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{{Infobox ET}}
{{Infobox ET}}
The '''1178 equal tuning''' divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also 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 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]].  
{{ED intro}}
 
== 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]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}
{{Harmonics in equal|1178|columns=11}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
Since 1178 = 2 × 19 × 31, 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 {{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]]
! 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 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! 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]]
|-
| 31
| 581\1178<br />(11\1178)
| 591.851<br />(11.205)
| 936/665<br />(?)
| [[31st-octave temperaments#217 & 1178|217 & 1178]]
|-
| 38
| 260\1178<br />(12\1178)
| 264.86<br />(12.22)
| 500/429<br />(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178<br />(7\1178)
| 498.13<br />(7.13)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp; 1178}} and enneadecal in 1178edo tuning


[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
[[Category:Enneadecal]]
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]
[[Category:Semihemienneadecal]]
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