Mike Battaglia @battaglia01@qoto.org

@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 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]

@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 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 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]

@keenanpepper @johncarlosbaez heh, I was just thinking of tweeting this at some prominent math people on here. Yes, magically the Riemann Zeta Function, known for its role in the Riemann hypothesis, happens to magically tell you how well different equal temperaments approximate the harmonic series (and the rational numbers). Of course from this standpoint we care about the peaks rather than the zeros.

Any idea how we can interpret some of the other special functions in number theory within this paradigm?

(Also I joined here on QOTO; should I join on Mathstodon instead?)