Octagar temperaments: Difference between revisions

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Octagar temperaments temper out the octagar comma, {{monzo|5 -4 3 -2}} = 4000/3969.
This is a collection of temperaments tempering out the octagar comma, {{monzo| 5 -4 3 -2 }} = [[4000/3969]].


Temperaments discussed elsewhere include [[Garibaldi temperament|garibaldi]], [[Augmented family #Augene|augene]], [[Tetracot family #Octacot|octacot]], [[Orwellismic temperaments #Quartonic|quartonic]], [[superkleismic]], [[Meantone family #Injera|injera]], [[Diaschismic family #Bidia|bidia]], [[Kleismic family #Hemikleismic|hemikleismic]], [[Mirkwai clan #Pluto|pluto]], [[Porcupine family #Nautilus|nautilus]], [[Avicennmic temperaments #Roman|roman]], [[Ripple family|ripple]], and [[Amity family #Hamity|hamity]].
Temperaments discussed elsewhere include:
* ''[[Ripple]]'', {36/35, 2560/2401} → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Nautilus]]'', {49/48, 250/243} → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Injera]]'', {50/49, 81/80} → [[Meantone family #Injera|Meantone family]]
* ''[[Augene]]'', {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Octacot]]'', {245/243, 2401/2400} → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Roman]]'', {525/512, 3125/3024} → [[Wesley family #Roman|Wesley family]] and [[Avicennmic temperaments #Roman|Avicennmic temperaments]]
* ''[[Superkleismic]]'', {875/864, 1029/1024} → [[Shibboleth family #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Quartonic]]'', {1728/1715, 4000/3969} → [[Orwellismic temperaments #Quartonic|Orwellismic temperaments]]
* ''[[Bidia]]'', {2048/2025, 3136/3125} → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hamity]]'', {2430/2401, 4000/3969} → [[Amity family #Hamity|Amity family]]
* ''[[Hemikleismic]]'', {4000/3969, 6144/6125} → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Quintupole]]'', {4000/3969, 458752/455625} → [[Quintaleap family #Quintupole|Quintaleap family]]
 
Considered below are slendi and slithy.  


== Slendi ==
== Slendi ==
Line 85: Line 101:
Badness: 0.028492
Badness: 0.028492


== Quintupole ==
[[Category:Regular temperament theory]]
: ''See also: [[16ed5/2 #Regular temperaments]] or [[34ed7 #Regular temperaments]]''
 
The ''quintupole'' temperament tempers out the octagar comma and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means ''five poles'', but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the [[Porwell temperaments #Polypyth|polypyth]] fourth (~4/3, about 495.8 cents). Proper pronunciation of the word "quintupole" is /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". ''Not to be confused with [[Marvel temperaments #Quintapole|quint<u>'''a'''</u>pole]] temperament (12&amp;85).''
 
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4000/3969, 458752/455625
 
[[Mapping]]: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]
 
{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}
 
[[POTE generator]]: ~135/128 = 99.175
 
{{Vals|legend=1| 12, 97, 109, 121 }}
 
[[Badness]]: 0.111620
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 896/891, 1375/1372, 4375/4356
 
Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]
 
POTE generator: ~35/33 = 99.156
 
Vals: {{Val list| 12, 109, 121, 351bde, 472bdee }}
 
Badness: 0.056501
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 364/363, 625/624, 2704/2695
 
Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]
 
POTE generator: ~35/33 = 99.165
 
Vals: {{Val list| 12f, 109, 121 }}
 
Badness: 0.038431
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
 
Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]
 
POTE generator: ~18/17 = 99.172
 
Vals: {{Val list| 12f, 109, 121 }}
 
Badness: 0.028721
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
 
Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]
 
POTE generator: ~18/17 = 99.164
 
Vals: {{Val list| 12f, 109, 121 }}
 
Badness: 0.023818
 
[[Category:Temperament collection]]
[[Category:Temperament collection]]
[[Category:Octagar]]
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 15:40, 21 November 2021

This is a collection of temperaments tempering out the octagar comma, [5 -4 3 -2⟩ = 4000/3969.

Temperaments discussed elsewhere include:

Considered below are slendi and slithy.

Slendi

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]

Wedgie: ⟨⟨ 17 28 8 5 -35 -60 ]]

POTE generator: ~49/48 = 29.181

Template:Val list

Badness: 0.142903

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -17 -28 -8 -22]]

POTE generator: ~49/48 = 29.180

Vals: Template:Val list

Badness: 0.063217

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -17 -28 -8 -22 -1]]

POTE generator: ~49/48 = 29.125

Vals: Template:Val list

Badness: 0.043644

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 3 5 4], ⟨0 -19 -36 -16]]

Wedgie: ⟨⟨ 19 36 16 13 -28 -64 ]]

POTE generator: ~21/20 = 89.282

Template:Val list

Badness: 0.107482

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 5 4 8], ⟨0 -19 -36 -16 -61]]

POTE generator: ~21/20 = 89.291

Vals: Template:Val list

Badness: 0.045895

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 5 4 8 4], ⟨0 -19 -36 -16 -61 -4]]

POTE generator: ~21/20 = 89.286

Vals: Template:Val list

Badness: 0.028492

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