The octave and perfect fifth are mathematically incompatible, forcing the fifth’s compromise.

Evidence

This is close to 24.5 cents, or about a quarter of a semitone off. Not good! The defect is called the Pythagorean comma. Both the octave and the perfect fifth are indeed perfectly harmonious as intervals for two complex tones, and yet they don’t agree on what should happen when you climb up in frequency using one or the other. The two most universal and sonorous intervals reach two mutually sour notes that spoil the whole show. Somehow and in some way, the perfect fifth has to go if the octave is to be preserved! — Eric J. Heller, Why You Hear What You Hear- An…, 2378

Categorization

Category: Definitional Content

• DEFINE: Defines the Pythagorean comma, the ~24-cent discrepancy central to all tuning theory.

Related

• Twelve stacked perfect fifths fail to close the octave exactly (similarity 0.87)
• Twelve stacked fifths slightly overshoot seven octaves by the Pythagorean comma (similarity 0.84)
• The Pythagorean comma requires redistribution across fifths for multi-key tuning (similarity 0.80)
• Pure perfect fifths never close the octave, leaving a 24-cent Pythagorean comma gap. (similarity 0.80)
• Stacking perfect fifths yields the octave, fifth, and fourth intervals. (similarity 0.80)
• No tuning system can make all diatonic scale intervals perfectly consonant (similarity 0.79)
• Leaps of pure fifths fail to close because intervals and octaves are incommensurable. (similarity 0.79)
• The major and minor sixth ratios derive by complementing thirds within an octave (similarity 0.79)
• Pythagorean tuning generates scales purely from whole-number ratios of the fifth (similarity 0.79)

Broader Connections

• Twelve stacked perfect fifths fail to close the octave exactly #expands (0.95) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1012
• Uneven intervals make Just intonation scales impossible to transpose freely. #expands (0.80) — Eric J. Heller, Why You Hear What You Hear- An…, 2383