User:Moremajorthanmajor/16edX
This is the scale which occurs as the dominant major edX.
Intervals
| Degrees | Enneatonic | Minor mode Middletown temperament | Neutral mode Middletown temperament | Major mode Middletown temperament | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Aeolian-Dorian | Dorian | Dorian-Mixolydian | Mixolydian | Mixolydian-Ionian | |||||||||||||||||||
| Subpental | Pental | Superpental | Subpental | Pental | Superpental | ||||||||||||||||||
| Soft | Intense | ||||||||||||||||||||||
| Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | ||||
| 1 | 1#/2b | F#/Gb | 90 | 91.667 | 93.75 | 95.4545 | 96.429 | 97.5 | 98.4375 | 100 | 100.862 | 103.448 | 105 | ||||||||||
| 2 | 2 | G | 180 | 183.333 | 187.5 | 190.909 | 192.857 | 195 | 196.875 | 200 | 201.724 | 206.897 | 210 | ||||||||||
| 3 | 2#/3b | G#/Jb | G#/Ab | 270 | 275 | 281.25 | 286.364 | 289.286 | 292.5 | 295.3125 | 300 | 302.586 | 310.345 | 315 | |||||||||
| 4 | 3 | J | A | 360 | 366.667 | 375 | 381.818 | 385.714 | 390 | 393.75 | 400 | 403.448 | 413.793 | 420 | |||||||||
| 5 | 3#/4b | J#/Ab | A#/Bb | 450 | 458.333 | 468.75 | 477.273 | 482.143 | 487.5 | 492.1875 | 500 | 504.31 | 517.241 | 525 | |||||||||
| 6 | 4 | A | B | 540 | 550 | 562.5 | 572.727 | 578.571 | 585 | 590.625 | 600 | 605.172 | 620.69 | 630 | |||||||||
| 7 | 5 | B | H | 630 | 641.667 | 656.25 | 668.182 | 675 | 682.5 | 689.0625 | 700 | 706.0345 | 724.138 | 735 | |||||||||
| 8 | 5#/6b | B#/Hb | H#/Cb | 720 | 733.333 | 750 | 763.636 | 771.429 | 780 | 787.5 | 800 | 806.897 | 827.586 | 840 | |||||||||
| 9 | 6 | H | C | 810 | 825 | 843.75 | 859.091 | 867.857 | 877.5 | 885.9375 | 900 | 907.759 | 931.0345 | 945 | |||||||||
| 10 | 6#/7b | H#/Cb | C#/Db | 900 | 916.667 | 937.5 | 954.5455 | 964.286 | 975 | 984.375 | 1000 | 1008.621 | 1034.483 | 1050 | |||||||||
| 11 | 7 | C | D | 990 | 1018.333 | 1031.25 | 1050 | 1060.714 | 1072.5 | 1082.8125 | 1100 | 1109.483 | 1137.931 | 1155 | |||||||||
| 12 | 7#/8b | C#/Db | D#/Sb | 1080 | 1100 | 1125 | 1145.4545 | 1157.143 | 1170 | 1181.25 | 1200 | 1210.345 | 1241.379 | 1260 | |||||||||
| 13 | 8 | D | S | 1170 | 1191.667 | 1218.75 | 1240.909 | 1253.571 | 1267.5 | 1279.6875 | 1300 | 1311.207 | 1344.828 | 1365 | |||||||||
| 14 | 8#/9b | D#/Eb | S#/Eb | 1260 | 1283.333 | 1312.5 | 1336.364 | 1350 | 1365 | 1378.125 | 1400 | 1412.069 | 1448.276 | 1470 | |||||||||
| 15 | 9 | E | 1350 | 1375 | 1406.25 | 1431.818 | 1446.429 | 1462.5 | 1476.5625 | 1500 | 1512.931 | 1551.724 | 1575 | ||||||||||
| 16 | 1 | F | 1440 | 1466.667 | 1500 | 1527.273 | 1542.857 | 1560 | 1575 | 1600 | 1613.793 | 1655.172 | 1680 | ||||||||||
Most harmonic tunings
| Degrees | Enneatonic | Minor mode Middletown temperament | Neutral mode Middletown temperament | Major mode Middletown temperament | ||
|---|---|---|---|---|---|---|
| 7/3±13¢ | 22/9±13¢ | 5/2±13¢ | ||||
| 1 | 1#/2b | F#/Gb | 90.867-92.492 | 95.9005-97.5255 | 98.332-99.957 | |
| 2 | 2 | G | 181.734-184.984 | 191.801-195.051 | 196.664-199.914 | |
| 3 | 2#/3b | G#/Jb | G#/Ab | 272.601-277.476 | 287.7015-292.5765 | 294.996-299.871 |
| 4 | 3 | J | A | 363.468-369.968 | 383.602-390.102 | 393.328-399.828 |
| 5 | 3#/4b | J#/Ab | A#/Bb | 454.335-462.46 | 479.5025-487.6275 | 491.6605-499.7855 |
| 6 | 4 | A | B | 545.202-554.952 | 575.403-585.153 | 589.993-599.743 |
| 7 | 5 | B | H | 636.0685-647.4435 | 671.3035-382.6785 | 688.325-699.7 |
| 8 | 5#/6b | B#/Hb | H#/Cb | 726.9355-739.9355 | 767.204-780.204 | 786.657-799.657 |
| 9 | 6 | H | C | 817.802-832.427 | 863.1045-877.7295 | 884.989-899.614 |
| 10 | 6#/7b | H#/Cb | C#/Db | 908.669-924.919 | 959.005-975.255 | 983.321-999.571 |
| 11 | 7 | C | D | 999.536-1017.411 | 1054.9055-1072.7205 | 1081.653-1099.528 |
| 12 | 7#/8b | C#/Db | D#/Sb | 1090.403-1109.903 | 1150.806-1170.306 | 1179.985-1199.485 |
| 13 | 8 | D | S | 1181.27-1202.395 | 1246.7065-1267.8315 | 1278.317-1299.442 |
| 14 | 8#/9b | D#/Eb | S#/Eb | 1272.137-1294.887 | 1342.607-1365.357 | 1376.6495-1399.3995 |
| 15 | 9 | E | 1363.004-1387.378 | 1438.507-14661.882 | 1474.982-1499.366 | |
| 16 | 1 | F | 1453.871-1479.871 | 1534.408-1560.408 | 1573.314-1599.314 | |
Golden tunings
Minor mode Middletown temperament
| Degrees | Enneatonic | Aeolian-Dorian | Dorian | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| -13¢ | Mean | -13¢ (s5D) | Subpental | Pental | Superpental | ||||||||||
| +13¢ (A-D) | Mean | +13¢ | -13¢ | -13¢ (S5) | Mean | Mean | +13¢ (5) | -13¢ (D-M) | |||||||
| 1 | 1#/2b | F#/Gb | 90.428 | 91.241 | 91.303 | 92.053 | 92.116 | 92.928 | 94.329 | 94.917 | 95.142 | 95.7295 | 95.943 | 96.369 | |
| 2 | 2 | G | 180.856 | 182.481 | 182.606 | 184.106 | 184.231 | 185.856 | 188.2585 | 189.834 | 190.2835 | 191.459 | 191.9085 | 192.728 | |
| 3 | 2#/3b | G#/Jb | G#/Ab | 271.2845 | 273.722 | 273.909 | 276.159 | 276.347 | 278.784 | 282.988 | 284.751 | 285.425 | 287.1885 | 287.863 | 289.092 |
| 4 | 3 | J | A | 361.712 | 364.962 | 365.212 | 368.213 | 368.462 | 371.712 | 377.317 | 379.668 | 380.567 | 382.918 | 383.817 | 385.456 |
| 5 | 3#/4b | J#/Ab | A#/Bb | 452.141 | 456.203 | 456.5155 | 460.266 | 460.578 | 464.6405 | 471.646 | 474.585. | 475.709 | 478.6475 | 479.771 | 481.82 |
| 6 | 4 | A | B | 542.569 | 547.444 | 547.819 | 552.319 | 552.694 | 557.569 | 565.974 | 569.502 | 570.851 | 574.377 | 575.726 | 578.184 |
| 7 | 5 | B | H | 632.997 | 638.685 | 639.122 | 644.372 | 644.809 | 650.497 | 660.305 | 664.419 | 665.992 | 670.106 | 671.68 | 674.548 |
| 8 | 5#/6b | B#/Hb | H#/Cb | 723.425 | 729.925 | 730.425 | 736.425 | 736.925 | 743.425 | 754.634 | 759.336 | 761.134 | 765.836 | 767.634 | 770.912 |
| 9 | 6 | H | C | 813.854 | 821.166 | 821.728 | 828.479 | 829.04 | 836.353 | 848.963 | 854.253 | 856.276 | 861.565 | 863.588 | 867.2755 |
| 10 | 6#/7b | H#/Cb | C#/Db | 904.282 | 912.407 | 913.031 | 920.532 | 921.156 | 920.281 | 943.293 | 949.17 | 951.418 | 957.295 | 959.543 | 963.6935 |
| 11 | 7 | C | D | 994.71 | 1003.6475 | 1004.334 | 1012.585 | 1013.272 | 1022.209 | 1037.622 | 1044.087 | 1046.559 | 1053.024 | 1055.497 | 1060.0035 |
| 12 | 7#/8b | C#/Db | D#/Sb | 1085.138 | 1094.888 | 1095.637 | 1104.638 | 1105.387 | 1115.137 | 1131.951 | 1139.004 | 1141.701 | 1148.753 | 1151.451 | 1156.367 |
| 13 | 8 | D | S | 1175.566 | 1186.129 | 1186.94 | 1106.691 | 1197.502 | 1208.065 | 1226.38 | 1233.921 | 1236.843 | 1244.483 | 1247.405 | 1252.731 |
| 14 | 8#/9b | D#/Eb | S#/Eb | 1265.9945 | 1277.3965 | 1278.2435 | 1288.7445 | 1289.6185 | 1300.9935 | 1320.61 | 1328.838 | 1331.985 | 1340.213 | 1343.36 | 1349.095 |
| 15 | 9 | E | 1356.423 | 1368.61 | 1369.547 | 1380.798 | 1381.734 | 1393.922 | 141.939 | 1423.755 | 1427.126 | 1435.942 | 1439.314 | 1445.459 | |
| 16 | 1 | F | 1446.851 | 1459.851 | 1460.85 | 1472.851 | 1473.85 | 1486.85 | 1509.268 | 1518.672 | 1522.268 | 1531.672 | 1535.268 | 1541.823 | |
Neutral (Dorian-Mixolydian) mode Middletown temperament
| Degrees | Enneatonic | +13¢ (S5D) | Mean | -13¢ (s5M) | ||
|---|---|---|---|---|---|---|
| 1 | 1#/2b | F#/Gb | 96.542 | 97.176 | 97.364 | |
| 2 | 2 | G | 193.084 | 194.353 | 194.7275 | |
| 3 | 2#/3b | G#/Jb | G#/Ab | 289.626 | 291.529 | 292.091 |
| 4 | 3 | J | A | 386.168 | 388.706 | 389.455 |
| 5 | 3#/4b | J#/Ab | A#/Bb | 482.71 | 485.882 | 486.819 |
| 6 | 4 | A | B | 579.252 | 580.235 | 584.183 |
| 7 | 5 | B | H | 675.794 | 725.584 | 681.546 |
| 8 | 5#/6b | B#/Hb | H#/Cb | 771.336 | 777.412 | 779.41 |
| 9 | 6 | H | C | 868.878 | 874.588 | 876.274 |
| 10 | 6#/7b | H#/Cb | C#/Db | 965.42 | 971.7645 | 973.638 |
| 11 | 7 | C | D | 1061.962 | 1068.941 | 1071.0015 |
| 12 | 7#/8b | C#/Db | D#/Sb | 1158.503 | 1166.117 | 1168.365 |
| 13 | 8 | D | S | 1255.046 | 1263.293 | 1265.729 |
| 14 | 8#/9b | D#/Eb | S#/Eb | 1351.588 | 1360.47 | 1363.093 |
| 15 | 9 | E | 1448.13 | 1457.645 | 1460.457 | |
| 16 | 1 | F | 1544.672 | 1554.823 | 1557.82 | |
Major mode Middletown temperament
| Degrees | Enneatonic | Mixolydian | Mixolydian-Ionian | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Subpental | Pental | Superpental | Mean | +13¢ | ||||||||||||||
| -13¢ (5) | +13¢ (D-M) | Mean | Mean | +13¢ (s5) | +13¢ | -13¢ (Soft s5) | Soft | Intense | ||||||||||
| +13¢ (Soft s5) | -13¢ | Mean | -13¢ (M-I) | +13¢ | ||||||||||||||
| 1 | 1#/2b | F#/Gb | 97.787 | 97.989 | 98.176 | 98.6 | 98.989 | 99.412 | 99.998 | 100.81 | 101.623 | 101.648 | 102.46 | 102.785 | 103.273 | 103.598 | 104.41 | |
| 2 | 2 | G | 195.5745 | 195.978 | 196.3525 | 197.1995 | 197.9775 | 198.8245 | 199.9955 | 201.6205 | 203.2455 | 203.296 | 204.921 | 205.57 | 206.546 | 207.195 | 208.82 | |
| 3 | 2#/3b | G#/Jb | G#/Ab | 293.362 | 293.967 | 294.529 | 295.799 | 296.966 | 298.237 | 299.993 | 302.431 | 304.868 | 304.944 | 307.381 | 308.356 | 309.819 | 310.793 | 313.231 |
| 4 | 3 | J | A | 391.149 | 391.956 | 392.705 | 394.399 | 395.955 | 397.649 | 399.991 | 403.241 | 406.491 | 406.592 | 409.842 | 411.141 | 413.092 | 414.391 | 417.641 |
| 5 | 3#/4b | J#/Ab | A#/Bb | 488.936 | 489.945 | 490.881 | 492.999 | 494.944 | 497.061 | 499.989 | 504.051 | 508.114 | 508.24 | 512.302 | 513.926 | 516.365 | 517.9885 | 522.051 |
| 6 | 4 | A | B | 586.724 | 587.934 | 589.058 | 591.599 | 593.933 | 596.474 | 599.987 | 599.987 | 609.737 | 609.888 | 614.763 | 616.711 | 619.638 | 621.586 | 626.461 |
| 7 | 5 | B | H | 684.511 | 685.923 | 687.234 | 690.198 | 692.921 | 695.886 | 699.984 | 705.672 | 711.359 | 711.5355 | 717.223 | 719.496 | 722.9105 | 725.184 | 730.871 |
| 8 | 5#/6b | B#/Hb | H#/Cb | 782.298 | 783.912 | 785.41 | 788.798 | 791.91 | 795.298 | 799.982 | 799.982 | 812.982 | 813.1835 | 819.6835 | 822.282 | 826.1835 | 828.781 | 835.282 |
| 9 | 6 | H | C | 880.085 | 881.9005 | 883.5865 | 887.398 | 890.899 | 894.71 | 899.98 | 907.292 | 914.605 | 914.831 | 922.144 | 925.067 | 929.456 | 932.379 | 939.692 |
| 10 | 6#/7b | H#/Cb | C#/Db | 977.872 | 979.8895 | 981.763 | 985.998 | 989.888 | 994.123 | 999.978 | 1008.103 | 1016.228 | 1016.479 | 1024.604 | 1027.852 | 1032.729 | 1035.977 | 1044.102 |
| 11 | 7 | C | D | 1075.66 | 1087.8785 | 1079.939 | 1084.597 | 1088.8765 | 1093.535 | 1099.975 | 1108.913 | 1117.85 | 1118.127 | 1127.065 | 1130.637 | 1136.002 | 1139.575 | 1148.512 |
| 12 | 7#/8b | C#/Db | D#/Sb | 1173.447 | 1175.867 | 1178.115 | 1183.197 | 1189.865 | 1192.947 | 1199.973 | 1209.723 | 1219.473 | 1219.775 | 1229.525 | 1233.422 | 1239.275 | 1243.172 | 1252.922 |
| 13 | 8 | D | S | 1271.234 | 1273.856 | 1276.292 | 1281.797 | 1286.854 | 1292.359 | 1299.971 | 1310.533 | 1321.096 | 1321.423 | 1331.986 | 1336.208 | 1342.548 | 1346.77 | 1357.333 |
| 14 | 8#/9b | D#/Eb | S#/Eb | 1369.022 | 1371.845 | 1374.468 | 1380.397 | 1385.843 | 1391.772 | 1399.968 | 1411.344 | 1422.719 | 1423.071 | 1434.446 | 1438.993 | 1445.821 | 1450.368 | 1461.743 |
| 15 | 9 | E | 1455.809 | 1469.834 | 1472.644 | 1478.996 | 1484.832 | 1491.184 | 1499.9665 | 1512.154 | 1524.3415 | 1524.719 | 1536.9065 | 1541.778 | 1549.094 | 1553.9655 | 1566.153 | |
| 16 | 1 | F | 1564.596 | 1567.96 | 1570.82 | 1577.596 | 1583.82 | 1590.596 | 1599.964 | 1612.964 | 1625.964 | 1626.366 | 1639.366 | 1644.563 | 1652.366 | 1657.563 | 1670.563 | |
By a surprising coincidence, neutral 16edX turns out to be a false Pelogic temperament with a very pure 5:4 (mistuned by no more than 3.596 cents), and tuning 5:3 pure creates a tenth almost exactly equal to 38/29edo or almost exactly the Golden Subpental Mixolydian step or every ninth step of 110edo, and tuning 4:3 pure creates an almost exact 43/74edo fifth. Also, the Golden Soft Superpental Mixolydian step is almost exactly (25/24)^(10/7) and tuning 12:7 pure creates almost exactly the Golden Mixolydian-Ionian step.