Re: Interval borders

> [Carl Lumma, TD 291.5]
>
>> [Dan Stearns, TD 290]
>>
>> 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.

IMO, the complexity thread was the hardest to follow in the
entire history of the Tuning List so far, and I think Carl's
summary is superb. I'd like to include these explanations
of 'tolerance' and 'span' in my Tuning Dictionary. Any
objections/ additions/ comments?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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[Paul H. Erlich:]
> Dan, I suspect you mean interval borders!

Yes, it was a (spelling) typo.

>What does your algorithm give for ratios without powers of two in the
denominator?

Here's a couple 4/3 & 5/3 examples:

3:4 -24 +27 474 498 525
3:8 -14 +15 484 498 513
3:16 -10 +10 488 498 508
3:32 -7 +7 491 498 505
3:64 -5 +5 493 498 503
3:128 -4 +4 494 498 502

3:5 -20 +21 864 884 906
3:10 -13 +13 872 884 897
3:20 -9 +9 876 884 893
3:40 -6 +6 878 884 891
3:80 -5 +5 879 884 889
3:160 -4 +4 880 884 889

>For really complex ratios?

Here's some examples of augmented fourths & diminished fifths:

512:729 -0.1400 +0.1401 612 612 612
729:1024 -0.0995 +0.0996 588 588 588

32:45 -2.3 +2.3 588 590 592
45:64 -1.6 +1.6 608 610 611

5:7 -14 +15 568 583 597
7:10 -10 +10 607 617 628

12:17 -6 +6 597 603 609
17:24 -4 +4 593 597 601

>Do the borders ever overlap?

Yes -- and by changing the parameters of "a" and "b" in the algorithm,
one could set the borders as coarsely as I'd suspect one would care
to -- but as it's set up now (where a=N*D and b=N+D)* there is some
overlap, but it appears to occurs were it would seem reasonable that
it should start to occur, for example:

1 : 2 - 52 + 57 1148 1200 1257
2 : 3 - 33 + 36 669 702 738
3 : 4 - 24 + 27 474 498 525
4 : 5 - 19 + 21 367 386 408
5 : 6 - 16 + 18 300 316 333
6 : 7 - 14 + 15 253 267 282
7 : 8 - 12 + 13 219 231 244
8 : 9 - 11 + 12 193 204 216
9 : 10 - 9 + 11 173 182 193

1 : 3 - 40 + 41 1862 1902 1943
2 : 4 - 26 + 28 1174 1200 1228
3 : 5 - 20 + 21 864 884 906
4 : 6 - 17 + 18 685 702 720
5 : 7 - 14 + 15 568 583 597
6 : 8 - 12 + 13 486 498 511
7 : 9 - 11 + 12 424 435 447
8 : 10 - 10 + 10 376 386 397
9 : 11 - 9 + 9 338 347 357
10 : 12 - 8 + 9 307 316 324
11 : 13 - 8 + 8 282 289 297
12 : 14 - 7 + 7 260 267 274
13 : 15 - 7 + 7 241 248 255
14 : 16 - 6 + 6 225 231 238
15 : 17 - 6 + 6 211 217 223
16 : 18 - 5 + 6 198 204 210
17 : 19 - 5 + 5 187 193 198

1 : 4 - 33 + 34 2367 2400 2434
2 : 5 - 23 + 23 1564 1586 1610
3 : 6 - 18 + 18 1182 1200 1218
4 : 7 - 15 + 15 954 969 984
5 : 8 - 13 + 13 801 814 827
6 : 9 - 11 + 12 691 702 714
7 : 10 - 10 + 10 607 617 628
8 : 11 - 9 + 9 542 551 561
9 : 12 - 8 + 9 490 498 507
10 : 13 - 8 + 8 447 454 462
11 : 14 - 7 + 7 410 418 425
12 : 15 - 7 + 7 380 386 393
13 : 16 - 6 + 6 353 359 366
14 : 17 - 6 + 6 330 336 342
15 : 18 - 5 + 6 310 316 321
16 : 19 - 5 + 5 292 298 303
17 : 20 - 5 + 5 276 281 286
18 : 21 - 5 + 5 262 267 272
19 : 22 - 4 + 5 249 254 258
20 : 23 - 4 + 4 238 242 246
21 : 24 - 4 + 4 227 231 235
22 : 25 - 4 + 4 217 221 225
23 : 26 - 4 + 4 208 212 216
24 : 27 - 4 + 4 200 204 208

etc.

Dan
____________
*And {([x2*y)*2]+1}:{[(x1*y)*2]+1} taken as the 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
the plus margin.