Tuesday, July 13, 2021

The Subjectivity Of Everything: Music And Math Edition

Recently from a combination of world-weariness, pandemic-weariness, and me-weariness, I was seized by a desire to think about something abstract, useless, inert, and intellectually challenging. "I know what I'll do," I thought. "I'll buy a book about math."

I went to a small bookstore and asked, "Do you have a math section?" No. But there among the science books I came upon Music By the Numbers by math historian Eli Maor. I thought it would be about music theory and the mathematical underpinnings of classical music structures -- something I vaguely remember being interested in when I read about it in Gödel, Escher, Bach as a teenager forty years ago. But it was more about the fundamental mechanics of western music's organization of sound. Where did all these notes come from?

I knew that with a string, half as long means an octave higher. What I learned first is that the Pythagoreans and their followers based an entire scale on their idea of pleasing fractional intervals. The full story is a bit complicated, but taking fourths and fifths as a starting point, you can create a scale with a version of the "whole tone" interval we're familiar with (C to D, for example) based on a 9/8 ratio of a note to its predecessor (and a half-tone with ratio 256/243).

Maor says that while this way of creating a scale is mathematically elegant, it is out of step with the pitches produced in harmonic overtones -- and thus with the fundamentals of acoustics itself. From a philosophical point of view, Maor argues that their obsession with formal beauty led Pythagorean followers into a self-circular mathematical maze: by insisting on mathematical simplicity, pleasing ratios, and no irrational numbers, they "subject[ed] the laws of nature to their ideals of beauty."

But the more amazing moment for me came later, when I learned about tempering. As I kid, I knew about Bach's Well-Tempered Clavier, and I used to enjoy imagining that the well-tempered clavier had replaced a sour and irritable keyboard instrument known as "the ill-tempered clavier." But in all these years, I never learned what tempering is.

The background for tempering in the Western context happened around 1550, when a new scale was created bringing in intervals of thirds and their inversions. This "just-intonation" scale has intervals close to the ones we'd find on a modern piano. Maor argues that as it is based on the first six members of the natural harmonic series, and has pitches corresponding to the natural harmonics of musical instruments, it is acoustically and empirically superior to the Pythagorean scale. You could say that it has formalism that is not so much mathematical but rather musical.

However: a crucial feature of the just-intonation scale is that not all the tone ratios are the same. For example, in a C-major scale, the ratio of C to D would be 9/8 and from D to E, 10/9.

From a musical point of view, Maor argues, this is as it should be. It's from the practical and social point of view that these differences became a problem. For instruments with keyboards and fixed holes, a just-intonation tuning in one key will have notes at slightly different pitches from those in another key. Increasingly, people wanted to play together, with multiple instruments all at once. What if a group wants to play one piece in one key and another in another? The workarounds were complicated: early harpsichords had multiple keyboards, each tuned to a different key, for pieces written in different key signatures.

"Tempering," then, is creating a scale with even divisions, all the same. You just divide the octave into 12 equal semitones. Now all the instruments can play together, and playing in C is the same set of notes as playing in any other key.

Math people may already see that, unlike the just-intonation scale, the ratio of a note to its predecessor in the tempered scale is based on irrational numbers -- numbers that cannot be expressed as fractions. Having even divisions requires dividing the scale into equal parts -- in this case, 12 equal half-tones -- and to be equal, each ratio between a note and its predecessor in the sale would have to be the twelfth root of 2 to 1.

Maor: "This irrational number would have been regarded with horror by the Pythagoreans, as it cannot be written as a ratio of integers." !!!

Leaving aside Pythagorean worries, Maor describes tempering as "an acceptable compromise" between "the dictates of musical harmony" and "the practicality of playing a piece on the keyboard." The difference between the tempered semi-tone and the just-intonation semi-tone is just barely within what human ears can discern, a difference "most musicians were willing to live with." Maor alludes to a suggestion that Bach's Well-Tempered Clavier was partly PR: written to convince his fellow musicians of the benefits of this sociable way of organizing everything.

I found myself astonished by idea that the structural simplicity I associate with piano keys is the result of an "acceptable compromise" to solve practical problems of musicians playing together.

From a personal point of view, I guess I thought that the Western division of the octave into 12 equal semitones was somehow linked to the fundamental structure of music and sound. Not that it was the only way to exploit that structure -- I've always known that non-Western music was organized differently -- but that it was one way. Now this book says not only is that not so, but it's even weirder: there is structure of music and sound, and those equal piano divisions are really, deeply, not it.

From a philosophical point of view, the analysis provides an interesting reminder about the subjectivity of concepts like simplicity and elegance. For the Pythagoreans, these concepts translated to rationality. For the just-intonation fans, they relate to harmonics and ratios. For the temperers, it's like brute force to make it work -- hey, make 12 that are exactly the same, whether they are rational or not.

From a mathematical point of view, it's striking to see how slippery the difference between natural numbers and other numbers can be. The nineteenth-century mathematician Leopold Kroeneker is famous for having said "God made the integers; all else is the work of man." But when you can describe the tempered scale in terms of "12 equal divisions" or "12 irrational tone ratios" -- well, things start to seem less clear. Even in math, simplicity and elegance can mean one thing in one context and another in another.

From a sociological point of view, I found myself wondering: did books like this used to be more common, or more talked about, or something? I feel like Gödel, Escher Bach was a bit splash of a book, right around the time of other math books like the Gleick book Chaos. But now my bookstore doesn't even have a math section. Did the world change, or is it something about me that's different?


thefringthing said...

Michael Rubinstein of the Pure Math department has a couple of interesting pages up on these matters:

- Well v.s. Equal Temperament
- Why 12 notes to the Octave?

Stefan Kac said...

One way the world may have changed (I'm speculating here but based on ample personal experience) is that most "professional" musicians know somewhat less (perhaps *considerably* less) about all of this than you've just demonstrated in your post; hence they become hysterical when made aware of the highly contingent and contrived nature of something they've taken wholly for granted their entire lives.