5L 3s

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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

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