67ed5
| ← 66ed5 | 67ed5 | 68ed5 → |
Division of the 5th harmonic into 67 equal parts (67ed5) is related to 29edo, but with the 5/1 rather than the 2/1 being just. The octave is about 6.0164 cents stretched and the step size is about 41.5868 cents.
Theory
67ed5 has a generally sharp tendency for harmonics up to 28. Unlike 29edo, it is only consistent up to the 8-integer-limit, with discrepancy for the 9th harmonic. As an equal temperament, it tempers out 49/48 in the 7-limit; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.
Prime harmonics
Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.
The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.0 | +11.0 | +0.0 | -0.3 | +7.4 | +9.3 | +2.3 | +17.7 | +19.6 | -7.4 | +1.9 |
| Relative (%) | +14.5 | +26.5 | +0.0 | -0.7 | +17.7 | +22.3 | +5.5 | +42.5 | +47.1 | -17.9 | +4.5 | |
| Steps (reduced) |
29 (29) |
46 (46) |
67 (0) |
81 (14) |
100 (33) |
107 (40) |
118 (51) |
123 (56) |
131 (64) |
140 (6) |
143 (9) | |
29edo for comparison:
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +1.5 | -13.9 | -17.1 | -13.4 | -12.9 | +19.2 | -7.9 | -7.6 | +4.9 | +13.6 |
| Relative (%) | +0.0 | +3.6 | -33.6 | -41.3 | -32.4 | -31.3 | +46.4 | -19.0 | -18.3 | +11.9 | +32.8 | |
| Steps (reduced) |
29 (0) |
46 (17) |
67 (9) |
81 (23) |
100 (13) |
107 (20) |
119 (3) |
123 (7) |
131 (15) |
141 (25) |
144 (28) | |
67ed5 as a generator
67ed5 can also be thought of as a generator of the 2.3.5.7.11.19 subgroup temperament which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a cluster temperament with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by 29edo, 202edo, and 231edo.
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 41.5868 | ||
| 2 | 83.1735 | 22/21, 21/20 | |
| 3 | 124.7603 | 3375/3136 | |
| 4 | 166.3471 | 11/10 | |
| 5 | 207.9339 | 150/133 | |
| 6 | 249.5206 | 800/693, 231/200 | |
| 7 | 291.1074 | 45/38 | |
| 8 | 332.6942 | 40/33 | |
| 9 | 374.2809 | 4455/3584 | |
| 10 | 415.8677 | 80/63, 14/11 | |
| 11 | 457.4545 | ||
| 12 | 499.0413 | 4/3 | |
| 13 | 540.6280 | ||
| 14 | 582.2148 | 7/5 | |
| 15 | 623.8016 | 1125/784 | |
| 16 | 665.3883 | 22/15, 147/100 | |
| 17 | 706.9751 | 200/133 | pseudo-3/2 |
| 18 | 748.5619 | 77/50 | |
| 19 | 790.1487 | 30/19 | |
| 20 | 831.7354 | 160/99 | |
| 21 | 873.3222 | 63/38 | |
| 22 | 914.9090 | 95/56, 56/33 | |
| 23 | 956.4958 | ||
| 24 | 998.0825 | 16/9, 57/32 | |
| 25 | 1039.6693 | ||
| 26 | 1081.2561 | 28/15 | |
| 27 | 1122.8428 | 375/196 | |
| 28 | 1164.4296 | 49/25 | |
| 29 | 1206.0164 | 800/399, 225/112 | pseudo-octave |
| 30 | 1247.6032 | 154/75 | |
| 31 | 1289.1899 | 40/19 | |
| 32 | 1330.7767 | 640/297 | |
| 33 | 1372.3635 | 495/224, 42/19 | |
| 34 | 1413.9502 | 95/42, 224/99 | |
| 35 | 1455.5370 | 297/128 | |
| 36 | 1497.1238 | 19/8 | |
| 37 | 1538.7106 | 375/154 | |
| 38 | 1580.2973 | 112/45, 399/160 | pseudo-5/2 |
| 39 | 1621.8841 | 125/49 | |
| 40 | 1663.4709 | 196/75 | |
| 41 | 1705.0576 | 75/28 | |
| 42 | 1746.6444 | ||
| 43 | 1788.2312 | 160/57, 45/16 | |
| 44 | 1829.8180 | ||
| 45 | 1871.4047 | 165/56, 56/19 | |
| 46 | 1912.9915 | 190/63 | |
| 47 | 1954.5783 | 99/32 | |
| 48 | 1996.1650 | 19/6 | |
| 49 | 2037.7518 | 250/77 | |
| 50 | 2079.3386 | 133/40 | pseudo-10/3 |
| 51 | 2120.9254 | 500/147, 75/22 | |
| 52 | 2162.5121 | 784/225 | |
| 53 | 2204.0989 | 25/7 | |
| 54 | 2245.6857 | ||
| 55 | 2287.2725 | 15/4 | |
| 56 | 2328.8592 | ||
| 57 | 2370.4460 | 55/14, 63/16 | |
| 58 | 2412.0328 | 3584/891 | |
| 59 | 2453.6195 | 33/8 | |
| 60 | 2495.2063 | 38/9 | |
| 61 | 2536.7931 | 1000/231, 693/160 | |
| 62 | 2578.3799 | 133/30 | |
| 63 | 2619.9666 | 50/11 | |
| 64 | 2661.5534 | 3136/675 | |
| 65 | 2703.1402 | 100/21 | |
| 66 | 2744.7269 | ||
| 67 | 2786.3137 | exact 5/1 | just major third plus two octaves |