114edt: Difference between revisions
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== Theory == | == Theory == | ||
114edt is related to [[72edo]], but with the 3/1 rather than the 2/1 being just | 114edt is related to [[72edo]], but with the [[3/1|perfect twelfth]] rather than the [[2/1|octave]] being just. The octave is stretched by about 1.23 cents. Like 72edo, 114edt is [[consistent]] to the [[integer limit|18-integer-limit]]. While its approximations to 2, [[7/1|7]] and [[11/1|11]] are sharp, it significantly improves on 72edo's approximation to [[13/1|13]]. | ||
=== Harmonics === | === Harmonics === | ||
Revision as of 19:52, 10 April 2025
| ← 113edt | 114edt | 115edt → |
114 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 114edt or 114ed3), is a nonoctave tuning system that divides the interval of 3/1 into 114 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of 31/114, or the 114th root of 3.
Theory
114edt is related to 72edo, but with the perfect twelfth rather than the octave being just. The octave is stretched by about 1.23 cents. Like 72edo, 114edt is consistent to the 18-integer-limit. While its approximations to 2, 7 and 11 are sharp, it significantly improves on 72edo's approximation to 13.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.23 | +0.00 | +2.47 | -0.12 | +1.23 | +1.30 | +3.70 | +0.00 | +1.12 | +2.95 | +2.47 |
| Relative (%) | +7.4 | +0.0 | +14.8 | -0.7 | +7.4 | +7.8 | +22.2 | +0.0 | +6.7 | +17.7 | +14.8 | |
| Steps (reduced) |
72 (72) |
114 (0) |
144 (30) |
167 (53) |
186 (72) |
202 (88) |
216 (102) |
228 (0) |
239 (11) |
249 (21) |
258 (30) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | +2.54 | -0.12 | +4.94 | +0.09 | +1.23 | +7.73 | +2.35 | +1.30 | +4.19 | -6.03 | +3.70 |
| Relative (%) | -15.8 | +15.2 | -0.7 | +29.6 | +0.5 | +7.4 | +46.4 | +14.1 | +7.8 | +25.1 | -36.2 | +22.2 | |
| Steps (reduced) |
266 (38) |
274 (46) |
281 (53) |
288 (60) |
294 (66) |
300 (72) |
306 (78) |
311 (83) |
316 (88) |
321 (93) |
325 (97) |
330 (102) | |
Subsets and supersets
Since 114 factors into primes as 2 × 3 × 19, 114edt contains subset edts 2, 3, 6, 19, 38, and 57.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 16.7 | 11.4 | |
| 2 | 33.4 | 22.8 | |
| 3 | 50.1 | 34.2 | 34/33, 35/34, 36/35 |
| 4 | 66.7 | 45.6 | 26/25, 27/26 |
| 5 | 83.4 | 57 | 21/20, 43/41 |
| 6 | 100.1 | 68.4 | 18/17, 35/33 |
| 7 | 116.8 | 79.8 | 31/29, 46/43 |
| 8 | 133.5 | 91.2 | 27/25, 40/37 |
| 9 | 150.2 | 102.6 | 12/11 |
| 10 | 166.8 | 114 | 11/10 |
| 11 | 183.5 | 125.4 | 10/9 |
| 12 | 200.2 | 136.8 | 46/41 |
| 13 | 216.9 | 148.2 | 17/15 |
| 14 | 233.6 | 159.6 | |
| 15 | 250.3 | 171.1 | 37/32 |
| 16 | 266.9 | 182.5 | 7/6 |
| 17 | 283.6 | 193.9 | 33/28 |
| 18 | 300.3 | 205.3 | 25/21, 44/37 |
| 19 | 317 | 216.7 | 6/5 |
| 20 | 333.7 | 228.1 | 40/33 |
| 21 | 350.4 | 239.5 | |
| 22 | 367 | 250.9 | 21/17, 47/38 |
| 23 | 383.7 | 262.3 | |
| 24 | 400.4 | 273.7 | 29/23, 34/27 |
| 25 | 417.1 | 285.1 | 14/11 |
| 26 | 433.8 | 296.5 | 9/7 |
| 27 | 450.5 | 307.9 | 35/27, 48/37 |
| 28 | 467.1 | 319.3 | |
| 29 | 483.8 | 330.7 | 37/28, 41/31, 45/34 |
| 30 | 500.5 | 342.1 | |
| 31 | 517.2 | 353.5 | 31/23 |
| 32 | 533.9 | 364.9 | 34/25 |
| 33 | 550.6 | 376.3 | 11/8 |
| 34 | 567.2 | 387.7 | 25/18, 43/31 |
| 35 | 583.9 | 399.1 | 7/5 |
| 36 | 600.6 | 410.5 | 41/29 |
| 37 | 617.3 | 421.9 | 10/7 |
| 38 | 634 | 433.3 | |
| 39 | 650.7 | 444.7 | 16/11 |
| 40 | 667.4 | 456.1 | 25/17, 47/32 |
| 41 | 684 | 467.5 | 43/29, 46/31 |
| 42 | 700.7 | 478.9 | 3/2 |
| 43 | 717.4 | 490.4 | |
| 44 | 734.1 | 501.8 | 26/17 |
| 45 | 750.8 | 513.2 | 37/24 |
| 46 | 767.5 | 524.6 | |
| 47 | 784.1 | 536 | 11/7 |
| 48 | 800.8 | 547.4 | 27/17, 46/29 |
| 49 | 817.5 | 558.8 | |
| 50 | 834.2 | 570.2 | 34/21 |
| 51 | 850.9 | 581.6 | 18/11 |
| 52 | 867.6 | 593 | 33/20 |
| 53 | 884.2 | 604.4 | 5/3 |
| 54 | 900.9 | 615.8 | 32/19, 37/22 |
| 55 | 917.6 | 627.2 | 17/10 |
| 56 | 934.3 | 638.6 | 12/7 |
| 57 | 951 | 650 | 26/15, 45/26 |
| 58 | 967.7 | 661.4 | 7/4 |
| 59 | 984.3 | 672.8 | 30/17 |
| 60 | 1001 | 684.2 | 41/23 |
| 61 | 1017.7 | 695.6 | 9/5 |
| 62 | 1034.4 | 707 | 20/11 |
| 63 | 1051.1 | 718.4 | 11/6 |
| 64 | 1067.8 | 729.8 | |
| 65 | 1084.4 | 741.2 | 43/23 |
| 66 | 1101.1 | 752.6 | 17/9 |
| 67 | 1117.8 | 764 | 21/11 |
| 68 | 1134.5 | 775.4 | |
| 69 | 1151.2 | 786.8 | 35/18 |
| 70 | 1167.9 | 798.2 | |
| 71 | 1184.6 | 809.6 | |
| 72 | 1201.2 | 821.1 | 2/1 |
| 73 | 1217.9 | 832.5 | |
| 74 | 1234.6 | 843.9 | |
| 75 | 1251.3 | 855.3 | 33/16, 35/17 |
| 76 | 1268 | 866.7 | |
| 77 | 1284.7 | 878.1 | 21/10 |
| 78 | 1301.3 | 889.5 | |
| 79 | 1318 | 900.9 | 15/7 |
| 80 | 1334.7 | 912.3 | |
| 81 | 1351.4 | 923.7 | 24/11 |
| 82 | 1368.1 | 935.1 | |
| 83 | 1384.8 | 946.5 | |
| 84 | 1401.4 | 957.9 | |
| 85 | 1418.1 | 969.3 | 34/15 |
| 86 | 1434.8 | 980.7 | |
| 87 | 1451.5 | 992.1 | 37/16 |
| 88 | 1468.2 | 1003.5 | 7/3 |
| 89 | 1484.9 | 1014.9 | 33/14 |
| 90 | 1501.5 | 1026.3 | |
| 91 | 1518.2 | 1037.7 | |
| 92 | 1534.9 | 1049.1 | 17/7 |
| 93 | 1551.6 | 1060.5 | |
| 94 | 1568.3 | 1071.9 | 47/19 |
| 95 | 1585 | 1083.3 | 5/2 |
| 96 | 1601.6 | 1094.7 | |
| 97 | 1618.3 | 1106.1 | 28/11 |
| 98 | 1635 | 1117.5 | 18/7 |
| 99 | 1651.7 | 1128.9 | |
| 100 | 1668.4 | 1140.4 | |
| 101 | 1685.1 | 1151.8 | 45/17 |
| 102 | 1701.7 | 1163.2 | |
| 103 | 1718.4 | 1174.6 | 27/10 |
| 104 | 1735.1 | 1186 | 30/11 |
| 105 | 1751.8 | 1197.4 | 11/4 |
| 106 | 1768.5 | 1208.8 | 25/9 |
| 107 | 1785.2 | 1220.2 | |
| 108 | 1801.9 | 1231.6 | 17/6 |
| 109 | 1818.5 | 1243 | 20/7 |
| 110 | 1835.2 | 1254.4 | 26/9 |
| 111 | 1851.9 | 1265.8 | 35/12 |
| 112 | 1868.6 | 1277.2 | |
| 113 | 1885.3 | 1288.6 | |
| 114 | 1902 | 1300 | 3/1 |