RE: [tuning] Re: tetradic model [harmonic entropy]

John deL: I'll contact you off-list about the coding.

>Surely 4:5:6:7 would float toward the
>top compared to the myriad of chords that "beat" it in the binary
>summation calculation!

Certainly. But remember, most of those chords didn't really beat it in a
very meaningful way, but mainly due to their wider intervals. For example,
if we had constrained the outer interval to be no more than 1000¢ (rather
than 1650¢), only seven chords would have "beat" 4:5:6:7 and its mirror
inversion. These are: 12:14:18:21, which has two perfect fifths; 5:6:8:9 and
its mirror inversion, each of which has a perfect fifth and a perfect
fourth; 6:8:9:10 and its mirror inversion, each of which has a perfect fifth
and a perfect fourth; and, just by a hair, and the slightly tempered
1/1:6/5:3/2:5/3 and its mirror inversion, each of which has a perfect fifth,
a major third, and a major sixth . . . . Concerning this last chord, there
was an error in my post:

>The 18:25s in these chords are actually expanded relative to JI, to help
them >approximate the concordant 5:7.

That's right, but

>These are thus pretty close to inversions of the 7-odd-limit tetrads

is not the case -- the corresponding interval in the inversions of the
7-odd-limit tetrads would be a more discordant 7:10, rather than a 5:7.

Anyway, the six intervals in 1/1:6/5:3/2:5/3 (or its mirror inversion) are a
perfect fifth, a minor third, a major third, a 5:7, a major sixth, and a
9:10. Compared with 4:5:6:7 (or its mirror inversion), which has the same
first four intervals, a 4:7, and a 6:7, the model chooses the former chord,
because the harmonic entropy values are

major sixth -- 3.755
4:7 -- 3.879
6:7 -- 4.344
10:9 -- 4.424

and 3.755+4.424=8.179 < 3.879+4.344=8.223 (just barely, but with enough
leeway to allow the tempering to take place).