Twelve stacked perfect fifths fail to close the octave exactly
Evidence
The Pythagorean scale is built up from the perfect fifth. Starting, for example, from the note C and going up in 12 steps of a perfect fifth produces the “circle of fifths”: C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, c. The final note after 12 steps around the circle of fifths, shown as c, has a frequency ratio to the starting note, C, of the frequency ratio of the perfect fifth (3:2) multiplied by itself 12 times, or: — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1012
Categorization
Category: Definitional Content
- DEFINE: Explains the construction of the Pythagorean scale via the circle of fifths and sets up the comma problem that motivated temperament systems.
- QUESTION: Worth confirming: the exact value of the Pythagorean comma that emerges from this construction (the 23 cents gap) vs equal temperament’s divided semitone.
Related
- Twelve stacked fifths slightly overshoot seven octaves by the Pythagorean comma (similarity 0.90)
- The octave and perfect fifth are mathematically incompatible, forcing the fifth’s compromise. (similarity 0.87)
- Pure perfect fifths never close the octave, leaving a 24-cent Pythagorean comma gap. (similarity 0.86)
- Pythagorean scales are constructed by stacking perfect fifths (similarity 0.84)
- The Pythagorean comma requires redistribution across fifths for multi-key tuning (similarity 0.83)
Broader Connections
- Ascending and descending circles of fifths produce different pitches
#expands(0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1015 - Pythagorean scales are constructed by stacking perfect fifths
#expands(0.75) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1017 - Equal temperament makes combination tones slightly out of tune
#expands(0.75) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1581