32 best 5-limit linear temperaments redux

27/25 limmal

map [[0, -2, -3], [1, 2, 3]]

generators 268.0564391 1200

badness 358.9821660 rms 35.60923982 g 2.160246899

ets
4
5
9
14

16/15 fourth-thirds

map [[0, -1, 1], [1, 2, 2]]

generators 442.1793558 1200

badness 129.0161774 rms 45.61410700 g 1.414213562

ets
1
2
3
5
6
8

135/128 pelogic

map [[0, -1, 3], [1, 2, 1]]

generators 522.8623453 1200

badness 461.2348421 rms 18.07773298 g 2.943920288

ets
2
7
9
11
16
23

25/24 neutral thirds

map [[0, 2, 1], [1, 1, 2]]

generators 350.9775007 1200

badness 81.60548797 rms 28.85189698 g 1.414213562

ets
3
4
6
7
10
13
17
20

648/625 diminished

map [[0, -1, -1], [4, 8, 11]]

generators 505.8656425 300

badness 385.3013916 rms 11.06006024 g 3.265986323

ets
4
8
12
16
24
28
36
40
52
64

250/243 porcupine

map [[0, -3, -5], [1, 2, 3]]

generators 162.9960265 1200

badness 359.5570529 rms 7.975800816 g 3.559026083

ets
7
8
15
22
29
30
37
44
51
59
66

128/125 augmented

map [[0, -1, 0], [3, 6, 7]]

generators 491.2018553 400

badness 142.2320613 rms 9.677665980 g 2.449489743

ets
3
6
9
12
15
18
21
24
27
30
33
36
39
42

3125/3072 small diesic

map [[0, 5, 1], [1, 0, 2]]

generators 379.9679494 1200

badness 239.3635979 rms 4.569472316 g 3.741657387

ets
3
6
16
19
22
25
35
38
41
44
57
60
63
66
76
79
82
85
104
107

81/80 meantone

map [[0, -1, -4], [1, 2, 4]]

generators 503.8351546 1200

badness 107.6110644 rms 4.217730124 g 2.943920288

ets
5
7
12
19
24
26
31
36
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43
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100
105
117
129

2048/2025 diaschismic

map [[0, -1, 2], [2, 4, 3]]

generators 494.5534684 600

badness 210.7220901 rms 2.612822498 g 4.320493799

ets
2
10
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14
20
22
24
32
34
36
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66
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112
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136
148
160

78732/78125 tiny diesic

map [[0, 7, 9], [1, -1, -1]]

generators 442.9792974 1200

badness 345.5378445 rms 1.157498146 g 6.683312553

ets
8
19
27
38
46
57
65
73
76
84
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111
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363
382
390
409
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447
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493
539
558
623

393216/390625 wuerschmidt

map [[0, 8, 1], [1, -1, 2]]

generators 387.8196732 1200

badness 251.1018953 rms 1.071950166 g 6.164414003

ets
3
6
28
31
34
37
62
65
68
71
93
96
99
102
127
130
133
136
158
161
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167
192
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223
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232
257
260
263
266
288
291
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297
322
325
328
331
353
356
359
362
365
387
390
393
421
452

2109375/2097152 orwell

map [[0, 7, -3], [1, 0, 3]]

generators 271.5895996 1200

badness 305.9258786 rms .8004099292 g 7.257180353

ets
9
13
22
31
40
44
53
62
66
75
84
93
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106
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128
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305
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327
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358
371
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402
411
424
433
455
464
486
517
570

15625/15552 kleismic

map [[0, 6, 5], [1, 0, 1]]

generators 317.0796754 1200

badness 96.73525308 rms 1.029625097 g 4.546060566

ets
4
15
19
23
30
34
38
49
53
57
68
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367
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507
526
545
560
579
613
632
666
719

1600000/1594323 amt

map [[0, -5, -13], [1, 3, 6]]

generators 339.5088258 1200

badness 305.5372197 rms .3831037874 g 9.273618495

ets
7
39
46
53
60
92
99
106
113
145
152
159
166
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244
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827
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880
919
926
933
972
979
986

1224440064/1220703125 parakleismic

map [[0, -13, -14], [1, 5, 6]]

generators 315.2509133 1200

badness 372.7314879 rms .2766026501 g 11.04536102

ets
19
38
42
57
61
76
80
99
118
137
156
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194
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217
236
255
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491
510
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552
571
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628
647
651
670
689
708
727
746
765
769
788
807
826
845
864
887
906
925
944
963
982

6115295232/6103515625 semisuper

map [[0, 7, 3], [2, -3, 2]]

generators 528.8539366 600

badness 190.1507467 rms .1940181460 g 9.933109620

ets
16
18
34
50
68
84
100
102
118
134
136
152
168
186
202
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236
252
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270
286
304
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338
354
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658
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742
758
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776
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810
826
844
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876
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894
910
928
944
962
978
994
996

19073486328125/19042491875328 enneadecal

map [[0, -1, -1], [19, 38, 52]]

generators 497.9709056 1200/19

badness 391.2170134 rms .1047837215 g 15.51343504

ets
19
38
57
76
95
114
133
152
171
190
209
228
266
285
304
323
342
361
380
399
418
437
456
475
494
513
532
551
570
589
608
627
646
665
684
703
722
760
779
798
817
836
855
874
893
931
950
969
988

32805/32768 shismic

map [[0, -1, 8], [1, 2, -1]]

generators 498.2724869 1200

badness 54.89487859 rms .1616904714 g 6.976149846

ets
12
17
24
29
36
41
53
65
77
82
89
94
101
106
118
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142
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867
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903
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932
944
956
961
968
973
985
997

582076609134674072265625/581595589965365114830848 heptadecal

map [[0, 2, 1], [17, 16, 34]]

generators 386.2716180 1200/17

badness 477.6214948 rms .3437099513e-1 g 24.04163055

ets
34
68
102
136
153
187
221
255
289
323
357
391
425
476
510
544
578
612
646
680
714
748
765
799
833
867
901
935
969

274877906944/274658203125 hemithird

map [[0, -15, 2], [1, 4, 2]]

generators 193.1996149 1200

badness 137.9992271 rms .6082244804e-1 g 13.14026890

ets
25
31
56
62
87
93
112
118
143
149
174
180
205
211
230
236
261
267
292
298
323
329
348
354
379
385
410
416
441
447
466
472
497
503
528
534
559
584
590
615
621
646
652
671
677
702
708
733
739
764
770
789
795
820
826
851
857
882
888
907
913
938
944
969
975
1000

50031545098999707/50000000000000000

map [[0, 17, 35], [1, -1, -3]]

generators 182.4660890 1200

badness 386.2264718 rms .2546863438e-1 g 24.75210428

ets
46
79
92
125
171
217
250
263
296
342
388
434
467
480
513
559
605
638
651
684
730
776
809
822
855
901
947
980
993

7629394531250/7625597484987

map [[0, -2, -3], [9, 19, 28]]

generators 315.6754868 400/3

badness 188.0842271 rms .2559261582e-1 g 19.44222209

ets
27
45
72
99
126
144
171
198
243
270
297
315
342
369
414
441
468
486
513
540
567
585
612
639
684
711
738
756
783
810
855
882
909
927
954
981

2475880078570760549798248448/2474715001881122589111328125

map [[0, -31, 12], [1, 5, 1]]

generators 132.1945105 1200

badness 463.2524095 rms .1499283745e-1 g 31.37939876

ets
9
109
118
127
218
227
236
245
336
345
354
363
454
463
472
481
572
581
590
599
690
699
708
717
808
817
826
835
926
935
944
953

9010162353515625/9007199254740992

map [[0, -8, 5], [2, 9, 1]]

generators 437.2581077 600

badness 113.0912513 rms .1772520822e-1 g 18.54723699

ets
22
44
52
66
74
96
118
140
162
170
184
192
206
214
236
258
280
288
302
310
324
332
354
376
398
406
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442
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494
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538
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560
568
590
612
634
656
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686
708
730
752
774
782
796
804
826
848
870
892
900
914
922
936
944
966
988

116450459770592056836096/116415321826934814453125

map [[0, -33, -25], [1, 17, 14]]

generators 560.5469696 1200

badness 178.7704200 rms .1239024539e-1 g 24.34474618

ets
15
30
122
137
152
167
274
289
304
319
411
426
441
456
471
563
578
593
608
700
715
730
745
760
852
867
882
897

444089209850062616169452667236328125/444002166576103304796646509039845376

map [[0, -51, -52], [1, 15, 16]]

generators 315.6478750 1200

badness 346.6194848 rms .4659979284e-2 g 42.05551886

ets
19
38
57
76
365
384
403
422
441
460
479
498
517
825
844
863
882
901
920
939
958

450359962737049600/450283905890997363

map [[0, -2, -37], [1, 2, 10]]

generators 249.0184480 1200

badness 146.1980313 rms .5736733648e-2 g 29.42787794

ets
53
106
159
188
212
241
265
294
318
347
371
400
424
453
506
559
612
665
718
771
800
824
853
877
906
930
959
983

162285243890121480027996826171875/162259276829213363391578010288128

map [[0, 14, -47], [1, -1, 11]]

generators 221.5678655 1200

badness 326.5508398 rms .3538891098e-2 g 45.18849411

ets
65
130
195
222
260
287
325
352
390
417
482
547
612
677
742
807
834
872
899
937
964

22300745198530623141535718272648361505980416/
22297583945629639856633730232715606689453125

map [[0, 47, -22], [1, -2, 4]]

generators 91.53102125 1200

badness 352.1515837 rms .2843331166e-2 g 49.84643083

ets
13
105
118
131
223
236
249
341
354
367
459
472
485
590
603
708
721
826
839
944
957

381520424476945831628649898809/381469726562500000000000000000

map [[0, -35, -62], [1, 11, 19]]

generators 322.8013866 1200

badness 256.7937928 rms .3022380142e-2 g 43.96210490

ets
26
145
171
197
316
342
368
487
513
658
684
803
829
855
974
1000

17763568394002504646778106689453125/17763086495282268024161967871623168

map [[0, 49, 15], [1, -6, 0]]

generators 185.7541789 1200

badness 34.16161967 rms .7631994496e-3 g 35.50586806

ets
71
84
155
168
239
252
323
407
478
491
562
575
646
659
730
801
814
885
898
969
982

I don't think we are going to make any progress on this unless we can
get beyond a badness measure that says the best 5-limit temperament is
one that takes 49 generators before we get a single fifth (because it
has such teensy weensy errors).

This badness measure also says that meantone is only 7th best (or
thereabouts) and thinks that a temperament whose perfect fifth is 758
cents and whose major third is 442 cents is only slightly worse than
meantone (because it only needs 2 generators to get one of these
supposed 1:3:5 chords).

Does anyone really believe this stuff?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I don't think we are going to make any progress on this unless we can
> get beyond a badness measure that says the best 5-limit temperament is
> one that takes 49 generators before we get a single fifth (because it
> has such teensy weensy errors).

This is just like saying we should not regard 2460 as a super-good
5-limit scale because its errors are so small that it could make no practical difference if they were larger, and that given the choice between 53 tones and 2460, 53 seems much more practical. This misses the point, which is that 2460 is very, very good compared to other things *in its size range*. If you compare wildly different values of "g", you are getting into apples and elephants.

> This badness measure also says that meantone is only 7th best (or
> thereabouts) and thinks that a temperament whose perfect fifth is 758
> cents and whose major third is 442 cents is only slightly worse than
> meantone (because it only needs 2 generators to get one of these
> supposed 1:3:5 chords).

> Does anyone really believe this stuff?

Paul has pointed out that ultra-funky scales may have more possibilities than is at first apparent. Again, why compare apples with e coli?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I don't think we are going to make any progress on this unless we
can
> > get beyond a badness measure that says the best 5-limit
temperament is
> > one that takes 49 generators before we get a single fifth (because
it
> > has such teensy weensy errors).
>
> This is just like saying we should not regard 2460 as a super-good
> 5-limit scale because its errors are so small that it could make no
practical difference if they were larger, and that given the choice
between 53 tones and 2460, 53 seems much more practical. This misses
the point, which is that 2460 is very, very good compared to other
things *in its size range*. If you compare wildly different values of
"g", you are getting into apples and elephants.
>
> > This badness measure also says that meantone is only 7th best (or
> > thereabouts) and thinks that a temperament whose perfect fifth is
758
> > cents and whose major third is 442 cents is only slightly worse
than
> > meantone (because it only needs 2 generators to get one of these
> > supposed 1:3:5 chords).
>
> > Does anyone really believe this stuff?
>
> Paul has pointed out that ultra-funky scales may have more
possibilities than is at first apparent. Again, why compare apples
with e coli?

Why indeed? But that's exactly what you're doing. You only gave one
list, in which a single badness metric compares all the temperamants
against each other. You didn't give a list of e coli, a list of apples
and a list of elephants, but only the "32 best 5-limit linear
temperaments".

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

You didn't give a list of e coli, a list of apples
> and a list of elephants, but only the "32 best 5-limit linear
> temperaments".

Is it the subject line you object to?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I don't think we are going to make any progress on this unless we
can
> > get beyond a badness measure that says the best 5-limit
temperament is
> > one that takes 49 generators before we get a single fifth (because
it
> > has such teensy weensy errors).
>
> This is just like saying we should not regard 2460 as a super-good
> 5-limit scale because its errors are so small that it could make no
practical difference if they were larger, and that given the choice
between 53 tones and 2460, 53 seems much more practical. This misses
the point, which is that 2460 is very, very good compared to other
things *in its size range*. If you compare wildly different values of
"g", you are getting into apples and elephants.
>
> > This badness measure also says that meantone is only 7th best (or
> > thereabouts) and thinks that a temperament whose perfect fifth is
758
> > cents and whose major third is 442 cents is only slightly worse
than
> > meantone (because it only needs 2 generators to get one of these
> > supposed 1:3:5 chords).
>
> > Does anyone really believe this stuff?
>
> Paul has pointed out that ultra-funky scales may have more
possibilities than is at first apparent. Again, why compare apples
with e coli?

I assume we are making these lists of temperaments for people who are
considering making practical musical use of them (as opposed to
say theoretical mathematical, or practical engineering use). Such a
person, searching in a list entitled "5-limit linear temperaments",
can be presumed to want two things:
1. 5-limit harmony, and
2. temperament.

The former implies that s/he wants intervals that _sound_ like some
kind of approximation of ratios of 1, 3 and 5, and their octave
equivalents and inversions.

The latter implies that s/he doesn't want to have to deal with as many
pitches as would be required in a 5-limit _rational_ tuning giving the
same numbers of harmonies.

The two temperaments I singled out would be of no interest to such a
person (except as curiosities). The one, because it isn't 5-limit and
the other because it isn't a temperament, for any practical purpose.

The one has both its 4:5 approximation and its 3:4 approximation
sounding exactly the same as each other. They both sound like 7:9s.
The other requires _more_ notes than a 5-limit rational scale for any
reasonable number of 5-limit harmonies, assuming we use a contiguous
chain of generators.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> The one has both its 4:5 approximation and its 3:4 approximation
> sounding exactly the same as each other. They both sound like 7:9s.
> The other requires _more_ notes than a 5-limit rational scale for any
> reasonable number of 5-limit harmonies, assuming we use a contiguous
> chain of generators.

What you're saying is that the search was too broad, it seems. That could be rectified if there was general agreement it is so by the simple expedient of leaving off the extremes.

>What you're saying is that the search was too broad, it seems. That
>could be rectified if there was general agreement it is so by the
>simple expedient of leaving off the extremes.

I for one do not understand steps*cents for linear temperaments.
Shouldn't it be g*cents?

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > The one has both its 4:5 approximation and its 3:4 approximation
> > sounding exactly the same as each other. They both sound like
7:9s.
> > The other requires _more_ notes than a 5-limit rational scale for
any
> > reasonable number of 5-limit harmonies, assuming we use a
contiguous
> > chain of generators.
>
> What you're saying is that the search was too broad, it seems. That
could be rectified if there was general agreement it is so by the
simple expedient of leaving off the extremes.

We've been over this before. That's exactly what I'm trying to
convince everyone to accept, not with sharp cutoffs, but gradual
rolloffs.

By choosing suitable values for the parameters k and p in

badness = gens * EXP((cents/k)^p)

we can start with a badness measure that gives exactly the same
ranking as your current measure, and then by tweaking these parameters
we can make those objectionable extreme cases fall off the bottom of
any length best-of list we choose to make.

I don't have time to check it at the moment, but I think if you set
p = 0.24 and k = 1 cent
you will get pretty much the same ranking as for gens^3 * cents.

In fact if you divide it by e (~=2.718) and cube it, you will get a
good approximation to your actual badness numbers. i.e.

gens^3 * cents
~=
(gens * EXP((cents/1)^0.24) / e)^3

but of course dividing by e and cubing doesn't change the ranking, so
they can be omitted. Then gradually increase p and k until both of
those objectionable temperaments (e-coli and elephant) just go off the
bottom of your list of 32 best and see how the rest of them are then
ranked. I guarantee it will make a lot more sense to most people, with
meantone much closer to the top for one thing, and you probably won't
need as many as 32 in the list to cover all the historical ones.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> You didn't give a list of e coli, a list of apples
> > and a list of elephants, but only the "32 best 5-limit linear
> > temperaments".
>
> Is it the subject line you object to?

No it's the badness measure. I actually _want_ a badness measure that
compares _all_ 5-limit temperaments irrespective of their e-coli-ness
or elephant-ness, but actually takes into account that e-coli and
elephants are inherently less interesting or useful than apples (but
in a continuous manner not a discrete one as the analogy of e-coli,
apples and elephants would suggest).

For 5-limit I'm currently using:

badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

Where wtd_rms_gens are weighted by log of odd limit.

I think this approaches your badness in the limit where the power (0.5
above) goes to zero and the 7.4 cents does something else (I forget
what).

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >What you're saying is that the search was too broad, it seems. That
> >could be rectified if there was general agreement it is so by the
> >simple expedient of leaving off the extremes.
>
> I for one do not understand steps*cents for linear temperaments.
> Shouldn't it be g*cents?

Yes Gene is using gens*cents for linear temperaments, or rather gens^n
* cents in general (different n>=1 for different limits). I don't like
abbreviating the number of generators to "g". That's being
unnecessarily obscure.

>>I for one do not understand steps*cents for linear temperaments.
>>Shouldn't it be g*cents?
>
>Yes Gene is using gens*cents for linear temperaments, or rather gens^n
>* cents in general (different n>=1 for different limits). I don't like
>abbreviating the number of generators to "g". That's being
>unnecessarily obscure.

Okay, thanks, this does answer my question -- he's using the number of
gens in one instance of the map, as opposed to the number in the et
that provides a near-optimal generator size, or something.

Though my particular suggestion was not only this; g is the rms of
gens in a map, or something. I'm still not clear exactly how it's
calculated, or if it's different from what Graham calls complexity.
For the record, my preferred complexity measure is...

(/ (- (max map) (min map)) (card map))

...but anything that levels the field for different limits is
fine, and Gene's already been using g, so...

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Okay, thanks, this does answer my question -- he's using the number of
> gens in one instance of the map, as opposed to the number in the et
> that provides a near-optimal generator size, or something.

What's the difference?

> Though my particular suggestion was not only this; g is the rms of
> gens in a map, or something. I'm still not clear exactly how it's
> calculated, or if it's different from what Graham calls complexity.

It's a different measure of complexity.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)
>
> Where wtd_rms_gens are weighted by log of odd limit.

This strikes me as completely ad hoc. Why not a non-fuzzy version, with a sharp cutoff?

>>Okay, thanks, this does answer my question -- he's using the number of
>>gens in one instance of the map, as opposed to the number in the et
>>that provides a near-optimal generator size, or something.
>
>What's the difference?

Meantone has a very compact 5-limit map. You only need 4 gens. In
listening tests I've preferred a generator close to that of 69-et,
though the rms optimum is closer to 31-et IIRC. In either case, why
should we penalize meantone because it takes 31 or 69 gens to yield
an et with the optimum generator?

>>Though my particular suggestion was not only this; g is the rms of
>>gens in a map, or something. I'm still not clear exactly how it's
>>calculated, or if it's different from what Graham calls complexity.
>
>It's a different measure of complexity.

What is different from what??

Once again, I'll list my preferred map complexity measure in unambiguous
mathematical notation. Why don't you and Graham give yours for the
record, so Dave can tell us which one he likes best?

Carl's preferred map complexity measure:
(/ (- (max map) (min map)) (card map))

Gene's preferred map complexity measure:
________________________________________

Graham's preferred map complexity measure:
________________________________________

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)
> >
> > Where wtd_rms_gens are weighted by log of odd limit.
>
> This strikes me as completely ad hoc.

Well it isn't. It is designed to fit what I've learned over the years
by corresponding with people on the tuning lists, regarding the
relative usefulness (or interest) in those 5-limit temperaments that
have been known for a long time. For example, I think meantone must be
in the top 3, or the badness measure is nonsense.

I recently adjusted the cents parameter upwards to take into account
Paul Erlich's suggestion that pelog just might exist in the real world
because it is a MOS of a rough 5-limit temperament.

I'd be happy to have lots of people play with those parameters to try
to make the list come out with the temperaments they are familiar
with, in the order they expect.

Another reason it isn't ad hoc. The perceptual "pain" caused by
mistuning is not directly proportional to the error in cents. Even the
best microtonal ear on the planet apparently experiences essentially
zero pain with a 0.5 cent mistuning. Most people aren't significantly
bothered by a 3c mistuning (depending on the interval and how long it
is sustained). But a 30 cent mistuning is so bad that a 40 cent one
could hardly be much worse.

So you could think of the
EXP((error/7.4_cents)^0.5)
part as the "mistuning pain", except that would leave the
"number-of-notes pain" as simply gens, which we both agree isn't
right.

If I take the "number-of-notes pain" as gens^2 then the "mistuning
pain" is
EXP((error/7.4_cents)^0.5) ^2
= EXP((error/1.85_cents)^0.5)

I also designed it so that it would have a choice of parameters that
gave a result very close to the "log flat" measure, so you could use
that as a starting point.

> Why not a non-fuzzy version, with a sharp cutoff?

Because that's not how musicians or composers relate to temperaments.
A temperament doesn't suddenly become of zero interest because it has
overstepped some sharp boundary of harmonic error or number of
generators.

By the way, I think my 3 previous messages in this thread arrived on
the list in the reverse order to the order I sent them. Thanks Yahoo.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Once again, I'll list my preferred map complexity measure in
unambiguous
> mathematical notation. Why don't you and Graham give yours for the
> record, so Dave can tell us which one he likes best?
>
> Carl's preferred map complexity measure:
> (/ (- (max map) (min map)) (card map))

This isn't unambiguous mathematical notation. It's Lisp. Took me a
while to figure that out.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Meantone has a very compact 5-limit map. You only need 4 gens. In
> listening tests I've preferred a generator close to that of 69-et,
> though the rms optimum is closer to 31-et IIRC. In either case, why
> should we penalize meantone because it takes 31 or 69 gens to yield
> an et with the optimum generator?

I don't; "g" has nothing to do with ets per se, and only measures
complexity.

> What is different from what??

Graham uses max error, and I use rms error.

> Once again, I'll list my preferred map complexity measure in
unambiguous
> mathematical notation. Why don't you and Graham give yours for the
> record, so Dave can tell us which one he likes best?

> Gene's preferred map complexity measure:

rms generator steps, times the number of periods in an octave. This
only works for linear temperaments, so I'm not that happy with it.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

For example, I think meantone must be
> in the top 3, or the badness measure is nonsense.

This is like saying 12-et must be in the top three. It's great in its
size range--if that happens to be the range you are interested in. If
it doesn't suit your requirements then it isn't great, whatever
number you come up with for it. What's top or not top depends on what
tone group you are looking at (5-limit, 7-limit?) and what sort of
accuracy you want.

Dave wrote...
>> Carl's preferred map complexity measure:
>> (/ (- (max map) (min map)) (card map))
>
>This isn't unambiguous mathematical notation. It's Lisp. Took me
>a while to figure that out.

It's generic prefix (Polish) notation with normal grouping by
parens. Graham knows scheme now, and you have a paper on the
lambda calculus on your web page, and Gene's a clever guy.

Gene wrote...
>>Meantone has a very compact 5-limit map. You only need 4 gens. In
>>listening tests I've preferred a generator close to that of 69-et,
>>though the rms optimum is closer to 31-et IIRC. In either case, why
>>should we penalize meantone because it takes 31 or 69 gens to yield
>>an et with the optimum generator?
>
>I don't; "g" has nothing to do with ets per se, and only measures
>complexity.

Good!

>>Once again, I'll list my preferred map complexity measure in
>>unambiguous mathematical notation. Why don't you and Graham
>>give yours for the record, so Dave can tell us which one he
>>likes best?
>>
>>Gene's preferred map complexity measure:
>
>rms generator steps, times the number of periods in an octave.

What made you go to rms? Isn't it over-kill?

>This only works for linear temperaments, so I'm not that happy with it.

What else do you want it to work for? Planar temperaments?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> It's generic prefix (Polish) notation with normal grouping by
> parens. Graham knows scheme now, and you have a paper on the
> lambda calculus on your web page, and Gene's a clever guy.

Not clever enough to figure out why you want to use prefix notation.

> What made you go to rms? Isn't it over-kill?

It's less sensitive to outliers; if a temperament does a lot of
things well and some badly, it still get credit for it.

> >This only works for linear temperaments, so I'm not that happy
with it.
>
> What else do you want it to work for? Planar temperaments?

Of course.

Dave wrote...
>Another reason it isn't ad hoc. The perceptual "pain" caused by
>mistuning is not directly proportional to the error in cents. Even the
>best microtonal ear on the planet apparently experiences essentially
>zero pain with a 0.5 cent mistuning. Most people aren't significantly
>bothered by a 3c mistuning (depending on the interval and how long it
>is sustained). But a 30 cent mistuning is so bad that a 40 cent one
>could hardly be much worse.

Gene's "cents" are already rms, which we've long ago decided is the
best single error measure. Chopping off anything less than .5 is a
hack, and hopefully an un-necessary one.

Gene wrote...
>>For example, I think meantone must be in the top 3, or the badness
>>measure is nonsense.
>
>This is like saying 12-et must be in the top three. It's great in its
>size range-- if that happens to be the range you are interested in. If
>it doesn't suit your requirements then it isn't great, whatever
>number you come up with for it. What's top or not top depends on what
>tone group you are looking at (5-limit, 7-limit?) and what sort of
>accuracy you want.

Dave's just saying that you're not weighting the span of the map
enough, since he considers musical history to be a worthy badness
measure in its own right -- one that selected meantone, diminished,
augmented, over the infinity of temperaments bigger than schismic.

You want something that exposes the pattern in the series of
best temperaments.

It would be nice to see a list with a much stronger penalty
for size, but I can live with a flat measure with a sharp cutoff.

-Carl

>>It's generic prefix (Polish) notation with normal grouping by
>>parens. Graham knows scheme now, and you have a paper on the
>>lambda calculus on your web page, and Gene's a clever guy.
>
>Not clever enough to figure out why you want to use prefix notation.

It's trivial one way or the other, though I personally find it
easier to parse when dealing with ASCI.

>>What made you go to rms? Isn't it over-kill?
>
>It's less sensitive to outliers; if a temperament does a lot of
>things well and some badly, it still get credit for it.

That's exactly what I _don't_ want. I want to know the size of
the chain I need to complete my map. The only reason I divide
by (card map) is so I can compare temperaments at different
limits.

>>>This only works for linear temperaments, so I'm not that happy
>>>with it.
>>
>>What else do you want it to work for? Planar temperaments?
>
>Of course.

There's nothing I want more than to get to planar temperaments,
but we should finish with LTs first! I say this simply because
I suspect if we can't handle LTs, we don't stand a ghost of a
chance with PTs!

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> For example, I think meantone must be
> > in the top 3, or the badness measure is nonsense.
>
> This is like saying 12-et must be in the top three. It's great in
its
> size range--if that happens to be the range you are interested in.

So you're gonna make different lists for different size ranges, are
you?

I think meantone is one of the 3 best overall.

> If
> it doesn't suit your requirements then it isn't great, whatever
> number you come up with for it. What's top or not top depends on
what
> tone group you are looking at (5-limit, 7-limit?) and what sort of
> accuracy you want.

Of course I was meaning 5-limit for meantone. That's what this thread
is about.

I'd rather a single list that takes into account a typical tradeoff
between accuracy number of notes. You can then go looking for your
"size range" within that.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> So you're gonna make different lists for different size ranges, are
> you?

I've tried listing them in order of size, which makes sense to me, though I think Paul didn't like it. What do you think?

genewardsmith wrote:

> Graham uses max error, and I use rms error.

I use RMS error, but max width for complexity.

Graham

dkeenanuqnetau wrote:

> By choosing suitable values for the parameters k and p in
>
> badness = gens * EXP((cents/k)^p)
>
> we can start with a badness measure that gives exactly the same
> ranking as your current measure, and then by tweaking these parameters
> we can make those objectionable extreme cases fall off the bottom of
> any length best-of list we choose to make.
>
> I don't have time to check it at the moment, but I think if you set
> p = 0.24 and k = 1 cent
> you will get pretty much the same ranking as for gens^3 * cents.

Have you tried running <http://microtonal.co.uk/temper/linear.html> with
width*math.exp((error/1200.0)**0.24) as the figure of demerit? It might
be the wrong width, but it's worth a try.

Graham

Carl Lumma wrote:

> Once again, I'll list my preferred map complexity measure in unambiguous
> mathematical notation. Why don't you and Graham give yours for the
> record, so Dave can tell us which one he likes best?
>
> Carl's preferred map complexity measure:
> (/ (- (max map) (min map)) (card map))

That is ambiguous because you haven't defined what map or card mean.

> Graham's preferred map complexity measure:

This is the actual code:

complexity = self.getWidth(self.getWidestInterval(
consonances))*self.mapping[0][0]

Depending on what your map is, it may give the same results up to the 7
limit and ignoring card. Beyond that, the widest interval may not be a
column in the map. So I have a method for finding that, which is a bit
uglier. It either loops over all the consonances and takes the largest
absolute number of generator steps, or loops over the harmonics of a
tonality diamond and takes the max-min. That mapping[0][0] is the number
of periods to the equivalence interval because getWidth() doesn't take
account of that.

Graham

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Dave wrote...
> >Another reason it isn't ad hoc. The perceptual "pain" caused by
> >mistuning is not directly proportional to the error in cents. Even
the
> >best microtonal ear on the planet apparently experiences
essentially
> >zero pain with a 0.5 cent mistuning. Most people aren't
significantly
> >bothered by a 3c mistuning (depending on the interval and how long
it
> >is sustained). But a 30 cent mistuning is so bad that a 40 cent one
> >could hardly be much worse.
>
> Gene's "cents" are already rms, which we've long ago decided is the
> best single error measure.

Whether it is rms or max-absolute is irrelevant to my argument. I'm
happy to use minimum rms error as the input to badness (though I think
it's good to know the minimum max-absolute as well). If you're
suggesting that using rms somehow removes the need to apply a
nonlinear "pain" function, then you're mistaken. You've put "cents" in
scare quotes above as if you think the units aren't really cents when
it's rms. They are.

> Chopping off anything less than .5 is a
> hack, and hopefully an un-necessary one.

It is definitely unnecessary and I do not propose to chop anything
off. Gene's use of straight cents gives far too much credit to
temperaments that have very small sub-half-cent errors. It is allowed
to compensate so much for large gens that a temperament with a
complexity of 35.5 gens (rms) can be considered the best 5-limit
temperament! If you were to take a 0.5 cent threshold into account in
a discontinuous manner, you would treat any temperament whose rms
error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not
proposing we do that. My pain function is non-linear but smooth. A
temperament still gets some credit for being sub-half-cent, just not
so much.

> Gene wrote...
> >>For example, I think meantone must be in the top 3, or the badness
> >>measure is nonsense.
> >
> >This is like saying 12-et must be in the top three.

I don't have a problem with that, of course 12-ET is in the top 3 ETs
for 5-limit.

> It's great in its
> >size range-- if that happens to be the range you are interested in.
If
> >it doesn't suit your requirements then it isn't great, whatever
> >number you come up with for it. What's top or not top depends on
what
> >tone group you are looking at (5-limit, 7-limit?) and what sort of
> >accuracy you want.
>
> Dave's just saying that you're not weighting the span of the map
> enough, since he considers musical history to be a worthy badness
> measure in its own right -- one that selected meantone, diminished,
> augmented, over the infinity of temperaments bigger than schismic.

That's a good way of putting it, except I wouldn't say that Gene's not
weighting the complexity enough, I'd say he's failing to level off
(asymptote) with sub-cent errors and not weighting super-20-cent
errors enough.

> You want something that exposes the pattern in the series of
> best temperaments.
>
> It would be nice to see a list with a much stronger penalty
> for size, but I can live with a flat measure with a sharp cutoff.

You don't _have_ to live with it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
>
> > So you're gonna make different lists for different size ranges,
are
> > you?
>
> I've tried listing them in order of size, which makes sense to me,
though I think Paul didn't like it. What do you think?

I'd rather they were listed in increasing order of "badness", assuming
"badness" actually means something, like badness. Then if I'm looking
for the best temperament whose error is in a particular range of sizes
I'll just go down the list until I find the first one _in_ that range.
The same goes for any other property that might be important to me at
the time, like having a half-octave period or having a single
generator making the perfect fifth.

If you think it's ok to have a badness measure that does not allow
comparison between temperaments whose errors are in different size
ranges, would you also think it ok to have one that didn't allow
comparison between say temperaments whose generators were in different
size ranges. i.e. you couldn't compare temperaments generated by
approximate fourths with those generated by approximate thirds.

If you agree that this would be a somewhat defective badness measure,
then you should know that this is how I view the badness measure you
are using.

--- In tuning-math@y..., graham@m... wrote:
> Have you tried running <http://microtonal.co.uk/temper/linear.html>
with
> width*math.exp((error/1200.0)**0.24) as the figure of demerit? It
might
> be the wrong width, but it's worth a try.

That's awesome! I had no idea it was that easy!

You might mention on that page that "error" is in octaves, not cents.
Or better still make "error" be in cents.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)
> >
> > Where wtd_rms_gens are weighted by log of odd limit.
>
> This strikes me as completely ad hoc. Why not a non-fuzzy version,
with a sharp cutoff?

If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?) I
think you'd be quite safe to ignore temperaments whose rms complexity
is more than 12 gens or whose rms error is more than 30 cents. That
will get rid of all of what I consider junk from your list (all the
unnamed ones, limmal, fourth-thirds and the two that end in "decal").

However, it won't solve the problem that the remaining temperaments
will not be sensibly ranked by complexity^3*error, and it won't solve
the problem that you are missing some temperaments that are IMHO at
least as good as those that would remain.

These temperaments are apparently rejected for the crime of having
medium errors (around 5 cents) and medium complexity (around 6 gens).
badness = complexity^3*error favours the extremes over these.

The four I know of have an octave period. Two of these might never be
generated by your algorithm, since you seemed to want to deny that
they were anything but a repeat of meantone last time they were
mentioned, despite the fact that they have different size MOS and work
in different ETs.

Here are the missing four.

Mapping Gen
to 3,5 (cents) Description MOS or ET sizes (improper)
------------------------------------------------------------------
[ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...)
[-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...)
[ 2, 8] 348.1 half meantone-fifth 7 (10 17 24) 31 (38 69 100)
[-2,-8] 251.9 half meantone-fourth 5 (9 14) 19 (24 43 62) 81 (100
...)

It turns out that complexity*EXP((error/1_cent)^0.24) does not produce
the same ranking as complexity^3*error. Even as gentle a rollof as
that sends most of the junk to where it belongs and puts meantone on
the top of the list where _it_ belongs.

I've made a spreadsheet so you can play with sorting Gene's list (plus
the four above) according to Gene's badness, my badness with variable
parameters, or any badness you care to calculate from the given
information.

http://dkeenan.com/Music/5LimitTemp.xls.zip
14KB zipped Excel spreadsheet with macros

Gene,

Could you humor me and temporarily use badness =
complexity*EXP((error/7.4_cents)^0.5)
in your program and see if we get any others that come out better than
pelogic by this measure but aren't in the list that started this
thread (or my recent spreadsheet).

There may be some with fractional-octave periods that I haven't found.

Also these should just squeak in.
Mapping Gen
to 3,5 (cents) Description MOS ET sizes
-------------------------------------------------------------------
[ 12, 10] 158.5 c half kleismic-minor-third 7 8 15 (23 38) 53 (68 121
...)
[-12,-10] 441.5 c half kleismic-major-sixth (5) 8 11 19 (30 49 68) 87
106

>Whether it is rms or max-absolute is irrelevant to my argument. I'm
>happy to use minimum rms error as the input to badness (though I think
>it's good to know the minimum max-absolute as well). If you're
>suggesting that using rms somehow removes the need to apply a
>nonlinear "pain" function, then you're mistaken. You've put "cents" in
>scare quotes above as if you think the units aren't really cents when
>it's rms. They are.

That's true. If you want pain, it should just be ms.

>> Chopping off anything less than .5 is a
>> hack, and hopefully an un-necessary one.
>
>It is definitely unnecessary and I do not propose to chop anything
>off. Gene's use of straight cents gives far too much credit to
>temperaments that have very small sub-half-cent errors. It is allowed
>to compensate so much for large gens that a temperament with a
>complexity of 35.5 gens (rms) can be considered the best 5-limit
>temperament! If you were to take a 0.5 cent threshold into account in
>a discontinuous manner, you would treat any temperament whose rms
>error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not
>proposing we do that. My pain function is non-linear but smooth. A
>temperament still gets some credit for being sub-half-cent, just not
>so much.

((rms_error/7.4_cents)^0.5)

Well, this just has too many constants for my taste.

>That's a good way of putting it, except I wouldn't say that Gene's not
>weighting the complexity enough, I'd say he's failing to level off
>(asymptote) with sub-cent errors and not weighting super-20-cent
>errors enough.

I think I'd rather have a smooth pain function, like ms, and a
stronger exponent on complexity.

>> You want something that exposes the pattern in the series of
>> best temperaments.
>>
>> It would be nice to see a list with a much stronger penalty
>> for size, but I can live with a flat measure with a sharp cutoff.
>
>You don't _have_ to live with it.

True. But it will take more learning and more dissatisfaction with
Gene and Graham than I'm currently experiencing to make me cook my
own list.

-Carl

>I'd rather they were listed in increasing order of "badness", assuming
>"badness" actually means something, like badness. Then if I'm looking
>for the best temperament whose error is in a particular range of sizes
>I'll just go down the list until I find the first one _in_ that range.
>The same goes for any other property that might be important to me at
>the time, like having a half-octave period or having a single
>generator making the perfect fifth.

I could be wrong, but I don't think you can have it both ways. If
you want small temperaments to be better, your list will be finite.
That's what happened with steps^3cents and ets (right, Gene?).

I think I can feel a two-list paper cooking. First, we say, "in some
meaningful sense, 2971 (or whatever) is as good as 31... there's a
periodicity here... here's a list of the best 20 temperaments up to
10,000 (or something)...". Then, "but increasing the exponent on
steps to yield a finite list meaningful for physical instruments such
as guitars and pianos, we have ...".

-Carl

>http://dkeenan.com/Music/5LimitTemp.xls.zip
>14KB zipped Excel spreadsheet with macros

Hooray! :)

-Carl

>> Carl's preferred map complexity measure:
>> (/ (- (max map) (min map)) (card map))
>
>That is ambiguous because you haven't defined what map or card mean.

That's true. Actually, card is standard set theory stuff.
I just need to define map. I was originally only referring
to the map of the generator, but as Gene points out, you need
to take into account if your other generator is a fraction
of your ie. I'll see if I can't schlep something together.
Now, it's off to brunch!

-C.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?)

Because we know what they mean--we haven't hidden assumptions in a function; we've got it right out in the open.

I
> think you'd be quite safe to ignore temperaments whose rms complexity
> is more than 12 gens or whose rms error is more than 30 cents.

Well, I went up to 50 gens and 50 cents, which seems to be a lot of your complaint.
> The four I know of have an octave period. Two of these might never be
> generated by your algorithm, since you seemed to want to deny that
> they were anything but a repeat of meantone last time they were
> mentioned, despite the fact that they have different size MOS and work
> in different ETs.

And you can't get from one note to the other using 5-limit consonances, so that they aren't authentic 5-limit temperaments. Why complain about me introducing "junk" and then insist on this? Toss 'em, and consider them again at higher limits, where they make sense.

> Here are the missing four.
>
> Mapping Gen
> to 3,5 (cents) Description MOS or ET sizes (improper)
> ------------------------------------------------------------------
> [ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...)
> [-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...)

Theese both have a badness under 1000; do you think a search in the range g < 12, rms < 30 and badness < 1000 would be a good idea? It seems you are saying that would be more relevant.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> I could be wrong, but I don't think you can have it both ways. If
> you want small temperaments to be better, your list will be finite.
> That's what happened with steps^3cents and ets (right, Gene?).

You can tweak it a little and not get a finite list, but only a little, and log-flat seems like the right place for an infinite list.
Dave's objections in good measure are that he doesn't *want* an infinite list, with or without microtemperaments on the actual list, because he doesn't want an infinity of micro-micro-micro-temperaments with no practical meaning being theoretically wonderful according to some measure. To me, that shows the measure is working, to Dave, that it's broken.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
>
> For example, I think meantone must be
> > in the top 3, or the badness measure is nonsense.
>
> This is like saying 12-et must be in the top three. It's great in its
> size range--if that happens to be the range you are interested
in. If
> it doesn't suit your requirements then it isn't great, whatever
> number you come up with for it. What's top or not top depends
on what
> tone group you are looking at (5-limit, 7-limit?) and what sort of
> accuracy you want.

dave is trying to make an important subjective decision for
musicians. gene, by insisting on a log-flat measure, best
permits the musician to make this decision for him/herself.
going too far in both directions doesn't hurt anyone. gene, you
may note, has given his lists in order of complexity (or similar,
but it should be in order of complexity), but not as an overall
ranking. an overall ranking is pretty meaningless outside of a
single musician's desiderata.

so i'm completely with gene on this one.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> > What made you go to rms? Isn't it over-kill?
>
> It's less sensitive to outliers; if a temperament does a lot of
> things well and some badly, it still get credit for it.

and it seems that the rms measure even agrees with the
heuristic reasonably well -- though i'd prefer a weighted
measure, as i've mentioned before. dave keenan has given the
exact formula for the weighting i want.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> That's exactly what I _don't_ want. I want to know the size of
> the chain I need to complete my map.

why insist so vehemently on completeness? you can do
wonderful musical things with an incomplete map.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "genewardsmith"
<genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> > > --- In tuning-math@y..., "genewardsmith"
<genewardsmith@j...>
> wrote:
> >
> > > So you're gonna make different lists for different size
ranges,
> are
> > > you?
> >
> > I've tried listing them in order of size, which makes sense to
me,
> though I think Paul didn't like it. What do you think?
>
> I'd rather they were listed in increasing order of "badness",
assuming
> "badness" actually means something, like badness. Then if
I'm looking
> for the best temperament whose error is in a particular range
of sizes
> I'll just go down the list until I find the first one _in_ that range.

sounds like a terrible idea.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:

> And you can't get from one note to the other using 5-limit
>consonances, so that they aren't authentic 5-limit
>temperaments. Why complain about me introducing "junk" and
>then insist on this? Toss 'em, and consider them again at
>higher limits, where they make sense.

i vote for just making a brief mention of the phenomenon, as
anyone can easily calculate them (and their badness values)
from the "real" ones if desired.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> > --- In tuning-math@y..., "genewardsmith"
<genewardsmith@j...> wrote:
>
> > So you're gonna make different lists for different size ranges,
are
> > you?
>
> I've tried listing them in order of size, which makes sense to
>me, though I think Paul didn't like it.

hmm? listing in order of complexity? works wonderfully. no need
to make separate lists, dave.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> dave is trying to make an important subjective decision for
> musicians.

This subjective/objective false-dichotomy again? Haven't read enough
Wilber yet? I'd like us to agree on a badness measure that best
represents the collective subjective experiences of lots of musicians.
In the same manner as my favourite definition of "just".

> gene, by insisting on a log-flat measure, best
> permits the musician to make this decision for him/herself.
> going too far in both directions doesn't hurt anyone.

How can you say that!?

I just showed that Gene's list failed to include at least two
temperaments that are _way_ more interesting and useful than
"fourth-thirds" and the one with 49 gens to the fifth. Namely
minimal diesic and the unnamed one with the [-4,3] map and the 126.2
cent generator. They were discriminated against purely because they
were middle-of-the-road in error and complexity.

No matter how far Gene goes with his badness cutoff, I will always be
able to show that he has left off temperaments which any sane musician
would find to be much better than some of those he has included,
unless his error and complexity cutoffs can be adjusted (ad hoc) to
avoid it.

> gene, you
> may note, has given his lists in order of complexity (or similar,
> but it should be in order of complexity), but not as an overall
> ranking. an overall ranking is pretty meaningless outside of a
> single musician's desiderata.

Some overall rankings are a lot more meaningful than others. If you're
going to give a single list and you're going to put the badness
numbers in there, then why _wouldn't_ readers assume they were
applicable overall? I don't think that, not putting the list in
badness order, will be enough.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> > I'd rather they were listed in increasing order of "badness",
> assuming
> > "badness" actually means something, like badness. Then if
> I'm looking
> > for the best temperament whose error is in a particular range
> of sizes
> > I'll just go down the list until I find the first one _in_ that
range.
>
> sounds like a terrible idea.

Would you mind saying why?

I'm just as likely to be looking for the best temperament whose
generator size is in a particular range or whose period is a
particular fraction of an octave, or whose rms error is in a
particular range, as I am to be looking for one whose number of gens
is in a particular range. Why favour any one of these (and thereby
make the others much more difficult to find) by sorting the list on
it?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > dave is trying to make an important subjective decision for
> > musicians.
>
> This subjective/objective false-dichotomy again? Haven't read
enough
> Wilber yet? I'd like us to agree on a badness measure that
best
> represents the collective subjective experiences of lots of
musicians.
> In the same manner as my favourite definition of "just".

ok. let our paper have two sets of lists then.

> > gene, by insisting on a log-flat measure, best
> > permits the musician to make this decision for him/herself.
> > going too far in both directions doesn't hurt anyone.
>
> How can you say that!?
>
> I just showed that Gene's list failed to include at least two
> temperaments that are _way_ more interesting and useful
than
> "fourth-thirds" and the one with 49 gens to the fifth.

not in their respective ranges of complexity -- and there were
other reasons for excluding "fourth-thirds", as i recall.

> > gene, you
> > may note, has given his lists in order of complexity (or
similar,
> > but it should be in order of complexity), but not as an overall
> > ranking. an overall ranking is pretty meaningless outside of a
> > single musician's desiderata.
>
> Some overall rankings are a lot more meaningful than others.
If you're
> going to give a single list and you're going to put the badness
> numbers in there,

*don't* put the badness numbers in there. the important part of
the paper should be about how temperaments work and why
they're important, not about how to *rank* them!

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> > > I'd rather they were listed in increasing order of "badness",
> > assuming
> > > "badness" actually means something, like badness. Then
if
> > I'm looking
> > > for the best temperament whose error is in a particular
range
> > of sizes
> > > I'll just go down the list until I find the first one _in_ that
> range.
> >
> > sounds like a terrible idea.
>
> Would you mind saying why?

because the systems of similar complexity should be next to
each other, so a musician who's interested in a particular
complexity range can immediately compare and contrast the
systems in that range.

> I'm just as likely to be looking for the best temperament whose
> generator size is in a particular range

why? seems overly specific. who is likely to have that as their
priority?

> or whose period is a
> particular fraction of an octave,

ditto.

temperaments are just mappings of ji. it's more important,
compositionally, *which* commas vanish than what the
generator is or what the period is. of course a fifth generator is
preferable and that's why i cling to my unpopular idea of a
*weighted* complexity calculation.

> or whose rms error is in a
> particular range,

you can always introduce more error by other means. really one
wants the tunings that *minimize* the error within the musician's
preferred sphere of 'complexity'.

> Why favour any one of these (and thereby
> make the others much more difficult to find) by sorting the list
on
> it?

hopefully this makes it clear. maybe this is why you didn't
understand my liking of log-flat badness -- it's because i've
assumed this is how you'd present the tunings.

>> That's exactly what I _don't_ want. I want to know the size of
>> the chain I need to complete my map.
>
>why insist so vehemently on completeness? you can do
>wonderful musical things with an incomplete map.

I agree, but then it's just a good temperament at a different
limit. If we use the concept of limit, then we should. If
a _few_ temperaments come up as really good except for one
identity, they could be mentioned in a footnote or something.
Otherwise, folks interested in stuff like 7:9:11 can go to
Graham's site.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> That's exactly what I _don't_ want. I want to know the size of
> >> the chain I need to complete my map.
> >
> >why insist so vehemently on completeness? you can do
> >wonderful musical things with an incomplete map.
>
> I agree, but then it's just a good temperament at a different
> limit.

not necessarily.

> If we use the concept of limit, then we should.

?

i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},
{3,5,7}, and {2,3,5,7}. isn't that right?

>> I agree, but then it's just a good temperament at a different
>> limit.
>
>not necessarily.

Example?

>> If we use the concept of limit, then we should.
>
>?

What it says.

>i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},
>{3,5,7}, and {2,3,5,7}. isn't that right?

Oh- I didn't know. Okay then, you're not using limit. So you should
really have no reason to list these temperaments, barring the example
requested above.

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> ok. let our paper have two sets of lists then.

OK. The mathematicians list and the musicians list. But if we're not
including any badness measure and we're not listing the temperaments
in order of any badness measure, then that may not be necessary. I've
just given cutoffs for Gene, so that he is guaranteed to include my
list in his (assuming my list doesn't grow when Gene finds out what
went wrong with his attempt to generate my list).

> > > gene, by insisting on a log-flat measure, best
> > > permits the musician to make this decision for him/herself.
> > > going too far in both directions doesn't hurt anyone.
> >
> > How can you say that!?
> >
> > I just showed that Gene's list failed to include at least two
> > temperaments that are _way_ more interesting and useful
> than
> > "fourth-thirds" and the one with 49 gens to the fifth.
>
> not in their respective ranges of complexity

That's not relevant. My point is that it _does_ hurt someone. It hurts
the person interested in middle-of-the-road temperaments.

> -- and there were
> other reasons for excluding "fourth-thirds", as i recall.

OK. Just substitute "pelogic" for "fourth-thirds" in the above.

> *don't* put the badness numbers in there. the important part of
> the paper should be about how temperaments work and why
> they're important, not about how to *rank* them!

OK. I can accept that.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Would you mind saying why?

Supposing you saw a list which rated calculators, portable PCs, desktops, workstations, servers, mainframes and supercomputers with a numerical score, which combined price with performance. Would it make much sense to find cheap $15 calculaors next to supercomputers?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
> > Would you mind saying why?
>
> because the systems of similar complexity should be next to
> each other, so a musician who's interested in a particular
> complexity range can immediately compare and contrast the
> systems in that range.
>
> > I'm just as likely to be looking for the best temperament whose
> > generator size is in a particular range
>
> why? seems overly specific. who is likely to have that as their
> priority?

e.g. Folks considering making a LT guitar and wanting to keep close to
standard open string tuning would favour a generator that was a
fourth.

Fourth/fifth generators are favoured in general.

Or someone might be specifically looking for generators that will be
steps of the MOS.

That reminds me, we must list _all_ possible generators less than an
octave, for each temperament. i.e gen, period-gen, period+gen,
2*period-gen, etc.

> > or whose period is a
> > particular fraction of an octave,
>
> ditto.

e.g. Someone might prefer not to have a temperament with multiple
chains, because dealing with that can be an added complexity in
itself.

> temperaments are just mappings of ji. it's more important,
> compositionally, *which* commas vanish than what the
> generator is or what the period is.

It may be so to you, but you can't predict that everyone will find it
so.

> of course a fifth generator is
> preferable and that's why i cling to my unpopular idea of a
> *weighted* complexity calculation.

I support that idea, but that won't help me with actually _finding_
the best fourth/fifth generators in the list.

> > or whose rms error is in a
> > particular range,
>
> you can always introduce more error by other means. really one
> wants the tunings that *minimize* the error within the musician's
> preferred sphere of 'complexity'.

I have done this very thing already, looking thru graham's lists for
microtemperaments suitable for fretting a guitar for some JI scale. I
wanted errors in the range of 2 to 3 cents, no more no less. Keeping
the error above 2 c served to limit the complexity, below 3c limited
the mistuning pain.

>
> > Why favour any one of these (and thereby
> > make the others much more difficult to find) by sorting the list
> on
> > it?
>
> hopefully this makes it clear.

Well a little. But it doesn't matter. If the only way we can move
forward is to agree not to list them in order of _any_ badness,
then I'm happy for them to be listed in order of complexity.

> maybe this is why you didn't
> understand my liking of log-flat badness -- it's because i've
> assumed this is how you'd present the tunings.

Er no. I still don't understand it, except as a mathematicians cutoff
vs a musicians cutoff.

Anyway, it doesn't matter now, if everyone else agrees to the
following.

1. No badness shown.

2. Listed in order of increasing complexity, which is:

neutral thirds
augmented
meantone
pelogic
diminished
porcupine
small diesic
diaschismic
kleismic
16875/16384
twin meantone
half meantone-fifth
half meantone-fourth
wuerschmidt
minimal diesic
1990656/1953125
tiny diesic
schismic
orwell
twin kleismic
half kleismic-minor-third
half kleismic-major-sixth
amt
semisuper
parakleismic
hemithird
and maybe a few others

3. Smith badness < 861
rms complexity < 13.2 gens
rms error < 28.9 cents

You can of course round the numbers up so it doesn't look so contrived
provided that doesn't add more than 2 or 3 extra temperaments.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},
> {3,5,7}, and {2,3,5,7}. isn't that right?

That's ok by me. But if we're listing these separately (which seems
the right thing to me), then the maximum interval width (which is also
the width of the complete chord - Carl's measure) seems more relevant
to me than the rms interval width.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> I agree, but then it's just a good temperament at a different
> >> limit.
> >
> >not necessarily.
>
> Example?

orwell with 9 or 13 notes. still 11-limit.

and what if the subset of 'primes' you're thinking about is not
really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:

> That's not relevant. My point is that it _does_ hurt someone. It
hurts
> the person interested in middle-of-the-road temperaments.
>
> > -- and there were
> > other reasons for excluding "fourth-thirds", as i recall.
>
> OK. Just substitute "pelogic" for "fourth-thirds" in the above.

then i'd definitely disagree with you.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Would you mind saying why?
>
> Supposing you saw a list which rated calculators, portable PCs,
desktops, workstations, servers, mainframes and supercomputers with a
numerical score, which combined price with performance. Would it make
much sense to find cheap $15 calculaors next to supercomputers?
>

If the analogy is say

dollars <-> complexity
megaflop.gigabytes <-> 1/error

then badness would be in dollars per megaflop.gigabyte.

Well if that's what your badness is, 1/value-for-money, then yes there
might well be a calculator next to a supercomputer.

Anyway. I'm willing to accept a list ordered by complexity provided
smith-badness is not shown anywhere.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Well if that's what your badness is, 1/value-for-money, then yes
there
> might well be a calculator next to a supercomputer.

I meant to add that that seems perfectly logical to me.

> Anyway. I'm willing to accept a list ordered by complexity provided
> smith-badness is not shown anywhere.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> then the maximum interval width (which is also
> the width of the complete chord - Carl's measure) seems more
relevant
> to me than the rms interval width.

and shouldn't it be the maximum or rms *weighted* interval
width?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > i thought the paper was going to concern {2,3,5}, {2,3,7},
{2,5,7},
> > {3,5,7}, and {2,3,5,7}. isn't that right?
>
> That's ok by me. But if we're listing these separately (which
>seems
> the right thing to me),

yes,

> then the maximum interval width (which is also
> the width of the complete chord - Carl's measure) seems more
> relevant
> to me than the rms interval width.

why is that?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:

> Anyway. I'm willing to accept a list ordered by complexity
provided
> smith-badness is not shown anywhere.

phew. a consensus emerges?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
>
> > That's not relevant. My point is that it _does_ hurt someone. It
> hurts
> > the person interested in middle-of-the-road temperaments.
> >
> > > -- and there were
> > > other reasons for excluding "fourth-thirds", as i recall.
> >
> > OK. Just substitute "pelogic" for "fourth-thirds" in the above.
>
> then i'd definitely disagree with you.

Perhaps that got a bit muddled. I mean that using a smith badness
cutoff without small enough cutoffs on error and complexity, I would
always find temperaments that any sane musician would consider better
than either pelogic or 49-gens-to-the-fifth. That would be
disadvantaging someone.

Anyway, if Gene uses the right cutoffs on eror and complexity this is
now a moot point.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
> > then the maximum interval width (which is also
> > the width of the complete chord - Carl's measure) seems
more
> relevant
> > to me than the rms interval width.
>
> and shouldn't it be the maximum or rms *weighted* interval
> width?

by the way, my now-famous heursitic for complexity would sort
the 5-limit temperaments by the size of the numerators (or
denominators, or n*d) of the commas. kees van prooijen
webpages seem to suggest clearly that this should be
expressible as a distance measure of some kind. no one else
seems interested in pursuing this obversation, however :(

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> and what if the subset of 'primes' you're thinking about is not
> really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

{2,3,5/3} defines the same subgroup as {2,3,5}, but {2,3,7/5} is a good example of a subgroup which doesn't fit the missing prime paradigm. I posted a list of such subgroups a while back.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> > wrote:
> >
> > > That's not relevant. My point is that it _does_ hurt someone.
It
> > hurts
> > > the person interested in middle-of-the-road temperaments.
> > >
> > > > -- and there were
> > > > other reasons for excluding "fourth-thirds", as i recall.
> > >
> > > OK. Just substitute "pelogic" for "fourth-thirds" in the above.
> >
> > then i'd definitely disagree with you.
>
> Perhaps that got a bit muddled. I mean that using a smith
badness
> cutoff without small enough cutoffs on error and complexity,

you mean large enough?

> I would
> always find temperaments

you mean with even larger error and complexity?

> that any sane musician would consider better
> than either pelogic or 49-gens-to-the-fifth.

you lost me.

> That would be
> disadvantaging someone.
>
> Anyway, if Gene uses the right cutoffs on eror and complexity
this is
> now a moot point.

what are the right cutoffs? i favor going well into uninteresting
territory on both ends, to demonstrate how the algorithm
functions (once it's truly an algorithm -- it seems it may be
missing some things currently). best to be explicit about it!

>Supposing you saw a list which rated calculators, portable PCs, desktops,
>workstations, servers, mainframes and supercomputers with a numerical
>score, which combined price with performance. Would it make much sense to
>find cheap $15 calculaors next to supercomputers?

That depends on the price/performance function!

You could be saying so yourself, in fact. There's nothing in your
analogy keeping price to what you want it to be. Dave might think
price is like size (complexity).

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> by the way, my now-famous heursitic for complexity would sort
> the 5-limit temperaments by the size of the numerators (or
> denominators, or n*d) of the commas. kees van prooijen
> webpages seem to suggest clearly that this should be
> expressible as a distance measure of some kind. no one else
> seems interested in pursuing this obversation, however :(

Height functions can be thought of as distance measures, but I'm not getting your point.

>If the analogy is say
>
>dollars <-> complexity
>megaflop.gigabytes <-> 1/error

I swear I did not read this before I sent my reply. :)

-Carl

>> i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7},
>> {3,5,7}, and {2,3,5,7}. isn't that right?
>
>That's ok by me. But if we're listing these separately (which seems
>the right thing to me), then the maximum interval width (which is also
>the width of the complete chord - Carl's measure) seems more relevant
>to me than the rms interval width.

That is a good one, but mine is that divided by the number of elements
in your chord (I said "card map", which is the same if you define map
like me, and remember to mulitply by periods in your interval of
equivalence at the end), which would be 3, 3, 3, 3, and 4 above,
respectively.

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > by the way, my now-famous heursitic for complexity would
sort
> > the 5-limit temperaments by the size of the numerators (or
> > denominators, or n*d) of the commas. kees van prooijen
> > webpages seem to suggest clearly that this should be
> > expressible as a distance measure of some kind. no one
else
> > seems interested in pursuing this obversation, however :(
>
> Height functions can be thought of as distance measures, but
>I'm not getting your point.

a _lattice_ distance function.

>orwell with 9 or 13 notes. still 11-limit.

I show orwell with a 9-tone MOS, and this map:

[0 7 -3 8 2]
[1 0 0 0 0]

'zthat right? That covers 10 gens in the 5-limit, 11-gens in
the 7-limit, and 11 gens in the 11-limit. You're saying...

[0 -3 2]
[1 0 0]
[1 5 11]

...only takes 5 notes? I know I said "limit", but I meant it
figuratively, not literally. It's an abuse of terminology to
call [1 5 11] the "[1 5 11]-limit", for example, but that's what
I meant.

So if your paper's really going to cover all these (by that I
mean have a list for each one), I'd suggest ranking by
complexity, with a sharp cutoff. Just show the most accurate
three temperaments for each integer of complexity up to 15 or
so. You could seed this with whatever badness measure you wanted,
as long as you let it go up to 500. By then, even the kind that
Dave can always add wouldn't make it up into my list

You can tell I don't care about sharp cutoffs. :)

>and what if the subset of 'primes' you're thinking about is not
>really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?

Fine. (I don't understand what you thought I thought... Maybe
I'm still thinking it! :)

-Carl

>e.g. Folks considering making a LT guitar and wanting to keep close to
>standard open string tuning would favour a generator that was a
>fourth.

Ah, mere mention of the days of Keenan GuitarFrettings^TM make me
shudder. In 6 months, will Dave be cooking up linear-tempered
guitars... or maybe by then, planar-tempered guitars? Mmmmmmm....

Two things where I've always thought Dave Keenan kicks butt: notation
systems, and guitar frettings. Well, I can't really speak to the
latter. Paul- should the people at home start cracking eachother's
skulls open and feasting on the sweet goo inside?

(simpsons reference -- Answer: Yes, Kent.)

Did you ever manage to automate the process at all, Dave?

-Carl

>> wrote:
>>> then the maximum interval width (which is also
>>> the width of the complete chord - Carl's measure) seems more
>>> relevant to me than the rms interval width.
>>
>> and shouldn't it be the maximum or rms *weighted* interval
>> width?
>
>by the way, my now-famous heursitic for complexity would sort
>the 5-limit temperaments by the size of the numerators (or
>denominators, or n*d) of the commas.

Sounds interesting, but I'll have to read up again on your
heuristic a bit to even know. I have the original post here
somewhere, I think....

-Carl

>> and what if the subset of 'primes' you're thinking about is not
>> really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?
>
>{2,3,5/3} defines the same subgroup as {2,3,5}, but {2,3,7/5} is a
>good example of a subgroup which doesn't fit the missing prime
>paradigm. I posted a list of such subgroups a while back.

Hmmm. You once told me it had to be primes. Does this have anything
to do with that?

-Carl

>>Height functions can be thought of as distance measures, but
>>I'm not getting your point.
>
>a _lattice_ distance function.

Does that mean a taxicab one? Whatever it is, if it can do
n*d, who cares?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>Height functions can be thought of as distance measures,
but
> >>I'm not getting your point.
> >
> >a _lattice_ distance function.
>
> Does that mean a taxicab one? Whatever it is, if it can do
> n*d, who cares?

i care because it gives us the natural lattice-based complexity
measure for linear temperaments in the 7-limit and higher,
where a single comma won't do, and a sort of 'angle' between
the two is at work.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> (simpsons reference -- Answer: Yes, Kent.)

And people let their children watch that crap.

> Did you ever manage to automate the process at all, Dave?

Only parts of it and not in a user friendly way.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Hmmm. You once told me it had to be primes. Does this have anything
> to do with that?

What had to be primes? The generators? Nope.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> i care because it gives us the natural lattice-based complexity
> measure for linear temperaments in the 7-limit and higher,
> where a single comma won't do, and a sort of 'angle' between
> the two is at work.

Any metric on R^n allows you to define lattices, so I don't see what your angle is.

>> (simpsons reference -- Answer: Yes, Kent.)
>
>And people let their children watch that crap.

Oh, sorry Dave. I forgot you weren't a fan.
I'll stop... it wasn't cool anyway.

-Carl

>>Hmmm. You once told me it had to be primes. Does this have anything
>>to do with that?
>
>What had to be primes? The generators? Nope.

The identities (column vectors of the map, I think).

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
> > then the maximum interval width (which is also
> > the width of the complete chord - Carl's measure) seems more
> relevant
> > to me than the rms interval width.
>
> and shouldn't it be the maximum or rms *weighted* interval
> width?

You also asked why max abs? I'll answer that first.

One of the reasons given for using rms was that if there was an
outlier the LT still got credit for the intervals it did well. But if
we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is
very wide, I don't think it is a good {1,3,5,7} temperament. Let it be
found to be a good {2,5,7} temperament or whatever.

As far as the weighting by odd-limit thing goes. I'm reasonably in
favour. But it's a very musician-type "subjective" thing to do and not
at all a mathematician type thing. And I'm not prepared to argue over
it, mainly because I expect the good temperaments will still make it
on the list without it. Also, I have no easy way of finding optimum
generators (which you'd have to do all over again before you can find
the error). Gene would have to do it.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> you lost me.

Never mind. It doesn't matter now.

> > Anyway, if Gene uses the right cutoffs on eror and complexity
> this is
> > now a moot point.
>
> what are the right cutoffs? i favor going well into uninteresting
> territory on both ends, to demonstrate how the algorithm
> functions (once it's truly an algorithm -- it seems it may be
> missing some things currently). best to be explicit about it!

As I posted twice before:

Smith badness < 861
rms complexity < 13.2 gens
rms error < 28.9 cents

You can of course round the numbers up so it doesn't look so
contrived, provided that doesn't add more than 2 or 3 extra
temperaments.

I wrote...
>I show orwell with a 9-tone MOS, and this map:
>
>[0 7 -3 8 2]
>[1 0 0 0 0]
>
>'zthat right?

Whoops, I think it should be more something like this:

[2] [0 1]
[3] [7 0]
[5] [-3 2]
[7] [8 1]
[11] [2 3]

Is the column out front a way to write that? Certainly we wouldn't
put the identities inside the map, as I first tried?

Gene, when you showed me this, you gave:

[0 1]
[1 1]
[4 0]

for meantone. but shouldn't the last row be:

[4 2]

Graham's script gives:

[0 1]
[1 0]
[4 -4]

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >orwell with 9 or 13 notes. still 11-limit.
>
> I show orwell with a 9-tone MOS,

and 13 too . . .

and this map:
>
> [0 7 -3 8 2]
> [1 0 0 0 0]
>
> 'zthat right? That covers 10 gens in the 5-limit, 11-gens in
> the 7-limit, and 11 gens in the 11-limit. You're saying...
>
> [0 -3 2]
> [1 0 0]
> [1 5 11]
>
> ...only takes 5 notes?

you lost me.

>I know I said "limit", but I meant it
> figuratively, not literally. It's an abuse of terminology to
> call [1 5 11] the "[1 5 11]-limit", for example, but that's what
> I meant.

is [2 3 7/5] a limit too?

>
> So if your paper's really going to cover all these (by that I
> mean have a list for each one), I'd suggest ranking by
> complexity, with a sharp cutoff. Just show the most accurate
> three temperaments for each integer of complexity up to 15 or
> so.

this is flat badness. i think log-flat is better, and it's what gene's
using.

>>I know I said "limit", but I meant it
>> figuratively, not literally. It's an abuse of terminology to
>> call [1 5 11] the "[1 5 11]-limit", for example, but that's what
>> I meant.
>
>is [2 3 7/5] a limit too?

In the sense that you can give a list for it or tell the reader
to try Graham's script with input X, so that you don't have to
use rms of the "highest interval width" (Graham's term for the
length of the chain).

>> So if your paper's really going to cover all these (by that I
>> mean have a list for each one), I'd suggest ranking by
>> complexity, with a sharp cutoff. Just show the most accurate
>> three temperaments for each integer of complexity up to 15 or
>> so.
>
>this is flat badness.

Which badness formulas produce flat?

>i think log-flat is better, and it's what
>gene's using.

Ok. Is steps^n*cents log-flat for any n > 1?

-Carl

Dave wrote:

> orwell

From George Orwell?

Manuel

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> > wrote:
> > > then the maximum interval width (which is also
> > > the width of the complete chord - Carl's measure) seems
more
> > relevant
> > > to me than the rms interval width.
> >
> > and shouldn't it be the maximum or rms *weighted* interval
> > width?
>
> You also asked why max abs? I'll answer that first.
>
> One of the reasons given for using rms was that if there was
an
> outlier the LT still got credit for the intervals it did well. But if
> we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is
> very wide, I don't think it is a good {1,3,5,7} temperament. Let it
be
> found to be a good {2,5,7} temperament or whatever.

it might not -- again, what if it's really a good {2, 3, 7/5}
temperament?

>
> As far as the weighting by odd-limit thing goes. I'm reasonably
in
> favour. But it's a very musician-type "subjective" thing to do and
not
> at all a mathematician type thing. And I'm not prepared to argue
over
> it, mainly because I expect the good temperaments will still
make it
> on the list without it. Also, I have no easy way of finding
optimum
> generators (which you'd have to do all over again before you
can find
> the error).

huh? the optimum generator would be exactly the same as
before, since we're not weighting the 'cents' part of it, just the
'steps' part of it . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> it might not -- again, what if it's really a good {2, 3, 7/5}
> temperament?

Ok let's stick with rms error. We seem to have the most agreement on
that.

> huh? the optimum generator would be exactly the same as
> before, since we're not weighting the 'cents' part of it, just the
> 'steps' part of it . . .

You're right. I got confused. So ask gene to do a run with weighted
rms gens to see if anything new comes up.

Oh and Manuel, Yes that's from George Orwell. The generator for that
temperament is essentially 19/84 oct. Gene's idea.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>I know I said "limit", but I meant it
> >> figuratively, not literally. It's an abuse of terminology to
> >> call [1 5 11] the "[1 5 11]-limit", for example, but that's what
> >> I meant.
> >
> >is [2 3 7/5] a limit too?
>
> In the sense that you can give a list for it or tell the reader
> to try Graham's script with input X, so that you don't have to
> use rms of the "highest interval width" (Graham's term for the
> length of the chain).

so do you want to propose a modification of the list of cases to be
studied in our paper?

> >> So if your paper's really going to cover all these (by that I
> >> mean have a list for each one), I'd suggest ranking by
> >> complexity, with a sharp cutoff. Just show the most accurate
> >> three temperaments for each integer of complexity up to 15 or
> >> so.
> >
> >this is flat badness.
>
> Which badness formulas produce flat?

this is a question gene can surely answer (if only to say, 'none').

> >i think log-flat is better, and it's what
> >gene's using.
>
> Ok. Is steps^n*cents log-flat for any n > 1?

no. n depends on the number of independent intervals in the map,
i.e., the number of 'terms' in the 'limit'.

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Dave wrote:
>
> > orwell
>
> From George Orwell?

gene calls it that because the generator is about 19/84 octave.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > it might not -- again, what if it's really a good {2, 3, 7/5}
> > temperament?
>
> Ok let's stick with rms error. We seem to have the most agreement
on
> that.
>
> > huh? the optimum generator would be exactly the same as
> > before, since we're not weighting the 'cents' part of it, just
the
> > 'steps' part of it . . .
>
> You're right. I got confused. So ask gene to do a run with weighted
> rms gens to see if anything new comes up.

ok. gene, once again, this means that in the 'gens' calculation, the
number of generators in the 3:1 should be multiplied by log(3), the
number of generators in the 5:3 should be multiplied by log(5), the
number of generators in the 5:1 should be multiplied by log(5). this
will cause temperaments generated by the fifth to look better than
they currently do, relative to those that aren't. this is important
since augmented (especially) and diminished (a little less so) are
far harder for the ear to understand than meantone, even when all are
*tuned* in 12-equal, and the badness values would no longer put
meantone as the 'best'.

>> In the sense that you can give a list for it or tell the reader
>> to try Graham's script with input X, so that you don't have to
>> use rms of the "highest interval width" (Graham's term for the
>> length of the chain).
>
>so do you want to propose a modification of the list of cases to be
>studied in our paper?

I'm just saying I think rms(complexity) smooths over information
I want. You objected that mean(complexity) might be unfair to
temperaments that only did one thing badly. I suggested such
temperaments be listed separately, as doing everything well over
a restricted set of identities.

If I were to say some temperaments are good over some list of
identities, you'd expect I meant all the identities, wouldn't
you? Isn't the idea behind a "limit" that you need the lower
identities to make the higher ones work (not that I think this
is correct)?

>> >i think log-flat is better, and it's what
>> >gene's using.
>>
>> Ok. Is steps^n*cents log-flat for any n > 1?
>
>no. n depends on the number of independent intervals in the map,
>i.e., the number of 'terms' in the 'limit'.

Oh!! Howso?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> In the sense that you can give a list for it or tell the reader
> >> to try Graham's script with input X, so that you don't have to
> >> use rms of the "highest interval width" (Graham's term for the
> >> length of the chain).
> >
> >so do you want to propose a modification of the list of cases to
be
> >studied in our paper?
>
> I'm just saying I think rms(complexity) smooths over information
> I want. You objected that mean(complexity) might be unfair to
> temperaments that only did one thing badly. I suggested such
> temperaments be listed separately, as doing everything well over
> a restricted set of identities.

by identities, you really mean all the consonant intervals, right?

well, i think if you look on a case-by-case basis, you won't be able
to object to what rms does. because only doing 'one thing badly'
isn't really possible in this context. find an example, if you can!

> >> >i think log-flat is better, and it's what
> >> >gene's using.
> >>
> >> Ok. Is steps^n*cents log-flat for any n > 1?
> >
> >no. n depends on the number of independent intervals in the map,
> >i.e., the number of 'terms' in the 'limit'.
>
> Oh!! Howso?

gene derived this from diophantine approximation theory.

>by identities, you really mean all the consonant intervals, right?

Right.

>well, i think if you look on a case-by-case basis, you won't be able
>to object to what rms does. because only doing 'one thing badly'
>isn't really possible in this context. find an example, if you can!

Well, you just gave Orwell, so let's look at it. I still have no
idea about the map. My best guess was:

[2] [0 1]
[3] [7 0]
[5] [-3 2]
[7] [8 1]
[11] [2 3]

Which disagrees with Gene:

>[ 0 1]
>[ 7 0]
>[-3 3]

Going by my map:

max int width card(map) me g
5-limit 10 2 5 7.257
11-limit 11 4 2.75 ??

I have no idea how Gene got 7.257, so I can't fill
g in for the 11-limit.

Still don't know why we can't include 9 in the map.

>>>> Ok. Is steps^n*cents log-flat for any n > 1?
>>>
>>>no. n depends on the number of independent intervals in the map,
>>>i.e., the number of 'terms' in the 'limit'.
>>
>>Oh!! Howso?
>
>gene derived this from diophantine approximation theory.

I remember now.

-Carl

>Going by my map:
>
> max int width card(map) me g
>5-limit 10 2 5 7.257
>11-limit 11 4 2.75 ??
>
>I have no idea how Gene got 7.257, so I can't fill
>g in for the 11-limit.

Dave's getting 7.3 too, with this:

SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13

Anybody care to explain why this isn't total rubbish? Putting
both the individual gens per idenitity (E13 and F13) and the
total width of the chain (E13-F13) together into the rms calc???

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >Going by my map:
> >
> > max int width card(map) me g
> >5-limit 10 2 5 7.257
> >11-limit 11 4 2.75 ??
> >
> >I have no idea how Gene got 7.257, so I can't fill
> >g in for the 11-limit.
>
> Dave's getting 7.3 too, with this:
>
> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13
>
> Anybody care to explain why this isn't total rubbish? Putting
> both the individual gens per idenitity (E13 and F13) and the
> total width of the chain (E13-F13) together into the rms calc???

carl, if 'identity' is defined as 'consonant interval', then the
*only* thing going in here is the individual gens per identity.
that's all. E13-F13 is the major sixth or minor third.

>carl, if 'identity' is defined as 'consonant interval', then the
>*only* thing going in here is the individual gens per identity.
>that's all. E13-F13 is the major sixth or minor third.

Oh. Well, I'm not sure how that's significant, since in regular
temperaments it will always be the difference of the 3 and 5
mappings.

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> ok. gene, once again, this means that in the 'gens' calculation, the
> number of generators in the 3:1 should be multiplied by log(3), the
> number of generators in the 5:3 should be multiplied by log(5), the
> number of generators in the 5:1 should be multiplied by log(5).

And to make the result meaningful (i.e. comparable to the unweighted
values) then after you take the RMS of these weighted values you
should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Dave's getting 7.3 too, with this:
>
> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13
>
> Anybody care to explain why this isn't total rubbish? Putting
> both the individual gens per idenitity (E13 and F13) and the
> total width of the chain (E13-F13) together into the rms calc???

That's unweighted rms complexity which is

SQRT((gens(1:3)^2+gens(1:5)^2+gens(3:5)^2)/3)*1200/period

where period is in cents and

gens(3:5) = gens(1:5)-gens(1:3), where the gens are signed quantities,
not absolute values. So it's not necessarily the total width.

Max-absolute complexity (your total width) is
MAX(ABS(gens(1:3),ABS(gens(1:5),ABS(gens(3:5))*1200/period
which is equivalent to
ABS(MAX(gens(1:3),gens(1:5))-MIN(gens(1:3),gens(1:5)))*1200/period

By expanding gen(3:5)^2, rms complexity could be calculated as
SQRT((gens(1:3)^2+gens(1:5)^2-gens(1:3)*gens(1:5))*2/3)*1200/period
but who cares.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >carl, if 'identity' is defined as 'consonant interval', then the
> >*only* thing going in here is the individual gens per identity.
> >that's all. E13-F13 is the major sixth or minor third.
>
> Oh. Well, I'm not sure how that's significant, since in regular
> temperaments it will always be the difference of the 3 and 5
> mappings.

right -- so what's your objection?

>--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
>> >carl, if 'identity' is defined as 'consonant interval', then the
>> >*only* thing going in here is the individual gens per identity.
>> >that's all. E13-F13 is the major sixth or minor third.
>>
>> Oh. Well, I'm not sure how that's significant, since in regular
>> temperaments it will always be the difference of the 3 and 5
>> mappings.
>
>right -- so what's your objection?

None anymore, except that people aren't considering max-absolute
complexity and ms error, with an exponent on complexity that makes
the list finite or one that makes it log-flat supplemented by a
sharp cutoff.

-Carl

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > ok. gene, once again, this means that in the 'gens' calculation,
the
> > number of generators in the 3:1 should be multiplied by log(3),
the
> > number of generators in the 5:3 should be multiplied by log(5),
the
> > number of generators in the 5:1 should be multiplied by log(5).
>
> And to make the result meaningful (i.e. comparable to the
unweighted
> values) then after you take the RMS of these weighted values you
> should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

why do you want them to be comparable to the unweighted values?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > ok. gene, once again, this means that in the 'gens' calculation,
> the
> > > number of generators in the 3:1 should be multiplied by log(3),
> the
> > > number of generators in the 5:3 should be multiplied by log(5),
> the
> > > number of generators in the 5:1 should be multiplied by log(5).
> >
> > And to make the result meaningful (i.e. comparable to the
> unweighted
> > values) then after you take the RMS of these weighted values you
> > should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).
>
> why do you want them to be comparable to the unweighted values?

Obviously it doesn't matter as far as choosing lists, but I like for a
human (e.g. me) to be able to look at the error values and have them
mean something. i.e. to actually be in cents. So when you see 5 you
know kinda what a 5c mistuning sounds like.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > > ok. gene, once again, this means that in the 'gens'
calculation,
> > the
> > > > number of generators in the 3:1 should be multiplied by log
(3),
> > the
> > > > number of generators in the 5:3 should be multiplied by log
(5),
> > the
> > > > number of generators in the 5:1 should be multiplied by log
(5).
> > >
> > > And to make the result meaningful (i.e. comparable to the
> > unweighted
> > > values) then after you take the RMS of these weighted values
you
> > > should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).
> >
> > why do you want them to be comparable to the unweighted values?
>
> Obviously it doesn't matter as far as choosing lists, but I like
for a
> human (e.g. me) to be able to look at the error values and have
them
> mean something. i.e. to actually be in cents. So when you see 5 you
> know kinda what a 5c mistuning sounds like.

but the units here are not cents, they're gens. what does 5 gens
sound like? faggeddabbouddit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> but the units here are not cents, they're gens. what does 5 gens
> sound like? faggeddabbouddit.

Oh dear I keep doing that don't I. Must be the Alzheimer's. :-)

In-Reply-To: <a6jdeq+58k4@eGroups.com>
paulerlich wrote:

> ok. gene, once again, this means that in the 'gens' calculation, the
> number of generators in the 3:1 should be multiplied by log(3), the
> number of generators in the 5:3 should be multiplied by log(5), the
> number of generators in the 5:1 should be multiplied by log(5). this
> will cause temperaments generated by the fifth to look better than
> they currently do, relative to those that aren't. this is important
> since augmented (especially) and diminished (a little less so) are
> far harder for the ear to understand than meantone, even when all are
> *tuned* in 12-equal, and the badness values would no longer put
> meantone as the 'best'.

When was it decided these temperaments were "far harder for the ear to
understand"? Even if so, augmented is more complex than meantone if you
measure by the simplest MOS to contain a consonant chord (8 compared to 5
notes).

Still, for whatever reason you want to privilege fifths. The rule you
give, generalized to higher limits, will favour simpler ratios in general.
That'll give the usual bias towards temperaments that work with the
limit lower than the one you asked for. And it'll miss temperaments where
a simple interval is close enough to use in modulations, but isn't the
defined approximation to 3:2.

Graham

In-Reply-To: <a6iupi+ksb1@eGroups.com>
Dave K:
> > One of the reasons given for using rms was that if there was
> an
> > outlier the LT still got credit for the intervals it did well. But if
> > we're looking at say {1,3,5,7} then if _any_ {1,3,5,7} interval is
> > very wide, I don't think it is a good {1,3,5,7} temperament. Let it
> be
> > found to be a good {2,5,7} temperament or whatever.

Paul:
> it might not -- again, what if it's really a good {2, 3, 7/5}
> temperament?

There shouldn't be anything special about 1.3.5 and 1.3.7 compared to
1.3.7:5 as defining chords. So far my script only works with the former,
but don't make any plans assuming that will always be the case. It should
even be possible to automatically find and evaluate the simplest subset if
that's all you want.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a6jdeq+58k4@e...>
> paulerlich wrote:
>
> > ok. gene, once again, this means that in the 'gens' calculation,
the
> > number of generators in the 3:1 should be multiplied by log(3),
the
> > number of generators in the 5:3 should be multiplied by log(5),
the
> > number of generators in the 5:1 should be multiplied by log(5).
this
> > will cause temperaments generated by the fifth to look better
than
> > they currently do, relative to those that aren't. this is
important
> > since augmented (especially) and diminished (a little less so)
are
> > far harder for the ear to understand than meantone, even when all
are
> > *tuned* in 12-equal, and the badness values would no longer put
> > meantone as the 'best'.
>
> When was it decided these temperaments were "far harder for the ear
to
> understand"? Even if so, augmented is more complex than meantone
if you
> measure by the simplest MOS to contain a consonant chord (8
compared to 5
> notes).

augmented is 6, diminished is 8 . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > ok. gene, once again, this means that in the 'gens' calculation,
> the
> > > number of generators in the 3:1 should be multiplied by log(3),
> the
> > > number of generators in the 5:3 should be multiplied by log(5),
> the
> > > number of generators in the 5:1 should be multiplied by log(5).
> this
> > > will cause temperaments generated by the fifth to look better
> than
> > > they currently do, relative to those that aren't.

Paul, don't you mean "divided by". log(3) is smaller than log(5) so if
you want to favour fifths ...

Here's the formula I'm using:

SQRT(((gens(1:3)/LN(3))^2+(gens(1:5)/LN(5))^2+(gens(3:5)/LN(5))^2)/(1/
LN(3)^2+1/LN(5)^2+1/LN(5)^2))*1200/period_in_cents

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > > ok. gene, once again, this means that in the 'gens'
calculation,
> > the
> > > > number of generators in the 3:1 should be multiplied by log
(3),
> > the
> > > > number of generators in the 5:3 should be multiplied by log
(5),
> > the
> > > > number of generators in the 5:1 should be multiplied by log
(5).
> > this
> > > > will cause temperaments generated by the fifth to look better
> > than
> > > > they currently do, relative to those that aren't.
>
> Paul, don't you mean "divided by". log(3) is smaller than log(5) so
if
> you want to favour fifths ...

oh yeah, i meant divided by . . . i said it the right way last time,
about two months ago . . .

and i don't buy graham's objections one bit. yes, it favors those
systems which are *aspects* of lower-limit systems -- but very often
you get more than one such aspect per lower-limit system! this is a
very positive feature, not a bug -- it sits nicely with a view of
higher-limit systems often evolving out of lower-limit ones. and of
course it reflects musical reality much better.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> oh yeah, i meant divided by . . . i said it the right way last time,
> about two months ago . . .

So if were to do yet another search, what should be the limits?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > oh yeah, i meant divided by . . . i said it the right way last
time,
> > about two months ago . . .
>
> So if were to do yet another search, what should be the limits?

what do you mean (who and what limits)?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> what do you mean (who and what limits)?

I don't want to do another search, this time weighted, unless you tell me what you want to look at.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > what do you mean (who and what limits)?
>
> I don't want to do another search, this time weighted, unless you
>tell me what you want to look at.

i think dave needs to answer this question (or repeat his answer).

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >
> > > what do you mean (who and what limits)?
> >
> > I don't want to do another search, this time weighted, unless you
> >tell me what you want to look at.
>
> i think dave needs to answer this question (or repeat his answer).

well, failing that, just do it with the same bounds as before, or
still more inclusive bounds if you prefer, and with the applied
badness measure given for each entry.

after that's done, we should strive for consensus and move on to the
7-limit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > >
> > > > what do you mean (who and what limits)?
> > >
> > > I don't want to do another search, this time weighted, unless
you
> > >tell me what you want to look at.
> >
> > i think dave needs to answer this question (or repeat his answer).

Yes I have already posted them, and they are also in my spreadsheet.
But here they are again. 35 cents rms, 12 weighted-rms generators (or
10 for the shorter list) and max 625 for Gene's badness (using
weighted complexity).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes I have already posted them, and they are also in my spreadsheet.
> But here they are again. 35 cents rms, 12 weighted-rms generators (or
> 10 for the shorter list) and max 625 for Gene's badness (using
> weighted complexity).

Here's some examples of weighted badness, using weighted complexity. Can you check your limits in terms of this?

81/80: 103.257
15625/15552: 187.010
250/243: 463.641
78732/78125: 525.371

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Yes I have already posted them, and they are also in my
spreadsheet.
> > But here they are again. 35 cents rms, 12 weighted-rms generators
(or
> > 10 for the shorter list) and max 625 for Gene's badness (using
> > weighted complexity).
>
> Here's some examples of weighted badness, using weighted complexity.
Can you check your limits in terms of this?
>
> 81/80: 103.257
> 15625/15552: 187.010
> 250/243: 463.641
> 78732/78125: 525.371

In terms of these, the badness limit would need to be around 915. You
could have checked them yourself against my spreadsheet.
http://dkeenan.com/Music/5LimitTemp.xls.zip
Are you able to read Excel spreadsheets?

Somehow your badnesses are a factor of 1.46131 times those I
calculate. How could that be? Are you still using complexity^3*error.

Here are the weighted-rms complexities and errors I have for those
temperaments (in case you can't read the spreadsheet). In the order
above.
error complexity
4.218 2.559
1.030 4.991
7.976 3.414
1.157 6.772

So Gene,

How's the 5-limit list coming along.

Also do you have any objections or suggestions for Graham's or my
'standard generators and mapping' proposals. Surely we want canonical
forms that are musically meaningful and convenient where possible.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> So Gene,
>
> How's the 5-limit list coming along.

It isn't; I've been writing my paper instead.

> Also do you have any objections or suggestions for Graham's or my
> 'standard generators and mapping' proposals. Surely we want canonical
> forms that are musically meaningful and convenient where possible.

I modifed my proposal and got it to work, so I suppose I should explain that. The next few days are teaching days, but then we have spring break.