144edo: Difference between revisions

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== Theory ==

144edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for 13 and 17. It is [[~~contorted~~]] 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; [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]] in the 17-limit; 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663 in the 19~~-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.

The ~~144gh~~ val is ~~a tuning~~ for ~~semihemisecordite temperament~~, ~~while~~ 144eff val ~~is a tuning for~~ hemimiracle. 144ee ~~is a tuning for~~ oracle. 144cf ~~val~~ supports necromanteion.

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]]. It [[support]]s [[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.

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 ===

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=== 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. In addition, 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" | [[Mapping]]

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5.7.11.13

| 225/224, 243/242, 385/384, 847/845, 1875/1859

| {{mapping| 144 228 334 404 498 533 }}

| +0.560

| 0.595

| 7.13

|-

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

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

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

| style="border-top: double;" | +0.653

| style="border-top: double;" | 0.596

| style="border-top: double;" | 7.15

|-

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

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

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

| style="border-top: double;" | +0.362

| style="border-top: double;" | 0.734

| style="border-top: double;" | 8.80

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator (Reduced)

! Cents (Reduced)

! Associated Ratio

! Temperament

|-

| 1

| 7\144

| 58.33

| 27/26

| [[Hemisecordite]] (144)

|-

| 2

| 7\144

| 58.33

| 27/26

| [[Semihemisecordite]] (144g)

|}

== Music ==

* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] ~~by [[5 hideya]]~~

; [[Amano Hideya]]

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

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for 13 and 17. It is [[contorted]] 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; [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]] in the 17-limit; 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663 in the 19-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.
-The 144gh val is a tuning for semihemisecordite temperament, while 144eff val is a tuning for hemimiracle. 144ee is a tuning for oracle. 144cf val supports necromanteion.
+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]]. It [[support]]s [[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.
+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 ===
@@ Line 16: / Line 18: @@
 === 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. In addition, 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" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5.7.11.13
+| 225/224, 243/242, 385/384, 847/845, 1875/1859
+| {{mapping| 144 228 334 404 498 533 }}
+| +0.560
+| 0.595
+| 7.13
+|-
+| style="border-top: double;" | 2.3.5.7.11.13.17
+| style="border-top: double;" | 170/169, 225/224, 243/242, 289/288, 375/374, 385/384
+| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 588 }} (144g)
+| style="border-top: double;" | +0.653
+| style="border-top: double;" | 0.596
+| style="border-top: double;" | 7.15
+|-
+| style="border-top: double;" | 2.3.5.7.11.13.17
+| style="border-top: double;" | 225/224, 243/242, 273/272, 325/324, 847/845, 875/867
+| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 589 }} (144)
+| style="border-top: double;" | +0.362
+| style="border-top: double;" | 0.734
+| style="border-top: double;" | 8.80
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+Table of rank-2 temperaments by generator
+! Periods<br>per 8ve
+! Generator<br>(Reduced)
+! Cents<br>(Reduced)
+! Associated<br>Ratio
+! Temperament
+|-
+| 1
+| 7\144
+| 58.33
+| 27/26
+| [[Hemisecordite]] (144)
+|-
+| 2
+| 7\144
+| 58.33
+| 27/26
+| [[Semihemisecordite]] (144g)
+|}
 == Music ==
-* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] by [[5 hideya]]
+; [[Amano Hideya]]
+* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022)