1178edo: Difference between revisions
if there is a music plz add to listen |
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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 basis for its 19-limit [[comma]]s | 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 === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|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 | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-1867 1178}} | | {{monzo| -1867 1178 }} | ||
|{{ | | {{mapping| 1178 1867 }} | ||
| 0.0276 | | +0.0276 | ||
| 0.0276 | | 0.0276 | ||
| 2.71 | | 2.71 | ||
|- | |- | ||
|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 }} | ||
| 0.0522 | | +0.0522 | ||
| 0.0415 | | 0.0415 | ||
| 4.07 | | 4.07 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|4375/4374, 703125/702464, | | 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }} | ||
|{{ | | {{mapping| 1178 1867 2735 3307 }} | ||
| 0.0450 | | +0.0450 | ||
| 0.0380 | | 0.0380 | ||
| 3.73 | | 3.73 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|3025/3024, 4375/4374, | | 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }} | ||
|{{ | | {{mapping| 1178 1867 2735 3307 4075 }} | ||
| 0.0484 | | +0.0484 | ||
| 0.0347 | | 0.0347 | ||
| 3.41 | | 3.41 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|3025/3024, 4225/4224, | | 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893 | ||
|{{ | | {{mapping| 1178 1867 2735 3307 4075 4359 }} | ||
| 0.0457 | | +0.0457 | ||
| 0.0322 | | 0.0322 | ||
| 3.16 | | 3.16 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|3025/3024, 4225/4224, | | 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872 | ||
|{{ | | {{mapping| 1178 1867 2735 3307 4075 4359 4815 }} | ||
| 0.0403 | | 0.0403 | ||
| 0.0327 | | 0.0327 | ||
| 3.21 | | 3.21 | ||
|- | |- | ||
|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 | ||
|{{ | | {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }} | ||
| 0.0370 | | 0.0370 | ||
| 0.0318 | | 0.0318 | ||
| 3.12 | | 3.12 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|337\1178 | | 337\1178 | ||
|343.29 | | 343.29 | ||
|8000/6561 | | 8000/6561 | ||
|[[Raider]] | | [[Raider]] | ||
|- | |- | ||
|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]] | ||
|- | |- | ||
|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]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
; | ; [[Eliora]] | ||
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023) | * [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 & 1178 and enneadecal in 1178edo tuning | ||
[[Category:Enneadecal]] | [[Category:Enneadecal]] | ||
[[Category:Hemienneadecal]] | [[Category:Hemienneadecal]] | ||
[[Category:Semihemienneadecal]] | [[Category:Semihemienneadecal]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||