1178edo

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← 1177edo1178edo1179edo →
Prime factorization 2 × 19 × 31
Step size 1.01868¢
Fifth 689\1178 (701.868¢)
Semitones (A1:m2) 111:89 (113.1¢ : 90.66¢)
Consistency limit 21
Distinct consistency limit 21
Special properties

1178 equal divisions of the octave (abbreviated 1178edo), or 1178-tone equal temperament (1178tet), 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 of 1178edo represents a frequency ratio of 21/1178, or the 1178th root of 2.

Theory

1178edo is a very strong 19-limit system, and is a zeta peak, integral and gap edo. It is also distinctly consistent through to the 21-odd-limit, and is the first edo past 742 with a lower 19-limit relative error. A basis for its 19-limit commas consists of 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, and 4914/4913. It supports and provides a great tuning for semihemienneadecal.

Prime harmonics

Approximation of prime harmonics in 1178edo
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 -9 -23 -6 -21 -12 -3 -6 +24 +30 -4
Steps
(reduced)
1178
(0)
1867
(689)
2735
(379)
3307
(951)
4075
(541)
4359
(825)
4815
(103)
5004
(292)
5329
(617)
5723
(1011)
5836
(1124)

Subsets and supersets

Since 1178 = 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, 1664000/1663893 [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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 337\1178 343.29 8000/6561 Raider
19 489\1178
(7\1178)
498.13
(7.13)
4/3
(225/224)
Enneadecal
31 581\1178
(11\1178)
591.851
(11.205)
936/665
(?)
217 & 1178
38 260\1178
(12\1178)
264.86
(12.22)
500/429
(144/143)
Semihemienneadecal
38 489\1178
(7\1178)
498.13
(7.13)
4/3
(225/224)
Hemienneadecal

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Eliora
  • Listening (2023) – 217 & 1178 and enneadecal in 1178edo tuning