101ed7
| ← 100ed7 | 101ed7 | 102ed7 → |
101 equal divisions of the 7th harmonic (abbreviated 101ed7) is a nonoctave tuning system that divides the interval of 7/1 into 101 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of 71/101, or the 101st root of 7.
Theory
101ed7 is closely related to 36edo (sixth-tone tuning), but with the 7th harmonic rather than the octave being just. The octave is stretched by about 0.770 cents. Like 36edo, 101ed7 is consistent to the 8-integer-limit.
Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 subgroup, with slightly better 3, 7, 13 and 17, and a slightly worse 2 versus 36edo. Using the patent val, the 5 is also less accurate. Overall this means 36edo is still better in the 5-limit, but 101ed7 is better in the 13- and 17-limit, especially when treating it as a dual-5 dual-11 tuning.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -0.7 | +1.5 | +15.5 | +0.0 | +0.0 | +2.3 | -1.5 | +16.3 | -15.3 | +0.8 |
| Relative (%) | +2.3 | -2.2 | +4.6 | +46.4 | +0.1 | +0.0 | +6.9 | -4.4 | +48.7 | -46.0 | +2.4 | |
| Steps (reduced) |
36 (36) |
57 (57) |
72 (72) |
84 (84) |
93 (93) |
101 (0) |
108 (7) |
114 (13) |
120 (19) |
124 (23) |
129 (28) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.4 | +0.8 | +14.7 | +3.1 | -1.8 | -0.7 | +5.8 | -16.3 | -0.7 | -14.6 | +8.5 | +1.6 |
| Relative (%) | -13.0 | +2.3 | +44.2 | +9.2 | -5.4 | -2.1 | +17.3 | -49.0 | -2.2 | -43.7 | +25.6 | +4.7 | |
| Steps (reduced) |
133 (32) |
137 (36) |
141 (40) |
144 (43) |
147 (46) |
150 (49) |
153 (52) |
155 (54) |
158 (57) |
160 (59) |
163 (62) |
165 (64) | |
Subsets and supersets
101ed7 is the 26th prime ed7, so it does not contain any nontrivial subset ed7's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 33.4 | |
| 2 | 66.7 | 27/26 |
| 3 | 100.1 | 18/17 |
| 4 | 133.4 | 41/38 |
| 5 | 166.8 | 43/39 |
| 6 | 200.1 | 37/33 |
| 7 | 233.5 | 8/7 |
| 8 | 266.8 | 7/6 |
| 9 | 300.2 | 44/37 |
| 10 | 333.5 | |
| 11 | 366.9 | 21/17 |
| 12 | 400.3 | 29/23, 34/27 |
| 13 | 433.6 | 9/7 |
| 14 | 467 | 38/29 |
| 15 | 500.3 | 4/3 |
| 16 | 533.7 | |
| 17 | 567 | 43/31 |
| 18 | 600.4 | 41/29 |
| 19 | 633.7 | |
| 20 | 667.1 | |
| 21 | 700.4 | 3/2 |
| 22 | 733.8 | 26/17, 29/19 |
| 23 | 767.2 | 14/9 |
| 24 | 800.5 | 27/17 |
| 25 | 833.9 | 34/21 |
| 26 | 867.2 | 38/23 |
| 27 | 900.6 | 32/19, 37/22 |
| 28 | 933.9 | 12/7 |
| 29 | 967.3 | 7/4 |
| 30 | 1000.6 | 41/23 |
| 31 | 1034 | |
| 32 | 1067.4 | |
| 33 | 1100.7 | 17/9 |
| 34 | 1134.1 | |
| 35 | 1167.4 | |
| 36 | 1200.8 | 2/1 |
| 37 | 1234.1 | |
| 38 | 1267.5 | 27/13 |
| 39 | 1300.8 | 36/17 |
| 40 | 1334.2 | |
| 41 | 1367.5 | |
| 42 | 1400.9 | |
| 43 | 1434.3 | |
| 44 | 1467.6 | 7/3 |
| 45 | 1501 | |
| 46 | 1534.3 | 17/7 |
| 47 | 1567.7 | 42/17 |
| 48 | 1601 | |
| 49 | 1634.4 | 18/7 |
| 50 | 1667.7 | |
| 51 | 1701.1 | |
| 52 | 1734.4 | |
| 53 | 1767.8 | |
| 54 | 1801.2 | 17/6 |
| 55 | 1834.5 | 26/9 |
| 56 | 1867.9 | |
| 57 | 1901.2 | 3/1 |
| 58 | 1934.6 | |
| 59 | 1967.9 | |
| 60 | 2001.3 | |
| 61 | 2034.6 | |
| 62 | 2068 | |
| 63 | 2101.3 | 37/11 |
| 64 | 2134.7 | 24/7 |
| 65 | 2168.1 | 7/2 |
| 66 | 2201.4 | |
| 67 | 2234.8 | |
| 68 | 2268.1 | |
| 69 | 2301.5 | 34/9 |
| 70 | 2334.8 | 27/7 |
| 71 | 2368.2 | |
| 72 | 2401.5 | 4/1 |
| 73 | 2434.9 | |
| 74 | 2468.2 | |
| 75 | 2501.6 | |
| 76 | 2535 | |
| 77 | 2568.3 | |
| 78 | 2601.7 | 9/2 |
| 79 | 2635 | |
| 80 | 2668.4 | 14/3 |
| 81 | 2701.7 | |
| 82 | 2735.1 | 34/7 |
| 83 | 2768.4 | |
| 84 | 2801.8 | |
| 85 | 2835.2 | 36/7 |
| 86 | 2868.5 | 21/4 |
| 87 | 2901.9 | |
| 88 | 2935.2 | |
| 89 | 2968.6 | |
| 90 | 3001.9 | 17/3 |
| 91 | 3035.3 | |
| 92 | 3068.6 | |
| 93 | 3102 | 6/1 |
| 94 | 3135.3 | |
| 95 | 3168.7 | |
| 96 | 3202.1 | |
| 97 | 3235.4 | |
| 98 | 3268.8 | |
| 99 | 3302.1 | |
| 100 | 3335.5 | |
| 101 | 3368.8 | 7/1 |