Chen Yi: Design as Narration, Narrative as Totem

T    he quest for music that elevates the mind is an essential part of the mainstream of Chinese music. Over 8,000 years of the Chinese musical civilization, all types of influences have been experienced... These include the adoption and jettisoning—of various temperament systems, of a wide range of formal designs, and of wide ranges of acoustics. Global consumerism, however, is one influenc it had never encountered [before the 20^th Century.”
  —  Sin-Yan Shen, Chinese Music in the 20th Century, p.189.

T    o my mind, the origin of music should not be sought in linguistic communication. Of course, the drum and song have long been carriers of linguistic meaning. But there is no convincing theory of music as language. The attempts that have been made in that direction are no more than camouflages for a kind of naturalism or the most mundane kind of pedantry. The musical message has no meaning, even if one artificially assigns a (necessarily rudimentary) signification to certain sounds, a move that is almost always associated with a hierarchical discourse.”
  —  Jacques Attali, Noise, p. 25.

I    t is true that I use compositional techniques from all of the cultures I have experienced and all the teachers I have learned from—Russian, French, German, Chinese, American. I do adapt Western classical techniques to express Chinese [sensibilities] and so on. ‘From the Path of Beauty’ does take Chinese folk songs as its primary material, but it [secondarily/indirectly] generates ideas that are universal. Music is, I think, a universal language.”
  —  Chen Yi, 17-JAN-2009, Friends of Chamber Music Concert.

I suppose that the main controversy has perennially been not whether music may evoke substantially the same emotions in all listeners/performers, but rather whether music can express propositions or specific questions or perform other representational linguistic functions that are associated with other textual or oral languages. In other words, the idea that musical gestures and harmonic colors and rhythmic textures can serve as signs and symbols that carry subjective emotional meaning is not tremendously controversial. But whether it can do more than this still is a source of disagreement.



T he performance at Friends of Chamber Music last night by Shanghai Quartet and Chanticleer, of Chen Yi’s ‘From the Path of Beauty’ for mixed ensemble was a good opportunity to revisit that topic. The performance was flawless, extraordinarily beautiful. The balance between the voices and the strings (who often were played with mutes on) was exceptionally good.


T he 7 movements of this piece include one a capella portion, one movement for string quartet alone, and the remaining 5 with the combined, mixed ensemble. The vocal parts have accompanying Chinese poems; however, the syllables did not signify anything to us who do not speak Chinese, and it was not clear whether the parts were wordless vocalizes or actual texts of poems.

• The Bronze Taotie (Shang Dynasty, 1600 – 1100 BCE; choir)
  
• The Dancing Ink (Tang Dynasty, 618 – 907 CE)
  
• The Ancient Totems
  
• The Rhymed Poems (Song Dynasty, 960 – 1279 CE; str qt)
  
• The Clay Figurines (Han Dynasty, 206 BCE – 220 CE)
  
• The Secluded Melody (Six Dynasties, 497 – 590 CE)
  
• The Village Band

T he punctuation and textures of the vocal parts and the string parts closely resembled each other through much of the work—pizzicatos, glissandos, melismata, and so on. On account of this, the ‘discursive’ quality of the interaction between the voices and the instruments was even more prominent than it otherwise would have been.

T he third movement is entitle ‘Totems’, but, to me, the entire piece felt ‘totemic’, mantra-like, invocative, monumental. Totems, after all, are mnemonics—conveniences for social identification, invocation, and attachment. ‘From the Path of Beauty’ is like an elaborate ‘pole’ of totemic figures, stacked one after the other. There is no hierarchy; no particular ‘arc’ to the 7 movements—with the exception that the seventh and last one does feel like it is approaching a conclusion. The rest appear, one by one, in a sequence that seems like it is pretty malleable/flexible; each leaves its impression, conveys what it has to say, be it humorous or meditative or celebratory, and gives way to the next.

T he concept of mantra goes back to the pre-Vedic and early Buddhist traditions and to the primitive cults of magic, animism—and totemism. It has since been a continuing element one way or another in the religious traditions of the world and traces of it pervade to this day among the most modern of them. Practitioners of those believe that mantras have power over the deity and can make it confer the desired benefit. Is this piece, commissioned by Chanticleer, informed by a commission specification that aims at historical survey [of Chinese musical/poetical idioms], or is it instead Chen Yi’s invention as Chinese diaspora/teacher?

G urmantras, mantras imparted by gurus or teachers, are made meaningful by be whispered into the ear of the disciple by the guru. We disciples [in the audience; on the stage] repeat Chen Yi’s gurmantra … to achieve enlightenment.

T here are parts of ‘From the Path of Beauty’ that are luxuriant (esp. mvt. #6), but there are other parts that are arid, desolate, ascetic. In the Buddhist form of asceticism, there is no metaphysical dualism of God and the world, or of soul and the body. Phenomenal existence is viewed as characterized by suffering, impermanence and not-self. The aim of ascetic culture is to go beyond this sphere of conditioned phenomena. The keynote of the ascetic culture of Chen Yi is moderation; self-mortification is rejected altogether. Singers, string players, and audience members can simultaneously appreciate ascetic beauty and enjoyment.

T he community created in the space of the piece is not one of any essentialist definition of ‘chineseness’—taking Chen Yi’s introductory words about ‘universal language’ and assimilating compositional gestures from all over the world, I sense that this is as much an embodiment of diaspora culture as it is a celebration of Chinese music history.

S till, the practice of composing is presently something much more counter-cultural than other public acts, and the community created in Chen Yi’s composition is a quintessential exposition of cultural inversions, turning taboos into totems. Conspiratorial counter-tenors!

I t raises up the space of the ensemble to the realization of the China of the dream of racial and gender and class inclusion.

S yntactic music, notation-centric, rich with norms, expectations, surprises, tensions and resolutions, repetitions. Middleton (p.145ff) once wrote about this, as did Gilles Deleuze in Difference and Repetition (1994). Difference is the distance moved from a repeat and a repeat being the smallest difference. Difference is quantitative and qualitative — how far different and what type of difference.

C hen Yi’s movements are variations on classical form, revealing highly individual choices of tonal planning, melodic repetition, digression and reminiscence. Her techniques enable her to undertake compositional strategies quite different from the models of her contemporaries. Her construction is taut, and her subjects exhibit sharp polarities. A detailed account of digressions and reminiscences at different hierarchical levels in composing can be found in Lewin (1986).

S uch gestures of ‘poeticizing sound’ inflect the tonal design and expressive action of the songs—her primary folk song materials—and their repetitions. The ‘modus operandi’ of disjunct keys by side-slipping to a remote tonal area in the songs—the ambiguities of the modulations up a half-step; the choral dissonances—is not only replayed in the 7 movements of the suite, but is recontextualized within the paradigm of its overall form. It appears as a series of tonal digressions that delays the subsequent subjects on the large-scale. The tonal path shifts unexpectedly and repeatedly, via interrupted cadences, into other keys. The expressive effect is one of ‘remoteness’ and ‘out-of-timeness’—very apropos in view of the Dynasties theme. This is of course not unknown in Western canon—for example in Schubert.

T    he year the mercury drops to the bottom line.
How I miss that year—
My wool jacket at the pawnshop,
Doors and windows sealed tight by a snowstorm.
Thenceforth even good friends were like
Mercury, dropping
To zero.”
  —  Zhen Chouyu, Thinking of a Friend at Year’s End.

• Chen Yi page at UMKC Conservatory
  
• Chanticleer website
  
• Shanghai Quartet website
  
• Society for Asian Music
  
• Central Conservatory of Music, Beijing
  
• CUHK Music Dept
  
• Modern Chinese Poetry Research Institute
  
• Journal of Music in China. Editor: Zhou Qinru. Published twice a year in Los Angeles. website (http://www.MusicInChina.org) seems to be defunct.
  
• de Andrade M. Poesias completas. Univ de São Paulo, 1987. (totemic poetics as a feature of writings by diasporic/displaced people)
  
• Attali J. Noise: The Political Economy of Music. Univ Minnesota, 1985.
  
• Cobussen M. Deconstruction in Music.
  
• Deleuze G. Difference and Repetition. Columbia Univ, 1995.
  
• Freud S. Totem & Taboo. Routledge, 2001 (1913).
  
• Jiayan Mi. Self-fashioning And Reflexive Modernity In Modern Chinese Poetry, 1919-1949. Mellen, 2004.
  
• Lerdahl F. Cognitive constraints on compositional systems. Contemp Mus Rev 1992; 6:97-121.
  
• Lewin D. Studies in Music with Text. Oxford Univ, 2006.
  
• Lewin D. Music, theory, phenomenology, and modes of perception. Music Perception 1986; 3:327-92.
  
• Liu C-C. A Critical History of New Music in China. Chinese Univ, 2009.
  
• Rees H, ed. Lives in Chinese Music. Univ Illinois, 2009.
  
• Sin-Yan Shen. Chinese Music in the 20th Century. Chinese Music Society, 2001.
  
• Yeh M, ed. & tr. Anthology of Modern Chinese Poetry. Yale, 1994.

Dissonance and ‘Frisson’: Harmonic Spectral Complexity and Rate of Change of Complexity

I   t does feel like the brighter vowels are placed in chords that will benefit from the ‘frisson’ of their higher overtones. (This compositional device also resonates with the use of solfege as an intonational device—the built-in brightening of 3^rd and 7^th degrees of the scale from their syllables.)”
  —  Liz Garnett, BCU, commenting on choral [and barbershop] orchestration in previous CMT blog post.

F   ris•son´ [free-sohn]
{1770–80; < F: shiver, shudder, OF friçons (pl.) < LL frictiōnem, acc. of frictiō shiver (taken as deriv. of frīgēre to be cold)}
n.: a sudden, passing sensation of excitement; a shudder of emotion; a thrill.”

T he previous CMT post led composer/vocal coach/musicologist Liz Garnett at the Birmingham Conservatoire to comment on the multi-factorial origins of perceptual interest that’s aroused by harmonically complex polyphony. I especially liked Liz’s expression, ‘frisson’, when she was describing the sizzle of close-harmony and beat-frequencies among intertwining harmonics. The word ‘frisson’ exactly captures the sense that I experienced when I was singing tenor in a barbershop quartet for four years in the early 1990s. Maybe the word is vivid and accurate for you as well.

L iz averred that she had not studied the nature of the effect systematically [yet] but is interested in doing so. And she and I both agree that the effect is real and that it deserves systematic study. The matter is of interest (a) to composers who, by knowing additional dimensions of how the underlying acoustics and cognitive processes of music work, will be better able to reliably write music that efficiently utilizes those processes; (b) to performers and conductors who, by more fully understanding how a piece operates, will be better able to interpret and deliver the effects that the piece has to offer; (c) to recording engineers and sound-reinforcement engineers and architects who, through new insights into the harmonic properties of the sounds they are helping to produce, will be better able to create ‘systems’ and performance venues that do justice to the live aesthetics; and (d) to musicologists and music theorists, who want to enable us to make sense of it all. Maybe this somewhat esoteric topic even has some interest to regular audience members—at least science-friendly, numerate ones like you.

T here are a variety of wonderful recent books on the cognitive psychology of music; many of those have been cited in CMT posts over the past two years. But none of them, so far as I am aware, has specifically and quantitatively examined mechanisms of attention-capture and –retention by music as a function of its harmonic spectral content and trajectory.

B ut how to go about doing this? Well, first we need to have a collection of examples of compositions and performances that illustrate the effect we’re wanting to study—the ‘frisson’ of harmonic/spectral trajectories. Many barbershop and other close-harmony choral pieces are ideal examples, insofar as they generate most of their perceptual interest precisely in the way that Liz and I were discussing. The Eric Whitacre ‘Lux Aurumque’ piece is perfect: it is harmonically rich, but it is rhythmically and textually very simple—elegant. Many pieces of instrumental chamber music will also be analyzable by the methods I have in mind, and contain passages where much of the cognitive content is generated by close-harmony polyphony. So, in response to Liz Garnett’s suggestion and with an interest in continuity with the CMT post that stimulated it, I thought I’d take the Dale Warland Singers’ recording of ‘Lux Aurumque’ and see what I could make of it with some digital spectrum analysis.

I   write bars, for the musicians—because they have to be ‘together’. It’s difficult when you have multiple different speeds going on at the same time. But don’t think that this is mathematics. There are some [mathematical] constructions in it [i.e., in my compositional design], but the whole thing is ‘music’. In my piano concerto I developed the polyphony to a much higher level of complexity. I think in harmony and rhythm I have found ‘it’. But in melody, I still search…”
  —  György Ligeti, interview with Dorle Soria, 1987, Musical America 107(4).

A s I often try to do here on CMT, I invoke other disciplines outside of music itself, to see what techniques and analytical formalisms they may have to offer us. The cognitive robotics and artificial intelligence (AI) communities, for example, are not ordinarily ‘collaborators’ with music theory people, except on ‘music information retrieval’ (MIR) and a few other subjects. Consider, though, the recent work of Jürgen Schmidhuber and colleagues at TUM and IDSIA.

I  nterestingness’ becomes the first derivative [with respect to time] of ‘subjective beauty’ [as a function of time]. As the learning agent improves its data compression algorithm, formerly apparently-random data parts become subjectively more regular and ‘beautiful’. Such progress in [adaptive, dynamic] data compression is measured and maximized by the ‘curiosity drive’ [that is exhibited by the agent’s behavior]: create action sequences that extend the observation history and that yield previously unknown, unpredictable-but-quickly-learnable algorithmic regularity. That is how beauty arises and how our interest is captured.”
  —  Jürgen Schmidhuber, Professor of Cognitive Robotics, Technische Univ Munich and Co-Director, IDSIA, Lugano, Switzerland, 2007.

J ürgen Schmidhuber is not the first to have linked aesthetic ‘interest’ with the time-derivative (the rate-of-change of ‘beauty’), nor is he the first to have associated ‘beauty’ with ‘complexity’. Fred Lerdahl at Columbia and many others have had similar thoughts in the past. But Jürgen takes the thought further, perhaps, than others have done, in terms of his analysis of the detailed adaptive encoding/compression behaviors that accompany cognitive ‘interest’—the ‘origination-persistence-dissipation’ life-cycle of it.

A nd, although the first-order time-derivative (slope) of harmonic complexity c(t) is not by any means the only predictor of musical interest or subjective ‘frisson’ or ‘sizzle’, it is one of the things that we can readily examine, to investigate how good or poor a correlate of ‘frisson’ it is. That’s what I set about to do this past week. The preliminary results of my initial exploring appear further down this page...

C   omplexity that works is built up out of modules that work perfectly, layered one over the other.”
  —  Kevin Kelly, ‘Out of Control: The New Biology of Machines, Social Systems, & the Economic World’, 1995.

Y ou may like to know that there are a variety of ‘freeware’ software applications that can help you do this sort of thing, in case you wish to undertake some investigation of these things yourself. SFS from the Phonetics Dept at UCL is the tool that I happened to use, although there are several other purpose-built spectral analysis programs for harmonic and formant analysis of speech, which allow importing .mp3 or .wav audio files. Or you can use the signal-processing modules of MATLAB or other full-function analytics packages.

T he more challenging thing, in my view, is to select a quantitative complexity index (or, alternatively, devise a new, entirely original complexity metric) that adequately represents the harmonic complexity of the polyphony at any instant in time and that enables us to analyze the timeseries of serial complexity indices of the music in sync with the music’s score. The mathematics and computer science literature on Kolmogorov complexity is especially extensive.

B ut Kolmogorov is computationally more difficult to implement than I wanted to do right now. So I considered Lempel-Ziv complexity (LZC), which is another complexity metric that has received much attention over the past 30 years. I have used LZC in a recent project in my software-engineering day-job, and I have coded it in C and C++ languages in the past. But the LZC is a ‘left-to-right reading’ or ‘directional’ complexity metric, which is undesirable for what we are aiming to do here. We don’t want a complexity metric that gives different or undue emphasis or deemphasis to harmonics that are on the ‘left’ of the input string (the low-frequency bands, bass) vs. the ‘middle’ (the mid-range registers, tenor and alto) vs. the ‘right’ (the high-frequency bands, soprano, etc.).

F or my initial study, I therefore settled on a simple, non-directional ‘compositional complexity’ metric, with a separate ‘channel’ for each of [any number of] the significant harmonics that exist (in the context of the chord that prevails at each moment). My complexity metric has no fancy adjustments for pitch error or jitter or phasing or nonlinear Fletcher-Munson- or Robinson-Dadson-type aural sensitivity curves or other important things. It’s a basic, exploratory tool, nothing more. It is not meant to be a definitive representation of the sound or the waveform or the neurophysiology of our hearing of the sound. It is not the product of some extensive, scholarly project. It’s just ‘lunch’—featuring left-over things I know from other analytics projects I’ve done, plus a few hours of Java coding and testing. In other words, my complexity metric is just a quick, simple method to see whether the interpretive path that I’m on makes sense and whether it might be productive or not. Productive for me … or sufficiently intriguing for you or others to pursue it further.

T his ‘compositional complexity’ metric is designed so as to assign the same complexity value to a given polyphony/harmonics ‘pattern’ regardless of its performance dynamics. That is, the string A=‘1020000000000000’ and B=‘2070000000000000’ have the same pitches sounded, for example. The pitches have different sound intensities and different relative loudnesses with respect to each other, so their timbres are actually substantially different. But for the preliminary purposes of this experiment, A and B are harmonically ‘the same’, according to our simple ‘compositional complexity’ metric, in terms of the pitches and beat-frequencies, etc., that they contain. They each have only two loudness levels instantiated and they each have only two harmonics instantiated: therefore, their complexity metrics are identical.

W hat next? The sound intensity needs to be determined at each moment in time (in elapsed milliseconds, in the .mp3), with each moment/sample tagged with the corresponding notation in the music score. For simplicity’s sake, I arbitrarily chose to do 3-bit quantization of the sound intensity (8 levels, coded as ‘0’ to ‘7’), corresponding roughly to qualitative ‘nil’, ‘ppp’, ‘pp’, ‘p’, ‘mp’, ‘mf’, ‘f’, and ‘fff’ dynamics. I measured the sound-level of each harmonic, ascertained from the SFS digital spectrograms of the Warland ‘Lux Aurumque’ .mp3 at each point in time. In other words, my sound intensity quantization of this .mp3 did not just assign one dynamics code at each moment but instead assigned a multi-frequency ‘vector’ comprised of up to 16 three-bit dynamics codes at each moment, one for each harmonic present [plus the ‘0’=‘nil’ sound intensity code for any harmonic that was ‘implied’ by the prevailing chord structure but un-voiced or absent in the sampled audio].


Here is a jpeg SFS screenshot of one of the ‘Lux’ vowels in the piece...


I  then wrote a simple Java program to accept these 16-vectors as input and calculate the compositional complexity for each one. Now, we don’t want a coarser or finer representation of the frequency domain (EQ bands) to cause us to compute wildly different complexity index values. We want our complexity metric to be ‘normed’ and vary, say, on a range between 0.0 and 1.0, regardless how many or few bands or channels we use to measure the polyphony and harmonics. The n_j are the numbers of each harmonic 'band' in the sample. C varies from values close to zero for simple, flute-like, near sine-wave sounds up to 1.0 for all-hell-broke-loose dissonance.

T he equation that governs the ‘normalization’ of the computed complexity index is a function of the number of ‘bands’—the EQ spectrogram ‘length’ L (expressed in bins, by note-by-note fundamental in bin #1, its octave in bin #2, and so on). The normalization also a function of the number of sound-intensity levels, which was N=8 for my 3-bit quantization, as shown in this equation.


T he normalization calculation is done automatically in my little Java program. If you like, you can click on the jpeg of the Java code below and download a copy to play with.


Y ou can download the free copy of the Java Development Kit (JDK) from Sun Microsystems here. Extract it to your hard-drive. Then compile the complexity.java program in a command window with the ‘javac’ compiler command:

     C:\>\Program Files\Java\jdk1.6.0_11\bin\javac complexity.java

To run the program, use the java command:

     C:\>\Program Files\Java\jdk1.6.0_11\bin\java complexity

Y ou don’t need to be a software developer/programmer to do this. It’s easy. Entering some intensity vectors from the keyboard as strings gives you output like what you see below…


T hen we take that timeseries of complexity values computed from our spectral analysis measurements of the .mp3 music file and use difference-equations (in Excel or whatever you like) to estimate the first derivative (slope, rate-of-change) of the complexity at each time point. When we plot it, it looks like this…


W hat we see here is that the moments that have that ‘frisson’ that Liz and I were referring to do have high complexity levels. And the heart-pounding, emotionally-evocative compositional structures/moments tend to be ones where the first derivative of complexity has high positive or negative values—moments of dramatic change in polyphony harmonics and dynamics.

T   he art of ‘simplicity’ is a puzzle of ‘complexity’.”
  —  Douglas Horton, Harvard Divinity School, 1955.

W hen the performance is delivered in an echoic space like a cathedral, you get significantly non-zero spectral complexities that linger hundreds of milliseconds after a note is released. The ‘shock’ value of this is almost as great as the ‘voix celeste’ beat-frequency implied-pitch effects we were discussing before. The effect is somewhat like SONAR: your voices are acoustically ‘imaging’ the geometry of the performance hall, somewhat like a bat would do … flying around in the upper reaches of Chartres Cathedral at Christmas Eve midnight Mass. The composer has created a gesture that causes you to acoustically plumb your ‘situatedness’ and physicality as a human being in the cathedral, much like a bat. The polyphony—the frisson and its decay—causes you to realize your physical humanness in a very dramatic, palpable way. You are performatively ‘present’ more vividly than at almost any other time in your life. Through the ‘frisson’ you become intensely aware of your own personhood and that of the other singers (or instrumental players) around you. It is breath-taking to hear it; tremendously exciting or emotionally unnerving to perform it.

Incidentally (but unsurprisingly), expressively-effective close-harmony ‘frisson’ doesn’t come suddenly out of ‘nowhere’. It has a foundation—a methodical, deliberate architecture. We look at the c(t) and dc(t)/dt curves, and we see that the effect builds progressively, logically, over a phrase or a portion of a phrase. You feel this happening when you sing it, of course. These spectral analysis and complexity timeseries plots only confirm quantitatively/objectively what most of us feel intuitively/subjectively. Same thing when you perform a similarly orchestrated close-harmony piece of instrumental music. The acclaim that Eric Whitacre has accrued over the past 10 years redounds to his skill in engineering these kinds of harmonic gestures. The ambiguity of the C#m [I] → C#m7/9 (D#m7) [II⁶] V-of-V supertonic and other things Whitacre’s got going on in here are impressive.

S   implicity does not precede complexity but instead follows it.”
  —  Alan J. Perlis, ‘Software Metrics’, 1981.

A  more extended, time-consuming effort will be required to statistically evaluate the correlation of the spectral ‘brightening’ (more acoustic energy in the higher harmonics) with certain vowel and consonant sounds and short time-scale structures associating text phonetics-to-timbre and –pitch/–spectrum. It’s more work than I have time to devote to right now. But this CMT post is meant to convince you that studying effects like these quantitatively is surely feasible—feasible even using available free-ware and simple complexity metrics that you can code for free in Java. In other words, there is a real possibility for you to do this without a big expenditure for software or expensive computers. If you are a student in a conservatory, though, you probably will want to get copies of several of the books in the list below ...


W hat else? Well, it’s important to note that you can’t write a piece that has constant low levels of complexity—or constant, excessive complexity—and expect people to sit still for it. Nor can you whip-saw people with changes in dc(t)/dt that are too frequent or too fast. You either won’t capture the artists’/audience’s attention at all, or you will rapidly lose it, annoying them in the process. Beauty and interest may indeed be functions of the first time-derivative, as Jürgen Schmidhuber says, but that doesn’t mean that ‘more is better’ or that ‘all frisson, all the time’ is a categorical aesthetic good. You need to write with a storyline—compose/arrange with a good ‘plot’ in mind. Of course, this comes as no surprise. For me, the surprise is being able now to quantatively see what we already knew or believed—understanding the mechanisms of musical ‘interest’ in terms of spectral complexity timeseries. It’s a new way to understand the ‘mechanics’ of how good writing works, and to identify the faults and reasons why other compositions don’t work. Simple. And pretty useful, I think.


T hanks for your interest in this topic! Please feel free to post a comment or email me anytime, about this or other aspects concerning how we transmit/receive/perceive musical aesthetic value. If there’s enough interest, I’ll continue with more posts in this vein. And thanks especially to you, Liz, for the dialogue.

T   he complexity of things—the things within things—is endless. Nothing is easy; nothing is simple... I was looking forward to telling the truth, or some of it, in all its complexity, to a person who would not be surprised or outraged by it.”
  —  Alice Munro, ‘Carried Away’, 2006.

• Schmidhuber J. Simple algorithmic principles of discovery, subjective beauty, selective attention, curiosity and creativity. Lecture Notes Comp Sci 2007; 4755:26-38.
  
• Jürgen Schmidhuber page at TUM
  
• Schmidhuber recurrent neural network page at IDSIA
  
• HelpingYouHarmonise.com (performance coaching and arranging clinics by Liz Garnett)
  
• Eric Whitacre website
  
• Complexity Digest
  
• SpeechAndHearing.net
  
• Univ Amsterdam (Paul Boersma and David Weenink) PRAAT phonetics analysis software [1.9MB self-extracting .exe, unpacks to 6.9MB application]
  
• PRAAT ‘shimmer’ measurement functions
  
• UCL Dept of Phonetics & Linguistics: software
  
• UCL Dept of Phonetics & Linguistics: phonetics fonts (Doulos and Charis, SIL)
  
• UCL Dept of Phonetics & Linguistics: SFS
  
• SIL software catalog
  
• Bakken R, Orlikoff R. Clinical Measurement of Speech & Voice. 2e. Singular, 1999.
  
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• Lerdahl F. Cognitive constraints on compositional systems. In J. Sloboda, ed., Cognitive Processes in Music. Oxford Univ, 1988, p. 231-59.
  
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I   ndy, why does the floor move?”
  —  Sallah (John Rhys-Davies) to Indiana Jones, upon opening the Well of Souls, ‘Raiders of the Lost Ark’, Lucasfilm, 1981.

I  t is our inimitable frisson of complexity—all writhing and beauteous complexity down here.”
  —  Snakes’ reply, to Sallah and Indiana, from the bottom of Well of Souls, ‘Raiders of the Lost Ark’, Lucasfilm, 1981.

The Physical Properties of Sound Itself: Whitacre & Choral Dissonance

A   ural relationships at times bear little relationship to harmonic analysis, or even harmonic function. Stated another way, singing in tune is the result of complex aural/acoustic relationships between and among tones. Those relationships assume a life all their own when placed within the context of vertical harmonic structures. While those relationships appear to defy traditional theoretical knowledge—especially with respect to chromatic movement within harmonic structures—it is clear that choral intonation is a function of the relationship of the tone sung to the harmonic context at that moment. It must be realized that we are using the solfege syllables of Guido d’Arezzo as an intonational device—just as the violinist uses the fingers to ‘place’ a note correctly within its proper intonational context.”
  —  James Jordan, preface to Matthew LaPine, Prognosticating the ensemble’s ear. in ‘Evoking Sound: The Choral Rehearsal,’ GIA, 2007. (Ch. 16, pp. 147-55).

L   ux,
calida gravisque pura velut aurum.
et canunt angeli moliter
modo natum.

Light,
warm and heavy as pure gold.
And the angels sing softly
to the new-born babe.”
  —  Edward Esch (original poem in English), Charles Anthony Silvestri (Latin translation).

H ow exactly does Eric Whitacre’s ‘Lux Aurumque’ work? How can it move us so deeply, so mysteriously? It does this in part through Whitacre’s extensive, deliberate use of beat-frequencies between the parts. For me—and maybe for you as well—the effect is not just the richness of a clashing pair of notes; it’s also the ‘phantom’ note that the beat-frequency of a note-pair implies: the ‘implied’ note that we hear when the dissonance is sounded. That implied or ‘phantom’ note has a pitch equal to the frequency difference between the two dissonant notes—the beat-frequency of the difference in frequency. Close-harmony like this is like a human analogue of ‘voix celeste’ on an organ—phantom notes expressly created by the dissonance beat-frequency. Yes, a choral tune is set as a cantus firmus—but it isn’t purely horizontal: there are all of the vertical/harmonic implications as well as the contrapuntal associations...


M athematically, we should hear the sine wave (f₁ + f₂)/2 as the average of the two pitches. But the sine part of the right side function alternates between negative and positive values many times during one period of the cosine part, and when this happens only the absolute value of the envelope is perceived. So the beat-frequency we hear is just the difference between the two frequencies: (f₁ − f₂).

T he violinist Giuseppe Tartini was the first to document this musical effect in the 18^th Century, calling it il Terzo Suono (Italian for ‘third sound’). Playing pure harmonies (say, a frequency pair whose members have a simple proportional relation to each other, like 4/5 or 5/6, as in the ‘just intonation’ major and minor third respectively) on the two upper strings, such as the C5 against an open E-string E5, gives the ‘phantom’ C3 two octaves lower.

T he major second (M2) interval of Soprano 2 singing C#5 against Soprano 1 singing D#5 in mm. 2, 4, etc. of ‘Lux Aurumque’ gives the implied C#2 three octaves lower.

T he minor second (m2) between Tenors singing D#4 and E4 in those same bars gives pulsations too low to hear as a pitch per se. This is because, when the dissonance involves two pitches that are close together and pretty low, the frequency of the cosine of the right side of the expression, (f₁−f₂)/2, is usually too low for us to perceive it as a pitch. Instead, we perceive it as a tremolo-pulsating change in the volume of the underlying pitches. The cosine term creates an amplitude-modulated ‘envelope’ for the sine wave (f₁ + f₂)/2, the average of the two pitches. So the dissonances in the tenor and bass parts mainly create a wonderful, shimmering tremolando texture. This is even more dramatic in cathedrals or other echoic spaces, where the reverberation and phase relationships between the parts can add yet more richness and acoustic complexity.



    [50-sec clip, Dale Warland Singers, Eric Whitacre, ‘Lux Aurumque’, 1.2MB MP3]

J ames Jordan’s student, Matthew LaPine, offers guidance on how to get singers to properly perceive and deliver these close intervals. He does not talk about beat-frequencies or ‘phantom’ beat-frequency pitches. But Matthew’s discovery that ‘resetting’ what ‘do’ is in your solfege—is a closely-related thing. Our minds associate each solfege syllable and its relation to other syllables/intervals with relative frequency ratios. Resetting where ‘do’ is enables us to better anticipate and accurately produce the dissonances that achieve the proper beat-frequency Terzo Suono.



    [50-sec clip, Dale Warland Singers, Eric Whitacre, ‘Lux Aurumque’, 1.2MB MP3]


T he Dale Warland Singers’ recording is my favorite performance of ‘Lux Aurumque’ to-date. Here is another nice performance, on YouTube:





[ King’s College Choir, Cambridge, performing Eric Whitacre’s ‘Lux Aurumque’ ]

T he tight harmonies need to be carefully balanced by a well-trained a cappella choir. Despite this, the descending half-steps tend to be over-shot. The solfege in mm. 13-15 should not be ‘fa-mi-ri, re-re-di-do’. And alternates ‘do-ti...’ and ‘li-la...’ also cause overshoot. Instead, try resetting ‘do’ three times: ‘fa-mi/do-ti, do-do-ti-do’. In other words, treat these as two entirely separate descending chromatic phrases.

C ool. It works. Not just with this piece but with many others. LaPine recommends recasting your solfege with ‘erosive’ resets whenever you are overshooting or whenever you are missing the proper beat-frequency implied pitch. Doing this isn’t rewriting the composition or arguing with the composer regarding what the key signature or mode really are. The whole point is simply to trick your mind and voice into doing the right thing. He says to specifically avoid solfege syllables that would, in your mind, denote a cadence.

D issonances are especially poignant and richly colored when it’s you who’s singing them; they are never as intense for members of the audience. When singing difficult chords with critical dissonances, I like it when singers face each other: that way the phasing of the dissonant notes (the beat-frequency) is even more obvious, easier to hear, and more intensely beautiful.

T he whole phenomenon is, for me, very much like experiencing the music of Alvin Lucier, who also has methodically explored beat-frequency and the physics of sonic-space phasing in his compositions and installation pieces.

If you like, you can click on the screen-shot below and download an Excel spreadsheet where I’ve calculated m2 and M2 combinations in the relevant vocal registers.


T he Bach chorale-like longer ‘note-against-note’ and shorter ‘figurated’ dissonances with binary divisions of the beat—these, too, are much to be admired in Whitacre’s writing. These interesting dissonances; the graceful, fluent motion from one chord to the next revealing the notes that successive chords have in common; the enriching harmonies with 7ths and 9ths … all without any sacrifice of melodic logic.

L ux Aurumque’ was originally commissioned by the Master Chorale of Tampa Bay, and first arranged for men’s voices for the Gay Men’s Chorus of Los Angeles.

• Eric Whitacre website
  
• Eric whitacre blog
  
• Eric Whitacre page at WikiMusicGuide
  
• Dale Warland Singers. Lux Arumque. (Gothic, 2007.)
  
• Lux Aurumque score at Boosey & Hawkes
  
• Lux Aurumque score at Walton Music
  
• Beat-frequency Java applet at MIT
  
• Garnett L. Choral Conducting and the Construction of Meaning: Gesture, Voice, Identity. Ashgate, 2009.
  
• Garnett L. The British Barbershopper: A Study in Socio-Musical Values. Ashgate, 2005.
  
• Higgins H, Kahn D, eds. Mainframe Experimentalism. Univ California, 2009.
  
• Jordan J, O'Regan T, eds. Evoking Sound: The Choral Rehearsal. GIA, 2007.
  
• Alvin Lucier site at Wesleyan Univ
  
• Alvin Lucier interview with pianist Thomas Moore, JAN-1983.