Magrathean diacritics

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

[Originally posted Tue May 26, 2020 9:59 am]

@Ash9903b4 I repeat your table here for my convenience. I deleted some spaces between columns so it doesn't require horizontal scrolling when on a wide enough screen.

Code: Select all

0 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  0.00  1n           [  0   0   0   0   0   0   0   0 ⟩               1               0.000000    1    0
  0.05  11:228095n   [  8   4  -1  -4   1   0   0  -1 ⟩         228096/228095         0.007590   19   63
  0.07  323:6125n    [  9  -3  -3  -2   0   0   1   1 ⟩         165376/165375         0.010469   19   65
  0.10  169:1925n    [ -6   6  -2  -1  -1   2   0   0 ⟩         123201/123200         0.014052   13   54
  0.12  104975n      [  4   8  -2   0   0  -1  -1  -1 ⟩         104976/104975         0.016492   19   59
  0.14  30685n       [ 26  -7  -1   0   0   0  -1  -2 ⟩       67108864/67108095       0.019838   19   60
  0.19  1990625n     [ 13   5  -5  -2   0  -1   0   0 ⟩        1990656/1990625        0.026960   13   52
  0.22  581875n      [ 35 -10  -4  -2   0   0   0  -1 ⟩    34359738368/34359136875    0.030307   19   53

0.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  0.27  22370117n    [-26   1   0   5   3   0   0   0 ⟩       67110351/67108864       0.038360   11   68
  0.31  17:2717n     [-20   8   0   0   1   1  -1   1 ⟩       17826237/17825792       0.043218   19   60
  0.33  17:9295n     [  2  -7   1   0   1   2  -1   0 ⟩          37180/37179          0.046564   17   59
  0.35  55859375n    [ 40  -9  -8   0  -1  -1   0   0 ⟩  1099511627776/1099480078125  0.049677   13   64
  0.42  121:38675n   [ 21  -8  -2  -1   2  -1  -1   0 ⟩      253755392/253746675      0.059472   17   69
  0.52  5:1574573n   [  4   9   1  -1  -3  -2   0   0 ⟩        1574640/1574573        0.073665   13   71
  0.53  289:1463n    [ -4   4   0  -1  -1   0   2  -1 ⟩          23409/23408          0.073957   19   71
  0.63  5:1771561n   [ -1 -11  -1   0   6   0   0   0 ⟩        1771561/1771470        0.088931   11   71
  0.69  3125:31213n  [ 16  -8   5  -4   0  -1   0   0 ⟩      204800000/204788493      0.097275   13   66
  0.73  13:4675n     [-18   6   2   0   1  -1   1   0 ⟩        3408075/3407872        0.103123   17   51

1 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  0.76  19:23375n    [  4  -9   3   0   1   0   1  -1 ⟩         374000/373977         0.106470   19   62
  0.80  169:12005n   [-22  10   1   4   0  -2   0   0 ⟩      708883245/708837376      0.112025   13   59
  0.82  247:60025n   [  0  -5   2   4   0  -1   0  -1 ⟩          60025/60021          0.115371   19   70
  0.83  49:3179n     [ 13 -12   0  -2   1   0   2   0 ⟩       26042368/26040609       0.116938   17   59
  0.83  7:221n       [-24  12   0  -1   0   1   1   0 ⟩      117448461/117440512      0.117175   17   37
  0.86  25:14399n    [  6   2   2  -1  -2   0  -1   0 ⟩          14400/14399          0.120229   17   56
  0.87  55:4693n     [  8  -1   1   0   1  -1   0  -2 ⟩          14080/14079          0.122961   19   67
  0.88  19882681n    [ 29  -3   0  -6   0  -2   0   0 ⟩      536870912/536832387      0.124235   13   68
  0.90  65:833n      [  9  -8  -1   2   0  -1   1   0 ⟩         426496/426465         0.125840   17   49
  0.92  209:4459n    [ -6   1   0   3  -1   1   0  -1 ⟩          13377/13376          0.129424   19   64
  0.93  19:207515n   [ 15  -1  -1  -3  -2   0   0   1 ⟩         622592/622545         0.130697   19   67
  0.97  13:12635n    [  2   5  -1  -1   0   1   0  -2 ⟩          12636/12635          0.137013   19   63
  0.98  226474325n   [ 23   3  -2  -7  -1   0   0   0 ⟩      226492416/226474325      0.138287   11   70
  1.00  1375:1547n   [  3  -2  -3   1  -1   1   1   0 ⟩          12376/12375          0.139892   17   63
  1.02  7:113135n    [ 33 -12  -1   1  -3   0  -1   0 ⟩    60129542144/60124577535    0.142946   17   62
  1.07  196625n      [-16  -1   3   0   2   1   0   0 ⟩         196625/196608         0.149687   13   50
  1.09  289:3125n    [ 14 -11   5   0   0   0  -2   0 ⟩       51200000/51195483       0.152741   17   59
  1.11  7:653125n    [  7   6  -5   1  -1   0   0  -1 ⟩         653184/653125         0.156384   19   62
  1.13  17:660275n   [-20   3   2   4   1   0  -1   0 ⟩       17827425/17825792       0.158589   17   66
  1.16  1183:1331n   [  3  -2   0  -1   3  -2   0   0 ⟩          10648/10647          0.162595   13   66
  1.17  7:120835n    [-22   5   1  -1   1   3   0   0 ⟩       29362905/29360128       0.163739   13   62
  1.18  323:605n     [-12   7   1   0   2   0  -1  -1 ⟩        1323135/1323008        0.166179   19   63
  1.20  5:10241n     [-11   0  -1   2   1   0   0   1 ⟩          10241/10240          0.169058   19   49
  1.23  17:57967n    [-26   9   0   3   0   2  -1   0 ⟩     1140964461/1140850688     0.172641   17   64

1.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  1.26  121:1225n    [ -3   4  -2  -2   2   0   0   0 ⟩           9801/9800           0.176648   11   46
  1.28  184093n      [ 27  -6   0  -2   0  -1  -2   0 ⟩      134217728/134203797      0.179701   17   61
  1.29  85:117649n   [  7 -11  -1   6   0   0  -1   0 ⟩       15059072/15057495       0.181306   17   64
  1.30  125:22477n   [-17   6  -3   1   0   2   0   1 ⟩       16385733/16384000       0.183110   19   67
  1.32  539:27625n   [ -7   8  -3   2   1  -1  -1   0 ⟩        3536379/3536000        0.185549   17   70
  1.33  25:15379n    [  5  -9  -2   1   0   3   0   0 ⟩         492128/492075         0.186456   13   56
  1.34  539:8075n    [ 15  -7  -2   2   1   0  -1  -1 ⟩       17661952/17660025       0.188896   19   71
  1.40  11:13671875n [-25   3   9   1  -1   0   0   0 ⟩      369140625/369098752      0.19639    1    63
  1.42  11:3125n     [ 24 -10  -5   0   1   0   0   0 ⟩      184549376/184528125      0.199364   11   36
  1.49  2197:15625n  [ -6   2   6   0   0  -3   0   0 ⟩         140625/140608         0.209300   13   69
  1.52  169:546875n  [ 18  -4  -7  -1   0   2   0   0 ⟩       44302336/44296875       0.213417   13   68
  1.59  625:1001n    [-12   8   4  -1  -1  -1   0   0 ⟩        4100625/4100096        0.223352   13   51
  1.61  1463:3125n   [ 10  -7   5  -1  -1   0   0  -1 ⟩        3200000/3199581        0.226698   19   62
  1.65  19:145775n   [-24   7   2   3   0   0   1  -1 ⟩      318809925/318767104      0.232547   19   67
  1.68  5:1399489n   [  7   7   1  -2   0  -4   0   0 ⟩        1399680/1399489        0.236260   13   71
  1.69  425:3773n    [ 19 -10   2  -3  -1   0   1   0 ⟩      222822400/222791877      0.237167   17   59
  1.73  5:2023n      [-15   4  -1   1   0   0   2   0 ⟩         163863/163840         0.243015   17   46

2 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  1.75  35:1573n     [ 15  -6   1   1  -2  -1   0   0 ⟩        1146880/1146717        0.246069   13   47
  1.80  931931n      [ 23  -2   0  -3  -1  -1   0  -1 ⟩        8388608/8387379        0.253659   19   64
  1.85  119:30875n   [-11  12  -3   1   0  -1   1  -1 ⟩       63241479/63232000       0.259507   19   71
  1.85  13:6655n     [  9   0  -1   0  -3   1   0   0 ⟩           6656/6655           0.260121   13   51
  1.92  119:40625n   [-10   1   5  -1   0   1  -1   0 ⟩         121875/121856         0.269916   17   62
  1.96  2519825n     [ -7  -9   2   2   2   0   1   0 ⟩        2519825/2519424        0.275527   17   63
  2.00  343:6175n    [ -1  -2   2  -3   0   1   0   1 ⟩           6175/6174           0.280384   19   63
  2.02  13:6470695n  [-11  -5   1   6   1  -1   0   0 ⟩        6470695/6469632        0.284429   13   71
  2.03  169:361n     [ 10  -7   0   0   0  -2   0   2 ⟩         369664/369603         0.285703   19   64
  2.06  187:665n     [ -5   2   1   1  -1   0  -1   1 ⟩           5985/5984           0.289286   19   59
  2.06  221221n      [-13  -3   0   1   1   2   1   0 ⟩         221221/221184         0.289579   17   61
  2.08  247:5929n    [ -3  -1   0   2   2  -1   0  -1 ⟩           5929/5928           0.292019   19   68
  2.11  5831n        [  3   6   0  -3   0   0  -1   0 ⟩           5832/5831           0.296876   17   38
  2.12  5:218491n    [-17   1  -1   5   0   1   0   0 ⟩         655473/655360         0.298481   13   53
  2.13  361:1925n    [  4  -1  -2  -1  -1   0   0   2 ⟩           5776/5775           0.299755   19   66
  2.15  1625:2057n   [  6  -4  -3   0   2  -1   1   0 ⟩         131648/131625         0.302487   17   67
  2.18  7:206635n    [ -1  10  -1   1  -1  -1  -2   0 ⟩         413343/413270         0.305778   17   70
  2.18  475:1001n    [ -9   5  -2   1   1   1   0  -1 ⟩         243243/243200         0.306071   19   60
  2.19  19:2100875n  [ 12   3  -3  -5   0   0   0   1 ⟩        2101248/2100875        0.307345   19   69
  2.20  7:60401n     [ 21  -5   0   1  -1   0  -2  -1 ⟩       14680064/14677443       0.309125   19   71
  2.21  13:420175n   [ 34 -12  -2  -5   0   1   0   0 ⟩   223338299392/223298222175   0.310691   13   58

2.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  2.27  163625n      [ 30  -8  -3  -1  -1   0  -1   0 ⟩     1073741824/1073543625     0.319593   17   50
  2.32  343:15625n   [ -4   6  -6   3   0   0   0   0 ⟩         250047/250000         0.325441    7   51
  2.39  17:65219n    [-23   7   0   2   3   0  -1   0 ⟩      142633953/142606336      0.335237   17   64
  2.46  125:25289n   [-14   4  -3   0   3   0   0   1 ⟩        2048409/2048000        0.345705   19   67
  2.48  25:17303n    [  8 -11  -2   0   3   1   0   0 ⟩        4429568/4428675        0.349052   13   56
  2.51  91:4913n     [  1   3   0   1   0   1  -3   0 ⟩           4914/4913           0.352342   17   71
  2.58  85:121n      [ -9   6   1   0  -2   0   1   0 ⟩          61965/61952          0.363244   17   44
  2.65  7:859375n    [-28   7   7  -1   1   0   0   0 ⟩     1879453125/1879048192     0.373039   11   53
  2.73  43681n       [ 17  -1   0   0  -2   0   0  -2 ⟩         131072/131043         0.383082   19   60

3 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  2.79  13:16625n    [-23   8   3   1   0  -1   0   1 ⟩      109076625/109051904      0.392409   19   54
  2.82  4375n        [ -1  -7   4   1   0   0   0   0 ⟩           4375/4374           0.395756    7   27
  2.85  275:4459n    [-15  12   2  -3   1  -1   0   0 ⟩      146146275/146112512      0.399999   13   55
  2.86  15714545n    [-35   7   1   5   1   0   1   0 ⟩    34367709915/34359738368    0.401604   17   68
  2.87  1375:6517n   [  7  -3   3  -3   1   0   0  -1 ⟩         176000/175959         0.403346   19   66
  2.89  7:1615n      [  8 -10   1  -1   0   0   1   1 ⟩         413440/413343         0.406224   19   48
  2.91  55:2197n     [ 18  -8   1   0   1  -3   0   0 ⟩       14417920/14414517       0.408664   13   55
  2.91  19:14399n    [-27  11   0   1   2   0   1  -1 ⟩     2550739653/2550136832     0.409194   19   65
  2.92  11:4225n     [ -7  -1   2   0  -1   2   0   0 ⟩           4225/4224           0.409808   13   47
  2.93  175:4199n    [  3   1   2   1   0  -1  -1  -1 ⟩           4200/4199           0.412248   19   66
  2.94  187:16807n   [ 16  -6   0  -5   1   0   1   0 ⟩       12255232/12252303       0.413814   17   63
  2.95  143:6517n    [  4  -6   0   3  -1  -1   0   1 ⟩         104272/104247         0.415126   19   64
  3.01  455n         [ 12  -2  -1  -1   0  -1   0   0 ⟩           4096/4095           0.422716   13   25
  3.03  13:3553n     [ -3   7   0   0  -1   1  -1  -1 ⟩          28431/28424          0.426300   19   60
  3.11  13:67375n    [  6   4  -3  -2  -1   1   0   0 ⟩          67392/67375          0.436768   13   53
  3.22  248897n      [-10  -5   0   0   4   0   1   0 ⟩         248897/248832         0.452175   17   61
  3.24  289:1925n    [-17   9   2   1   1   0  -2   0 ⟩       37889775/37879808       0.455465   17   62

3.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  3.27  13:1890625n  [ 33 -10  -6   0  -2   1   0   0 ⟩   111669149696/111639515625   0.459485   13   65
  3.32  209:595n     [ -8   6  -1  -1   1   0  -1   1 ⟩         152361/152320         0.465934   19   59
  3.34  119:143n     [ 14  -9   0  -1   1   1  -1   0 ⟩        2342912/2342277        0.469280   17   48
  3.38  125:539539n  [-20   5  -3   3   2   1   0   0 ⟩      131107977/131072000      0.475129   13   71
  3.40  343:1445n    [ 10  -5  -1   3   0   0  -2   0 ⟩         351232/351135         0.478182   17   60
  3.44  125:102487n  [ -3   8   3  -1  -4   0   0   0 ⟩         820125/819896         0.483473   11   66
  3.48  133:10625n   [ 19  -8  -4   1   0   0  -1   1 ⟩       69730304/69710625       0.488651   19   63
  3.57  847:390625n  [ -7  10  -8   1   2   0   0   0 ⟩       50014503/50000000       0.502089   11   69
  3.60  7:161051n    [ 24  -6   0   1  -5   0   0   0 ⟩      117440512/117406179      0.506189   11   62
  3.65  138125n      [-25   5   4   0   0   1   1   0 ⟩       33564375/33554432       0.512931   17   50
  3.67  2057:15625n  [  5  -5   6   0  -2   0  -1   0 ⟩         500000/499851         0.515985   17   69
  3.71  11:300125n   [-29   9   3   4  -1   0   0   0 ⟩     5907360375/5905580032     0.521833   11   54
  3.74  935:1183n    [ -6   4   1  -1   1  -2   1   0 ⟩          75735/75712          0.525839   17   66
  3.75  2375:5929n   [ 14  -8   3  -2  -2   0   0   1 ⟩       38912000/38900169       0.526453   19   70

4 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  3.79  11:15827n    [-20   6   0   2  -1   0   1   1 ⟩       11537883/11534336       0.532301   19   61
  3.79  361:1625n    [  1  -2   3   0   0   1   0  -2 ⟩           3250/3249           0.532769   19   66
  3.80  25:1294139n  [ 22  -4   2  -6  -1   0   0   0 ⟩      104857600/104825259      0.534043   11   63
  3.81  343:2197n    [-10   8   0   3   0  -3   0   0 ⟩        2250423/2249728        0.534741   13   60
  3.81  11:54145n    [  2  -9   1   2  -1   1   1   0 ⟩         216580/216513         0.535648   17   60
  3.83  1715:3211n   [ 12  -7   1   3   0  -2   0  -1 ⟩        7024640/7022457        0.538088   19   71
  3.84  17:245n      [-12  10  -1  -2   0   0   1   0 ⟩        1003833/1003520        0.539891   17   36
  3.86  5:1310309n   [ 18   0   1  -2  -2  -1  -1   0 ⟩        1310720/1310309        0.542945   17   71
  3.87  221:931n     [ 10  -5   0  -2   0   1   1  -1 ⟩         226304/226233         0.543238   19   63
  3.87  121:117649n  [ -2  -5   0   6  -2   0   0   0 ⟩         117649/117612         0.544550   11   64
  3.90  119:4225n    [ 21 -10  -2   1   0  -2   1   0 ⟩      249561088/249482025      0.548556   17   60
  3.93  49:1045n     [  6  -1  -1   2  -1   0   0  -1 ⟩           3136/3135           0.552140   19   49
  3.98  442225n      [ 14   3  -2  -2   0   0   0  -2 ⟩         442368/442225         0.559730   19   62
  4.00  17:6875n     [ 15  -4  -4   0  -1   0   1   0 ⟩         557056/556875         0.562608   17   48
  4.02  8960875n     [-12  -7   3   3   1   0   0   1 ⟩        8960875/8957952        0.564813   19   66
  4.05  91:11495n    [-26  12   1  -1   2  -1   0   1 ⟩     6108914295/6106906624     0.569057   19   66
  4.07  7:3025n      [ -4  -3   2  -1   2   0   0   0 ⟩           3025/3024           0.572403   11   39
  4.12  786695n      [-18  -1   1   2   0   2   0   1 ⟩         786695/786432         0.578866   19   64
  4.14  221:18865n   [ -8   1   1   3   1  -1  -1   0 ⟩          56595/56576          0.581305   17   67
  4.17  49:1859n     [-10   3   0  -2   1   2   0   0 ⟩          50193/50176          0.586456   13   51
  4.21  325:1463n    [  1  -2  -2   1   1  -1   0   1 ⟩           2926/2925           0.591774   19   60
  4.24  85:637n      [-14   7  -1   2   0   1  -1   0 ⟩        1393119/1392640        0.595357   17   49

4.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  4.26  121:557375n  [  9   2  -3  -3   2  -1   0   0 ⟩         557568/557375         0.599364   13   71
  4.31  247:625n     [ -5   4  -4   0   0   1   0   1 ⟩          20007/20000          0.605826   19   52
  4.33  125:169n     [ 17 -11  -3   0   0   2   0   0 ⟩       22151168/22143375       0.609172   13   41
  4.41  143:390625n  [-13   1   8   0  -1  -1   0   0 ⟩        1171875/1171456        0.619108   13   64
  4.43  11:1421875n  [ 36 -12  -6  -1   1  -1   0   0 ⟩   755914244096/755642671875   0.622081   13   61
  4.43  265625n      [ -1  12  -6   0   0   0  -1   0 ⟩         531441/531250         0.622318   17   47
  4.53  13:19140625n [ 30  -6  -8  -2   0   1   0   0 ⟩    13958643712/13953515625    0.63613    1    67
  4.55  1770125n     [-16  -3   3   2   0   0   2   0 ⟩        1770125/1769472        0.638771   17   63
  4.60  125:91091n   [  0   6   3  -2  -1  -2   0   0 ⟩          91125/91091          0.646068   13   66
  4.62  93347n       [ -7  -6   0   0   0   0   3   1 ⟩          93347/93312          0.649239   19   70
  4.64  1331:3325n   [-14   8   2   1  -3   0   0   1 ⟩       21815325/21807104       0.652530   19   69
  4.66  247:4165n    [-12   5   1   2   0  -1   1  -1 ⟩        1012095/1011712        0.655263   19   68
  4.67  1331:11375n  [  8  -7   3   1  -3   1   0   0 ⟩        2912000/2910897        0.655877   13   68
  4.70  5:41503n     [ -6  12   1  -3  -2   0   0   0 ⟩        2657205/2656192        0.660120   11   48
  4.74  289:325n     [ -3   2  -2   0   0  -1   2   0 ⟩           2601/2600           0.665731   17   57

5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  4.76  20449n       [ 27  -8   0   0  -2  -2   0   0 ⟩      134217728/134165889      0.668785   13   48
  4.86  232925n      [ 21  -2  -2  -1  -3   0   0   0 ⟩        2097152/2096325        0.682837   11   50
  4.92  55883275n    [-40   9   2   6   0   0   0   1 ⟩  1099950501825/1099511627776  0.690890   19   71
  4.93  625:833n     [  2  -1   4  -2   0   0  -1   0 ⟩           2500/2499           0.692632   17   51
  4.94  191179625n   [-18  -6   3   6   0   1   0   0 ⟩      191179625/191102976      0.694237   13   70
  4.95  11:9025n     [  3  -8   2   0  -1   0   0   2 ⟩          72200/72171          0.695511   19   59
  4.97  13:71995n    [  5 -11   1   1   2  -1   1   0 ⟩        2303840/2302911        0.698243   17   64
  4.99  19:175175n   [-10  -2   2   2   1   1   0  -1 ⟩         175175/175104         0.701827   19   67
  5.00  95:2401n     [ 11  -4   1  -4   0   0   0   1 ⟩         194560/194481         0.703101   19   52
  5.04  25:3553n     [-23  10  -2   0   1   0   1   1 ⟩      209801097/209715200      0.708949   19   57
  5.06  11:5764801n  [ 19   0   0  -8   1   0   0   0 ⟩        5767168/5764801        0.710691   11   67
  5.07  19:2431n     [  7   0   0   0  -1  -1  -1   1 ⟩           2432/2431           0.712002   19   60
  5.07  5:2431n      [ -1  -5  -1   0   1   1   1   0 ⟩           2431/2430           0.712295   17   46
  5.09  19:15379n    [-16   4   0   1   0   3   0  -1 ⟩        1245699/1245184        0.715879   19   65
  5.12  2653105n     [ 15   4  -1  -4   0  -1  -1   0 ⟩        2654208/2653105        0.719592   17   63
  5.13  25:2401n     [ -5  -1  -2   4   0   0   0   0 ⟩           2401/2400           0.721197    7   38
  5.14  775523n      [ 37 -11   0  -4   0   0  -1  -1 ⟩   137438953472/137381572881   0.722939   19   64
  5.19  11:2375n     [  3   3  -3   0   1   0   0  -1 ⟩           2376/2375           0.728787   19   45

5.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  5.28  885115n      [-15  -3   1   1   3   0   0   1 ⟩         885115/884736         0.741461   19   64
  5.32  49:1015625n  [  8   4  -7   2   0  -1   0   0 ⟩        1016064/1015625        0.748158   13   62
  5.33  343:14641n   [ -7   1   0  -3   4   0   0   0 ⟩          43923/43904          0.749051   11   65
  5.35  49:296875n   [ 30 -11  -6   2   0   0   0  -1 ⟩    52613349376/52590515625    0.751504   19   63
  5.40  289:1235n    [  9  -7   1   0   0   1  -2   1 ⟩         632320/632043         0.758567   19   71
  5.40  265837n      [  1 -12   0   0   2   3   0   0 ⟩         531674/531441         0.758860   13   61
  5.40  121:371875n  [-10  -1   5   1  -2   0   1   0 ⟩         371875/371712         0.759000   17   71
  5.47  537109375n   [-29   0  11   0   1   0   0   0 ⟩      537109375/536870912      0.768795   11   66
  5.49  1573:2125n   [-17  11  -3   0   2   1  -1   0 ⟩      278652231/278528000      0.772005   17   67
  5.52  24565n       [ 13   1  -1   0   0   0  -3   0 ⟩          24576/24565          0.775058   17   56
  5.59  17:11011n    [  3   4   0  -1  -2  -1   1   0 ⟩          11016/11011          0.785960   17   59
  5.62  85:16093n    [ 25 -11   1  -1  -2   0   1  -1 ⟩     2852126720/2850826671     0.789307   19   70
  5.66  637:171875n  [-16   5   6  -2   1  -1   0   0 ⟩       41765625/41746432       0.795755   13   68

6 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  5.76  343:8125n    [-22  11   4  -3   0   1   0   0 ⟩     1439319375/1438646272     0.809807   13   54
  5.80  169:475n     [-11   6   2   0   0  -2   0   1 ⟩         346275/346112         0.815126   19   55
  5.80  174845n      [-19   1   1   0   2   0   2   0 ⟩         524535/524288         0.815418   17   61
  5.82  13:125n      [ 11  -9   3   0   0  -1   0   0 ⟩         256000/255879         0.818472   13   28
  5.85  55:405769n   [ -3  10   1  -4   1  -2   0   0 ⟩        3247695/3246152        0.822715   13   70
  5.87  13:448987n   [-23   5   0   4   1  -1   1   0 ⟩      109103841/109051904      0.824320   17   69
  5.90  323:637n     [ 20 -12   0  -2   0  -1   1   1 ⟩      338690048/338527917      0.828941   19   63
  5.90  19:77n       [-17  12   0  -1  -1   0   0   1 ⟩       10097379/10092544       0.829178   19   37
  5.92  11:199927n   [ 30 -10   0  -1   1  -4   0   0 ⟩    11811160064/11805489423    0.831380   13   70
  5.92  65:77n       [  5  -3   1  -1  -1   1   0   0 ⟩           2080/2079           0.832524   13   36
  5.94  5:54587n     [ 15  -1   1   0   0  -2  -1  -1 ⟩         163840/163761         0.834964   19   67
  5.97  25:75803n    [-29  11  -2   3   0   1   1   0 ⟩    13428274041/13421772800    0.838372   17   61
  5.98  13:319333n   [ 13   1   0  -5   0   1   0  -1 ⟩         319488/319333         0.840114   19   67
  5.99  343:2057n    [  1   1   0   3  -2   0  -1   0 ⟩           2058/2057           0.841426   17   60
  6.02  207025n      [ 24  -4  -2  -2   0  -2   0   0 ⟩       16777216/16769025       0.845432   13   50
  6.04  124355n      [  9   5  -1  -1  -1   0  -1  -1 ⟩         124416/124355         0.849016   19   59
  6.06  133:15125n   [ 10  -2  -3   1  -2   0   0   1 ⟩         136192/136125         0.851895   19   63
  6.12  2358125n     [ 18   2  -4  -3  -1   0   0   0 ⟩        2359296/2358125        0.859484   11   52
  6.13  17:26125n    [ -9  -1   3   0   1   0  -1   1 ⟩          26125/26112          0.861690   19   62
  6.20  18883865n    [-21  -2   1   4   2   1   0   0 ⟩       18883865/18874368       0.870885   13   68
  6.21  49:3971n     [  0  -4   0  -2   1   0   0   2 ⟩           3971/3969           0.872158   19   63
  6.25  55:3757n     [ -5   7   1   0   1  -1  -2   0 ⟩         120285/120224         0.878182   17   63

6.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  6.25  19:17303n    [-13   2   0   0   3   1   0  -1 ⟩         155727/155648         0.878474   19   65
  6.28  66171875n    [ 45 -12  -7  -1  -2   0   0   0 ⟩ 35184372088832/35166447421875 0.882201   11   64
  6.35  11:4165n     [ 26 -11  -1  -2   1   0  -1   0 ⟩      738197504/737817255      0.891997   17   47
  6.39  625:5929n    [ -8   3  -4   2   2   0   0   0 ⟩         160083/160000         0.897845   11   56
  6.41  49:93925n    [ 22  -7  -2   2   0  -1  -2   0 ⟩      205520896/205413975      0.900898   17   71
  6.58  19:109375n   [-22   6   6   1   0   0   0  -1 ⟩       79734375/79691776       0.925179   19   56
  6.66  7:2125n      [-13   3   3  -1   0   0   1   0 ⟩          57375/57344          0.935647   17   39
  6.72  143:8575n    [-17   7   2   3  -1  -1   0   0 ⟩       18753525/18743296       0.944549   13   55
  6.75  209:42875n   [  5  -8   3   3  -1   0   0  -1 ⟩        1372000/1371249        0.947896   19   66

7 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  6.78  5:1729n      [ -9  11   1  -1   0  -1   0  -1 ⟩         885735/885248         0.952139   19   44
  6.80  25:2527n     [ 13  -4   2  -1   0   0   0  -2 ⟩         204800/204687         0.955486   19   55
  6.81  49:142805n   [  2   6  -1   2   0  -4   0   0 ⟩         142884/142805         0.957457   13   71
  6.82  11:119n      [ 14 -11   0   1  -1   0   1   0 ⟩        1949696/1948617        0.958364   17   35
  6.85  17:111475n   [  0   8  -2  -3   0  -1   1   0 ⟩         111537/111475         0.962608   17   61
  6.87  17:32585n    [ 22  -7  -1  -3   0   0   1  -1 ⟩       71303168/71263395       0.965954   19   62
  6.89  1890625n     [ -5 -10   6   0   2   0   0   0 ⟩        1890625/1889568        0.968159   11   52
  6.91  17:274625n   [ 33 -12  -3   0   0  -3   1   0 ⟩   146028888064/145946984625   0.971272   17   71
  6.91  7:232375n    [ -4  12  -3   1  -1  -2   0   0 ⟩        3720087/3718000        0.971509   13   59
  6.94  7:67925n     [ 18  -3  -2   1  -1  -1   0  -1 ⟩        1835008/1833975        0.974856   19   60
  6.97  7:339625n    [-33  11   3  -1   1   1   0   1 ⟩    60163549875/60129542144    0.978865   19   65
  6.99  7:1161875n   [-11  -4   4  -1   1   2   0   0 ⟩        1161875/1161216        0.982211   13   64
  7.03  13:256025n   [  0  -9   2   2   1  -1   0   1 ⟩         256025/255879         0.987530   19   67
  7.05  17:557375n   [-15   0   3   3   0   1  -1   0 ⟩         557375/557056         0.991113   17   66
  7.08  605:637n     [  8  -5   1  -2   2  -1   0   0 ⟩         154880/154791         0.995120   13   54
  7.10  5:2527n      [-28  12  -1   1   0   0   0   2 ⟩     1342951407/1342177280     0.998235   19   50
  7.13  1729n        [ -6  -3   0   1   0   1   0   1 ⟩           1729/1728           1.001582   19   39
  7.14  539:2873n    [  4  -1   0   2   1  -2  -1   0 ⟩           8624/8619           1.004021   17   68
  7.18  143:1715n    [  2   1  -1  -3   1   1   0   0 ⟩           1716/1715           1.009172   13   50
  7.20  5:1090397n   [ 32  -9   1  -3  -1   0  -2   0 ⟩    21474836480/21462284151    1.012225   17   71
  7.22  209:21125n   [ 13  -4  -3   0   1  -2   0   1 ⟩        1712128/1711125        1.014490   19   71
  7.24  7:425n       [ -2   5  -2   1   0   0  -1   0 ⟩           1701/1700           1.018074   17   34

7.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  7.25  55:28561n    [ 10 -12  -1   0  -1   4   0   0 ⟩       29246464/29229255       1.018980   13   68
  7.27  91:1615n     [ 20 -10  -1   1   0   1  -1  -1 ⟩       95420416/95364135       1.021420   19   61
  7.32  19:21875n    [  7   2  -5  -1   0   0   0   1 ⟩          21888/21875          1.028542   19   51
  7.51  3125:5929n   [ -7   5   5  -2  -2   0   0   0 ⟩         759375/758912         1.055876   11   61
  7.53  669921875n   [ 42  -8  -9  -3   0   0   0   0 ⟩  4398046511104/4395357421875  1.058849    7   66
  7.55  425:2299n    [ 15 -11  -2   0   2   0  -1   1 ⟩       75333632/75287475       1.061054   19   68
  7.55  13:50575n    [ -4  -5   2   1   0  -1   2   0 ⟩          50575/50544          1.061487   17   64
  7.65  95:17303n    [ -2   6   1   0  -3  -1   0   1 ⟩          69255/69212          1.075246   19   70
  7.65  11:3757n     [-10   1   0   0  -1   1   2   0 ⟩          11271/11264          1.075539   17   58
  7.67  119:3211n    [  0   3   0   1   0  -2   1  -1 ⟩           3213/3211           1.077979   19   69
  7.68  25:1331n     [ 20  -9   2   0  -3   0   0   0 ⟩       26214400/26198073       1.078593   11   43
  7.70  245:3971n    [-15  12   1   2  -1   0   0  -2 ⟩      130203045/130121728      1.081563   19   68
  7.71  3776773n     [  6  10   0  -4  -2  -1   0   0 ⟩        3779136/3776773        1.082836   13   63
  7.74  119:390625n  [  1  -8   8  -1   0   0  -1   0 ⟩         781250/780759         1.088388   17   64

8 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  7.77  17:5225n     [ -6   9  -2   0  -1   0   1  -1 ⟩         334611/334400         1.092031   19   57
  7.79  11790625n    [-33   6   5   3   1   0   0   0 ⟩     8595365625/8589934592     1.094236   11   57
  7.82  343:59375n   [ 10 -11   5  -3   0   0   0   1 ⟩       60800000/60761421       1.098857   19   65
  7.84  133:75625n   [-25  10   4  -1   2   0   0  -1 ⟩     4465580625/4462739456     1.101826   19   68
  7.86  621775n      [-24   3   2   1   1   0   1   1 ⟩       16787925/16777216       1.104705   19   64
  7.88  2127125n     [ -2 -12   3   1   1   1   1   0 ⟩        2127125/2125764        1.108051   17   63
  7.89  1035125n     [-39  12   3   2   0   2   0   0 ⟩   550107865125/549755813888   1.108288   13   55
  7.91  343:10285n   [-16   7   1  -3   2   0   1   0 ⟩       22493295/22478848       1.112295   17   65
  7.94  125:75803n   [ 14  -3   3  -3   0  -1  -1   0 ⟩        2048000/2046681        1.115348   17   66
  7.95  420175n      [ -6  -8   2   5   0   0   0   0 ⟩         420175/419904         1.116953    7   45
  7.96  54587n       [-30   9   0   0   0   2   1   1 ⟩     1074435921/1073741824     1.118757   19   62
  7.98  13:77n       [-20  11   0   1   1  -1   0   0 ⟩       13640319/13631488       1.121197   13   31
  7.98  186745n      [ -8  -6   1   0   0   3   1   0 ⟩         186745/186624         1.122103   17   61
  8.00  19:385n      [  2  -4   1   1   1   0   0  -1 ⟩           1540/1539           1.124543   19   42
  8.01  19:218491n   [ 23  -6   0  -5   0  -1   0   1 ⟩      159383552/159279939      1.125817   19   67
  8.02  5:110789n    [  3 -11  -1   3   0   0   1   1 ⟩         886312/885735         1.127422   19   62
  8.05  11:405769n   [-12  -2   0   4  -1   2   0   0 ⟩         405769/405504         1.131005   13   65
  8.08  175:187n     [ 11  -7  -2  -1   1   0   1   0 ⟩         382976/382725         1.135012   17   45
  8.10  34969n       [  4   7   0   0  -2   0  -2   0 ⟩          34992/34969          1.138302   17   56
  8.10  95:169n      [ -4   2  -1   0   0   2   0  -1 ⟩           1521/1520           1.138595   19   50
  8.14  343:1625n    [  7  -3  -3   3   0  -1   0   0 ⟩          43904/43875          1.143913   13   49
  8.19  11:1080625n  [ 15   1  -4  -1   1  -1   0  -1 ⟩        1081344/1080625        1.151503   19   70
  8.24  637:34375n   [  1   3  -5   2  -1   1   0   0 ⟩          34398/34375          1.157966   13   63

8.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  8.28  13:25289n    [ -3  -5   0   0   3  -1   0   1 ⟩          25289/25272          1.164177   19   65
  8.31  17:55055n    [-18   4   1   1   2   1  -1   0 ⟩        4459455/4456448        1.167761   17   64
  8.38  175:29887n   [ -9   1  -2  -1   2   1   0   1 ⟩          89661/89600          1.178229   19   71
  8.39  122825n      [-28   7   2   0   0   0   3   0 ⟩      268618275/268435456      1.178662   17   61
  8.41  19:2023n     [ 21  -9   0  -1   0   0  -2   1 ⟩       39845888/39818709       1.181283   19   60
  8.43  107421875n   [ 14   8 -10   0  -1   0   0   0 ⟩      107495424/107421875      1.184926   11   61
  8.50  121:74375n   [ -5   9  -4  -1   2   0  -1   0 ⟩        2381643/2380000        1.194721   17   66
  8.57  1715:14641n  [ -8   7   1   3  -4   0   0   0 ⟩        3750705/3748096        1.204670   11   70
  8.58  49:5078125n  [-23   4   8  -2   0   1   0   0 ⟩      411328125/411041792      1.205563   13   67
  8.63  177023n      [  0  11   0  -1  -3   0   0  -1 ⟩         177147/177023         1.212260   19   59
  8.66  77:1328125n  [ 22  -5  -7   1   1   0  -1   0 ⟩      322961408/322734375      1.217438   17   70
  8.72  11:11875n    [-18   5   4   0  -1   0   0   1 ⟩        2885625/2883584        1.224934   19   50
  8.74  11:40625n    [  4 -10   5   0  -1   1   0   0 ⟩         650000/649539         1.228280   13   49

9 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  8.77  125:2401n    [-10   9   3  -4   0   0   0   0 ⟩        2460375/2458624        1.232524    7   43
  8.78  13265525n    [-30   4   2   4   0   1   1   0 ⟩     1074507525/1073741824     1.234128   17   68
  8.81  95:15379n    [  1   4   1  -1   0  -3   0   1 ⟩          15390/15379          1.237842   19   70
  8.83  25:1183n     [ 23 -11   2  -1   0  -2   0   0 ⟩      209715200/209564901      1.241188   13   43
  8.84  19:12155n    [-22   8   1   0   1   1   1  -1 ⟩       79748955/79691776       1.241718   19   65
  8.86  125:3553n    [  8  -2   3   0  -1   0  -1  -1 ⟩          32000/31977          1.244772   19   62
  8.88  121:4655n    [  9  -9   1   2  -2   0   0   1 ⟩        2383360/2381643        1.247650   19   60
  8.90  19:2401n     [-26  12   0   4   0   0   0  -1 ⟩     1275989841/1275068416     1.250620   19   47
  8.91  19:35035n    [ -5  10  -1  -2  -1  -1   0   1 ⟩        1121931/1121120        1.251894   19   62
  8.93  539n         [ 17  -5   0  -2  -1   0   0   0 ⟩         131072/130977         1.255240   11   25
  8.97  125:833n     [-17   9  -3   2   0   0   1   0 ⟩       16395939/16384000       1.261089   17   46
  8.99  11176655n    [ 25  -1  -1  -6   0   0   0  -1 ⟩       33554432/33529965       1.262830   19   66
  9.00  49:133705n   [ 13  -1  -1   2  -2  -1  -1   0 ⟩         401408/401115         1.264142   17   71
  9.04  1164625n     [-36  10   3   1   3   0   0   0 ⟩    68769941625/68719476736    1.270884   11   55
  9.17  289:1625n    [-12   6   3   0   0   1  -2   0 ⟩        1184625/1183744        1.287990   17   62
  9.19  17:1484375n  [ 18  -1  -7   0   0   0   1  -1 ⟩        4456448/4453125        1.291395   19   71
  9.21  41503n       [ -9  -4   0   3   2   0   0   0 ⟩          41503/41472          1.293601   11   43
  9.24  247:833n     [ -3   3   0  -2   0   1  -1   1 ⟩           6669/6664           1.298458   19   63

9.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  9.26  665:1331n    [ -1   0  -1  -1   3   0   0  -1 ⟩           1331/1330           1.301191   19   64
  9.26  833:845n     [ 19 -12   1  -2   0   2  -1   0 ⟩      443023360/442690353      1.301805   17   62
  9.31  25:91091n    [-15   2  -2   2   1   2   0   0 ⟩         819819/819200         1.307653   13   61
  9.36  354025n      [  1  11  -2  -2   0   0  -2   0 ⟩         354294/354025         1.314950   17   58
  9.40  847:40625n   [  4   1  -5   1   2  -1   0   0 ⟩          40656/40625          1.320561   13   67
  9.47  1328125n     [-14  -4   7   0   0   0   1   0 ⟩        1328125/1327104        1.331403   17   52
  9.50  143:78125n   [ -2   7  -7   0   1   1   0   0 ⟩         312741/312500         1.334613   13   59
  9.52  9934375n     [ 28  -3  -5   0  -1   0  -2   0 ⟩      268435456/268228125      1.337667   17   70
  9.59  247:3125n    [-10   4   5   0   0  -1   0  -1 ⟩         253125/252928         1.347895   19   57
  9.62  361:15625n   [ 12 -11   6   0   0   0   0  -2 ⟩       64000000/63950067       1.351241   19   68
  9.67  425:637n     [ -1   1   2  -2   0  -1   1   0 ⟩           1275/1274           1.358363   17   54
  9.69  1463:1625n   [-16  10   3  -1  -1   1   0  -1 ⟩       95954625/95879168       1.361947   19   65
  9.73  245:1859n    [ -5   5   1   2  -1  -2   0   0 ⟩          59535/59488          1.367265   13   56

10 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
  9.75  1225:2717n   [ 17 -10   2   2  -1  -1   0  -1 ⟩      160563200/160436133      1.370612   19   67
  9.77  221:3773n    [ -7   7   0  -3  -1   1   1   0 ⟩         483327/482944         1.372416   17   62
  9.78  157339n      [  3   9   0  -2   0  -2   0  -1 ⟩         157464/157339         1.374855   19   59
  9.81  5:229957n    [ 25  -6   1  -2   0  -1   0  -2 ⟩      167772160/167638653      1.378202   19   70
  9.83  7:605n       [-11  11  -1   1  -2   0   0   0 ⟩        1240029/1239040        1.381317   11   34
  9.85  91:2299n     [ 11  -4   0   1  -2   1   0  -1 ⟩         186368/186219         1.384664   19   61
  9.90  91:378125n   [  7 -12   5  -1   2  -1   0   0 ⟩       48400000/48361131       1.390875   13   67
  9.90  17:34375n    [-30  12   5   0   1   0  -1   0 ⟩    18268284375/18253611008    1.391113   17   53
  9.92  2211125n     [-29   5   3   2   0   0   0   2 ⟩      537303375/536870912      1.393991   19   67
  9.92  15104375n    [  8  10  -4   0  -1  -3   0   0 ⟩       15116544/15104375       1.394226   13   70
  9.94  7564375n     [ -7 -10   4   2   0   1   0   1 ⟩        7564375/7558272        1.397338   19   66
  9.94  4415125n     [ 30  -5  -3   0  -1  -2   0  -1 ⟩     1073741824/1072875375     1.397572   19   71
  9.97  49:5225n     [-21   9   2  -2   1   0   0   1 ⟩      102843675/102760448      1.401581   19   54
 10.00  49:17875n    [  1  -6   3  -2   1   1   0   0 ⟩          35750/35721          1.404928   13   53
 10.04  169:7315n    [ 12 -11   1   1   1  -2   0   1 ⟩       29962240/29937843       1.410246   19   68
 10.04  1310309n     [-33   8   0   2   2   1   1   0 ⟩     8596937349/8589934592     1.410776   17   66
 10.06  17:1225n     [ -3  -2   2   2   0   0  -1   0 ⟩           1225/1224           1.413829   17   41
 10.09  121:57967n   [ 20  -7   0  -3   2  -2   0   0 ⟩      126877696/126773829      1.417836   13   69
 10.10  343:10985n   [ -5   0   1  -3   0   3   0   0 ⟩          10985/10976          1.418980   13   65
 10.10  5:1294139n   [-37  12  -1   6   1   0   0   0 ⟩   687758524299/687194767360   1.419678   11   58
 10.11  325:361n     [-16  10  -2   0   0  -1   0   2 ⟩       21316689/21299200       1.420951   19   61
 10.11  55:15827n    [  5   2   1  -2   1   0  -1  -1 ⟩          15840/15827          1.421419   19   66
 10.14  5:19n        [  6  -5  -1   0   0   0   0   1 ⟩           1216/1215           1.424298   19   24
 10.16  187:1183n    [ -9   4   0   1  -1   2  -1   0 ⟩          95823/95744          1.427882   17   61
 10.19  11:60025n    [ 14  -1  -2  -4   1   0   0   0 ⟩         180224/180075         1.431888   11   49

10.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 10.25  27625n       [ 10   3  -3   0   0  -1  -1   0 ⟩          27648/27625          1.440790   17   45
 10.28  8075n        [ 32 -12  -2   0   0   0  -1  -1 ⟩     4294967296/4291386075     1.444136   19   46
 10.46  25:102487n   [-12   0  -2   1   4   0   0   0 ⟩         102487/102400         1.470248   11   61
 10.56  169:1445n    [  8  -7   1   0   0  -2   2   0 ⟩         369920/369603         1.484203   17   65
 10.58  143:7225n    [ 12  -4  -2   0   1   1  -2   0 ⟩         585728/585225         1.487354   17   68
 10.59  209:2975n    [ -7   2   2   1  -1   0   1  -1 ⟩          26775/26752          1.487787   19   64
 10.66  289:385n     [  2  -1  -1  -1  -1   0   2   0 ⟩           1156/1155           1.498255   17   57
 10.68  5:121121n    [ 32 -11   1  -1  -3  -1   0   0 ⟩    21474836480/21456221787    1.501309   13   58
 10.70  7:51623n     [ -3  10   0   1  -1  -1   0  -2 ⟩         413343/412984         1.504279   19   69

11 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 10.79  13:336875n   [-21   4   4   2   1  -1   0   0 ⟩       27286875/27262976       1.516952   13   58
 10.82  19:1684375n  [  1 -11   5   2   1   0   0  -1 ⟩        3368750/3365793        1.520299   19   69
 10.87  13:17765n    [-12   1   1   0   1  -1   1   1 ⟩          53295/53248          1.527421   19   65
 10.89  2275n        [-27  10   2   1   0   1   0   0 ⟩      134336475/134217728      1.531005   13   30
 10.92  19:147875n   [ -5  -5   3   1   0   2   0  -1 ⟩         147875/147744         1.534351   19   67
 10.96  13:12005n    [  6 -10   1   4   0  -1   0   0 ⟩         768320/767637         1.539669   13   46
 10.97  35:4199n     [-18   7  -1  -1   0   1   1   1 ⟩        9183213/9175040        1.541473   19   61
 10.99  11:845n      [ -8   9  -1   0   1  -2   0   0 ⟩         216513/216320         1.543913   13   42
 10.99  95:14399n    [ 12  -3   1  -1  -2   0  -1   1 ⟩         389120/388773         1.544527   19   70
 10.99  7:2873n      [  4  -8   0  -1   0   2   1   0 ⟩          45968/45927          1.544820   17   50
 11.01  11:247n      [ 14  -6   0   0   1  -1   0  -1 ⟩         180224/180063         1.547259   19   43
 11.03  275:6517n    [-22  11  -2   3  -1   0   0   1 ⟩     1154466999/1153433600     1.550375   19   61
 11.04  3142909n     [ 20   1   0  -5  -1   0  -1   0 ⟩        3145728/3142909        1.552117   17   63
 11.06  55:4459n     [  0  -4  -1   3  -1   1   0   0 ⟩           4459/4455           1.553721   13   50
 11.08  187:79625n   [ 23  -9  -3  -2   1  -1   1   0 ⟩     1568669696/1567258875     1.557728   17   70
 11.11  13:3325n     [  8   0  -2  -1   0   1   0  -1 ⟩           3328/3325           1.561311   19   49
 11.15  49:105625n   [ 19  -5  -4   2   0  -2   0   0 ⟩       25690112/25666875       1.566630   13   60
 11.18  218405n      [-32   9   1   0   2   0   0   2 ⟩     4298865615/4294967296     1.570639   19   65
 11.20  747175n      [-10  -6   2   0   2   1   0   1 ⟩         747175/746496         1.573985   19   64
 11.25  7:171875n    [ 13   1  -6   1  -1   0   0   0 ⟩         172032/171875         1.580682   11   48

11.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 11.32  17:121n      [ -6   2   0   0   2   0  -1   0 ⟩           1089/1088           1.590477   17   39
 11.39  2299:6125n   [  3  -1  -3  -2   2   0   0   1 ⟩          18392/18375          1.600945   19   70
 11.42  1859:2975n   [-12   8  -2  -1   1   2  -1   0 ⟩       12196899/12185600       1.604529   17   71
 11.48  539:2125n    [ 21 -12  -3   2   1   0  -1   0 ⟩     1130364928/1129312125     1.613194   17   57
 11.52  85:65219n    [  8   1   1  -2  -3   0   1   0 ⟩          65280/65219          1.618484   17   69
 11.58  49:190333n   [  4   5   0   2  -4  -1   0   0 ⟩         190512/190333         1.627386   13   71
 11.59  343:78125n   [-11   2   7  -3   0   0   0   0 ⟩         703125/702464         1.628279    7   56
 11.72  1001:2375n   [ -6   3   3  -1  -1  -1   0   1 ⟩          64125/64064          1.647650   19   65
 11.75  77:625n      [ 16 -12   4  -1  -1   0   0   0 ⟩       40960000/40920957       1.650996   11   38

12 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 11.77  361:9625n    [-19   9   3   1   1   0   0  -2 ⟩      189448875/189267968      1.653966   19   71
 11.78  25:218491n   [  2   7   2  -5   0  -1   0   0 ⟩         218700/218491         1.655240   13   58
 11.79  29155n       [-18   2   1   3   0   0   1   0 ⟩         262395/262144         1.656845   17   43
 11.80  125:319333n  [ 24  -8   3  -5   0   0   0  -1 ⟩     2097152000/2095143813     1.658586   19   69
 11.84  5:221221n    [ -2  11   1  -1  -1  -2  -1   0 ⟩         885735/884884         1.664142   17   66
 11.84  133:187n     [-10   6   0  -1   1   0   1  -1 ⟩         136323/136192         1.664434   19   54
 11.86  25:38437n    [ -9  -1  -2   1   0   0   2   1 ⟩          38437/38400          1.667313   19   70
 11.88  17:823543n   [ 33 -11   0  -7   0   0   1   0 ⟩   146028888064/145888171821   1.669055   17   66
 11.89  133:1573n    [ 21 -11   0   1  -2  -1   0   1 ⟩      278921216/278652231      1.670367   19   61
 11.91  343:1235n    [-14  10  -1   3   0  -1   0  -1 ⟩       20253807/20234240       1.673336   19   58
 11.93  343:361n     [  8  -5   0   3   0   0   0  -2 ⟩          87808/87723          1.676683   19   59
 11.94  245245n      [ 29  -7  -1  -3  -1  -1   0   0 ⟩      536870912/536350815      1.677957   13   50
 11.98  119:8125n    [ -5   7  -4   1   0  -1   1   0 ⟩         260253/260000         1.683805   17   57
 12.01  119:2375n    [ 17  -8  -3   1   0   0   1  -1 ⟩       15597568/15582375       1.687151   19   58
 12.02  181575625n   [-10 -11   4   4   2   0   0   0 ⟩      181575625/181398528      1.689357   11   70
 12.05  13:33275n    [-24   8   2   0   3  -1   0   0 ⟩      218317275/218103808      1.693600   13   56
 12.08  19:166375n   [ -2  -7   3   0   3   0   0  -1 ⟩         166375/166212         1.696946   19   67
 12.08  289:109375n  [ 26 -11  -6  -1   0   0   2   0 ⟩    19394461696/19375453125    1.697620   17   71
 12.10  4659655n     [-38  10   1   3   1   1   0   1 ⟩   275147968095/274877906944   1.700062   19   69
 12.10  14141875n    [ 19   3  -4   0  -3   0  -1   0 ⟩       14155776/14141875       1.700911   17   70
 12.12  15940925n    [-16  -5   2   3   1   2   0   0 ⟩       15940925/15925248       1.703409   13   68
 12.15  49:22627n    [  7 -10   0  -2   3   0   1   0 ⟩        2896256/2893401        1.707415   17   64
 12.17  25:2023n     [  0   4   2  -1   0   0  -2   0 ⟩           2025/2023           1.710706   17   51
 12.21  65:5929n     [  3  -6  -1   2   2  -1   0   0 ⟩          47432/47385          1.716317   13   54
 12.22  1399489n     [-22   1   0   2   0   4   0   0 ⟩        4198467/4194304        1.717461   13   66
 12.25  19:29155n    [  9   1  -1  -3   0   0  -1   1 ⟩          29184/29155          1.721174   19   62

12.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 12.31  125:1001n    [ -3   0  -3   1   1   1   0   0 ⟩           1001/1000           1.730369   13   46
 12.33  7:15895n     [ 27 -10  -1   1  -1   0  -2   0 ⟩      939524096/938583855      1.733423   17   57
 12.41  2197:3125n   [ -9   6  -5   0   0   3   0   0 ⟩        1601613/1600000        1.744421   13   64
 12.48  91:265625n   [ -2  -6   6  -1   0  -1   1   0 ⟩         265625/265356         1.754119   17   67
 12.50  11:2734375n  [ 10   5  -8  -1   1   0   0   0 ⟩        2737152/2734375        1.757329   11   58
 12.65  121:6125n    [-12   4   3   2  -2   0   0   0 ⟩         496125/495616         1.777073   11   51
 12.67  7:48828125n  [ 37  -9 -11   1   0   0   0   0 ⟩   962072674304/961083984375   1.780046    7   62
 12.67  85:57967n    [ 11  -1   1  -3   0  -2   1   0 ⟩         174080/173901         1.781080   17   69
 12.70  25:10241n    [ -4   8   2  -2  -1   0   0  -1 ⟩         164025/163856         1.784663   19   54
 12.72  121:323n     [ -3   1   0   0  -2   0   1   1 ⟩            969/968            1.787542   19   58
 12.74  7:24167n     [  7   3   0   1  -1  -3   0   0 ⟩          24192/24167          1.789981   13   57
 12.75  1331:5915n   [-18  10   1   1  -3   2   0   0 ⟩      349274835/348913664      1.791125   13   71

13 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 12.76  35:35321n    [ 29 -12   1   1  -1  -2   0  -1 ⟩    18790481920/18771027561    1.793328   19   68
 12.77  17:132055n   [  5   5  -1  -4  -1   0   1   0 ⟩         132192/132055         1.795132   17   61
 12.79  7:3265625n   [-22   2   6  -1   1   0   0   1 ⟩       29390625/29360128       1.797337   19   67
 12.81  1331:2401n   [ 15 -10   0   4  -3   0   0   0 ⟩       78675968/78594219       1.799790   11   61
 12.84  39325n       [  1   9  -2   0  -2  -1   0   0 ⟩          39366/39325          1.804034   13   45
 12.86  11495n       [ 23  -6  -1   0  -2   0   0  -1 ⟩        8388608/8379855        1.807380   19   46
 12.88  17:5359375s  [ -4  -9   6   3   0   0  -1   0 ⟩        5359375/5353776        1.809585   17   68
 12.91  1547:6875s   [-18  10   4  -1   1  -1  -1   0 ⟩      405961875/405536768      1.813829   17   68
 12.93  13:63175s    [-17   3   2   1   0  -1   0   2 ⟩        1705725/1703936        1.816707   19   68
 12.95  16625s       [  5 -12   3   1   0   0   0   1 ⟩         532000/531441         1.820054   19   41
 12.98  1045:4459s   [ -9   7   1  -3   1  -1   0   1 ⟩        2285415/2283008        1.824297   19   69
 13.01  275:343s     [ 13  -8   2  -3   1   0   0   0 ⟩        2252800/2250423        1.827644   11   42
 13.03  11:4693s     [-23   9   0   0  -1   1   0   2 ⟩       92372319/92274688       1.830759   19   62
 13.04  133:2873s    [-13  11   0   1   0  -2  -1   1 ⟩       23560551/23535616       1.833199   19   69
 13.05  5:14399s     [-21   6  -1   1   2   0   1   0 ⟩       10496871/10485760       1.833492   17   51
 13.05  11:16055s    [ -1  -6   1   0  -1   2   0   1 ⟩          16055/16038          1.834106   19   61
 13.07  35:221s      [  9  -4   1   1   0  -1  -1   0 ⟩          17920/17901          1.836546   17   42
 13.08  40353607s    [-11  -9   0   9   0   0   0   0 ⟩       40353607/40310784       1.838150    7   63
 13.11  169:2401s    [  7  -2   0  -4   0   2   0   0 ⟩          21632/21609          1.841696   13   54
 13.11  325:184877s  [-25  10  -2   5   1  -1   0   0 ⟩    10916801973/10905190400    1.842394   13   69
 13.13  95:184877s   [ -3  -5  -1   5   1   0   0  -1 ⟩         184877/184680         1.845740   19   70
 13.14  19:2275s     [ 18  -7  -2  -1   0  -1   0   1 ⟩        4980736/4975425        1.847014   19   49
 13.15  125:31603s   [-27  12  -3   0   1   2   1   0 ⟩    16795129923/16777216000    1.847544   17   69
 13.17  13:935s      [  3   2  -1   0  -1   1  -1   0 ⟩            936/935            1.850598   17   46
 13.21  625:64141s   [  6  -8  -4   3   1   0   1   0 ⟩        4105024/4100625        1.856209   17   69

13.5 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 13.26  12569375s    [ 22   1  -4  -1   0  -2  -1   0 ⟩       12582912/12569375       1.863506   17   70
 13.28  17935225s    [-13  -7   2   2   4   0   0   0 ⟩       17935225/17915904       1.866004   11   68
 13.38  1574573s     [-19  -1   0   1   3   2   0   0 ⟩        1574573/1572864        1.880056   13   66
 13.48  11:36125s    [  1  -8   3   0  -1   0   2   0 ⟩          72250/72171          1.894011   17   60
 13.59  11:252875s   [ 24  -6  -3  -1   1   0  -2   0 ⟩      184549376/184345875      1.910070   17   67
 13.59  85:2717s     [  5   0   1   0  -1  -1   1  -1 ⟩           2720/2717           1.910503   19   65
 13.71  9953125s     [-28   3   6   2   0   1   0   0 ⟩      268734375/268435456      1.926760   13   57

14 tinas
tinas   comma        monzo                                      interval              cents   limit  pop
 13.78  524875s      [-19   0   3   0   0   1   1   1 ⟩         524875/524288         1.937229   19   64
 13.80  169:9625s    [ -9   2   3   1   1  -2   0   0 ⟩          86625/86528          1.939669   13   59
 13.83  323:30625s   [-24  11   4   2   0   0  -1  -1 ⟩     5425126875/5419040768     1.943252   19   70
 13.85  11:1140475s  [-23   4   2   4  -1   0   0   1 ⟩       92378475/92274688       1.946131   19   68
 13.87  11:3901625s  [ -1 -11   3   4  -1   1   0   0 ⟩        3901625/3897234        1.949477   13   67
 13.90  5s           [-15   8   1   0   0   0   0   0 ⟩          32805/32768          1.953721    5    5

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Re: Magrathean diacritics

Post by cmloegcmluin »

Dave Keenan wrote: Tue May 26, 2020 9:53 am
Ash9903b4 wrote: Tue May 26, 2020 4:31 am I've only used tina diacritics in one context, and that was to make it clear that the mina accent represents different schisminas at the Olympian level, not just the 455n. Maybe we could check the differences between existing symbols with and without mina accents and just default to those values, since, after all, Sagittal already supports those ratios, though imprecisely at the Magrathean level.
That's brilliant! Thanks. It's obvious once you say it. But it hadn't occurred to me, despite the fact that cmloegcmluin was also pushing in that direction when he wrote:
cmloegcmluin wrote: Mon May 25, 2020 3:51 pm In determining commas for the tinas, is it not also very important to consider what these commas produce when combined with the existing commas in Sagittal? This question would seem to become increasingly pressing with each new precision level.
Sorry I summarily dismissed that, cmloegcmluin.
I didn't feel dismissed! I just assumed that what I'd said had flung open the gates to the abyss and, in exposing you to the horrific enormity of the perfectionism possible in this effort, you'd recoiled.

I wrote my post in haste. I hadn't really understood @Ash9903b4 's suggestion. And while it's brilliant and obvious to you now, I admit I still don't get it. Nor do I get how it connects with my concerns. What I was thinking was more like: should we looking at, like, a 37-limit tonality diamond and trying to cross off the simpler pitches we can't exactly notate yet? Or something like that? I know there is a link, of course, between picking relatively simple commas and receiving generally common combinations with existing commas, but why not just shoot straight for them? At least, if we're choosing between two close candidates, this should be weighed in the mix.
So we've nailed down 3, 4, 6, 8 and 9 tinas. To complete SMuFL, we only need 1, 2, 5 and either 0.5 or 0.33.
Have some work on it, but first priority is to reshape other stuff briefly lost yesterday.
Any thoughts about whether the dot should be ½ tina or ⅓ tina?
I think it's fertile territory to explore. Of course binary and trinary division is familiar. But we've already got the precedent of tetrary/pentary-ish division from the schisma to the mina. We could have it do something like that. I could imagine having two little dots.
Dave Keenan wrote: Sat May 23, 2020 10:14 pmIs there an accurate EDO that's approximately 3 × 8539edo?
None yet. I haven't looked into edos beyond 8539, mainly because I don't know how to find them. I'm aware of the Riemann zeta function and I have a basic idea about what it means in the context of tuning, but I don't yet know how to use it. One particular equal temperament that's mentioned on the xen.wiki is 16808, which would imply splitting the tina into almost two, but I don't know if a finer gradation could be better.
http://x31eq.com/cgi-bin/more.cgi?r=1&l ... error=0.01
confirms 16808, but that is not close to 3 × 8539 = 25617. Graham's calculator doesn't seem to offer suggestions much higher than that. At least, it seems to include suggestions of overall higher and higher EDOs as you increase the prime limit, but it doesn't go any higher than 53-limit, and it only lets you increase the result size once and then you're stuck with whatever 30 or so suggestions it makes.

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Re: Magrathean diacritics

Post by Ash9903b4 »

I've done a (rather naïve) search for good low-error edos around 10000, and they seem to match up with the ones listed on the Riemann zeta article on the wiki:

3395, 6079, 7033, 8269, 8539, 11664, 12770, 13112, 14348, 15230, 16808, 19461, 20203, 27730, 28472, 28742, 30631, 33315, 34691, 36269, 41584, 41854, 45861, 49039, 50434, 51499, 51940, 54624, 57578, 58973...

Note that there are no low-error edos near 25000; third-tinas simply aren't accurate enough. Half-tinas (16808edo) and seventh-tinas (58973edo) look promising, but the latter are way too precise, even for Magrathean.

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Re: Magrathean diacritics

Post by cmloegcmluin »

I've cross-referenced my results with @Ash9903b4 's. Ours are a bit different because he cut off his prime limit at 19, while I cut off the total number of primes > 3 lower than he did. As you could imagine, most of the difference between our two result sets is in the fringes — the commas unlikely to be candidates anyway. Although it's interesting to note that neither of our results contained George's original suggestion for the half tina, 49:9765625n; in Ash9903b4's case I think this is due to his 720720 odd limit. So it's possible that there's better ones lurking out there, but we're not letting perfection be the enemy of the good here.

Here are what I think are probably the best candidates for the remaining wholes at least. Columns are tina fraction, cents, monzo, ratio, comma, limit, SoPF.

1:

0.834 0.117 [ -24 12 0 -1 0 1 1 ⟩ 117448461/117440512 221/7n 17 37
1.07 0.150 [ -16 -1 3 0 2 1 ⟩ 196625/196608 196625/1n 13 50

2:

2.11 0.297 [ 3 6 0 -3 0 0 -1 ⟩ 5832/5831 1/5831n 17 38

Low SoPF, and a 3-exponent of 6, which when subtracted from the 5-schisma's 8 will be quite nice for the 12-tina.

5:

5.13 0.721 [ -5 -1 -2 4 ⟩ 2401/2400 2401/25n 7 38
4.93 0.693 [ 2 -1 4 -2 0 0 -1 ⟩ 2500/2499 625/833n 17 51

7:

7.13 1.00 [ -6 -3 0 1 0 1 0 1 ⟩ 1729/1728 1729/1n 19 39

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Re: Magrathean diacritics

Post by cmloegcmluin »

Ash9903b4 wrote: Wed May 27, 2020 2:34 am Note that there are no low-error edos near 25000; third-tinas simply aren't accurate enough. Half-tinas (16808edo) and seventh-tinas (58973edo) look promising, but the latter are way too precise, even for Magrathean.
Half-tina it is after all, then? Now with more reason?

I'll re-widen my scope to 1/4-tina error.

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Re: Magrathean diacritics

Post by cmloegcmluin »

I forgot — amidst the browser-cache-spelunking hullabaloo (perhaps I'm now driven by instinct to include strange and hard-to-forget sequences of words in my posts so I can excavate them later!) — that I did the work @Dave Keenan asked to check @Ash9903b4 's list against some criteria. But then I cast it aside once Dave posted saying that the list I had processed was in-exhaustive (inhaustive? haustive?). But perhaps it could still be of some use. So I redid it, including the new ones I also found. It might not include ones Ash9903b4 found later... I'm not sure (update: it does now! I was only missing the 125:833n).

The 5-limit comma I suggested above is the same one as George suggested.

Re: the criteria for the 0.5 tina, all commas were 37-limit (19-limit, in fact) so I removed that column.

Re: the criteria for whole tinas,
  • all have a prime exponent vector <=12 so I left that one out (I note that George's suggestion for the half-tina has 13, though)
  • one of George's own commas did not satisfy the <0.25 tina error criterion so I shifted it up to <=0.26 and now it does; by that criterion, all of these commas pass, so I removed that column
  • that left only one useful criterion, which is being <= 51 popularity rank
I wasn't sure whether Dave meant for the criteria he described for the 0.5 tina to also apply to the whole tinas, because otherwise, his criteria for whole tinas does not involve being better than George's commas in any way. Probably Dave just wanted to gather up anything reasonable and compare them together. But I figured it would be important to evaluate not just whether they pass some sieve, but also whether they're worth considering as replacements for George's for any reason.
So I kept a column which checks whether they are at least as good as George's by at least one criterion.

I also added two criteria to Dave's list:
1. limit ALAGA (at least as good as) George's?
2. error ALAGA George's?

There were 6 commas which met Dave's criteria yet were not better than George's in at least one way.

Re: commas, I eliminated George's alternate 5-tina comma as it did not satisfy Dave's criteria. This was the only comma of George's that was eliminated.

There was no comma found which was better than George's in every respect and thus a shoe-in.

I have highlighted in yellow the ones Dave claims as "nailed down", and in pale yellow the ones I'm voting for.

                                        		SoPF    3exp    SoPF    limit   error   >=1     worth	actual
                                       		tina	ALAGA   ALAGA   <=      ALAGA   ALAGA   ALAGA   consid-	mina	
tina	comma		abs3exp	limit	SoPF	error	George  George	51	George	George  George  eration	count	monzo						ratio	                whose

0.5 tinas
0.50	49:9765625n     13	7	64	0.00	yes	yes	n/a	yes	yes	yes	yes		[   3 -13  10  -2                             ⟩	78125000/78121827	George
0.73	13:4675n        6	17	51	0.23	yes	yes	n/a			yes	yes		[ -18   6   2   0   1  -1   1                 ⟩	3408075/3407872		Cmloegcmluin
0.33	17:9295n        7	17	59	0.17	yes	yes	n/a			yes	yes		[   2  -7   1   0   1   2  -1                 ⟩	37180/37179		Ash9903b4

1 tina
1.26	121:1225n       4	11	46	0.26	yes	yes	yes	yes	yes	yes	yes		[  -3   4  -2  -2   2                         ⟩	9801/9800		George
0.90	65:833n         8	17	49	0.10			yes		yes	yes	yes		[   9  -8  -1   2   0  -1   1                 ⟩	426496/426465		Ash9903b4
0.83	7:221n          12	17	37	0.17	yes		yes		yes	yes	yes		[ -24  12   0  -1   0   1   1                 ⟩	117448461/117440512	Cmloegcmluin
1.07	196625n         1	13	50	0.07		yes	yes		yes	yes	yes		[ -16  -1   3   0   2   1                     ⟩	196625/196608		Ash9903b4
1.20	5:10241n        0	19	49	0.20		yes	yes		yes	yes	yes		[ -11   0  -1   2   1   0   0   1             ⟩	10241/10240		Ash9903b4

2 tinas
1.85	13:6655n        0	13	51	0.15	yes	yes	yes	yes	yes	yes	yes		[   9   0  -1   0  -3   1                     ⟩	6656/6655		George
1.75	35:1573n        6	13	47	0.25	yes		yes	yes		yes	yes		[  15  -6   1   1  -2  -1                     ⟩	1146880/1146717		Cmloegcmluin
1.82	49:145n         11	29	48	0.18	yes		yes			yes	yes		[  19 -11  -1   2   0   0   0   0   0  -1     ⟩	25690112/25686315	Cmloegcmluin
1.87	1705n           9	31	47	0.13	yes		yes		yes	yes	yes		[ -25   9   1   0   1   0   0   0   0   0   1 ⟩	33559515/33554432	Cmloegcmluin
2.11	5831n           6	17	38	0.11	yes		yes		yes	yes	yes	2	[   3   6   0  -3   0   0  -1                 ⟩	5832/5831		Ash9903b4

3 tinas
3.01	455n            0	13	25	0.01	yes	yes	yes	yes	yes	yes	yes	21	[  12  -2  -1  -1   0  -1                     ⟩	4096/4095		Sagittal

4 tinas
4.07	7:3025n         3	11	39	0.07	yes	yes	yes	yes	yes	yes	yes	3	[  -4  -3   2  -1   2                         ⟩	3025/3024		George
3.84	17:245n         10	17	36	0.16	yes		yes			yes	yes	2	[ -12  10  -1  -2   0   0   1                 ⟩	1003833/1003520		Cmloegcmluin
4.24	85:637n         7	17	49	0.24			yes						[ -14   7  -1   2   0   1  -1                 ⟩	1393119/1392640		Cmloegcmluin
3.93	49:1045n        1	19	49	0.07		yes	yes		yes	yes	yes	1	[   6  -1  -1   2  -1   0   0  -1             ⟩	3136/3135		Ash9903b4
4.00	17:6875n        4	17	48	0.00			yes		yes	yes	yes		[  15  -4  -4   0  -1   0   1                 ⟩	557056/556875		Ash9903b4
4.17	49:1859n        3	13	51	0.17		yes	yes			yes	yes		[ -10   3   0  -2   1   2                     ⟩	50193/50176		Ash9903b4

5 tinas
5.13	25:2401n        1	7	38	0.13	yes	yes	yes	yes	yes	yes	yes	2	[  -5  -1  -2   4                             ⟩	2401/2400		George
4.76	20449n          8	13	48	0.24			yes						[  27  -8   0   0  -2  -2                     ⟩	134217728/134165889	Cmloegcmluin
4.86	232925n         2	11	50	0.14			yes						[  21  -2  -2  -1  -3                         ⟩	2097152/2096325		Cmloegcmluin
4.93	625:833n        1	17	51	0.07		yes	yes		yes	yes	yes		[   2  -1   4  -2   0   0  -1                 ⟩	2500/2499		Ash9903b4
5.07	5:2431n         5	17	46	0.07			yes		yes	yes	yes		[  -1  -5  -1   0   1   1   1                 ⟩	2431/2430		Ash9903b4
5.19	11:2375n        3	19	45	0.19			yes						[   3   3  -3   0   1   0   0  -1             ⟩	2376/2375		Ash9903b4

6 tinas
5.92	65:77n          0	13	36	0.08	yes	yes	yes	yes	yes	yes	yes		[   5  -3   1  -1  -1   1                     ⟩	2080/2079		Sagittal

7 tinas
7.18	143:1715n       1	13	50	0.18	yes	yes	yes	yes	yes	yes	yes		[   2   1  -1  -3   1   1                     ⟩	1716/1715		George
6.78	5:1729n         11	19	44	0.22	yes		yes			yes	yes		[  -9  11   1  -1   0  -1   0  -1             ⟩	885735/885248		Cmloegcmluin
6.82	11:119n         11	17	35	0.18	yes		yes		yes	yes	yes	1	[  14 -11   0   1  -1   0   1                 ⟩	1949696/1948617		Cmloegcmluin
7.10	5:2527n         12	19	50	0.10	yes		yes	yes	yes	yes	yes		[ -28  12  -1   1   0   0   0   2             ⟩	1342951407/1342177280	Cmloegcmluin
7.13	1729n           3	19	39	0.13	yes		yes		yes	yes	yes		[  -6  -3   0   1   0   1   0   1             ⟩	1729/1728		Ash9903b4
7.24	7:425n          5	17	34	0.24	yes		yes			yes	yes	2	[  -2   5  -2   1   0   0  -1                 ⟩	1701/1700		Ash9903b4

8 tinas
7.95	420175n         8	7	45	0.05	yes	yes	yes	yes	yes	yes	yes		[  -6  -8   2   5                             ⟩	420175/419904		George
7.82	5:253n          4	23	39	0.18	yes	yes	yes	yes		yes	yes	3	[ -12   4  -1   0   1   0   0   0   1         ⟩	20493/20480		Cmloegcmluin
7.98	13:77n          11	13	31	0.02	yes		yes	yes	yes	yes	yes	2	[ -20  11   0   1   1  -1                     ⟩	13640319/13631488	Cmloegcmluin
8.00	19:385n         4	19	42	0.00	yes	yes	yes		yes	yes	yes	1	[   2  -4   1   1   1   0   0  -1             ⟩	1540/1539		Ash9903b4
8.08	175:187n        7	17	45	0.08	yes	yes	yes			yes	yes	1	[  11  -7  -2  -1   1   0   1                 ⟩	382976/382725		Ash9903b4
8.10	95:169n         2	19	50	0.10		yes	yes			yes	yes		[  -4   2  -1   0   0   2   0  -1             ⟩	1521/1520		Ash9903b4
8.14	343:1625n       3	13	49	0.14		yes	yes			yes	yes		[   7  -3  -3   3   0  -1                     ⟩	43904/43875		Ash9903b4

9 tinas
8.93	539n            5	11	25	0.07	yes	yes	yes	yes	yes	yes	yes		[  17  -5   0  -2  -1                         ⟩	131072/130977		George
8.77	125:2401n       9	7	43	0.23			yes	yes		yes	yes		[ -10   9   3  -4                             ⟩	2460375/2458624		Cmloegcmluin
8.83	25:1183n        11	13	43	0.17			yes						[  23 -11   2  -1   0  -2                     ⟩	209715200/209564901	Cmloegcmluin
8.90	19:2401n        12	19	47	0.10			yes						[ -26  12   0   4   0   0   0  -1             ⟩	1275989841/1275068416	Cmloegcmluin
9.21	41503n          4	11	43	0.21		yes	yes	yes		yes	yes		[  -9  -4   0   3   2                         ⟩	41503/41472		Ash9903b4
8.97	125:833n        9	17	46	0.03			yes		yes	yes	yes		[ -17   9  -3   2   0   0   1                 ⟩	16395939/16384000	Ash9903b4

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Re: Magrathean diacritics

Post by Dave Keenan »

cmloegcmluin wrote: Wed May 27, 2020 1:03 am I didn't feel dismissed! I just assumed that what I'd said had flung open the gates to the abyss and, in exposing you to the horrific enormity of the perfectionism possible in this effort, you'd recoiled.
This deliciously-florid prose cracked me up. Mainly because it is so true, but I hadn't realised it myself. Thanks. :lol:
I wrote my post in haste. I hadn't really understood @Ash9903b4 's suggestion. And while it's brilliant and obvious to you now, I admit I still don't get it. Nor do I get how it connects with my concerns.
When I imagine stacking a Magrathean/Insane layer on top of the existing Olympian/Extreme, I imagine that the Olympian primary commas may be assigned to Magrathean symbols that deviate by ±1 tina from their Olympian symbols. i.e. a comma that has an Olympian symbol with a 1 mina accent might have a Magrathean symbol with either a 2, 3 or 4 tina accent, depending on whether the mina accent was representing a schismina that was small, near-equal or large in relation to 455n.

You have already worked out what non-455n schisminas the mina accents are currently representing in Olympian. So it would make sense to have the most common of these be exactly notated as 2 or 4 tinas where their size allows.

The same applies to 2 minas in relation to 5, 6 or 7 tinas. And maybe even zero minas with -1, 0 or 1 tina.
What I was thinking was more like: should we looking at, like, a 37-limit tonality diamond and trying to cross off the simpler pitches we can't exactly notate yet? Or something like that? I know there is a link, of course, between picking relatively simple commas and receiving generally common combinations with existing commas, but why not just shoot straight for them? At least, if we're choosing between two close candidates, this should be weighed in the mix.
I believe that is what we have been doing. A prime-limited tonality diamond is only one kind of scale. We are instead favouring ratios with low SoPF>3 as a way of estimating how likely a ratio is to need notating. We leave the primes 2 and 3 out of that popularity metric so that we are selecting for simplicity in the ratios-to-be-notated rather than in the commas themselves. We care about the comma's 3-exponent for a different reason, namely that the required nominals plus sharps or flats should not be too distant from those of the 1/1.

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Re: Magrathean diacritics

Post by cmloegcmluin »

Dave Keenan wrote: Wed May 27, 2020 10:34 am When I imagine stacking a Magrathean/Insane layer on top of the existing Olympian/Extreme, I imagine that the Olympian primary commas may be assigned to Magrathean symbols that deviate by ±1 tina from their Olympian symbols. i.e. a comma that has an Olympian symbol with a 1 mina accent might have a Magrathean symbol with either a 2, 3 or 4 tina accent, depending on whether the mina accent was representing a schismina that was small, near-equal or large in relation to 455n.
Ah ok, maybe I get it...?

So consider :`::)|: . Its mina represents 3136/3135 which is 0.552, about a tina higher than a mina. So you're saying that in the Insane precision level, the comma :`::)|: represents, the 49/55s, would get assigned to :@4::)|: (maybe one day that'll turn into a Magrathean smilie for 4-tinas!)? But wouldn't that mean that we'd have to change the comma for :`::)|: to something else?

This feels kind of ass-backwards. Surely the other precision levels weren't developed this way. I thought lower levels, once locked in, were locked in. Unless the Extreme precision layer is different because it's the first layer where we don't really care enough about the commas we found for it and don't mind if they get booted to a later (higher) precision level. Or am I still not getting this?

I just thought that we were stuck with :`::)|: being the 49/55s, and even if that was more like 4 tinas than 3 tinas off from :)|: 's 19s, that's how it was, and we'd still be using it as our basis from which we'd deviate by tinas up and down.
You have already worked out what non-455n schisminas the mina accents are currently representing in Olympian. So it would make sense to have the most common of these be exactly notated as 2 or 4 tinas where their size allows.

The same applies to 2 minas in relation to 5, 6 or 7 tinas. And maybe even zero minas with -1, 0 or 1 tina.
Back on the consistent 37 limit thread, you asked for me to "add commas that correspond to zero minas (preferably without requiring horizontal scroll bars)" to my table of attested values for minas. I can't figure out what you would mean by this.
What I was thinking was more like: should we looking at, like, a 37-limit tonality diamond and trying to cross off the simpler pitches we can't exactly notate yet? Or something like that? I know there is a link, of course, between picking relatively simple commas and receiving generally common combinations with existing commas, but why not just shoot straight for them? At least, if we're choosing between two close candidates, this should be weighed in the mix.
I believe that is what we have been doing. A prime-limited tonality diamond is only one kind of scale. We are instead favouring ratios with low SoPF>3 as a way of estimating how likely a ratio is to need notating. We leave the primes 2 and 3 out of that popularity metric so that we are selecting for simplicity in the ratios-to-be-notated rather than in the commas themselves. We care about the comma's 3-exponent for a different reason, namely that the required nominals plus sharps or flats should not be too distant from those of the 1/1.
This makes sense to me. We could say that SoPF is the closest thing we have to primacy here. Tonality diamonds, objective-ish as they are, are still kinda arbitrary.

I guess I was just worried that somehow just haphazardly shooting toward simplicity metrics might lead to suboptimal redundancies.

Or maybe "suboptimal" is basically the call of Chthulu in this context. Turn away!

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Re: Magrathean diacritics

Post by cmloegcmluin »

I added one more column to my previous chart: for the count of occurrences of the comma being a value the minas actually represent (the difference between the primary comma of the mina-accented symbol and the un-mina-accented symbol).

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

cmloegcmluin wrote: Wed May 27, 2020 5:51 am Half-tina it is after all, then? Now with more reason?
Thanks @Ash9903b4 and @cmloegcmluin.

Unfortunately, a good EDO doesn't necessarily mean a good EDA. The two happen to be almost indistinguishable in the case of 8539edo and 809-EDA. But that is not the case for 16808edo and 1592-EDA. 1592-EDA is more like 16804edo and is nothing special.

I dusted off my old EDA search spreadsheet (attached) and pushed it out to 3000-EDA. It's somewhat quirky since it only looks at errors in the commas commonly represented by Sagittal elements (flags and accents).

Between 809-EDA and 3000-EDA, it suggests the following possibilities:
1067
1105
1914 (nearest to 1/2 tina)
1915
2227
2348 (nearest to 1/3 tina)
2613
2626
2627
2723

At the end of that spreadsheet I came across something I'd completely forgotten about. You've heard of the mina and the tina. Meet the "neutrina":
One degree of 30723-EDA, approximately 1/38th of a tina, discovered by Gene Ward Smith ca 2007. Relative errors less than 25.3% of a step, out to some large SoPF limit. Sorry I don't have the details. Mathematically interesting, but not relevant to our quest here.
Attachments
EDA errors for sagittal elements.xlsx
(1.51 MiB) Downloaded 47 times

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