The following two links provide a brief introduction to Pythagorean comma:
Pythagorean Comma - Indiana University
The "Pythagorean Comma" - Jody Nagel
The wikipedia article on this subject is also quite good.

I will continue referring to Jody Nagel's link for the rest of this post. Let me define two more terms before I proceed. I define an Octave Jump as multiplication by 2 and a Fifth Jump as multiplication by 3/2.

If you start from a frequency of 100 Hz, after 7 Octave Jumps you reach 12800 Hz (i.e. 100 Hz * 2^7). Likewise if you start from the same place, after 12 Fifth Jumps you reach 12974.634 Hz (i.e. 100 Hz * (3/2)^12). The numbers 12800 and 12974.634 are very close and the difference between them is very small. This is as good an illustration of the Pythagorean commas any I guess. But what's the significance of this error?

I was intrigued because Euclid seems to have analyzed this. But it's obvious that powers of 3/2 and powers of 2 cannot ever converge at the same point. That much, I am sure, was obvious to Euclid as well (centuries ahead of his contemporaries that he was). So why was this being analyzed? What's the musical significance of the Pythagorean comma?

  • 4
    Well, one significant result of this discrepancy is that if you stick to the Pythaorean definition of octaves and fifths, then you cannot have the usual concept of "circle of fifths" (at least not precisely). What you get instead is a "spiral of fifths". So, for instance if you keep changing the key by going up by a perfect fifth starting from C major, then you never get back to C major.
    – Old John
    Jun 20 '15 at 18:28

It is significant when you are trying to tune an instrument by ear, using the purity of intervals as your guide. You (and the pages you link) refer to jumping up 7 octaves vs. 12 fifths, but don't forget that any notes you reach that way can also be brought down by one or more octaves as well.

To illustrate this, let's bring all the notes down into the same octave. For the sake of simplicity, let's define a new compound interval: a Whole Tone is obtained by ascending two perfect fifths, then descending an octave (e.g. C to D). In terms of ratios, this is equivalent to multiplying a frequency by:

  • (3/2) x (3/2) x (1/2) = (9/8)

Since cents are logarithmic units, the same formula can be expressed as an addition:

  • 702 cents + 702 cents - 1200 cents = ~204 cents

In a moment, we'll define a whole tone scale, but first let's take a look at the line of fifths, just to refresh our memories of how these notes are named. Specifically, notice how each successive fifth must be named with the letter that's five letters "higher" (counting both endpoints), even if it has to be modified via a sharp or flat. This will be important when we start naming notes. Also note that the line is infinite.

B♭♭ - F♭ - C♭ - G♭ - D♭ - A♭ - E♭ - B♭ - F - C - G - D - A - E - B - F♯ - C♯ - G♯ - D♯ - A♯ - E♯ - B♯ - F♯♯

Now we're ready to create a scale from whole tones. Since each whole tone is two fifths away, we're using the note name two places to the right along the line of fifths (which always will use the next letter in the alphabetic sequence). I won't calculate any actual frequencies here, just the frequency ratio and number of cents difference between each note and our starting note.

  • C = (9/8)0 = +0 cents
  • D = (9/8)1 = +204 cents
  • E = (9/8)2 = +408 cents
  • F# = (9/8)3 = +612 cents
  • G# = (9/8)4 = +816 cents
  • A# = (9/8)5 = +1020 cents
  • B# = (9/8)6 = +1224 cents

Here you can see that our "one-octave" scale actually overshot an octave by one Pythagorean Comma (caution: I've been slightly careless with rounding the cents values: 1224 should actually be 1200 + P. Comma). There's a slight, but distinct, difference between the sound of a B♯ and a C -- if you try to substitute one for another it will sound distinctly out of tune.

It may not seem like a big deal to cut out a B♯ -- no one uses that note much anyway, right? -- but if you follow this process backwards, you can go down 204 cents at a time from the high C and get the following notes:

  • C' = 2x(9/8)0 = +1200 cents
  • B♭ = 2x(9/8)-1 = +996 cents
  • A♭ = 2x(9/8)-2 = +792 cents
  • G♭ = 2x(9/8)-3 = +588 cents
  • F♭ = 2x(9/8)-4 = +384 cents
  • E♭♭ = 2x(9/8)-5 = +180 cents
  • D♭♭ = 2x(9/8)-6 = -24 cents

Comparing the two charts, we now have a problem because every note in our scale has at least one alternately-named doppelganger, separated by a Pythagorean comma (e.g. F♯ at 612 cents vs G♭ at 588 cents). Some of these might not seem like they make a lot of sense (it would be a rare piece that needs to use D♭♭ or B♯ instead of C, though it could theoretically come up). But some cases come up quite frequently. If you're playing in the key of A minor, then G♯ (816 cents) comes up frequently as a leading tone. But if you want to play in a key like C minor, F minor, or any key on the "flat" side of the circle, then you're going to be sorely missing that A♭ (792 cents), which will be painfully out of tune if you try to substitute the G♯ (maybe you could play in the key of G♯ instead of A♭, but then you're going to need that B♯ instead of C, which we already said we wouldn't use...).

If your instrument is capable of playing any possible pitch (e.g. human voice & unfretted strings) you don't care, you just adjust as needed. But if you're designing an instrument with a fixed number of pitches, like a keyboard, you have to provide a way to play both, or else just never use one of them.

There are a few possible solutions:

  • Just completely avoid the bad chords. This is why you never see, for example, A♭ or Fm chords in early music.
  • Use a keyboard with split keys, so that, for example, both G♯ and A♭ exist as separate keys -- such keyboards were built historically, but never really caught on. Theoretically, this splitting process could go on forever, since the circle never closes. In practice, there can only be so many playable keys, so you have to pick an arbitrary stopping point. You may have increased the number of playable keys, but you still reach a limit to what you can play: you haven't closed the circle and you can't easily get from one side to the other without a horrible sounding interval somewhere.
  • Detune everything slightly, called tempering, so that no single interval sounds too bad. There are actually lots of ways to do this that were experimented with in the Renaissance through Classical eras; modern "equal temperament" only caught on relatively recently (end of the 19th century, I think). Before that, the dissonance was spread around unequally, so that each key was usable, but had a subtle but unique "color" to it, determined by where the dissonances were located in the tempered scale. Bach's "well tempered" clavier used such an unequal temperament, and each key had its own color.

As a historical side note, this non-closure of the circle of fifths didn't really seem to bother composers in the Renaissance and early Baroque; they were mostly happy writing in a limited number of keys (though, the increased use of chromaticism in the Baroque did start to exacerbate the problem). What was seen as a much worse problem with Pythagorean tuning was its out-of-tune thirds: C-E is (9/8)2, or 408 cents, while a "pure" major third (per the harmonic series) has the ratio (5/4), or 386 cents. This difference, which ends up being about 21.5 cents, is called a Syntonic Comma (distinct from the Pythagorean Comma). Composers and theorists at the time were in love with thirds, so a popular tuning system used in the Renaissance and early Baroque was Quarter Comma Meantone. This system made all the fifths too flat by 1/4 of the Syntonic Comma -- more than was needed to close the circle -- so that the major thirds would be perfectly in tune. This actually overcompensated for the non-closure issue and, in fact, made it worse. For example, using three pure major thirds, a B♯ ends up being considerably less than C:

  • C = (5/4)0 = +0 cents
  • E = (5/4)1 = +386 cents
  • G♯ = (5/4)2 = +772 cents
  • B♯ = (5/4)3 = +1158 cents

Notice here that our Meantone B♯ undershoots C by about twice the amount that our Pythagorean B♯ overshot it by. This system has thus all the same non-closure issues (and possible solutions) that we discussed above, only worse, and in the opposite direction.

It was a tradeoff that made sense at the time, but eventually composers wanted to do more. Bach, for example, is said to have tormented his organ-builder by playing that terribly out-of-tune A♭ chord at full volume.

  • This answer pretty much touches all the questions I had going on in the back of my mind; I just couldn't articulate it properly and went ahead with what I could articulate. Thanks for the comprehensive answer (esp. for the split and tempered keys part)! Jun 23 '15 at 1:00

To complete what Caleb is saying in his answer, I had added this, few years ago to the French Wikipedia article about commas, which might help shed light on your question as well :

In fact, when we spread the Pythagorean comma over, for example, 4 fifths (do-sol-re-la-mi), then the interval of third do-mi is truncated by a Pythagorean comma. Truncated by a syntonic comma, this third do-mi would be pure (ratio 5/4)... But given the quasi-equivalence between the two Pythagorean and syntonic commas (which is mathematically remarkable), this is fine in temperament calculations, which, being physically concerned with dividing the Pythagorean comma over the circle of fifths, are in fact mainly interested in reducing the falsity of thirds, linked to the syntonic comma!

(because the falsity of fifths themselves, if spread regularly over the circle, would not by themselves be a problem: As you probably guessed, a 12th of comma difference in a fifth is never really a topic in itself. But by contrast, a full or half comma difference, in a third, or any interval, is a real topic.)

(source of the idea: "Musique et tempérament", Pierre-Yves Asselin, Éditions JOBERT, 2000, 236 p. (ISBN 2-905335-00-9) )

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