Music and Math

The latest Seed has a very interesting article on the complicated geometry underlying Western music, and the intuitive mathematical understanding demonstrated by composers:

The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It turns out that superficially different styles--Renaissance music, classical and Romantic music, jazz, rock, and other popular forms--all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit understanding of their mathematical contemporaries.

Perhaps my next book should be Chopin was a Topologist. But the article also left me with lots of questions. Ultimately, music works not by mirroring complex mathematical functions but by titillating hair cells in the cochlea and exciting neural networks in the auditory cortex. Why does the mind find such obscure geometrical orders so alluring? One possibility is that mathematical music hijacks our brain's penchant for patterns, allowing us to extract some predictable sequences from the cacophony of soundwaves. The equations might be obscure, but the cortex is still able to hear them. As I write in my book:

A work of music is not simply a set of individual notes arranged in time. Music really begins when the separate pitches are melted into a pattern. This is a consequence of the brain's own limitations. Music is the pleasurable overflow of information. Whenever a noise exceeds our processing abilities - we can't decipher all the different sound waves hitting our hair cells - the mind surrenders. It stops trying to understand the individual notes, and seeks instead to understand the relationships between the notes. Our auditory cortex pulls off this feat by using its short term memory for sound (in the left posterior hemisphere) to uncover patterns at the larger level of the phrase, motif and movement. This new approximation lets us extract order from all these notes haphazardly flying through space, and the brain is obsessed with order. We need our sensations to make sense.

It is this psychological instinct - this desperate neuronal search for a pattern, any pattern - that is the source of music. When we listen to a symphony, we hear a noise in motion, each note blurring into the next. The sound seems continuous. Of course, the physical reality is that each sound wave is really a separate thing, as discrete as the notes written in the score. But this isn't the way we experience the music. We continually abstract on our own inputs, inventing patterns in order to keep pace with the onrush of noise. And once our brain finds a pattern, we immediately start to make predictions, imagining what notes will come next. We project our imaginary order into the future, transposing the melody we have just heard into the melody we expect. By listening for patterns, by interpreting every note in terms of our expectations, we turn the scraps of sound into the ebb and flow of a symphony.

PS. Nature has put a great collection of articles on neuroscience and music online.

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" Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords."

I hope the authors do not believe this is some sort of profound insight on their part. Chords in western music arose from polyphonic writing in scales so of course they are related. Chords as we think of them did not exist until the baroque period. The "chords" in medieval and renaissance music are byproducts of contrapuntal rules. In music theory 101 today you are still tought to link chords as though each note was sung by a section of the choir

And then there are sad individuals like myself who are utterly tone deaf, thus resigning music to, at best, a pleasant background noise. I sometimes wonder what it must sound like to other people who seem to get such an intense emotional response that I have never managed to achieve...

"In music theory 101 today you are still tought to link chords as though each note was sung by a section of the choir"

It was certainly true 30 years ago when I took it. I still remember doing four part harmony. The prof would write out a 16 bar melody and say something like "Write the other three parts using only I, III, IV, and V chords...and no parallel perfect fifths!"

Tymoczko had the first ever article in "Science" on Music Theory. This is wonderful stuff.

Tymoczko has created a QuickTime movie of a particularly tricky Chopin piano prelude in E-minor (Opus 28, No. 4) to illustrate how the orbifold works.

"This prelude is mysterious," he explained in a Princeton news release. "While it uses traditional harmonies, they are connected with nonstandard chord progressions that people have had trouble describing. However, when you plot the chord movement in geometric space, you can see Chopin is moving along very short lines, staying primarily within one region."

On Tymoczko's Web site, you can find additional resources, including his ChordGeometries software, a version of his Science paper and a series of four QuickTime video files that provide further audiovisual explanation. There's even a QuickTime depiction of the famous chords from Deep Purple's "Smoke on the Water."

The scheme works less well for musical styles that don't have the Western notion of chord progression. But even for non-Western styles - say, the rhythms of African music - "you can use my geometric model to think about how you evolve from one rhythmic pattern to another."

I think the math is interesting but do not think it illuminates anything materially new. Every sophmore music theory class analyzes that em Chopin prelude and comes to the same conclusion as Tumoczko - that the nonstandard chords are a function of the chromatic voice leading. You don't need a model of geometric space to see this, you can just look at the score. The intervallic relationships between all possible chords of a particular number of notes have been plotted out by Elliott Carter, Allan Forte and others 40+ years ago. Just like theorist Heinrich Shenker reflected the reductionist zeitgeist of early 20th century when he reduced all music to a fundamental line, I wonder if Tumoczko just reflects a contemporary infatuation with pretty computer graphics and mathematical jargon.

I have just started actually singing and I find the less I do the better it comes out. I just listen to the music and as long as I don't over-think it I can usually hit the notes. It's nice the way our brain wants things to work.
P.S. I would definitely read the "Chopin Was A..." book.

Jonah - What was wrong with my post about the Lucy Scale and Jimmy Hotz and LightDancer?

By GrayGaffer (not verified) on 11 Jul 2008 #permalink

Comment 3 is interesting. The writer talks about lines/voices and THEN says his tutor set the limits in CHORDS! Just shows how weird this subject is. In my harmony class the tutor said the same "This old music is not made our of chords as we know them...' Then he set down the harmonies wanted in Roman Numerals!!!!!! Online Blumberg's Cipher specifically states that new research and discussion establish 'unequivocally' that chords were in use in Medieval times. the Dolmetsch site has a section called Chord Structure in Medieval Music!
So where are we now?

Sequal to comment 9 here. Groves says that the Roman numbers in music (I, VI, V etc) started with Gottfried Weber (c 1820). So we are debating using a system devised in the 1800s and applying it in retrospect even prior to Bach/Handel and the rest. Is that it?