Twelve stacked fifths slightly overshoot seven octaves by the Pythagorean comma
Evidence
Thus 12 perfect fifths (C to c) is slightly sharp compared with seven octaves (C to c’) of the so-called “Pythagorean comma” which has a frequency ratio: — David Howard, Jamie Angus, coustics and Psychoacoustics, 1035
Categorization
Category: Factual Claims
- QUESTION: The Pythagorean comma’s mismatch is a foundational tuning problem; worth chasing how equal temperament resolves it for a future piece on musical tuning.
- XREF: The tuning imperfection resonates with the reader’s interest in how perceptual systems tolerate minute deviations, echoing themes of acoustic imperfection.
Related
- Twelve stacked perfect fifths fail to close the octave exactly (similarity 0.90)
- Pure perfect fifths never close the octave, leaving a 24-cent Pythagorean comma gap. (similarity 0.87)
- The octave and perfect fifth are mathematically incompatible, forcing the fifth’s compromise. (similarity 0.84)
- The Pythagorean comma requires redistribution across fifths for multi-key tuning (similarity 0.84)
- Pythagorean scales are constructed by stacking perfect fifths (similarity 0.79)
Broader Connections
- Twelve stacked perfect fifths fail to close the octave exactly
#corroborates(0.95) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1012 - Pythagorean scales are constructed by stacking perfect fifths
#expands(0.90) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1017