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1178edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1178}}
{{ED intro}}


== 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 ==
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! 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
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|-
|-
| 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
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| 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
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| 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
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| 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
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| 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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Music ==
== Music ==
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