11-limit rank 2 using only wedgies

This is the result of a search done for 11-limit rank two wedgies starting <<i j k l...|| in the range 0 <= i <= 35, |j| <= 51, |k| <= 62, |l| <= 76; the numbers were taken from the 22-et val. The method is the same as before, except to test for whether an 11-limit candidate is a bival involves five polynomials in the coefficients rather than one. Below I give the results, starting somewhat arbitrarily with the <<1 2 3 4...|| temperament, in order of complexity. The low complexity "junk" temperaments were moved to the back of the bus.

The number after the wedgie is wedgie logflat badness, for which I used a cutoff of 1/30. I used RMS normalization for the norm; that is, I divided the sum of squares of the weighted coefficients by the dimension of the multival being evaluated. This has various useful features:

(1) ||JIP|| = 1 always
(2) For bivals for the "same" temperament in different prime limits, the complexity roughly corresponds
(3) Badness figures from one prime limit to another roughly correspond
(4) Graham calls all this stuff "RMS" anyway

<<1 2 3 4 1 2 3 1 2 1] .02254936788
<<0 0 3 0 0 5 0 7 0 -10] .03266613161
<<1 -1 0 -2 -4 -3 -7 3 -1 -6] .03311691780
<<0 0 0 4 0 0 6 0 9 11] .02253963361
<<0 2 2 4 3 3 6 -1 2 4] .02755088377
<<2 1 3 2 -3 -1 -4 4 1 -5] .02248847709
<<1 -1 -2 -1 -4 -6 -5 -2 1 4] .02195658953
<<0 0 3 3 0 5 5 7 7 -2] .02559225555
<<1 -1 -2 1 -4 -6 -2 -2 6 10] .02492788192
<<2 3 1 3 0 -4 -2 -6 -3 5] .02220252249
<<2 3 1 2 0 -4 -4 -6 -6 2] .02418353086
<<1 -1 3 1 -4 2 -2 10 6 -8] .02382322654
<<2 1 -1 1 -3 -7 -5 -5 -1 6] .02907128342
<<2 1 3 5 -3 -1 1 4 8 4] .01985376693
<<1 -1 3 -1 -4 2 -5 10 1 -13] .03079469661
<<0 0 0 5 0 0 8 0 12 14] .02478522816
<<2 -1 1 3 -6 -4 -2 5 10 5] .03225888860
<<1 -1 -2 -4 -4 -6 -10 -2 -6 -4] .03283132281
<<1 -1 3 4 -4 2 3 10 13 1] .02058916969
<<2 1 -1 -2 -3 -7 -10 -5 -8 -2] .02498782323
<<1 -3 -2 -1 -7 -6 -5 4 8 4] .02682804122
<<1 4 3 6 4 2 6 -4 0 6] .02450604432
<<2 1 3 -2 -3 -1 -10 4 -8 -16] .02711421676
<<2 1 -1 5 -3 -7 1 -5 8 17] .02565999778
<<1 4 3 -1 4 2 -5 -4 -16 -13] .02516659533
<<2 3 1 7 0 -4 4 -6 6 16] .02279941727
<<2 3 1 -2 0 -4 -10 -6 -15 -9] .02264908627
<<1 4 5 6 4 5 6 0 0 0] .03252075307
<<2 1 6 5 -3 4 1 11 8 -7] .02236564256
<<2 -2 -2 0 -8 -9 -7 1 7 7] .02919265057
<<1 -3 -4 -1 -7 -9 -5 -1 8 11] .02275335295
<<1 -1 3 -4 -4 2 -10 10 -6 -22] .02815315008
<<3 5 1 4 1 -7 -4 -12 -8 8] .02678951162
<<1 4 -2 -1 4 -6 -5 -16 -16 4] .02614071303
<<0 5 0 0 8 0 0 -14 -17 0] .03193414415
<<4 4 4 8 -3 -5 -1 -2 5 9] .02716388253
<<2 1 -4 -2 -3 -12 -10 -12 -8 8] .03171868983
<<0 5 0 5 8 0 8 -14 -6 14] .03088327491
<<4 2 2 0 -6 -8 -14 -1 -7 -7] .03238507966
<<2 6 6 8 5 4 6 -3 -2 2] .03146793001
<<2 1 6 -2 -3 4 -10 11 -8 -26] .03223933563
<<3 0 6 6 -7 1 -1 14 14 -4] .02019130094
<<1 4 -2 6 4 -6 6 -16 0 24] .02197777803
<<0 0 7 0 0 11 0 16 0 -24] .02352356326
<<4 4 4 0 -3 -5 -14 -2 -14 -14] .02213232992
<<1 -3 5 -1 -7 5 -5 20 8 -20] .02721116574
<<2 1 -4 5 -3 -12 1 -12 8 28] .03067952576
<<3 5 9 4 1 6 -4 7 -8 -20] .02232531388
<<2 6 6 0 5 4 -7 -3 -21 -21] .03023466549
<<2 -4 -4 0 -11 -12 -7 2 14 14] .02379786448
<<5 3 7 4 -7 -3 -11 8 -1 -13] .02605034314
<<4 2 2 10 -6 -8 2 -1 16 21] .02671190571
<<0 5 0 -5 8 0 -8 -14 -29 -14] .02464119287
<<4 4 4 12 -3 -5 5 -2 14 20] .02657422761
<<1 4 -2 -6 4 -6 -13 -16 -28 -10] .02417969987
<<4 2 2 -4 -6 -8 -20 -1 -16 -18] .03145593439
<<1 4 10 6 4 13 6 12 0 -18] .02142298045
<<4 2 9 10 -6 3 2 15 16 -3] .03295662006
<<0 5 0 10 8 0 16 -14 6 28] .02920022543
<<2 -6 1 -2 -14 -4 -10 19 16 -9] .02853528796
<<4 -3 2 5 -14 -8 -6 13 22 7] .02619038361
<<3 0 9 9 -7 6 4 21 21 -6] .03117080207
<<2 8 8 12 8 7 12 -4 0 6] .02312376270
<<4 -3 2 -4 -14 -8 -20 13 1 -18] .03061680055
<<6 0 3 3 -14 -12 -16 7 7 -2] .02615175347
<<0 0 0 12 0 0 19 0 28 34] .03053585670
<<1 4 10 -1 4 13 -5 12 -16 -37] .03153931830
<<2 8 1 -2 8 -4 -10 -20 -32 -9] .02851047590
<<3 0 -6 -6 -7 -18 -20 -14 -14 4] .01961264729
<<2 8 1 12 8 -4 12 -20 0 30] .02894695095
<<6 5 3 13 -6 -12 0 -7 13 26] .02764846617
<<3 -5 -6 -1 -15 -18 -12 0 15 18] .03103574745
<<6 10 10 8 2 -1 -8 -5 -16 -12] .02309527228
<<6 5 3 -2 -6 -12 -24 -7 -22 -16] .02741039306
<<3 5 -6 4 1 -18 -4 -28 -8 32] .02156244376
<<2 -4 -4 -12 -11 -12 -26 2 -14 -20] .02034318368
<<6 10 3 8 2 -12 -8 -21 -16 12] .02602330724
<<2 -4 -4 10 -11 -12 9 2 37 42] .02834894667
<<5 1 12 14 -10 5 5 25 29 -2] .02710913682
<<7 9 13 4 -2 1 -18 5 -22 -34] .02868049607
<<1 9 -2 -6 12 -6 -13 -30 -45 -10] .03276775418
<<3 5 16 4 1 17 -4 23 -8 -44] .02726834859
<<1 4 10 18 4 13 25 12 28 16] .01702684705
<<2 -9 -4 5 -19 -12 1 16 43 28] .03286268418
<<7 -3 8 2 -21 -7 -21 27 15 -22] .01523093554
<<8 6 6 -4 -9 -13 -34 -3 -30 -32] .03205771555
<<1 9 -2 16 12 -6 22 -30 6 52] .02497560698
<<4 16 9 10 16 3 2 -24 -32 -3] .02163634826
<<1 -13 -2 -6 -23 -6 -13 32 31 -10] .03220318743
<<9 5 -3 7 -13 -30 -20 -21 -1 30] .01668692244
<<5 1 12 -8 -10 5 -30 25 -22 -64] .02035194624
<<1 4 10 -13 4 13 -24 12 -44 -71] .02154275985
<<2 8 -11 5 8 -23 1 -48 -16 52] .02606397402
<<1 -8 -14 -18 -15 -25 -32 -10 -14 -2] .02355567706
<<11 13 17 12 -5 -4 -19 3 -17 -25] .02562068068
<<2 8 20 5 8 26 1 24 -16 -55] .02551560762
<<3 12 -1 -8 12 -10 -23 -36 -60 -19] .02564183306
<<8 18 11 20 10 -5 4 -25 -16 18] .02407796431
<<10 9 7 25 -9 -17 5 -9 27 46] .01684171424
<<9 10 -3 2 -5 -30 -28 -35 -30 16] .02565855359
<<8 1 18 20 -17 6 4 39 43 -6] .02441311023
<<6 -7 -2 15 -25 -20 3 15 59 49] .01068445535
<<3 12 -1 23 12 -10 26 -36 12 68] .03133422109
<<7 9 13 31 -2 1 25 5 41 42] .02948606565
<<4 -8 14 -2 -22 11 -17 55 23 -54] .02648914689
<<6 -7 -2 -16 -25 -20 -46 15 -13 -38] .03294555529
<<5 -11 -12 -3 -29 -33 -22 3 31 33] .02365870234
<<4 9 26 10 5 30 2 35 -8 -62] .03133237410
<<2 -4 -16 -24 -11 -31 -45 -26 -42 -12] .02503394875
<<11 -6 10 7 -35 -15 -27 40 37 -15] .03130868343
<<5 1 12 33 -10 5 35 25 73 51] .03070587478
<<12 5 -9 1 -20 -48 -40 -35 -15 34] .03307815855
<<16 2 5 9 -34 -37 -41 6 14 8] .02926993072
<<3 17 -1 -13 20 -10 -31 -50 -89 -33] .02309315537
<<6 -12 10 -14 -33 -1 -43 57 9 -74] .02598700821
<<17 6 15 27 -30 -24 -16 18 42 24] .02247588148
<<7 -3 8 33 -21 -7 28 27 87 65] .03143814834
<<2 25 13 5 35 15 1 -40 -75 -31] .02349765869
<<1 -8 -14 23 -15 -25 33 -10 81 113] .02739627000
<<12 -2 20 -6 -31 -2 -51 52 -7 -86] .01853857519
<<13 -10 6 17 -46 -27 -18 42 74 27] .02534215223
<<0 12 24 36 19 38 57 22 42 18] .02223519823
<<15 -2 -5 22 -38 -50 -17 -6 58 79] .01900326112
<<6 5 22 -21 -6 18 -54 37 -66 -135] .02184888494
<<12 34 20 30 26 -2 6 -49 -48 15] .01586719974
<<7 26 25 -3 25 20 -29 -15 -97 -95] .02984386015
<<12 22 -4 -6 7 -40 -51 -71 -90 -3] .01547939572
<<18 27 18 45 1 -22 9 -34 11 64] .02034656577
<<18 15 -6 9 -18 -60 -48 -56 -31 46] .01933342476
<<16 2 5 40 -34 -37 8 6 86 95] .02106940074
<<6 -19 -26 -21 -44 -58 -54 -7 17 31] .02896455529
<<12 10 44 30 -12 36 6 74 35 -68] .02931897034
<<6 5 22 51 -6 18 60 37 101 67] .03042153376
<<18 -9 18 9 -56 -22 -48 67 52 -37] .03042588313
<<6 17 46 15 13 56 3 59 -24 -117] .03168530657
<<24 32 40 24 -5 -4 -45 3 -55 -71] .01409731507
<<22 -5 3 24 -59 -57 -38 21 73 57] .03017142067
<<3 -24 -1 28 -45 -10 34 65 148 82] .02664831932
<<6 41 22 15 51 18 3 -64 -107 -34] .02795861183
<<23 -1 13 42 -55 -44 -13 33 101 73] .01984511706
<<6 29 -2 -21 32 -20 -54 -86 -149 -52] .02846714993
<<30 25 38 39 -30 -24 -42 18 4 -22] .02551496700
<<10 38 36 54 37 29 51 -23 -6 27] .03046127011
<<4 -32 -15 10 -60 -35 2 55 134 80] .02930650389
<<24 20 16 -12 -24 -42 -102 -19 -97 -89] .02340595233
<<18 39 42 9 20 16 -48 -12 -114 -120] .02212584813
<<16 -10 34 -8 -53 9 -68 107 16 -140] .03207955110
<<30 13 14 3 -49 -62 -99 -4 -38 -40] .03222499068
<<1 33 27 -18 50 40 -32 -30 -156 -144] .02621922176
<<21 -9 -7 37 -63 -70 -14 9 117 128] .02980739136
<<6 46 10 44 59 -1 49 -106 -57 89] .02824010103
<<18 27 18 -27 1 -22 -105 -34 -156 -138] .03112267679
<<2 -16 -40 -60 -30 -69 -102 -48 -84 -30] .02815987312
<<18 -14 30 -20 -64 -3 -94 109 2 -160] .02461886951
<<5 13 -17 62 9 -41 81 -76 99 233] .03150609199
<<30 49 14 39 8 -62 -42 -105 -79 61] .03087452342
<<10 26 -34 -28 18 -82 -79 -152 -155 39] .03130726348
<<29 -8 11 57 -80 -64 -10 48 160 122] .02762121371
<<9 -7 -61 -10 -32 -122 -47 -122 1 183] .02963779494
<<5 -40 -29 33 -75 -60 35 45 215 193] .03154872160
<<6 -48 10 -50 -90 -1 -100 158 50 -175] .03185714120
<<22 48 -38 -34 25 -122 -130 -223 -245 36] .01285972851

Low accuracy "temperaments"

<<0 0 0 1 0 0 2 0 2 3] .01764625288
<<0 0 1 0 0 2 0 2 0 -3] .02727378055
<<0 0 1 1 0 2 2 2 2 -1] .03098796258
<<0 1 0 1 2 0 2 -3 -1 3] .02713003262
<<0 1 1 1 2 2 2 0 -1 -1] .03231118192
<<1 1 1 1 -1 -1 -2 0 -1 -1] .02143249952
<<1 0 1 1 -2 -1 -2 2 2 -1] .02477374166
<<1 0 1 0 -2 -1 -3 2 0 -3] .02422282590
<<0 0 0 2 0 0 3 0 5 6] .02025028723
<<1 0 0 1 -2 -3 -2 0 2 3] .03277515697
<<1 1 0 1 -1 -3 -2 -3 -1 3] .02060835682
<<1 1 2 1 -1 0 -2 2 -1 -4] .02341451468
<<1 2 1 2 1 -1 0 -3 -2 2] .02094253408
<<0 0 2 0 0 3 0 5 0 -7] .02470241631
<<1 2 1 1 1 -1 -2 -3 -5 -1] .02491468151
<<1 1 2 3 -1 0 1 2 4 2] .03115757418
<<0 0 2 2 0 3 3 5 5 -1] .03181070804
<<0 2 0 0 3 0 0 -6 -7 0] .03171538105
<<0 0 0 3 0 0 5 0 7 8] .02649562767
<<0 2 2 2 3 3 3 -1 -2 -1] .01899285806
<<1 2 0 1 1 -3 -2 -6 -5 3] .02320684630
<<1 2 0 2 1 -3 0 -6 -2 6] .03075984113
<<1 -1 0 1 -4 -3 -2 3 6 3] .02004874438
<<1 -1 0 -1 -4 -3 -5 3 1 -3] .02695193034
<<0 2 2 0 3 3 0 -1 -7 -7] .02180987118
<<1 2 3 1 1 2 -2 1 -5 -8] .02616965635
<<1 2 1 -1 1 -1 -5 -3 -9 -6] .03324993964

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> The number after the wedgie is wedgie logflat badness, for which I used a cutoff of 1/30.

If we use 1/50 instead, we get a far more exclusive club:

Dicot
> <<2 1 3 5 -3 -1 1 4 8 4] .01985376693

Meantone
> <<1 4 10 18 4 13 25 12 28 16] .01702684705

Orwell
> <<7 -3 8 2 -21 -7 -21 27 15 -22] .01523093554

Valentine
> <<9 5 -3 7 -13 -30 -20 -21 -1 30] .01668692244

Myna
> <<10 9 7 25 -9 -17 5 -9 27 46] .01684171424

Miracle
> <<6 -7 -2 15 -25 -20 3 15 59 49] .01068445535

Wizard
> <<12 -2 20 -6 -31 -2 -51 52 -7 -86] .01853857519

Luna
> <<15 -2 -5 22 -38 -50 -17 -6 58 79] .01900326112

Harry
> <<12 34 20 30 26 -2 6 -49 -48 15] .01586719974

Unidec
> <<12 22 -4 -6 7 -40 -51 -71 -90 -3] .01547939572

Minorsemi
> <<18 15 -6 9 -18 -60 -48 -56 -31 46] .01933342476

Name?
> <<24 32 40 24 -5 -4 -45 3 -55 -71] .01409731507

Name?
> <<23 -1 13 42 -55 -44 -13 33 101 73] .01984511706

Name?
> <<22 48 -38 -34 25 -122 -130 -223 -245 36] .01285972851

genewardsmith wrote:
> > Name?
>> <<24 32 40 24 -5 -4 -45 3 -55 -71] .01409731507

octoid

> Name?
>> <<23 -1 13 42 -55 -44 -13 33 101 73] .01984511706

grendel, voodoo

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> genewardsmith wrote:
> >
> > Name?
> >> <<24 32 40 24 -5 -4 -45 3 -55 -71] .01409731507
>
> octoid
>
>
> > Name?
> >> <<23 -1 13 42 -55 -44 -13 33 101 73] .01984511706
>
> grendel, voodoo
>

They both looked familiar; why didn't I find them on your Xenwiki page, I wonder? They both must be there.

That leaves:

Name?
> <<22 48 -38 -34 25 -122 -130 -223 -245 36] .01285972851

Quite a strong system according to the badness figure, but of course a microtempering. Period half an octave, generator something dubious
such as 640/567 or 704/441, the latter being shy of 8/5 by 441/440. 46, 224, 270 and 494 support it, and 764 or 1798 can be used for a tuning.

I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@> wrote:
> >
> > genewardsmith wrote:
> > > ...
> That leaves:
>
> Name?
> > <<22 48 -38 -34 25 -122 -130 -223 -245 36] .01285972851
>
> Quite a strong system according to the badness figure, but of course a microtempering. Period half an octave, generator something dubious
> such as 640/567 or 704/441, the latter being shy of 8/5 by 441/440.

I looked for the common generator for 224, 270, and 494 around 5 years ago and found that it's approximated by 44/39 with a period of 1/2 octave.

> 46, 224, 270 and 494 support it, and 764 or 1798 can be used for a tuning.
>
> I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things.

Unless you can come up with something based on 44/39.

--George

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> They both looked familiar; why didn't I find them on your Xenwiki page, I wonder?

Because the sign on the mappings can vary. I usually search wedgies, where the first nonzero coefficient is positive, so it's unique.

--- In tuning-math@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:

> I looked for the common generator for 224, 270, and 494 around 5 years ago and found that it's approximated by 44/39 with a period of 1/2 octave.

And it's a very strong system in the 13-limit as well.
(640/457)/(44/39) = 2080/2079, I see. I have a vague memory this could have been discussed before.

> > I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things.
>
> Unless you can come up with something based on 44/39.

Scala doesn't list it, and Kyle Gann lists it but doesn't name it.