Music theory, medieval & modern

I went to grad school to study math and computer science, but I happened to be at a school (UCSD) with a top-notch experimental-music department, so I took a year's worth of electronic music and a year's worth of computer music. In those courses, there was an occasional discussion of alternative scales: rather than the 12 equal intervals that make up the modern Western-culture octave, what if you used a different number, or if you used unequal intervals, or if you didn't base things on an octave at all? In particular, somebody pointed out that a 19-step equal-tempered scale gives pretty nice-sounding intervals; why don't we use that?

"Nice-sounding", or "pure", is shorthand for "small-integer frequency ratios", which the human ear seems to like. A unison is two exactly-equal frequencies; if you play two notes that are almost in tune, say 440 Hz and 441 Hz, you can hear a "wow-wow-wow" at the frequency of the difference, in this case 1 beat per second; guitar players routinely use this to confirm that two strings are really in tune with one another.
A pure octave is a 2:1 frequency ratio, e.g. 440 Hz and 880 Hz. A pure fifth is ideally a 3:2 frequency ratio, a pure fourth a 4:3 frequency ratio, a pure major third a 5:4 frequency ratio, and a pure minor third a 6:5 frequency ratio.

Pythagorean tuning, which as its name suggests is thousands of years old, is based on fifths and octaves: go up a pure fifth, then another pure fifth, and you're a little over an octave from where you started, so go down a pure octave, and repeat. Unfortunately, if you tune everything by pure fifths and octaves, you end up almost but not quite where you started: 12 pure fifths is a frequency ratio of 129.746, while 7 pure octaves takes you back down by a ratio of 128, leaving you at a frequency 1.36% (the "Pythagorean comma") higher than where you started. This is enough to be audible, even to the un-trained musician: all the intervals sound beautifully in tune except this one real clunker. Which means when you tune an instrument, you have to decide in advance which intervals you want to sound good, and which one can be horribly out of tune; in practice, you decide what keys you're most likely to play in, and pick an interval that's unlikely to show up in those keys.

Another problem with Pythagorean tuning is that getting all the fifths right gives you terrible thirds: a Pythagorean major third is a ratio of 1.266, over a percent "wider" than the ideal 1.25, while a Pythagorean minor third is 1.185, over a percent "narrower" than the ideal 1.2.

People have tried various ways to solve the problem: split the comma in thirds, so there are a bunch of beautiful intervals and three less-bad ones ("third-comma meantone"), or split it in fourths ("quarter-comma meantone"), or split it in sixths ("sixth-comma meantone"), or split it in twelfths so all the intervals are equally slightly out of tune. That's "equal temperament", the way most pianos are tuned today. In equal temperament, a fifth is a frequency ratio of 1.4983, about 0.1% narrower than a Pythagorean fifth; a fourth is likewise about 0.1% wider than a Pythagorean fourth; a major third is 1.26, almost a percent too wide; and a minor third is 1.19, almost a percent too narrow.

In the 15th-17th centuries there was a different approach, called "just tuning": rather than insisting on fifths über alles, maybe we should aim for getting thirds in tune. Combining a pure major third (5:4) and a pure minor third (6:5) actually gives you a 3:2 pure fifth, so you haven't lost that after all, and the thirds sound much better than in Pythagorean tuning. But tuning the thirds still doesn't help you get all the way around the scale and end up back where you started; some intervals will be beautifully in tune, while others are horrible, and you just try to place the horrible ones in out-of-the-way corners of the keys you usually play in.

By the way, modern singers, and players of un-fretted stringed instruments like violins, are taught to use just intonation, adjusting vocal pitch or finger position slightly so that each third, fourth, or fifth is in tune when you get to it, even though it may not be in tune with where the piece started. Since most musical pieces don't modulate very far from their original key, you can usually get away with this and end up in the same key you started in, but theoretically if you played on the violin a long piece that modulated all the way around the circle of fifths, you'd end up off by a Pythagorean comma from where you started; in particular, the notes you played on open strings at the beginning would have to be fretted at the end.

I was in an early-music workshop on Friday, and the teacher mentioned the "Berkeley manuscript", a 1370's-vintage music theory text currently owned by UC Berkeley. We had a copy (in facing-page translation) at home, so I've been reading through it in the last few days. In the first section the author describes the hexachord (or deducciones -- the word "hexachord" was invented retroactively by 19th-century musicologists) system, how the C, G, and F hexachords overlap, how and why one switches from one hexachord to another, how intervals that should be a whole tone in the gamut can be narrowed to a semitone by explicitly marking one note "mi" or "fa" (what we would now call "sharp" or "flat"), etc.

But in the fifth section of the treatise (which I haven't gotten to yet; this is what the teacher and the editor say) the author says a mi-fa semitone really shouldn't be half a tone, but two-thirds of a tone (a "major semitone"; the remaining third is a "minor semitone"). If each of the five whole tones in an octave is three units, and each of the two semitones is two units, that makes... 19!

So how well does this actually work, using modern mathematical techniques (roots and exponents)?

• A major third, ut-mi or fa-la, is two whole tones, or 6/19 of an octave, which is a ratio of 1.24469, less than half a percent off from a just major third at 1.25 -- about equally bad as a third-comma-meantone major third, although narrow where that was wide.


• A minor third, re-fa or mi-sol, is a whole tone and a major semitone, or 5/19 of an octave, which is a ratio of 1/2001, less than a hundredth of a percent off from a just minor third.


• A fourth, ut-fa or re-sol or mi-la, is two whole tones and a major semitone, 8/19 of an octave, a ratio of 1.3389, about half a percent off from a Pythagorean fourth.


• Similarly, a fifth, ut-sol or re-la, is three whole tones and a major semitone, or 11/19 of an octave, a ratio of 1.49376, again about half a percent off. (This had to be true, since the perfect fourth and perfect fifth are one another's inverses: if one was half a percent off, the other would have to be likewise.)

So this system has reasonably-good fifths, fourths, and major thirds, and very good minor thirds, and it has the conceptual advantage of equal temperament that you don't need to decide which key you want to sound best: any piece of music can be transposed into any other key and its intervals will sound the same as they did before.

If you wanted to build a keyboard in this system, it would probably have the usual seven white keys (A, B, ... G), with two black keys between A and B, C and D, D and E, F and G, and G and A, and one black key between B and C and between E and F.