1178edo
| ← 1177edo | 1178edo | 1179edo → |
1178 equal divisions of the octave (abbreviated 1178edo or 1178ed2), also called 1178-tone equal temperament (1178tet) or 1178 equal temperament (1178et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1178 equal parts of about 1.02 ¢ each. Each step represents a frequency ratio of 21/1178, or the 1178th root of 2.
Theory
1178edo is a very strong 19-limit system, distinctly consistent through to the 21-odd-limit, and is the first edo past 742 with a lower 19-limit relative error. It is a zeta peak, integral and gap edo.
As an equal temperament, it tempers out [-14 -19-19⟩ (enneadeca) in the 5-limit, 4375/4374 (ragisma) and 703125/702464 (meter) in the 7-limit, so that it supports enneadecal. In the 11-limit it tempers out 3025/3024, 9801/9800, and 234375/234256, supporting hemienneadecal, and in the 13-limit 4225/4224 and 10648/10647, supporting and providing a great tuning for semihemienneadecal. It further tempers out 2500/2499, 4914/4913 in the 17-limit; 3250/3249, 4200/4199 in the 19-limit; and 2025/2024 among others in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.087 | -0.236 | -0.065 | -0.214 | -0.120 | -0.032 | -0.060 | +0.249 | +0.304 | -0.044 |
| Relative (%) | +0.0 | -8.6 | -23.1 | -6.4 | -21.0 | -11.8 | -3.1 | -5.9 | +24.4 | +29.8 | -4.3 | |
| Steps (reduced) |
1178 (0) |
1867 (689) |
2735 (379) |
3307 (951) |
4075 (541) |
4359 (825) |
4815 (103) |
5004 (292) |
5329 (617) |
5723 (1011) |
5836 (1124) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.269 | -0.200 | -0.142 | -0.311 | -0.499 | +0.251 | -0.416 | +0.150 | -0.410 | +0.394 | +0.149 |
| Relative (%) | +26.4 | -19.6 | -14.0 | -30.6 | -49.0 | +24.6 | -40.9 | +14.7 | -40.2 | +38.7 | +14.6 | |
| Steps (reduced) |
6137 (247) |
6311 (421) |
6392 (502) |
6543 (653) |
6747 (857) |
6930 (1040) |
6986 (1096) |
7146 (78) |
7244 (176) |
7292 (224) |
7426 (358) | |
Subsets and supersets
Since 1178 factors into primes as 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are 2, 19, 31, 38, 62, and 589.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1867 1178⟩ | [⟨1178 1867]] | +0.0276 | 0.0276 | 2.71 |
| 2.3.5 | [-14 -19-19⟩, [-99 61 1⟩ | [⟨1178 1867 2735]] | +0.0522 | 0.0415 | 4.07 |
| 2.3.5.7 | 4375/4374, 703125/702464, [-52 -5 -2 23⟩ | [⟨1178 1867 2735 3307]] | +0.0450 | 0.0380 | 3.73 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 234375/234256, [-27 3 -4 10 1⟩ | [⟨1178 1867 2735 3307 4075]] | +0.0484 | 0.0347 | 3.41 |
| 2.3.5.7.11.13 | 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360 | [⟨1178 1867 2735 3307 4075 4359]] | +0.0457 | 0.0322 | 3.16 |
| 2.3.5.7.11.13.17 | 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872 | [⟨1178 1867 2735 3307 4075 4359 4815]] | +0.0403 | 0.0327 | 3.21 |
| 2.3.5.7.11.13.17.19 | 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913 | [⟨1178 1867 2735 3307 4075 4359 4815 5004]] | +0.0370 | 0.0318 | 3.12 |
| 2.3.5.7.11.13.17.19.23 | 2025/2024, 2500/2499, 3025/3024, 3060/3059, 3250/3249, 3520/3519, 4200/4199, 4375/4374 | [⟨1178 1867 2735 3307 4075 4359 4815 5004 5329]] | +0.0292 | 0.0436 | 4.28 |
- 1178et holds the record of lowest relative error in the 19-limit, being the first to beat 742 in relative error and 954h in absolute error, before getting superseded by 1578 in both.
- It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, 1106 and before 1236, 1236, and 1308, respectively.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 337\1178 | 343.29 | 8000/6561 | Raider |
| 19 | 7\1178 | 7.13 | 225/224 | Enneadecal |
| 38 | 12\1178 | 12.22 | 144/143 | Semihemienneadecal |
| 38 | 7\1178 | 7.13 | 225/224 | Hemienneadecal |
Music
- Listening (2023) – 217 & 1178 and enneadecal in 1178edo tuning