Designing scales from upper harmonics yields ratios no Pythagorean scheme can reach

Evidence

These ratios can also be inferred from knowledge of the musical intervals and the harmonic series. Figure 3.3 shows that the major sixth is the interval between the fifth and third harmonics—in this example, these are G4 and E5—and therefore their frequency ratio is 5:3. Similarly, the interval of a minor sixth is the interval between the fifth and eighth harmonics, in this case E5 and C6; therefore, the frequency ratio for the minor sixth is 8:5. Knowledge of the notes of the harmonic series is both musically and acoustically useful and is something that all brass players and organists who understand mutation stops (see Section 5.4) are particularly aware of. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 811 (#contradicts, via linked claim)

Scientific and engineering knowledge of communication processes has developed over the last hundreds, even thousands, of years. First there were beliefs that sounded rational, but a closer investigation found them to be false or only partially true. Some early thinkers and experimenters, such as Pythagoras, turned out to be very successful and influential (Hunt, 1992). Pythagoras studied the behavior of strings and understood the inverse relation between pitch and string length in ancient Greece during the sixth century B.C. — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 96 (#responds-to, via linked claim)

Categorization

• XREF: connects to: Pythagorean whole-number ratios led to the doctrine of the music of the spheres; Twelve stacked fifths slightly overshoot seven octaves by the Pythagorean comma; Musical intervals trace directly to harmonic-series frequency ratios — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 811 (+1 more)

Broader Connections

• Musical intervals trace directly to harmonic-series frequency ratios #contradicts (0.80) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 811
• Twelve stacked fifths slightly overshoot seven octaves by the Pythagorean comma #qualifies (0.75)
• Pythagoras established the inverse relation between pitch and string length #responds-to (0.75) — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 96