143ed11
| ← 142ed11 | 143ed11 | 144ed11 → |
143 equal divisions of the 11th harmonic (abbreviated 143ed11) is a nonoctave tuning system that divides the interval of 11/1 into 143 equal parts of about 29 ¢ each. Each step represents a frequency ratio of 111/143, or the 143rd root of 11.
Theory
Why 143ed11 matters
143ed11 is extremely strong in the 5.7.8.9.11.13.17.23 subgroup.
All ratios in the 5.7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error.
According to the finite Euler product with sigma = 1, the maxima in the 5.7.8.9.11.13.17.23 subgroup is about 41.33601-ED2 (or 29.03038 ¢).
The Tenney–Euclidean regular temperement in the 5.7.8.9.11.13.17.23 subgroup mapped with [⟨96 116 124 131 143 153 169 187]] gives 41.33650-ED2 (or 29.03005 ¢). The finite Euler product with sigma = 1/2 gives about the same result.
143ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 5.7.8.9.11.13.17.23 in this area.
It serves as a slightly stretched version of 96ed5.
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Todo: cleanup |
Intervals and approximation to JI
| 143ed11 degree |
Cents | Ratios in the 5.7.8.9.11.13.17.23 subgroup |
Error (abs, ¢) | Error (rel, %) |
|---|---|---|---|---|
| 5 | 145.2 | 25/23 | 0.798 | 2.7 |
| 7 | 203.2 | 9/8 | -0.699 | -2.4 |
| 8 | 232.2 | 8/7 | 1.067 | 3.7 |
| 10 | 290.3 | 13/11 | 1.092 | 3.8 |
| 12 | 348.4 | 11/9 | 0.954 | 3.3 |
| 15 | 435.5 | 9/7 | 0.369 | 1.3 |
| 16 | 464.5 | 17/13 | 0.055 | 0.2 |
| 18 | 522.5 | 23/17 | -0.775 | -2.7 |
| 19 | 551.6 | 11/8 | 0.256 | 0.9 |
| 20 | 580.6 | 7/5 | -1.908 | -6.6 |
| 22 | 638.7 | 13/9 | 2.047 | 7.1 |
| 23 | 667.7 | 25/17 | 0.022 | 0.1 |
| 26 | 754.8 | 17/11 | 1.148 | 4.0 |
| 27 | 783.8 | 11/7 | 1.323 | 4.6 |
| 28 | 812.8 | 8/5 | -0.841 | -2.9 |
| 29 | 841.9 | 13/8 | 1.348 | 4.6 |
| 34 | 987.0 | 23/13 | -0.720 | -2.5 |
| 35 | 1016.1 | 9/5 | -1.539 | -5.3 |
| 37 | 1074.1 | 13/7 | 2.415 | 8.3 |
| 38 | 1103.1 | 17/9 | 2.102 | 7.2 |
| 39 | 1132.2 | 25/13 | 0.078 | 0.3 |
| 44 | 1277.3 | 23/11 | 0.372 | 1.3 |
| 45 | 1306.4 | 17/8 | 1.403 | 4.8 |
| 47 | 1364.4 | 11/5 | -0.585 | -2.0 |
| 49 | 1422.5 | 25/11 | 1.170 | 4.0 |
| 53 | 1538.6 | 17/7 | 2.471 | 8.5 |
| 56 | 1625.7 | 23/9 | 1.327 | 4.6 |
| 57 | 1654.7 | 13/5 | 0.507 | 1.7 |
| 61 | 1770.8 | 25/9 | 2.124 | 7.3 |
| 63 | 1828.9 | 23/8 | 0.628 | 2.2 |
| 68 | 1974.1 | 25/8 | 1.426 | 4.9 |
| 71 | 2061.1 | 23/7 | 1.695 | 5.8 |
| 73 | 2119.2 | 17/5 | 0.563 | 1.9 |
| 76 | 2206.3 | 25/7 | 2.493 | 8.6 |
| 91 | 2641.7 | 23/5 | -0.213 | -0.7 |
| 96 | 2786.9 | 5/1 | 0.585 | 2.0 |
| 116 | 3367.5 | 7/1 | -1.323 | -4.6 |
| 124 | 3599.7 | 8/1 | -0.256 | -0.9 |
| 131 | 3803.0 | 9/1 | -0.954 | -3.3 |
| 143 | 4151.3 | 11/1 | 0.000 | 0.0 |
| 153 | 4441.6 | 13/1 | 1.092 | 3.8 |
| 169 | 4906.1 | 17/1 | 1.148 | 4.0 |
| 187 | 5428.6 | 23/1 | 0.372 | 1.3 |
| 192 | 5573.8 | 25/1 | 1.170 | 4.0 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.8 | +14.0 | +9.5 | +0.6 | +4.3 | -1.3 | -0.3 | -1.0 | -9.2 | +0.0 | -5.5 | +1.1 |
| Relative (%) | -33.6 | +48.4 | +32.7 | +2.0 | +14.7 | -4.6 | -0.9 | -3.3 | -31.6 | +0.0 | -18.9 | +3.8 | |
| Steps (reduced) |
41 (41) |
66 (66) |
83 (83) |
96 (96) |
107 (107) |
116 (116) |
124 (124) |
131 (131) |
137 (137) |
143 (0) |
148 (5) |
153 (10) | |
| Harmonic | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.1 | -14.4 | -10.0 | +1.1 | -10.7 | +11.8 | +10.1 | +12.7 | -9.8 | +0.4 | +13.8 | +1.2 |
| Relative (%) | -38.2 | -49.6 | -34.5 | +4.0 | -36.9 | +40.7 | +34.8 | +43.8 | -33.6 | +1.3 | +47.5 | +4.0 | |
| Steps (reduced) |
157 (14) |
161 (18) |
165 (22) |
169 (26) |
172 (29) |
176 (33) |
179 (36) |
182 (39) |
184 (41) |
187 (44) |
190 (47) |
192 (49) | |
Commas of the 5.7.8.9.11.13.17.23 subgroup tempered out in 143ed11
Commas with numerator < 100000 :
392/391, 441/440, 576/575, 729/728, 833/832, 847/845, 936/935, 1001/1000, 1089/1088, 1127/1125, 1225/1224, 1288/1287, 1377/1375, 1449/1445, 1496/1495, 1575/1573, 1771/1768, 1863/1859, 2024/2023, 2025/2023, 2025/2024, 2200/2197, 2205/2197, 2277/2275, 2401/2392, 2601/2600, 2695/2691, 2875/2873, 2880/2873, 3128/3125, 3136/3125, 3185/3179, 3520/3519, 3584/3575, 3773/3757, 4096/4095, 4232/4225, 4235/4232, 4459/4455, 4761/4760, 4928/4913, 5103/5083, 5544/5525, 5632/5625, 5635/5632, 5824/5819, 5831/5819, 5832/5819, 5832/5831, 6561/6545, 6655/6647, 6656/6647, 6656/6655, 6664/6655, 6877/6875, 6885/6877, 7371/7360, 7497/7475, 7616/7605, 7744/7735, 7889/7865, 8019/8000, 8281/8280, 9000/8993, 9009/8993, 9317/9315, 9801/9775, 9801/9800, 10304/10285, 10647/10625, 10648/10625, 10648/10647, 11016/10985, 11016/11011, 11583/11560, 11781/11776, 11979/11960, 12005/11968, 12005/11979, 12167/12155, 12168/12167, 12376/12375, 12397/12393, 13041/13000, 13013/13005, 13689/13685, 13720/13689, 13923/13915, 14175/14144, 14161/14157, 14399/14375, 14400/14399, 14641/14625, 14651/14641, 14875/14872, 14904/14875, 16807/16731, 16807/16767, 17595/17576, 17920/17901, 18515/18496, 18515/18513, 19136/19125, 19208/19125, 20480/20449, 20493/20480, 21609/21505, 21879/21875, 21952/21879, 22295/22264, 22680/22627, 23040/23023, 24219/24200, 24696/24565, 25047/25000, 25025/25024, 25088/25047, 25515/25432, 25921/25857, 25921/25920, 26411/26325, 27209/27200, 27225/27209, 27783/27625, 28611/28561, 28672/28561, 28672/28611, 30613/30600, 30625/30613, 31185/31096, 31752/31603, 31752/31625, 32768/32725, 32805/32768, 33327/33275, 33327/33280, 33957/33800, 33856/33813, 33880/33813, 33957/33856, 34391/34375, 34391/34385, 34496/34385, 34496/34425, 34983/34969, 35000/34969, 35000/34983, 35640/35581, 36179/36125, 36288/36125, 36288/36179, 39325/39304, 39375/39304, 39445/39304, 40131/40000, 40733/40625, 40817/40625, 40824/40625, 41400/41327, 41472/41327, 41503/41400, 41472/41405, 41503/41472, 42840/42757, 42875/42757, 42875/42849, 43659/43520, 44275/44217, 45056/44965, 45927/45760, 46529/46475, 46648/46475, 46656/46475, 46656/46529, 46592/46575, 46648/46575, 46656/46585, 47320/47311, 47385/47311, 47432/47385, 49000/48841, 49005/48841, 49049/48875, 49049/48960, 50575/50531, 50625/50531, 50688/50531, 50688/50575, 51597/51425, 52488/52325, 53248/53125, 53361/53125, 53235/53176, 53361/53176, 53361/53248, 54080/54043, 55223/54925, 55223/55000, 55125/55016, 55223/55080, 56000/55913, 57024/56875, 57967/57800, 59049/58880, 60025/59823, 60025/59904, 60928/60835, 60984/60835, 61952/61893, 61965/61952, 62951/62920, 64000/63869, 64009/63869, 64152/63869, 64009/64000, 64141/64000, 65219/65000, 65219/65025, 66248/66125, 66339/66125, 67392/67375, 69632/69575, 71001/70720, 72128/71825, 72171/71825, 72072/71875, 72171/71875, 72171/71944, 72171/72128, 73125/73117, 73304/73125, 75712/75625, 75803/75625, 75816/75625, 75735/75712, 78336/78125, 78408/78125, 83655/83521, 83720/83521, 83835/83521, 83853/83521, 84035/83521, 83835/83776, 85169/85000, 85293/85000, 85184/85169, 85293/85169, 85293/85184, 86625/86411, 86751/86515, 86625/86528, 86751/86528, 86779/86528, 88209/87880, 88088/87975, 89600/89401, 91125/91091, 92664/92575, 94208/93925, 94325/94208, 95823/95744, 95832/95795, 97461/97336, 97405/97344, 98923/98865, 99127/98865, 99176/98865, 99225/98923, 99127/99000