editing disabled

5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of 5edo = 480¢) to 3\8 (three degrees of 8edo = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this:
generator
tetrachord
g in cents
2g
3g
4g
Comments
2\5




1 0 1
480.000
960.000
240.00
720.000

21\53




10 1 10
475.472
950.943
226.415
701.887
Vulture/Buzzard is around here
19\48




9 1 9
475
950
225
700

17\43




8 1 8
474.419
948.837
223.256
697.674

15\38




7 1 7
473.684
947.368
221.053
694.737

13\33




6 1 6
472.727
945.455
218.181
690.909

11\28




5 1 5
471.429
942.857
214.286
685.714

9\23




4 1 4
469.565
939.130
208.696
678.261
L/s = 4





pi 1 pi
467.171
934.3425
201.514
668.685
L/s = pi
7\18




3 1 3
466.667
933.333
200.000
666.667
L/s = 3





e 1 e
465.535
931.069
196.604
662.139
L/s = e

19\49



8 3 8
465.306
930.612
195.918
661.2245



50\129


21 8 21
465.116
930.233
195.349
660.465




131\338

55 21 55
465.089
930.1775
195.266
660.335





212\547
89 34 89
465.082
930.1645
195.247
660.329




81\209

34 13 34
465.072
930.1435
195.215
660.287



31\80


13 5 13
465
930
195
660


12\31



5 2 5
464.516
929.032
193.549
658.065

5\13




2 1 2
461.538
923.077
184.615
646.154






√3 1 √3
459.417
918.8345
178.252
637.669


13\34



5 3 5
458.824
917.647
176.471
635.294



34\89


13 8 13
458.427
916.854
175.281
633.708




89\233

34 21 34
458.369
916.738
175.107
633.473





233\610
89 55 89
458.361
916.721
175.082
633.443
Golden father



144\377

55 34 55
458.355
916.711
175.066
633.422



55\144


21 13 21
458.333
916.666
175
633.333


21\55



8 5 8
458.182
916.364
174.545
632.727






pi 2 pi
457.883
915.777
173.665
631.553

8\21




3 2 3
457.143
914.286
171.429
628.571
Optimum rank range (L/s=3/2) father
11\29




4 3 4
455.172
910.345
165.517
620.690

14\37




5 4 5
454.054
908.108
162.162
616.216

17\45




6 5 6
453.333
906.667
160
613.333

20\53




7 6 7
452.83
905.66
158.491
611.321

23\61




8 7 8
452.459
904.918
157.377
609.836

26\69




9 8 9
452.174
904.348
156.522
608.696

29\77




10 9 10
451.948
903.896
155.844
607.792

3\8




1 1 1
450.000
900.000
150.000
600.000

The only notable harmonic entropy minimum is Vulture/Buzzard, in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.

By a weird coincidence, the other generator for this MOS will generate the same pattern within a tritave equivalence. By yet another weird coincidence, this MOS belongs to a temperament which has Bohlen-Pierce as its index-2 subtemperament. In addition to being harmonious, this tuning of the MOS gives an L/s ratio between 3/1 and 3/2, which is squarely in the middle of the range, being thus neither too exaggerated nor too equalized to be recognizable as such, unlike in octaves, where the only notable harmonic entropy minimum is near a greatly exaggerated 10/1 L/s ratio.

generator
tetrachord
g in cents
2g
3g
4g
Comments
2\5




1 0 1
760.782
1521.564
380.391
1141.173

27\68




13 1 13
755.188
1510.376
363.609
1118.797






~6626 515 6626
755.132
1510.265
363.442
1118.574
2g=12/5 minus quarter comma
25\63




12 1 12
754.744
1509.488
362.277
1117.021

23\58




11 1 11
754.2235
1508.447
360.716
1114.939

21\53




10 1 10
753.605
1507.21
358.859
1112.464

19\48




9 1 9
752.857
1505.714
356.617
1109.474

17\43




8 1 8
751.936
1503.871
353.852
1105.788

15\38




7 1 7
750.771
1501.543
350.36
1101.132


28/71



13 2 13
750.067
1500.1335
348.245
1098.312


41\104



19 3 19
749.809
1466.618
347.4725
1097.282
3g=11/3 near here
13\33




6 1 6
749.255
1498.51
345.81
1095.065


24\61



11 2 11
748.31
1496.62
342.976
1091.286


35\89



16 3 16
747.96
1495.92
341.924
1089.884


46\117



21 4 21
747.777
1495.554
341.377
1089.154


57\145



26 5 26
747.665
1495.33
341.04
1088.705






5+√29 2 5+√29
747.648
1495.297
340.99
1088.638


68\173



31 6 31
747.589
1495.178
340.813
1088.402



147\374


67 13 67
747.56
1495.12
340.725
1088.285
4g=45/8 near here

79\201



36 7 36
747.535
1495.069
340.649
1088.183

11\28




5 1 5
747.197
1494.393
339.635
1086.831


20\51



9 2 9
745.865
1491.729
335.639
1081.50


29\74



13 3 13
745.361
1490.721
334.127
1079.488


38/97



17 4 17
745.096
1490.192
333.332
1078.428






2+√5 1 2+√5
754.051
1490.101
333.197
1078.247


47\120



21 5 21
744.932
1489.865
332.842
1077.7745

9\23




4 1 4
744.243
1488.487
330.775
1075.018
L/s = 4

43\110



19 5 19
743.4915
1486.983
328.5195
1072.011



77\197


34 9 34
743.404
1486.807
328.256
1071.66
4g=39/7 near here

34\87



15 4 15
743.293
1486.586
327.923
1071.216


25\64



11 3 11
742.951
1485.902
326.899
1069.85


16\41



7 2 7
742.226
1484.453
324.724
1066.95


23\59



10 3 10
741.44
1482.88
322.365
1063.805






3+√13 2 3+√13
741.289
1482.577
321.911
1063.20


30\77



13 4 13
741.021
1482.043
321.109
1062.131






pi 1 pi
740.449
1480.898
319.392
1056.841
L/s = pi
7\18




3 1 3
739.649
1479.298
316.992
1056.642
L/s = 3

89\229



38 13 38
739.188
1478.376
315.608
1054.796
3g=18/5 near here

82\211



35 12 35
739.148
1478.297
315.49
1054.639


75\193



32 11 32
739.102
1478.203
315.35
1054.452


68\175



29 10 29
739.045
1478.091
315.181
1054.227


61/157



26 9 26
738.976
1477.952
314.973
1053.949


54\139



23 8 23
738.889
1477.778
314.712
1053.601


47\121



20 7 20
738.776
1477.552
314.373
1053.149


40\103



17 6 17
738.623
1477.247
313.915
1052.538


33\85



14 5 14
738.406
1476.812
313.263
1051.669


26\67



11 4 11
738.072
1476.144
312.261
1050.333






e 1 e
737.855
1478.71
311.61
1049.465
L/s = e

19\49



8 3 8
737.493
1474.986
310.523
1048.016




164\423

69 26 69
737.401
1474.802
310.248
1047.649
3g=18/5 minus quarter comma near here



145\374

61 23 61
737.389
1474.778
310.212
1047.601




126\325

53 20 53
737.373
1474.747
310.165
1047.538




107\276

45 17 45
737.352
1474.704
310.101
1047.453




88\227

37 14 37
737.322
1474.644
310.01
1047.332




69\178

29 11 29
737.275
1474.549
309.869
1047.144



50\129


21 8 21
737.192
1474.384
309.621
1046.812




131\338

55 21 55
737.148
1474.296
309.49
1046.638





212\547
89 34 89
737.138
1474.276
309.459
1046.597




81\209

34 13 34
737.121
1474.243
309.409
1046.53



31\80


13 5 13
737.008
1474.015
309.068
1046.075


12\31



5 2 5
736.241
1472.481
306.767
1043.007






1+√2 1 1+√2
735.542
1471.084
304.6715
1040.214
Silver false father

17\44



7 3 7
734.846
1469.693
302.584
1037.41


22\57



9 4 9
734.088
1468.176
300.309
1034.397


27\70



11 5 11
733.611
1467.222
298.879
1032.49



59\153


24 11 24
733.434
1466.867
298.346
1031.779


32\83



13 6 13
733.284
1466.568
297.897
1031.181
2g=7/3 near here
5\13




2 1 2
731.521
1463.042
292.609
1024.13


53\138



21 11 21
730.461
1460.922
289.428
1018.889



101\263


40 21 40
730.409
1460.817
289.271
1019.679
3g=39/11 near here

48\125



19 10 19
730.35
1460.701
289.097
1019.448


43\112



17 9 17
730.215
1460.43
288.69
1018.905


38\99



15 8 15
730.043
1460.087
288.175
1018.218



71\185


28 15 28
729.9395
1459.879
287.8635
1017.803



104\271


41 22 41
729.902
1459.803
287.75
1017.651
4g=27/5 near here

33\86



13 7 13
729.82
1459.64
287.505
1017.325


28\73



11 6 11
729.547
1459.034
286.596
1016.113


23\60



9 5 9
729.083
1458.1655
285.293
1014.376



41\107


16 9 16
728.7865
1457.573
284.4045
1013.191
3g=99/28 near here


59\154


23 13 23
728.671
1457.342
284.058
1012.729



77\201


30 17 30
728.61
1457.219
283.874
1012.483



95\248


37 21 37
728.5715
1457.143
283.7595
1012.331
Golden BP is index-2 near here

18\47



7 4 7
728.408
1456.817
283.27
1011.678






√3 1 √3
728.159
1456.318
282.522
1010.6815



49\128


19 11 19
728.092
1456.184
282.321
1010.413
4g=27/5 minus third comma near here


31\81


12 7 12
727.909
1455.817
281.771
1009.68


13\34



5 3 5
727.218
1454.436
279.699
1006.917



34\89


13 8 13
726.59
1453.179
277.814
1004.403




89\233

34 21 34
726.498
1452.996
277.538
1004.036





233\610
89 55 89
726.4845
1452.969
277.4985
1003.983
Golden false father



144\377

55 34 55
726.476
1452.952
277.473
1003.95



55\144


21 13 21
726.441
1452.882
277.368
1003.809


21\55



8 5 8
726.201
1452.402
276.468
1002.849






pi 2 pi
725.736
1451.472
275.252
1000.988

8\21




3 2 3
724.554
1449.109
271.708
996.226
Optimum rank range (L/s=3/2) false father





~543 361 543
724.511
1449.022
271.579
996.09
4g=16/3

27\71



10 7 10
723.279
1446.557
267.881
991.16



46\121


17 12 17
723.057
1446.115
267.217
990.274



65\171


24 17 24
722.965
1445.931
266.941
989.907
3g=7/2 near here

19\50



7 5 7
722.743
1445.486
266.274
989.017

11\29




4 3 4
721.431
1442.862
262.338
983.77


25\66



9 7 9
720.4375
1440.875
259.3575
979.795



64\169


23 18 23
720.267
1440.534
258.848
979.113




167\441

60 47 60
720.2415
1440.483
258.7965
979.001





437\1154
157 123 157
720.238
1440.475
258.758
978.996




270\713

97 76 97
720.235
1440.471
258.751
978.987



103\272


37 29 37
720.226
1440.451
258.722
978.947


39\103



14 11 14
720.158
1440.315
258.518
978.676

14\37




5 4 5
719.659
1439.317
257.021
976.679


31\82



11 9 11
719.032
1438.064
255.14
974.172



79\209


28 23 28
718.921
1437.842
254.807
973.728




206\545

73 60 73
718.904
1437.808
254.757
973.661





539\1426
191 117 191
718.902
1437.803
254.75
973.652




333\881

118 97 118
718.90
1437.80
254.745
973.6455



127\336


45 37 45
718.893
1437.787
254.726
973.619


48\127



17 14 17
718.849
1437.698
254.592
973.441

17\45




6 5 6
718.516
1437.032
253.549
972.11

20\53




7 6 7
717.719
1435.438
251.202
968.9205






~401 344 401
717.695
1435.3905
251.131
968.826
4g=21/4
23\61




8 7 8
717.131
1434.261
249.437
966.567






~6682 5875 6682
716.9925
1433.985
249.0225
966.015
6g=12
26\69




9 8 9
716.679
1433.357
248.081
964.76

29\77




10 9 10
716.321
1432.641
247.007
963.328

32\85




11 10 11
716.03
1432.06
246.135
962.1655

35\93




12 11 12
715.7895
1431.759
245.4135
961.203

38/101




13 12 13
715.587
1431.174
244.806
960.393
2g=16\7 near here
3\8




1 1 1
713.233
1426.466
237.744
950.9775

.