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.