7315edo
7315 equal divisions of the octave (abbreviated 7315edo or 7315ed2), also called 7315-tone equal temperament (7315tet) or 7315 equal temperament (7315et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 7315 equal parts of about 0.164 ¢ each. Each step represents a frequency ratio of 21/7315, or the 7315th root of 2.
| ← 7314edo | 7315edo | 7316edo → |
7315edo is consistent up to the 27-odd-limit. 7315 = 11 × 665, and 7315edo shares its excellent approximation to harmonic 3 with 665edo. It notably demonstrates 3-2 telicity, and is in fact the largest multiple of 665edo to do so. It has a sharp tendency, with most lower harmonics tuned sharp. In the 13-limit it tempers out 123201/123200; in the 17-limit, 14400/14399 and 194481/194480; in the 19-limit, 14080/14079, 23409/23408, 27456/27455, 89376/89375; in the 23-limit, 23276/23275, 52326/52325 among others.
It can be used in higher limits, notably the 31-odd-limit, and even the no-41 59-odd-limit, due to a large number of the harmonics being either sharp, or not too negative. The inconsistencies here largely happen with primes 29, 37, and 43, which are slightly flat; namely: 49/29, 49/37, 49/43, 29/19, 37/19, 43/38, 59/58, 59/37, 59/43, and octave complements.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0001 | +0.0157 | +0.0326 | +0.0423 | +0.0465 | +0.0343 | +0.0673 | +0.0237 | -0.0215 | +0.0089 |
| Relative (%) | +0.0 | -0.1 | +9.6 | +19.9 | +25.8 | +28.3 | +20.9 | +41.0 | +14.4 | -13.1 | +5.4 | |
| Steps (reduced) |
7315 (0) |
11594 (4279) |
16985 (2355) |
20536 (5906) |
25306 (3361) |
27069 (5124) |
29900 (640) |
31074 (1814) |
33090 (3830) |
35536 (6276) |
36240 (6980) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.0248 | -0.0809 | -0.0208 | +0.0299 | +0.0430 | +0.0765 | -0.0564 | -0.0726 | -0.0656 | +0.0711 | -0.0256 |
| Relative (%) | -15.1 | -49.3 | -12.7 | +18.3 | +26.2 | +46.6 | -34.4 | -44.2 | -40.0 | +43.3 | -15.6 | |
| Steps (reduced) |
38107 (1532) |
39190 (2615) |
39693 (3118) |
40632 (4057) |
41900 (5325) |
43032 (6457) |
43383 (6808) |
44373 (483) |
44985 (1095) |
45279 (1389) |
46112 (2222) | |
Subsets and supersets
Since 7315 factors into 5 × 7 × 11 × 19, 7315edo contains subset edos 5, 7, 11, 19, 35, 55, 77, 95, 133, 209, 385, 665, 1045, and 1463.