The circle of fifths is a beautiful thing, fundamental to music theory.
Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since
(3/2)¹² ≈ 128 = 2⁷
when you're done your note vibrates about 2⁷ times as fast as when you started! We say it's 7 octaves higher.
Notes have letter names, and the notes you played form this 12-pointed star: the circle of fifths!
It's great! But....
(1/n)
I said
(3/2)¹² ≈ 128
but this is just approximate! In reality
(3/2)¹² = 129.746....
so the circle of fifths does not precisely close - see below!
This is called the Pythagorean comma, and you can hear the problem here:
https://en.wikipedia.org/wiki/Pythagorean_comma
As a result, the equal-tempered 12-tone scale now used on most pianos doesn't have 'perfect fifths' - frequency ratios of 3/2.
People have dealt with this in many, many ways. No solution makes everyone happy.
(2/2)
@johncarlosbaez Oh boy, math people talking about microtonal music and tuning theory! Maybe my chance to say something useful. (Also @johncarlosbaez I am a huge fan!)
This story about the Pythagorean comma is often the way that people introduce microtonality. It's one of those things that's not technically wrong, but like, it should be wrong. Kind of.
I will clarify that in a minute because from a purely mathematical standpoint, everything you said is correct - the math of how you temper out the Pythagorean comma to get 12-EDO, I mean - but still, in a deeper way, something about this story is wrong. Let me explain. [1/n]
@johncarlosbaez First, as you said, the octave is a 2/1 frequency ratio, and a perfect fifth is a 3/2 frequency ratio.
If these are literally the only musical intervals you care about, and simple combinations of them, then 12 tone equal temperament is an absolute mathematical miracle. The fifths of 12-equal are about as "perfect" as a "perfect fifth" can get.
The fifths of 12-equal are exactly 700 cents, and a just 3/2 ratio is 701.955 cents. The difference of 1.955 cents is far too small for human beings to hear in virtually any situation, and way within the margin of error of how precisely you can tune or play most physical instruments. You would need people whose hearing is 10 standard deviations above the mean to even be able to differentiate between these two stimuli in most real world situations.
If for whatever reason you only care about fourths and fifths, then 12 is basically as good as it gets for a small equal temperament. You ever hear how nice power chords sound with super high gain on an electric guitar? You probably just take this for granted. Well, you don't get anything that sounds quite as nice again until you get up to 29 notes per octave! (Except for 24, which is just 2*12. And 17 and 19 and 22 and maybe 27 are decent enough, but still not as good as 12.)
OK, I cannot possibly emphasize this enough: the problem with 12-equal is in no way that the fifths and fourths are too far from just intonation. So what is the "problem" with it then? [2/n]
@johncarlosbaez The problem is that we care about intervals besides just fourths and fifths.
If you just tune a standard 12-tone keyboard to a chain of 12 perfect fifths, you get something called "Pythagorean tuning." On a standard keyboard, different keys will sound different in this tuning. And people are sometimes very surprised to hear that on most instruments, chords in Pythagorean tuning sound *absolutely awful!* A Pythagorean tuned C major chord of C-E-G sounds *much* harsher than in 12-equal, particularly on a piano or keyboard instrument.
Why is this?
Because the "major third" of Pythagorean tuning is a pretty complex 81/64 ratio, which sounds kind of crappy. Our ears like simple frequency ratios, and 81/64 is not simple. And this brings us to the second reason why this whole story about the Pythagorean comma "should be wrong": [3/n]
@johncarlosbaez Europe was *very* focused on this comma and built an entire style of music built on the idea that it should be tempered out. The resulting temperament is called "meantone temperament," and basically corresponds to a deep mathematical property about Western music.
It literally cannot be overstated how important to the mathematical structure of Western music it is that this comma be tempered out. I mean, for basically any Western style of music you can think of - classic rock, jazz, classical music, R&B, whatever - there will be chord progressions that literally do not work *unless* this comma is tempered out. The thing that most people call "music theory" should probably be called "meantone theory."
The thing is that the fifths of your typical meantone temperament are even *flatter* than 12-equal, and even further from a just 3/2 ratio - but that's alright, because the thirds are closer to 5/4 and 6/5. We can do a kind of least-squares error optimization to get the "best" meantone tuning, which splits the difference of the approximation error between 3/2 and 5/4 and so on, and it turns out to be very close to 31 tone equal temperament - also something people were studying hundreds of years ago.
As a result, major chords in 31-equal sound "crunchier" than they do in 12-equal, as they are much closer to the idealized 4:5:6 ratio (even though the fifths are a little further off; they're about 697 cents rather than 700 cents). [5/n]
@johncarlosbaez OK, this is long enough, and I didn't expect it'd be that long, but hopefully people who are interested in microtonality will think this is interesting.
The TL;DR:
1. As far as small equal temperaments go, 12-equal has about as good of an approximation to the perfect fifth and fourth as you can get.
2. The "problem" with it is the other intervals, not the fifths and fourths.
3. The Pythagorean comma isn't even the comma most people were focused on, historically, at least in the setting of Western polyphonic music.
4. The real comma of interest in that regard is the syntonic comma, and tempering it out gives you meantone temperament.
5. 31-equal is about as good as meantone gets; chords sound much crunchier.
If you want to hear what any of this sounds like, here are some musical examples in 31 equal:
Sweet Lorraine: https://www.youtube.com/watch?v=RGZ0JlMwZpY
Speak Like a Child:
https://www.youtube.com/watch?v=gKT3W2aF4LA&t=54s
Infant Eyes: https://www.youtube.com/watch?v=uIYg8b2p8JY
Anyway, this was fun though longer than I expected! End rant. [6/6]
@johncarlosbaez The second reason is that from a historical standpoint, the Pythagorean comma *is the wrong comma!* Or at least for Western polyphonic music. The comma that the Europeans were focused on, for hundreds of years, is called the *syntonic comma*: (https://en.wikipedia.org/wiki/Syntonic_comma)
This comma is the difference between the "major third" you get if you stack four fifths minus two octaves (81/64), and the "major third" that appears earliest on in the harmonic series (5/4). It's about 20 cents (similar in size to the Pythagorean comma).
Why is this important? Because the entire reason that your ears like fourths and fifths is that they like frequencies being played in simple rational relationships. So if you're in Pythagorean tuning, and you have some bizarre major chord of which that third is 81/64, there is a HUGE tendency to flatten that third some 20 cents toward the much simpler 5/4, making the entire major chord a *very* nice 4:5:6 ratio.
This realization goes back to the mathematician Claudius Ptolemy in the second century, who noted that real-world musicians often deviated significantly from a strict Pythagorean chain of justly-tuned fifths to do things like this. [4/n]