144edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
144edo's step size is called a '''farab''' when used as an [[interval size unit]].<ref>http://tonalsoft.com/enc/f/farab.aspx</ref> | |||
== Theory == | == Theory == | ||
144edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for [[13/1|13]] and [[17/1|17]]. It is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[225/224]], [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]]. Using the [[patent val]], it tempers out [[847/845]], [[1188/1183]], 1701/1690, 1875/1859, and [[4225/4224]] in the 13-limit. It [[support]]s [[hemisecordite]], the 41 & 103 temperament, though [[103edo]] is better suited for this purpose. | 144edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for [[13/1|13]] and [[17/1|17]]. It is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[225/224]], [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]]. Using the [[patent val]], it tempers out [[847/845]], [[1188/1183]], 1701/1690, 1875/1859, and [[4225/4224]] in the 13-limit. It [[support]]s [[hemisecordite]], the {{nowrap|41 & 103}} temperament, though [[103edo]] is better suited for this purpose. | ||
Although the patent val comes out on top accuracy in the 13-limit, in the 17-limit 144 falls behind to 144g. The 144g val tempers out [[170/169]], [[289/288]], [[375/374]], [[561/560]], [[595/594]]. It supports [[semihemisecordite]], the 62 & 82f temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]], supporting 17-limit hemisecordite. In the 19-limit the patent val tempers out 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663. | Although the patent val comes out on top accuracy in the 13-limit, in the 17-limit 144 falls behind to 144g. The 144g val tempers out [[170/169]], [[289/288]], [[375/374]], [[561/560]], [[595/594]]. It supports [[semihemisecordite]], the {{nowrap|62 & 82f}} temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]], supporting 17-limit hemisecordite. In the 19-limit the patent val tempers out 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663. | ||
Besides all these, the 144eff val supports hemimiracle, the 41 & 103e temperament. 144ee supports oracle, the 31 & 113e temperament. 144cf supports necromanteion, the 31 & 113cf temperament. | Besides all these, the 144eff val supports hemimiracle, the {{nowrap|41 & 103e}} temperament. 144ee supports oracle, the {{nowrap|31 & 113e}} temperament. 144cf supports necromanteion, the {{nowrap|31 & 113cf}} temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
144edo is the square of world-dominant [[12edo]] | 144edo is the square of world-dominant [[12edo]]. | ||
Since 144 factors into 2<sup>4</sup> × 3<sup>2</sup>, 144edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72 }}. | Since 144 factors into 2<sup>4</sup> × 3<sup>2</sup>, 144edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72 }}. | ||
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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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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| 0.595 | | 0.595 | ||
| 7.13 | | 7.13 | ||
|-style="border-top: double;" | |- style="border-top: double;" | ||
| 2.3.5.7.11.13.17 | |||
| 225/224, 243/242, 273/272, 325/324, 847/845, 875/867 | |||
| {{mapping| 144 228 334 404 498 533 589 }} (144) | |||
| +0.362 | |||
| 0.734 | |||
| 8.80 | |||
|- style="border-top: double;" | |||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 170/169, 225/224, 243/242, 289/288, 375/374, 385/384 | | 170/169, 225/224, 243/242, 289/288, 375/374, 385/384 | ||
| Line 44: | Line 54: | ||
| 0.596 | | 0.596 | ||
| 7.15 | | 7.15 | ||
|} | |} | ||
=== 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 | |+ 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> | ! Associated<br />ratio* | ||
! | ! Temperaments | ||
|- | |- | ||
| 1 | | 1 | ||
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| [[Substitute harmonic#Dotcom|Dotcom]] | | [[Substitute harmonic#Dotcom|Dotcom]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
; [[Hideya]] | ; [[Hideya]] | ||
* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022) | * [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022) | ||
== References == | |||
<references /> | |||
== External links == | == External links == | ||