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1 ARCHITECTURE IN MOTION: A MODEL FOR MUSIC COMPOSITION By JORGE ELIAS VARIEGO 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 2011
2 2011 Jorge Elias Variego
3 To my family
4 ACKNOWLEDGMENTS I thank the University of Florida School of Music and the College of Fine Arts for their continuous support and to my com mittee members for their ideas and contributions to this study.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABL ES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 DELIMITATION OF THE OBJECT OF STUDY ................................ ...................... 13 Overview ................................ ................................ ................................ ................. 13 Adagio for Strings Op. 11 and String Quartet No. 4 ................................ ................................ 19 Volume of Orchestration and Chorus Effect ................................ ..................... 20 Addition of the Double Bass and Expansion of the Frequency Range ............. 26 Volume of Orchestration and Tempo ................................ ................................ 27 The Proposed Model and Spectral Composition ................................ ..................... 28 Equivalences and Terminology ................................ ................................ ............... 29 2 INTERRRELATIONS BETWEEN MUSIC AND AURAL ARCHITECTURE THROUGH HISTORY ................................ ................................ ............................. 33 Overview ................................ ................................ ................................ ................. 33 Polio Vitruvius, the Ten Books of Architecture (1 st c. BCE) ................................ ..... 33 Coro Spezzato, Polychoral Style al (16 th c. ACE) .................. 38 Claude Debussy Color, Shapes and Proportions in La Mer (1909) and Other Works ................................ ................................ ................................ .................. 48 Le Corbusier Le Modulor (1948) ................................ ................................ ........... 56 Alvin Lucier Chambers (1968), I am sitting in a room (1969) ................................ 59 Iannis Xenakis Metastasis (1954), the Philips Pavi lion (1958) and Other Works .. 65 Daniel Libeskind The Jewish Museum of Berlin (1999) and Aesthetic Considerations on Other Works ................................ ................................ ........... 85 3 THE MODEL ................................ ................................ ................................ ........... 93 Overview ................................ ................................ ................................ ................. 93 Data Collection ................................ ................................ ................................ ....... 93 Structural and Acoustical Considerations ................................ ......................... 94 Parameters from Room Acoustics and their Application ................................ ....... 101 Pitch Frequency Response and Centroid ................................ .................... 101 Room Formants Specific Resonances: ................................ .................. 102 Orchestrating the Centroid Values ................................ ........................... 102
6 ................................ ...................... 107 Time I Reverberation RT ................................ ................................ ............. 110 Orchestrating the Reverberation ................................ .............................. 113 Time II Rhythm of the Early Reflections ................................ ...................... 116 Orchestrating the Rhythm of the Early Reflections ................................ .. 118 Amplitude (Dynamics) the G factor ................................ .............................. 120 Orchestrating the G Factor ................................ ................................ ....... 121 4 APPLICATION OF THE MODEL ................................ ................................ .......... 127 Personal Motivations and Application of the Proposed Model in a Musical Composition ................................ ................................ ................................ ....... 127 Data Collection Details ................................ ................................ .......................... 134 ................. 135 APPENDIX A APPLICATION OF THE MODEL IN A MUSIC COMPOSITION COL ORS FOR ORCHESTRA ................................ ................................ ................................ ....... 136 B APPLICATION OF THE MODEL IN A MUSIC COMPOSITION ETUDE FOR STRING QUARTET ................................ ................................ .............................. 178 LIST OF REFERENCES ................................ ................................ ............................. 180 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 183
7 LIST OF TABLES Table page 1 1 Adding sound sources of equal l evel. ................................ ................................ 22 1 2 Dynamics equivalences in dB. ................................ ................................ ............ 30 1 3 Instruments dynamic range in dB. ................................ ................................ ...... 30 2 1 Rhythmic density ratios as proposed by Xenakis in Formalized Music .............. 79 2 2 Rhythmic density ratios in m. 96 116 from Metastasis. ................................ .... 81 3 1 Data collected from the Baughman Center ................................ ........................ 94 3 2 Room formants and centroid values for the different S and R locations. .......... 102 3 3 Note rhythmic values and their respective weight. ................................ ............ 110 3 4 Delay times of the first 10 early reflections. ................................ ...................... 117 3 5 Values of the first 10 early reflections. ................................ .............................. 117
8 LIST OF FIGURES Figure page 1 1 The human proportions according to Le Co Le Modulor. ...................... 14 1 2 Excerpt from measures 1 Adagio for strings Op. 11. ................................ ................................ ................................ ................ 21 1 3 Exc erpt from measures 1 Adagio for strings Op. 11, reworked to show a gradual increase in the orchestrational volume. ..... 22 1 4 Excerpt from measures 1 2 of th Adagio for strings Op. 11, reworked to show a non gradual increase in the orchestrational volume. ................................ ................................ ................................ ............... 23 1 5 Volume of orchestration: Giacinto Scelsi String quartet No 4 m. 5 9. ............. 23 1 6 Sonogram of an F2 on a cello without chorus (single instrument). ..................... 25 1 7 Sonogram of an F2 on a cello with chorus (cello section). ................................ 26 1 8 Larger instrumentation influence in tempo and duration. ................................ .... 27 2 1 Tuning of the catapult, from th e Ten Books of Architecture. ............................... 34 2 2 Tuning system for the vessels. ................................ ................................ ........... 36 2 3 Theater ceiling and audience area suggested by Vitru vius. ............................... 37 2 4 Example of basso sequente according to Zarlino. ................................ ............. 42 2 5 Excerpt from measures 5 Domine Probas ti. ........................ 43 2 6 Excerpt from the final Amen, measures 222 Domine Probasti. ................................ ................................ ................................ ............. 44 2 7 Verb um bonum. ................................ .............................. 46 2 8 Cum invocarem. ................................ ............................. 47 2 9 Spiraled form of La Mer according to Howat ................................ ...................... 50 2 10 Katsushika Hokusaki's The hollow of the wave off Kanagawa .......................... 51 2 11 Measures 1 3 from Prlude l'aprs midi d'un faune. ................................ ....... 53 2 12 Measures 1 2 from ................................ ................................ ..... 53
9 2 13 Measures 1 2 from Syrinx ................................ ................................ ................ 53 2 14 Iannis Xenakis, Le Sacrifice (1953). Source: Andre Baltensperger, Iannis Xenak is und die Stochastische Musik. ................................ ............................... 68 2 15 Iannis Xenakis, Metastasis (1954) measures 309 314. ................................ ... 70 2 16 Excerpt of the graph paper design of the string glissandi in Metastasis measures 309 314. ................................ ................................ ............................. 71 2 17 Images of the first model (left ) of the Philips Pavilion and its finalized version (right) at the 1958. ................................ ............................ 74 2 18 Measures 94 116 from Metastasis. ................................ ................................ .. 81 2 19 Table with progressions of rectangles with increasing widths drawn from Le Modulor ................................ ................................ ................................ ............. 82 2 20 The poly metrical section from m.104 from Metastasis and a graphical represent ation of its rhythmical density. ................................ ............................. 83 2 21 Musical note as a vectorial multiplicity. ................................ ............................... 84 2 22 Daniel Libeskind, floor plan of the Jewish Musem of Berlin (1999). .................... 92 3 1 Diagram of acoustic consequences of parallel walls and vaulted ceilings. ......... 99 3 2 Baughma n Center Meditation Pavilion floor plan with exact location of the Sources (S) and Receivers (R) utilized ................................ ............................ 100 3 3 Orchestrated centroid (I). ................................ ................................ .................. 103 3 4 Orchestrated centroid (II). ................................ ................................ ................. 103 3 5 Weighted mean formula. ................................ ................................ .................. 103 3 6 Orchestrated centroid (III), with w eighted dynamics. ................................ ........ 104 3 7 Weighting dynamics and varying centroid value (I). ................................ ......... 104 3 8 Weighting dynamics and varying centroid v alue (II). ................................ ........ 105 3 9 Example of a single centroid value (E5 or 659 Hz) through a series of complexes at the same dynamic level. ................................ ............................. 105 3 10 Weighting dynamics and varying centroid value (III). ................................ ....... 106 3 11 Duration as a weighting factor. ................................ ................................ ......... 109
10 3 12 Graphic repres entation of a 60 dB decay envelope. ................................ ......... 111 3 13 Graphic of two successive tones performed in rooms with different RT envelopes. ................................ ................................ ................................ ........ 112 3 14 Orchestrated RT (I). ................................ ................................ .......................... 113 3 15 Orchestrated RT (II). ................................ ................................ ......................... 113 3 16 Orchestrated RT (III). ................................ ................................ ........................ 114 3 17 Orchestrated RT per frequency band (I). ................................ .......................... 115 3 18 Orchestrated RT per frequency band (II). ................................ ......................... 116 3 19 Rhythmic structure extracted from the early reflections. ................................ ... 118 3 20 Orchestration of the rhythmic structure extracted from the early reflections (I). 119 3 21 Orchestration of the rhythmic structure extracted from the early reflections (II). ................................ ................................ ................................ .................... 119 3 22 Orchestration of the rhythmic structure extracted from the early reflections ............ 120 3 23 G factor calculation. ................................ ................................ .......................... 121 3 24 Impact of the G factor in the listener. ................................ ................................ 121 3 25 Combinatorial example (I). ................................ ................................ ............... 122 3 26 Combinatorial example (II). ................................ ................................ .............. 123 3 27 Combinatorial orchestration example of the G factor for string ensemble. ....... 125 4 1 Example of orchestrated resonances in C olors measures 10 12. .................... 130 4 2 Example of orchestrated resonances from measures 97 101. ......................... 131 4 3 Polyrhythmic monophonic texture in Colors as an example of an illusory increment of the G value. ................................ ................................ ................. 133
11 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 ARCHITECTURE IN MOTION: A MODEL FOR MUSIC COMPOSITION By Jorge Elias Variego August 2011 Chair : James Paul Sain Major : Music S peculations regarding the relationship between music and architecture go back to the ve ry origins of these disciplines. Through out history, these lin ks have always reaffirmed that music and a rchitecture are analogous art forms that only diverge in their object of study. In the 1 st c. BC E i the arts, having a vast knowledge in history, music and philosophy. In the 18th c., the German thinker Johann Wolfgang von Goethe musi More recently, in the 20 th c., Iannis Xenakis studied the similar structuring composition based on mathematical principles and geometric constructions. The goal of this do cument is to propose a compositional method that will function as a translator between the acoustic al properties of a room and music, to facilitate the creation of musical works that will not only happen within an enclosed space but will also intentionally interact wit h the space Acoustical measurements of rooms such as reverberation time, frequency response and volume will be measured and
12 systematically organized in correspondence with orchestration al parameters. The musical compositions created after the proposed model are evocative of the spaces on which they are based. T hey are mean t to be performed in any space, not exclusively in the one where the acoustical measurements were obtained. The visual component of architectural design is disregarded; the room is considered a musical instrument, with its particular sound qualities and resonances. Compositions using the proposed model will not result as sonified shapes, they will be Architecture in m otion is an attempt to adopt scientific research to the service of a creative activity and to let the aural properties of enclosed spaces travel through music. Numbers. Through their effect there is a plu rality of individuals: sympathy, order harmony, beauty, etc. [...] in short, everything that is of mind. In the spatial world the images of the numerical world are projected, first by nature itself, then by m en and above all by artists. It can be said that our duty on earth and during the whole of our life consists precisely in this projection of forms issued forth from numbers, and that you, the artists, fulfill that moral law to the highest degree. Not only is it possible to appeal simultaneously to geometry and to numbers, but to do so is the true Andreas Speiser
13 CHAPTER 1 DELIMITATION OF THE OBJECT OF STUDY Overview This document proposes a model for musical composition a set of ru les that govern the creative process. Those rules are established prior to the music composition and are the framework for the imagination and creativity of the music composer. This systematic organization of rules by no means intends to limit the creativ ity of the artist, instead, it is pr oposed as an the elaboration of a piece of music. Those guidelines are particularly elaborated and based upon data collected from the acoustical properties of enclosed spaces The architectural acoustic properties of rooms provide constant values that afterwards become dynamic through music. In Iannis Xenakis words in 1 a view the moving architecture In the same line of thought but in the realm of architecture Le Corbusier 2 proposed in Le Modulor section contained in human proportions. The pillar values of his system are (in man; also, according to the model, 113 183 226 define the occupation of space by a man six feet high. 3 Those values have a golden mean relationship and follow the Fibonacci series. 1 Le Corbusier (1955), 326 327 2 Le Corbusier (1948), 3 Ibid. 65 66.
14 Figure 1 Le Modulor. This model was a new measurement system that was not based on the meter nor the foot and their su bdivisions. Le Corbusier proposed a model to standardize the design of buildings for humans based on human proportions H e believed that the traditional metr meter, indifferent to the st ature of man, divisible into half meters and quarter meters, decimeters, centimeters and millimeters, any number of measures, but all indifferent to the stature of man, for there is no such thing as a one meter or a two 4 Le tion is not a matrix but a set of guidelines of scale and proportion for the standardization of building construction. The proposed model in this study is not a indiffer ent to architectural acoustics. The cre ative process begins with data collection that is then translated through the model into musical terms, which are organized in time as a musical comp osition meant to be performed. The uniqueness of this sequence resid es in the fact that its first and last steps are essentially aural; it begins by 4 Ibid. 21 22.
15 quantizing a sonic experience that at the end of the process becomes sound again, the model goes from sound (measure) to sound (performance) The acoustical measurements are the starting point of the model and need only to be taken once; the application of those measurements could vary. For example, a number of different chords result from the same frequency centroid value obtained from a room As in any model, the artistry is revealed by its application, not by the model itself. This initial set of numbers is musically meaningless before it is projected through the model. Likewise, Le Corbusier initiates his reasoning by accepting the concept of an desired measurements His model and the buildings constructed after it would be completely different if he had chos being with a dissimilar body size and proportions from the one he ultimately chose for his model In his anthropomorphic approach to architectural design, Le Corbusier first had to decide on the most suitable representation of a human being, measure it, and construct a model after it. It is here important to state that Vitruvius, in the 1 st c. BC E, had already thought the or der of classical architecture connected with man its relations with buildings and with the cosmos 5 He also based his theory on the human figure, creating the well known which would later become popular in the Renaissance through a drawing by Leonardo da Vinci. application and goal to standard ize the constructions of houses. Vitruvius instead had a purely ideal objective, which was to perpetuate through architecture the perfectly 5 Pollio Vitruvius, translation by Ingrid Rowland (1999).
16 harmonic proportions of man as a scaled representation of the harmony between the planets. The aim of the study I propose is not to find the room with the finest acoustical properties for music compositi on and to create works af ter it. On the contrary, the purpose is to provide the composer with a set of versatile tools to translate acoustical Space and sound are inseparably bound, the music finds its meaning in the space which, at the s ame time, beco mes a unique musical source. The not only conceiv ing a piece of music as a mere succession or progression of sonic events through time but also as a spatial journey. The structu re of the composition becomes three dimensional letting the artist apply data from one or more spaces, suggest ing a transit between them by means of their acoustical properties or even utilizing computer software to create their own ad s. Far from being a literal translation of an architectural plan into music, the proposed compositional mod el has specific characteristics. I n architecture, the model is an actual scaled representation of the finished construction, an id entical copy of th e final piece. T hat scaled representation of the original can only generate more versions that merely replicate the original. On the other hand, the composition al model I prop ose operates in a different way; identical acoustical data can be implemented by the same composer to generate an infinite number of works that do not resem ble each other. It is an open algorithm that can produce diverse results depending on the decisions made by the composer Similarly, a single twelve tone row could be the model for many compositions.
17 Like a piece of music, a prospective building in an architectural plan could change its character depending up on the context where it is constructed (performed). Imagine the same studio room built identically in two different environmen ts, one in the epicenter of an over crowded city area, and the other in a rural area. Despite the correspondence between the two constructions, their T he former is an efficient administration of space where the architectural design successfully balances the specific issues of housing, location and size of the room, the latter being a wasteful one in which the housing needs and room size do not address its location. The ideas of context and re contextualizatio n take part in the proposed model; compositions created using this model re contextualize the acoustical space by means of the elements of a foreign space incorporated in the music. A composition based on acoustical measurements from a room x is performed in room x, in that case being this same composition is performed in room y then the acoustical properties of that room get re interpreted and re contextualized through the music that has embedded prope rties of a different space in a x w h ere the music is performed, filtered through the attributes of room y tha t are incorporated in the music. In 6 terms, the liste That illumination takes place when the physical processes of a space get excited. When processes such as interference, reflections, shadowing, dispersion, absorption, diffrac tion and reverberation are activated by means of a sonic impulse (music, noise, 6 Barry Blesser and Linda Ruth Salter (2007), 17.
18 people talking, etc.) the listener receives multiple acoustic cues to aurally visualize objects and the spatial geometry. 7 A simple wall, for example is perceived as a visual boundary of the enclosed space and also as a sound reflective surface that provides the listener with aural cues for location (within the space) volume, and materials of the room. The dichotomy for the architect is that he needs to handle both architectures simultaneously as they are interdependent. A window could add luminosity and enhance the sensation of volume of a space but at the same time could have undesired aural consequences by letting outside sounds filter through it. The architect decides whether to add that window or not in his plan a visual tool that merely projects the geometric properties of a space though it is unable to reproduce the R oom response can be calculated precisely but its aural representation is elusive for the architect who mostly relies on visual cues. communicate the artistic, social, emotional and historical context of a space, architects almost exclusively consider the visual aspects of a structure. Only rarely do they consider the acoustic aspects. The native ability of human beings to sense space is 8 Consequently, a natural question arises: is an architectural plan the most efficient representation of the aural architecture? The architectural pla n is in the realm of visual architecture; proportions, distribution of the elements within the construction, design (shapes) and measurements are perceived and represented visually. The role of the 7 Ibid. 17. 8 Ibid. 1.
19 architectural designer here is in general terms to cre ate and administer a physical space. On the other hand, the aural architecture refers exclusively to the huma n experience of a sonic process. As to the properties of a space that can be experienced by 9 The architect is in charge of that listening experience and is also solely responsible for it. Therefore, the plan becomes a deficient tool to project the acoustic properties of a space. In this proposed model the aural element is the key f actor; architecture is deprived of its tangible and visual components becoming a discipline of pure sonic perception, blurring even more its differences with music. That listening experience within a room is quantized and re sonified by the composer who b composer employs only the visual queues and values that are relevant to measure, explain and reproduce the aural experience. Adagio for Strings Op. 11 and Giacinto String Quartet No. 4 Can the music composer effectively represent space in a musical score? A musical score has the same limit ation as the architectural plan: it provides a limited visual representation of the succession of the sonic events in time. If we consid er Adagio for strings in its original setting for string quartet (1936) 10 and its later version for string orchestra (1938) 11 we can clearly appreciate that the differences between the two scores are min imal, which by no means reflect the s ingularity of each version. In other words, it is impossible to visualize in the musical 9 Ibid. 5. 10 Samuel Barber (1936), G. Schirmer. 11 Samuel Barber (1938), G. Schirmer.
20 orchestration. The larger orchestration al volume of the version for string orchestra elongates the resonances a nd multiplies the dynamic range of the ensemble suggesting a l music. The apparent simplicity of the re orchestration of the Adagio has two relevant consequence s that deserve further analysis. Volume of Orchestration and Chorus E ff ect V olume of orchestration 12 could be defined as the amount of input sources in a chorus effect that modu late reciprocally. In Barber's Adagio the duplication of the individual voices (in a string orchestra) has that specific result, which occurs when indi vidual sounds with roughly the same timbre and nearly the same pitch (due to performance imperfections) conv erge and are perceived as one. No matter how well trained a group of string players may be, they do not tune nor vibrate identically and they never perform the same pitch in impeccable unison. S o the random, unpredictable phase cancellations that occur as a result of these slight pitch differences are the source of the chorus effect. The volume of orchestration is, then, a way to measure the amount of sound sources used to generate a chorus effect. The version of the Adagio f or string orchestra produces this effect acoustically as instruments of the orchestra). Re garding the generation of the chorus it is important to mention that the effect can also be simulated using electronic effects or a signal processing device. It is achieved by combining multiple copies of a sound each one delayed and pitch shifted slight ly differently and mixed with the original undelayed 12 Dante G. Grela H., (to be publ ished by the editorial of the Universidad Nacional del Litoral, Argentina).
21 sound. This can be done by continual slight random modulation of the delay time of two or more different delay taps. It is clear that the electronic simulation of the effect approximately reenacts the physicality of the original acoustic result. Regardless of the technology or form factor, the processor achieves the effect by taking an audio signal and mixing it with one or more delayed, pitch modulated copies of itself. The pitch of the added voi ces is typically modulated by a L ow F requency O scillator (LFO) Similarly to the concept of physical volume, the volume of an orchestration reflects which is considered as a three dimensional object and the amount of space it occupi Adagio for Strings would have an orchestration al volume of one if its was performed with string quartet, or of eight if that same line was played by a string orchestra with eight first violins. F igure 1 2. Excerpt from measures 1 Adagio for strings Op. 11. Th is concept becomes relevant when considering the consequences in dB of the addit ion of sources of equal level. The level of the previous excerpt performed with a solo violin ( volume 1) increases in 9 dB when it is performed by a string section of eight pla yers (volume 8) with the same dynamic. The orchestration al volume can be a dynamic value if it changes through time or non dynamic if it changes abruptly. That change can be achieved gradually (by means
22 of a crescendo or a decrescendo) or abruptly by simply adding an extra instrument with the same dynamic indication. Table 1 1 Adding sound sources of equal level. 13 Number of sources of equal level Level inc rease in dB Orchestrational volume 1 0 1 2 3 2 3 4.8 3 4 6 4 5 7 5 6 7.8 6 7 8.5 7 8 9 8 9 9.5 9 10 10 10 The following examples show a practical application of those possibilities. The first excerpt shows a dynamic orchestration al volume value that increases from 1 to 2 by means of a crescendo. Figure 1 3. Excerpt from measures 1 Adagio for strings Op. 11, reworked to show a gradual increase in the orchestrational volume. 13 23.
23 The second example shows a non dynamic change of orchestration al volume (from 1 to 2) by the simple addition of one instrument to the line. Figure 1 4. Excerpt from measures 1 Adagio for strings Op. 11, reworked to show a non gradual increase in th e orchestrational volume. Chorus is the resulting effect of the doublings and the levels of volume of orchestration are a numerical way to quantize them. Regarding this topic, it is inevitable to consider the work of Giacinto Scelsi. Figure 1 5. Volume of orchestration: Giacinto Scelsi String quartet No. 4 m. 5 9.
24 The excerpt on Figure 1 5, from measure 5 9 of his String Quartet 4, exemplifies his extremely limited pitch vocabulary and his deep commitment to spectral variations through the orchestra tional volume. The whole passage is built throughout upon two groups of instruments (violins I and II, and violin II and viola) orchestrated with volume 2. The two groups are an octave apart on a Cb4 (in the violins) and a Cb3 (in the violin II and the vio la), both with a very active polyrhythmic texture. scordatura traditional tuning for the strings in order to achieve a unique color due to the unusual tensio n given to the strings. Each string is conceived as a different musical instrument with its own tone. None of the open strings of the two violins match in pitch; furthermore, the composer specifically assigns an individual stave per string to perform speci fic passages. The use of orchestrational volume is the core of this work, where the listener is required to be extremely alert in order to appreciate sounds that evolve within other sounds of near identical pitch. Those pitch centers together with the spec trum changes controlled by the use of micro tonality and volume of orchestration, define the overall formal arc of the composition Xenakis also considered the subject in his Formalized Music where he dissects all p roperties of sound in order to, po ssibly, reconstruct it using electronic means. Those small variations of spectral lines in frequen cy as well as in amplitude are, i n his point of view, of great importance as they make the difference between a lifeless sound made up of a sum of harmonics p roduced by a frequency generator and a sound of the same sum of harmonics played on an orchestral instrument. 14 14 Iannis Xenakis, (1971), 244.
25 The essential components of the electronically generated chorus tend to capture and replicate the inconsistencies of the human performers throug h the consistency of electronic media. Those elements are: a) random delay time, between 20 ms and 50 ms so the original and delayed sources are aurally perceived as a single source; b) frequency shift (LFO); c) amplitude modulation. Those microscopic imp erfections appear ubiquitously in the string version of the Adagio, creating a new sonic dimension unique in every performance that can be hardly quantized but can be analyzed and visually appreciated through a sonogram. The following examples are sono grams of the spectral changes of a single p itch with and without a chorus. The two examples could easily correspond to the two Adagio. The partials of the fundamental tone are well defined in both examples, but the amount of inha rmonicy (spectral components that do not correspond to a harmonic partial) is clearly dissimilar. Figure 1 6. Sonogram of an F2 on a cello without chorus (single instrument).
26 Figure 1 7. Sono gram of a n F2 on a cello with chorus (cello section). The increased richness in the spectrum and the irregularity of the wave (as a result of the modulation) are the most evident consequences that can be visual ly appreciated in the sonograms. Delay time is also one of the main components of this effect. In a large ensemble, those concatenated delay lines inevitably blur the attacks and note changes compromising the overall intelligibilit y of the musical material. As will be discussed later, a slower tempo is the nat ural solution for conductors to regain clarity. Add ition of the Double Bass and Expansion of the Frequency R ange Doubling the bass line with double basses and cellos is a common resource that Barber utilized in his orchestration. For the mo s t part, the do uble bass reads the same part as the cello sounding an octave below. That basic re orchestration has relevant spectral consequences. First, the frequ ency range is considerably expanded : Barber asks for a double bass with an extended range to the low C1 w hich opens the spectrum
27 a whole octave below the cello range roughly covering the whole range of human hearing (from 20 Hz to 20 MHz). Despite that fact, Barber avoids the open C1 in the basses (he never uses it throughout the work ) in order to stay away from the roughness of the open string and to keep the drama alive through his orchestration. Secondly, the bass does n ot double the entire cello part. I n m. 13 14 the two cello lines in divisi and the double bass play in unison reinforcing the stepwise descending line, momentarily "thickening" the cello section. In addition to that, the section without basses has a thinner, more "airy" texture full of drama (m. 28) that somehow prepares the climactic moment of m. 54 in the extreme high register. Paradoxi cally, the basses rest during that section and the string orchestra becomes a string quartet with an increased orchestral volume, achieving the utmost tension of the work not with a tutti but with an extreme rise of the spectral content of the ensemble. Volume of Orchestration and Tempo It is also important to state that all of the recorded versions of the Adagio for string orchestra are at least 50% longer than the versions recorded with string quartet, reaffirming the idea that the orchestration al volum e added by the large ensemble embeds an impression of a larger physical space into the music. Figure 1 8. Larger instrumentation influence in tempo and duration. qualities i mplied by the larger volume of orchestration: a larger space suggests a bigger
28 physical volume that generates a longer reverberation time 15 which consequentially lowers the intelligibility index. In order to recove r that intelligibility, the conductor decides on a slower performance tempo This is the ultimate reason for the longer duration of the version for string orchestra. Th is link between orchestrational volume and musical tempo is an example of the implied re lations between orchestration and time, but is not utilized as a parameter in the proposed model. However, the proposed model is not indifferent to the RT values, which are consistently trans lated into duration and enveloping in the music domain. The Propo sed Model and Spectral Composition The paradigm of spectral music composition is similar to the proposed model. In both cases the compositional decisions are preceded by the analysis of a sound source. Spectral composers focus on the qualities of timbre an d utilize mathematical analysis techniques, like the Fast Fourier T ransform among others, in order to obtain a clear description of the components of the sound source. Despite its many aesthetic variants, the core of spectral composition (for traditional i nstruments and/or electronic media) is the focus on timbre and its spectral components. The similarities with the proposed model are clear; both systems begin with a type of sound analysis, which provides the platform for music al artistry. In spite of tha t analogy, the systems are dissimilar regarding the ir final objective On the one hand, spectral composers search for the combination of qualities of a complex sound that define its color and make it unique; on the other hand composers utilizing the propos ed system can seek the color of a room manifest ed through sound, and for all other 15 establishes a direct relation between room volume and RT. T = 0.05 V/a
29 properties of an enclosed space that somehow influence the sonic event. Those properties come from various sources: impulse response studies, architectural plan, form and st ructure of the space, materials utilized in the construction, and many others. Equivalences and Terminology One of the first issues of this model is the tr anslation between the perceived c hange of the dynamic of a sound (in the music domain ) to its parall el in the realm of room acoustics, a variation in sound pressure level measured in decibels. For practical reasons each dynamic indication is assumed to be perceived by the listener as tw ice as loud as its previous one: mf is perceived as twice as loud as mp which is perceived as twice as loud as p That relation is accurate with in performance 16 practice and facilitates the translation of musical dynamics into decibels This is supported by the widely spread t heory of psycho acoustic pioneer Stanley Smith St evens 17 It indicates that the doubling or halving the sensation of loudness corresponds to a level difference of 10 dB or one dynamic level. The sof able 1 2 correspond to the lowest and highest levels of sound pressure that those instruments can produce. The samples are from instruments with an extended dynamic range that also represent e ach of the instrument families. Those equivalencies represent the frame for the translation of sound pressure values from room acoustics into musical terms (that are non numerical). As an example, a hypothetical room with a G factor (the ratio in dB between the energy of the direct 16 (2007). 71. 17 Stanley Smith Stev Psychological Review 64(3): 153 181
30 sound and the energy of the reverberated one ) of 60 dB would result in a passage or chestrated in p Table 1 2. Dynamics equivalences in dB. Dynamic indication (musical notation) Meaning Decibels ppp Pianissi ssi mo extremely soft 40 pp Pianissimo very soft 50 p Piano soft 60 mp Mezzo piano medium soft 70 mf Mezzo forte medi um loud 80 f Forte loud 90 ff Fortissimo very loud 100 fff Fortissi ssi mo extremely loud 11 0 The following table shows the dynamic range (expressed as sound level pressure in dB) of a representative group of musical instruments. The measurements w ere taken from a distance of 10 ft. 18 Table 1 3. Instruments dynamic range in dB. Instrument Softest (in dB) Loudest (in dB) Cymbal 40 110 Organ 35 110 Piano 60 100 Trumpet 55 108 Violin 42 90 Overall range 35 40 108 110 18 Online resources: Marshall Chasin, http://www.generalhearing.com/explore.cfm/chasin/ and Geoff Husband http://www.tnt audio.com/topics/frequency_e. html
31 It is important to menti on that the sound pressure level range of musical instruments (in the domain of architectural acoustics) is translated to its most analogous parallel in the realm of music composition (as a dynamic range). The application of the proposed composition al mode l is discussed in depth in Chapter III The following classification and definitions are based on the ones proposed by Leo Beranek 19 1 Dimensional: V = volume of the hall in cubic feet or meters. H = average room height, this value is needed to calculat e the time of the first ceiling reflection. W = average width, important to determine the intimacy of the room. L = average room length, useful to determine the magnitude of the decrease of sound with distance. 2 Acoustical: RT = reverberation time in s econds. EDT = early decay time in seconds. Bass ratio: ratio between the RT at octave center frequencies of 125 Hz and 250 Hz, and octave center frequencies 500 Hz and 1000 Hz (between low and mid high frequency bands). IACC = interaura l cross correlation coefficient Measures the difference in the sounds arriving at the two ears of the listener. That coefficient can be calculated with or without frequency weighting. 19 Leo Beranek (2004), 580.
32 ITDG = initial time delay gap, is the amount of time, measured in milliseconds, between t he direct sound and the arrival of the first reflections. C80 = clarity factor. Ratio between the energy perceived by the listener within the first 80 msec. of an impulse and the remaining energy of the sound after the 80 msec. It is expressed in dB. LG = lateral energy. Measures the ratio between the direct sound and the one reflected from the sides and is also expressed in dB. G = the ratio in dB between the energy of the direct sound and the energy of the
33 CHAPTER 2 INTERRRELATIONS BETW EEN MUSIC AND AURAL ARCHITECTURE THROUGH HISTORY Overview This section is a chronological study of the interrelations between music and architecture through the examination of paradig matic works. Concrete applications implemented by artists both architects and music composers are studied as well as their influence on other creative disciplines. The chosen works respond to the main object of this study as they are somehow related to properties of aural architecture that influenced music composition as well as mus ical parameters that influenced architectural design. Polio Vitruvius, the Ten Books of Architecture (1 st c. BC E ) 20 In Book I of his Ten Book of Architecture Vitruvius discus ses the importance of the integral education of the architect who not only should be a capable draftsman but also well versed in many other disciplines like medicine, music and philosophy. This e Greek arkhitekton arkhi "chief" + tekton "builder, carpenter" ; this R broader than the one we conceive today. A modern architect is exclusively dedicated to the design, plan and construction of buildings; in the 1 st c. BC E the architect was also a broad spectrum technician who was well versed in fields that spanned from urban planning to military engineering. the ma s ter craft sman. To be educated, he must be an experienced draftsman, well 20 Pollio Vitruvius, translation by Ingrid Rowland (1999).
34 versed in geometry, familiar with history, a diligent student of philosophy, know music, have some acquaintance with medicine, understand the rulings of legal experts, and have a clear grasp o 21 That general instruction had very specific applications. Vitruvius served as a b allista (artilleryman) where musical instruction became very practical when aiming catapults. The tension of the cords in the catapults w mathematical relations, and besides that, to calibrate ballis tae, catapult s and scorpions. In the headpieces of war machines there are hemitone spring holes, right and left, through which the twisted sinew cords are pulled tight by windlass and handspikes; these cords should not be wedged in place or fastened down unless they give off a particular and identical sound to the ears of the catapult maker. For when the arms of the catapult have been cocked to these tensions, upon release they should deliver an ide ntical and equivalent thrust; if they are not tuned ident ically, they will 22 Figure 2 1. Tuning of the catapult, from the Ten Books of Architecture. 21 Ibid. 22. 22 Ibid. 23.
35 capitals, and carried a cross to the other side, and then they are fastened around the windlasses and wound around them, so that when the ropes are stretched over them by the levers, when struck with the hand, each of them d with handspikes on windlasses until they make an identical sound, and in this way catapults are adjusted to tone by propping with wedges according to the musical 23 In addition to that, in Book V Vitruvius applied Pythagorean principle s of harmonic bronze or earthen vessels called echea placed under the seats of theatres, to assist, by their resonance, the voices of the performers. This is one of the first examples of an enclosed space conceived as a musical instrument w h ere its design features are intended to enhance specific sounds that take place within it. The V itruvian theaters interact with the performers, amplifying and ounds of the ir on stage performances. In figure 2 2 we can clearly appreciate that the tuning of the room is not indifferent to its design; small and large theaters were tuned differently. sympho niae in Greek, and number six: diastesseron (fourth), diapente (fifth), diapason (octave), and disdiatesseron (octave + fourth), disdiapente (octave + fifth) and disdiapason (double 24 the ones the Gre eks call echea which are enclosed underneath the seats, are placed according to mathematical principle based on their pitch. The vessels are grouped in sections around the circle of the theater to create intervals of a fourth a fifth and so on up to a d ouble octave. As a 23 Ibid. 23. 24 Ibid. 67.
36 result, the speaker design that when it strikes the echea it will be amplified on impact, reaching the ears of 25 Figure 2 2. Tu ning system for the vessels. The vessel tuning suggested by Vitruvius follows the progression of the first four partials after a fundamental tone, which are also separated by a fifth, a fourth and the resulting double octave. B y reinforcing the first partials of an implied fundamental he provided the space with a characteristic resonance. L ike in brass instruments that fundamental is the lowest possible resonance that can be obtained from an instrument in this c ase an enclosed space related to its siz e and length of the sound waves. T hat resonance is not aurally present 26 but it is the basis for the harmonic series. Vitruvius design follows that same principle, using the vessels tuned to the first partials of a harmonic series to provide the space with a particular formant. However that approach depends almost exclusively on the richness of the spectrum of the sound performed on stage. T he pure tone of a lyre would rarely provide enough spectral 25 Ibid. 68. 26 The Acoustic (1969), 220 221.
37 energy to stimul ate the vibration of the vessels and on the other hand a strong baritone voice would take true advantage of them. Following a basic acoustical principle, the V itruvian vessels can reinforce properties already existing in a sound source but they are un able to generate them as there is no acoustical design capable of lyre, who, when they want to sing in a higher key, turn toward the stage doors and thus avail themselv es of the harmonic support that these can provide for their voices. When, however, theaters are constructed of more solid material, that is, of masonry, stone, or marble, which can not resonate, then they should be outfitted with echea for just that reason 27 Figure 2 3. T heater ceiling and audience area suggested by Vitruvius It is clear that the Roman master was foreseeing two core acoustical issues of theater design: reinforcement of sounds events happening on stage and the frequency response of the space. His solution came from a combination between the steep angle of the audience seats and the design of the ceiling (sound reinforcement) and the resonating vesse ls system (frequency response). onstage, po ured forth from stage as it were, from the center of the theater and 27 Pollio Vitruvius, 68.
38 circling outward, strikes the hollows of the individual vessels on contact, stirring up an 28 Vitruvius was already consid ering in his designs the quality of the sound perceived and desire d to provide every member of the audience with the same aural experience. Those concerns are very much current today when la rge audiences are the norm. In the design of theaters, aural architects have to simultaneously deal with the general acoustic properties of the space and their perception, reassuring that they propagate evenly throughout the audience area. Parameters of balance, warmth, brightness and clarity are of standard use by architects in order to measure the desirable qualities of a space desire. Sound was his main co ncern in theatre design, and he strived to achieve an uld please the audience members. Vitruvius translated the general harmonic proportions of sound into theater design In other words, he made a theater a Coro S pezzato Polychoral S tyle th c. ACE ) The Vitruvian science of so und was still current in the 16 th c., being based on the Greek model of sound propagation, similar to the concentric waves generated by water after dropping a stone in it. Concepts of wavelength and geometrical acoustics arrived in late 17 th c. with the studies of Gaston Pardies and Isaac Newton. Until then, buildings and specifically churches were built following the lineaments of the Vitruvian four att ributes of acoustica l quality. According to them, the spaces could be : 1) dis sonantes 28 Ibid. 68.
39 (spaces that partially reflect the sound waves) 2) con sonantes (spaces in which the environment facilitates the circulation of sound waves ) 3) circum sonantes (spaces in which the sound waves reflected by curved surfaces return to their starting point creating a reverberation ) and 4) re sonantes (spaces that create echo es) Those attributes were purely subjective and intended for the design of open spaces like Greco Roman theatres, show ing limitations when applied to the acoustics of enclosed spaces. 29 St. Mark Cathedral follows the canons of Byzantine architecture with a complex geometric design, mosaics with sophisticated patterns, high domes, decoration with public figures and a sl ightly illuminated interior. The building has two singing galleries, or pergoli that were erected by the Venetian chief architect Jacopo Sansovino between 1536 and 1544 pergoli are decorated with bronze reliefs of the Marcian legends by Sansovino, in each case three on the front panels and one on the end towards the rood screen. Whereas the first series displays familiar scenes from the life of the apostle, already well known to Venetians, the second presents obscure miracles supposedly performed by a saint at sites in the lagoon. This fact strongly suggests that the second pergolo was an afterthought, requiring considerable inventiveness to devise an iconographic programme In th arranged for divided choirs or coro spezzato in these very years, it may be inferred that the left hand pergolo was added in order to provide a second location for singers, spatially separated from the first. 30 pergoli inferred that the architectural design responded to a musical need. It is possible that pergolo was the natural response 29 Deborah Howard and Laura Moretti Musica (2006). 30 Sound an d Space in Renaissance Venice: Architecture, Music, A (2009), 37.
40 experiments with split choirs, providing an architectural solution to obtain a spatial balance between the performing forces. According to Carver a polychoral work or passage is one in which the ensemble is consistently split into two or more groups, each retaining its own identity, which sing separated and together within a through composed framework in which ]. 31 In the 16 th c. many composers in Venice applied that compositional resource which would later be one of the trademarks of the Venetian style. Among those composers, the names of Adrian Willaert and Andrea and Giovanni Gabrieli stand out as the most in fluential of the time. Wil l maestro di capella at St. Mark on December 12, 1527, was a relevant event in the development of the cori spezzati style as he probably introduced it as a performance practice in Venice. 32 This type of comp osition was a real gaining popularity not only in Italy but also throughout cori spezzati psalms were printed as a collection in 1550 by Antonio ation known in the style. 33 Gioseffo Zarlino published in 1558 a treatise in music theory called Le intitutioni harmoniche type salms in chori spezzati are arranged and divided into two choirs, or even three, each in four parts; the choirs sing one after another, in turn, and sometimes 31 Anthony P. Carver, (1988), preliminary notes 1. 32 Sound an d Space in Renaissance Venice: A rchitectur e, Music, A (2009), 27 28. 33 Anthony P. Carver, Ibid. 35.
41 (depending on the purpose) all together, especially at the end, which works very well. a 34 Zarlino continues addressing the interrelation between the split choirs: composer should be warned (so that he is not displeased by dissonance amongst the parts of any of them) to compo se the piece in this manner, that each choir be consonant; that is that the parts of one choir be ordered in such a way as if they were composed simply for four voices without considering the other choirs, having nevertheless regard in the placing of the p arts, that they accord with one another, and there be no dissonance. So that [with] the choirs composed in such a manner, each one would be able to sing separately, and one would not hear anything, which might offend the hearer. This advise is not to be de spised, because it is very 35 Those compositional recommendations made by Zarlino in his treatise are not ributes of St. acoustical space for which those compositions were intended. The utilization of acoustical space was not a mere coincidence; according to Zarlino each choir sh ould be harmonically self sufficient and consonant, complementing each other to produce a pleasant sensation on the listener. In addition to that, Zarlino in his work makes specific reference to the treatment of the bass parts in relation to the spacing of the choirs. The spatial separation of the choirs could easily gener ate undesired chord inversions, like 6/4 for example, due to the fact that singers might not be able to perceive the true bass of t he harmony. In order to counter balance this potential pro blem, Zarlino makes a series of suggestions for the composer, which tend 34 Gioseffo Zarlino, armoniche", (1558), 268. 35 Anthony Carver, Ibid. 10.
42 to overcome the arising problems of two groups of singers that perform in synchronicity but are placed far from each other. T he following example shows how, according to Zarlino, th e bass lines for a composition for two spatially separated choirs should be treated. The intention here is to conceive the bass line as a single line (for the most part) that works as the basis for both choirs, which are independent from each other. The co mparison between the two basses is clear enough, octaves and unisons work together outlining and reinforcing the actual single bass line that unifies both choirs. Figure 2 4. Example of basso sequente according to Zarlino. Like Vitruvius, composers in t he 16 th c. particularly Willaert were also concerned utilized clear harmonies and sophisticated textures based on a thoughtful spacing between the individual voi ces. That approach can be seen in the following example from Domine Probasti Me (Ps. 138) by Willaert for two choirs. The first excerpt shows an economic use of counterpoint, with a primarily homorhythmic texture and an antiphonal interaction between the t wo groups. The overlapping between the two choirs is only two beats and with the same chord (with almost identical voicing), which facilitates the crossover between separated ensembles. T he bass line is also composed following
43 Figure 2 5. Excerpt from measures 5 Domine Probasti. The next example, also from Domine Probasti, corresponds to the final Alleluia of the psalm The contrapuntal activity here is noticeably increased but not obtrusive, each line pr eserves its individuality within the texture. The very few non harmonic tones that appear contribute to create momentum towards the end of the work. The bass line is a basso sequente between the two choirs, mainly built upon octaves and unisons between the two groups, also doubling, in the very end, the cantus firmus that is carried by the tenors of choir I. Tenors and basses from both choirs emphatically affirm the plagal cadence by moving almost exclusively within the sonorities of F and C major. In Wila pitch contrast is not large, and indeed is more a question of chord spacing, since choir one always contains the highest and lowest voices, giving choir two a closer spacing. One might speculate that it was the latter to which the four sol o singers were assigned
44 in S ain obtain. 36 Figure 2 6. Excerpt from the final Amen, measures 222 Domine Probasti. Carver suggests that the blending wi thin the choirs was also considered by Willaert in his works, who supported the idea that a large ensemble of voices was most likely to produce a fused sound despite the inter nal separation of the voices. O n the other hand choirs with one singer per part n eeded to be scored in closer position in order to achieve a unified sound. 37 According to Howard and Moretti, the average RT in S aint sec onds, 38 36 Ibid. 37. 37 Ibid. 35 37. 38 Deborah Howard and Laura Moretti, Ibid. 244
45 fingerprints w ere homophony cally contrasted with note against note polyphony, concise imitative points, syllabic writing with controlled melisma towards the cadence 39 On the other hand, Rufin ssisi who served as maestro di capella at the Padua Cathedral in 1510 h ad a different approach to the use of spaced choirs and with d ifferent results. In his Missa Verbum bonum there is an unbalanced relation between tuttis and antiphony. 40 The large overlapping between the two choirs produces many parallel octaves as well as dynamic peaks instead of smooth cross fades between the successive entrances. That fact becomes even more relevant if we consider the reverberation decay: Willaert used it [RT] as a smoothing device between groups, on the contrary, Ruffino actually compos ed the cross fades between the entrances. the addition of sound sources of equal intensity in dB. If we hypothetically consider that each choir has 4 members (one on a pa rt) each of them singing at 75 dB, we have a level of 81 dB per choir. When the choirs overlap (like m. 4 of the example) the overall level is increased in 3 dB for three beats as a consequence of the addition of the voices of the second choir. This acoust ical consequence is perceivable by the listener and has relevant compositional implications: each of the choir entrances creates a dynamic composition. This fact added to the poor quality of the part writing and the RT of the space result s in a mostly blurred and unclear musical result. 39 Anthony Carver, Ibid. 39. 40 Ibid. 24.
46 Figure 2 7 Excerpt from Verbum bonum to a certain extent, the opposite. His eight polych oral settings in I salmi are scored for two separate choirs, the first of which delimits the overall range of the work while the second fills in the middle of the texture. The psalm text is distributed equally between the two choirs, which present alternat e verses or half verses. Compared to the more exuberant settings of ear lier as well as later composers [Ruffino, Gabrielis respectively] adopting the expressive character, mode, melodic material and c adential articulations of the plainsong psalm tones. The two choruses sing together only rarely, primarily near the cadences that mark the verse endings or in the final doxology. 41 Cum invocarem is a perfect example of how the spatial perfo rmance of the work complements the simplicity of the writing. exchanges] reveal a sensibility to the text where A f ructu frumenti, vini et olei hich 41 Michele Fromson, Oxford Music Online
47 reveal a consummate grasp of the expressive possibilities of simple triads alternated by groups of subtly differing composition. 42 The aural stereophonic effect is reinforced by a onic binaural experience in the listener. The progressions are self sufficient in each choir but not necessarily create a smooth voice leading between the two ensembles The voice leading from the choral exchanges is quite angular and generate s parallel su ccessions (i.e. the parallel fifths resulting from the successive entrances of the choirs in measure 3 of the excerpt). It can also be appreciated how Willaert solved the issue of continuity and intonation between the two choirs using a basso sequente stru ctured upon the roots of the tonic and the dominant chords of the relations between musical composition and aural architecture is undeniable. His intuitive contribution was based on the clarity and economy of means in his music, Figure 2 8 Excerpt from Cum invocarem. 42 Ibid. 39.
48 After death, in 1562, Andrea Gabrieli was appointed organist at St. hich for some authors demarked the beginning of the Venetian style. Rich sonorities, colorful harmonies and the lively interplay of forces that started to appear in Wilaert f inally emerge in the works by Andres and Giovanni Gabrieli. Claud e Debussy Color Shapes and P roportions in La Mer (1909) and Other W orks Theories that assert Debussy's intentional use of mathematical proportions in his works remain as mere speculations for many reasons. Works like La Mer and Estampes show distinctive structural attr ibutes that suggest Debussy's unequivocal use of mathematical ratios such as the Golden Section and the Fibonacci series. However, those composition s are not obvious regarding the intentional or intuitive approach by the composer. These two systems of bala nce and proportion are pillars of the French Symbolist movement with which Debussy was very much associated towards the end of his student days. At this time, the composer spent more time among writers and painters than with fellow musicians. 43 As a reacti on against naturalism and realism, symbolism was among the anti idealistic movements, which attempted to capture reality in its rough distinctiveness, and to elevate the humble and the ordinary over the ideal. Symbolism began with that reaction, favoring s pirituality, the imagination and dreams. Symbolist poets believed that art should aim to capture more absolute truths, which could only be accessed by indirect methods. Thus, they wrote in a highly metaphorical and suggestive manner, endowing particular im ages or objects with s Manifesto, Jean Moras, who published the 43 Ray Howat, (1983), 163.
49 document in 1886, proclaimed that symbolism was adverse to "plain meanings, declama tions, false sentimentality and matter of fact description", and that its goal instead was to "clothe [sic] the i dea l in a perceptible form whose goal was not in itself, but whose s ole purpose was to express the i deal". 44 They conceived the physical w orld as a collection of symbols, a language that needed to be deciphered by th e spectator, where nothing has a plain meaning, everything evokes an deeper image, a metaphor The Symbolist influence on Debussy is palpable, and of course Prlude l'aprs midi d'un faune was i nspired by Mallarm's poem L'aprs midi d'un faune. Those pr inciples also influenced Debussy's musical perception, where the architectural design of a work and its expressivity were inseparably bound. Every note had a meaning with an enormous expressive potential. In his review on Paul Duka Piano Concerto from 19 01, Debussy writes: "[...] you could say that the emotions themselves are a structural force, for the piece evokes a beauty comparable to the most perfect lines found in architecture." 45 Music is a series of perceptible surfaces created to represent esote ric affinities, which Symbolists used to evoke their primordial i deals. 46 In Debussy, sounds and their succession in time are indirect methods that conceal deeper meanings. Spirals are present La Mer as a formal device that the composer utilizes in order to revisit certa in material from the past that, and at the same time, transform it into new musical ideas that continue to develop in the same fashion. In this work Debussy 44 (1886). 45 Ray Howat, Ibid. 173. 46 Jean Moreas, Ibid.
50 reveals his natural tendency towards repeated visits to the same musical territor y, a characteristic fixation upon specific sounds, patterns, textures, harmonic structures, sonorities, even absolute pitches and melodic fragments, resulting in what we may call aural images. One comes to feel that Debussy's aural images are psychologica l links between certain works of his; they are signs of his unremitting perfectionism, in that, always doubting his own accomplishment, he may have attempted to pursue some forever elusive musical idea by resurrecting it in another work turning his whole c ompositional output into a spiral. 47 Musical proportions and numerical experiences could have also come to Debussy through his Baudelaire readings, e specially his essay Du vin et du hachish in which the poet describes a particularly vivid experience of mu sic as numbers, intimately related to La Mer's spiraled construction. Figure 2 9 Spiraled form of La Mer according to Howat The first edition of La M er appeared with a reproduction on the cover, at Debussy's request, from Katsushika Hokusaki's print The hollow of the wave off Kanagawa a copy 47 Mark De Voto, of T (2004), 24.
51 of which also hung on Debussy's study wall. The dominating motive of the print is the wave, whose lower outline curves in logarithmic spiral, admittedly broader than Debus sy's variety In addition, the golden s ection divisio ns indicated around the picture show s how close the c omposition comes to overall GS, especially if we consider the upper extremity of the wave, the side of its lower curve, and the top of Mount Fuji. 48 Those curves a nd extreme points of the waves are actual mathematical functi ons (if x and y axis are added, for example) that Debussy transplanted into form and the development of his musical ideas. A similar concept of translations of functions into music will be seen l ater on in this c hapter, in the section about Xennakis. Figure 2 10 Katsushika Hokusaki's The hollow of the wave off Kanagawa Despite those facts, none of Debussy' s surviving manuscripts contain any signs of numerical calculations concerning structure. This however is inconclusive, and also not surprising. Most of these manuscripts are the final copies given to the engraver, an artist as meticulous as Debussy was over the vis ual presentation of his scores, both manuscript and prin ted, would hardly have been so unprofessional as to deliver his finished product 48 Ray Howat, Ibid. 178.
52 with scaffolding still attached. In any case these final copies are mostly third or fourth drafts of the works concerned, by which stage their forms would be well established Apart from these final copies, only a very small number of sketches have survived. Debussy is known to have destroyed the large majority of his sketches, and, while that proves neither side of the question, it could be conjectured that the few sketches w hich remain are those that divulge no secrets [...] No firm conclus ion can be drawn from the 49 Debussy was deeply aware of the numerical connotations of music composition, but it is still uncertain if he used them intentionally or not. Music is a mysterious mathematical process whose elements are a part of infinity 50 In addition to that, it can be said that Debussy was completely unaw are of his proportional systems. His subconscious judgment was responsible for organizing them with such precise logic and he would later have had to completely avoid the possibility of such occurrences in his late r works. 51 That awareness is confirmed if we compare the similar use of proportional schemes that appear in La Mer and La C ath drale E ngloutie two works th Additionally, the use of a center of resonance as a compositional device is another Prlude l'aprs midi d'un faune from 1895 (one of his most remarkable early works) undoubtedly circles around the pitch class C#; L'isle joyeuse from 1905 (from his middle period), recurrently plays around the initial pitch class C#; Syrinx fro m 1913 (in his late period) is, for the mos t part, constructed around the initial pitch class Bb. These three works have a very similar initial harmonic and gestural structure and spanned 49 Ray Howat, Ibid. 6. 50 Ibid. 171. Debussy ( 1977 ) 199. 51 Ibid. 162.
53 throughout all Debu material seems to be revisited and reinterprete ndency towards self recycling. This arabesque ornamentation around a given pitch class is on e trademarks. 1 Excerpt from a solo piano version the first bars of Prlude l'aprs midi d'un faune (1895). Figure 2 11 Measures 1 3 from Prlude l'aprs midi d'un faune. 2 Excerpt from the first bars of (1905). Figure 2 12 Measures 1 2 from 3 Excerpt from Syrinx for solo flute (1913). Figure 2 13 Measu res 1 2 from Syrinx
54 Similarly, the idea of Debussy using such scientific means of formal regulation (consciously or not) is quite in compatible with his known distaste for musical formulas which at the same time were concept plain meaning s quite unwelcome among the Symbolists. Literally, a formula is a prescribed method, convention or recipe, nothing metaphoric a definition applicable to such construction as fugue, sonata form and so forth. 52 Debussy desired to evoke images and spiritua lity through his music and these self explanatory processes were not a suitable tool. Even though Debussy's search for perfection in his scores is documented in some of his letters where he specifi cally expresses concern about golden section proportions. In a letter of August 1903 from Debussy to his publisher Jacques Durand, returning the corrected proofs of the Estampes Debussy writes: "You'll see, on page 8 of 'Jardins sous la pluie' that there's a bar missing my mistake, besides, as it's not in th e manuscript. However, it's necessary, as regards number; the divine number, as Plato and Mlle Liane de Pougy would say, each admittedly for different reasons". 53 The difference in proportion between the final score and the Sibley manuscript of De l'aube a midi sur la mer make s clear that the music was not composed to fit rigid plans impervious to any subsequent modification. If Debussy was applying GS consciously, the plans could evidently be remodeled according to other musical demands, many of which ma y have been primarily instinctive ones, however consciously carried out and perfected eventually. The point again is that Debussy would never have set his intellect on the rampage without simultaneously applying his intuitive judgment. If alternatively, he was completely unconscious of the proportions 52 Ibid. 9. 53 Ibid. 7.
55 just seen, we are left with awkward logic. This is because the Sibley manuscript, even in its final state, does not have overall GS coherence and the final score has. This would mean, therefore, that Debussy's proportional intuition failed him entirely with the large scale dimensions in the Sibley manuscript, and then suddenly brought the form to virtually maximum happily provided exactl y the necessary dimensional adjustment. 54 Even though, few analyses of Debussy's music consider dynamic shape at all, and those that do tend to focus only on isolated aspects such as the principal climatic point of a work 55 Nevertheless dynamics are a vital structural element of Debussy's mature music. The tidal flow of swelli ng intensities of the dynamics, specifically in works like La Mer reveals a novel programmatic method with an outstanding dramatic outcome. The tides, undulations of the waves, the wind and its shape, are metaphorically symbolized through the dynamic structure of the work which evokes particular states of mind and aural images that invite the listener to deciph er them. As a vital component for the completion of Debussy's music the listener is constantly challenged to resolve aural puzzles. What is given appears to be incomplete; the symbols need the spectator to become meaningful. In 'Reflets dans l'eau' the composer evokes the concentric propagation of sound waves, not only are many of the sequences [...] visibly reflected round some central musical turning point; but also their reflected portions (or images) tend to be compressed in size, giving an effect of refraction another aspect of reflection (or deflection) in water." 56 54 Ibid. 91. 55 Ibid. 12. 56 Ibid. 28.
56 Somehow anticipating Le Corbusier, Debussy was also bound to the intrinsic properties of sound explained by Pythagoras and their numerical implications. Le Corbusier used the Pythagore an theory in order to justify his own set of ideas in which he desires to import the universality of the proportions of the individual components (harmonics) of a given sound into a model for architectural design not based on the properties of sound but on the human figure and its scope. On the other han d, Debussy finds a more poetic that music should be reduced to a combination of numbers: it is the 'arithmetic of sound' just as optics 57 Interestingly his use of prop ortions never becomes formulaic, as it is never used in the same way twice The s imilarities between pieces are always offset by a sharp contrast In Debussy the presence of spirals is always recurrent. Le Corbusier Le Modulor (1948) The Second World War left Europe with an incalculable housing deficit. Le Corbusier, in 1948, published a method for standardized construction based on the proporti ons of the human body. He called it Le Modulor 58 and defined it as a "range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things." With this work Le Corbusier carried on with the tradition started in 1 st c. BCE by the Roman arch itect Polio Vitruvius and continued by Lenardo Da Vinci in the 16 th c. who sought to discover the human proportions applicable to architecture. 57 Ibid. 171. Debussy ( 1977 ) 255. 58 to develop a scale of visual measures that would unite two virtually incompatible systems: the Anglo Saxon foot and inch and the French Metric system.
57 This section of the study comple ments the explanation given in c hapter I (delimitation of the object of study method is shown. Here, specific references to music and sound phenomena in Le and their relations (ratios) was one of the i nspirations for Le Modulor. natural qualities of sound in order to make music permanently transmissible in another way than from mouth to ear. Sound is a continuous phenomenon, an uninterrupted transition from low to high. The voice can produce and modulate it and certain instruments, like the violin, can do the same. As there was no method available at the time to notate music, it was necessary to represent sound by elements, wh ich could be grasped, breaking up a continuous whole in accordance with a certain convention and making from it a series of progressions. These progressions, based on certain ratios, would then constitute the rungs of a scale, an artificial scale of sounds created by man. The question here was how to divide that perfect continuum of frequencies into breakpoints, cutting up sound in accordanc e with a rule acceptable to all. That rule should be efficient, flexible, adaptable, allowing for a wealth of nuances and yet simple, manageable and easy to communicate and understand. 59 Pythagoras solved the problem by taking two points of support capable of giving certainty and diversity: on the one hand, the human ear, the hearing of human beings, 59 Le Corbusier, Le Modulor (1948), 15.
58 on the other, num bers, that is to say mathematics in all its forms: Mathematica herself the daughter of the Universe. 60 To Pythagoras, music was one of the dependencies of the divine science of mathematics, and mathematical proportions inflexibly controlled its harmonies Pythagoreans professed that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. He wanted to kn ow why some musical intervals sounded more beautiful than others observing that when the lengths of vibrating strings are expressible as ratios of integers (e.g. 2 to 3, 3 to 4), the tones produced were harmonious. 61 After discovering these harmonic rat ios, Pythagoras gradually initiated his disciples into this, the supreme Arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. 62 The Greek mysteries included in their doctrines a magnificent concept of the relationship existing between music and form. The elements of architecture, for example, were considered as comparable to musical modes and notes, o r as having a musical counterpart. Consequently when a building was erected in which a number of these elements were combined, the structure was then likened to a musical chord, which was harmonic only when it fully satisfied the mathematical requirements of Ten books of 60 Ibid. 15 16. 61 Online resource: http://www.sacred texts.com /eso/sta/sta19.htm 62 Online resource: http://www.sacred texts.com/eso/sta/sta19.htm
59 Architecture when he addresses the importance of t he specific orientation of the t heaters in order to be in full harmony with the cosmos. In Book III 63 of his work, V itruvius states that other public places, [should be made] with a view to general convenience and utility. If the city is on the sea, we should choose ground close to the harbor as the pl ace where the forum is to be built; but if inland, in the middle of the town. For the temples, the sites for those of the gods under whose particular protection the state is thought to rest and for Jupiter, Juno, and Minerva, should be on the very highest point commanding a view of the greater part of the city. Mercury should be in the forum, or, like Isis and Serapis, in the emporium: Apollo and Father Bacchus near the theatre: Hercules at the circus in The beauty of Le Corbusier's method resides in the fact that he was able to translate the divine proportions (Golden Section and the Fibonacci series) into a fully pragmatic model for architectural design without giving away any of the essential components. The Pythagorean theory of the harmonic overtone s series and their ratios was the integrate idealism with pragmatic concepts overcoming the disparities betwee n the Anglo Saxon and the French metric systems. Alvin Lucier Chambers (1968), I am sitting in a room (1969) Alvin Lucier was born in New Hampshire, U S in 1931. In his early compositions, he incorporated the utilization of alpha waves to generate sub sonic inaudible sound 63 Pollio Vitruvius, Ibid. 77.
60 waves that made pe rcussion instruments vibrate. H is piece Music for solo performer represented his constant search for the roots, with an unstoppable desire to reveal the essence of life through music. In his own words : [ ... ] it's just an extension of what you do when you're a child at the beach and you put a shell up to your ear and hear the ocean. Then you stop. You don't do that as you grow older. Your ears stop doing that because you've got to think about other things, how to m ake a living and how speak to people, how to communicate verbally. I guess I'm trying to help people hold shells up to their ears and listen to the ocean again." 64 Brandon Labelle describes Lucier's work as an explorative pursuit of how sound works as phys ical phenomena [...] In his experimental compositions; Lucier explores auditory perception from a scientific point of view. Much of his work is influenced by the physical properties of sound itself. 65 Th e properties of sound that Lucier helps explore in h is works are always intended to appear in their most pure form where any type of sound manipulation gets in the way. His style could be defined as a search for the complexities within sound itself revealed through the apparently simplest and most unobtrusi ve compositional processes. space as a compositional device. T he two works chosen for this section of the study characterize his fascination with acoustic al space and its musical implicatio ns. In his piece Chambers the first performance di rection from the score reads: 66 Those environments are detailed immediately afterwards and span from seashells to bays, tombs and canyons. 64 Alvin Lucier, Douglas Simon, (1980), 19. 65 Seth Kim Cohen, (2009), 193. 66 Alvin Lucier, Douglas Simo n, Ibid. 3.
61 Curious ly, those items selected for the performance are classified as environments not just resonating things or vibrating bodies, environments with their own acoustical properties that become exposed during the performance of the piece. His main idea was to m ove sound environments of different sizes into other environments, to carry sounds from one place to another, changing them. 67 Those sounds are grafted in to foreign environments that operate as filters, with their own standing waves and resonating propertie s that re model the qualities of the original sound source. Chambers for publication I decided to expand it. I wanted to make it bigger in the sense that it would imply more, so I extended it to incl ude any resonant environment, large or small, that performers could use to produce or alter sounds in the same way that this room we're in alter our sounds. If a room can intrude its personality on whatever sounds occur in that room, then, any other size e nvironment can do the same thing, so for the sake of performing I decided that performers could collect resonant objects into which they could put sounds, and the acoustic characteristics of the object themselves shells, pots, pans and so forth would a lter the sounds with their own 68 In this way Lucier forces the listener to create new links between the space suggested in the sound source and the environment in which it is actually being performed Is that the sound of an orchestra comi ng from a bottle? Are the sound and shape of an object related? Why does this r oom sound as if it was bigger? compositions become complete only with the active participation of the listener establishing connections, imagining new sonic spaces, an d being inquisitive about what is been proposed in the artwork. The listener is constantly challenged with incomplete sonic puzzles. 67 (1995), 169. 68 Alvin Lucier, Douglas Simon, Ibid. 10.
62 A Beethoven symphony implies a large space, as the orchestra has a hundred players If the symphony is recorded in a big hall, and the recording played from a two inch loudspeaker, it's very strange, when you think about it. On the other hand, to try to recreate an environment and put it into another is like taking something that belongs somewhere and putting it somewhere e lse, so you make connections between things things that no one else would ever make. 69 their most pu re form. H is works offer ways to re discover the roots of music, sound and its properties. In order to achieve that, he keeps an intransigent passive position towards sound manipulation. His desire is just to let the natural properties of sound sources emerge through these environments that is reinforce certain acoustical properties. He is not interested i n pursuing that line of thought; he just wants to find out what those environments do to sounds. Made en vironment s would not fulfill that desire. T hat is why Chambers is performed using 70 that make of each performance a singular learning experience. performance environment element in Chambers remains unobtrusive; you can do almost anything in a performance of this piece as long as you think of it in terms of physical environments that alter sounds because of what they are. 71 The paradigm of I am sitting in a room let the a coustical pro perties of a room, a human habitable enclosed space, surface 69 Ibid. 13. 70 Ibid. 12. 71 Ibid. 11.
63 through the composition The performance of the work is very simple; several sentences of recorded speech are continuously played back into a room and re recorded there many times. As the repetit ive process continues, those sounds common to the original spoken statement and those implied by the structural dimens ions of the room are reinforced, being the other ones gradually eliminated. The space acts as a filter; the speech is transformed into pur e sound. Sound is always affected by the physical space in which it is heard. The size and shape of the room, the materials of the walls, floor and ceiling, the presence or absence of curtains and carpeting all exert an influence. Some frequencies fit n aturally in a given room and are therefore maintained with minimal degradation. Other frequencies, however, clash with the room and are canceled out. 72 The score of I am sitting in a room reads which you would like to evoke". 73 The instrumentation is the room, the resonant body is the room itself; the speech is a mere device that stimulates the acoustical properties of the chosen space. This initial statement describes the essence of the work, which is about evoking the musical qualities of enclosed spaces. When Barry Blesser makes the distinction between the proto instruments and the meta instruments 74 he is referring to the incompleteness of a musical instrument as a mere vibrating source. The direct vibration produ ced by the instrument (of any type) the proto instrument is completed with the reflected vibrations provided by the space in which that sound is performed the meta instrument The space of a concert hall is not simply a place for musicians and listen ers to gather, but also an extension of the musical instruments played within it. 72 Seth Kim Cohen, Ibid. 186. 73 Alvin Lucier, Douglas Simo n, Ibid. 33. 74 Barry Blesser and Linda Ruth Salter (2007), 135 136.
64 A musical instrument has two acoustically bound elements: a source of energy ( vibra tion) and a passive element ( resonance that provides a particular timbral color). Those c oncepts are inseparably bound, as we cannot have in an enclosed space only direct sound or just reflected sound waves. In I am sitting in a room the process leads to the unfolding of the most pure version of the meta instrument. Every iteration during the performance of the work represents a deeper look at the resonances provided by the space that simultaneously obliterate the clarity of the speech. Through the process, the text becomes gradually incompr ehensible, arriving at its climactic point when the s peech goes from intelligibility to unintelligibility, or from words to music. 75 letting the hidden melody of the room become audible. 76 The rooms then, have their own unique tunes that emerge through the process. That process is now the musical composition, music and process are inseparably bound in a new compositional model. Rooms are musical instruments, with their musical instrument has its own particular wavelength; the h igher the pitch, the shorter the wavelength. Actually, there's no such thing as "high" notes or "low" notes, we simply 77 Following this path, we can say that the structural p roperties of sound itself; sounds are conceived as measurable wavelengths ( instead of high or low musical notes ) subverting the idea of music as a metaphor into a calculable fact connected to architecture. 78 75 Ibid. 39. 76 Ibid. 37. 77 Ibid. 35. 78 Ibid. 36.
65 The work also makes evident the always encroachi ng semantic decay of speech. The voice is always fading away. In I am sitting in a room it plays a cruel game with the listener, repeating itself [speech] for semantic confirmation, while at the same time eroding the sonic clarity that makes meaning poss ible. 79 The artistic value of this work is unquestionable and resides in intention to expose the gradual disintegration of the speech. An acoustical engineer would have arrived at the same result in one step but Lucier was reluctant to this he was interested in the step by step process, in the progressive dissolution of the words and the reinforcement of the resonant frequencies of a room, in the gradual transformation of sound from pure energy to pure resonance. 80 Iannis Xenakis Meta stasis (1954), the Philips Pavilion (1958) and Other W orks His dual expertise as a composer and a professional architect make Iannis Xenakis one of the paradigmatic contemporary music figures of the second half of the 20 th c. Ianniks Xenakis was born in 1 922, of Greek parents in Braila, Rumania, on the Danube. His early musical studies were mainly with Aristotle Kundurov, pupil of Ippolitov Ivanov. At that time he was particularly interested in Greek traditional music that of the Byzantine Church and fol k music; and this inspired him to write some choral and instrumental works, which he later destroyed. But already in this modal music he had begun his explorations in timbre and sonority. Along with his musical studies Xenakis pursued a scientific educatio n that took him to the Polytechnic School in Athens 79 Seth Kim Cohen, Ibid. 191. 80 Alvin Lucier, Douglas Simon, Ibid. 34.
66 from which he graduated in 1947 with an engineering degree that opened the door to him for a brilliant career as an architect. 81 T he same year he left Athens for Paris where he continued his musical stud ies under Arthur Honegger and Darius Milhaud. H e attended Olivier Messiaen's courses in analysis and musical a esthetics at the Paris Conservatoire. 82 Also in 1 947, Xenakis was able to get a job at Le Corbusier's architectural studio. He worked as an engine ering assistant at first, but quickly rose to performing more important tasks, and eventually to collaborating with Le Corbusier on major projects. The most significant was the Philips Pavilion at Expo 58, despite the fact that it was actually completed by Xenakis himself, using a basic sketch by Le Corbusier. 83 music: important early compositions such as Metastasis and Le Sacrifice were based directly on architectural concepts In the latter, he experimented with numerical proportions in the same way Le Corbusier had done following a quite logical step in his line of thought, probably incentivized by the numerous references to music contained in Le Modulor. Despite the fact t hat he never conceptualized theoretically his way of linking music and architecture, Xenakis treated both arts from a scientif ic and mathematical perspective. H is Formalized Music (1971) is a perfect example of his desire to apply 81 Mario Bois, The man and his music. A conversation with the composer and a (1980), 3. 82 Ibid. 3. 83 Ibid. 3 4.
67 mathematic to every single parameter of music composition. As he 84 Xenakis early period reflects his utmost rigor towards music composition as a purely logical activity, therefore relevant for this section of the study. Every sonic event was the convergence of vectors taken from matrixes based on probabilities and proportions. The application of these vectors was strict, sometimes compromising the ultimate artistic result. In Le Sacrifice ( 1953), for example, this rigid algorithmic approach was not really successful, as the simple permutation of two sets of values was excessively simple to keep the attention. 85 Whereas in Zyia (1952) Xenakis draws freely on elements derived from traditional Greek practice, in Le Sacrifice he constructs an edifice worthy of the Eu ropean avant garde of the 1950s: In the manner of Messiaen's "modal" serialism, Xenakis bases his composition on a series of eight pitches fixed in register each linked to a duration derived from the Fibonacci series. These pitches are elaborated by neighboring notes and glissandi in between, characteristic features of later pieces [...] The deployment and repetition of the associated durations follows a mathematical pro cess [...] The work is an orchestral piece constructed on th e basis of a melodic series of eight pitches, associated with a scale of 8 durations whose values (in 16 th notes) were determined by the first 8 numbers of the Fibonacci series. 86 However, one yea n Metastasis Xenakis leaves no trace of how he views the relationship between the abstract serial structure of Le Sacrifice and his original inspiration from the Di onysian sacrifice of the bulls. That the text had been dropped 84 Ianniks Xenakis, Ibid. 162. 85 Sven Sterken, k of From (2007) Mikesch W. Muecke and Miriam S. Zach, editors. 24. 86 Ibid. 24.
68 from the music is certainly of some significance. 87 According to Harley, it is possible to hypothesize that Xenakis could only find a true expression of the ritual through an absolute abstraction 88 Figure 2 14 Iannis Xenakis, Le Sacrifice (1953). Source : Andre Baltensperger, Iannis Xenakis und die Stochastische Musik. Komposition im Spannungsfeld von Architektur und Mathematik (Bern: Haupt Verlag, 1996): 231. In propose d processes, the composer decides on the elements that complete the sets of v alues, their algebraic relations and how they influence the vectors that are ultimately translated into musical te rms. Music becomes organization that he an organization of elementary operations and relations between sonic entities or between f unctions of sonic entities. 89 This new approach to music composition obliterates the existence of musical material that is not unde r total control by the composer. The ideas of improvisation where the performer is at the same level in the cr eative proces s as the composer and indeterminacy, were inadmissible for Xenakis at the time. From his point of view, the composer is a self s ufficient builder who decides on every aspect of the composition until the utmost detail, determining foundations, dimensions, materials and design. However, he does not believe that the 87 James Harley, (2004), 5 6. 88 Ibid. 5 6. 89 Ianniks Xenaki s, (1971), 4.
69 d to the completion of the work as all decisions are fully calculated by the composer. T he performer gets in some way relegated to a less relevant function, as a mere operator with very little room to make creative the natural consequence of this implacable desire for total control. To make music is nothing but to express huma n intelligence by sonic means This intelligence is conceived in its broadest sense, which includes not only the peregrinati ons of pure logic logic' of emotions and intuition. In the end, a musical composition offers a collection of logic al sequences, which it wishes to be casual. 90 After various unfruitful attempts to find a mentor, Xenakis was finally accepted in 53. Messiaen immediately recognized Xenakis's talent: is of superior intelligence. [...] I did something horrible, which I should do with no other student, for I think one should study harmony and counterpoint. But this was a man so much out of the ordinary that I said... No, you are almost thirty, you have the good fortune of being Greek, of being an architect and having studied special mathematics. Take 91 Messiaen and his students studied music from a wide range of genres and styles, with particular attention to rhythm. Messiaen gave Xenakis some key advice, suggesting him to incorporate in his pieces the two central elements of his then daily routine, that is to say the Modulor an d the use of graph paper. He applied the Modulor to systematize 90 Ibid 178. 91 Sven Sterken, Ibid. 23.
7 0 the logical proportions and the graph paper in order to determine pitch content, densities and structure. 92 Figure 2 15 Iannis Xenakis, Metastasis (1954) measures 309 314. 92 Jean Boivin ( 1995 ). Paris: Christian Bourgois.
71 A c daily tools can be seen on his translation of the first bars of Matastasis from the graph paper in to traditional music notation. The scientific thought addressed in the design of the hyperbolic paraboloids can be clearly seen in the orthogonality of the score. Every sonic event is considered as a sum of vectors. T he x and y coordinates in thi s case represent pitch and time respectively, being the continuous linear glissandi in the strings, which creates the volume of the figure (as they actually do in the graph paper). The aural illusion resides on the fact that those massive straight lines of sound create curves that move in a three dimensional sonic space. Figure 2 16. Excerpt of the g raph paper design of the string glissandi in Metastasis measures 309 314 He also arrived to new architectural solutions that were influenced by his previous musical research. Metastasis which was a purely musical work, had pointed towards cer tain approac hes in architecture by means of which later in 1956 he was able to
72 design the Phillips Pavilion for the Brussels exhibition. The music of Metastasis from 1953, commanded the ideas for this project. 93 If glissandi are long and sufficiently interlaced, we obtain sonic spaces of continuous evolution. It is possible to produce ruled surfaces by drawing the glissandi as straight lines. 94 Parameters of speed rate, the distance (register) covered by the glissand i and dynamics are carefully calculated using stoc hastics. According to Xenakis, the data appears to be aleatory only at the first hearing. Afterwards, and during re hearings, the relations between the events predetermined by stochastics, start to take on a definite meaning to the listener, initiating a s pecial logical cohesion capable of scheme defined under this form of vector matrix is consequently capable of establishing a more or less self determined regulation o f the rare sonic events contained in a musical composition sample. It represents a compositional attitude, a fundamentally 95 The utilization of lines as a basic compositional tool in Xenakis is similar to wha t Daniel Libeskind manifests now in the realm of architectural design through h is works. Libeskind shows a deep interest in the profound relation, which exists between the intuition of geometric structure as it manifests itself in a pre objective sphe re of experience. 96 The materialization of those Libeskind creative processes start almost identically, both begin with 93 Mario Bois, Ibid. 5. 94 Ianniks Xenakis, Ibid. 10. 95 Ibid. 37. 96 Daniel Libeskind, (1991), 14.
73 orthogonal planes where simple lines become complex multidimensional figures. In Xenakis, the materialization is through sound, in Libeskind through a building. This difference gets blurred if we compare the graphical origins of the Philips Pavilion (based on musical experiments in Metastasis materialized project, the Jewish Museum in Berlin Libeskind himself has entitled the project b etween the lines implying that it is in this intermediate zone (when a design is completed but not materia lized) where the essential lies. The design of the museum is a project about two lines of thinking, organization and relationship. One is a straight line, but broken into many fragments; the other is a tortuous line but continuing indefinitely. These two lines develop architecturally and programmatically through a limited but definite dialogue. They also fall apart, become disengaged, and are seen as separated. 97 In architecture lines define the relationship between material and immaterial reality. Any tw o lines on the paper of an architectural plan will shape and delimit the empty space between them, and at the same time configure the solid, impenetrable masses of the projected structure. 98 Lines that separate, lines that intersect and become vectors, line s in graph paper, various densities of lines, thic k and thin, curved and straight, Xenakis and Libeskind never abandoned the recurrence to the pillars of architecture in their work chance in his works. Purely aleatoric events seem to be a truly simplistic ac tivity, 97 Ibid. 86 98 Bernhard Schneider, (1999), 36.
74 worthless to a musician. Consequently the calculation of the aleatory, that is to say stochastics, guarantees first that in a region of precise definitio n slips will not be made, and then furnishes a powerful method of reasoning and enrichment of the sonic processes. 99 In this sonic environment, the listener is required to participate intelligently in the musical discourse in order to esta blish its definit e meaning. including Metastasis have an undeniable graphical origin. Some of them as sets of coordinates in t he orthogonal plane and others like Terretektorh (1965) as a sketch for the spatial organization of the sound sou rces. Figure 2 17. Images of the first model (left) of the Philips Pavilion and its finalized version (right) at the 1958. More so, his experiments from Metastasis were a direct influence in the design of the Philips Pavilion in 1958. Those works were constructions based on hyperbolic 99 Ianniks Xenakis, Ibid. 39.
75 paraboloids, the first being in the sound domain and the latter in the realm of architecture. A straight line in a two dimensional space represents the continuous change of one dimension compared to the other. The same happens between the pitch and time domain s the straight line is the continuous change of pitch versus time. Therefore, musical space is homogeneous as its two dimensions are lengths and distances. Despite that fact, these two music al dimensions are alien in nature from one another and are connected only by their ordering structure. 100 Xenakis conceived those structures independently in what he called organizations in time and outside time 101 Paradoxically, Xenakis was conceptually unsy mpathetic to the use of graphic notation. H e sustained that the graphists exalted the value of the symbol above the sound of the music, which is valued according to the quality of the drawings and not to the intrinsic beauty of the sound. Absolute contro l and planning are architectural the score similar to the architectural plan is constituted by a detailed set of instructions for the performer builder. Through his meticulou sness, Xenakis intends to think like a painter who can deal directly with the existent reality of his own work, without this indirect and imprecise "translation stage" by the performer. Even scores that may appear similar may actually be extremely differe nt in their notational function as different notational systems can use the same symbols in much the same way that different languages can 100 Balint Andras Varga, (1996), 70. 101 Iannis Xenakis, Ibid. 193.
76 use some of the same letters in their alphabets 102 Performers can also interpret those symbols differently, which emp hasizes the clash control everything through his scores Not by chance his approach to aleatoric and chance music was analogous: called aleatoric music, which is an abuse of language act that graphical writing, whether it be symbolic, as in traditional notation, geometric, or numerical, should be no more than an image that is as faithful as possible to all the instructions the composer gives to the orchestra or to the machine. 103 Graphi c notation is essentially an incomplete set of instructions that delegate compositional responsibility to the performer, open in essence, undetermined but with an ide There must be a fixed (even flexible) sound content, to establish the character of the work, in order to be called 'open' or 'available' form. We recognize people regardless of what they are doing or saying o r how they are dressed if their basic identity has been established as a constant but flexible function of being alive. 104 Even still Xenakis admitted the use of graphical notation for the sake of continuity between musical thought and its visual represe ntation. The immediate mapping of a sonic ramification in the Cartesian system provides an exact picture of the sound or traditional musical 102 Theresa Sauer, Ibid. P 11, excer pt from "An introduction to the Scribing Sounds Exhibit" by Sylvia Smith. 103 Ianniks Xenakis, Ibid. 180. 104 Theresa Sauer, Ibid. 10, from Earle Brown and Ryan David, on Brown's available forms 1. 2006.
77 notation. 105 It is well known that notation has been a constant difficulty and frustration to composers, being a relatively inefficient and incomplete transcription of the infinite totality of what a 106 Xenakis and Brown seem to agree on the "vitality" provided by the graphi c notation and about the immediacy that it provide s between co mposer and sound represented on paper. E ven t hough, their main conceptual difference resides on the fact that graphic notation was, for Xenakis, just an initial practical step in his process tha t required further translation, for Brown, instead it represented a self process is the translation into traditional notation, which explains the complexity of his scores. His creative proc ess starts by establishing an overall view of the work, and afterwards by choosing th e materials and how they relate to each other, conjointly or independently, until they become organized. In this way his work resembles that of an architect, or more corre it must have a head and arms; it would be better to speak o f biology than architecture. In m usic space is not three dimensional, it is multi 107 Despite the fact that Xenakis never wrote a piece with acoustical space in mind, he composed the Concrt PH (for electronic media) specifically for the Phillips Pavilion The work was based on manipulations of the sound of burning charcoal and played through the 350 speakers located ac However, the main composition for the installation was Poeme Electronique by Varese 105 Balint Andras Varga, Ibid. 90. 106 Theresa Sauer, (2009), 40. 107 Mario Bois, Ibid. 13.
78 who drew up a detailed spatialization scheme for the entire piece which made great use of the physical layout of the pavili on In a similar scenario, the composer Steve Roden took a radically different approach. In his Pavilion Score (2005), the composer was asked to create a sound work related to an architectural structure. The building was also a pavilion, the Serpentine Gal lery's Summer Pavilion designed by Alvaro Siza, Eduardo Souto de Mora and Cecil Balmond. Despite that initial similarity, the fundamentally opposed approach taken by the composer is here remarkable. In this case Roden's process was relatively simple: eight different colored pencils (one for each note on an already color coded child's glockenspiel) where used to fill in the pavi lion plans with colors indicati n g musical notes. The intention was that these scores would be playable by anyone musician or not, mapping the space in sound so the audience could listen to a drawing in sound of the space in which they were located. 108 Xenakis also suggested a novel approach to space or spatialization of sound sources without electronic means in his work Terrekekto rh, were musicians are sprinkled around the audience, with the conductor at the centre on a circular podium. His desire was to place the orchestra among the audience in order to regain the 50% of the sound that usually gets lost when listening from the dis tance (as in a theatre), and to drown the listener with sound, as in a storm or rain. 109 This new idea also embedded inevitable unique experiences by the listener depending on his locati on within the performing forces. A n audience member sitting next to th e trombone would necessarily 108 Theresa Sauer, Ibid. 193. 109 Mario Boise, Ibid. 20.
79 have a radically different experience of the work from the one located near one of the violins. This new vision of the orchestra as a malleable sound source does not reveal a limited confidence in pure music on the contrary it expands its sonic horizons for the listener, the performer and the composer. 110 In Metastasis Xenakis also resorted to numerical proportions to determine the temporal structures. The section starting at measure 104 shows his new conception of the rhyt hmic microstructures. The section is marked quarter note = 50 which operates as the temporal unit for all (initial) 6 voices facilitating its performance. These tiny rhythmical structures are essentially poly metrical as a result of the different grades o f density of the individual lines. There is not a pol yphonic thought in this section; the intricate texture is the natural consequence of the appli cation of a matrix of densities similar to the one s he developed in Formalized Music In order to understa nd this process we first have to build a matrix reflecting all the values we want to control in the composition. In this case, we have an initial density matrix that works as a model Table 2 1. R hythmic density ratios as proposed by Xenakis in Formali zed Music Event Density = Sounds/measure 26 MM Sounds/sec zero 0 0 single 5 2.2 double 10 4.4 triple 15 6.6 quadruple 20 8.8 110 Ianniks Xenakis, Ibid. 181.
80 We must specify the unit events, whose frequencies were adjusted in the standard matrix. We shall take as a single even density sounds/sec ond. 10 sounds/sec ond is about the limit that a normal orchestra can play. We shall choose = 5 sounds/measure at MM 26, so that = 2.2 sounds/sec. 111 The specific tempo markings in m. 104 of Metast asis (MM = 50) as well as the poly metrical rhythmic structure (4/16, 3/8 and 5/16) slightly change the values of the matrix keeping the fundamental concept unaltered. That measure unit change (from MM = 26 to MM = 50) also has a practical consequence as i t facilitates the work of the performers and the conductor. The composer does not res ign any of his textural desires; he just absorbs through the notation all the unnecessary difficulties. Here Xenakis translates the original matrix, which becomes a sonica lly complex musical score that is relatively easy to perform. The musician in this section of the work only needs to keep an invariable subdivision in order to fulfill the role sought by the the performers, otherwise I would have written symphonic compositions for a single interpreter, for one piano. But I also take into account the fact that what is a limitation today may not be so 112 In the following matrix (with the specific data from the studied section from Metastasis) the sounds/measure is similar to the sounds/second. Xenakis presumably for practical reasons chose the rate of sound events/unit (measure). 111 Ibid. 32. 112 Balint Andras Varga, Ibid. 65.
81 Table 2 2 R hythmic density ratios in m. 96 116 from Metastasis. Eve nt Density = Sounds/measure 50 MM Sounds/sec zero 0 0 single 5 4.16 double 10 12 In this section of the work the 10 notes per measure are achieved as a result of the mathematical relation between the chosen meters; 3/8, 4/16 and 5/16 have the same measure unit (MM 50) which is divided by each met er in unequal parts, resulting if all the subdivisions are played in 10 consecutive attacks (as marked i n figure 2 20 ). The only simultaneity would happen at the beginning of each unit (measure) but Xen akis chooses to hide that element of synchronicity between the meters in order to keep the pulse unrevealed and the listener engaged with the varying sonic densities. These densities are controlled by the quantity of unit subdivisions performed by each ins trument. From the following excerpt we can appreciate that m. 106 has a density of 5 events/unit and m. 108 of 7 events/unit. Later on, in m. 144 145 the curve of rhythmic densities reaches its peak in two consecutive units (measures) with a density rati o of 10 events /unit. Figure 2 18. Measures 94 116 from Metastasis
82 The rhythmic structure in m. 104 of Metastasis is based on those calculations that are based on stochastic results where rhythm is not just a succession of events in time but the amount of sound events that occur during a fixed (length unit) period of time. This parallel between rhythm an density comes from Xenakis architectural experiments in his design for the Monastery of La Tourette (1953 56) where he applied width progression s based on Le Modulor 113 In this example from Metastasis Xenakis combi ned his own matrix, to contr ol the micro rhythm structures, w proportions, to manage their utilization in the overall macro form of the composition. Figure 2 19. (left) Table with progressions of rectangles with increasing widths drawn from Le Modulor The preliminary ideas for the Monastery of La Tourette (right) table of proportions from Le Modulor density both is musical rhythm and architectural design. This massively active texture of overlapped parts (in divisi ) that move at different rates is not only the result of Xenakis' compositional processes but also a natural consequence of his aesthetic vision towards polyphonic compo sition. He thought that linear polyphony was self destructive in its complexity, and it prevented the listener f rom 113 Sven Sterken, Ibid. 21 44.
83 following the tangled lines. According to Xenakis its [polyphonic composition] effect is one of unreasonable and gratuitous dispersion of s ounds over the whole sound spectrum 114 From his viewpoint, there is a contradiction between the linear polyphonic system and its audible result, which ends up being a surface, a mass. In order to avoid this inherent contradiction of polyphony, sounds need to become totally independent, forming a texture in which the linear combinations and their polyphonic superposition are no longer workable. In such environment, what matters is the statistical average of s at any given moment. 115 That is Xenakis' justification of the use of calculations in order to control the evolution of each of the components in textures that are not the result of interwoven individual lines (like in polyphony) but sound masses, vast gro ups of sound events governed by the principles of density, degree of order, and rate of change, which require definitions and realizations using probability theory. Figure 2 20. Th e poly metrical section from m. 104 fr om Metastasis and a graphical representation of its rhythmical density, which recalls the design and table of the Monastery of La Tourette 114 Ianniks Xenakis, Ibid. 182. 115 Ibid. 182.
84 I the connection between music and architecture has less to do with common features than with the existence of a third element that acts as an intermediate between both fields: mathematical proportions in the first case and the concept of space in the second. 116 Despite the fact that he never articulated theoretically his way of linking music and architecture, Xenakis treated both arts from a scientif ic and mathematical perspective. Formalized Music (1971) is a perfect example of his desire to apply mathematica l control to every single parameter of music composition. 117 If we consider ordered sets H (pitches in Hz), G (intensity intervals in dB), U (time intervals in seconds) and T (amount of space between sonic events in seconds), we could define the next event as: H = 440, G = 60, U = 4, T = 0. Figure 2 21. Musical note as a vectorial multiplicity. Those values, however, do not come from any acoustical consideration of an enclosed spa ce and are not related whatsoever with architectural acoustics. Xenakis did not consider space as a component of his matrixes nor had the desire to use the aura l and measurable properties of room s a s the basis for his compositions. 116 Sven Sterken, Ibid. 22. 117 Ianniks Xenakis, Ibid. 162.
85 In the preface to the b ook, Xenakis also expresses a profound need to understand and explain sonic events through a formal process. Understanding, logic, scientific thought, reasoned support, and human intelligence are key words in this section that etic vision. The following passage is the milestone of his early works, in which he justifies his need of a prior control and logical understanding of every musical parameter employed in a composition. His scientific thought unavoidably re conceptualized m usic as a product of human intelligence. causes, to dominate them, and then to use them in wanted constructions; the effort to materialize movements of thought through sounds, the n to test them in compositions; the effort to understand better the pieces of the past by searching for an underlying unit which would be identical with that of the scientific thought of our time; the effort to make "art" while "geometrizing", that is, by giving it a reasoned support less perishable than the impulse of the moment, and hence more serious, more worthy of the fierce fight which the human intelligence wages in all other domains all these efforts have led to a sort of abstraction and formaliz ation of the musical compositional 118 The constant rational questing in the arts is imperishable, it lies latent or it dominates according to the epoch is always there. 119 In musical world, compositions are reifications and formal structur es of abstract ideas not ends that are later incorporated into families of compositions. Daniel Libeskind The Jewish Museum of Berlin (1999) and Aesthetic Considerations on Other W orks Daniel Libeskind (b. Poland 1946) is one of the most prominent name s in 21 st c. architectural practice and urban design. Libeskind is also well known for introducing a new critical discourse into architecture and for his multidisciplinary approach. His 118 Ibid. Preface 9. 119 Mario Bois, Ibid. 10.
86 practice extends from build ing major cultural institutions ( including museums and concert halls, landscape and urban projects ), to stage design ( installations and exhibitions ) Unlike Xenakis direction was, ultimately, in the realm of architecture. However his background as a music performer is always present in his approach towards architecture which he conceives as an abstract form of art absolutely alienated from its materialization. He studied music in Israel on the America Israel Cultural Foundation Scholarship and later in New York where he was recognized as a virtuoso piano performer. 120 He eventually left his music studies to focus exclusively on the study of architecture obtaining his architectural degree at the Cooper Union for the Advancement of Science and Art in 1970. In 1990 Libeskind was awarded the first prize in the competition for the Berlin Museum with the Jewish Museum project, which in 1999 became his first materialized (built) architectural design. 121 From viewpoint, architecture does not necessarily depend on the activit y of building rchitecture is the plan, architecture is drawing on paper. His drawings and collages from Chamber W orks an area of architectu ral thinking, which is neither physics nor poetics of space where the ulti mate reality of architecture is not its material becoming. 122 His architectu ral explorations in the series Micromegas and the Three Lessons of Architecture ( machines) are studies that are not meant to do anything, they are just etudes that address specific architectural prob lems through creative works. The concepts of the se piece s are still as he calls them 120 Bernhard Schneider, (1999), 60. 121 Ibid. 60. 122 Daniel Libeskind, (1991), 15.
87 become the models for newer works 123 obliterating the difference between architec tural theory and practice. He approaches archite cture as a music composer, conceiving his architectural work a deep moment of introspection, a moment, as he calls it, 124 The ultimate work result, the object is naturally expected. T he architect is then responsible for its production and, therefore, for part of its realization Furthermore, m aking an architectural piece like composing a piece of music comprises all the people who are involved in the process, all those who are par t in the building composing trajectory, from its conception to its final result. 125 His Three L essons of Architecture and Chamber W orks manifest a desire to challenge the traditional foundations of the di scipline. Architecture is then, like music, an absolu tely abstract art of pure space for limitless imagination or a s he defines it: that divine luxury of faith, highest crystallization of the material liberty of through his discipline, he continues : product of necessity provided by the technicians of educational and monetary 126 In less than fifty years, Le Corbusier (in Le Modulor) and Libeskind at tributed to arch itecture a totally opposed meaning: a model for mass production for the first and anything but a model for mass production for the latter. Le Corbusier embedded his artistic identi ty within the model he created not th rough the constructio ns that resulted after its i mplementation. On the contrary, Libeskind reflects his artistic 123 Daniel Libeskind (2008), 11. 124 Fabio Oppici and Enrique Walker, (1998), 132. 125 Ibid. 132. 126 Daniel Libeskind, matrix (1997), 155.
88 personality in the individuality of all of his works, which cannot be successfully transplanted nor even imagined in a different environment. His works intentionall y capture the history and project it into the future. In his project Musicon Bremen, for the Phil h armonic Hall in Bremen, Germany, Libeskind proposes a hall where musical performance is clearly the central goal of the building, which, at the same time, cre ates deep connections with the history of the city where it is located. The visual queues are also present in the design, manifesting deep desire for assimilation. The new museum ( Jewish Museum in Berlin ) harmoniously integrates with the conti guous Collegienhaus and the other buildings of the surrounding area exhibiting a compelling spatial juxtaposition of key historical buildings and architectural styles. visitor can glimpse through the new construction and reestablish the link with famili ar spatial relatio nships and architectural scales. O n the other hand those constructions also serve to underscore the unsettling and exciting divergence of the new building from the rule. 127 Thus, t he new building gets integrated with the older surrounding constructions, in a sort of understanding of the past and its projections towards the future. The Jewish Museum does not negate the historical legacy of its surroundings; on the other hand, it inte grates it and projects it hereafter This apparent clash is somehow unexplainable but at the same time unique, people of all ages, walks of life, and cultural backgrounds appear to experience the drama and emotional force of this extraordinary spatial configuration immediately and instinctively. 128 127 Bernhard Schneider, Ibid. 57. 128 Ibid. 58.
89 Libeskind is part of a historical tradition in architecture whereby drawings (as well as other forms of communication) denote more than what can be embodied in frameworks of objective data (such as buildings) If we can go beyond the material carrier [building] into t he internal reality of the drawing, the reduction of the rep begins to appear as an extension of the reality The system ceases to be perceived as a prop whose coherence is supported by empty symbols, and reveals itsel f as a structure whose manifestation is only mediated in symbolism. 129 Like in Debussy, in Libeskind symbolism and functionality are intertwined. The Jewish Museum has several "voids" ( negative spaces of utter silence that are arranged along an absolutely s traight line through the entire structure ) 130 Those spaces are "holes" of silence, where aural architecture evokes the devastation of the holocaust. silence there is nothi ng more to say, I believe, on the contrary, that where there is silence there is most to be said. I've never believed in the silent space of architecture where the forms create the illusion that all has been said. I prefer buildings that don't an esthetize us but make u 131 The spiraled line of thought in Debussy was guided by his constant reinvention of his mus ical material through his works. His artistic identity continuously grows from the originality of his musical substance as well as from hi s endles s processes of transformation. In Libeskind, the idea of identity through his work is similar, with a basis 129 Daniel Libeskind, Ibid. 14. 130 Bernhard Schneider, Ibid. 51. 131 Daniel Libeskind (2008), 16.
90 expressed in his early experiments ( Chamber W orks, Three lessons in Architecture and Micromegas) (built) respond ing to specific needs The main concepts are always present and his desire to preserve freshness and individuality in his work stays untouched. The artistic originality is a need that always fights to overcome the demands of large amounts of work. 132 For Libeskind, poets, writers, and composers face the same challenge that is to avoid becoming formulaic. 133 Each of his works is necessarily unique; each of his buildings is extremely specific regarding its location and does not suggest that it could be easily transported to another setting. 134 In addition to the visual and aural components, the spatiality of a building has to be a part of the story it tries to communicate. According to Libeskind, an architectural plan is no t just a container to be filled; it is part of the sy symbol transports the visitor [ the listener in a music composition] beyond the material reality and, in architecture [or in music], toward s the realm of what cannot be articulated through language 135 Architectur al acoustics are also important for Libeskind. His design proposal for the Musicon Bremen shows a deep concern about the aural properties of the design. Despite that fact, the Musicon is not exclusively designed for musical performance, which necessarily a lters the approach of the desi gner into a more crossover idea: an acoustical design that fits several needs or so called The care for 132 Daniel Libeskind ibid. 8. 133 Ibid. 8. 134 Ibid. 9. 135 Ibid. 14.
91 acoustics in a room is not only for a music hall but also for every building. Interestingly, our sense of balance and relationship to gravity and to the ground lies in our ears and not in our eyes. A well balanced design devotes equal care to how a building looks like as well as how it sounds, if it enhances or becomes obtrusive of the sonic events that take place within it 136 An architectural drawing is as much a prospective unfolding of future possibilities as it is a recovery of a particular history to whose intentions it testifies and whose limits it always challenges. In any case, a drawing is more tha n the shadow of an object, more than a pile of lines, more than a resignation to the inertia of convention. 137 That non conventionality is what brings Libeskind thinking even closer to music. Like Xenakis, his ground rules stay unaltered. B oth artists sta rt from t he endless possibilities of the simple lines to evoke multidimensionality, space, volume, shape, density, and oth er converging concepts between music composition and architectural design Again, it is interesting to see that an architectur al desi gn can potentially become a piece of music, like in Metastasis or a building, like the Jewish Museum or the Philips Pavilion The artistry of the architect is within the design itself, not in its materialization; which, as it was discussed before, is not exclusively in the realm of architecture. Taking these ideas to the extreme, if we say that architectural design exists even without a building, and we consider music to be "complete" without a performance then the two disciplines are equal, and th e composer and the architect have interchangeable roles The music becomes such when it is composed and the architectural design when it is drawn. 136 (1991). 137 Ibid. 14.
92 Paradoxically, there is not a direct correlation at the moment of the materialization, as an architectural de sign could become sound and a musical score a building. Libeskind, showing his multidisciplinary approach, blurs even more the apparent differences between the two disciplines as he says: 't just see them as planes, two dimensional or three dimensional projections. I see them as a musical composition. I hear them acoustically. Architecture is a world of relationships that is very, very close to my experience as a performing musician. My own response is that architecture, the way it is produced and rece very note is exactly where it needs to be. It's extremely structured. Yet its impact is totally 138 Figure 2 22. Daniel Libeskind, f loor plan of the Jewish Musem of Berlin (1999). 138 Daniel Libeskind (2008), 12.
93 CHAPTER 3 THE MODEL Overview This section of the study is dedicated to the specific application of the compositional model, how the acoustical data is obtained, and how it is translated into musical terms. The applicati ons are organized per area of study: the first section describes how the parameters related to frequency (pitch) are utilized, the second section describes how information regarding time is used, and the third section utilizes amplitude as a musical parame ter translating acoustical measurements like the G factor. The latter also includes the possible utilization of energy ratios (i.e. lateral fraction or LF) for musical purposes. Data Collection The room selected for the initial application of the propos ed model is the Baughman Meditation Pavilion a t the University of Florida. This room has specific acoustical qualities that justify its use as an object of study. In order to fully understand and appreciate the design elements of this building, a basic kn owledge of physics and materials science is necessary. One of the most important of these topics is the physics involving the path of a sound wave from source to receiver. An enclosed space, like the Baughman Center provides an infinite number of differen t paths for the longitudinal sound wave to take in traveling from source to receiver. Depending on the properties of a surface, a sound wave will experience reflection, diffraction, diffusion, or absorption when contacting the surface. The reflection of a
94 surfaces, the material will actually absorb some of the energy. An absorption coefficient (normal ized values between 0 and 1) is used to evaluate the amount of sound absorpti on of a particular material in specific frequency bands. The coefficient is higher when the material is highly absorbent and most of the sound energy is not reflected, it is lower when the material is mostly reflective. Structura l and Acoustical Considerations The building structure was fabricated of steel tubing bent to the shapes of the design in a Gothic style. This fact, in addition to the considerably high ceiling, has a direc t acoustic consequence in the space, generating a longer RT. The shape of the ceiling (dome) as well as the lack of reflecting panels in the performance area, reduce the index of clarity and directionality of the space. According to the measurements taken from the space, the f ollowing tables show the index values according to the different locations of source (S) and receiver (R) as presented in the image of the floor plan. Each value is expressed in its own unit, related to time, amplitude or frequency (se conds, dB or Hz respectively). Table 3 1. Data collected from the Baughman Center S1 R1 F(Hz) 63 125 250 500 1000 2000 4000 8000 snr(dB) 3 22.5 30.1 30.7 26.7 30.7 35 39.8 edr(dB) 15.7 33.3 41.6 40.8 37.4 42.6 47.3 53 EDT(s) 1.38 1.55 2.62 2.55 2.17 1 .6 1.49 1.06 T30(s) 2.17 1.86 2.37 2.65 2.45 1.84 1.69 1.34 corr 0.99 0.996 0.999 1 1 0.999 1 0.999 T20(s) 2.17 1.72 2.38 2.68 2.41 1.76 1.65 1.28 corr 0.99 0.992 0.998 0.999 0.999 0.999 1 0.999 Tc(ms) 192 119 174 173 139 90 68 39 C80 (dB) 2.5 0.7 2 2.2 0.6 1.8 4 7.1 D50(%) 40 45 30 30 37 50 63 76 G(dB) 56.9 59.3 62.2 61.4 51.1 47.5 45.2 43
95 Table 3 1. Continued. S1 R3 F(Hz) 63 125 250 500 1000 2000 4000 8000 snr(dB) 7.9 24.1 30.2 28.1 29.5 35.2 36.7 38.1 edr(dB) 25.2 33.9 40.3 38.5 40.6 48.4 49.6 51.4 EDT(s) 1.18 1.37 2.25 2.61 2.09 1.82 1.63 1.33 T30(s) 1.02 2.22 2.48 2.77 2.38 1.99 1.71 1.39 corr 0.994 0.995 0.999 1 0.999 1 1 1 T20(s) 1.02 2.04 2.38 2.77 2.27 2 1.68 1.37 corr 0.995 0.989 0.999 1 1 1 1 1 Tc(ms) 106 108 168 202 166 142 130 102 C80(dB) 2.2 1.4 1.1 4.4 2.6 1.4 1.3 0.5 D50(%) 43 46 36 19 19 24 26 34 G(dB) 58.3 59 60 57.4 51.1 48.6 43.7 38.4 S2 R1 F(Hz) 63 125 250 500 1000 2000 4000 8000 snr(dB) 5.7 10.9 25.2 24.8 22.3 27.1 27.6 31 edr(dB) 15.4 24.2 36.6 36 34.4 39.8 40.3 45.5 EDT(s) 0.85 1.69 2.19 2.51 2.23 1.77 1.58 1.25 T30(s) 1 1.97 2.32 2.57 2.33 1.85 1.73 1.36 corr 0.986 0.998 0.999 0.999 1 1 1 1 T20(s) 1 1.97 2.4 2.57 2.38 1.8 1.69 1.36 corr 0.986 0.998 0.997 0.999 1 1 1 1 Tc(ms) 176 147 168 171 156 128 111 88 C80(dB) 1.5 1.4 2.1 2.5 1.3 0.2 0.2 1.6 D50(%) 54 32 16 27 29 31 37 44 G(dB) 60.9 56.8 59 56.7 46.7 43 38 34.1 S2 R2 F(Hz) 63 125 250 500 1000 2000 4000 8000 snr(dB) 7.8 23.3 31.9 31.3 29.9 36.8 39.8 42.8 edr(dB) 22.3 35.5 42.5 41.9 40.7 49.4 52.5 56.4 EDT(s) 1.6 1.57 2.18 2.69 2.13 1.47 1.33 0.99 T30(s) 1.62 1.9 2.29 2.65 2.51 1.8 1.64 1.33 corr 0.998 0.997 0.998 0.999 0.999 0.999 1 1 T20(s) 1.62 1.89 2.18 2.54 2.43 1.76 1. 6 1.31 corr 0.998 0.991 0.999 0.999 0.999 0.999 1 0.999 Tc(ms) 100 115 135 136 104 80 61 40 C80(dB) 4.3 0.4 0.3 1.2 1.6 2.6 4.8 6.9 D50(%) 68 37 41 44 55 55 65 76 G(dB) 58.7 57.2 58.8 54.7 45.4 43.6 41.1 37.4
96 Table 3 1. Continued. S2 R3 F( Hz) 63 125 250 500 1000 2000 4000 8000 snr(dB) 8.5 23.3 32 31.8 32.3 37.8 39 41.9 edr(dB) 25.4 37.3 43.4 43 43.5 51.2 51.9 55.5 EDT(s) 1.15 1.75 2.16 2.42 1.84 1.53 1.38 1.07 T30(s) 1.22 1.8 2.51 2.72 2.46 1.76 1.69 1.35 corr 0.994 0.999 0.999 1 0.999 1 1 1 T20(s) 1.22 1.82 2.53 2.7 2.32 1.72 1.66 1.3 corr 0.994 0.998 0.998 0.999 0.999 0.999 0.999 0.999 Tc(ms) 67 95 119 104 85 81 82 62 C80(dB) 5.3 3.6 1.4 2.4 3.7 3.2 2.7 4.5 D50(%) 77 56 49 60 62 58 52 61 G(dB) 56 56.1 57 55.9 47. 5 44.8 39.7 36 The collected data reflects an interesting behavior of the room regarding the indexes of bass r atio that is the comparison between the G factor in the low and mid frequency bands or (G 125 + G 250) / (G 500 + G 1000) and b ass strength, whi ch is the subtraction between the same two v alues. The calculation gives a b ass ratio of 1.22 which implies a strong primacy of the low frequenci es over the mid highs. This is a clear example on how the acoustical environment in which the music is perform ed could affect its tone color. In this case, the hall amplifies the low components of the spectrum, increasing the brittleness and muffled quality o f the music performed within it The performance space shapes vertical definition with regard to acoustical factors such as balance among the sounds of the various instruments as they reach the audience; the degree to which the tones from the different instruments in the stage enclosure blend together; the relative response of the hall at low, middle, and high frequencies. 139 Finally, it can be said that the Baughman Center colors the sounds events that occur 139 Leo Beranek, (2004), 26.
97 within it, with a significant reduction of the high frequencies above 1000 Hz and a simultaneous enhancement of the low bands below 250 Hz. Another relevant conclusion about the collected data is the influence that a long RT (of more t han 2 seconds) can have on the c larity index and a room's definition. The terms "definition" and "clarity" are synonyms for the same musical quality. They name the degree to whi ch a listener can distinguish sounds in a musical performance. Definition is discernible in two forms: horizontal, related to tones played in succession, and vertical, related to tones played simultaneously. In either case, d efinition results from a series of factor s, both musical and acoustical, pertaining to a certain piece of music, played in a certain way in a certain environment. Horizontal definition is usually defin ed by acousticians as the ratio expressed in decibels of the strength of the early sou nd to that of the reverberant sound. 140 The Baughman Center has the particularity of having the longest RT in the most sensitive frequenc y bands for the human ear. This fact prevents the listener from clearly hearing successive events (horizontal definition) which become su bsumed, as shown in Figure 3 14, into their own reflections. The front door is reminiscent of ancient Gothic doorways and is comprised of three types of wood, maple, cherry and mahogany. Wood is one of the main components of the construc tion (both inside and outside) with a low Noise reduction coefficient (NRC) 141 of 0.1. 140 Leo Beranek, Ibid. 25 26. 141 NRC is an average between the reduction coefficients from 6 octave bands in Hz (250, 500, 1000, 2000). It is a single number rating of the sound absorption coefficients of a material, an average that only includes the coeff icients only in the 250 Hz to 2000 Hz frequency range. That limitation creates discrepancies with the materials that have an identical NRC but have a radically different response to frequencies that are beyond
98 The side doors are constructed of identical materials and follow the same geometric patterns as the repetitive bay windows along the nave. The east Gothic (glass) window of the building covers the back section of the performance area providing a constant view of the lake. The NRC here is also very low 0.15, favoring the existence of poly directional sound reflections from the performance area. Those reflections therefore do not reinforce the sound sources on stage but increase the G factor of the space or the ratio between direct and reflected sound. The building is constructed of three main materials with similar acoustic properties: wood, glass and marble. These mater ials have radical acoustical results. In particular, the exterior walls are made of natural Florida cypress stained t o reflect its surroundings. The roof system is made of tongue and grooved yellow pine, rigid insulation and standing seam copper to refle ct the ancient materials of me dieval cathedrals. All glazing (glass) is comprised of low E, high energy efficient glass in three shades of green, creating an additional pattern in the glazing system. The interior materials are comprised primarily of paint ed structural steel, stained Southern yellow pine tongue in grooved roof planking organized in intricate lacework patterns creating a dramatic vertical space nearly 50 ft. tall. The properties of the ceiling have relevant acoustical consequences such as fo cal points and echoes. Acoustically, concave surfaces can cause problems as they may focus the sound in certain areas while leaving others with too little sound. Thus, vaulted ceilings as seen in Fig 3 1 are only acceptable if the radius of curvatur e is less than half the height of the room ( or the boundaries considered by the coefficient ( < 250 and 2000 <). David Egan, Architectural Acoustics, (1988), 50 53.
99 rather half the distance from peoples' heads to the ceiling) so that the focus center is placed high above the listeners. Similarly, the standing waves can be easily avoided by creating irregular designs on the par allel walls, which would also benefit the overall level of diffusion of the space. In this particular case, the ceiling is higher than necessary for a room that is designed for speech and music; consequently, substantial areas of the room surfaces should b e treated with sound absorbing material in order to control the reverberation time. However, excessive sound absorption could reduce the overall sound level (G factor) in the room. The implementation of ceiling panels to redirect the early reflections coul d be a proper solution to balance out the acoustical inaccuracies space. Diagram of the standing waves and echoes that can be generated by the existence of parallel walls in the design. This graphic represents the undesired focal point as a consequence of the vaulted ceiling. That design feature also favors the uneven distribution of the reflections within the audience area. Figure 3 1. Diagram of acoustic conse quences of parallel walls and vaulted ceilings. The floor material is travertine marble organized in a geometric pattern that reflects the structural logic of the building using three shades of travertine, a device reminiscent of the ancient cathedrals af ter which it was patterned. Building acoustics have been enhanced by faceting the side windows to disperse sound waves, thereby
100 "softening" the overall acoustics. The fixed seating was custom designed and locally fabricated of maple and cherry with deep bl ue fabric seats and backs for com fort, visual interest and color. Figure 3 2 Baughman Center Meditation Pavilion floor plan with exact location of the Sources (S) and Receivers (R) utilized for the impulses and the data collection. Plan by Zona Humburg & Associates Architects.
101 Parameters from Room Acoustics and their A pplication These parameters are organized according to the pitch, time and amplitude domains. This is the section of the study in which the values of room acoustics actually be come music. Pitch Frequency Response and C entroid The frequency response of a room is the measure of the spectrum output of its response to a signal inpu t (impulse). T he centroid is where the center of mass of that spectrum is located, calculated as a w e ighted mean of the frequencies and their co rresponding amplitudes In this section of the c hapter, the fundamental frequency that corresponds to a tone is the only element taken in to account for the calculation of the weighted mean then implemented throu gh orchestration. All the remaining partials of the spectrum are disregarded. In that case, the center of weight between two pitches (tones) at the same amplitude with respective frequencies of 200 Hz and 400 Hz will have a center of 300 Hz as only the fun damentals are used for the calculation. As mentioned before, the ultimate purpose of this study is to elaborate a composition al method through models for orchestration based on architectural acoustic measurements. For that purpose, the instruments are onl y conceived as sine tones (a single x fundamental frequency at an y amplitude with a z duration ) in order to facilitate the implementation of the model when numerous sound sources are present and to control the multiple relations between center of weight dynamics and volume of orchestration. The collected data shows that the spectrum content of any source performed on stage is perceived differently depending on the location of the receiver.
102 Room centroid values obtained from the Baughman Center as an a verage between the centr oids of eight different impulse locations within the room (everything according to the S an d R locations in Table 3 2 ). Room Formants Specific R esonances: Table 3 2 Room formants and centroid values for the different S and R loc ations. 142 Location Formant peak (in Hz) Centroid (in Hz) S1 R1 366 357 S1 R2 279 399 S1 R3 259 315 S2 R1 298 336 S2 R2 298 262 S2 R3 429 310 S3 R1 236 241 S3 R2 214 326 Average 298 319 Orchestrating the Centroid V alues The centroid value obtained from the impulses in the Baughman Center is less than 10 Hz off the 329 Hz or E4. Some of the following examples utilize different values in order to clarify the examples. obtained values from the Baug hman Center. As mentioned before, the centroid is calculated as a we ighted medium. If the spectrum had all frequencies of equal amplitude, then the centroid would be identical as the median fr equency. That situation is rare and is mainly reserved for elec tronically synthesized sounds. A spectrum is characterized by the different amplitude of each of 142 Measurements obtained using PVC, sound analysis and processing software, created by Paul Koonce.
103 its component frequencies, which need to be weighed according ly when making the centroid calculation. This example shows an ideal situation where the median o f this chord 143 is also Figure 3 3 Orchestrated centroid (I). In the next example, despite its apparent complexity, the cen troid equals the median because all the frequencies evaluated are at the same amplitude. Figure 3 4 Orchestrated centroid (II). The calculation gets more complex when each pitch has a different amplitude and d uration; for that case we will use the formula of the weighted mean in order to obtain a precise result Figure 3 5 Weighted mean formula. 143 e concept and are, therefore, interchangeable: group/s of two or more pitches that occur simultaneously.
104 In this case the centroid will increase in the direction of the pit ch with higher amplitude. The high A at 880 Hz is four times louder than the lower one pulling the centroid upwards to 693 Hz, getting to an absolute frequency that is almost a half step above (F4) from the previous example with 2 notes at the same dyamic Figure 3 6 Orchestrated centroid (III), with weighted dynamics. Those orchestrational concepts can also be a pplied in a more dynamic way by following the curve reflected by the values obtained in the actual roo m. We can use orchestration to gradually change (increse or decrease) the centroid. The following excerpt shows how the increasing amplitude of the lower pitch has a direct influence on the value of the centroid, pulling it down. The inverse effect would o ccur if the higher pitch increased in amplitude. Figure 3 7 Weighting dynamics and varying centroid value (I). process of orchestrating the ce ntroid values can be expanded even more. The center frequency of a room can be used to create cho rds that, despite their different configuration, share the same centroid. Those
105 naturally expand the resonance of the room. The f ollowing excerpt shows a series of chords that have the same centroid of 440 Hz (with a maximum margin of error of 10 Hz) as if they were orchestrated using the cent roid of a hypotetical room. Each note has its own dynamic indication for the sake of clarit y and consistency of the example. Figure 3 8 Weighting dynamics and varying centroid value (II). This concept could be further developed by calculating the centroid of a sub group of the components of the r esulting chords in a sort of ch ( ) is a compositional device that focuses on the gravitational point of a note, frequency or array of them; pitches are consirered as a whole along with their amplitudes (dynamics) that influence the location of the center of mass of a spectrum or a chord. In the following example, the initial centroid value is 660 Hz (equivalent to E5). Figure 3 9 Example of a single centroid value (E5 or 659 Hz) through a serie s of complexes at the same dynamic level. The upper and lower pitches ( 880 Hz and 440 Hz respectively) are at the same dynamic and do not alter this value. If we consider these last two pitche s as new
106 centroid values, we could think about them as new cent roids and orchestrate chords around them and, therefore, still keep the overall 660 Hz of the complex unchanged. It is important to clarify that the centroid of the centroid needs to be calculated prior to the calculation of the centroid value of the overa ll co mplex. Otherwise the values do, in fact, change. This approach becomes more intricate when each pitch component is performed at a different dynamic (amplitude). Ea ch pit c h gets weighted by 10 dB per dynamic indication 144 In order to keep the same centr oid value on an E5, the complexes in the following example justify the implementation of micro tunning. The development of the complex is similar but with a different interval configuration. Figure 3 10 Weighting dynamics and varying centroid value (III). T he frequencies that correspond to the center of weight of the complexes respond to the weighting values coming from the amplitude domain. In addition to that it is a remarkable fact that those centers correspond to pitches that are not present in the 144 Accor ding to Tables 1 2 and 1 3
107 complexes, E5 (659 Hz) in the first one and G5 (774 Hz) in the last one. This compositional system facilitates the elaboration of pieces around pitch centers that are, potentially, n ever revealed. Unlike musics in which the center of gr avity establish es its aesthetic primacy as it emerges this model proposes the opposite approach, where the center of weight can, potentially, never surface and still remain the essential foundation of the musical composition. ps of two or more pitches that occur simultaneously and are governed by the weight of their components. The weight of each of the components is calculated considering its frequency, duration, and amplitude (dynamics). These comple xes are not built upon any interval vector or functionality within a key, but rather they are b uilt around a center of weight given by a frequency centroid value obtained from an enclosed space. This single value represents the implicit core and gravitatio nal center of th e complexes built around it, and is based on the centroid value that, in this system, operates like the key center in a tonal environment. A musical composition could be structured upon centers of weight based on any frequency values. These centers of grav ity could be easily implemented, for example, to determine the overall form of a work with a hypotetical A section built upon a centroid value of 200 t i on with a centroid of 500 Hz to finish in a C section com posed around 660 Hz. Unlike tonal music, these centers of weight can be effectively present or just implied. In the first case the pitch corresponding to the center of weight (i.e. 200 Hz) needs to be present as a written G3. I n the second situation, that center is the implied resulting mean between two pitches (i.e. two pitches with frequencies of 100 Hz and 300 Hz respectively that are at the
108 same dynamic level). The latter situation represents a particular ity as it favors the construction of a musical c omposition around pitches that could potentially never appear in the work U nlike tonal music, the center of weight can remain unreveiled throughout a composition and still be its gravitational center. This system justifies the use of microtonality as a na tural consequence of the numerical calculations between frequency values that do not necessarily correspond to tempered pitches. The method also e stablishes a direct interdependence between the chosen instrumentation and the center of weight of the composi tion. The extreme (lowest and highest) pitches of a complex fix the boun daries of its center weight In the previous example, the pitches that correspond to 100 Hz and 300 Hz determine the low and high boudaries for the center. In the same line of thought, the instrumentation chosen for a composition has a direct influence on its center of weight. The initial limitation is given by the extreme registers of the performing forces. In a piece composed for flute duo (instruments with a B foot ) for example, the center of weight would ne ver go above 23 46 Hz or below 246 Hz. A piece for one or two pianos would necessarily be framed between 27.5 Hz and 4186 Hz In a tonal composition, the instrumentation does not have an influence on the tonal scheme of the work, a piece for any instrumentation could potentially go from any key to any other key. Thinking in terms of center of weight, the relations between instrumentation and center of weight is radically different as the instrumentation is the boudary for the cent ro id. A composition for two flutes can not have a center weight at a frequency that exceeds the low and high extreme registers of the flute itself. The choice of instrumentation is, then, the first limitation to the mobility of the centroid.
109 At first sight, it can be seen that symmetric complexes (i.e. diminished and augmented) already have their center of weight in their axis of symmetry. That concept is initially valid if we do not consider the amplitudes of each of the components or if the amplitudes are all equal. Symmetric complexes with components of unequal amplitude have a centroid that does not correspond to their axis of symmetry. In one of the previous examples each of the components of a complex were ing in this way the centroid value. Another important weighting element is the duration of each of the components of a complex. We not only need to consider amplitude, but also the duration and its influence on the centroid. If we apply the same weighting principle to the durations, a longer component becomes more influential in the value of the centro id. A complex with two pitches at 220 Hz and 660 Hz with the same duration and amplitude has a centroid value of 440 Hz or A4. If the higher pitch is louder t ha n the lowest by three dynamic indications ( mp to ff or 30 dB) the resulting centroid value is then raised to 455 Hz. Weighting further, if that high pitch at 660 Hz is not only 30 dB louder but also two times longer than the lowest pitch at 220 Hz, the centroid value is pulled up even more to 485 Hz (8 Hz flat from B4) as a natural result of the weights (amplitude and duration) added to the component. The follow ing example shows a musical application of the duration of an event as a weighting factor, whe Figure 3 11 Duration as a weighting factor.
110 The durations are weighted in a similar way as the amplitude, thereby establishing a parall el with music al notation. In this way amplitude and duration are equally influential and can be utilized compositionally to counter balance each other. The following chart shows how the musical durations are considered in the proposed model, in relation to their weighted values. The system simply maintains the duration al relation between figures, which are multiplied by 10 in order to balance the way dB values are paralleled with dynamics. Table 3 3 Note rhythmic values and their respective weight. No te value Duration (in beats) Weighting value whole note 4 40 half note 2 20 quarter note 1 10 eight note 5 sixteenth note 2.5 Time I R everberation RT The reverberation (R60) of a room is the amount of time that it takes for a sound source t o become inaudible after that source has been removed. Generally, in acoustics, a sound is considered inaudible when its amplitude has decreased by 60 d B from the orig inal. In some cases, the high level of ambience noise reduces the measuring range to less than 60 dB. In those situations, the RT is measured using only the initial decay time in a narrower amplitude range of 15 dB or 30 dB The following example shows an ideal case A where the sound decay of 60 d B can be measured
111 case B in which the 60 d B can only be projected using the slope of the early decay time Figure 3 12 Graphic representation of a 60 dB decay envelope. Wallace Sabine conducted the pioneering research concerning room reverberation and the amplitude of sound. When working at Harvard, he determined that there was a direct relation between the volume of the room and the reverberation time: T = 0.05 Volume / A (absorption coefficient in sabins) 145 Other acoustical properties that can influenc e this criterion are 1) the shape and proportion of the room ; 2) the absorption coefficients of materials in interior walls, floor, seating area and ceiling; and 3) the seating capacity of the hall the latter being an important factor in making measuremen ts for a design. The following list further explicates these three concepts with respect to the Baughman Center. Shape and proportion: the Baughman Center has a high dome that reinforces the G factor of the room by increasing the amount of reflections. Co nversely, the parallel walls facilitate the existence of standing waves and/or echoes. T he rectangular shape is not the most recommended by acousticians Moreover, the sound rein forcement structure of the room does not reduce the ITDG as it would be desire d for musical performance. Absorption coefficients of the interior materials: the interior walls, ceiling and floor are mainly wood and glass, highly reverberant in most frequency bands. High 145 M. David Egan, Architectural acoustics 63.
112 ceiling s and parallel walls with reflective materials are syno nym s with long reverberation time and low intelligibility. In addition to that, they do not distri bute energy evenly throughout the listening space. The seating capacity of the hall : the audience area is limited to no more than 100 members. Th is can potentially reduce the RT, but by no means control it. T he Baughman Center has an average reverberation time (between frequency bands and different locations of source and receiver) of 2 seconds. If we take the bands where the human ear is most sen sitive (250 Hz 4000 Hz), the averaged result is slightly longer 2.2 seconds. This is not a minor fact, especially if we consider that when a musician per forms, the speed at which the performer plays has a vital relationship with the acoustics of the hall In particular, the speed at which successive tone s follow one another interacts with reverberation time and thus shapes what the audience hears. The faster the musicians play, the more notes pile up under the same RT envelope Figure 3 13 Graphi c of two successive tones performed in rooms with different RT envelopes. W h ere the speed of successive notes as represented in Figure 3 13 increase s the second tone would move to the left because the time between the two notes would become shorter. If t he speed is high enough the second tone will fall below the reverberation envelope of the first tone E ven for a RT of less than a second, the attack
113 of the second tone would become inaudible. If RT = 0, the notes will stand out clearly whether played qui ckly or slow ly just as with music outdoors. 146 Orchestrating the R everberation In this musical example, we can see the performed sound and the actual result with a 2.2 second reverberation time that corresponds to the values obtained from the Baughman Cente r Figure 3 14 Orchestrated RT (I). The tempo marking of (quarter note = 60) facilitates the translation between musical values and time The RT value obtained from the Baughman Center can now be embedded in the music through the orchestrat ion. Figure 3 15 Orchestrated RT (II). 146 Leo Beranek, Ibid. 24.
114 When two chords are performed in succession, meaning the attack of the second one within the decay tail of the first one, the result is of particular interest. The excerpt in Figure 3 15 shows a standard accompanim ent pattern for string players and its actual aural result when performed in a room with a RT of 2.2 seconds. The second chord of the pattern is performed 1 second after the first one, in the middle of its decay tail where the initial chord still sounds b ut 2 times softer. Figure 3 16 Orchestrated RT (III). The musical consequences of that have an impact upon rhythm, dynamics, harmony and texture : Rhythm: its definition gets blurred. Succesive attacks are atenuated by the prolonged ring of the previous ones. Dynamics: the intensity gets increased. The first attack is mf the following one is mf plus the reverberation of the previous one. The succesive increments are particularly perceivable when the same chord is played several times wit hin the total duration of the reverberation time. The phenomenon is explained clearly by Blesser 147 when he shows in a diagram how, i n very reverberant spaces, the reverberant sound can overpower the direct one. Harmony: the harmonic changes can become uncl ear b ecause of the fact that in some situations two different chords will ring simultaneously. This quality can be used as a compositional device, where harmony could be the result of the overlapping reverberation of single notes. 147 Barry Blesser and Linda Ruth Salter, 142 143.
115 Texture: the written p assage shows a monorhythmic polyphonic texture, which becomes polyrhythmic polyphonic due to the RT. The excerpt in Figure 3 16 the T60 of 2.2 second facilitates the aural illusion of an 8 note cluster on the second beat of t he measure, which is somethin g, presumably, never intended by the composer. The excerpt in Figure 3 17 is a simultaneous complex that is performed at the same dynamic and that is equally spread along all the frequency bands measured. As those bands decay at different times, the resul t becomes evocative of the space. The 329 Hz or E5 combining data from both the time and frequency domains. The tempo is MM = 60. The whole passag e recreates in musical notation the acou stical data taken from the Baughman Center This exa mple could be developed further if we utilize different reverberation times per frequency band Figure 3 17 Orchestrated RT per frequency band (I). The following example considers not only the cen troid value obtained from the room, but also the RT per frequency band. Now the decay times respond not to the average RT value, but to the specific values as shown in the initial chart from this Chapter (Figure 3.1)
116 Figure 3 18 Orchestrated RT p er frequency band (II). Due to the particularities of the collected data and for orchestration purposes, the RT values are averages from all impulses in three main band subgroups (63 to 125, 250 to 1000, 2000 to 8000). The values per group are (in seconds) 1.7, 2.5 and 1.65 respectively. The orchestration example was composed using the s ame centroid value with different pitch distribut ion, and independent RT per band. A ll the values are according to the measurements obtained from the Baughman Center. Time I I Rhythm of the Early Reflections The rhythmic implication of the RT is only limited to duration and enveloping. Rhythm, as the succession of sonic events in time, is not taken into account. Even ic identity e extracted from the temporal succession of the early reflections of an impulse. The early reflections were chosen for the proposed model because they unequivocally define the dimensions of the enclosed space As t hey travel a longer path, the amount of ti me it takes the first reflected sounds to reach our ears give us clues as to the size and nature of the listening environment.
117 For the proposed model, the impulse S1 R1 was utilized. The Acoustic Tools software helped to measure the time between the first 10 reflections after the direct source. Those reflections were separated by the following times (in milliseconds): Table 3 4. Delay times of the first 10 early reflections. Reflection 1 2 3 4 5 6 7 8 9 10 Delay time (ms) 3 4 5 8 9 12 22 24 25 29 These minute time delays are initially irrelevant for the human ear, which cannot differentiate events happening between the first 25 ms. For echoes delayed less than 25 ms., there is almost complete subjective integration of signal and echo. Although they are not heard as discrete echoes, their energy is not lost and contributes materially to the apparent level, quality and intelligibility of the sound. In fact, the ear performs this integration of the direct and reflected sound Reflected energy arriving at th e ear within 25 ms. is integrated with the direct sound and is perceived as part of the direct sound as opposed to reverberant sound. These early reflections increase the loudness of the sound 148 Despite that fact, the proportions and placement in time of t he attacks of each of the reflections can easily become a relevant compositional consideration. First, in order to fully extract the rhythm of the reflections, we need to remove the initial time reference to the direct sound, starting from 1. That will giv e us a new set of values that are completely isolated from their source and original unit. Table 3 5. Values of the first 10 early reflections. Reflection 1 2 3 4 5 6 7 8 9 10 Value 1 2 3 6 7 10 20 22 23 27 148 F. Alton Everest (2008),
118 The sequence of values corresponds to the arc hitectural design of the Baughman Center and the location of the S1 R1 impulse. The first five values come from the closer side walls and the stage (in relation to the location S1, see Figure 3 2), the values after those are increasingly more separated cor responding to the reflections coming from the this particular room. Orchest rating the Rhythm of the Early R eflections The translation of the obtained values into rhythm can be achieved in different ways. If the values originally come from delay times in ms., the musical translation of them could simply be v 1000. In that way, w e keep the time proportion between events as well as their temporal origin. Thus, this 10 values 27 seconds rhythmic pattern is acoustically related to the studied room and at the same time musically significant. Figure 3 19. Rhythmic structure ext racted from the early reflections. Another approach is to only keep the temporal proportions between the obtained values without any reference to the original unit of time. The composer can now freely choose a time unit with the corresponding dissimilar r esult; the ten events (corresponding to the reflections) are placed in a varying time spam. Those events occur at the same ratio as the early reflections but are expanded or compressed
119 ly decided provided that they do not compromise the placement of the attacks. The excerpt on Figure 3 20 is built upon the pitch classes of an ascending chromatic scale (F# to D#) that are organized in time according to the proportions extracted from the early reflections of the Baughman Center. Figure 3 20. Orchestration of the rhythmic structure extracted from the early reflections (I). Figure 3 21. Orchestration of the rhythmic structure extracted from the early reflections (II). The integral app lication of the rhythmic elements results in a musical example that takes into account the placement of the attacks of the sonic events (coming from the early reflections) as well as their resonances, which come from the RT value.
120 Figure 3 22. Orchestrat ion of the rhythmic structure extracted from the early reflections Amplitude (Dynamics) the G factor This criterion differentiates the amount of sound pressure coming directly from the source from the one that is added by the room. This addition of sound pressure amount of sound reflections that it generates. 149 An impulse response can be used to determine the G factor of a room in several ways: The level of the quiet reflects by itself the lowest threshold of ambience noise in a ro om (a room with a quiet of 35 dB is louder one with a level of 30 dB ). This criterion is very dynamic and can ref lect di fferences within the same space. I n t he University Auditorium at UF, for example, the value increases by a 30% when the decibel meter is located towards the back of the room. This level also changes dep ending on the acoustic calendar. A t night it i s considerably lower considering the reduced traffic and amount of people in the area. 149 Leo Beranek, Ibid. 509.
121 An impulse response of a source in an anechoic room (SPL free) can be compared to one of the same source in the room (SPL hall), the difference between them is the lou dness level, or G factor. The equation calculates the difference between sound pressure level in the hall and in a free field (anechoic). Figure 3 23 G factor calculation. The following graphic shows how reverberation builds up the level of l oudness to the listener who perceives direct sound + reflections. Figure 3 24 Impact of the G factor in the listener. Sound absorption coefficients have an inverse relation with the loudness criteria. If a room has a lo w absorption coefficient in the surfaces of interior walls, ceiling and floor, the loudness will increase. In addition to that, mitigating undesired sounds that increase the level ambience can reduce the loudness level as well. Orchestrating the G Factor The G level is clo sely related to the concept of definition or c larity of a room. The terms definition and clarity are synonyms for the same musical quality. They name the degree to which a listener can distinguish sounds in a musical performance. Defi nition is discernible in two forms: horizontal related to tones played in succession,
122 and vertical related to tones played simultaneously. In either case, definition results from a complex of factor s, both musical and acoustical, a certain piece of music played in a certain way, in a certain environment. Horizontal definition is usually defined by acousticians as the ratio expressed in decibels of the strength of the early sound to that of the reverberant sound. Thus, if the definition, in decibels, is a positive quantity, the early sound dominates. If negative, the strength of the reverberant sound dominates. If zero, they are alike. 150 The implementation of the G factor in the proposed model is the ratio between the G values of certain frequencies. Speci fically, the mid low bands 63, 125, 250 and 500 divided by the mid high ones 1000, 2000, 4000 and 8000. That operation returns a ratio the o verall G factor. This choice was made spectral color, which in this particular case is clearly oriented towards the mid low bands. Figure 3 25 Combinatorial example (I). 150 Ibid. 25 26.
123 The previous example shows an application of the preceding pr inciple. The problem here is that the equivalencies between amplitudes and dynamic indications established initially for the model are structured in doubling values ( p is twice as loud as pp or 10 dB more) and the ratio shown by the room measurements are l ess than 2. In this case, and to make the translation musically interesting, the G value between mid lows and mid high is rounded to 2, so the frequencies below 500 Hz are orchestrated twice as loud in relation to the high ones. Figure 3 26 Combinatoria l example (II). Another feasible and musically relevant application would be to consider the G factor more strictly related to the direct sound, sugge sting through the orchestration the idea of the reflected sound ending somehow embedded in the direct sour ce. In order to
124 achieve that effect, a poly rhythmic and monophonic texture is necessary as well as an accurate dynamic relation between direct source and reflected one. The example on Figure 3 26 shows one of the three possible scenarios. The excerpt orch estrates a negative G value ( louder direct source doubles the amplitude of the reverberated sound). Despite that fact, the G factor could be positive (the reverberated sound is louder ), or equal 0 ( when dire ct and reverberated are equal). That data could be easily translated into the same excerpt by adapting the dynamic indications. This particular example embeds a n unambiguous aural representation of a direct sound source and its posterior reflection. The tempo is MM = 60 and it could be a potential fragm ent of a composition for two pianos, or string ensemble. The ITDG evoked (between the direct source and the first reflection) is of 1 and 1.5 seconds respectively; that value rarely occurs as an actual acoustical measurement, but it is implemented to sugge st the existence of a reflection within the composition. If actual values of ITDG (normally around 80 ms in concert halls) were used, every orchestration attempt to translate them into a musical passage would be unproductive as the human ear unifies musica l events separated by 80 ms or less. In addition to that, it would also be impractical to provide the performers with 80 ms rests. Regarding ITDG, the musical relevance prevails over the dogmatic numerical accuracy. We naturally arrive at a creative appli cation of the model, in which the acoustical data is no longer evident but still present. The resonances are orchestrated with a decreasing G factor also taking advantage of the idiomatic orchestration resources of the string ensemble; the evocation of the decreasing reflections is also achieved by the
125 non vibrato effect, which helps differentiate them from the original source. The b owings also help to reinforce the idea of what is direct source and what is not, the down bows are utilized to give more empha sis to the direct sources and up bows to suggest the milder attack of the resonances. Color contrasts are also reinforced with the sul ponticello effect. Figure 3 27. Combinatorial orchestration example of the G factor for string ensemble. The initial centroid value example (as seen in Figure 3 3) developed into a musical excerpt of 10 seconds of duration that could continue to evolve in complexity and
126 musical interest with the implementation of other simultaneous complexes and volumes of orches tration.
127 CHAPTER 4 APPLICATION OF THE M ODEL Personal M otivations and A pp lication of the Proposed Model in a Musical C omposition As a composer of music, I am interested in developing creative processes that evolve from concepts inherent in sound itself. The idea of going from sound to sound through the process of music composition is one of the main goals of this study. Alvin Lucier, Tristan Murail Jean Claude Risset and Olivier Messiaen, are just a few examples of artists that, deliberately or not, had a similar intent In the case of the proposed method, it is my desir e to continue a path that Iannis Xenakis did not explore He explicitly mentions in his Formalized M usic that he did not take into account elements from architectural ac oustics i nto his compositions. One can appreciate his matrixes and probability studies and their application into music; his need of an initial sketching stage of graphic notation in order not to interrupt the flow of ideas is also remarkable. Despite those facts, I believe that he left one path uncharted ; all the initial inputs that allow the generation of stochastic studies and ma trices are human decisions, those numbers do not come from any mind That initial input data was that wh ich I wanted to connect with sound and more particularly, with the properties of sou nd occurring in enclosed spaces. By doing this, I intend to elaborate a method that is based on vectors and numbers that are not a t rather specific values taken from an acoustical measurement s In the initial stage of the study I had to determine which parameters of room acoustics could be musically relevant. The initial broad classification was done according to the three main do mains of music: time, pitch, and dynamics, that had a
128 direct correspondence with fundamental concepts of room acoustics: time > reverberation time (RT), pitch > frequency and dynamics > amplitude. The se analogies clarified the scenario and supported th e meaningfulness of the method. Another concept that influenced this study is the concept of proto and meta instruments proposed by Barry Blesser and Linda Salter in Spaces speak, are you listening? This concept is based on the interdependence of the direc t vibrations of musical instruments and the indirect ones provided by the space through which these vibrations are propagated Going further, this idea reinterprets the concept the sound of a musical instrument A tone of any musical ins trument is provided by its direct vibration and also by the vibrations provided by the room If we think in those terms, the daily practice of a musical instrument becomes an incomplete activity, as there is a decisive spectral component to the tone qualit y that is usually beyond the control of the instrumentalist C onsequentially, the pra ctice of an instrument becomes quite complex : the performer must address the specifi c technical difficulties of her instrument, and work on tone quality considering (at th e same time) the added acoustical features provided by the room. Practicing becomes even more complicated if it is done in an open field. In that particular case (like marching bands, for example) the instrument remains as an incomplete proto instrument a nd the performer must adapt herself accordingly. Following Blesser and Salter, the room is then, a passive component of any it gets excited. Music is composed for instruments that, when perfo rmed indoors, necessarily embody the acoustical qualities of the space. If we accept this fact, then the idea of composing
129 music that is not indifferent to room acoustics becomes a natural line of thinking. An additional motivation f or this study was the fact that architectural acoustics and music share similar concern s with respect to sound and its components. Moreover, the two activities are almost identical processes if we consider them before their respective materializations. Th e acoustical design of a hall, for example, is an actual orchestration, were the architect decides which frequencie s to reinforce and for how long. I n the reverse example, the composer is the acous tician of a piece of music that through the orchestration, h andles sound sources and their reverberations. It is not by mere chance that Xenakis and Libeskind had such a similar system through which their work was conceived and constructed. As mentioned before, it is important to reiterate that one of the purposes of this study is to complete the circle that starts by analyzing properties of sound. That whole progression ends with a sonic manifestation : a piece of music. That piece is the final section of the study and represents a creative application of the compos itional model. The piece is entitled Colors and is written for full symphonic orchestra. As expressed in the program notes, the work is an exploration of the endless orchestration mixtures offered by the ensemble and their unique timbres Resonances are obtained through orchestration in numerous places throughout the work. In the excerpt, on Figure 4 1 the brass section performs a resonance to the abrupt ending of the sustained trill in the woodwinds. The chosen pitches as well as the even volume of the o rchestration help achieve the desired effect. In the excerpt on Figure 4 2, the resonance is orchestrated as a selection of partials from a fundamental tone that in the example appears as the
130 lowest note. In addition to that, the pitches that represent the resonant frequencies (implying a room response) have a different decay time in a similar fashion as suggested in the examples of the proposed model. Figure 4 1 Example of orchestrated resonance s in Colors measures 10 12. A similar phenomenon occurs in measures 47 48 where the tutti operates as an impulse that bursts on beat three of measure 47, this being a polyrhythmical pattern in the strings and its consequent room response. In this case, the room decay is orchestrated as a gradual fade out follow ing the acoustically appropriate spectral envelope, where the high frequencies decay faster than the lower ones. This idea is
131 also treated more freely in the manner of where the lower frequencies decay first. That thought results in a particular design interest as it represents a palindromic construction of resonances around a fundamental pitch, which produces illusory resonances above and below it. There is a specific example of this idea in m. 92 where the resonance is orchestrated below the high G6 harmonic in the solo violin. The section of the work that develops after that moment combines illusory resonances with the drama of the increasing reflections. Figure 4 2 Example of orchestrated resonance s from measures 97 101 In th is piece, the application of the model is approached with the utmost creativity, favoring the flexibility of the implementation of the obtained data. In the previous example, the pitch content is not related to the centroid value obtained from the room, bu t rather to the harmonic series of a given fundamental tone. Despite that, the RT times suggest the ones obtained from the measurements. The volume of orchestration in the work was arrived at by applying the model. Measures 110 113 are constructed upon th e idea of a constantly increasing G value which is made manifest through a monophonic polyrhythmic texture with an increasing orchestrational volume. The successive unison entrances on a D4 evoke non decaying reflections that initially recall
132 the loudness obtained from the Baughman Center That texture gradually transforms into an illusory aural space where the reflections become louder than the sound source (as if the room could become more and more reflective). The metaphoric approach of this particular s ection puts forward a personal aesthetic vision in the application of the model; the drama comes from the utilization of the G value, not from the value itself. My main interest here was to create the illusion of a morphing space that is only revealed by i Every enclosed space offers a unique acoustical configuration, thus, every piece of music composed according to the application of this model and based on those distinctive qualiti es, would have its own identity and therefore "sound" different. That being so it must be stated that not all rooms have the same musical potential. Paradoxically, the spaces that have the most acoustical "imperfections" are the most suitable for the appl ication of the proposed model. A room with a "perfect" acoustical design has the following general properties: Is w ell balanced (every audience member perceives the sources at the same level). H as a natural frequency response that enhances the inherent qu alities of the sources without adding o r subtracting anything to them. Its reverberation time is between 1.5 and 2 seconds and is considerably even all throughout the frequency spectrum (this is also r elated to the room's response). Its G factor is negati ve (meaning that the reflections are always softer than the direct source). Its LF coefficient is well distributed between left and right, the ITDG for music is below 80 ms (the sound source coming from the pe rformance area gets reinforced) T he design avoi ds parallel walls and triangular domes that could add echoes and uneven distribution of the reflections.
133 Th e se cha racteristics are examples of what a reasonable acoustical design should fulfill, and many other properties could be added to the preceding li st. Figure 4 3. Polyrhythmic monophonic texture in Colors as an example of an illusory increment of the G value. On the other hand, an "imperfect" acou stical design can have the following general characteristics: Is uneven meaning that its frequen cy response is too colorful (reinforces certain frequency bands over others). Has an extreme RT value of more than 2 .2 seconds or less than 1 second (even with different times within the same space). Its G factor could be positive (where the reflections source). Has an unevenly distributed LF index between left and right.
134 Its design favors the existence of echoes or uneven reflections (due to parallel walls or vaulted ceilings, for example). I previously mentioned that acous tically imperfect rooms better suit the application of the model because of the variety of the data they offer, they are less uniform and varied, all qualities that are acoustically questionable but musically desirable The Baughman Center is acoustically source for this study. Room responses have the particularity of having similar amplitude envelopes. Earlier or later, they inevitably decay; like the piano, a room response offers very little control of it s amplitude envelope after it is e xcited. That is the reason why, in a compositional method that uses archit ectural acoustics as its basis, having the option of an increasi ng G factor becomes so relevant. It allows the composer to consider the indispensabl e idea of a crescendo that otherwise would be completely negated. A model that does not offer this option would embed its own restrictions in the music. Data Collection D etails The acoustical data from the Baughman Center included in all the tables in cha pter III were obtained using different types of impulses (sources or S): balloons, dodecaphonic speakers, and frequency sweeps. Those sources were recorded (receivers or R) using an M Audio Microtrack II portable recorder with an E lectr et stereo microphone The recorded sound files were processed and analyzed using the software Ac oustic Tools and WinMLS. The latter software was specifically employed to generate maximum length sequence signal sweeps, which were amplified by a JBL EON15 G2 Port able Powered Lo udspeaker System and captured by an Earthworks microphone
135 This study is far from being exhaustive, engendering a great deal of potential for further development. The more I research the s ubject, the more questions arise. The following topics are tangential curiosities. They could become objects of study of the relations between music and architecture and, more specifically, about musical composition based on acoustical measurements: The an alysis of the music of Giacinto Scelsi in terms of G factor values and room volume. Architectural designs based on values taken from musical scores This topic suggests the reverse approach offered in this study, where the musical score is strictly consid ered as an architectural plan. Preludes Algorithmic composition around a given centroid value. Elaboration of a system that can generate music weighted around room formants. Centroid and music cognition. Research about the aural perception of centers of weight in music. Do we perceive the centroid value as the gravity center of complexes constructed around it? Inclusion of the meta instrument in the daily practice for instrumentalists. Study o f the necessary acoustical qualities for a reliable practice room. their acoustical properties and translate them into music. Use of the system of weights as a tool to analyze works of the so called total serialism Similarities and differences between the proposed model and the techniques proposed by the total serialism.
136 APPENDIX A APPLICATION OF THE M ODEL IN A MUSIC COMP OSITION COLORS FOR ORCHESTRA
178 APPENDIX B APPLICATION OF THE M ODEL IN A MUSIC COMP OSITION ETUDE FOR STRING QUARTET
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183 BIOGRAPHICAL SKETCH Jorge Elias Variego is a clarinetist and composer born in Rosario, Argentina, in clarinet performance. He received his PhD from the University of Florida in the spring of 2011. He has performed as soloist with the most renowned orchestras in Argentina and his works have been performed throughout the world. Among other distinctions, he has de Msica Argentina 2007, ASCAP Award, Society of Composers Students Commission Regional Competition. He has received prestigious scholarships from Fondo Nacional de las Artes (Argentina), Antorchas Foundation (Argentina), Fulbright Commission (Argentina) Carnegie Mellon University (USA), Universidad de Santiago de Compostela (Spain), Pi Kappa Lambda Honors Music Society, among others. He has been a resident artist at the Pittsburgh Center for th e A rts for the last five years