1178edo

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1178edo

1179edo →

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

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 (%)	+0.0	-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)

Subsets and supersets

Since 1178 factors into 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 distinct

Music

Eliora

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