Alternate music history

I probably should not try to say anything about ancient music history, because really I know very little about it (and what I “know” isn’t necessarily right). Still, I can’t resist the following look at a parallel universe…

In our own universe, the ancient Greeks thought about music theory in terms of tetrachords: sets of four notes, the first and fourth of which are required to make a perfect fourth, while the two in between can be varied. There were several categories and lots of particular tetrachords. I tend to assume there was a connection with 4-string lyres here, though maybe that’s not a great assumption.

For talking about (or tuning) larger numbers of notes, spanning a larger range, they combined two or more tetrachords. So, in particular, an octave could be thought of in terms of two tetrachords, where the first and last notes of the second tetrachord were a perfect fifth above those of the first. Then between the last note of the first tetrachord and the first note of the second tetrachord was a whole tone.

One particular such combination of two tetrachords can be constructed by taking seven consecutive notes on the circle of fifths — say F-C-G-D-A-E-B. Put in order starting with C they make a tetrachord (C-D-E-F) followed by a whole tone (F-G) followed by another tetrachord (G-A-B-C) with the same intervallic structure as the first. And that of course is a diatonic scale.

So… what if the Greeks were thinking more of five string lyres?

Then they might have talked about pentachords — still spanning a perfect fourth, but with five notes. For more notes and more range, two pentachords a perfect fifth apart, with a whole tone between them.

But… okay, here’s where it starts getting a little ludicrous, maybe, but let’s do it. A whole tone between two tetrachords is okay. But when you have pentachords, with four intervals adding up to a perfect fourth, those intervals average 1.25 semitones each and that whole tone starts to seem a little large. So let’s say one more note gets added, within the whole tone, which thereby gets divided into two semitones. Not, probably, equal ones.

Again there are lots of ways to tune the pentachords’ inner notes, and now also there are different ways to tune the note splitting the whole tone. But if you take ten consecutive notes on the circle of fifths — like F-C-G-D-A-E-B-F♯-C♯-G♯ — and put them in order starting with C, guess what? You have one pentachord (C-C♯-D-E-F) followed by two semitones (F-F♯-G) followed by another pentachord (G-G♯-A-B-C) with the same structure as the first. A scale of ten notes, a decatonic scale. It’s made of eight semitones, both diatonic and chromatic, and two whole tones.

(The notation makes sense for diatonic scales and not so much for decatonic ones; in our alternate universe the note names would be different, but to avoid confusion I’ll use the familiar names here.)

Now this would have been an entirely reasonable scale, I think, for Western music to have based itself upon in this parallel universe. In this tuning, it would’ve been fine for music in unisons and octaves, and still fine when they started writing music in fourths and fifths. And like the Pythagorean-tuned diatonic scale, it would’ve run into trouble once they pushed on to thirds and sixths. After all, it has the same Pythagorean major thirds, frequency ratio 81/64, more than 20 cents wider than just.

Then they might have tried just intonation, but as problematic as that is for diatonic scales, it’s even worse for decatonic. Here’s a way to think about just intonation in diatonic scales: Instead of a Pythagorean tuned chain of fifths, F-C-G-D-A-E-B, where “-” represents a just perfect fifth, we replace one of those fifths with one tempered by a full syntonic comma, denoted ~: F-C-G-D~A-E-B. A major third is four upward fifths, so there are only three of them in a seven note scale (on the first three notes), and by putting the ~ between D and A it gets used in all three of those major thirds and makes them just. There are two problems, though. First, there are four minor thirds (made using three downward fifths, starting from D, A, E, and B) and putting the ~ after D means the minor third from D to F is still Pythagorean, not just. And second, there’s a very bad fifth from D to A. (The minor third on D can be repaired by putting the ~ after G, but then we have a bad fifth on G. In C major, having the bad fifth on D is probably preferable. Not only that, but we’ve now broken the minor third from B to D.)

Things get even worse when you try to add accidentals so you can play in other keys, but let’s not go there right now.

Instead consider what’s going on in the decatonic scale. With ten notes, now there are six Pythagorean major thirds, not three. The ones on F, C, G, and D can be repaired by putting a ~ after D, but that doesn’t help the ones on A and E. To fix them you’d need a second ~:

F-C-G-D~A-E-B-F♯~C♯-G♯

Again, you haven’t repaired the minor third from D to F. But there’s also a bad minor third from F♯ to A. In this case you can repair both of them by shifting both tempered fifths:

F-C-G~D-A-E-B~F♯-C♯-G♯

which breaks the B-D third but nothing else. Again, though, you might prefer a bad fifth from D to A over one from G to D. But you have to have one or the other, and a second bad fifth later on too.

And there’s one more problem. There are two more major thirds which aren’t four fifths up, but eight fifths down: From C♯ to F and G♯ to C. Here, in Pythagorean tuning, the syntonic comma in one direction is almost exactly canceled by a Pythagorean comma in the other, so these fifths are very nearly just. But if you try imposing just intonation, they both get widened by two commas and become unusable!

So just intonation really falls apart for decatonic scales.

Maybe going halfway is a better idea. Consider this:

F-C-G-D-A~E-B-F♯-C♯-G♯

There’s only one bad fifth, and I’ve put it after A for a couple of reasons. One, it pushes it further up the circle of fifths where maybe it’ll be less problematic. Two, it preserves the pentachordal structure, with the two pentachords the same intervals.

With only one ~, only four of the six Pythagorean major thirds are repaired (and only three of the seven Pythagorean minor thirds). And the nearly just major thirds on C♯ and G♯ get broken again. But this time they’re broken only by one syntonic comma. They’re practically the same as the unrepaired Pythagorean major thirds. So they’re bad, but not that bad.

But as in our universe, someone sooner or later will think up quarter comma meantone. Will it be as popular there as it was here? Maybe not…

Use ten notes separated by quarter comma tempered fifths, and now all your major thirds composed of four upward fifths become just. Huzzah! But the thirds on C♯ and G♯ again get broken, badly. Can’t have everything.

What did in quarter comma meantone in our universe was, pretty much, modulation: there’s a two comma difference between A♭ and G♯. You can tune your clavier to have, for instance, two flats (B♭ and E♭) and three sharps (F♯, C♯, and G♯), but as soon as you’re in a key that needs A♭ or D♯, you’re in trouble. There are six such major keys (C major, and the ones using up to three sharps or two flats)… too restrictive for Herr Bach.

But there are only three decatonic keys that don’t use flats below E♭ or sharps above G♯! That wouldn’t do at all.

So of course you can go to equal temperament, probably a lot sooner than it happened in our universe. Use narrowly tempered fifths to make A♭ and G♯ the same note, and so on, and then you can play your decatonic scales in any key. And as in our universe, if you go that route, you have to give up on very good thirds.

Or…

Instead of using one scale degree for the eight semitones and two for the two whole tones, hence twelve notes to the octave, how about two degrees for the semitones and three for the whole tones? Giving you 22 notes to the octave. The parallel universe’s Guillaume Costeley and Francisco de Salinas might have pushed for that. In that tuning the thirds are significantly better than in 12-equal while the fifths are slightly worse, but still good.

And, bonus, the harmonic seventh — frequency ratio 7/4 — is pretty good, much more usable than anything in 12-equal. In fact there are good approximations to the septimal minor third (7/6) and lesser septimal tritone (7/5), too, so that usable 7-limit major and minor tetrads (4:5:6:7 and 1/4:1/5:1/6:1/7) exist — and there are some not just in the 22-equal scale but in the decatonic subset as well. So perhaps our alternate friends would have started exploring septimal harmony by the early Baroque.

In fact, this whole alternate universe fantasy has led us to Paul Erlich’s 22-equal-based pentachordal decatonic scales, constructed in our own universe specifically for support of 7-limit harmony. And to which my initial reaction was, well, this seems pretty artificial. But you know… maybe if the Greeks had used five-string lyres, it’s what we all would be singing and playing now.