1178edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
Line 70: Line 62:
| 0.0318
| 0.0318
| 3.12
| 3.12
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 88: Line 74:
|-
|-
| 19
| 19
| 489\1178<br>(7\1178)
| 489\1178<br />(7\1178)
| 498.13<br>(7.13)
| 498.13<br />(7.13)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|-
|-
| 31
| 31
| 581\1178<br>(11\1178)
| 581\1178<br />(11\1178)
| 591.851<br>(11.205)
| 591.851<br />(11.205)
| 936/665<br>(?)
| 936/665<br />(?)
| [[31st-octave temperaments#217 & 1178|217 & 1178]]
| [[31st-octave temperaments#217 & 1178|217 & 1178]]
|-
|-
| 38
| 38
| 260\1178<br>(12\1178)
| 260\1178<br />(12\1178)
| 264.86<br>(12.22)
| 264.86<br />(12.22)
| 500/429<br>(144/143)
| 500/429<br />(144/143)
| [[Semihemienneadecal]]
| [[Semihemienneadecal]]
|-
|-
| 38
| 38
| 489\1178<br>(7\1178)
| 489\1178<br />(7\1178)
| 498.13<br>(7.13)
| 498.13<br />(7.13)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Music ==
== Music ==
; [[Eliora]]
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 & 1178 and enneadecal in 1178edo tuning
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) &ndash; 217 & 1178 and enneadecal in 1178edo tuning


[[Category:Enneadecal]]
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]

Revision as of 04:21, 16 November 2024

← 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 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 = 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

Music

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