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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 92 factors into primes as 2<sup>2</sup> × 23, 92edt contains subset edts {{EDs|equave=t| 2, 4, 23, and 46 }}. | Since 92 factors into primes as {{nowrap| 2<sup>2</sup> × 23 }}, 92edt contains subset edts {{EDs|equave=t| 2, 4, 23, and 46 }}. | ||
== Intervals == | == Intervals == | ||
Revision as of 16:24, 18 March 2025
| ← 91edt | 92edt | 93edt → |
92 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 92edt or 92ed3), is a nonoctave tuning system that divides the interval of 3/1 into 92 equal parts of about 20.7 ¢ each. Each step represents a frequency ratio of 31/92, or the 92nd root of 3.
Theory
92edt is related to 58edo, but with the 3/1 rather than the 2/1 being just. The octave is about 0.9414 cents compressed. Like 58edo, 92edt is consistent to the 18-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.94 | +0.00 | -1.88 | +4.60 | -0.94 | +0.94 | -2.82 | +0.00 | +3.66 | +4.04 | -1.88 |
| Relative (%) | -4.6 | +0.0 | -9.1 | +22.2 | -4.6 | +4.6 | -13.7 | +0.0 | +17.7 | +19.5 | -9.1 | |
| Steps (reduced) |
58 (58) |
92 (0) |
116 (24) |
135 (43) |
150 (58) |
163 (71) |
174 (82) |
184 (0) |
193 (9) |
201 (17) |
208 (24) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.26 | +0.00 | +4.60 | -3.77 | -5.35 | -0.94 | +8.82 | +2.72 | +0.94 | +3.10 | +8.84 | -2.82 |
| Relative (%) | +20.6 | +0.0 | +22.2 | -18.2 | -25.9 | -4.6 | +42.7 | +13.1 | +4.6 | +15.0 | +42.7 | -13.7 | |
| Steps (reduced) |
215 (31) |
221 (37) |
227 (43) |
232 (48) |
237 (53) |
242 (58) |
247 (63) |
251 (67) |
255 (71) |
259 (75) |
263 (79) |
266 (82) | |
Subsets and supersets
Since 92 factors into primes as 22 × 23, 92edt contains subset edts 2, 4, 23, and 46.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 20.7 | 14.1 | |
| 2 | 41.3 | 28.3 | 40/39, 41/40, 42/41, 43/42 |
| 3 | 62 | 42.4 | 28/27, 29/28 |
| 4 | 82.7 | 56.5 | 21/20, 22/21, 43/41 |
| 5 | 103.4 | 70.7 | 17/16, 35/33 |
| 6 | 124 | 84.8 | 29/27, 43/40 |
| 7 | 144.7 | 98.9 | 25/23, 37/34, 38/35 |
| 8 | 165.4 | 113 | 11/10 |
| 9 | 186.1 | 127.2 | 39/35 |
| 10 | 206.7 | 141.3 | |
| 11 | 227.4 | 155.4 | 41/36 |
| 12 | 248.1 | 169.6 | 15/13 |
| 13 | 268.8 | 183.7 | 7/6 |
| 14 | 289.4 | 197.8 | 13/11 |
| 15 | 310.1 | 212 | 43/36 |
| 16 | 330.8 | 226.1 | 23/19, 40/33 |
| 17 | 351.4 | 240.2 | 38/31 |
| 18 | 372.1 | 254.3 | 26/21, 31/25, 36/29 |
| 19 | 392.8 | 268.5 | |
| 20 | 413.5 | 282.6 | 33/26 |
| 21 | 434.1 | 296.7 | 9/7 |
| 22 | 454.8 | 310.9 | 13/10 |
| 23 | 475.5 | 325 | 25/19 |
| 24 | 496.2 | 339.1 | 4/3 |
| 25 | 516.8 | 353.3 | 27/20, 31/23, 35/26 |
| 26 | 537.5 | 367.4 | 15/11 |
| 27 | 558.2 | 381.5 | 29/21, 40/29 |
| 28 | 578.9 | 395.7 | |
| 29 | 599.5 | 409.8 | 24/17, 41/29 |
| 30 | 620.2 | 423.9 | 10/7 |
| 31 | 640.9 | 438 | 29/20, 42/29 |
| 32 | 661.5 | 452.2 | 22/15, 41/28 |
| 33 | 682.2 | 466.3 | 40/27, 43/29 |
| 34 | 702.9 | 480.4 | 3/2 |
| 35 | 723.6 | 494.6 | 38/25, 41/27 |
| 36 | 744.2 | 508.7 | 20/13, 43/28 |
| 37 | 764.9 | 522.8 | 14/9 |
| 38 | 785.6 | 537 | |
| 39 | 806.3 | 551.1 | 35/22, 43/27 |
| 40 | 826.9 | 565.2 | 29/18 |
| 41 | 847.6 | 579.3 | 31/19 |
| 42 | 868.3 | 593.5 | 33/20, 38/23, 43/26 |
| 43 | 889 | 607.6 | |
| 44 | 909.6 | 621.7 | 22/13 |
| 45 | 930.3 | 635.9 | |
| 46 | 951 | 650 | 26/15 |
| 47 | 971.7 | 664.1 | |
| 48 | 992.3 | 678.3 | 39/22 |
| 49 | 1013 | 692.4 | |
| 50 | 1033.7 | 706.5 | 20/11 |
| 51 | 1054.3 | 720.7 | |
| 52 | 1075 | 734.8 | 41/22 |
| 53 | 1095.7 | 748.9 | 32/17 |
| 54 | 1116.4 | 763 | 40/21 |
| 55 | 1137 | 777.2 | 27/14 |
| 56 | 1157.7 | 791.3 | 39/20, 41/21, 43/22 |
| 57 | 1178.4 | 805.4 | |
| 58 | 1199.1 | 819.6 | 2/1 |
| 59 | 1219.7 | 833.7 | |
| 60 | 1240.4 | 847.8 | 41/20, 43/21 |
| 61 | 1261.1 | 862 | 29/14 |
| 62 | 1281.8 | 876.1 | 21/10 |
| 63 | 1302.4 | 890.2 | 17/8 |
| 64 | 1323.1 | 904.3 | 43/20 |
| 65 | 1343.8 | 918.5 | 37/17 |
| 66 | 1364.4 | 932.6 | 11/5 |
| 67 | 1385.1 | 946.7 | 20/9 |
| 68 | 1405.8 | 960.9 | 9/4 |
| 69 | 1426.5 | 975 | 41/18 |
| 70 | 1447.1 | 989.1 | 30/13 |
| 71 | 1467.8 | 1003.3 | 7/3 |
| 72 | 1488.5 | 1017.4 | 26/11 |
| 73 | 1509.2 | 1031.5 | 43/18 |
| 74 | 1529.8 | 1045.7 | 29/12 |
| 75 | 1550.5 | 1059.8 | |
| 76 | 1571.2 | 1073.9 | |
| 77 | 1591.9 | 1088 | |
| 78 | 1612.5 | 1102.2 | 33/13 |
| 79 | 1633.2 | 1116.3 | 18/7 |
| 80 | 1653.9 | 1130.4 | 13/5 |
| 81 | 1674.5 | 1144.6 | |
| 82 | 1695.2 | 1158.7 | |
| 83 | 1715.9 | 1172.8 | 35/13 |
| 84 | 1736.6 | 1187 | 30/11 |
| 85 | 1757.2 | 1201.1 | |
| 86 | 1777.9 | 1215.2 | |
| 87 | 1798.6 | 1229.3 | |
| 88 | 1819.3 | 1243.5 | 20/7 |
| 89 | 1839.9 | 1257.6 | |
| 90 | 1860.6 | 1271.7 | 41/14 |
| 91 | 1881.3 | 1285.9 | |
| 92 | 1902 | 1300 | 3/1 |