Thursday, February 05, 2009

Hierarchic asymmetry

Back when I was teaching my first music theory class as a graduate student, I was caught completely off guard by a student's complaint. We were talking about the difference between simple meter and compound meter, which I regarded as a very basic topic. In one the beat is divided into two equal parts, and in the other the beat is divided into three equal parts. The exemplars for each of these would be 4/4 and 6/8. The quarter note beat in 4/4 is divided into two eighth notes, and the dotted-quarter beat in 6/8 is divided into three eighth notes. Easy, right? Well, this student, who was quite bright, said that 6/8 wasn't really a compound meter, because the eighth notes themselves didn't divide into three equal parts. I thought he was making a common mistake of assuming that eighth notes were the beat in 6/8, but he wasn't . He expected that if the beat is divided into three equal parts, then each subsequent lower level in the metric hierarchy would also divide by threes. This is actually a very rare phenomenon, not even covered by normal metric notation. There is no time signature that indicates a beat that will be divided into 9 equal parts, much less 27 or 81 parts. The basic assumption in Western notated music is for beats to be grouped by 2's or 3's (or some combination thereof) and to be divided by 2 or 3, but that all lower levels of subdivision are by 2.

I never really thought about this lack of symmetry, until I was recalling an argument about chord and key relationships. Petr Janata and his colleagues have done some interesting work on brain imaging and tonal sensitivity. But at one conference he presented a torus mapping of key relations, playing along a chord sequence that modulated from one key to the next. This mapping was meant to represent brain activity, showing how the brain interprets different modulations as close or far. As expected, the modulations followed a circle of fifths pattern, but what I found quite disturbing was that mode was not taken into account. At one point the progression went from C minor to G major. I pointed out to him that this was a very distant key relationship in music theory, which he found surprising. After all, C minor chords go to G major chords all the time. We then started a debate about the difference between chord relationships and key relationships, with Carol Krumhansl coming up to keep us from punching each other. (Not really, but it was an interesting and sprited debate, including the aforementioned Dr. Krumhansl). Carol did the seminal work on tonal hierarchies, using probe tones to determine the closest cognitive notes to a given tonic, and helped develop a key-finding algorithm that might mimic how our brains determine what key we are in. What I realized that I knew to be false, but that these scientists had assumed to be true, was that the hierarchy of relationships between notes and chords would extend to keys without change.

Why is it that Western music* does not encourage complete symmetry among all levels of a given feature? Could it be like the asymmetric design of the diatonic scale, working as a signpost to help us identify where we are? The assymmetry between keys and chords could help distinguish the scope of relations we are perceiving, so we don't get the two mixed up. Likewise with meter, to help identify the beat more clearly.

*If anyone knows about any other music traditions that do include non-binary metric symmetry or note/harmony/tonality symmetry, let me know.


Dan B. said...

I think part of it may be a distinction between symmetry and equality. Beats, keys, or pitches don't have to be equal to be symmetric (I-V-vi-V-I is symmetric, but the individual elements still remain hierarchically not-equivalent).

And of course, true symmetry is rare in music (see any arch-shaped melody), or anywhere actually. But close counts here. It's just that "close only counts in horseshoes and hand-grenades" is pithier than "horseshoes and musical symmetry in a tonal system."

Daniel Wolf said...

I'm not aware of any traditional examples but Lou Harrison's Threnody for Carlos Chavez (for viola and gamelan degung) uses perfect (triple) subdivisions at every level: sections, phrases, greater and lesser moods, time, prolation and one smaller division for seven levels total. I believe that Lou, long an admirer of the description of rhythmic modes in Morley's Plain & Easy Introduction was to create an example of one of the moods which Morley indicates in a table but does not describe in his text proper.

Triple metres (and triplet subdivisions within metres) of any sort actually appear to be rather rare in traditional musics. In the west, most triple divisions — for example Gigues and Gigs or Minuets and Walzes — are housed in larger binary groups and subdivide binarily, which is reasonable given the fact that a foot raised in dancing will eventually have to come down. (The Sarabande, in a very slow three is more likely an alternation between a short and a long beat than three beats). Likewise, in Korean music, which has an unusually large triple metre repertoire, the three is usually in a compound metric context.

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Anonymous said...

I don't know about harmony or chords, but research by e.g. Patel (Patel & Daniele 2003) provide empirical evidence for the hypothesis that speech rhythm is reflected in musical rhythm, at least for Western music, but possibly for other music as well. One explanation they give for this is that language rhythm becomes part of your system at a very early age, and that composers consciously or unconsciously use these patterns in their music.

Since most languages (perhaps all?) seem to favour systems with prominence/background, stress/non-stress, fortis/lenis oppositions, metric symmetry is not something you'd expect to occur (although, like Dan B., it's perhaps not lack of symmetry that I'm talking about, but lack of equality - a lack of even distribution of prominence or stress over syllables, feet, clitic groups etc in an utterance).

If it's not there in language, it's perhaps also less likely to occur in music, unless you make a conscious effort, as contemporary composers did/do.

It's funny, my percussion teacher asked something similar last week; Way back, I studied phonology & phonetics, and we spend a good deal of my timpani lessons discussing the link between language and music. There's so much similarity between the two; pity that they are taught separately.