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, the [[5/1|5]] and [[17/1|17]] are nearly pure, and the [[13/1|13]] is significantly improved compared to 72edo, although the [[19/1|19]] becomes much worse. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|114|3|1|intervals=integer|columns=11}} | {{Harmonics in equal|114|3|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}} | {{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}} | ||
=== Subsets and supersets === | |||
Since 114 factors into primes as {{nowrap| 2 × 3 × 19 }}, 114edt contains subset edts {{EDs|equave=t| 2, 3, 6, 19, 38, and 57 }}. | |||
== Intervals == | == Intervals == | ||
| Line 15: | Line 18: | ||
* [[72edo]] – relative edo | * [[72edo]] – relative edo | ||
* [[186ed6]] – relative ed6 | * [[186ed6]] – relative ed6 | ||
* [[258ed12]] – relative ed12 | |||
Latest revision as of 11:32, 22 May 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, the 5 and 17 are nearly pure, and the 13 is significantly improved compared to 72edo, although the 19 becomes much worse.
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 |