@@ Line 1: / Line 1: @@
-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 ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
@@ Line 23: / Line 36: @@
 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
@@ Line 36: / Line 49: @@
 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
 == Slithy ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
@@ Line 64: / Line 75: @@
 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
@@ Line 77: / Line 88: @@
 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
-== Quintupole ==
+=== 17-limit ===
-: ''See also: [[16ed5/2 #Regular temperaments]] or [[34ed7 #Regular temperaments]]''
+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
-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).''
+=== 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
-Subgroup: 2.3.5.7
+Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}
-[[Comma list]]: 4000/3969, 458752/455625
+Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288
-[[Mapping]]: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]
+{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
-{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}
+Badness: 0.016408
-[[POTE generator]]: ~135/128 = 99.175
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
-{{Vals|legend=1| 12, 97, 109, 121 }}
+Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
-[[Badness]]: 0.111620
+Mapping: {{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|legend=0| 94, 121i, 215 }}
+Badness: 0.014946
+== Tridecatonic ==
+{{See also| 13th-octave temperaments #Tridecatonic }}
+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).
+[[Subgroup]]: 2.3.5.7
+[[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 ===
 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: ~35/33 = 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 ===
 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: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]
+Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}
-POTE generator: ~35/33 = 99.165
+Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573
-Vals: {{Val list| 12f, 109, 121 }}
+{{Optimal ET sequence|legend=0| 26, 39df, 65d }}
-Badness: 0.038431
+Badness: 0.037851
-=== 17-limit ===
+== Dhaivatic ==
-Subgroup: 2.3.5.7.11.13.17
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4000/3969, 2100875/2097152
+{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~6/5 = 319.198
+{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
+[[Badness]]: 0.168373
-Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]
+Comma list: 385/384, 1331/1323, 4000/3969
-POTE generator: ~18/17 = 99.172
+Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}
-Vals: {{Val list| 12f, 109, 121 }}
+Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197
-Badness: 0.028721
+{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}
-=== 19-limit ===
+Badness: 0.051216
-Subgroup: 2.3.5.7.11.13.17.19
-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]]

Octagar temperaments: Difference between revisions