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.
  
• Benade A. Fundamentals of Musical Acoustics. 2e. Dover, 1990.
  
• Bod R. A unified model of structural organization in language and music. J Artif Intell Res 2002; 17:289-308. [1MB pdf]
  
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• Catford J. A Practical Introduction to Phonetics. 2e. Oxford Univ, 2002.
  
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• Croft J. Musical memory, complexity, and Lehrdahl's cognitive constraints. M.Mus thesis, 1999, Univ Sheffield. [1MB pdf]
  
• Cilibrasi R, Vitanyi P, de Wolf R. Algorithmic clustering of music. [1MB pdf]
  
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• Frenken K. Innovation, Evolution and Complexity Theory. Elgar, 2006.
  
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• Garnett L. Harmonic ‘charge’ and voicing. 06-JAN-2009.
  
• Hashemi V, Ramezani S. Inducing meter for a rhythm and measures for rhythm complexity. Univ Amsterdam, 2008.
  
• Hochreiter S, Schmidhuber J. Long short-term memory. Neural Computation 1997; 9:1735-80.
  
• Honingh A, Bod R. Convexity and the well-formedness of musical objects. J New Music Res 2005; 34:293-303. [1MB pdf]
  
• Huckvale M, et al. Acoustics of speech and hearing. Univ College London Dept of Phonetics & Linguistics. [1MB pdf]
  
• Kaspar F, Schuster H. Easily calculable measure for the complexity of spatiotemporal patterns. Physical Review A 1987; 36:842-8.
  
• Kirilyuk A. Universal Dynamic Complexity, Its Conservation and Development as Unified Order of the World. Eighth European Conference on Science and Theology, Lyon, 14-19 APR-2000). [1MB pdf]
  
• Ladefoged P. Vowels and Consonants. 2e. Wiley, 2005.
  
• Ladefoged P. Vowels and Consonants: An Introduction to the Sounds of Languages. Wiley, 2000.
  
• Ladefoged P. Elements of Acoustic Phonetics. Univ Chicago, 1995.
  
• Lerdahl F. Cognitive constraints on compositional systems. In J. Sloboda, ed., Cognitive Processes in Music. Oxford Univ, 1988, p. 231-59.
  
• Li M, Vitanyi P. An Introduction to Kolmogorov Complexity and Its Applications. 3e. Springer, 2008.
  
• MacDuff R. Music and fractal landscapes. [500KB pdf]
  
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• Roetzheim W. Why Things Are: How Complexity Theory Answers Lifes Toughest Questions. Level4, 2007.
  
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• DSM. Bronfman: Is virtuosity 'necessary'? Is sublimity 'measurable'? CMT blog, 08-DEC-2007.

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.
  
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• Alvin Lucier interview with pianist Thomas Moore, JAN-1983.

Glagolitic Ekphrasis and Melisma in Tondal

DSM: Dialogos uses such—What would we say?—onomatopoetical sounds. If Tondal’s spirit is leaving his body, then what better way to represent that movement than by the splitting of vocal lines—the divergence from unison to close harmonies and dissonances, diminished seconds, and so on—and the re-convergence of those lines back into unison! The whole thing is melismatic (that is, many notes per syllable of text)! The extended, braided melismas distort our sense of time! A fantastic effect!

CMT: Yes. In Tondal, all six of the Dialogos members’ voices execute extensive melismas—sometimes in the upper range to express aspiration or anxiety or other emotions; sometimes low in their vocal range to express a pensive rumble. I think that long melismas are such a vivid musical rendering of the self, of the spirit. Incidentally, Bach often does this on the word ‘geistlich,’ and he even works in some irony in his choral settings—for example, when the alto sings the chorale while the soprano and tenor parts accompany—in his Cantatas and in his smaller works (Singet dem Herrn ein neues Lied; Komm, Jesu, komm; Lobet den Herrn; Jesu, meine Freude; and Fürchte dich nicht). The singers bid farewell to life’s vices and passions to gain eternal life, and you would expect Bach to emphasize the joy and eagerness of those expecting salvation. But instead the musical figure of the soprano voices—the way in which they reluctantly release their held notes down a half step—clearly represents regret and doubt. You hear a predominating sorrow—an aporia. Much of Dialogos’ account of Tondal’s Vision is like this—these aporietic melismas!

DSM: Buelow notes Bach’s use of melismatic gestures, which he says are “contradictory to all Baroque theory of recitative.” But maybe Bach was harking back to an early tradition. Not only poets may respond to a work of visual art with a creative act in their own medium, transposing the style and structure, the message and metaphors from the visual to the verbal. Composers routinely engage in interartistic transfers like this. Although the musical medium is reputedly abstract, composers, just like poets, can respond in many different ways to a visual representation. They transpose structure and content. They interpret or play with some of the suggestive elements of the original image. This is basically the musical equivalent of what literary theorists call ‘ekphrasis’. Central questions concern the definition of musical ekphrasis in relation to music’s ability to represent extramusical realities or narratives—to relate to reality by way of musical mimesis or symbolic reference.

CMT: You know, listeners don’t have unbounded capacity to track multiple concurrent lines of sound. David Huron found that as the number of concurrent voices in a polyphonic texture increases, expert listeners are both slower to recognize the addition of new voices, and more prone to underestimate the number of voices present. Here is Huron’s plot of experimental data from five expert musicians. This shows voice-tracking errors while listening to polyphonic music. The data show that when listening to polyphonic textures employing relatively homogeneous timbres, tracking confusions are common when more than three voices are present.


DSM: The solid columns show mean estimation errors for polyphonic textures of varying textural densities; the shaded columns indicate mean error rates for single-voice entries. For polyphonic textures employing relatively homogeneous timbres, the accuracy of identifying the number of concurrent voices drops markedly at the point where a three-voice texture is augmented to four voices. Beyond three voices, tracking confusions are commonplace. In a study of isolated chords, Parncutt (1993) found similar results. When asked to count the number of tones in chords constructed using tones with octave-spaced partials (Shepard tones), Parncutt found that listeners make significant errors (underestimation) once the number of chromas exceeds three.

CMT: Limitations in the perception of auditory numerosity may be symptomatic of a broader perceptual limitation, since the results of these studies parallel the results of similar studies in vision. Using a dishabituation paradigm, Strauss and Curtis showed that performance degrades when discriminating three from four items, and reaches chance performance when discriminating four from five items. The work of Strauss and Curtis is especially significant since pre-verbal infants can be expected to possess no explicit knowledge of counting. This implies that the perceptual confusion arising for visual and auditory fields containing more than three items is a low-level constraint that is not mediated by cognitive skills in counting.

DSM: These results are consistent with reports of musical experience offered by musicians themselves including composer Paul Hindemith. It strikes me that this is much like the close harmonies and complex dissonances of barbershop quartet singing. Your mind naturally follows and anticipates the divergences and convergences. In fact, that is a major source of the power and charm of such music—the textural complexities.

CMT: Dialogos compounds this further when they circle themselves on the stage. The choral effect is as though the dissonances are massed—one single aggregate collective spirit, despite the continued multi-vocality of it. Then the antiphonal qualities are restored when they disperse from the circle and distribute themselves across the stage—the individualistic contrast is especially apparent when they are performing in a large cathedral, with all of the reverberations that are characteristic of a church.


DSM: Temperley addresses a fundamental question about music cognition: how do we extract basic kinds of musical information, such as meter, phrase structure, counterpoint, pitch spelling, harmony, and key from music as we hear it? Taking a computational approach, Temperley develops models for generating these aspects of musical structure. The models he proposes are based on preference rules, which are criteria for evaluating a possible structural analysis of a piece of music. A preference rule system evaluates many possible interpretations and chooses the one that best satisfies the rules.

CMT: Temperley has addressed some of this in his book, as has Taruskin in his. There is the ontology of musical structure: meter, phrase structure, contrapuntal structure, harmony, and key, as well as pitch spelling (the labeling of pitch events with spellings such as A-flat or G-sharp). Musical structures are inferred, but there is also musical ambiguity, “look-back” retrospective revision of our concepts as we continue listening, and sweet anticipation—‘expectation’ in Huron’s sense. Dialogos is masterful in interweaving all of these to make their performances powerfully evocative!

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