144edo: Difference between revisions

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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 ==

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.

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 ~~[[support]]s~~ [[semihemisecordite]], the 62 & 82f temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]]~~. It supports [[~~hemisecordite~~]], the 41 & 103 temperament, though [[103edo]] is better suited for this purpose~~. 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 ===

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=== Subsets and supersets ===

144edo is the square of world-dominant [[12edo]]~~. One step of it is called ''[[farab]]''~~.

144edo is the square of world-dominant [[12edo]].

Since 144 factors into 24 × 32, 144edo has subset edos {{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 phi|logarithmic golden ratio]] at 89 steps. ~~In addition~~, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.

144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. Coincidentally, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 0.595

| 7.13

|-

|- style="border-top: double;"

| style="border-top: double;" | 2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

| ~~style="border-top: double;" | 170/169,~~ 225/224, 243/242, ~~289~~/~~288~~, ~~375~~/~~374~~, ~~385~~/~~384~~

| 225/224, 243/242, 273/272, 325/324, 847/845, 875/867

~~| style="border-top: double;"~~ | {{mapping| 144 228 334 404 498 533 ~~588~~ }} (~~144g~~)

| {{mapping| 144 228 334 404 498 533 589 }} (144)

~~| style="border-top: double;"~~ | +0.~~653~~

| +0.362

~~| style="border-top: double;"~~ | 0.~~596~~

| 0.734

| ~~style="border-top: double;" | 7~~.15

| 8.80

|-

|- style="border-top: double;"

| style="border-top: double;" | 2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

| ~~style="border-top: double;" |~~ 225/224, 243/242, ~~273~~/~~272~~, ~~325~~/~~324~~, ~~847~~/~~845, 875/867~~

| 170/169, 225/224, 243/242, 289/288, 375/374, 385/384

~~| style="border-top: double;"~~ | {{mapping| 144 228 334 404 498 533 ~~589~~ }} (~~144~~)

| {{mapping| 144 228 334 404 498 533 588 }} (144g)

~~| style="border-top: double;"~~ | +0.~~362~~

| +0.653

~~| style="border-top: double;"~~ | 0.~~734~~

| 0.596

| ~~style="border-top: double;" | 8~~.80

| 7.15

|}

=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! ~~Temperament~~

! Associated ratio*

! Temperaments

|-

| 1

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| 27/26

| [[Semihemisecordite]] (144g)

|-

| 12

| 1\144

| 8.33

| 129/128

| [[Substitute harmonic#Dotcom|Dotcom]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==

; [[~~Amano~~ Hideya]]

; [[Hideya]]

* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022)

== References ==

== External links ==

* [http://tonalsoft.com/enc/number/144edo.aspx 144-tone equal-temperament / 144-edo] on [[Tonalsoft Encyclopedia]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|144}}
+{{ED intro}}
+edo's step size is called a '''farab''' when used as an [[interval size unit]].<ref>http://tonalsoft.com/enc/f/farab.aspx</ref>
 == Theory ==
-edo 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.
+edo 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 &amp; 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 [[support]]s [[semihemisecordite]], the 62 & 82f temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]]. It supports [[hemisecordite]], the 41 & 103 temperament, though [[103edo]] is better suited for this purpose. 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 &amp; 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 &amp; 103e}} temperament. 144ee supports oracle, the {{nowrap|31 &amp; 113e}} temperament. 144cf supports necromanteion, the {{nowrap|31 &amp; 113cf}} temperament.
 === Prime harmonics ===
@@ Line 13: / Line 15: @@
 === Subsets and supersets ===
-edo is the square of world-dominant [[12edo]]. One step of it is called ''[[farab]]''.
+edo 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 }}.
 === Approximation to φ ===
-edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. In addition, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.
+edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. Coincidentally, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 37: / Line 40: @@
 | 0.595
 | 7.13
-|-
+|- style="border-top: double;"
-| style="border-top: double;" | 2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-| style="border-top: double;" | 170/169, 225/224, 243/242, 289/288, 375/374, 385/384
+| 225/224, 243/242, 273/272, 325/324, 847/845, 875/867
-| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 588 }} (144g)
+| {{mapping| 144 228 334 404 498 533 589 }} (144)
-| style="border-top: double;" | +0.653
+| +0.362
-| style="border-top: double;" | 0.596
+| 0.734
-| style="border-top: double;" | 7.15
+| 8.80
-|-
+|- style="border-top: double;"
-| style="border-top: double;" | 2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-| style="border-top: double;" | 225/224, 243/242, 273/272, 325/324, 847/845, 875/867
+| 170/169, 225/224, 243/242, 289/288, 375/374, 385/384
-| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 589 }} (144)
+| {{mapping| 144 228 334 404 498 533 588 }} (144g)
-| style="border-top: double;" | +0.362
+| +0.653
-| style="border-top: double;" | 0.734
+| 0.596
-| style="border-top: double;" | 8.80
+| 7.15
 |}
 === Rank-2 temperaments ===
 {| 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
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
-! Temperament
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 73: / Line 77: @@
 | 27/26
 | [[Semihemisecordite]] (144g)
+|-
+| 12
+| 1\144
+| 8.33
+| 129/128
+| [[Substitute harmonic#Dotcom|Dotcom]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Music ==
-; [[Amano Hideya]]
+; [[Hideya]]
 * [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022)
+== References ==
+<references />
+== External links ==
+* [http://tonalsoft.com/enc/number/144edo.aspx 144-tone equal-temperament / 144-edo] on [[Tonalsoft Encyclopedia]]
+[[Category:Listen]]