the 75 "best" 7-limit ETs below 100,000-tET

Out of the consistent ones:

rank ET "badness"
1 171 0.20233
2 18355 0.25034
3 31 0.30449
4 84814 0.33406
5 4 0.33625
6 270 0.34265
7 99 0.35282
8 3125 0.35381
9 441 0.37767
10 6691 0.42354
11 72 0.44575
12 3566 0.45779
13 10 0.46883
14 5 0.47022
15 11664 0.48721
16 41 0.48793
17 12 0.49554
18 342 0.50984
19 21480 0.51987
20 68 0.52538
21 3395 0.53654
22 19 0.53668
23 15 0.53966
24 27 0.54717
25 140 0.5502
26 9 0.55191
27 6 0.55842
28 22 0.56091

Up through this point in the list, most of the results tend to
be "small" ETs . . . hereafter, they don't.

29 1578 0.56096
30 6520 0.56344
31 14789 0.56416
32 39835 0.5856
33 612 0.58643
34 33144 0.59123
35 202 0.59316
36 130 0.59628
37 1547 0.59746
38 5144 0.5982
39 11835 0.62134
40 63334 0.63002
41 36710 0.63082
42 2954 0.63236
43 53 0.63451
44 2019 0.63526
45 3296 0.6464
46 44979 0.65123
47 8269 0.65424
48 51499 0.67526
49 301 0.68163
50 1376 0.68197
51 51670 0.68505
52 1848 0.68774
53 66459 0.68876
54 14960 0.68915
55 103 0.69097
56 16 0.69137
57 33315 0.69773
58 1749 0.70093
59 1407 0.70125
60 46 0.71008
61 37 0.71553
62 26624 0.732
63 4973 0.73284
64 1106 0.73293
65 239 0.73602
66 472 0.75857
67 30019 0.76
68 26 0.76273
69 9816 0.76717
70 62 0.76726
71 58 0.77853
72 1718 0.77947
73 15230 0.7845
74 25046 0.78996
75 58190 0.79264

What seems to be happening is that whatever effect is creating 60
equally-spaced "waves" in the data starts to dominate the result
after about the top 30 or so; or after the cutoff 'badness' value
exceeds approximately e times its "global" minimum value, the
logarithmic character of the results begins to loosen it grip . . .
is there anything to this observation, Gene?

Assuming a "critical exponent" of 3/2 for this case (is that right?)
:
Out of the consistent ones:

rank ET "badness"
1 4296
0.20153554902775
2 78005
0.253840852090173
3 118
0.298051576414275
4 3
0.325158891374691
5 53
0.361042754847595
6 1783
0.376704792560154
7 2513
0.38157807050998
8 25164
0.410594002644579
9 19
0.410991123902702
10 12
0.417509911542676
11 612
0.436708226862349
12 730
0.440328484445999
13 34
0.458833616575689
14 171
0.461323498406156
15 20868
0.462440101460723
16 7
0.479263869467813
17 4
0.517680428544775
18 441
0.525786933473794
19 1171
0.54066707734392
20 8592
0.570028613470703
21 65
0.580261609859836
22 52841
0.584468600555837
23 73709
0.592105848504379
24 6809
0.613067695688349
25 15
0.644650341848039
26 5
0.654939089766412
27 31
0.659243117645396
28 289
0.666113665527379
29 22
0.713295533690924
30 1342
0.734143972117584
31 16572
0.736198397866562
32 323
0.744599492497238
33 559
0.763541910323762
34 152
0.785452598431966
35 9
0.804050483021927
36 29460
0.806162085936717
37 98873
0.808456458619207
38 1053
0.816063953343609
39 10
0.831348880236421
40 27677
0.840139252565266
41 236
0.843017163303497
42 6079
0.854618300478436
43 87
0.855517482964681
44 1901
0.875919286932322
45 8
0.885030392786763
46 3684
0.886931822414785
47 48545
0.889578653724097
48 11105
0.89024911748373
49 84
0.908733078006219
50 46
0.91773712251282
51 6
0.919688228216577
52 99
0.929380204600093
53 205
0.94016876679068
54 103169
0.941197892471069
55 57137
0.942202383987665
56 12276
0.953572058507306
57 31973
0.95740445574325
58 494
0.959416685644995
59 270
0.962515005704479
60 23381
0.982018213968414
61 5467
0.999256787729657
62 2954
1.01217901495476
63 130846
1.01554445408754
64 16
1.02089438881021
65 106
1.02118312100403
66 3125
1.02398156287357
67 7980
1.03541593154556
68 12888
1.04720943128646
69 46032
1.06067411398872
70 3566
1.06548205329903
71 5026
1.07926576483874
72 41
1.08648217274233
73 72
1.09019587056164
74 2395
1.12961831514844
75 82301
1.14050108729716

Hey guys. This is _tuning_ math remember. It's a serious stretch of my
imagination to think that ETs above 2000 have anything to do with
tuning, let alone 100,000!

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Up through this point in the list, most of the results tend to
> be "small" ETs . . . hereafter, they don't.

Doesn't that suggest your badness measure isn't flat?

How about generating the corresponding charts for steps*cents for
comparison with those you did for steps^(4/3)*cents?

1/badness seems to show it best and it still looks to me like
steps^(4/3)*cents has gradually falling goodness.

And when you give these "best of" lists, please quote your goodness or
badness function.

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

> Hey guys. This is _tuning_ math remember. It's a serious stretch of
my
> imagination to think that ETs above 2000 have anything to do with
> tuning, let alone 100,000!

These high numbers have their uses, and Paul seems to have discovered
something of considerable number-theoretic interest, so give us a
break, please.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>> Hey guys. This is _tuning_ math remember. It's a serious stretch
>> of my imagination to think that ETs above 2000 have anything to
>> do with tuning, let alone 100,000!
>
> These high numbers have their uses, and Paul seems to have
> discovered something of considerable number-theoretic interest,
> so give us a break, please.

It's a fine thread, but it's also the first one on this list that I
can remember for which I can see no musical application, and it's
probably good somebody pointed this out. If it leads to more
powerful methods for finding good temperaments, that could be used
in something like Graham's temperament finder, then I'll take this
back.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Assuming a "critical exponent" of 3/2 for this case (is that right?)

Correct, but could you give the exact definition of badness you are using? I also could not find the FFT data you said you posted.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Assuming a "critical exponent" of 3/2 for this case (is that right?)
>
> Correct, but could you give the exact definition of badness you are
using?

badness = steps^(3/2)*sqrt((err(3/2)^2 + err(5/3)^2 + err(5/4)^2)/3)

I tried both ignoring consistency and setting badness to infinity for
inconsistent ETs; either way, the pattern (power spectrum of
1/badness) looked about the same.

>I also could not find the FFT data you said you posted.

I simply mentioned that the result looked about the same as the
7-limit case but with the peak at 612 . . . if you like I can post the
graph when I get back to the office . . .

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

> badness = steps^(3/2)*sqrt((err(3/2)^2 + err(5/3)^2 + err(5/4)^2)/3)
>
> I tried both ignoring consistency and setting badness to infinity for
> inconsistent ETs; either way, the pattern (power spectrum of
> 1/badness) looked about the same.

Consistency matters only to the bad ets, so it isn't going to make a difference.