The Pythagorean scale is built from stacked perfect fifths (3:2) but cannot close back to the octave exactly

The Pythagorean scale generates its pitches by repeatedly multiplying by 3/2 (a perfect fifth) and reducing by 2/1 (an octave) to stay within one octave. Starting from C: C, G, D, A, E, B, F#, C#, G#, D#, A#, F## — 12 stacked perfect fifths nearly returns to C, but not exactly: the result is 531441/524288 ≈ 1.0136, which is 23.46 cents (the Pythagorean comma) sharp of the target. This means a closed 12-note Pythagorean system is mathematically impossible — some fifth must be flattened (the ‘wolf fifth’) or the scale must spiral outward infinitely. The Pythagorean scale gives very pure fifths and fourths but significantly wide major thirds (408 cents vs. the just 386 cents).

Examples

A 12-tone Pythagorean keyboard with one wolf fifth between G# and Eb. Music avoiding that fifth sounds with pure fifths; accidentally landing on the wolf fifth produces a jarring flat-sounding interval. The circle of fifths diagram in tonal theory reflects this Pythagorean spiral structure.

Assessment

Explain why 12 perfect fifths cannot stack to produce a perfect 7-octave span. Define the Pythagorean comma and give its value in cents. How does equal temperament ‘solve’ the Pythagorean comma problem, and what does it sacrifice?