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Breedsmic temperaments: Difference between revisions

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This page discusses miscellaneous rank-2 temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.
{{Technical data page}}
This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -5 -1 -2 4 }} = 2401/2400. This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.


The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that 49/40 × 10/7 = 7/4 and 49/40 × (10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Decimal]]'' → [[Dicot family #Decimal|Dicot family]] ({25/24, 49/48})
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' → [[Archytas clan #Beatles|Archytas clan]] ({64/63, 686/675})
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* [[Squares]] → [[Meantone family #Squares|Meantone family]] ({81/80, 2401/2400})
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] → [[Starling temperaments #Myna|Starling temperaments]] ({126/125, 1728/1715})
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] ({225/224, 1029/1024})
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' → [[Tetracot family #Octacot|Tetracot family]] ({245/243, 2401/2400})
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]] ({405/392, 1323/1280})
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quasitemp]]'' → [[Keemic temperaments #Quasitemp|Keemic temperaments]] ({875/864, 2401/2400})
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' → [[Diaschismic family #Quadrasruta|Diaschismic family]] ({2048/2025, 2401/2400})
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrimage]]'' → [[Magic family #Quadrimage|Magic family]] ({2401/2400, 3125/3072})
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]] and [[Hemimean clan #Hemiwürschmidt|hemimean clan]] ({2401/2400, 3136/3125})
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* [[Ennealimmal]] → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]] ({2401/2400, 4375/4374})
* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Quadritikleismic]]'' → [[Kleismic family #Quadritikleismic|Kleismic family]] ({2401/2400, 15625/15552})
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* [[Harry]] → [[Gravity family #Harry|Gravity family]] ({2401/2400, 19683/19600})
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Sesquiquartififths]]'' → [[Schismatic family #Sesquiquartififths|Schismatic family]] ({2401/2400, 32805/32768})
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' → [[Amity family #Amicable|Amity family]] ({2401/2400, 1600000/1594323})
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' → [[Gammic family #Neptune|Gammic family]] ({2401/2400, 48828125/48771072})
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Eagle]]'' → [[Vulture family #Eagle|Vulture family]] ({2401/2400, 10485760000/10460353203})
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


== Hemififths ==
== Hemififths ==
{{Main| Hemififths }}
{{Main| Hemififths }}


Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo|99EDO]] and [[140edo|140EDO]] providing good tunings, and [[239edo|239EDO]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5s, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7s. It may be called the 41&amp;58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS{{clarify}}.
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.


By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 5120/5103
[[Comma list]]: 2401/2400, 5120/5103


[[Mapping]]: [{{val| 1 1 -5 -1 }}, {{val| 0 2 25 13 }}]
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


{{Multival|legend=1| 2 25 13 35 15 -40 }}
: mapping generators: ~2, ~49/40


[[POTE generator]]: ~49/40 = 351.477
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo|1/5 0 1/25}}
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: [{{monzo|1 0 0 0}}, {{monzo|7/5 0 2/25 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}]
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: Eigenmonzos: 2, 5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Algebraic generator]]: (2 + sqrt(2))/2
Line 49: Line 56:
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}


[[Badness]]: 0.022243
[[Badness]] (Smith): 0.022243


=== 11-limit ===
=== 11-limit ===
Line 56: Line 63:
Comma list: 243/242, 441/440, 896/891
Comma list: 243/242, 441/440, 896/891


Mapping: [{{val| 1 1 -5 -1 2 }}, {{val| 0 2 25 13 5 }}]
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


POTE generator: ~11/9 = 351.521
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99e }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Badness: 0.023498
Badness (Smith): 0.023498


==== 13-limit ====
==== 13-limit ====
Line 69: Line 78:
Comma list: 144/143, 196/195, 243/242, 364/363
Comma list: 144/143, 196/195, 243/242, 364/363


Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


POTE generator: ~11/9 = 351.573
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


Badness: 0.019090
Badness (Smith): 0.019090


=== Semihemi ===
=== Semihemi ===
Line 82: Line 93:
Comma list: 2401/2400, 3388/3375, 5120/5103
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 
: mapping generators: ~99/70, ~400/231


POTE generator: ~49/40 = 351.505
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047


{{Optimal ET sequence|legend=1| 58, 140, 198 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Badness: 0.042487
Badness (Smith): 0.042487


==== 13-limit ====
==== 13-limit ====
Line 95: Line 110:
Comma list: 352/351, 676/675, 847/845, 1716/1715
Comma list: 352/351, 676/675, 847/845, 1716/1715


Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


POTE generator: ~49/40 = 351.502
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


{{Optimal ET sequence|legend=1| 58, 140, 198, 536f }}
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}


Badness: 0.021188
Badness (Smith): 0.021188


=== Quadrafifths ===
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense.  
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 110: Line 127:
Comma list: 2401/2400, 3025/3024, 5120/5103
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: [{{val| 1 1 -5 -1 8 }}, {{val| 0 4 50 26 -31 }}]
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 
: Mapping generators: ~2, ~243/220


POTE generator: ~243/220 = 175.7378
Optimal tunings:
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378


{{Optimal ET sequence|legend=1| 41, 157, 198, 239, 676b, 915be }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.040170
Badness (Smith): 0.040170


==== 13-limit ====
==== 13-limit ====
Line 123: Line 144:
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: [{{val| 1 1 -5 -1 8 10 }}, {{val| 0 4 50 26 -31 -43 }}]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


POTE generator: ~72/65 = 175.7470
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


{{Optimal ET sequence|legend=1| 41, 157, 198, 437f, 635bcff }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Badness: 0.031144
Badness (Smith): 0.031144


== Tertiaseptal ==
== Tertiaseptal ==
{{Main| Tertiaseptal }}
{{Main| Tertiaseptal }}


Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 65625/65536
[[Comma list]]: 2401/2400, 65625/65536


[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}


{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
: Mapping generators: ~2, ~256/245


[[POTE generator]]: ~256/245 = 77.191
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191


{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
Line 155: Line 178:
Comma list: 243/242, 441/440, 65625/65536
Comma list: 243/242, 441/440, 65625/65536


Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}


POTE generator: ~256/245 = 77.227
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171, 202 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}


Badness: 0.035576
Badness: 0.035576
Line 168: Line 191:
Comma list: 243/242, 441/440, 625/624, 3584/3575
Comma list: 243/242, 441/440, 625/624, 3584/3575


Mapping: [{{val| 1 3 2 3 7 1 }}, {{val| 0 -22 5 -3 -55 42 }}]
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}


POTE generator: ~117/112 = 77.203
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}


Badness: 0.036876
Badness: 0.036876
Line 181: Line 204:
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


Mapping: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}


POTE generator: ~68/65 = 77.201
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201


{{Optimal ET sequence|legend=1| 31, 109eg, 140e, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}


Badness: 0.027398
Badness: 0.027398
Line 194: Line 217:
Comma list: 385/384, 1331/1323, 1375/1372
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}


POTE generator: ~22/21 = 77.173
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173


{{Optimal ET sequence|legend=1| 31, 109, 140, 171e, 311e }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}


Badness: 0.030171
Badness: 0.030171
Line 207: Line 230:
Comma list: 352/351, 385/384, 625/624, 1331/1323
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}


POTE generator: ~22/21 = 77.158
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158


{{Optimal ET sequence|legend=1| 31, 109, 140, 311e, 451ee }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}


Badness: 0.028384
Badness: 0.028384
Line 220: Line 243:
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714


Mapping: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}


POTE generator: ~22/21 = 77.162
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162


{{Optimal ET sequence|legend=1| 31, 109g, 140, 311e, 451ee }}
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}


Badness: 0.022416
Badness: 0.022416
Line 233: Line 256:
Comma list: 2401/2400, 6250/6237, 65625/65536
Comma list: 2401/2400, 6250/6237, 65625/65536


Mapping: [{{val|1 3 2 3 -4}}, {{val|0 -22 5 -3 116}}]
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}


POTE generator: ~256/245 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}


Badness: 0.056926
Badness: 0.056926
Line 246: Line 269:
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


Mapping: [{{val|1 3 2 3 -4 1}}, {{val|0 -22 5 -3 116 42}}]
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}


POTE generator: ~117/112 = 77.168
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168


{{Optimal ET sequence|legend=1| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}


Badness: 0.027474
Badness: 0.027474
Line 259: Line 282:
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197


Mapping: [{{val|1 3 2 3 -4 1 1}}, {{val|0 -22 5 -3 116 42 48}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}


POTE generator: ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.018773
Badness: 0.018773
Line 272: Line 295:
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197


Mapping: [{{val|1 3 2 3 -4 1 1 11}}, {{val|0 -22 5 -3 116 42 48 -105}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}


POTE generator: ~68/65 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}


Badness: 0.017653
Badness: 0.017653
Line 285: Line 308:
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3}}, {{val|0 -22 5 -3 116 42 48 -105 117}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}


POTE generator: ~23/22 = 77.168
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfgg }}
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}


Badness: 0.015123
Badness: 0.015123
Line 298: Line 321:
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1}}, {{val|0 -22 5 -3 116 42 48 -105 117 60}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}


POTE generator: ~23/22 = 77.167
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfggj }}
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}


Badness: 0.012181
Badness: 0.012181
Line 311: Line 334:
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1 11}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}


POTE generator: ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.012311
Badness: 0.012311
Line 324: Line 347:
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1 11 0}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94 81}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}


POTE generator: ~23/22 = 77.170
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.010949
Badness: 0.010949
Line 337: Line 360:
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1 11 0 6}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}


POTE generator: ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Badness: 0.009825
Badness: 0.009825
Line 350: Line 373:
Comma list: 2401/2400, 3025/3024, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}


POTE generator: ~45/44 = 38.596
: Mapping generators: ~2, ~45/44


{{Optimal ET sequence|legend=1| 31, 280, 311, 342 }}
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}


Badness: 0.015633
Badness: 0.015633
Line 363: Line 388:
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}


POTE generator: ~45/44 = 38.588
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588


{{Optimal ET sequence|legend=1| 31, 280, 311, 964f, 1275f, 1586cff }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}


Badness: 0.033573
Badness: 0.033573
Line 376: Line 401:
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}


POTE generator: ~45/44 = 38.589
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589


{{Optimal ET sequence|legend=1| 31, 280, 311, 653f, 964f }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}


Badness: 0.025298
Badness: 0.025298
Line 389: Line 414:
Comma list: 2401/2400, 9801/9800, 65625/65536
Comma list: 2401/2400, 9801/9800, 65625/65536


Mapping: [{{val|2 6 4 6 1}}, {{val|0 -22 5 -3 46}}]
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}


POTE generator: ~256/245 = 77.193
: Mapping generators: ~99/70, ~256/245


{{Optimal ET sequence|legend=1| 62e, 140, 202, 342 }}
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
 
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}


Badness: 0.025790
Badness: 0.025790


== Quasiorwell ==
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = {{monzo|22 -1 -10 1}}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7s, or 384<sup>1/38</sup>, giving pure fifths.
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.


Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 29360128/29296875
[[Comma list]]: 2401/2400, 29360128/29296875


[[Mapping]]: [{{val|1 31 0 9}}, {{val|0 -38 3 -8}}]
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}


{{Multival|legend=1| 38 -3 8 -93 -94 27 }}
: Mapping generators: ~2, ~875/512


[[POTE generator]]: ~1024/875 = 271.107
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107


{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
Line 421: Line 448:
Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 2401/2400, 3025/3024, 5632/5625


Mapping: [{{val|1 31 0 9 53}}, {{val|0 -38 3 -8 -64}}]
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}


POTE generator: ~90/77 = 271.111
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111


{{Optimal ET sequence|legend=1| 31, 208, 239, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}


Badness: 0.017540
Badness: 0.017540
Line 434: Line 461:
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}]
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}


POTE generator: ~90/77 = 271.107
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107


{{Optimal ET sequence|legend=1| 31, 239, 270, 571, 841, 1111 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.017921
Badness: 0.017921
== Decoid ==
{{see also| Qintosec family #Decoid }}
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14 equal-step tuning|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&amp;270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].
Subgroup: 2.3.5.7
[[Comma list]]: 2401/2400, 67108864/66976875
[[Mapping]]: [{{val| 10 0 47 36 }}, {{val| 0 2 -3 -1 }}]
Mapping generators: ~15/14, ~8192/4725
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
[[POTE generator]]: ~16/15 = 111.099
{{Optimal ET sequence|legend=1| 10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
[[Badness]]: 0.033902
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 5632/5625, 9801/9800
Mapping: [{{val| 10 0 47 36 98 }}, {{val| 0 2 -3 -1 -8 }}]
POTE generator: ~16/15 = 111.070
{{Optimal ET sequence|legend=1| 10e, 130, 270, 670, 940, 1210, 2150c }}
Badness: 0.018735
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Mapping: [{{val| 10 0 47 36 98 37 }}, {{val| 0 2 -3 -1 -8 0 }}]
POTE generator: ~16/15 = 111.083
{{Optimal ET sequence|legend=1| 10e, 130, 270, 940, 1210f, 1480cf }}
Badness: 0.013475


== Neominor ==
== Neominor ==
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 177147/175616
[[Comma list]]: 2401/2400, 177147/175616


[[Mapping]]: [{{val|1 3 12 8}}, {{val|0 -6 -41 -22}}]
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}


{{Multival|legend=1|6 41 22 51 18 -64}}
: Mapping generators: ~2, ~189/160


[[POTE generator]]: ~189/160 = 283.280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Line 511: Line 491:
Comma list: 243/242, 441/440, 35937/35840
Comma list: 243/242, 441/440, 35937/35840


Mapping: [{{val|1 3 12 8 7}}, {{val|0 -6 -41 -22 -15}}]
Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}


POTE generator: ~33/28 = 283.276
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}


Badness: 0.027959
Badness: 0.027959
Line 524: Line 504:
Comma list: 169/168, 243/242, 364/363, 441/440
Comma list: 169/168, 243/242, 364/363, 441/440


Mapping: [{{val|1 3 12 8 7 7}}, {{val|0 -6 -41 -22 -15 -14}}]
Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}


POTE generator: ~13/11 = 283.294
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294


{{Optimal ET sequence|legend=1| 72, 161f, 233f }}
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}


Badness: 0.026942
Badness: 0.026942


== Emmthird ==
== Emmthird ==
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 14348907/14336000
[[Comma list]]: 2401/2400, 14348907/14336000


[[Mapping]]: [{{val|1 -3 -17 -8}}, {{val|0 14 59 33}}]
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}


{{Multival|legend=1|14 59 33 61 13 -89}}
: Mapping generators: ~2, ~2187/1372


[[POTE generator]]: ~2744/2187 = 392.988
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
Line 554: Line 534:
Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 243/242, 441/440, 1792000/1771561


Mapping: [{{val|1 -3 -17 -8 -8}}, {{val|0 14 59 33 35}}]
Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}


POTE generator: ~1372/1089 = 392.991
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.052358
Badness: 0.052358
Line 567: Line 547:
Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 243/242, 364/363, 441/440, 2200/2197


Mapping: [{{val|1 -3 -17 -8 -8 -13}}, {{val|0 14 59 33 35 51}}]
Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}


POTE generator: ~180/143 = 392.989
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.026974
Badness: 0.026974
Line 580: Line 560:
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Mapping: [{{val|1 -3 -17 -8 -8 -13 9}}, {{val|0 14 59 33 35 51 -15}}]
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


POTE generator: ~64/51 = 392.985
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.023205
Badness: 0.023205


== Quinmite ==
== Quinmite ==
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth".
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 1959552/1953125
[[Comma list]]: 2401/2400, 1959552/1953125


[[Mapping]]: [{{val|1 -7 -5 -3}}, {{val|0 34 29 23}}]
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}


{{Multival|legend=1|34 29 23 -33 -59 -28}}
: Mapping generators: ~2, ~42/25


[[POTE generator]]: ~25/21 = 302.997
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997


{{Optimal ET sequence|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}


[[Badness]]: 0.037322
[[Badness]]: 0.037322


== Unthirds ==
== Unthirds ==
The generator for unthirds temperament is undecimal major third, 14/11.
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.


Subgroup: 2.3.5.7
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 68359375/68024448
[[Comma list]]: 2401/2400, 68359375/68024448


[[Mapping]]: [{{val|1 -13 -14 -9}}, {{val|0 42 47 34}}]
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}


{{Multival|legend=1|42 47 34 -23 -64 -53}}
: Mapping generators: ~2, ~6125/3888


[[POTE generator]]: ~3969/3125 = 416.717
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
Line 627: Line 609:
Comma list: 2401/2400, 3025/3024, 4000/3993
Comma list: 2401/2400, 3025/3024, 4000/3993


Mapping: [{{val|1 -13 -14 -9 -8}}, {{val|0 42 47 34 33}}]
Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}


POTE generator: ~14/11 = 416.718
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}


Badness: 0.022926
Badness: 0.022926
Line 640: Line 622:
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: [{{val|1 -13 -14 -9 -9 -47}}, {{val|0 42 47 34 33 146}}]
Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}


POTE generator: ~14/11 = 416.716
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716


{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}


Badness: 0.020888
Badness: 0.020888


== Newt ==
== Newt ==
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].
Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 33554432/33480783
[[Comma list]]: 2401/2400, 33554432/33480783


[[Mapping]]: [{{val|1 1 19 11}}, {{val|0 2 -57 -28}}]
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}


{{Multival|legend=1|2 -57 -28 -95 -50 95}}
: mapping generators: ~2, ~49/40


[[POTE generator]]: ~49/40 = 351.113
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}


[[Badness]]: 0.041878
[[Badness]]: 0.041878
Line 670: Line 652:
Comma list: 2401/2400, 3025/3024, 19712/19683
Comma list: 2401/2400, 3025/3024, 19712/19683


Mapping: [{{val|1 1 19 11 -10}}, {{val|0 2 -57 -28 46}}]
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}


POTE generator: ~49/40 = 351.115
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}


Badness: 0.019461
Badness: 0.019461
Line 683: Line 665:
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095


Mapping: [{{val|1 1 19 11 -10 -20}}, {{val|0 2 -57 -28 46 81}}]
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}


POTE generator: ~49/40 = 351.117
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117


{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}


Badness: 0.013830
Badness: 0.013830
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}


== Septidiasemi ==
== Septidiasemi ==
Line 696: Line 689:
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 2152828125/2147483648
[[Comma list]]: 2401/2400, 2152828125/2147483648


[[Mapping]]: [{{val| 1 -1 6 4 }}, {{val| 0 26 -37 -12 }}]
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


{{Multival|legend=1|26 -37 -12 -119 -92 76}}
: Mapping generators: ~2, ~28/15


[[POTE generator]]: ~15/14 = 119.297
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
Line 717: Line 710:
Comma list: 243/242, 441/440, 939524096/935859375
Comma list: 243/242, 441/440, 939524096/935859375


Mapping: [{{val| 1 -1 6 4 -3 }}, {{val| 0 26 -37 -12 65 }}]
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}


POTE generator: ~15/14 = 119.279
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}


Badness: 0.090687
Badness: 0.090687
Line 730: Line 723:
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Mapping: [{{val| 1 -1 6 4 -3 4 }}, {{val| 0 26 -37 -12 65 -3 }}]
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}


POTE generator: ~15/14 = 119.281
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 835eeff }}
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}


Badness: 0.045773
Badness: 0.045773
Line 743: Line 736:
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575


Mapping: [{{val| 1 -1 6 4 -3 4 2 }}, {{val| 0 26 -37 -12 65 -3 21 }}]
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}


POTE generator: ~15/14 = 119.281
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 503ef, 835eeff }}
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Badness: 0.027322
Badness: 0.027322


== Maviloid ==
== Maviloid ==
{{see also| Ragismic microtemperaments #Parakleismic }}
{{See also| Ragismic microtemperaments #Parakleismic }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 1224440064/1220703125
[[Comma list]]: 2401/2400, 1224440064/1220703125


[[Mapping]]: [{{val| 1 31 34 26 }}, {{val| 0 -52 -56 -41 }}]
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}


{{Multival|legend=1|52 56 41 -32 -81 -62}}
: Mapping generators: ~2, ~1296/875


[[POTE generator]]: ~1296/875 = 678.810
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810


{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
Line 771: Line 764:
{{See also| Luna family }}
{{See also| Luna family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 274877906944/274658203125
[[Comma list]]: 2401/2400, 274877906944/274658203125


[[Mapping]]: [{{val| 1 19 0 6 }}, {{val| 0 -60 8 -11 }}]
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


{{Multival|legend=1|60 -8 11 -152 -151 48}}
: Mapping generators: ~2, ~57344/46875


[[POTE generator]]: ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301


{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}


[[Badness]]: 0.045792
[[Badness]]: 0.045792
Line 788: Line 781:
{{See also| Metric microtemperaments #Geb }}
{{See also| Metric microtemperaments #Geb }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 31381059609/31360000000
[[Comma list]]: 2401/2400, 31381059609/31360000000


[[Mapping]]: [{{val| 1 13 33 21 }}, {{val| 0 -32 -86 -51 }}]
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}


{{Multival|legend=1|32 86 51 62 -9 -123}}
: Mapping generators: ~2, ~2800/2187


[[POTE generator]]: ~2800/2187 = 428.066
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


[[Badness]]: 0.028307
[[Badness]]: 0.028307


== Gorgik ==
== Gorgik ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 28672/28125
[[Comma list]]: 2401/2400, 28672/28125


[[Mapping]]: [{{val| 1 5 1 3 }}, {{val| 0 -18 7 -1 }}]
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


{{Multival|legend=1|18 -7 1 -53 -49 22}}
: Mapping generators: ~2, ~8/7


[[POTE generator]]: ~8/7 = 227.512
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
Line 822: Line 815:
Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: [{{val| 1 5 1 3 1 }}, {{val| 0 -18 7 -1 13 }}]
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


POTE generator: ~8/7 = 227.500
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness: 0.059260
Badness: 0.059260
Line 835: Line 828:
Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 176/175, 196/195, 364/363, 512/507


Mapping: [{{val| 1 5 1 3 1 2 }}, {{val| 0 -18 7 -1 13 9 }}]
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}


POTE generator: ~8/7 = 227.493
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bcef, 211bccdeeff }}
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness: 0.032205
Badness: 0.032205


== Fibo ==
== Fibo ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 341796875/339738624
[[Comma list]]: 2401/2400, 341796875/339738624


[[Mapping]]: [{{val| 1 19 8 10 }}, {{val| 0 -46 -15 -19 }}]
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}


{{Multival|legend=1|46 15 19 -83 -99 2}}
: Mapping generators: ~2, ~125/96


[[POTE generator]]: ~125/96 = 454.310
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310


{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}


Badness: 0.100511
Badness: 0.100511
Line 863: Line 856:
Comma list: 385/384, 1375/1372, 43923/43750
Comma list: 385/384, 1375/1372, 43923/43750


Mapping: [{{val| 1 19 8 10 8 }}, {{val| 0 -46 -15 -19 -12 }}]
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


POTE generator: ~100/77 = 454.318
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


Badness: 0.056514
Badness: 0.056514
Line 876: Line 869:
Comma list: 385/384, 625/624, 847/845, 1375/1372
Comma list: 385/384, 625/624, 847/845, 1375/1372


Mapping: [{{val| 1 19 8 10 8 9 }}, {{val| 0 -46 -15 -19 -12 -14 }}]
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}


POTE generator: ~13/10 = 454.316
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


Badness: 0.027429
Badness: 0.027429


== Mintone ==
== Mintone ==
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&amp;103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -3 11 -5 -1 }} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 177147/175000
[[Comma list]]: 2401/2400, 177147/175000


[[Mapping]]: [{{val| 1 5 9 7 }}, {{val| 0 -22 -43 -27 }}]
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


{{Multival|legend=1|22 43 27 17 -19 -58}}
: Mapping generators: ~2, ~10/9


[[POTE generator]]: ~10/9 = 186.343
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
Line 906: Line 899:
Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 43923/43750


Mapping: [{{val| 1 5 9 7 12 }}, {{val| 0 -22 -43 -27 -55 }}]
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}


POTE generator: ~10/9 = 186.345
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586b, 747bc }}
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}


Badness: 0.039962
Badness: 0.039962
Line 919: Line 912:
Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 243/242, 351/350, 441/440, 847/845


Mapping: [{{val| 1 5 9 7 12 11 }}, {{val| 0 -22 -43 -27 -55 -47 }}]
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}


POTE generator: ~10/9 = 186.347
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


Badness: 0.021849
Badness: 0.021849
Line 932: Line 925:
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Mapping: [{{val| 1 5 9 7 12 11 3 }}, {{val| 0 -22 -43 -27 -55 -47 7 }}]
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}


POTE generator: ~10/9 = 186.348
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


Badness: 0.020295
Badness: 0.020295


== Catafourth ==
== Catafourth ==
{{see also| Sensipent family }}
{{See also| Sensipent family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 78732/78125
[[Comma list]]: 2401/2400, 78732/78125


[[Mapping]]: [{{val| 1 13 17 13 }}, {{val| 0 -28 -36 -25 }}]
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


{{Multival|legend=1| 28 36 25 -8 -39 -43 }}
: Mapping generators: ~2, ~250/189


[[POTE generator]]: ~250/189 = 489.235
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
Line 962: Line 955:
Comma list: 243/242, 441/440, 78408/78125
Comma list: 243/242, 441/440, 78408/78125


Mapping: [{{val| 1 13 17 13 32 }}, {{val| 0 -28 -36 -25 -70 }}]
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}


POTE generator: ~250/189 = 489.252
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252


{{Optimal ET sequence|legend=1| 103, 130, 233, 363, 493e, 856be }}
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}


Badness: 0.036785
Badness: 0.036785
Line 975: Line 968:
Comma list: 243/242, 351/350, 441/440, 10985/10976
Comma list: 243/242, 351/350, 441/440, 10985/10976


Mapping: [{{val| 1 13 17 13 32 9 }}, {{val| 0 -28 -36 -25 -70 -13 }}]
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}


POTE generator: ~65/49 = 489.256
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256


{{Optimal ET sequence|legend=1| 103, 130, 233, 363 }}
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}


Badness: 0.021694
Badness: 0.021694


== Cotritone ==
== Cotritone ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 390625/387072
[[Comma list]]: 2401/2400, 390625/387072


[[Mapping]]: [{{val| 1 -13 -4 -4 }}, {{val| 0 30 13 14 }}]
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}


{{Multival|legend=1|30 13 14 -49 -62 -4}}
: Mappping generators: ~2, ~10/7


[[POTE generator]]: ~7/5 = 583.385
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
Line 1,003: Line 996:
Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: [{{val| 1 -13 -4 -4 2 }}, {{val| 0 30 13 14 3 }}]
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}


POTE generator: ~7/5 = 583.387
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}


Badness: 0.032225
Badness: 0.032225
Line 1,016: Line 1,009:
Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 169/168, 364/363, 385/384, 625/624


Mapping: [{{val| 1 -13 -4 -4 2 -7 }}, {{val| 0 30 13 14 3 22 }}]
Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}


POTE generator: ~7/5 = 583.387
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


{{Optimal ET sequence|legend=1| 37, 72, 109, 181f }}
Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}


Badness: 0.028683
Badness: 0.028683
Line 1,027: Line 1,020:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 3645/3584
[[Comma list]]: 2401/2400, 3645/3584


[[Mapping]]: [{{Val|1 1 9 6}}, {{Val|0 2 -23 -11}}]
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
 
: Mapping generators: ~2, ~49/40


[[POTE generator]]: ~49/40 = 348.603
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
Line 1,044: Line 1,039:
Comma list: 243/242, 441/440, 1815/1792
Comma list: 243/242, 441/440, 1815/1792


Mapping: [{{Val|1 1 9 6 2}}, {{Val|0 2 -23 -11 5}}]
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


POTE generator: ~11/9 = 348.639
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639


{{Optimal ET sequence|legend=1| 31, 86ce, 117ce, 148bce }}
Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}


Badness: 0.046181
Badness: 0.046181
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: Mapping generators: ~2, ~3828125/2985984
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
[[Badness]] (Smith): 0.0597
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 766656/765625
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
Badness (Smith): 0.0262
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
Badness (Smith): 0.0160
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Badness (Smith): 0.0210
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 549755813888/533935546875
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
: mapping generators: ~2, ~7/4
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
[[Badness]] (Sintel): 4.40


== Surmarvelpyth ==
== Surmarvelpyth ==
Line 1,059: Line 1,157:
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}


[[Mapping]]: {{val| 1 43 -74 -25 }}, {{val| 0 -70 129 47 }}
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}


Mapping generators: ~2, ~675/448
: Mapping generators: ~2, ~675/448


[[Optimal tuning]] ([[CTE]]): ~675/448 = 709.9719
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}


[[Badness]]: 0.202
[[Badness]]: 0.202249


=== 11-limit ===
=== 11-limit ===
Line 1,074: Line 1,172:
Comma list: 2401/2400, 820125/819896, 2097152/2096325
Comma list: 2401/2400, 820125/819896, 2097152/2096325


Mapping: {{val| 1 43 -74 -25 36 }}, {{val| 0 -70 129 47 -55 }}
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}


Optimal tuning (CTE): ~675/448 = 709.9720
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795 }}
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.0523
Badness: 0.052308


=== 13-limit ===
=== 13-limit ===
Line 1,087: Line 1,185:
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167


Mapping: {{val| 1 43 -74 -25 36 25 }}, {{val| 0 -70 129 47 -55 -36 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}


Optimal tuning (CTE): ~98/65 = 709.9723
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795f }}
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}


Badness: 0.0325
Badness: 0.032503


=== 17-limit ===
=== 17-limit ===
Line 1,100: Line 1,198:
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619


Mapping: {{val| 1 43 -74 -25 36 25 -103 }}, {{val| 0 -70 129 47 -55 -36 181 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}


Optimal tuning (CTE): ~98/65 = 709.9722
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.0325
Badness: 0.020995


=== 19-limit ===
=== 19-limit ===
Line 1,113: Line 1,211:
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984


Mapping: {{val| 1 43 -74 -25 36 25 -103 -49 }}, {{val| 0 -70 129 47 -55 -36 181 90 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722


Optimal tuning (CTE): ~98/65 = 709.9722
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Badness: 0.013771


Badness: 0.0138
== Notes ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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