Pure perfect fifths never close the octave, leaving a 24-cent Pythagorean comma gap.

Evidence

We pick a point on the circle, arbitrarily calling it C. As we progress clockwise around the circle, the logarithm of the frequency increases in proportion to the angle, finally reaching the C′ above where we have come full circle ideally at an angle of 2π. But C and C′ are equivalent, so the circle contains all the information we need. We plot our progress so far on such a circle using fifths. Beginning at C, we take leaps of 2π log2(3/2) in angle, which correspond to subtending an angle of about 210.587 degrees. We assign new letters of the alphabet to the points at which we arrive on the circle. After four such jumps, we have the the situation shown in figure 26.15. The circle of fifths makes it clear how we can continue this process to generate more notes. We continue around the circle making leaps of 210.587 degrees, until we come back to C. Unfortunately, because of the Pythagorean comma, we won’t quite make it back, but instead will miss by about 24 cents, as seen in figure 26.16. If we keep going with jumps of 3/2 in frequency, bringing them always down into the first octave, we first generate the note B, followed by F♯, C♯, G♯, E♭, B♭ and finally F. F corresponds to n = 11, m = 17, and our formula 3n/2m gives 177147/131072 = 1.35152, or 21 cents above the modern equal temperament F (see the following), and 24 cents above the harmonious ratio 4/3. — Eric J. Heller, Why You Hear What You Hear- An…, 2387

Categorization

• DEFINE: Defines the Pythagorean comma concretely as the roughly 24-cent shortfall that keeps the circle of fifths from closing.
• CANDY: The fact that stacking pure perfect fifths misses home by 24 cents is a fun, tweet-grade music fact.