John Carlos Baez on Nostr: Western music has long used chords with fractions built from the primes 2, 3, and 5. ...

Western music has long used chords with fractions built from the primes 2, 3, and 5. What about 7?

I'll paraphrase Gann's book The Arithmetic of Listening:

Four hundred years ago, the possibility of using seven as a tuning ratio was still up in the air in Europe. Zarlino, in his Le institutioni harmoniche of 1558, had enshrined the number six as the “sonorous number” beyond which no consonances. Over the next couple of centuries, though, various theorists argued for septimal (seven-based) intervals, only to retreat in the face of common practice.

The mathematician Mersenne claimed in 1636, “I have not the slightest doubt that the dissonant intervals of which I have spoken . . . i.e., the ratios 7:6 and 8:7 that subdivide the fourth—may become pleasing if one accustoms oneself to hearing and bearing them . . . for diverse effects that ordinary music lacks.” Later, however, he changed his tune and his tuning, writing that because 7/6 “is neither a consonance nor a difference of consonances, nature—which is harmonic—rejects it and prefers to interrupt its series of intervals and melodies than to move through an interval that serves no purpose except to wound the ear and the spirit.”

The astronomer and scientist Huygens likewise wrote that the number seven “is not unable to produce a consonance” but then dismissed 7/6 and 8/7 as being “incompatible with the consonances already established."

And so it went, writer after writer admitting that septimal intervals could please the ear before deciding that employing them was inconvenient. It was as though if you came out publicly for the number seven, a couple of thugs from the Six-Based Mafia showed up at your office for a little persuasion.