Octagar temperaments: Difference between revisions

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Octagar temperaments temper out the octagar comma, {{monzo|5 -4 3 -2}} = 4000/3969.
{{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).


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 considered below are slendi and slithy. Discussed elsewhere are:
* ''[[Ripple]]'' (+36/35) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* [[Garibaldi temperament|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|Shibboleth family]] and [[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]]
* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Pluto]]'' (+10976/10935) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]


== 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}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 29.181


[[POTE generator]]: ~49/48 = 29.181
{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
 
{{Val list|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}


[[Badness]]: 0.142903
[[Badness]]: 0.142903
Line 23: Line 36:
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 = 1200.000, ~49/48 = 29.180


Vals: {{Val list| 41, 206e }}
{{Optimal ET sequence|legend=0| 41, 206e }}


Badness: 0.063217
Badness: 0.063217
Line 36: Line 49:
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 = 1200.000, ~49/48 = 29.125


Vals: {{Val list| 41, 124ef, 165ef }}
{{Optimal ET sequence|legend=0| 41, 124ef, 165ef }}


Badness: 0.043644
Badness: 0.043644


== 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}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 89.282


[[POTE generator]]: ~21/20 = 89.282
{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
 
{{Val list|legend=1| 27, 94, 121, 215, 336d }}


[[Badness]]: 0.107482
[[Badness]]: 0.107482
Line 64: Line 75:
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 = 1200.000, ~21/20 = 89.291


Vals: {{Val list| 94, 121, 215, 336de, 551bde }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336de, 551bde }}


Badness: 0.045895
Badness: 0.045895
Line 77: Line 88:
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 = 1200.000, ~21/20 = 89.286


Vals: {{Val list| 94, 121, 215, 336def }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336def }}


Badness: 0.028492
Badness: 0.028492


== Quintupole ==
=== 17-limit ===
: ''See also: [[16ed5/2 #16ed5/2 as a generator]] or [[34ed7 #34ed7 as a generator]]''
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
 
Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286
 
{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
 
Badness: 0.019811
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
 
Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288
 
{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
 
Badness: 0.016408
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483


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).''
Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}


Subgroup: 2.3.5.7
Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297


[[Comma list]]: 4000/3969, 458752/455625
{{Optimal ET sequence|legend=0| 94, 121i, 215 }}


[[Mapping]]: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]
Badness: 0.014946


{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}
== Tridecatonic ==
{{See also| 13th-octave temperaments #Tridecatonic }}


[[POTE generator]]: ~135/128 = 99.175
The tridecatonic temperament ({{nowrap| 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 ({{nowrap| 26 & 39d }}, tempering out 4000/3969), [[ragismic microtemperaments #Trideci|trideci]] ({{nowrap| 26 & 39 }}, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] ({{nowrap| 26 & 130 }}, tempering out 321489/320000).


{{Vals|legend=1| 12, 97, 109, 121 }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.111620
[[Comma list]]: 4000/3969, 8748/8575
 
{{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}
 
[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077, ~3/2 = 701.9605
 
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
 
[[Badness]]: 0.139213


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 896/891, 1375/1372, 4375/4356
Comma list: 99/98, 1944/1925, 2560/2541


Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]
Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}


POTE generator: ~132/125 = 99.156
Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.7028


Vals: {{Val list| 12, 109, 121, 351bde, 472bdee }}
{{Optimal ET sequence|legend=0| 26, 39d, 65d }}


Badness: 0.056501
Badness: 0.063257


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 364/363, 625/624, 2704/2695
Comma list: 99/98, 144/143, 275/273, 1040/1029
 
Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}
 
Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573
 
{{Optimal ET sequence|legend=0| 26, 39df, 65d }}
 
Badness: 0.037851


Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]
== Dhaivatic ==
[[Subgroup]]: 2.3.5.7


POTE generator: ~55/52 = 99.165
[[Comma list]]: 4000/3969, 2100875/2097152


Vals: {{Val list| 12f, 109, 121 }}
{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}


Badness: 0.038431
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~6/5 = 319.198


=== 17-limit ===
{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
[[Badness]]: 0.168373


Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~18/17 = 99.172
Comma list: 385/384, 1331/1323, 4000/3969


Vals: {{Val list| 12f, 109, 121 }}
Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}


Badness: 0.028721
Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197


=== 19-limit ===
{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}
Subgroup: 2.3.5.7.11.13.17.19
 
Badness: 0.051216


Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]
Comma list: 275/273, 325/324, 385/384, 1331/1323


POTE generator: ~18/17 = 99.164
Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}


Vals: {{Val list| 12f, 109, 121 }}
Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165


Badness: 0.023818
{{Optimal ET sequence|legend=0| 15, 49bceff, 64cf, 79, 94 }}


[[Category:Temperament collection]]
Badness: 0.037468
[[Category:Octagar]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Octagar| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:28, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the octagar comma (monzo: [5 -4 3 -2⟩, ratio: 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]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~21/20 = 89.286

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

Badness: 0.028492

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 5 4 8 4 9], ⟨0 -19 -36 -16 -61 -4 -66]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286

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

Badness: 0.019811

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 3 5 4 8 4 9 7], ⟨0 -19 -36 -16 -61 -4 -66 -37]]

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288

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

Badness: 0.016408

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 3 5 4 8 4 9 7 11], ⟨0 -19 -36 -16 -61 -4 -66 -37 -87]]

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297

Optimal ET sequence: 94, 121i, 215

Badness: 0.014946

Tridecatonic

See also: 13th-octave 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 = 92.3077, ~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 = 92.3077, ~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 = 92.3077, ~3/2 = 701.3573

Optimal ET sequence: 26, 39df, 65d

Badness: 0.037851

Dhaivatic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 2100875/2097152

Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.198

Optimal ET sequence: 15, 64c, 79, 94, 203, 297

Badness: 0.168373

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 4000/3969

Mapping: [⟨1 -4 -3 6 0], ⟨0 21 20 -12 13]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197

Optimal ET sequence: 15, 64c, 79, 94, 203, 297c

Badness: 0.051216

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 275/273, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -4 -3 6 0 -8], ⟨0 21 20 -12 13 44]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165

Optimal ET sequence: 15, 49bceff, 64cf, 79, 94

Badness: 0.037468

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