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> | 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 === | ||
| Line 24: | Line 24: | ||
== 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]] (¢) | ||
| Line 39: | Line 40: | ||
| 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 46: | 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 | ||
| Line 82: | Line 84: | ||
| [[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 == | ||
| Line 89: | Line 91: | ||
== References == | == References == | ||
<references/> | <references /> | ||
== External links == | == External links == | ||
Latest revision as of 17:57, 19 February 2025
| ← 143edo | 144edo | 145edo → |
144 equal divisions of the octave (abbreviated 144edo or 144ed2), also called 144-tone equal temperament (144tet) or 144 equal temperament (144et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 144 equal parts of about 8.33 ¢ each. Each step represents a frequency ratio of 21/144, or the 144th root of 2.
144edo's step size is called a farab when used as an interval size unit.[1]
Theory
144edo is closely related to 72edo, but the patent vals differ on the mapping for 13 and 17. It is 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 supports hemisecordite, the 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.
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.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -2.98 | -2.16 | -3.91 | -1.32 | +1.14 | +3.40 | +3.38 | +2.49 | -4.11 | -3.27 |
| Relative (%) | -23.5 | -35.8 | -25.9 | -46.9 | -15.8 | +13.7 | +40.8 | +40.5 | +29.8 | -49.4 | -39.3 | |
| Steps (reduced) |
228 (84) |
334 (46) |
404 (116) |
456 (24) |
498 (66) |
533 (101) |
563 (131) |
589 (13) |
612 (36) |
632 (56) |
651 (75) | |
Subsets and supersets
144edo is the square of world-dominant 12edo.
Since 144 factors into 24 × 32, 144edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72.
Approximation to φ
144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the logarithmic golden ratio at 89 steps. Coincidentally, it also excellently represents the acoustic golden ratio by 100 steps.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11.13 | 225/224, 243/242, 385/384, 847/845, 1875/1859 | [⟨144 228 334 404 498 533]] | +0.560 | 0.595 | 7.13 |
| 2.3.5.7.11.13.17 | 225/224, 243/242, 273/272, 325/324, 847/845, 875/867 | [⟨144 228 334 404 498 533 589]] (144) | +0.362 | 0.734 | 8.80 |
| 2.3.5.7.11.13.17 | 170/169, 225/224, 243/242, 289/288, 375/374, 385/384 | [⟨144 228 334 404 498 533 588]] (144g) | +0.653 | 0.596 | 7.15 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 7\144 | 58.33 | 27/26 | Hemisecordite (144) |
| 2 | 7\144 | 58.33 | 27/26 | Semihemisecordite (144g) |
| 12 | 1\144 | 8.33 | 129/128 | Dotcom |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Music
- Like an endless uphill (2022)