FW: [tuning] triadic harmonic entropy

Dan wrote (privately, but he asked that I forward this),

>Wow, this really looks great.

Thanks. Here's one using N=100 instead of N=64:

http://www.egroups.com/files/tuning/triad100.jpg

I found the local minima -- the cells where the total pairwise H.E. is lower
than inin the eight neighboring cells -- within this region. The
corresponding chords are:

low mid
low mid upp : :
¢ ¢ ¢ mid upp

0 314 584 ~5:6|6:7 (otonal 7-limit)
0 432 702 ~7:9|6:7 (utonal 9-limit)
0 313 626 ~5:6|5:6 (maybe both squished so outer int. apprx. 7:10)
0 270 584 ~6:7|5:6 (utonal 7-limit)
0 387 702 ~4:5|5:6 (otonal 5-limit)
0 498 813 ~3:4|5:6 (utonal 5-limit)
0 315 702 ~5:6|4:5 (utonal 5-limit)
0 428 815 ~7:9|5:4 (7:9 squished so outer int. apprx. 5:8)
0 498 885 ~3:4|4:5 (otonal 5-limit)
0 390 780 ~5:4|5:4 (stretched so outer interval nears 5:8*)
0 387 815 ~5:4|7:9 (7:9 squished so outer int. apprx. 5:8)
0 270 702 ~6:7|7:9 (otonal 9-limit)
0 442 884 ~7:9|7:9 (both stretched so outer int. apprx. 3:5)
0 315 813 ~5:6|3:4 (otonal 5-limit)
0 387 885 ~4:5|3:4 (utonal 5-limit)
0 498 996 ~3:4|3:4 (symmetrical 9-limit)

*the interval 9:14 is too complex to get anything like a local minimum on
the harmonic entropy curve with s=1%. Instead there is a steep downward
slope from about 750¢ to about 810¢.

For those people who like dissonance, the local maxima in this region are:

lower middle upper
¢ ¢ ¢

0 353 604
0 458 744
0 544 835
0 355 648
0 331 662
0 413 752
0 449 792
0 251 604
0 293 648
0 368 737
0 531 941
0 339 752
0 422 843
0 343 792
0 538 995
0 286 744
0 460 920
0 410 941
0 457 995
0 540 1080
0 291 835

Again, these calculations only take pairwise interactions into account --
they ignore harmonic/subharmonic distinctions, or any other
holistic/synergistic effects inherent to the triad as a whole.