the literature for the amplitude modulations in singing
vibrato. The values found vary from nonmeasurable to several decibels.
It is understood by most of the researchers that the curve shape for frequency
vibrato is almost sinusoidal (Horii, 1989, p.l).
Winckel (Winckel, 1953, p.252) considered nonsinusoidal vibrato curves as
characterizing poor vibrato. Programs used to synthesize the singing voice use sinusoidal
patterns for singing vibrato because there are no good mathematical models for vibrato.
The main objective of this study, therefore, was to investigate modulation rates,
extents, and rates of the fundamental frequency (Fo) increase and decrease for each


140
Parameter Statistics
Table 7.2 shows the value of the parameters for each of the analyzed samples. The
proposed mathematical model will emulate a particular vibrato sample if these values are
applied to it.
In addition, table 7.2 shows the statistics for the parameters. The most common
measures of central tendency, measures of dispersion, and statistical tests were looked at
and it was concluded that the following measures would give plenty of information about
the statistical properties of the parameters:
1. Mean value.
2. Maximum.
3. Minimum.
4. Standard deviation.
5. Confidence interval.
The confidence interval is calculated based on the central limit theorem. The
confidence interval in table 7.2 shows where 95 percent of the values for a specific
parameter fall.
The statistics were not calculated for all the parameters, only for those in which the
statistics would provide useful information. For example, it does not make sense to
calculate the statistics for a phase angle in the power spectrum since it can take many
different values. We can see from table 7.2 that all the parameters posses a relatively high
standard deviation, except for fmeaninst.


55
Figure 4.20. Instantaneous frequency vector.
Another module displays the frequency vector in a new window and calculates its
power spectrum. Figures 4.21 and 4.22 display the frequency vector and power spectrum
corresponding to figure 4.20.
Figure 4.21. Instantaneous frequency vector.