RE: [MusicTheory] Interval values

Hi Bob Helsloot.

Looks like you assumed (on the Music Theory list) 11:12 for the minor
second.

11:12 is actually halfway between a minor second and a major second.

A minor second in 12-tone equal temperament is almost exactly 18:17.

Looks like you made the same error for the major seventh too.

Anyway, this is what I get for 12-tone equal temperament with harmonic
entropy relative to the unison (for those on the harmonic entropy list, I'm
using s=1.2%).

unison: 0
minor second: 1.8019
major second: 1.7117
minor third: 1.6859
major third: 1.662
perfect fourth: 1.5475
tritone: 1.7047
perfect fifth: 1.3576
minor sixth: 1.7091
major sixth: 1.6349
minor seventh: 1.7047
major seventh: 1.7465
octave: 0.7173

(higher numbers more discordant)

The minor sixth was the last of the traditional "consonances" to be accepted
as such -- it was originally considered a _sharp_ dissonance in early
European musical theory (we're talking medieval times), and when it was
accepted, it was tuned more purely than today. I think our musical grammar
has developed so that we recognize the minor sixth as consonant when it's
the inversion of the major third; note that the augmented fifth is still
perceived as dissonant, even though it's the exact same interval!

As for the fourth being dissonant in jazz, that is: (a) in the context of a
larger chord -- here we're only talking about dyads; and (b) not really a
fourth but a fourth plus several octaves, since the "root" appears in the
bass and the "dissonant fourth" in the melody.