Trying to work through this post.

Oh, you'd love it. The first thing to know is that the Pythagorean comma

OK so first of all, "Pythagorean comma" refers to the difference between 12 "pure" perfect fifths and 7 octaves, right?

isn't really a problem - it's very small when distributed amount 12 fifths.

Because a "pure" perfect fifth is 7.020 equal-tempered semitones, and 0.020 is pretty damn small, right?

(From 12 log

_{2}(3/2) = 7.020.)

The bigger problem is the syntonic comma, which is almost as big, and is the difference between a circle of only four fifths (e.g. from C to the E, a major third higher) and the "pure" major third given by the ratio 4:5.

Ok, (3/2)

^{4} = 81/16, versus 80/16 (=5) for two octaves plus a "pure" major third.

In order to "purify" that third, you need to distribute all of it over just 4 fifths, and that means you are simultaneously OVER-compensating for the Pythagorean comma.

I'm guessing that you're referring to tuning systems in which you'd make a perfect fifth into a frequency ratio of 5

^{1/4}:1, so that four of those perfect fifths would then give you two octaves plus a "pure" major third.

Which means that not only are your fifths too small (and much smaller than if you distributed the Pythagorean comma over 12 fifths),

12 log

_{2}(5

^{1/4}) = 3 log

_{2}(5) = 6.966. So if I'm reading you right, then this makes sense: Such a fifth is smaller than the equal-tempered version, thus overcompensating for the Pythagorean comma.

but you end up with a "wolf" at the end to make up the extra you've taken off. Which you have to put somewhere where you hope you won't need it.

I'm not sure I get this "wolf" thing, but I'm going to take a wild guess, and you can tell me whether I'm on the right track.

Suppose we're aiming to have a "pure" major third, and we're building an entire tuning system based on that. Let's look at some of the frequencies we might get within one octave from C up to B.

Let's say C = 1 arbitrary frequency unit (AFU).

G = 5

^{1/4} AFU. D = 5

^{1/2}/2 AFU. A = 5

^{3/4}/2 AFU. E = 5/4 AFU.

B = 5

^{5/4}/4 AFU. F# = 5

^{3/2}/8 AFU. C# = 5

^{7/4}/16 AFU. G# = 25/16 AFU.

But working in the other direction, we get

F = 2/5

^{1/4} AFU. B♭ = 4/5

^{1/2} AFU. E♭ = 4/5

^{3/4} AFU. A♭ = 8/5 AFU.

(I hope the flat symbol shows up properly. It does on my screen, but different machines seem to treat special characters differently.)

Now comparing the G# to the A♭, they're apart by a difference of

12 log

_{2}((8/5)/(25/16)) = 12 log

_{2}(128/125) = 0.411 equal-tempered semitones

...which is big enough to be startling. I'm guessing that your word "wolf" has something to do with this difference.

(And of course the fact that I put it at G# was arbitrary.)

Systems in which you distribute the syntonic comma between four intervals of a fifth are called "quarter comma" mean tone. "Quarter comma" because you take 1/4 of a syntonic comma off each fifth, and "mean tone" because the resulting tones are the "mean" of the two "tones" given by the ratios of 9:8 and 10:9. The first is called the "major tone", and the second the "minor tone" (not a semitone, just a smaller "tone" than the "major tone".

OK, I tried to do something like this above. And yes, the tones are 5

^{1/2}:2, which is exactly the geometric mean between 9:8 and 10:9. (So you get an arithmetic mean when you work with the logarithms thereof.)

Having all the tones the same size is convenient for Western music, but mean tone tunings have two sizes of semitone (because of the syntonic comma distribution).

Chromatic semitone 5

^{7/4}:16

Diatonic semitone 8:5

^{5/4}Right?

So on a fretted instrument, the frets are of alternating size (as well as decreasing over all as you go up the string of course).

But that's weird, isn't it? If every second fret means going up a tone (5

^{1/2}:2), then the 12th fret wouldn't give you an octave. It'd give you six tones (125:64), which is an octave minus a "wolf", or something like that. No?