MUSIC THEORY FUNDAMENTALS - Chapter 3, Exercise 3 Solution
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Why the Pythagorean Comma Prevents a Perfectly Pure Circle of Fifths

PROBLEM
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Explain, in your own words, why the Pythagorean comma shows that a
perfectly "pure" (exact 3:2 ratio) circle of fifths cannot actually
close into a real circle at all.

SOLUTION
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For the circle of fifths to genuinely "close" - meaning that stacking
twelve real perfect fifths in a row lands you back on exactly the same
pitch class you started from, just seven octaves higher - two real
numbers would need to be exactly equal:

    stacking 12 real 3:2 perfect fifths = (3/2)^12
    rising 7 real octaves (each a 2:1 ratio) = 2^7

Calculating both:

    (3/2)^12 = 531441 / 4096 ≈ 129.746
    2^7 = 128

These two numbers are NOT equal - 129.746 does not equal 128. The real,
small gap between them is the Pythagorean comma: dividing one by the
other gives a ratio of about 1.0136, which converts to roughly 23.46
cents (about a quarter of a semitone).

ANSWER: A perfectly pure circle of fifths cannot close because twelve
real, exact 3:2 perfect fifths genuinely overshoot seven real octaves by
a small but real, nonzero amount (the Pythagorean comma) - it is a
mathematical fact about the two ratios themselves, not a flaw in tuning
technique or an error that could be corrected by playing more precisely.

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WHY THIS WORKS AS AN ANSWER

This is exactly the real calculation Chapter 3's own warn-box referenced:
(3/2)^12 is not exactly equal to 2^7, so the "circle" of fifths is really
a spiral that never quite lands back where it started using pure
ratios. This is precisely why equal temperament exists - it deliberately,
slightly shrinks every real perfect fifth by a tiny, evenly-distributed
amount so that the twelve fifths do land exactly on the octave, at the
cost of no single fifth being a perfectly pure 3:2 ratio anymore.