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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]] [[regular temperament|temperaments]] [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -5 -1 -2 4 }}, [[ratio]]: 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})
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Beatles]]'' → [[Archytas clan #Beatles|Archytas clan]] ({64/63, 686/675})
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
* [[Squares]] → [[Meantone family #Squares|Meantone family]] ({81/80, 2401/2400})
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* [[Myna]] → [[Starling temperaments #Myna|Starling temperaments]] ({126/125, 1728/1715})
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] ({225/224, 1029/1024})
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Octacot]]'' → [[Tetracot family #Octacot|Tetracot family]] ({245/243, 2401/2400})
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Greenwood]]'' → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]] ({405/392, 1323/1280})
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Quasitemp]]'' → [[Keemic temperaments #Quasitemp|Keemic temperaments]] ({875/864, 2401/2400})
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrasruta]]'' → [[Diaschismic family #Quadrasruta|Diaschismic family]] ({2048/2025, 2401/2400})
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* ''[[Quadrimage]]'' → [[Magic family #Quadrimage|Magic family]] ({2401/2400, 3125/3072})
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Hemiwürschmidt]]'' → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]] and [[Hemimean clan #Hemiwürschmidt|hemimean clan]] ({2401/2400, 3136/3125})
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* [[Ennealimmal]] → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]] ({2401/2400, 4375/4374})
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Quadritikleismic]]'' → [[Kleismic family #Quadritikleismic|Kleismic family]] ({2401/2400, 15625/15552})
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Harry]] → [[Gravity family #Harry|Gravity family]] ({2401/2400, 19683/19600})
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Sesquiquartififths]]'' → [[Schismatic family #Sesquiquartififths|Schismatic family]] ({2401/2400, 32805/32768})
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic 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})
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Eagle]]'' → [[Vulture family #Eagle|Vulture family]] ({2401/2400, 10485760000/10460353203})
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


== Hemififths ==
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].
{{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}}.
== Tertiaseptal ==
{{Main| Tertiaseptal }}


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.
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 {{nowrap| 31 & 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. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  


Subgroup: 2.3.5.7
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) 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.


[[Comma list]]: 2401/2400, 5120/5103
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 1 -5 -1 }}, {{val| 0 2 25 13 }}]
[[Comma list]]: 2401/2400, 65625/65536


{{Multival|legend=1| 2 25 13 35 15 -40 }}
{{Mapping|legend=1| 1 -19 7 0 | 0 22 -5 3 }}
: mapping generators: ~2, ~245/128


[[POTE generator]]: ~49/40 = 351.477
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979{{c}})
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899{{c}})
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
* [[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}}]
: Eigenmonzos: 2, 5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Badness]] (Sintel): 0.329
 
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}
 
[[Badness]]: 0.022243


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


Comma list: 243/242, 441/440, 896/891
Comma list: 243/242, 441/440, 65625/65536


Mapping: [{{val| 1 1 -5 -1 2 }}, {{val| 0 2 25 13 5 }}]
Mapping: {{mapping| 1 -19 7 0 -48 | 0 22 -5 3 55 }}


POTE generator: ~11/9 = 351.521
Optimal tunings:  
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99e }}
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Badness: 0.023498
Badness (Sintel): 1.18


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


Comma list: 144/143, 196/195, 243/242, 364/363
Comma list: 243/242, 441/440, 625/624, 3584/3575


Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
Mapping: {{mapping| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}


POTE generator: ~11/9 = 351.573
Optimal tunings:  
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}


Badness: 0.019090
Badness (Sintel): 1.52


=== Semihemi ===
==== 17-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 2401/2400, 3388/3375, 5120/5103
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]
Mapping: {{mapping| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}


POTE generator: ~49/40 = 351.505
Optimal tunings:  
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})


{{Optimal ET sequence|legend=1| 58, 140, 198 }}
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}


Badness: 0.042487
Badness (Sintel): 1.40


==== 13-limit ====
=== Tertia ===
Subgroup: 2.3.5.7.11.13
Subgroup:2.3.5.7.11


Comma list: 352/351, 676/675, 847/845, 1716/1715
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]
Mapping: {{mapping| 1 -19 7 0 -19 | 0 22 -5 3 24 }}


POTE generator: ~49/40 = 351.502
Optimal tunings:  
* WE: ~2 = 1200.2336{{c}}, ~21/11 = 1123.0454{{c}} (~22/21 = 77.1882{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8311{{c}} (~22/21 = 77.1689{{c}})


{{Optimal ET sequence|legend=1| 58, 140, 198, 536f }}
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}


Badness: 0.021188
Badness (Sintel): 0.997
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: [{{val| 1 1 -5 -1 8 }}, {{val| 0 4 50 26 -31 }}]
 
POTE generator: ~243/220 = 175.7378
 
{{Optimal ET sequence|legend=1| 41, 157, 198, 239, 676b, 915be }}
 
Badness: 0.040170


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


Comma list: 352/351, 847/845, 2401/2400, 3025/3024
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: [{{val| 1 1 -5 -1 8 10 }}, {{val| 0 4 50 26 -31 -43 }}]
Mapping: {{mapping| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}


POTE generator: ~72/65 = 175.7470
Optimal tunings:  
* WE: ~2 = 1200.1395{{c}}, ~21/11 = 1122.9727{{c}} (~22/21 = 77.1669{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8426{{c}} (~22/21 = 77.1574{{c}})


{{Optimal ET sequence|legend=1| 41, 157, 198, 437f, 635bcff }}
{{Optimal ET sequence|legend=0| 31, 78f, 109, 140 }}


Badness: 0.031144
Badness (Sintel): 1.17


== Tertiaseptal ==
==== 17-limit ====
{{Main| Tertiaseptal }}
Subgroup: 2.3.5.7.11.13.17


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.
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}


[[Comma list]]: 2401/2400, 65625/65536
Optimal tunings:  
* WE: ~2 = 1200.1655{{c}}, ~21/11 = 1122.9926{{c}} (~22/21 = 77.1729{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8376{{c}} (~22/21 = 77.1624{{c}})


[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]
{{Optimal ET sequence|legend=0| 31, 78fg, 109g, 140 }}


{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
Badness (Sintel): 1.14


[[POTE generator]]: ~256/245 = 77.191
=== Tertiaseptia ===
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.  


{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
[[Badness]]: 0.012995
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 65625/65536
Comma list: 2401/2400, 6250/6237, 65625/65536


Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]
Mapping: {{mapping| 1 -19 7 0 112 | 0 22 -5 3 -116 }}


POTE generator: ~256/245 = 77.227
Optimal tunings:  
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171, 202 }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness: 0.035576
Badness (Sintel): 1.88


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


Comma list: 243/242, 441/440, 625/624, 3584/3575
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


Mapping: [{{val| 1 3 2 3 7 1 }}, {{val| 0 -22 5 -3 -55 42 }}]
Mapping: {{mapping| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}


POTE generator: ~117/112 = 77.203
Optimal tunings:  
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}


Badness: 0.036876
Badness (Sintel): 1.14


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
 
Mapping: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]
 
POTE generator: ~68/65 = 77.201
 
{{Optimal ET sequence|legend=1| 31, 109eg, 140e, 171 }}
 
Badness: 0.027398
 
=== Tertia ===
Subgroup:2.3.5.7.11


Comma list: 385/384, 1331/1323, 1375/1372
Mapping: {{mapping| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}


Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]
Optimal tunings:  
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})


POTE generator: ~22/21 = 77.173
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


{{Optimal ET sequence|legend=1| 31, 109, 140, 171e, 311e }}
Badness (Sintel): 0.956


Badness: 0.030171
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23


==== 13-limit ====
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 385/384, 625/624, 1331/1323
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}


Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]
Optimal tunings:  
* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})


POTE generator: ~22/21 = 77.158
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}


{{Optimal ET sequence|legend=1| 31, 109, 140, 311e, 451ee }}
Badness (Sintel): 0.944


Badness: 0.028384
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23.29


==== 17-limit ====
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}


Mapping: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]
Optimal tunings:  
* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})


POTE generator: ~22/21 = 77.162
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}


{{Optimal ET sequence|legend=1| 31, 109g, 140, 311e, 451ee }}
Badness (Sintel): 0.858


Badness: 0.022416
=== Hemitert ===
 
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 6250/6237, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


Mapping: [{{val|1 3 2 3 -4}}, {{val|0 -22 5 -3 116}}]
Mapping: {{mapping| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
: mapping generators: ~2, ~88/45


POTE generator: ~256/245 = 77.169
Optimal tunings:  
* WE: ~2 = 1200.1008{{c}}, ~88/45 = 1161.5020{{c}} (~45/44 = 38.5988{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4053{{c}} (~45/44 = 38.5947{{c}})


{{Optimal ET sequence|legend=1| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
{{Optimal ET sequence|legend=0| 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee }}


Badness: 0.056926
Badness (Sintel): 0.517


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


Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Mapping: [{{val|1 3 2 3 -4 1}}, {{val|0 -22 5 -3 116 42}}]
Mapping: {{mapping| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}


POTE generator: ~117/112 = 77.168
Optimal tunings:  
* WE: ~2 = 1199.9822{{c}}, ~88/45 = 1161.3952{{c}} (~45/44 = 38.5871{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4123{{c}} (~45/44 = 38.5877{{c}})


{{Optimal ET sequence|legend=1| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
{{Optimal ET sequence|legend=0| 31, 280, 311 }}


Badness: 0.027474
Badness (Sintel): 1.39


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}


Mapping: [{{val|1 3 2 3 -4 1 1}}, {{val|0 -22 5 -3 116 42 48}}]
Optimal tunings:  
* WE: ~2 = 1200.0042{{c}}, ~88/45 = 1161.4149{{c}} (~45/44 = 38.5893{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4109{{c}} (~45/44 = 38.5891{{c}})


POTE generator: ~68/65 = 77.169
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f }}


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Badness (Sintel): 1.29


Badness: 0.018773
=== Semitert ===
Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 2401/2400, 9801/9800, 65625/65536
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Mapping: {{mapping| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
: mapping generators: ~99/70, ~693/512


Mapping: [{{val|1 3 2 3 -4 1 1 11}}, {{val|0 -22 5 -3 116 42 48 -105}}]
Optimal tunings:  
* WE: ~99/70 = 600.0548{{c}}, ~693/512 = 522.8547{{c}} (~256/245 = 77.2002{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~693/512 = 522.8069{{c}} (~256/245 = 77.1931{{c}})


POTE generator: ~68/65 = 77.169
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}


{{Optimal ET sequence|legend=1| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Badness (Sintel): 0.853


Badness: 0.017653
== Emmthird ==
Emmthird tempers out the [[scheme comma]] and may be described as the {{nowrap| 58 & 171 }} temperament. The generator for emmthird is flatter than [[81/64]] by a lee comma, [[177147/175616]], and sharper than [[5/4]] by the hemimage comma, [[10976/10935]]. The [[ploidacot]] for this temperament is delta-14-cot.  


==== 23-limit ====
The [[11-limit]] version, which tempers out [[243/242]] and [[441/440]], has much lower accuracy and is [[support]]ed by much fewer equal temperaments.  
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3}}, {{val|0 -22 5 -3 116 42 48 -105 117}}]
[[Comma list]]: 2401/2400, 14348907/14336000


POTE generator: ~23/22 = 77.168
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~2744/2187


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfgg }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}


Badness: 0.015123
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


==== 29-limit ====
[[Badness]] (Sintel): 0.424
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1}}, {{val|0 -22 5 -3 116 42 48 -105 117 60}}]
Comma list: 243/242, 441/440, 1792000/1771561


POTE generator: ~23/22 = 77.167
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfggj }}
Optimal tunings:
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}


Badness: 0.012181
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


==== 31-limit ====
Badness (Sintel): 1.73
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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}}]
Comma list: 243/242, 364/363, 441/440, 2200/2197


POTE generator: ~23/22 = 77.169
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal tunings:
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}


Badness: 0.012311
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


==== 37-limit ====
Badness (Sintel): 1.11
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37


Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


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}}]
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


POTE generator: ~23/22 = 77.170
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Optimal tunings:
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}


Badness: 0.010949
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


==== 41-limit ====
Badness (Sintel): 1.18
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41


Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
== Hemififths ==
{{Main| Hemififths }}


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}}]
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}}.


POTE generator: ~23/22 = 77.169
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.


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.009825
[[Comma list]]: 2401/2400, 5120/5103


=== Hemitert ===
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~2, ~49/40


Comma list: 2401/2400, 3025/3024, 65625/65536
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}
: [[error map]]: {{val| -0.259 +0.590 +0.021 -0.346 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}
: error map: {{val| 0.000 +0.979 +0.364 +0.246 }}


Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]
[[Minimax tuning]]:
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


POTE generator: ~45/44 = 38.596
[[Algebraic generator]]: (2 + sqrt(2))/2


{{Optimal ET sequence|legend=1| 31, 280, 311, 342 }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}


Badness: 0.015633
[[Badness]] (Sintel): 0.563


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 243/242, 441/440, 896/891


Mapping: [{{val| 1 3 2 3 6 1 }}, {{val| 0 -44 10 -6 -79 84 }}]
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


POTE generator: ~45/44 = 38.588
Optimal tunings:  
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}


{{Optimal ET sequence|legend=1| 31, 280, 311, 964f, 1275f, 1586cff }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Badness: 0.033573
Badness (Sintel): 0.777


==== 17-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Comma list: 144/143, 196/195, 243/242, 364/363


Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


POTE generator: ~45/44 = 38.589
Optimal tunings:  
* WE: ~2 = 1198.8875{{c}}, ~11/9 = 351.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5438{{c}}


{{Optimal ET sequence|legend=1| 31, 280, 311, 653f, 964f }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


Badness: 0.025298
Badness (Sintel): 0.789


=== Semitert ===
=== Semihemi ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 9801/9800, 65625/65536
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: [{{val|2 6 4 6 1}}, {{val|0 -22 5 -3 46}}]
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231


POTE generator: ~256/245 = 77.193
Optimal tunings:  
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}


{{Optimal ET sequence|legend=1| 62e, 140, 202, 342 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Badness: 0.025790
Badness (Sintel): 1.40


== Quasiorwell ==
==== 13-limit ====
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.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 352/351, 676/675, 847/845, 1716/1715


Subgroup: 2.3.5.7
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


[[Comma list]]: 2401/2400, 29360128/29296875
Optimal tunings:  
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}


[[Mapping]]: [{{val|1 31 0 9}}, {{val|0 -38 3 -8}}]
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}


{{Multival|legend=1| 38 -3 8 -93 -94 27 }}
Badness (Sintel): 0.876


[[POTE generator]]: ~1024/875 = 271.107
=== Quadrafifths ===
 
This has been catalogued as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}


[[Badness]]: 0.035832
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: [{{val|1 31 0 9 53}}, {{val|0 -38 3 -8 -64}}]
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220


POTE generator: ~90/77 = 271.111
Optimal tunings:  
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}


{{Optimal ET sequence|legend=1| 31, 208, 239, 270 }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.017540
Badness (Sintel): 1.33


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


Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


POTE generator: ~90/77 = 271.107
Optimal tunings:  
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}


{{Optimal ET sequence|legend=1| 31, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Badness: 0.017921
Badness (Sintel): 1.29


== Decoid ==
=== Cutefourths ===
{{See also| Quintosec family #Decoid }}
This extension splits the neutral third plus an octave in three, with a ploidacot signature of beta-hexacot. The generator is an acute fourth in size (but not representing [[27/20]]), hence the name.


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 [[quintosec]] temperament.
Subgroup: 2.3.5.7.11


[[Subgroup]]: 2.3.5.7
Comma list: 2401/2400, 4000/3993, 5120/5103


[[Comma list]]: 2401/2400, 67108864/66976875
Mapping: {{mapping| 1 -1 -30 -14 -28 | 0 6 75 39 73 }}
: mapping generators: ~2, ~66/49


{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
Optimal tunings:
* WE: ~2 = 1199.7345{{c}}, ~66/49 = 517.0436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1543{{c}}


: mapping generators: ~15/14, ~8192/4725
{{Optimal ET sequence|legend=0| 58, 181, 239, 1014bcee }}


{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
Badness (Sintel): 1.71


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099)
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
Comma list: 352/351, 847/845, 1575/1573, 2401/2400


[[Badness]]: 0.033902
Mapping: {{mapping| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}


Comma list: 2401/2400, 5632/5625, 9801/9800
{{Optimal ET sequence|legend=0| 58, 181, 239f }}


Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
Badness (Sintel): 1.45


Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070)
== Osiris ==


{{Optimal ET sequence|legend=1| 10e, 130, 270, 670, 940, 1210, 2150c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.018735
[[Comma list]]: 2401/2400, 31381059609/31360000000


=== 13-limit ===
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~2, ~2187/1400


Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0285{{c}}, ~2187/1400 = 771.9522{{c}}
: [[error map]]: {{val| +0.028 -0.025 +0.068 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1400 = 771.9343{{c}}
: error map: {{val| 0.000 -0.056 +0.039 -0.175 }}


Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083)
[[Badness]] (Sintel): 0.716


{{Optimal ET sequence|legend=1| 10e, 130, 270, 940, 1210f, 1480cf }}
== Quasiorwell ==
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 {{nowrap| 31 & 270 }} temperament, and its [[ploidacot]] is eta-38-cot (or omega-triseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). 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.


Badness: 0.013475
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


== Neominor ==
[[Subgroup]]: 2.3.5.7
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.


Subgroup: 2.3.5.7
[[Comma list]]: 2401/2400, 29360128/29296875


[[Comma list]]: 2401/2400, 177147/175616
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~1024/875


[[Mapping]]: [{{val|1 3 12 8}}, {{val|0 -6 -41 -22}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9403{{c}}, ~1024/875 = 271.0935{{c}}
: [[error map]]: {{val| -0.060 +0.018 +0.226 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1024/875 = 271.1064{{c}}
: error map: {{val| 0.000 +0.087 +0.367 +0.025 }}


{{Multival|legend=1|6 41 22 51 18 -64}}
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}


[[POTE generator]]: ~189/160 = 283.280
[[Badness]] (Sintel): 0.907
 
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
 
[[Badness]]: 0.088221


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


Comma list: 243/242, 441/440, 35937/35840
Comma list: 2401/2400, 3025/3024, 5632/5625


Mapping: [{{val|1 3 12 8 7}}, {{val|0 -6 -41 -22 -15}}]
Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}


POTE generator: ~33/28 = 283.276
Optimal tunings:  
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}


Badness: 0.027959
Badness (Sintel): 0.580


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


Comma list: 169/168, 243/242, 364/363, 441/440
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}
 
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}
 
Badness (Sintel): 0.741
 
== Quinmite ==
Quinmite may be described as the {{nowrap| 99 & 103 }} temperament. The generator for quinmite is the quasi-tempered minor third [[25/21]], sharper than [[32/27]] by the marvel comma, [[225/224]]. It is also generated by 1/5 of the 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>. Its [[ploidacot]] is eta-34-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1959552/1953125


Mapping: [{{val|1 3 12 8 7 7}}, {{val|0 -6 -41 -22 -15 -14}}]
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21


POTE generator: ~13/11 = 283.294
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9361{{c}}, ~25/21 = 302.9808{{c}}
: [[error map]]: {{val| -0.064 -0.162 +0.448 -0.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 302.9953{{c}}
: error map: {{val| 0.000 -0.116 +0.549 +0.065 }}


{{Optimal ET sequence|legend=1| 72, 161f, 233f }}
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}


Badness: 0.026942
[[Badness]] (Sintel): 0.945


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


Subgroup: 2.3.5.7
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit, and may be described as the {{nowrap| 10 & 171 }} temperament. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]), with a [[ploidacot]] of beta-26-cot. It is an excellent temperament for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.


[[Comma list]]: 2401/2400, 14348907/14336000
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 2152828125/2147483648


[[Mapping]]: [{{val|1 -3 -17 -8}}, {{val|0 14 59 33}}]
{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}
: mapping generators: ~2, ~15/14


{{Multival|legend=1|14 59 33 61 13 -89}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1043{{c}}, ~15/14 = 119.3076{{c}}
: [[error map]]: {{val| +0.104 -0.061 -0.070 -0.100 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 119.2971{{c}}
: error map: {{val| 0.000 -0.230 -0.307 -0.391 }}


[[POTE generator]]: ~2744/2187 = 392.988
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd }}


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
[[Badness]] (Sintel): 1.12


[[Badness]]: 0.016736
=== Sedia ===
The sedia temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out [[243/242]] and [[441/440]].


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


Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 243/242, 441/440, 939524096/935859375


Mapping: [{{val|1 -3 -17 -8 -8}}, {{val|0 14 59 33 35}}]
Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}


POTE generator: ~1372/1089 = 392.991
Optimal tunings:  
* WE: ~2 = 1199.9635{{c}}, ~15/14 = 119.2755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2791{{c}}


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.052358
Badness (Sintel): 3.00


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


Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Mapping: [{{val|1 -3 -17 -8 -8 -13}}, {{val|0 14 59 33 35 51}}]
Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}


POTE generator: ~180/143 = 392.989
Optimal tunings:  
* WE: ~2 = 1199.8922{{c}}, ~15/14 = 119.2700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2804{{c}}


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.026974
Badness (Sintel): 1.89


=== 17-limit ===
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}


Mapping: [{{val|1 -3 -17 -8 -8 -13 9}}, {{val|0 14 59 33 35 51 -15}}]
Optimal tunings:  
* WE: ~2 = 1199.9088{{c}}, ~15/14 = 119.2719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2808{{c}}


POTE generator: ~64/51 = 392.985
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef }}


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Badness (Sintel): 1.39


Badness: 0.023205
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}


== Quinmite ==
[[Subgroup]]: 2.3.5.7
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
[[Comma list]]: 2401/2400, 1224440064/1220703125


[[Comma list]]: 2401/2400, 1959552/1953125
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648


[[Mapping]]: [{{val|1 -7 -5 -3}}, {{val|0 34 29 23}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}


{{Multival|legend=1|34 29 23 -33 -59 -28}}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


[[POTE generator]]: ~25/21 = 302.997
[[Badness]] (Sintel): 1.46


{{Optimal ET sequence|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''


[[Badness]]: 0.037322
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  


== Unthirds ==
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 has a [[temperament merging|join]] 103 & 270, hence the name. The name was proposed in 2022 by [[Eliora]], who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  
The generator for unthirds temperament is undecimal major third, 14/11.


Subgroup: 2.3.5.7
Lockerbie also has a unique extension that adds the [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.


[[Comma list]]: 2401/2400, 68359375/68024448
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 -13 -14 -9}}, {{val|0 42 47 34}}]
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


{{Multival|legend=1|42 47 34 -23 -64 -53}}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~3828125/2985984


[[POTE generator]]: ~3969/3125 = 416.717
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9950{{c}}, ~3828125/2985984 = 431.1055{{c}}
: [[error map]]: {{val| -0.005 -0.024 +0.146 -0.120 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
: error map: {{val| 0.0000 -0.020 +0.155 -0.108 }}


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}


[[Badness]]: 0.075253
[[Badness]] (Sintel): 1.51


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


Comma list: 2401/2400, 3025/3024, 4000/3993
Comma list: 2401/2400, 3025/3024, 766656/765625


Mapping: [{{val|1 -13 -14 -9 -8}}, {{val|0 42 47 34 33}}]
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


POTE generator: ~14/11 = 416.718
Optimal tunings:  
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


Badness: 0.022926
Badness (Sintel): 0.865


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


Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
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:
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
 
Badness (Sintel): 0.662
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


Mapping: [{{val|1 -13 -14 -9 -9 -47}}, {{val|0 42 47 34 33 146}}]
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}


POTE generator: ~14/11 = 416.716
Optimal tunings:  
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}


{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


Badness: 0.020888
Badness (Sintel): 1.07


== Newt ==
== Unthirds ==
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].
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. It is generated by the interval of [[14/11]] (<u>un</u>decimal major <u>third</u>, hence the name) tuned less than a cent flat, 42 of which [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 14-sheared 42-cot. The 23-note [[mos]] from the generator serves as a well temperament of, of all things, [[23edo]]. The 49-note mos is needed to access the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s.


Subgroup: 2.3.5.7
The commas it tempers out in the 11-limit include 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).


[[Comma list]]: 2401/2400, 33554432/33480783
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 1 19 11}}, {{val|0 2 -57 -28}}]
[[Comma list]]: 2401/2400, 68359375/68024448


{{Multival|legend=1|2 -57 -28 -95 -50 95}}
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125


[[POTE generator]]: ~49/40 = 351.113
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}


[[Badness]]: 0.041878
[[Badness]] (Sintel): 1.90


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


Comma list: 2401/2400, 3025/3024, 19712/19683
Comma list: 2401/2400, 3025/3024, 4000/3993


Mapping: [{{val|1 1 19 11 -10}}, {{val|0 2 -57 -28 46}}]
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


POTE generator: ~49/40 = 351.115
Optimal tunings:  
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}


Badness: 0.019461
Badness (Sintel): 0.758


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


Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: [{{val|1 1 19 11 -10 -20}}, {{val|0 2 -57 -28 46 81}}]
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}


POTE generator: ~49/40 = 351.117
Optimal tunings:  
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}


{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}


Badness: 0.013830
Badness (Sintel): 0.863


== Septidiasemi ==
== Neominor ==
{{Main| Septidiasemi }}
Neominor tempers out [[177147/175616]] and may be described as the {{nowrap| 72 & 89 }} temperament. The generator is a neogothic minor third, which represents [[13/11]][[~]][[20/17]], or its [[octave complement]], which represents [[17/10]]~[[22/13]]. The latter stacked six times [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]], and the temperament has a [[ploidacot]] of delta-hexacot. [[72edo]] and [[89edo]] can be used as tunings.


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, 177147/175616


[[Comma list]]: 2401/2400, 2152828125/2147483648
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~320/189


[[Mapping]]: [{{val| 1 -1 6 4 }}, {{val| 0 26 -37 -12 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
: [[error map]]: {{val| +0.428 -0.955 +0.216 +0.224 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/189 = 916.7320{{c}}
: error map: {{val| 0.000 -1.563 -0.301 -0.722 }}


{{Multival|legend=1|26 -37 -12 -119 -92 76}}
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}


[[POTE generator]]: ~15/14 = 119.297
[[Badness]] (Sintel): 2.23
 
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
 
[[Badness]]: 0.044115
 
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.


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


Comma list: 243/242, 441/440, 939524096/935859375
Comma list: 243/242, 441/440, 35937/35840


Mapping: [{{val| 1 -1 6 4 -3 }}, {{val| 0 26 -37 -12 65 }}]
Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


POTE generator: ~15/14 = 119.279
Optimal tunings:  
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332 }}
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}


Badness: 0.090687
Badness (Sintel): 0.924


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


Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Comma list: 169/168, 243/242, 364/363, 441/440


Mapping: [{{val| 1 -1 6 4 -3 4 }}, {{val| 0 26 -37 -12 65 -3 }}]
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


POTE generator: ~15/14 = 119.281
Optimal tunings:  
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 835eeff }}
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}


Badness: 0.045773
Badness (Sintel): 1.11


==== 17-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363


Mapping: [{{val| 1 -1 6 4 -3 4 2 }}, {{val| 0 26 -37 -12 65 -3 21 }}]
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}


POTE generator: ~15/14 = 119.281
Optimal tunings:  
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 503ef, 835eeff }}
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}


Badness: 0.027322
Badness (Sintel): 0.918


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 1224440064/1220703125
[[Comma list]]: 2401/2400, 78732/78125


[[Mapping]]: [{{val| 1 31 34 26 }}, {{val| 0 -52 -56 -41 }}]
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125


{{Multival|legend=1|52 56 41 -32 -81 -62}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~189/125 = 710.7220{{c}}
: [[error map]]: {{val| -0.072 -0.656 +1.050 +0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/125 = 710.7626{{c}}
: error map: {{val| 0.000 -0.603 +1.139 +0.238 }}


[[POTE generator]]: ~1296/875 = 678.810
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}


{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
[[Badness]] (Sintel): 2.01


[[Badness]]: 0.057632
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Subneutral ==
Comma list: 243/242, 441/440, 78408/78125
{{See also| Luna family }}


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 -15 -19 -12 -38 | 0 28 36 25 70 }}


[[Comma list]]: 2401/2400, 274877906944/274658203125
Optimal tunings:  
* WE: ~2 = 1200.0219{{c}}, ~189/125 = 710.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/125 = 710.7487{{c}}


[[Mapping]]: [{{val| 1 19 0 6 }}, {{val| 0 -60 8 -11 }}]
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}


{{Multival|legend=1|60 -8 11 -152 -151 48}}
Badness (Sintel): 1.22


[[POTE generator]]: ~57344/46875 = 348.301
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
Comma list: 243/242, 351/350, 441/440, 10985/10976


[[Badness]]: 0.045792
Mapping: {{mapping| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}


== Osiris ==
Optimal tunings:
{{See also| Metric microtemperaments #Geb }}
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}


[[Comma list]]: 2401/2400, 31381059609/31360000000
Badness (Sintel): 0.896


[[Mapping]]: [{{val| 1 13 33 21 }}, {{val| 0 -32 -86 -51 }}]
== Cotritone ==
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1|32 86 51 62 -9 -123}}
[[Comma list]]: 2401/2400, 390625/387072


[[POTE generator]]: ~2800/2187 = 428.066
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
: mappping generators: ~2, ~7/5


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~7/5 = 583.5994{{c}}
: [[error map]]: {{val| +0.441 +0.289 -1.287 -0.200 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/5 = 583.3956{{c}}
: error map: {{val| 0.000 -0.086 -2.170 -1.287 }}


[[Badness]]: 0.028307
{{Optimal ET sequence|legend=1| 35, 37, 72, 181, 253, 325c }}


== Gorgik ==
[[Badness]] (Sintel): 2.49
Subgroup: 2.3.5.7
 
[[Comma list]]: 2401/2400, 28672/28125
 
[[Mapping]]: [{{val| 1 5 1 3 }}, {{val| 0 -18 7 -1 }}]
 
{{Multival|legend=1|18 -7 1 -53 -49 22}}
 
[[POTE generator]]: ~8/7 = 227.512
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]]: 0.158384


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


Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: [{{val| 1 5 1 3 1 }}, {{val| 0 -18 7 -1 13 }}]
Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}


POTE generator: ~8/7 = 227.500
Optimal tunings:  
* WE: ~2 = 1200.4058{{c}}, ~7/5 = 583.5845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3950{{c}}


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
{{Optimal ET sequence|legend=0| 35, 37, 72, 181, 253, 325c }}


Badness: 0.059260
Badness (Sintel): 1.07


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


Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 169/168, 364/363, 385/384, 625/624


Mapping: [{{val| 1 5 1 3 1 2 }}, {{val| 0 -18 7 -1 13 9 }}]
Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}


POTE generator: ~8/7 = 227.493
Optimal tunings:  
* WE: ~2 = 1200.6111{{c}}, ~7/5 = 583.6837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3987{{c}}


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bcef, 211bccdeeff }}
{{Optimal ET sequence|legend=0| 35f, 37, 72, 181f, 253ff }}


Badness: 0.032205
Badness (Sintel): 1.19


== 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 -27 -7 -9 | 0 46 15 19 }}
: mapping generators: ~2, ~192/125


{{Multival|legend=1|46 15 19 -83 -99 2}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2050{{c}}, ~192/125 = 745.8170{{c}}
: [[error map]]: {{val| +0.205 +0.094 -0.493 -0.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 745.6927{{c}}
: error map: {{val| 0.000 -0.092 -0.924 -0.665 }}


[[POTE generator]]: ~125/96 = 454.310
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}


{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}
Badness (Sintel): 2.54
 
Badness: 0.100511


=== 11-limit ===
=== 11-limit ===
Line 863: Line 997:
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 -27 -7 -9 -4 | 0 46 15 19 12 }}


POTE generator: ~100/77 = 454.318
Optimal tunings:  
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Badness: 0.056514
Badness (Sintel): 1.87


=== 13-limit ===
=== 13-limit ===
Line 876: Line 1,012:
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 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}


POTE generator: ~13/10 = 454.316
Optimal tunings:  
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Badness: 0.027429
Badness (Sintel): 1.13


== Mintone ==
== Quasimoha ==
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.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 177147/175000
[[Comma list]]: 2401/2400, 3645/3584
 
[[Mapping]]: [{{val| 1 5 9 7 }}, {{val| 0 -22 -43 -27 }}]


{{Multival|legend=1|22 43 27 17 -19 -58}}
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
: mapping generators: ~2, ~49/40


[[POTE generator]]: ~10/9 = 186.343
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5059{{c}}, ~49/40 = 348.0409{{c}}
: [[error map]]: {{val| +1.506 -2.367 -0.702 +0.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 348.5582{{c}}
: error map: {{val| 0.000 -4.839 -3.152 -2.966 }}


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 24c, 31, 117c, 148bc, 179bcd }}


[[Badness]]: 0.125672
[[Badness]] (Sintel): 2.80


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


Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 1815/1792


Mapping: [{{val| 1 5 9 7 12 }}, {{val| 0 -22 -43 -27 -55 }}]
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


POTE generator: ~10/9 = 186.345
Optimal tunings:  
* WE: ~2 = 1201.7630{{c}}, ~11/9 = 349.1510{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.6050{{c}}


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=0| 24c, 31, 86ce, 117ce, 148bce }}


Badness: 0.039962
Badness (Sintel): 1.53


=== 13-limit ===
== Mintone ==
Subgroup: 2.3.5.7.11.13
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 may be described as the {{nowrap| 58 & 103 }} temperament. It has a generator of [[~]][[10/9]], tuned to around [[49/44]]. Note that in the data below, the generator is its [[octave complement]], ~[[9/5]], so that 22 of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 18-sheared 22-cot. As one might expect, 25\161 makes for an excellent tuning choice.


Comma list: 243/242, 351/350, 441/440, 847/845
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 5 9 7 12 11 }}, {{val| 0 -22 -43 -27 -55 -47 }}]
[[Comma list]]: 2401/2400, 177147/175000


POTE generator: ~10/9 = 186.347
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
: mapping generators: ~2, ~9/5


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}


Badness: 0.021849
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}


=== 17-limit ===
[[Badness]] (Sintel): 3.18
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 5 9 7 12 11 3 }}, {{val| 0 -22 -43 -27 -55 -47 7 }}]
Comma list: 243/242, 441/440, 43923/43750


POTE generator: ~10/9 = 186.348
Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
Optimal tunings:
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}


Badness: 0.020295
{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}


== Catafourth ==
Badness (Sintel): 1.32
{{see also| Sensipent family }}


Subgroup: 2.3.5.7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 2401/2400, 78732/78125
Comma list: 243/242, 351/350, 441/440, 847/845


[[Mapping]]: [{{val| 1 13 17 13 }}, {{val| 0 -28 -36 -25 }}]
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}


{{Multival|legend=1| 28 36 25 -8 -39 -43 }}
Optimal tunings:
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}


[[POTE generator]]: ~250/189 = 489.235
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
Badness (Sintel): 0.903


Badness: 0.079579
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 11-limit ===
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 78408/78125
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}


Mapping: [{{val| 1 13 17 13 32 }}, {{val| 0 -28 -36 -25 -70 }}]
Optimal tunings:  
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}


POTE generator: ~250/189 = 489.252
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


{{Optimal ET sequence|legend=1| 103, 130, 233, 363, 493e, 856be }}
Badness (Sintel): 1.03


Badness: 0.036785
== Gorgik ==
{{See also| Llywelynsmic clan }}


=== 13-limit ===
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament.
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440, 10985/10976
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 13 17 13 32 9 }}, {{val| 0 -28 -36 -25 -70 -13 }}]
[[Comma list]]: 2401/2400, 28672/28125


POTE generator: ~65/49 = 489.256
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4


{{Optimal ET sequence|legend=1| 103, 130, 233, 363 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.5503{{c}}, ~7/4 = 971.3132{{c}} (~8/7 = 227.2371{{c}})
: [[error map]]: {{val| -1.450 +0.528 +2.896 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.4675{{c}} (~8/7 = 227.5325{{c}})
: error map: {{val| 0.000 +2.460 +6.414 +3.642 }}


Badness: 0.021694
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
== Cotritone ==
Subgroup: 2.3.5.7


[[Comma list]]: 2401/2400, 390625/387072
[[Badness]] (Sintel): 4.01
 
[[Mapping]]: [{{val| 1 -13 -4 -4 }}, {{val| 0 30 13 14 }}]
 
{{Multival|legend=1|30 13 14 -49 -62 -4}}
 
[[POTE generator]]: ~7/5 = 583.385
 
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
 
[[Badness]]: 0.098322


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


Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: [{{val| 1 -13 -4 -4 2 }}, {{val| 0 30 13 14 3 }}]
Mapping: {{mapping| 1 -13 8 2 14 | 0 18 -7 1 -13 }}


POTE generator: ~7/5 = 583.387
Optimal tunings:  
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness: 0.032225
Badness (Sintel): 1.96


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


Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 176/175, 196/195, 364/363, 512/507


Mapping: [{{val| 1 -13 -4 -4 2 -7 }}, {{val| 0 30 13 14 3 22 }}]
Mapping: {{mapping| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}


POTE generator: ~7/5 = 583.387
Optimal tunings:  
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})


{{Optimal ET sequence|legend=1| 37, 72, 109, 181f }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness: 0.028683
Badness (Sintel): 1.33


== Quasimoha ==
== Hemigoldis ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
Hemigoldis may be described as the {{nowrap| 68 & 89 }} temperament. Though fairly complex in the [[7-limit]], it does a lot better in badness metrics than pure [[5-limit]] [[goldis]], and yet again has many possible extensions to higher 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
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3645/3584
 
[[Mapping]]: [{{Val|1 1 9 6}}, {{Val|0 2 -23 -11}}]
 
[[POTE generator]]: ~49/40 = 348.603
 
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
 
[[Badness]]: 0.110820
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 1815/1792
[[Comma list]]: 2401/2400, 549755813888/533935546875


Mapping: [{{Val|1 1 9 6 2}}, {{Val|0 2 -23 -11 5}}]
{{Mapping|legend=1| 1 21 -9 2 | 0 24 -14 -1 }}
: mapping generators: ~2, ~8/7


POTE generator: ~11/9 = 348.639
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2264{{c}}, ~8/7 = 229.1679{{c}}
: [[error map]]: {{val| -0.774 +0.394 +1.468 -0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.3103{{c}}
: error map: {{val| 0.000 +1.491 +3.343 +1.864 }}


{{Optimal ET sequence|legend=1| 31, 86ce, 117ce, 148bce }}
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}


Badness: 0.046181
[[Badness]] (Sintel): 4.40


== Surmarvelpyth ==
== Surmarvelpyth ==
''Surmarvelpyth'' is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.
Surmarvelpyth can be described as the {{nowrap| 311 & 431 }} temperament, starting with the 7-limit to the 19-limit. Its [[ploidacot]] is 28-sheared 70-cot. It was named by [[Eliora]] in 2022 for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,059: Line 1,201:
[[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 -27 55 22 | 0 70 -129 -47 }}
: mapping generators: ~2, ~896/675


Mapping generators: ~2, ~675/448
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.0051{{c}}, ~896/675 = 490.0303{{c}}
[[Optimal tuning]] ([[CTE]]): ~675/448 = 709.9719
: [[error map]]: {{val| +0.005 +0.025 +0.063 -0.136 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/675 = 490.0282{{c}}
: error map: {{val| 0.000 +0.017 +0.052 -0.150 }}


{{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]] (Sintel): 5.12


=== 11-limit ===
=== 11-limit ===
Line 1,074: Line 1,219:
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 -27 55 22 -19 | 0 70 -129 -47 55 }}


Optimal tuning (CTE): ~675/448 = 709.9720
Optimal tunings:  
* WE: ~2 = 1199.9901{{c}}, ~896/675 = 490.0239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~896/675 = 490.0279{{c}}


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795 }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.0523
Badness (Sintel): 1.73


=== 13-limit ===
=== 13-limit ===
Line 1,087: Line 1,234:
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 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}


Optimal tuning (CTE): ~98/65 = 709.9723
Optimal tunings:  
* WE: ~2 = 1199.9701{{c}}, ~65/49 = 490.0155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0277{{c}}


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795f }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795f }}


Badness: 0.0325
Badness (Sintel): 1.34


=== 17-limit ===
=== 17-limit ===
Line 1,100: Line 1,249:
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 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}


Optimal tuning (CTE): ~98/65 = 709.9722
Optimal tunings:  
* WE: ~2 = 1199.9726{{c}}, ~65/49 = 490.0164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.0325
Badness (Sintel): 1.07


=== 19-limit ===
=== 19-limit ===
Line 1,113: Line 1,264:
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 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}
 
Optimal tunings:
* WE: ~2 = 1199.9756{{c}}, ~65/49 = 490.0176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}


Optimal tuning (CTE): ~98/65 = 709.9722
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Badness (Sintel): 0.838


Badness: 0.0138
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"