1178edo: Difference between revisions

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== Theory ==
== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]].  
1178edo is a very strong [[19-limit]] system, [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is a [[zeta peak edo|zeta peak]], [[zeta integral edo|integral]] and [[zeta gap edo|gap edo]].
 
As an equal temperament, it tempers out {{monzo| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464  ([[meter]]) in the [[7-limit]], so that it [[support]]s [[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 ===
=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}
{{Harmonics in equal|1178|columns=11}}
{{Harmonics in equal|1178|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1178edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 1178 factors into {{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 factors into primes as {{nowrap| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.


== Regular temperament properties ==
== Regular temperament properties ==
Line 17: Line 20:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -1867 1178 }}
| {{Monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| {{Mapping| 1178 1867 }}
| +0.0276
| +0.0276
| 0.0276
| 0.0276
Line 31: Line 34:
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{Monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| {{Mapping| 1178 1867 2735 }}
| +0.0522
| +0.0522
| 0.0415
| 0.0415
Line 39: Line 42:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| {{Mapping| 1178 1867 2735 3307 }}
| +0.0450
| +0.0450
| 0.0380
| 0.0380
Line 46: Line 49:
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| {{Mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| +0.0484
| 0.0347
| 0.0347
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|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| {{Mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| +0.0457
| 0.0322
| 0.0322
Line 60: Line 63:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| +0.0403
| 0.0327
| 0.0327
Line 67: Line 70:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| +0.0370
| 0.0318
| 0.0318
| 3.12
| 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
| {{Mapping| 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 [[742edo|742]] in relative error and [[954edo|954h]] in absolute error, before getting superseded by [[1578edo|1578]] in both.
* It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, [[1106edo|1106]] and before [[1236edo|1236]], 1236, and [[1308edo|1308]], respectively.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 19
| 19
| 489\1178<br />(7\1178)
| 7\1178
| 498.13<br />(7.13)
| 7.13
| 4/3<br />(225/224)
| 225/224
| [[Enneadecal]]
| [[Enneadecal]]
|-
| 31
| 581\1178<br />(11\1178)
| 591.851<br />(11.205)
| 936/665<br />(?)
| [[31st-octave temperaments#217 & 1178|217 & 1178]]
|-
|-
| 38
| 38
| 260\1178<br />(12\1178)
| 12\1178
| 264.86<br />(12.22)
| 12.22
| 500/429<br />(144/143)
| 144/143
| [[Semihemienneadecal]]
| [[Semihemienneadecal]]
|-
|-
| 38
| 38
| 489\1178<br />(7\1178)
| 7\1178
| 498.13<br />(7.13)
| 7.13
| 4/3<br />(225/224)
| 225/224
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Music ==
== Music ==

Latest revision as of 10:27, 13 July 2026

← 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

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

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)
Approximation of prime harmonics in 1178edo (continued)
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

Table of rank-2 temperaments by generator
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

* In minimal-generator form

Music

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