Musical scales universally favor intervals with small integer frequency ratios

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 (#corroborates, via linked claim)

The musical intervals of adjacent harmonics in the natural harmonic series starting with the fundamental or first harmonic, illustrated on a musical stave and as notes on a keyboard in Figure 3.3, are: octave (2:1), perfect fifth (3:2), perfect fourth (4:3), major third (5:4), minor third (6:5), flat minor third (7:6), sharp major second (8:7), a major whole tone (9:8) and a minor whole tone (10:9). … The frequency ratios for intervals between nonadjacent harmonics in the series can also be inferred from the figure. For example, the musical interval between the fourth harmonic and the fundamental is two octaves, and the frequency ratio is 4:1, equivalent to a doubling for each octave. Similarly, the frequency ratio for three octaves is 8:1 and for a twelfth (octave and a fifth) is 3:1. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 803 (#corroborates, via linked claim)

Categorization

• XREF: connects to: String length ratios explain musical intervals since Pythagoras; No pure integer-ratio system can tune musical intervals without compromise.; Simple integer frequency ratios minimize partial interference and thus dissonance (+12 more)

Broader Connections

• Musical intervals trace directly to harmonic-series frequency ratios #corroborates (0.92) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 811
• Adjacent harmonics of a fundamental produce every familiar musical interval #corroborates (0.88) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 803
• Adjacent scale intervals are found by dividing note frequency ratios #corroborates (0.82)
• Interval ratios between adjacent scale notes derive from division of C-based ratios #expands (0.80)
• Musical intervals between adjacent harmonics shrink as harmonic number rises #expands (0.78)
• Chord consonance is maximized when singers’ fundamentals are integer-related harmonics #corroborates (0.75)
• Harmonic tones are integer multiples of a fundamental frequency #expands (0.72) — Jennifer Lentz, Psychoacoustics- Perception of Normal…, 1180