Breedsmic temperaments: Difference between revisions

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{{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.
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.


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* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2401/2400) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
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* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


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{{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]] and [[140edo]] providing 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 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.
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: mapping generators: ~2, ~49/40
: mapping generators: ~2, ~49/40


{{Multival|legend=1| 2 25 13 35 15 -40 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.477
: [[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 list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 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 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Algebraic generator]]: (2 + sqrt(2))/2
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{{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 ===
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Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ====
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Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ===
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Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}


: mapping generators: ~99/70, ~49/40
: mapping generators: ~99/70, ~400/231


Optimal tuning (POTE): ~99/70 = 1\2, ~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 ====
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Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~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
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Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}


: mapping generators: ~2, ~243/220
: Mapping generators: ~2, ~243/220


Optimal tuning (POTE): ~2 = 1\1, ~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 ====
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Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


Optimal tuning (POTE): ~2 = 1\1, ~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]] 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
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{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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 310: Line 325:
Optimal tuning (POTE): ~2 = 1\1, ~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 323: Line 338:
Optimal tuning (POTE): ~2 = 1\1, ~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 336: Line 351:
Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}


: mapping generators: ~2, ~45/44
: Mapping generators: ~2, ~45/44


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596


{{Optimal ET sequence|legend=1| 31, 280, 311, 342 }}
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}


Badness: 0.015633
Badness: 0.015633
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Optimal tuning (POTE): ~2 = 1\1, ~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
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Optimal tuning (POTE): ~2 = 1\1, ~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 401: Line 416:
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}


: mapping generators: ~99/70, ~256/245
: Mapping generators: ~99/70, ~256/245


Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193


{{Optimal ET sequence|legend=1| 62e, 140, 202, 342 }}
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}


Badness: 0.025790
Badness: 0.025790
Line 412: Line 427:
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.
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
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{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
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Optimal tuning (POTE): ~2 = 1\1, ~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 452: Line 465:
Optimal tuning (POTE): ~2 = 1\1, ~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| Quintosec family #Decoid }}
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used as 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 [[quintosec]] temperament.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 67108864/66976875
{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
: mapping generators: ~15/14, ~8192/4725
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099)
{{Optimal ET sequence|legend=1| 10, 120, 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: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~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: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083)
{{Optimal ET sequence|legend=1| 10e, 130, 270, 940, 1210f, 1480cf }}
Badness: 0.013475


== Neominor ==
== Neominor ==
Line 512: Line 478:
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
Line 531: Line 495:
Optimal tuning (POTE): ~2 = 1\1, ~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 544: Line 508:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 557: Line 521:
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}


: mapping generators: ~2, ~2187/1372
: Mapping generators: ~2, ~2187/1372
 
{{Multival|legend=1| 14 59 33 61 13 -8 9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
Line 576: Line 538:
Optimal tuning (POTE): ~2 = 1\1, ~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 589: Line 551:
Optimal tuning (POTE): ~2 = 1\1, ~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 602: Line 564:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 615: Line 577:
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
Line 626: Line 586:


== 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.
 
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
[[Subgroup]]: 2.3.5.7
Line 634: Line 596:
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
Line 653: Line 613:
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


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


Badness: 0.022926
Badness: 0.022926
Line 666: Line 626:
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716


{{Optimal ET sequence|legend=1| 72, 239f, 311, 694, 1005c }}
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
Line 680: Line 640:


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}


[[Badness]]: 0.041878
[[Badness]]: 0.041878
Line 698: Line 656:
Optimal tuning (POTE): ~2 = 1\1, ~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 }}
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}


Badness: 0.019461
Badness: 0.019461
Line 711: Line 669:
Optimal tuning (POTE): ~2 = 1\1, ~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 }}
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 726: Line 695:
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
Line 747: Line 714:
Optimal tuning (POTE): ~2 = 1\1, ~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 760: Line 727:
Optimal tuning (POTE): ~2 = 1\1, ~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 773: Line 740:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 786: Line 753:
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
Line 805: Line 770:
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
Line 824: Line 787:
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
Line 841: Line 802:
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
Line 860: Line 819:
Optimal tuning (POTE): ~2 = 1\1, ~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 873: Line 832:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 884: Line 843:
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
Line 903: Line 860:
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318


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


Badness: 0.056514
Badness: 0.056514
Line 916: Line 873:
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316


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


Badness: 0.027429
Badness: 0.027429
Line 929: Line 886:
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
Line 948: Line 903:
Optimal tuning (POTE): ~2 = 1\1, ~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 961: Line 916:
Optimal tuning (POTE): ~2 = 1\1, ~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 974: Line 929:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 987: Line 942:
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
Line 1,006: Line 959:
Optimal tuning (POTE): ~2 = 1\1, ~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 1,019: Line 972:
Optimal tuning (POTE): ~2 = 1\1, ~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
Line 1,030: Line 983:
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
Line 1,049: Line 1,000:
Optimal tuning (POTE): ~2 = 1\1, ~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 1,062: Line 1,013:
Optimal tuning (POTE): ~2 = 1\1, ~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 1,075: Line 1,026:
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}


: mapping generators: ~2, ~49/40
: Mapping generators: ~2, ~49/40


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
Line 1,092: Line 1,043:
Optimal tuning (POTE): ~2 = 1\1, ~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 1,105: Line 1,159:
{{Mapping|legend=1| 1 43 -74 -25 | 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]]): ~2 = 1\1, ~675/448 = 709.9719
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
Line 1,111: Line 1,165:
{{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 1,122: Line 1,176:
Optimal tuning (CTE): ~2 = 1\1, ~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 1,135: Line 1,189:
Optimal tuning (CTE): ~2 = 1\1, ~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 1,148: Line 1,202:
Optimal tuning (CTE): ~2 = 1\1, ~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 1,161: Line 1,215:
Optimal tuning (CTE): ~2 = 1\1, ~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: 0.013771


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:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]