Octagar temperaments: Difference between revisions

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


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


== Slendi ==
== Slendi ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 84035/82944
[[Comma list]]: 4000/3969, 84035/82944


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -17 -28 -8 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}


{{Multival|legend=1| 17 28 8 5 -35 -60 }}
{{Multival|legend=1| 17 28 8 5 -35 -60 }}


[[POTE generator]]: ~49/48 = 29.181
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 29.181


{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
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Comma list: 100/99, 245/242, 46656/46585
Comma list: 100/99, 245/242, 46656/46585


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -17 -28 -8 -22 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}


POTE generator: ~49/48 = 29.180
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.180


{{Optimal ET sequence|legend=1| 41, 206e }}
{{Optimal ET sequence|legend=1| 41, 206e }}
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Comma list: 100/99, 144/143, 245/242, 343/338
Comma list: 100/99, 144/143, 245/242, 343/338


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -17 -28 -8 -22 -1 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -1 }}


POTE generator: ~49/48 = 29.125
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.125


{{Optimal ET sequence|legend=1| 41, 124ef, 165ef }}
{{Optimal ET sequence|legend=1| 41, 124ef, 165ef }}
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== Slithy ==
== Slithy ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 420175/419904
[[Comma list]]: 4000/3969, 420175/419904


[[Mapping]]: [{{val| 1 3 5 4 }}, {{val| 0 -19 -36 -16 }}]
{{Mapping|legend=1| 1 3 5 4 | 0 -19 -36 -16 }}


{{Multival|legend=1| 19 36 16 13 -28 -64 }}
{{Multival|legend=1| 19 36 16 13 -28 -64 }}


[[POTE generator]]: ~21/20 = 89.282
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 89.282


{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
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Comma list: 540/539, 2200/2187, 4375/4356
Comma list: 540/539, 2200/2187, 4375/4356


Mapping: [{{val| 1 3 5 4 8 }}, {{val| 0 -19 -36 -16 -61 }}]
Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}


POTE generator: ~21/20 = 89.291
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.291


{{Optimal ET sequence|legend=1| 94, 121, 215, 336de, 551bde }}
{{Optimal ET sequence|legend=1| 94, 121, 215, 336de, 551bde }}
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Comma list: 325/324, 352/351, 540/539, 1575/1573
Comma list: 325/324, 352/351, 540/539, 1575/1573


Mapping: [{{val| 1 3 5 4 8 4 }}, {{val| 0 -19 -36 -16 -61 -4 }}]
Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}


POTE generator: ~21/20 = 89.286
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.286


{{Optimal ET sequence|legend=1| 94, 121, 215, 336def }}
{{Optimal ET sequence|legend=1| 94, 121, 215, 336def }}
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== Tridecatonic ==
== Tridecatonic ==
{{see also| High badness temperaments #Tridecatonic }}
{{See also| High badness temperaments #Tridecatonic }}


The ''tridecatonic'' temperament (26&39) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo|-11 26 -13}} in the 5-limit. There are some extensions for 7-limit, including ''tridecatonic'' (26&39d, tempering out 4000/3969), ''[[Ragismic microtemperaments #Trideci|trideci]]'' (26&39, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] (26&130, tempering out 321489/320000).
The tridecatonic temperament (26 & 39) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 & 39d, tempering out 4000/3969), [[Ragismic microtemperaments #Trideci|trideci]] (26 & 39, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] (26 & 130, tempering out 321489/320000).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 8748/8575
[[Comma list]]: 4000/3969, 8748/8575


[[Mapping]]: [{{val|13 21 31 37}}, {{val|0 -1 -2 -1}}]
{{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}


[[POTE generator]]: ~3/2 = 701.9605
[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\13, ~3/2 = 701.9605


{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
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Comma list: 99/98, 1944/1925, 2560/2541
Comma list: 99/98, 1944/1925, 2560/2541


Mapping: [{{val|13 21 31 37 45}}, {{val|0 -1 -2 -1 0}}]
Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}


POTE generator: ~3/2 = 701.7028
Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.7028


{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
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Comma list: 99/98, 144/143, 275/273, 1040/1029
Comma list: 99/98, 144/143, 275/273, 1040/1029


Mapping: [{{val|13 21 31 37 45 49}}, {{val|0 -1 -2 -1 0 -2}}]
Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}


POTE generator: ~3/2 = 701.3573
Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.3573


{{Optimal ET sequence|legend=1| 26, 39df, 65d }}
{{Optimal ET sequence|legend=1| 26, 39df, 65d }}
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Octagar| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 08:53, 21 September 2023

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

Temperaments considered below are slendi and slithy. Discussed elsewhere are:

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.181

Optimal ET sequence: 41, 206, 247bc, 288bcd, 329bcd

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]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.180

Optimal ET sequence: 41, 206e

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]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.125

Optimal ET sequence: 41, 124ef, 165ef

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.282

Optimal ET sequence: 27, 94, 121, 215, 336d

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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.291

Optimal ET sequence: 94, 121, 215, 336de, 551bde

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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.286

Optimal ET sequence: 94, 121, 215, 336def

Badness: 0.028492

Tridecatonic

See also: High badness temperaments § Tridecatonic

The tridecatonic temperament (26 & 39) has a period of 1/13 octave. It tempers out the tridecalimma, [-11 26 -13⟩ in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 & 39d, tempering out 4000/3969), trideci (26 & 39, tempering out 4375/4374), and bosonic (26 & 130, tempering out 321489/320000).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 8748/8575

Mapping: [⟨13 21 31 37], ⟨0 -1 -2 -1]]

Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.9605

Optimal ET sequence: 26, 39d, 65d

Badness: 0.139213

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨13 21 31 37 45], ⟨0 -1 -2 -1 0]]

Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.7028

Optimal ET sequence: 26, 39d, 65d

Badness: 0.063257

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨13 21 31 37 45 49], ⟨0 -1 -2 -1 0 -2]]

Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.3573

Optimal ET sequence: 26, 39df, 65d

Badness: 0.037851

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