98edt: Difference between revisions
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{{Harmonics in equal|98|3|1|intervals=integer|columns=11}} | {{Harmonics in equal|98|3|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|98|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 98edt (continued)}} | {{Harmonics in equal|98|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 98edt (continued)}} | ||
=== Subsets and supersets === | |||
Since 98 factors into primes as {{nowrap| 2 × 7<sup>2</sup> }}, 98edt contains subset edts {{EDs|equave=t| 2, 7, 14, and 49 }}. | |||
== Intervals == | == Intervals == | ||
Latest revision as of 17:36, 30 March 2025
| ← 97edt | 98edt | 99edt → |
98 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 98edt or 98ed3), is a nonoctave tuning system that divides the interval of 3/1 into 98 equal parts of about 19.4 ¢ each. Each step represents a frequency ratio of 31/98, or the 98th root of 3.
Theory
98edt is related to 62edo, but with the twelfth rather than the octave being just. The octave is stretched by about 3.28 cents, same as in 49edt. Unlike 62edo, which is consistent to the 8-integer-limit, 98edt is only consistent to the 7-integer-limit. The prime harmonics 2 to 23 are all tuned sharp, except for 3.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.28 | +0.00 | +6.56 | +8.40 | +3.28 | +8.11 | -9.57 | +0.00 | -7.73 | +1.93 | +6.56 |
| Relative (%) | +16.9 | +0.0 | +33.8 | +43.3 | +16.9 | +41.8 | -49.3 | +0.0 | -39.9 | +9.9 | +33.8 | |
| Steps (reduced) |
62 (62) |
98 (0) |
124 (26) |
144 (46) |
160 (62) |
174 (76) |
185 (87) |
196 (0) |
205 (9) |
214 (18) |
222 (26) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.84 | -8.02 | +8.40 | -6.30 | +5.19 | +3.28 | +6.71 | -4.46 | +8.11 | +5.21 | +5.88 | -9.57 |
| Relative (%) | +19.8 | -41.3 | +43.3 | -32.4 | +26.8 | +16.9 | +34.6 | -23.0 | +41.8 | +26.8 | +30.3 | -49.3 | |
| Steps (reduced) |
229 (33) |
235 (39) |
242 (46) |
247 (51) |
253 (57) |
258 (62) |
263 (67) |
267 (71) |
272 (76) |
276 (80) |
280 (84) |
283 (87) | |
Subsets and supersets
Since 98 factors into primes as 2 × 72, 98edt contains subset edts 2, 7, 14, and 49.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 19.4 | 13.3 | |
| 2 | 38.8 | 26.5 | 43/42, 44/43 |
| 3 | 58.2 | 39.8 | |
| 4 | 77.6 | 53.1 | 23/22 |
| 5 | 97 | 66.3 | 18/17 |
| 6 | 116.4 | 79.6 | 31/29 |
| 7 | 135.9 | 92.9 | |
| 8 | 155.3 | 106.1 | 23/21, 35/32 |
| 9 | 174.7 | 119.4 | 21/19 |
| 10 | 194.1 | 132.7 | 19/17, 28/25 |
| 11 | 213.5 | 145.9 | 26/23, 43/38 |
| 12 | 232.9 | 159.2 | 8/7 |
| 13 | 252.3 | 172.4 | 22/19 |
| 14 | 271.7 | 185.7 | |
| 15 | 291.1 | 199 | 13/11 |
| 16 | 310.5 | 212.2 | |
| 17 | 329.9 | 225.5 | 23/19 |
| 18 | 349.3 | 238.8 | 11/9 |
| 19 | 368.7 | 252 | 26/21 |
| 20 | 388.2 | 265.3 | 5/4 |
| 21 | 407.6 | 278.6 | 19/15, 43/34 |
| 22 | 427 | 291.8 | 32/25 |
| 23 | 446.4 | 305.1 | 22/17 |
| 24 | 465.8 | 318.4 | 17/13 |
| 25 | 485.2 | 331.6 | 41/31 |
| 26 | 504.6 | 344.9 | |
| 27 | 524 | 358.2 | 23/17 |
| 28 | 543.4 | 371.4 | 26/19, 37/27 |
| 29 | 562.8 | 384.7 | 18/13 |
| 30 | 582.2 | 398 | 7/5 |
| 31 | 601.6 | 411.2 | 17/12, 41/29 |
| 32 | 621 | 424.5 | 43/30 |
| 33 | 640.5 | 437.8 | |
| 34 | 659.9 | 451 | |
| 35 | 679.3 | 464.3 | |
| 36 | 698.7 | 477.6 | |
| 37 | 718.1 | 490.8 | |
| 38 | 737.5 | 504.1 | 23/15, 26/17 |
| 39 | 756.9 | 517.3 | |
| 40 | 776.3 | 530.6 | 36/23 |
| 41 | 795.7 | 543.9 | 19/12 |
| 42 | 815.1 | 557.1 | 8/5 |
| 43 | 834.5 | 570.4 | 34/21 |
| 44 | 853.9 | 583.7 | 18/11 |
| 45 | 873.3 | 596.9 | 43/26 |
| 46 | 892.8 | 610.2 | |
| 47 | 912.2 | 623.5 | 22/13, 39/23 |
| 48 | 931.6 | 636.7 | 12/7 |
| 49 | 951 | 650 | 26/15 |
| 50 | 970.4 | 663.3 | 7/4 |
| 51 | 989.8 | 676.5 | 23/13, 39/22 |
| 52 | 1009.2 | 689.8 | 34/19, 43/24 |
| 53 | 1028.6 | 703.1 | 38/21 |
| 54 | 1048 | 716.3 | 11/6 |
| 55 | 1067.4 | 729.6 | |
| 56 | 1086.8 | 742.9 | 15/8 |
| 57 | 1106.2 | 756.1 | 36/19 |
| 58 | 1125.6 | 769.4 | 23/12 |
| 59 | 1145.1 | 782.7 | |
| 60 | 1164.5 | 795.9 | |
| 61 | 1183.9 | 809.2 | |
| 62 | 1203.3 | 822.4 | |
| 63 | 1222.7 | 835.7 | |
| 64 | 1242.1 | 849 | 43/21 |
| 65 | 1261.5 | 862.2 | |
| 66 | 1280.9 | 875.5 | 44/21 |
| 67 | 1300.3 | 888.8 | 36/17 |
| 68 | 1319.7 | 902 | 15/7 |
| 69 | 1339.1 | 915.3 | 13/6 |
| 70 | 1358.5 | 928.6 | |
| 71 | 1377.9 | 941.8 | |
| 72 | 1397.4 | 955.1 | |
| 73 | 1416.8 | 968.4 | 34/15 |
| 74 | 1436.2 | 981.6 | 39/17 |
| 75 | 1455.6 | 994.9 | 44/19 |
| 76 | 1475 | 1008.2 | |
| 77 | 1494.4 | 1021.4 | |
| 78 | 1513.8 | 1034.7 | 12/5 |
| 79 | 1533.2 | 1048 | |
| 80 | 1552.6 | 1061.2 | 27/11 |
| 81 | 1572 | 1074.5 | |
| 82 | 1591.4 | 1087.8 | |
| 83 | 1610.8 | 1101 | 33/13, 38/15 |
| 84 | 1630.2 | 1114.3 | |
| 85 | 1649.7 | 1127.6 | |
| 86 | 1669.1 | 1140.8 | 21/8 |
| 87 | 1688.5 | 1154.1 | |
| 88 | 1707.9 | 1167.3 | |
| 89 | 1727.3 | 1180.6 | 19/7 |
| 90 | 1746.7 | 1193.9 | |
| 91 | 1766.1 | 1207.1 | |
| 92 | 1785.5 | 1220.4 | |
| 93 | 1804.9 | 1233.7 | 17/6 |
| 94 | 1824.3 | 1246.9 | 43/15 |
| 95 | 1843.7 | 1260.2 | |
| 96 | 1863.1 | 1273.5 | 44/15 |
| 97 | 1882.5 | 1286.7 | |
| 98 | 1902 | 1300 | 3/1 |