Yahoo Tuning Groups Ultimate Backup harmonic_entropy New dyadic ordering for tetrads

Rather than just summing the exponential, N=10000, s=1% dyadic entropies for
the six intervals in the tetrad, I wanted to use a function that would give
more emphasis to the most dissonant intervals. So I raised the exponential
entropies to the power of 6*pi before summing them (kind of an arbitrary
choice -- 6*pi comes up in the formula for the probability of two numbers
being mutually prime). Here's the new ordering:

bass is always 0

rank tenor alto soprano
(sum(exp(entropy)^(6*pi)))^(1/(6*pi))
1. 498 886 1384 99.304
2. 492 980 1472 99.834
3-4. 184 498 886 101.63
3-4. 388 702 886 101.63
5. 316 702 1018 101.84
6-7. 502 1002 1390 102.22
6-7. 388 888 1390 102.22
8-9. 388 702 970 102.76
8-9. 268 582 970 102.76
10. 386 702 1088 103.12
11-12. 184 388 886 103.14
11-12. 498 702 886 103.14
13-14. 186 576 888 103.37
13-14. 312 702 888 103.37
15-16. 302 502 1004 103.72
15-16. 502 702 1004 103.72
17-18. 386 884 1088 104.14
17-18. 204 702 1088 104.14
19. 268 702 970 104.47
20-21. 318 816 1020 104.63
20-21. 204 702 1020 104.63
22-23. 384 588 1086 105.17
22-23. 498 702 1086 105.17
24. 500 816 1316 105.41
25-26. 202 702 974 105.64
25-26. 272 772 974 105.64
27-28. 502 1002 1320 105.75
27-28. 318 818 1320 105.75
29-30. 434 820 1320 105.87
29-30. 500 886 1320 105.87
31-32. 388 776 1090 105.95
31-32. 314 702 1090 105.95
33. 388 886 1274 107.41
34-35. 498 888 1282 107.64
34-35. 394 784 1282 107.64
36. 442 884 1326 108.08

For an ordering which cannot distinguish otonal and utonal, is this better
than the one we had before? I certainly like seeing those bottom four where
they are, the complete 7-limit tetrads move up to #8 and #9, and of course I
have a personal interest in seeing the 22-tET stacked-fourth tetrad move up
to #2 . . . seriously, what do you guys think?

🔗Paul H. Erlich <PERLICH@...>

I wrote,

>6*pi comes up in the formula for the probability of two numbers being
mutually prime

Actually that probability is 6/(pi^2). So the Farey series of order N
contains about 6*(N/pi)^2 fractions, since, of the N^2 possible fractions of
numbers up to N, a fraction ~6/(pi^2) of them are in lowest terms.

What if I calculate the harmonic entropy, using the "probablity is
proportional to 1/sqrt(n*d)" rule as before, but _not_ restricting the
possible fractions to be in lowest terms? This thought comes to me a lot
since a 3:2 can also be a 6:4 or a 9:6, etc., with lower fundamentals . . .

I think I'll try this out!

🔗Paul H. Erlich <PERLICH@...>

I wrote,

>What if I calculate the harmonic entropy, using the "probablity is
proportional to 1/sqrt(n*d)" rule as >before, but _not_ restricting the
possible fractions to be in lowest terms? This thought comes to me a >lot
since a 3:2 can also be a 6:4 or a 9:6, etc., with lower fundamentals . . .

>I think I'll try this out!

I did this and I was shocked by the result:

http://www.egroups.com/files/harmonic_entropy/dyadic/t3_01_13p2877.jpg

It's virtually the same shape as the _exponential_ as the harmonic entropy
curve obtained when the fractions _were_ restricted to be in lowest terms!
For those curves, see

http://www.egroups.com/files/tuning/perlich/tenney/

This is kind of a relief -- I was wondering why discordance seemed better
modeled by exponentials (or powers of exponentials) of entropy, rather than
entropy itself . . . but maybe all I had to do was include all the unreduced
ratios . . .

But of course the 1/sqrt(n*d) weighting becomes harder to justify in this
scheme, since the interpretation in terms the widths of mutually exclusive,
mutually exhaustive segments of interval space no longer applies . . .

🔗Paul H. Erlich <PERLICH@...>

I wrote,

>It's virtually the same shape as the _exponential_ as the harmonic entropy
>curve obtained . . .

that should read:

"It's virtually the same shape as the _exponential_ of the harmonic entropy
curve obtained . . ."