92edt: Difference between revisions
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== Theory == | == Theory == | ||
92edt is related to [[58edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 0. | 92edt is related to [[58edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 0.941 cents compressed. Like 58edo, 92edt is consistent to the [[integer limit|18-integer-limit]]. The [[prime harmonic]]s [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]], which are tuned sharp in 58edo, remain sharp here, but significantly less so. The [[17/1|17]], which is flat to begin with, becomes worse. | ||
=== Harmonics === | === Harmonics === | ||
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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 == | ||
Latest revision as of 15:18, 12 April 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 0.941 cents compressed. Like 58edo, 92edt is consistent to the 18-integer-limit. The prime harmonics 5, 7, 11, and 13, which are tuned sharp in 58edo, remain sharp here, but significantly less so. The 17, which is flat to begin with, becomes worse.
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 |