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'''Breedsmic temperaments''' are rank two 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.


It 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 chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 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 not discussed here include [[Dicot family #Decimal|decimal]], [[Archytas clan #Beatles|beatles]], [[Meantone family #Squares|squares]], [[Starling temperaments #Myna|myna]], [[Keemic temperaments #Quasitemp|quasitemp]], [[Gamelismic clan #Miracle|miracle]], [[Magic family #Quadrimage|quadrimage]], [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], [[Tetracot family #Octacot|octacot]], [[Kleismic family #Quadritikleismic|quadritikleismic]], [[Schismatic family #Sesquiquartififths|sesquiquartififths]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], and [[Gammic family #Neptune|neptune]].
Temperaments discussed elsewhere include:
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
* ''[[Neptune (temperament)|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)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie {{multival|2 25 13 35 15 -40}}, which tells us that it requires 25 generator steps to get to the class for major thirds, 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.
== 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
 
{{Val list|legend=1| 41, 58, 99, 239, 338, 1253bbc, 1591bbc }}
 
[[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}})


Vals: {{Val list| 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: {{mapping| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}
 
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=0| 31, 140e, 171, 373ef }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 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 49 | 0 22 -5 3 55 -42 -48 }}


POTE generator: ~11/9 = 351.573
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}})


Vals: {{Val list| 17c, 41, 58, 99ef }}
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}


Badness: 0.019090
Badness (Sintel): 1.40


=== Semihemi ===
=== Tertia ===
Subgroup: 2.3.5.7.11
Subgroup:2.3.5.7.11


Comma list: 2401/2400, 3388/3375, 9801/9800
Comma list: 385/384, 1331/1323, 1375/1372


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


POTE generator: ~49/40 = 351.505
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}})


Vals: {{Val list| 58, 140, 198, 734bc, 932bcd, 1130bcd }}
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}


Badness: 0.042487
Badness (Sintel): 0.997


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


Comma list: 352/351, 676/675, 847/845, 1716/1715
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]
Mapping: {{mapping| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}


POTE generator: ~49/40 = 351.502
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}})


Vals: {{Val list| 58, 140, 198, 536f, 734bcf, 932bcdf }}
{{Optimal ET sequence|legend=0| 31, 78f, 109, 140 }}


Badness: 0.021188
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
 
[[Comma list]]: 2401/2400, 65625/65536


[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]
Mapping: {{mapping| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}


{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
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}})


[[POTE generator]]: ~256/245 = 77.191
{{Optimal ET sequence|legend=0| 31, 78fg, 109g, 140 }}


{{Val list|legend=1| 31, 109, 140, 171 }}
Badness (Sintel): 1.14


[[Badness]]: 0.012995
=== 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.  


=== 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}})


Vals: {{Val list| 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}})


Vals: {{Val list| 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 }}]
Mapping: {{mapping| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}


POTE generator: ~68/65 = 77.201
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}})


Vals: {{Val list| 31, 109eg, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness: 0.027398
Badness (Sintel): 0.956


=== Tertia ===
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup:2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197
 
POTE generator: ~22/21 = 77.173
 
Vals: {{Val list| 31, 109, 140, 171e, 311e }}
 
Badness: 0.030171
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}


POTE generator: ~22/21 = 77.158
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}})


Vals: {{Val list| 31, 109, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}


Badness: 0.028384
Badness (Sintel): 0.944


==== 17-limit ====
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.23.29


Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155


Mapping: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}


POTE generator: ~22/21 = 77.162
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}})


Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}


Badness: 0.022416
Badness (Sintel): 0.858


=== Hemitert ===
=== Hemitert ===
Line 187: Line 226:
Comma list: 2401/2400, 3025/3024, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


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


POTE generator: ~45/44 = 38.596
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}})


Vals: {{Val list| 31, 280, 311, 342 }}
{{Optimal ET sequence|legend=0| 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee }}


Badness: 0.015633
Badness (Sintel): 0.517


==== 13-limit ====
==== 13-limit ====
Line 200: Line 242:
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


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


POTE generator: ~45/44 = 38.588
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}})


Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}
{{Optimal ET sequence|legend=0| 31, 280, 311 }}


Badness: 0.033573
Badness (Sintel): 1.39


==== 17-limit ====
==== 17-limit ====
Line 213: Line 257:
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Mapping: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]
Mapping: {{mapping| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}


POTE generator: ~45/44 = 38.589
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}})


Vals: {{Val list| 31, 280, 311, 653f, 964f }}
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f }}


Badness: 0.025298
Badness (Sintel): 1.29


== Harry ==
=== Semitert ===
{{main|Harry}}
Subgroup: 2.3.5.7.11
{{see also|Gravity family #Harry}}


Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;72 temperament, with wedgie {{multival|12 34 20 26 -2 -49}}. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.
Comma list: 2401/2400, 9801/9800, 65625/65536


Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is {{multival|12 34 20 30 ...}}.
Mapping: {{mapping| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
: mapping generators: ~99/70, ~693/512


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival|12 34 20 30 52 ...}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
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}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}


[[Comma list]]: 2401/2400, 19683/19600
Badness (Sintel): 0.853


[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]
== 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.


{{Multival|legend=1| 12 34 20 26 -2 -49 }}
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.


[[POTE generator]]: ~21/20 = 83.156
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
[[Comma list]]: 2401/2400, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~2744/2187
 
[[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]]: 0.034077
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
[[Badness]] (Sintel): 0.424


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


Comma list: 243/242, 441/440, 4000/3993
Comma list: 243/242, 441/440, 1792000/1771561


Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


POTE generator: ~21/20 = 83.167
Optimal tunings:  
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}


Vals: {{Val list| 14c, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.015867
Badness (Sintel): 1.73


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


Comma list: 243/242, 351/350, 441/440, 676/675
Comma list: 243/242, 364/363, 441/440, 2200/2197


Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}


POTE generator: ~21/20 = 83.116
Optimal tunings:  
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}


Vals: {{Val list| 58, 72, 130, 332f, 462ef }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.013046
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: 221/220, 243/242, 289/288, 351/350, 441/440
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


POTE generator: ~21/20 = 83.168
Optimal tunings:  
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}


Vals: {{Val list| 58, 72, 130, 202g }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.012657
Badness (Sintel): 1.18


== Quasiorwell ==
== Hemififths ==
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = {{monzo|22 -1 -10 1}}. It has a wedgie {{multival|38 -3 8 -93 -94 27}}. 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)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.
{{Main| Hemififths }}
 
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.
 
[[Subgroup]]: 2.3.5.7


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]]: 2401/2400, 5120/5103


Subgroup: 2.3.5.7
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~49/40


[[Comma list]]: 2401/2400, 29360128/29296875
[[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 31 0 9}}, {{val|0 -38 3 -8}}]
[[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]]: ~1024/875 = 271.107
[[Algebraic generator]]: (2 + sqrt(2))/2


{{Val list|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}


[[Badness]]: 0.035832
[[Badness]] (Sintel): 0.563


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


Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 243/242, 441/440, 896/891


Mapping: [{{val|1 31 0 9 53}}, {{val|0 -38 3 -8 -64}}]
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


POTE generator: ~90/77 = 271.111
Optimal tunings:  
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}


Vals: {{Val list| 31, 208, 239, 270 }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Badness: 0.017540
Badness (Sintel): 0.777


=== 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: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
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=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Sintel): 0.789


Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}]
=== Semihemi ===
Subgroup: 2.3.5.7.11


POTE generator: ~90/77 = 271.107
Comma list: 2401/2400, 3388/3375, 5120/5103


Vals: {{Val list| 31, 239, 270, 571, 841, 1111 }}
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231


Badness: 0.017921
Optimal tunings:  
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}


== Decoid ==
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
{{see also|Qintosec family #Decoid}}


Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14ths equal temperament|linus comma]], {{monzo|11 -10 -10 10}}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&amp;270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].
Badness (Sintel): 1.40


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


[[Comma list]]: 2401/2400, 67108864/66976875
Comma list: 352/351, 676/675, 847/845, 1716/1715


[[Mapping]]: [{{val|10 0 47 36}}, {{val|0 2 -3 -1}}]
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


{{Multival|legend=1|20 -30 -10 -94 -72 61}}
Optimal tunings:
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}


[[POTE generator]]: ~8/7 = 231.099
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}


{{Val list|legend=1| 10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
Badness (Sintel): 0.876


[[Badness]]: 0.033902
=== 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.  


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


Comma list: 2401/2400, 5832/5825, 9801/9800
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: [{{val|10 0 47 36 98}}, {{val|0 2 -3 -1 -8}}]
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220


POTE generator: ~8/7 = 231.070
Optimal tunings:  
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}


Vals: {{Val list| 10e, 130, 270, 670, 940, 1210, 2150c }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.018735
Badness (Sintel): 1.33


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


Comma list: 676/675, 1001/1000, 1716/1715, 4225/4224
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: [{{val|10 0 47 36 98 37}}, {{val|0 2 -3 -1 -8 0}}]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


POTE generator: ~8/7 = 231.083
Optimal tunings:  
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}


Vals: {{Val list| 10e, 130, 270, 940, 1210f, 1480cf }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Badness: 0.013475
Badness (Sintel): 1.29


== Neominor ==
=== Cutefourths ===
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as "<b>Neo-gothic minor third</b>".
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 4000/3993, 5120/5103
 
Mapping: {{mapping| 1 -1 -30 -14 -28 | 0 6 75 39 73 }}
: mapping generators: ~2, ~66/49
 
Optimal tunings:
* WE: ~2 = 1199.7345{{c}}, ~66/49 = 517.0436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1543{{c}}
 
{{Optimal ET sequence|legend=0| 58, 181, 239, 1014bcee }}
 
Badness (Sintel): 1.71
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1575/1573, 2401/2400
 
Mapping: {{mapping| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}
 
Optimal tunings:
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}
 
{{Optimal ET sequence|legend=0| 58, 181, 239f }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.45


[[Comma list]]: 2401/2400, 177147/175616
== Osiris ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 3 12 8}}, {{val|0 -6 -41 -22}}]
[[Comma list]]: 2401/2400, 31381059609/31360000000


{{Multival|legend=1|6 41 22 51 18 -64}}
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
: mapping generators: ~2, ~2187/1400


[[POTE generator]]: ~189/160 = 283.280
[[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 }}


{{Val list|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


[[Badness]]: 0.088221
[[Badness]] (Sintel): 0.716


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


Comma list: 243/242, 441/440, 35937/35840
Comma list: 2401/2400, 9801/9800, 1771561/1771470
 
Mapping: {{mapping| 2 -6 -20 -9 -14 | 0 32 86 51 73 }}
: mapping generators: ~99/70, ~243/220
 
Optimal tunings:
* WE: ~99/70 = 600.0152{{c}}, ~243/220 = 171.9383{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~243/220 = 171.9344{{c}}
 
{{Optimal ET sequence|legend=0| 328, 342, 1012, 1354, 1696 }}
 
Badness (Sintel): 0.703
 
== 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.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


Mapping: [{{val|1 3 12 8 7}}, {{val|0 -6 -41 -22 -15}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~33/28 = 283.276
[[Comma list]]: 2401/2400, 29360128/29296875


Vals: {{Val list| 72, 161, 233, 305 }}
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~1024/875


Badness: 0.027959
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 243/242, 364/363, 441/440
[[Badness]] (Sintel): 0.907


Mapping: [{{val|1 3 12 8 7 7}}, {{val|0 -6 -41 -22 -15 -14}}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~13/11 = 283.294
Comma list: 2401/2400, 3025/3024, 5632/5625


Vals: {{Val list| 72, 161f, 233f }}
Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}


Badness: 0.026942
Optimal tunings:  
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}


== Emmthird ==
{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Subgroup: 2.3.5.7
Badness (Sintel): 0.580


[[Comma list]]: 2401/2400, 14348907/14336000
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val|1 -3 -17 -8}}, {{val|0 14 59 33}}]
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


{{Multival|legend=1|14 59 33 61 13 -89}}
Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}


[[POTE generator]]: ~2744/2187 = 392.988
Optimal tunings:  
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}


{{Val list|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}


[[Badness]]: 0.016736
Badness (Sintel): 0.741


== Quinmite ==
== Quinmite ==
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 -7 -5 -3}}, {{val|0 34 29 23}}]
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21


{{Multival|legend=1|34 29 23 -33 -59 -28}}
[[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 }}


[[POTE generator]]: ~25/21 = 302.997
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}


{{Val list|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
[[Badness]] (Sintel): 0.945


[[Badness]]: 0.037322
== Septidiasemi ==
{{Main| Septidiasemi }}


== Unthirds ==
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.
The generator for unthirds temperament is undecimal major third, 14/11.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 68359375/68024448
[[Comma list]]: 2401/2400, 2152828125/2147483648


[[Mapping]]: [{{val|1 -13 -14 -9}}, {{val|0 42 47 34}}]
{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}
: mapping generators: ~2, ~15/14


{{Multival|legend=1|42 47 34 -23 -64 -53}}
[[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]]: ~3969/3125 = 416.717
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd }}


{{Val list|legend=1| 72, 167, 239, 311, 694, 1005c }}
[[Badness]] (Sintel): 1.12


[[Badness]]: 0.075253
=== 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: 2401/2400, 3025/3024, 4000/3993
Comma list: 243/242, 441/440, 939524096/935859375


Map: [{{val|1 -13 -14 -9 -8}}, {{val|0 42 47 34 33}}]
Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}


POTE generator: ~14/11 = 416.718
Optimal tunings:  
* WE: ~2 = 1199.9635{{c}}, ~15/14 = 119.2755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2791{{c}}


Vals: {{Val list| 72, 167, 239, 311, 1316c }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.022926
Badness (Sintel): 3.00


=== 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: 243/242, 441/440, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}
 
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=0| 10, 151, 161, 171, 332 }}
 
Badness (Sintel): 1.89
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.9088{{c}}, ~15/14 = 119.2719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2808{{c}}
 
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef }}
 
Badness (Sintel): 1.39
 
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1224440064/1220703125
 
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648
 
[[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 }}


Mapping: [{{val|1 -13 -14 -9 -9 -47}}, {{val|0 42 47 34 33 146}}]
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


POTE generator: ~14/11 = 416.716
[[Badness]] (Sintel): 1.46


Vals: {{Val list| 72, 311, 694, 1005c, 1699cd }}
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''


Badness: 0.020888
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.  


== Newt ==
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.  
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].


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, 33554432/33480783
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 1 19 11}}, {{val|0 2 -57 -28}}]
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


{{Multival|legend=1|2 -57 -28 -95 -50 95}}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~3828125/2985984


[[POTE generator]]: ~49/40 = 351.113
[[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 }}


{{Val list|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}


[[Badness]]: 0.041878
[[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, 19712/19683
Comma list: 2401/2400, 3025/3024, 766656/765625


Mapping: [{{val|1 1 19 11 -10}}, {{val|0 2 -57 -28 46}}]
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


POTE generator: ~49/40 = 351.115
Optimal tunings:  
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


Vals: {{Val list| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


Badness: 0.019461
Badness (Sintel): 0.865


=== 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: 1001/1000, 1716/1715, 3025/3024, 4225/4224


Mapping: [{{val|1 1 19 11 -10 -20}}, {{val|0 2 -57 -28 46 81}}]
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}


POTE generator: ~49/40 = 351.117
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: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
 
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=0| 103, 167, 270 }}
 
Badness (Sintel): 1.07
 
== Unthirds ==
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.


Vals: {{Val list| 41, 229, 270, 581, 851, 2283b, 3134b }}
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).


Badness: 0.013830
[[Subgroup]]: 2.3.5.7


== Amicable ==
[[Comma list]]: 2401/2400, 68359375/68024448
{{see also| Amity family }}


Subgroup: 2.3.5.7
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125


[[Comma list]]: 2401/2400, 1600000/1594323
[[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 }}


[[Mapping]]: [{{val|1 3 6 5}}, {{val|0 -20 -52 -31}}]
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}


{{Multival|legend=1|20 52 31 36 -7 -74}}
[[Badness]] (Sintel): 1.90


[[POTE generator]]: ~21/20 = 84.880
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 99, 212, 311, 410, 1131, 1541b }}
Comma list: 2401/2400, 3025/3024, 4000/3993


[[Badness]]: 0.045473
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


== Septidiasemi ==
Optimal tunings:
{{main|Septidiasemi}}
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
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.
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}


[[Comma list]]: 2401/2400, 2152828125/2147483648
Badness (Sintel): 0.758


[[Mapping]]: [{{val|1 -1 6 4}}, {{val|0 26 -37 -12}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1|26 -37 -12 -119 -92 76}}
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


[[POTE generator]]: ~15/14 = 119.297
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}


{{Val list|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}


[[Badness]]: 0.044115
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}


=== Sedia ===
Badness (Sintel): 0.863
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.


Subgroup: 2.3.5.7.11
== Neominor ==
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.  


Comma list: 243/242, 441/440, 939524096/935859375
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|1 -1 6 4 -3}}, {{val|0 26 -37 -12 65}}]
[[Comma list]]: 2401/2400, 177147/175616


POTE generator: ~15/14 = 119.279
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~320/189


Vals: {{Val list| 10, 151, 161, 171, 332 }}
[[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 }}


Badness: 0.090687
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}


==== 13-limit ====
[[Badness]] (Sintel): 2.23
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 441/440, 2200/2197, 3584/3575
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|1 -1 6 4 -3 4}}, {{val|0 26 -37 -12 65 -3}}]
Comma list: 243/242, 441/440, 35937/35840


POTE generator: ~15/14 = 119.281
Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


Vals: {{Val list| 10, 151, 161, 171, 332, 835eeff }}
Optimal tunings:  
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}


Badness: 0.045773
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}


==== 17-limit ====
Badness (Sintel): 0.924
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val|1 -1 6 4 -3 4 2}}, {{val|0 26 -37 -12 65 -3 21}}]
Comma list: 169/168, 243/242, 364/363, 441/440


POTE generator: ~15/14 = 119.281
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


Vals: {{Val list| 10, 151, 161, 171, 332, 503ef, 835eeff }}
Optimal tunings:  
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}


Badness: 0.027322
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}


== Maviloid ==
Badness (Sintel): 1.11
{{see also| Ragismic microtemperaments #Parakleismic }}


Subgroup: 2.3.5.7
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 2401/2400, 1224440064/1220703125
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363


[[Mapping]]: [{{val|1 31 34 26}}, {{val|0 -52 -56 -41}}]
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}


{{Multival|legend=1|52 56 41 -32 -81 -62}}
Optimal tunings:
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}


[[POTE generator]]: ~1296/875 = 678.810
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}


{{Val list|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
Badness (Sintel): 0.918


[[Badness]]: 0.057632
== Catafourth ==
{{See also| Sensipent family }}


== Subneutral ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 2401/2400, 274877906944/274658203125
[[Comma list]]: 2401/2400, 78732/78125


[[Mapping]]: [{{val|1 19 0 6}}, {{val|0 -60 8 -11}}]
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125


{{Multival|legend=1|60 -8 11 -152 -151 48}}
[[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]]: ~57344/46875 = 348.301
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}


{{Val list|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
[[Badness]] (Sintel): 2.01


[[Badness]]: 0.045792
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Osiris ==
Comma list: 243/242, 441/440, 78408/78125
{{see also|Metric microtemperaments #Geb}}


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


[[Comma list]]: 2401/2400, 31381059609/31360000000
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 13 33 21}}, {{val|0 -32 -86 -51}}]
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}


{{Multival|legend=1|32 86 51 62 -9 -123}}
Badness (Sintel): 1.22


[[POTE generator]]: ~2800/2187 = 428.066
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
Comma list: 243/242, 351/350, 441/440, 10985/10976
 
Mapping: {{mapping| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}


[[Badness]]: 0.028307
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}


== Gorgik ==
Badness (Sintel): 0.896
Subgroup: 2.3.5.7


[[Comma list]]: 2401/2400, 28672/28125
== Cotritone ==
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 5 1 3}}, {{val|0 -18 7 -1}}]
[[Comma list]]: 2401/2400, 390625/387072


{{Multival|legend=1|18 -7 1 -53 -49 22}}
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
: mappping generators: ~2, ~7/5


[[POTE generator]]: ~8/7 = 227.512
[[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 }}


{{Val list|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 181, 253, 325c }}


[[Badness]]: 0.158384
[[Badness]] (Sintel): 2.49


=== 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}}


Vals: {{Val list| 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}}


Vals: {{Val list| 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}}


[[POTE generator]]: ~125/96 = 454.310
[[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 }}


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


Badness: 0.100511
Badness (Sintel): 2.54


=== 11-limit ===
=== 11-limit ===
Line 721: Line 1,012:
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}}


Vals: {{Val list| 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 734: Line 1,027:
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 }}
 
Optimal tunings:
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}


POTE generator: ~13/10 = 454.316
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Vals: {{Val list| 37, 103, 140, 243e }}
Badness (Sintel): 1.13


Badness: 0.027429
== Quasimoha ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3645/3584
 
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
: mapping generators: ~2, ~49/40
 
[[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| 24c, 31, 117c, 148bc, 179bcd }}
 
[[Badness]] (Sintel): 2.80
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 1815/1792
 
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
 
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=0| 24c, 31, 86ce, 117ce, 148bce }}
 
Badness (Sintel): 1.53


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 5 9 7}}, {{val|0 -22 -43 -27}}]
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
 
: mapping generators: ~2, ~9/5
{{Multival|legend=1|22 43 27 17 -19 -58}}


[[POTE generator]]: ~10/9 = 186.343
[[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 }}


{{Val list|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}


[[Badness]]: 0.125672
[[Badness]] (Sintel): 3.18


=== 11-limit ===
=== 11-limit ===
Line 764: Line 1,097:
Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 43923/43750


Mapping: [{{val|1 5 9 7 12}}, {{val|0 -22 -43 -27 -55}}]
Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}


POTE generator: ~10/9 = 186.345
Optimal tunings:  
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}


Vals: {{Val list| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}


Badness: 0.039962
Badness (Sintel): 1.32


=== 13-limit ===
=== 13-limit ===
Line 777: Line 1,112:
Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 243/242, 351/350, 441/440, 847/845


Mapping: [{{val|1 5 9 7 12 11}}, {{val|0 -22 -43 -27 -55 -47}}]
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}


POTE generator: ~10/9 = 186.347
Optimal tunings:  
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}


Vals: {{Val list| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


Badness: 0.021849
Badness (Sintel): 0.903


=== 17-limit ===
=== 17-limit ===
Line 790: Line 1,127:
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Mapping: [{{val|1 5 9 7 12 11 3}}, {{val|0 -22 -43 -27 -55 -47 7}}]
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}


POTE generator: ~10/9 = 186.348
Optimal tunings:  
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}


Vals: {{Val list| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


Badness: 0.020295
Badness (Sintel): 1.03


== Catafourth ==
== Gorgik ==
{{see also| Sensipent family }}
{{See also| Llywelynsmic clan }}


Subgroup: 2.3.5.7
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. 


[[Comma list]]: 2401/2400, 78732/78125
[[Subgroup]]: 2.3.5.7


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


{{Multival|legend=1|28 36 25 -8 -39 -43}}
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4


[[POTE generator]]: ~250/189 = 489.235
[[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 }}


{{Val list|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


Badness: 0.079579
[[Badness]] (Sintel): 4.01


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


Comma list: 243/242, 441/440, 78408/78125
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: [{{val|1 13 17 13 32}}, {{val|0 -28 -36 -25 -70}}]
Mapping: {{mapping| 1 -13 8 2 14 | 0 18 -7 1 -13 }}


POTE generator: ~250/189 = 489.252
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}})


Vals: {{Val list| 103, 130, 233, 363, 493e, 856be }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness: 0.036785
Badness (Sintel): 1.96


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


Comma list: 243/242, 351/350, 441/440, 10985/10976
Comma list: 176/175, 196/195, 364/363, 512/507
 
Mapping: {{mapping| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}
 
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=0| 21, 37, 58, 153bcef, 211bccdeeff }}
 
Badness (Sintel): 1.33
 
== Hemigoldis ==
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
 
[[Comma list]]: 2401/2400, 549755813888/533935546875


Mapping:  [{{val|1 13 17 13 32 9}}, {{val|0 -28 -36 -25 -70 -13}}]
{{Mapping|legend=1| 1 21 -9 2 | 0 24 -14 -1 }}
: mapping generators: ~2, ~8/7


POTE generator: ~65/49 = 489.256
[[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 }}


Vals: {{Val list| 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}


Badness: 0.021694
[[Badness]] (Sintel): 4.40


== Cotritone ==
== Surmarvelpyth ==
Subgroup: 2.3.5.7
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.  


[[Comma list]]: 2401/2400, 390625/387072
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 -13 -4 -4}}, {{val|0 30 13 14}}]
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}


{{Multival|legend=1|30 13 14 -49 -62 -4}}
{{Mapping|legend=1| 1 -27 55 22 | 0 70 -129 -47 }}
: mapping generators: ~2, ~896/675


[[POTE generator]]: ~7/5 = 583.385
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0051{{c}}, ~896/675 = 490.0303{{c}}
: [[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 }}


{{Val list|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}


[[Badness]]: 0.098322
[[Badness]] (Sintel): 5.12


=== 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: 2401/2400, 820125/819896, 2097152/2096325


Mapping: [{{val|1 -13 -4 -4 2}}, {{val|0 30 13 14 3}}]
Mapping: {{mapping| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}


POTE generator: ~7/5 = 583.387
Optimal tunings:  
* WE: ~2 = 1199.9901{{c}}, ~896/675 = 490.0239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~896/675 = 490.0279{{c}}


Vals: {{Val list| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.032225
Badness (Sintel): 1.73


=== 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: 2401/2400, 4096/4095, 6656/6655, 24192/24167
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}
 
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=0| 120, 191, 311, 742, 1053, 1795f }}
 
Badness (Sintel): 1.34
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}
 
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=0| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness (Sintel): 1.07
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}


Mapping: [{{val|1 -13 -4 -4 2 -7}}, {{val|0 30 13 14 3 22}}]
Optimal tunings:  
* WE: ~2 = 1199.9756{{c}}, ~65/49 = 490.0176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}


POTE generator: ~7/5 = 583.387
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Vals: {{Val list| 37, 72, 109, 181f }}
Badness (Sintel): 0.838


Badness: 0.028683
== References ==


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