1178edo: Difference between revisions

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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.  
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-1867 1178}}
|{{val|1178 1867}}
| 0.0276
| 0.0276
| 2.71
|-
|2.3.5
|{{monzo|-14 -19-19}}, {{monzo|-99 61 1}}
|{{val|1178 1867 2735}}
| 0.0522
| 0.0415
| 4.07
|-
|2.3.5.7
|4375/4374, 703125/702464, 2460375/2458624
|{{val|1178 1867 2735 3307}}
| 0.0450
| 0.0380
| 3.73
|-
|2.3.5.7.11
|3025/3024, 4375/4374, 9801/9800, 43923/43904
|{{val|1178 1867 2735 3307 4075}}
| 0.0484
| 0.0347
| 3.41
|-
|2.3.5.7.11.13
|3025/3024, 4225/4224, 9801/9800, 91125/91091, 1664000/1663893
|{{val|1178 1867 2735 3307 4075 4359}}
| 0.0457
| 0.0322
| 3.16
|-
|2.3.5.7.11.13.17
|3025/3024, 4225/4224, 5832/5831, 9801/9800, 14875/14872, 31213/31212
|{{val|1178 1867 2735 3307 4075 4359 4815}}
| 0.0403
| 0.0327
| 3.21
|-
|2.3.5.7.11.13.17.19
|4200/4199, 5985/5984, 9801/9800, 10830/10829, 12636/12635, 14365/14364, 23409/23408
|{{val|1178 1867 2735 3307 4075 4359 4815 5004}}
| 0.0370
| 0.0318
| 3.12
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|337\1178
|343.29
|8000/6561
|[[Raider]]
|-
|19
|489\1178<br>(7\1178)
|498.13<br>(7.13)
|4/3<br>(225/224)
|[[Enneadecal]]
|-
|38
|489\1178<br>(7\1178)
|498.13<br>(7.13)
|4/3<br>(225/224)
|[[Hemienneadecal]]
|}


== Music ==
== Music ==

Revision as of 19:11, 4 November 2023

← 1177edo 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

Template:EDO intro

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 is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984. 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 = 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, 2460375/2458624 ⟨1178 1867 2735 3307] 0.0450 0.0380 3.73
2.3.5.7.11 3025/3024, 4375/4374, 9801/9800, 43923/43904 ⟨1178 1867 2735 3307 4075] 0.0484 0.0347 3.41
2.3.5.7.11.13 3025/3024, 4225/4224, 9801/9800, 91125/91091, 1664000/1663893 ⟨1178 1867 2735 3307 4075 4359] 0.0457 0.0322 3.16
2.3.5.7.11.13.17 3025/3024, 4225/4224, 5832/5831, 9801/9800, 14875/14872, 31213/31212 ⟨1178 1867 2735 3307 4075 4359 4815] 0.0403 0.0327 3.21
2.3.5.7.11.13.17.19 4200/4199, 5985/5984, 9801/9800, 10830/10829, 12636/12635, 14365/14364, 23409/23408 ⟨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
(reduced) 		Cents
(reduced) 		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
38 489\1178
(7\1178) 		498.13
(7.13) 		4/3
(225/224) 		Hemienneadecal

Music

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