124ed8
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| ← 123ed8 | 124ed8 | 125ed8 → |
124 equal divisions of the 8th harmonic (abbreviated 124ed8) is a nonoctave tuning system that divides the interval of 8/1 into 124 equal parts of about 29 ¢ each. Each step represents a frequency ratio of 81/124, or the 124th root of 8.
Theory
124ed8 is an alternative version of 143ed11, which is a strong 8.9.5.7.11.13.17.23 subgroup equal step tuning. 124ed8 is for people who absolutely need a locked pure 8/1 when using some software or hardware. As a trade-off, 124ed8 has a slightly sharp tendency above the optimal tuning for the subgroup. Comparared to 143ed11, 124ed8 stretches the 11th harmonic by about 0.295 ¢ and the mapping of the 8th harmonic (now pure) by about 0.256 ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.7 | +14.2 | +9.7 | +0.8 | +4.5 | -1.1 | +0.0 | -0.7 | -8.9 | +0.3 | -5.2 |
| Relative (%) | -33.3 | +48.8 | +33.3 | +2.7 | +15.5 | -3.7 | +0.0 | -2.4 | -30.6 | +1.0 | -17.8 | |
| Steps (reduced) |
41 (41) |
66 (66) |
83 (83) |
96 (96) |
107 (107) |
116 (116) |
124 (0) |
131 (7) |
137 (13) |
143 (19) |
148 (24) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.4 | -10.8 | -14.1 | -9.7 | +1.5 | -10.4 | +12.2 | +10.5 | +13.1 | -9.4 | +0.8 |
| Relative (%) | +4.8 | -37.1 | -48.5 | -33.3 | +5.2 | -35.7 | +41.9 | +36.0 | +45.1 | -32.3 | +2.6 | |
| Steps (reduced) |
153 (29) |
157 (33) |
161 (37) |
165 (41) |
169 (45) |
172 (48) |
176 (52) |
179 (55) |
182 (58) |
184 (60) |
187 (63) | |
Intervals and approximation to JI
| 124ed8 degree |
Cents | Ratios in the 8.9.5.7.11.13.17.23 subgroup |
Error (abs, ¢) | Error (rel, %) |
|---|---|---|---|---|
| 5 | 145.2 | 25/23 | 0.808 | 2.8 |
| 7 | 203.2 | 9/8 | -0.684 | -2.4 |
| 8 | 232.3 | 8/7 | 1.084 | 3.7 |
| 10 | 290.3 | 13/11 | 1.113 | 3.8 |
| 12 | 348.4 | 11/9 | 0.979 | 3.4 |
| 15 | 435.5 | 9/7 | 0.4 | 1.4 |
| 16 | 464.5 | 17/13 | 0.088 | 0.3 |
| 18 | 522.6 | 23/17 | -0.738 | -2.5 |
| 19 | 551.6 | 11/8 | 0.295 | 1.0 |
| 20 | 580.6 | 7/5 | -1.867 | -6.4 |
| 22 | 638.7 | 13/9 | 2.092 | 7.2 |
| 23 | 667.7 | 25/17 | 0.07 | 0.2 |
| 26 | 754.8 | 17/11 | 1.201 | 4.1 |
| 27 | 783.9 | 11/7 | 1.379 | 4.7 |
| 28 | 812.9 | 8/5 | -0.783 | -2.7 |
| 29 | 841.9 | 13/8 | 1.408 | 4.8 |
| 34 | 987.1 | 23/13 | -0.65 | -2.2 |
| 35 | 1016.1 | 9/5 | -1.467 | -5.1 |
| 37 | 1074.2 | 13/7 | 2.492 | 8.6 |
| 38 | 1103.2 | 17/9 | 2.18 | 7.5 |
| 39 | 1132.3 | 25/13 | 0.158 | 0.5 |
| 44 | 1277.4 | 23/11 | 0.463 | 1.6 |
| 45 | 1306.5 | 17/8 | 1.496 | 5.2 |
| 47 | 1364.5 | 11/5 | -0.488 | -1.7 |
| 49 | 1422.6 | 25/11 | 1.271 | 4.4 |
| 53 | 1538.7 | 17/7 | 2.58 | 8.9 |
| 56 | 1625.8 | 23/9 | 1.442 | 5.0 |
| 57 | 1654.8 | 13/5 | 0.625 | 2.2 |
| 61 | 1771.0 | 25/9 | 2.25 | 7.8 |
| 63 | 1829.0 | 23/8 | 0.758 | 2.6 |
| 68 | 1974.2 | 25/8 | 1.566 | 5.4 |
| 71 | 2061.3 | 23/7 | 1.842 | 6.3 |
| 73 | 2119.4 | 17/5 | 0.713 | 2.5 |
| 76 | 2206.5 | 25/7 | 2.65 | 9.1 |
| 91 | 2641.9 | 23/5 | -0.025 | -0.1 |
| 96 | 2787.1 | 5/1 | 0.783 | 2.7 |
| 116 | 3367.7 | 7/1 | -1.084 | -3.7 |
| 124 | 3600.0 | 8/1 | 0.0 | 0.0 |
| 131 | 3803.2 | 9/1 | -0.684 | -2.4 |
| 143 | 4151.6 | 11/1 | 0.295 | 1.0 |
| 153 | 4441.9 | 13/1 | 1.408 | 4.8 |
| 169 | 4906.5 | 17/1 | 1.496 | 5.2 |
| 187 | 5429.0 | 23/1 | 0.758 | 2.6 |
| 192 | 5574.2 | 25/1 | 1.566 | 5.4 |