The use of 53 frets per octave in Pythagoras' primary microtonal mode was a bit of an accident. I set out to tune strictly to the harmonic series and avoid any equal tunings, but got pushed into it. If all you do is line up your strings to just fourths (ie: 4/3 ratio rather than 2^(5/12) standard tuning), then that tiny little offset sets off a chain of consequences. I just drew lines straight up and down for 4/3 coming out of each octave, and the Just 4th offset caused the 53 frets per octave to appear. A Pythagorean tuning is by definition, merely a walk around the circle of Just Fifths. You can stop at 12 and pick those as piano notes, and just realize that when you try to modulate, the primary keys will sound excellent, but some keys will sound bad because the piano pitches are fixed but not equally distributed. You can go around the circle in both directions to get distinct sharps and flats for the different situations and add 5 more sharps to get a 17 note piano with duplicate (but slightly differently tuned) sharps and flats. You could go around the circle of fifths a little more to pick 22 notes and get a set of notes that is roughly the Indian Sruti scale.
The point is that this is something being dictated by Physics rather than musical numerology. Pitches are important in this case, so notes should have something to do with the spectrum itself. My hope is to get people out of the habit of thinking of tuning and pitch as something that you hire some piano tuner guy to think about (or just let a computer dictate it all for you). This is because it isn't really the exact pitches that are important in microtonality, but the very distinct Timbre that gets created by perfect ratios in chords and sympathetics, especially ones that sound new because they can not exist in the 12 tone system.
But only a few days ago I really understood what this meant. Besides saying that if you tune your strings using the harmonics, then if you had a 53 fret per octave guitar you would be basically perfectly in tune after that procedure. Not perfectly in tune in the sense as being in tune with an electronic piano, but with the harmonic series which the piano tuning tries to approximate (somewhat badly); octaves, fifths, fourths and major thirds specifically are very much in tune.
But the other thing that it meant is that this system which matches so nicely with the spectrum is very much like the Turkish Makam system. A Just "Whole Tone" is the pitch you get by going up two Just Fifths and back down one Just Octave. You can very nearly use an equal division system to deal with it, but it isn't as simple as a whole number of Just Whole Tones in an octave. This is what that "Wolf Interval" story you hear about tuning is all about.
In chromatic scales, a whole tone is two chromatic "frets". A Just Whole Tone is slightly larger. 6 chromatic whole tones make exactly 1 octave. These chromatic whole tones are "unphysical" in that their sound physics justification is very rough. The spectrum kind of roughly breaks up into 12 equal-ish parts when you take all combinations of fifths and octaves, but it is really rough. It doesn't line up all that well.
The real Just Whole Tone if it were split into 9 equal parts would have about 6 of these whole tones in an octave. Except six Whole Tones overshoot the octave by an about that is very nearly equal one of these 9 equal parts.
Six whole tones:
6 * 9 = 54
That's 54 "frets". There are 53 frets per octave. So we fretted the fret just above the octave on that sixth Just Whole Tone. When playing Pythagoras, seeing the simultaneous overlay of the 12tone system over the 53tone system will really reset what you know about music theory. So it's a walk around the Just circle of fifths plus one more step. Now the Wolf Interval story is just common sense.
Just as a side note to give you numerological nightmares. The next closest approximation to Just 5ths requires a much larger, but very curious number. After 665 steps of a Just 5th, the difference with octave is totally imperceptible on its own, let alone when the error is spread across 665 frets. So if you go one fret beyond that, you are at 666 Just Fifths. But you don't need to go there, because the approximation is so close that there is no practical reason to keep going. The extremely tiny interval that makes up this fret step is called "The Satanic Comma" for obvious reasons.
Every few days I pick it up to play, I get a new insight that I didn't have before I started.
To me, learning is whole point of playing music. It's why I would prefer to write an instrument than to just be a consumer of it. It's why I roll my eyes at the sort of musicianship that has no patience for doing anything from first principles. It's why I enthusiastically learned theory when I played guitar, and am enthusiastically trying to come up with something new with respect to pitch now that people are starting to play fretlessly on touch computers.
See the similarity to this article http://en.wikipedia.org/wiki/Makam. When I saw the image, which I posted at the top, I didn't have to read any more (as it's mostly ethno-musicology stuff and jargon that doesn't interest me). The day before, I had decided that my instrument had a natural coordinate system of ( (9/8)^x, (4/3)^floor(y) ), where 9/8 was chosen because it was the closest line in terms of the distance of the interval from the circle of fifths. When I saw this Makam article, I finally went to make Pythagoras widen so that I could hit every interval exactly, which I needed to run on the phone. Compare it to the picture of Pythagoras, which emerged as a happy accident of my attempt at drawing the full 3-limit on the screen.
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