Octagar temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|rank-2]] ~~[[regular temperament|~~temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).

This is a collection of [[rank-2 temperament|rank-2]] '''octagar temperaments''', which [[tempering out|temper out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).

Temperaments discussed elsewhere are:

* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]

* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]

* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]

* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]

* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]

* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]

* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]

* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]

* ''[[~~Injera~~]]'' (+50/~~49 or 81/80) → [[Meantone family #Injera|Meantone family]]~~

* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]

* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]

* [[Garibaldi]] (+225/224) → [[~~Schismatic~~ family #~~Garibaldi~~|~~Schismatic~~ family]]

* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]

* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]

* [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]

* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]

* ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]

* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family #Septimal quartonic|Quartonic family]]

* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]

* ''[[~~Quintupole~~]]'' (+~~458752~~/~~455625~~) → [[~~Quintaleap~~ family #~~Quintupole~~|~~Quintaleap~~ family]]

* ''[[Trisedodge]]'' (+110592/109375) → [[Trisedodge family #Septimal trisedodge|Trisedodge family]]

Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].

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Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[10/7]], and [[121edo|59\121]] is about perfect as a tuning.

Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[~]][[10/7]] [[tritone]], seven of which give the [[interval class]] of [[3/1|3]], and the [[ploidacot]] for this temperament is gamma-heptacot. [[121edo|59\121]] is about perfect as a tuning.

[[Subgroup]]: 2.3.5.7

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Comma list: 540/539, 896/891, 1375/1372

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 364/363, 540/539

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}

Optimal tunings:

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Comma list: 256/255, 325/324, 352/351, 364/363, 540/539

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}

{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}

Optimal tunings:

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Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}

{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}

Optimal tunings:

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Comma list: 144/143, 196/195, 275/273, 896/891

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}

Optimal tunings:

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Comma list: 100/99, 245/242, 10976/10935

~~Mapping:~~ {{~~mapping~~| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}

Optimal tunings:

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Comma list: 100/99, 196/195, 245/242, 729/728

~~Mapping:~~ {{~~mapping~~| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}

Optimal tunings:

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Comma list: 540/539, 2200/2187, 4375/4356

~~Mapping:~~ {{~~mapping~~| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 540/539, 1575/1573

~~Mapping:~~ {{~~mapping~~| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}

Optimal tunings:

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Comma list: 325/324, 352/351, 375/374, 540/539, 595/594

~~Mapping:~~ {{~~mapping~~| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}

{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}

Optimal tunings:

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Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594

~~Mapping:~~ {{~~mapping~~| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}

{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}

Optimal tunings:

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Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483

~~Mapping:~~ {{~~mapping~~| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}

{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}

Optimal tunings:

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== Tridecatonic ==

~~{{See also| 13th~~-~~octave~~ temperaments #Tridecatonic }}

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]]

Tridecatonic has a period of 1/13 octave, and may be described as {{nowrap| 26 & 39 }}. It tempers out [[devil's tridecalimma]], {{monzo| -11 26 -13 }} in the 5-limit. Its [[ploidacot]] is 13-ploid monocot.

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Comma list: 99/98, 1944/1925, 2560/2541

~~Mapping:~~ {{~~mapping~~| 13 0 -11 16 45 | 0 1 2 1 0 }}

Optimal tunings:

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Comma list: 99/98, 144/143, 275/273, 1040/1029

~~Mapping:~~ {{~~mapping~~| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}

Optimal tunings:

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Comma list: 100/99, 245/242, 46656/46585

~~Mapping:~~ {{~~mapping~~| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}

Optimal tunings:

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Comma list: 100/99, 144/143, 245/242, 343/338

~~Mapping:~~ {{~~mapping~~| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}

Optimal tunings:

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== References ==

[[Category:Octagar temperaments| ]]

[[Category:Temperament collections]]

[[Category:~~Octagar temperaments| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

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 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
+This is a collection of [[rank-2 temperament|rank-2]] '''octagar temperaments''', which [[tempering out|temper out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
 Temperaments discussed elsewhere are:
+* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
+* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
+* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
 * ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
+* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
 * ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
+* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
 * ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
-* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
+* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
-* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
-* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
 * ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
-* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
 * [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
-* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
-* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
 * ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
+* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family #Septimal quartonic|Quartonic family]]
 * ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
-* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
+* ''[[Trisedodge]]'' (+110592/109375) → [[Trisedodge family #Septimal trisedodge|Trisedodge family]]
 Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
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 {{Distinguish| Plutus }}
-Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[10/7]], and [[121edo|59\121]] is about perfect as a tuning.
+Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[~]][[10/7]] [[tritone]], seven of which give the [[interval class]] of [[3/1|3]], and the [[ploidacot]] for this temperament is gamma-heptacot. [[121edo|59\121]] is about perfect as a tuning.
 [[Subgroup]]: 2.3.5.7
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 Comma list: 540/539, 896/891, 1375/1372
-Mapping: {{mapping| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}
+{{Mapping|legend=0| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}
 Optimal tunings:
@@ Line 62: / Line 63: @@
 Comma list: 325/324, 352/351, 364/363, 540/539
-Mapping: {{mapping| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}
 Optimal tunings:
@@ Line 77: / Line 78: @@
 Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
-Mapping: {{mapping| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}
 Optimal tunings:
@@ Line 92: / Line 93: @@
 Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
-Mapping: {{mapping| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}
 Optimal tunings:
@@ Line 107: / Line 108: @@
 Comma list: 144/143, 196/195, 275/273, 896/891
-Mapping: {{mapping| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}
+{{Mapping|legend=0| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}
 Optimal tunings:
@@ Line 122: / Line 123: @@
 Comma list: 100/99, 245/242, 10976/10935
-Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}
+{{Mapping|legend=0| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}
 Optimal tunings:
@@ Line 137: / Line 138: @@
 Comma list: 100/99, 196/195, 245/242, 729/728
-Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}
+{{Mapping|legend=0| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}
 Optimal tunings:
@@ Line 172: / Line 173: @@
 Comma list: 540/539, 2200/2187, 4375/4356
-Mapping: {{mapping| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}
 Optimal tunings:
@@ Line 187: / Line 188: @@
 Comma list: 325/324, 352/351, 540/539, 1575/1573
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
 Optimal tunings:
@@ Line 202: / Line 203: @@
 Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}
 Optimal tunings:
@@ Line 217: / Line 218: @@
 Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}
 Optimal tunings:
@@ Line 232: / Line 233: @@
 Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}
 Optimal tunings:
@@ Line 243: / Line 244: @@
 == Tridecatonic ==
-{{See also| 13th-octave temperaments #Tridecatonic }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]]
 Tridecatonic has a period of 1/13 octave, and may be described as {{nowrap| 26 & 39 }}. It tempers out [[devil's tridecalimma]], {{monzo| -11 26 -13 }} in the 5-limit. Its [[ploidacot]] is 13-ploid monocot.
@@ Line 269: / Line 270: @@
 Comma list: 99/98, 1944/1925, 2560/2541
-Mapping: {{mapping| 13 0 -11 16 45 | 0 1 2 1 0 }}
+{{Mapping|legend=0| 13 0 -11 16 45 | 0 1 2 1 0 }}
 Optimal tunings:
@@ Line 284: / Line 285: @@
 Comma list: 99/98, 144/143, 275/273, 1040/1029
-Mapping: {{mapping| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
+{{Mapping|legend=0| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
 Optimal tunings:
@@ Line 321: / Line 322: @@
 Comma list: 100/99, 245/242, 46656/46585
-Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
 Optimal tunings:
@@ Line 336: / Line 337: @@
 Comma list: 100/99, 144/143, 245/242, 343/338
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
+{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
 Optimal tunings:
@@ Line 398: / Line 399: @@
 == References ==
+[[Category:Octagar temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Octagar temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Octagar temperaments: Difference between revisions

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