Interval boarders

I regret not following the dyadic complexity formula thread (of a
couple months back) more attentively, and I'm wondering if there was
any sort of a rough outcome, or general consensus - anyone who was on
top of it care to post a (synoptic) recap?

So in a process that I would suspect is probably covering some of the
same ground, I've been trying to look at some simple 'self-regulating'
algorithms that will set up some rough ways to look at these sorts of
interval boarders, of which this would be one such example:

If {([x2*y)*2]+1}:{[(x1*y)*2]+1} is taken as a minus margin, where x2
and x1 are [D-(D/a)/b] and [N-(D/a)/b], a and b are the
self-regulating factors @:a=N*D and b=N+D, and y is
[(logN-logD)*(12/log2)], and {[(x1*y)*2]-1}:{[(x2*y)*2]-1} is taken as
a plus margin, some harmonic series sequences would look like this:

(1:1 @ +/- 0)
1:2 @ - 52 + 57 1148 1200 1257

2:3 @ - 33 + 36 669 702 738
1:3 @ - 40 + 41 662 702 743

2:4 @ - 26 + 28 1174 1200 1228
1:4 @ - 33 + 34 1167 1200 1234

4:5 @ - 19 + 21 367 386 408
2:5 @ - 23 + 23 364 386 410
1:5 @ - 29 + 30 357 386 417

4:6 @ - 17 + 18 685 702 720
2:6 @ - 20 + 20 682 702 722
1:6 @ - 27 + 27 675 702 729

4:7 @ - 15 + 15 954 969 984
2:7 @ - 18 + 19 951 969 987
1:7 @ - 25 + 25 944 969 994

4:8 @ - 13 + 14 1187 1200 1214
2:8 @ - 17 + 17 1183 1200 1217
1:8 @ - 23 + 24 1177 1200 1224

8:9 @ - 11 + 12 193 204 216
4:9 @ - 12 + 12 192 204 216
2:9 @ - 16 + 16 188 204 220
1:9 @ - 22 + 23 182 204 227

8:10 @ - 10 + 10 376 386 397
4:10 @ - 11 + 12 375 386 398
2:10 @ - 15 + 15 371 386 401
1:10 @ - 21 + 22 365 386 408

8:11 @ - 09 + 09 542 551 561
4:11 @ - 11 + 11 541 551 562
2:11 @ - 14 + 14 537 551 566
1:11 @ - 21 + 21 531 551 572

8:12 @ - 08 + 09 693 702 711
2:12 @ - 13 + 14 688 702 716
4:12 @ - 10 + 10 692 702 712
1:12 @ - 20 + 20 682 702 722

8:13 @ - 08 + 08 833 841 849
4:13 @ - 10 + 10 831 841 850
2:13 @ - 13 + 13 828 841 854
1:13 @ - 19 + 20 821 841 860

8:14 @ - 07 + 08 961 969 976
4:14 @ - 09 + 09 960 969 978
2:14 @ - 13 + 13 956 969 981
1:14 @ - 19 + 19 950 969 988

8:15 @ - 07 + 07 1081 1088 1095
4:15 @ - 09 + 09 1080 1088 1097
2:15 @ - 12 + 12 1076 1088 1101
1:15 @ - 18 + 18 1070 1088 1107

8:16 @ - 07 + 07 1193 1200 1207
4:16 @ - 08 + 08 1192 1200 1208
2:16 @ - 12 + 12 1188 1200 1212
1:16 @ - 18 + 18 1182 1200 1218

(etc.)

But I'd really like to checkout (and understand...) what's already
been done along these lines, before I personally do (or especially
post) much more attempting to use this sort of a process (which I
would imagine that no matter how rigorously or simply it's argued or
presented, will only offer some roughly generalized - and doubtlessly
contestable - model).

Dan

>I regret not following the dyadic complexity formula thread (of a
>couple months back) more attentively, and I'm wondering if there was
>any sort of a rough outcome, or general consensus - anyone who was on
>top of it care to post a (synoptic) recap?

The conclusion was that ratios of small whole numbers are more consonant
that ratios of large whole numbers, and anything that measures the size of
the numbers in a ratio will reflect this. Size of the denominator,
arithmetic mean of numerator and denominator, and geometric mean of n and d
were three of the most popular, the last being my favorite (see TD 1216.4).
Largely dismissed were metrics based on prime factorization.

Aside from this, two useful concepts discussed and named (thanks to Dave
Keenan):

1. TOLERANCE- Irrational intervals, like those found in ETs, can be
consonant when they are close in size to rational intervals. A definition
of consonance based on properties of rational numbers (as above) can be
adapted to explain this by including a TOLERANCE funtion. Everybody's
favorite is Paul Erlich's Harmonic Entropy, which, along the way, explains
one of the reasons why small-numbered ratios are consonant in the first place.

2. SPAN- When intervals are very small, they can be highly dissonant
although they may be represented by small-numbered ratios. When intervals
are very large, they may have very low dissonance _and_ consonance. For
example, one can play almost any notes together 6 octaves apart on a piano
and they won't sound particularly consonant, but they won't clash either.
This runs contrary to the idea that consonance and dissonance represent
opposite ends of the same spectrum. No rigorous adjustment for SPAN was
worked out, although normalizing all intervals to the octave between 8/7
and 16/7 is one expedient workaround.

-C.

Dan, I suspect you mean interval borders!

>I regret not following the dyadic complexity formula thread (of a
>couple months back) more attentively, and I'm wondering if there was
>any sort of a rough outcome, or general consensus - anyone who was on
>top of it care to post a (synoptic) recap?

Dave Keenan and I came to the conclusion that primality was not an issue,
contrary to Euler's, Barlow's, and Wilson's complexity measures. To my mind,
no one had a counterargument of substance. We were left in a position where
several different complexity measures (n, d, n+d, n*d) made sense for
numbers up to about 13, beyond which tolerance effects became more
important, and techniques like harmonic entropy look necessary.

How did you derive your algorithm? It looks like you used 12-tone equal
temperament in some way. What does your algorithm give for ratios without
powers of two in the denominator? For really complex ratios? Do the borders
ever overlap? Any harmonic entropy calculation will come up with the most
likely ratio for any cents interval, which would automatically define
non-overlapping borders for a set of simple ratios (more complex ones would
never be recognized as such).