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Breedsmic temperaments: Difference between revisions

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


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


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


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


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


==5-limit==
[[Subgroup]]: 2.3.5.7
Comma: 858993459200/847288609443


POTE generator: ~655360/531441 = 351.476
[[Comma list]]: 2401/2400, 5120/5103


Map: [&lt;1 1 -5|, &lt;0 2 25|]
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
: mapping generators: ~2, ~49/40


Badness: 0.3728
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}


==7-limit==
[[Minimax tuning]]:
Commas: 2401/2400, 5120/5103
* [[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


7 and 9-limit minimax
[[Algebraic generator]]: (2 + sqrt(2))/2


[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}


Eigenvalues: 2, 5
[[Badness]] (Smith): 0.022243


Algebraic generator: (2 + sqrt(2))/2
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
Comma list: 243/242, 441/440, 896/891


EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Badness: 0.0222
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


==11-limit==
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
Commas: 243/242, 441/440, 896/891


POTE generator: ~11/9 = 351.521
Badness (Smith): 0.023498


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 7ccd, 17c, 41, 58, 99e
Comma list: 144/143, 196/195, 243/242, 364/363


Badness: 0.0235
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Smith): 0.019090
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Smith): 0.042487
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Smith): 0.021188
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 
: Mapping generators: ~2, ~243/220
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Smith): 0.040170
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Smith): 0.031144
 
== Tertiaseptal ==
{{Main| Tertiaseptal }}
 
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 65625/65536
 
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
 
: Mapping generators: ~2, ~256/245
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
 
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
 
[[Badness]]: 0.012995
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
 
Badness: 0.035576
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
 
Badness: 0.036876
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
 
Optimal ET sequence: {{Optimal ET sequence| 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 3 2 3 5 | 0 -22 5 -3 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
 
Optimal ET sequence: {{Optimal ET sequence| 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: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
 
Badness: 0.028384
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
 
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
 
Badness: 0.022416
 
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 6250/6237, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
 
Badness: 0.056926
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
 
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
 
Badness: 0.027474
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.018773
 
==== 19-limit ====
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| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
 
Badness: 0.017653
 
==== 23-limit ====
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
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
 
Badness: 0.015123
 
==== 29-limit ====
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
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
 
Badness: 0.012181
 
==== 31-limit ====
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
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.012311
 
==== 37-limit ====
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
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.010949
 
==== 41-limit ====
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
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.009825
 
=== Hemitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
 
: Mapping generators: ~2, ~45/44
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
 
Badness: 0.015633
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
 
Badness: 0.033573
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
 
Badness: 0.025298
 
=== Semitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 65625/65536
 
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
 
: Mapping generators: ~99/70, ~256/245
 
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
 
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
 
Badness: 0.025790
 
== 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 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 29360128/29296875
 
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
 
: Mapping generators: ~2, ~875/512
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
 
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
 
[[Badness]]: 0.035832
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5632/5625
 
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
 
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
 
Badness: 0.017540


=== 13-limit ===
=== 13-limit ===
Commas: 144/143, 196/195, 243/242, 364/363
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
 
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
 
Badness: 0.017921
 
== Neominor ==
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, 177147/175616
 
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
 
: Mapping generators: ~2, ~189/160


POTE generator: ~11/9 = 351.573
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}


EDOs: 7ccd, 17c, 41, 58, 99ef
[[Badness]]: 0.088221


Badness: 0.0191
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Semihemi ==
Comma list: 243/242, 441/440, 35937/35840
Commas: 2401/2400, 3388/3375, 9801/9800


POTE generator: ~49/40 = 351.505
Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}


Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276


EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}


Badness: 0.042487
Badness: 0.027959


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 676/675, 847/845, 1716/1715
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 243/242, 364/363, 441/440
 
Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
 
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}


POTE generator: ~49/40 = 351.502
Badness: 0.026942


Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]
== Emmthird ==
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
[[Subgroup]]: 2.3.5.7


Badness: 0.0212
[[Comma list]]: 2401/2400, 14348907/14336000


=Tertiaseptal=
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
Aside from the breedsma, [[tertiaseptal]] tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 31&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.


Commas: 2401/2400, 65625/65536
: Mapping generators: ~2, ~2187/1372


POTE generator: ~256/245 = 77.191
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


EDOs: 15, 16, 31, 109, 140, 171
[[Badness]]: 0.016736


Badness: 0.0130
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 243/242, 441/440, 1792000/1771561
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991


EDOs: 15, 16, 31, 171, 202
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.0356
Badness: 0.052358


=== 13-limit ===
=== 13-limit ===
Commas: 243/242, 441/440, 625/624, 3584/3575
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440, 2200/2197


POTE generator: ~117/112 = 77.203
Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}


Map: [&lt;1 3 2 3 7 1|, &lt;0 -22 5 -3 -55 42|]
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989


EDOs: 31, 140e, 171
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Badness: 0.0369
Badness: 0.026974


=== 17-limit ===
=== 17-limit ===
Commas: 243/242, 375/374, 441/440, 625/624, 3584/3575
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~68/65 = 77.201
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Map: [&lt;1 3 2 3 7 1 1|, &lt;0 -22 5 -3 -55 42 48|]
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


EDOs: 31, 140e, 171
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985


Badness: 0.0274
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


==Tertia==
Badness: 0.023205
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
== Quinmite ==
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", 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>.


Map: [&lt;1 3 2 3 5|, &lt;0 -22 5 -3 -24|]
[[Subgroup]]: 2.3.5.7


EDOs: 31, 109, 140, 171e, 311e
[[Comma list]]: 2401/2400, 1959552/1953125


Badness: 0.0302
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}


== Hemitert ==
: Mapping generators: ~2, ~42/25
Commas: 2401/2400, 3025/3024, 65625/65536


POTE generator: ~45/44 = 38.596
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997


Map: [&lt;1 3 2 3 6|, &lt;0 -44 10 -6 -79|]
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}


EDOs: 31, 280, 311, 342, 2021cde, 3731cde
[[Badness]]: 0.037322


Badness: 0.0156
== 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 in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.


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


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 &lt;&lt;12 34 20 30 ...||.
[[Comma list]]: 2401/2400, 68359375/68024448


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[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.
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}


Commas: 2401/2400, 19683/19600
: Mapping generators: ~2, ~6125/3888


[[POTE_tuning|POTE generator]]: ~21/20 = 83.156
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}


Wedgie: &lt;&lt;12 34 20 26 -2 -49||
[[Badness]]: 0.075253


EDOs: 14c, 58, 72, 130, 202, 534, 938
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0341
Comma list: 2401/2400, 3025/3024, 4000/3993


==11-limit==
Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
Commas: 243/242, 441/440, 4000/3993


[[POTE_tuning|POTE generator]]: ~21/20 = 83.167
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}


EDOs: 14c, 58, 72, 130, 202
Badness: 0.022926


Badness: 0.0159
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Commas: 243/242, 351/350, 441/440, 676/675


POTE generator: ~21/20 = 83.116
Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716


EDOs: 58, 72, 130, 462
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}


Badness: 0.0130
Badness: 0.020888


==17-limit==
== Newt ==
Commas: 221/220, 243/242, 289/288, 351/350, 441/440
Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better.


POTE generator: ~21/20 = 83.168
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 4 7 7 9 11 9|, &lt;0 -6 -17 -10 -15 -26 -6|]
[[Comma list]]: 2401/2400, 33554432/33480783


EDOs: 58, 72, 130, 202g
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}


Badness: 0.0127
: mapping generators: ~2, ~49/40


=Quasiorwell=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;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.


Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}


Commas: 2401/2400, 29360128/29296875
[[Badness]]: 0.041878


POTE generator: ~1024/875 = 271.107
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
Comma list: 2401/2400, 3025/3024, 19712/19683


EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}


Badness: 0.0358
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
Badness: 0.019461


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095


Badness: 0.0175
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
Badness: 0.013830


EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19


Badness: 0.0179
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400


=Decoid=
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
{{see also|Qintosec family #Decoid}}


In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of [[10edo]]. Either 8/7 or 16/15 can be used its generator. It may be described as the 10&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]].
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117


Commas: 2401/2400, 67108864/66976875
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}


POTE generator: ~8/7 = 231.099
== Septidiasemi ==
{{Main| Septidiasemi }}


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
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.


Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
[[Subgroup]]: 2.3.5.7


EDOs: 10, 120, 130, 270
[[Comma list]]: 2401/2400, 2152828125/2147483648


Badness: 0.0339
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


==11-limit==
: Mapping generators: ~2, ~28/15
Commas: 2401/2400, 5832/5825, 9801/9800


POTE generator: ~8/7 = 231.070
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}


EDOs: 130, 270, 670, 940, 1210
[[Badness]]: 0.044115


Badness: 0.0187
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.


==13-limit==
Subgroup: 2.3.5.7.11
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224


POTE generator: ~8/7 = 231.083
Comma list: 243/242, 441/440, 939524096/935859375


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}


EDOs: 130, 270, 940, 1480
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279


Badness: 0.0135
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}


=Neominor=
Badness: 0.090687
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Weggie: &lt;&lt;6 41 22 51 18 -64||
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}


EDOs: 72, 161, 233, 305
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


Badness: 0.0882
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}


==11-limit==
Badness: 0.045773
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575


EDOs: 72, 161, 233, 305
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}


Badness: 0.0280
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


==13-limit==
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
Commas: 169/168, 243/242, 364/363, 441/440


POTE generator: ~13/11 = 283.294
Badness: 0.027322


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}


EDOs: 72, 161f, 233f
[[Subgroup]]: 2.3.5.7


Badness: 0.0269
[[Comma list]]: 2401/2400, 1224440064/1220703125


=Emmthird=
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
: Mapping generators: ~2, ~1296/875


POTE generator: ~2744/2187 = 392.988
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810


Map: [&lt;1 11 42 25|, &lt;0 -14 -59 -33|]
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


Wedgie: &lt;&lt;14 59 33 61 13 -89||
[[Badness]]: 0.057632


EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
== Subneutral ==
{{See also| Luna family }}


Badness: 0.0167
[[Subgroup]]: 2.3.5.7


=Quinmite=
[[Comma list]]: 2401/2400, 274877906944/274658203125
Commas: 2401/2400, 1959552/1953125


POTE generator: ~25/21 = 302.997
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
: Mapping generators: ~2, ~57344/46875


Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301


EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}


Badness: 0.0373
[[Badness]]: 0.045792


=Unthirds=
== Osiris ==
Commas: 2401/2400, 68359375/68024448
{{See also| Metric microtemperaments #Geb }}


POTE generator: ~3969/3125 = 416.717
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
[[Comma list]]: 2401/2400, 31381059609/31360000000


Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}


EDOs: 72, 167, 239, 311, 694, 1005c
: Mapping generators: ~2, ~2800/2187


Badness: 0.0753
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066


==11-limit==
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
Commas: 2401/2400, 3025/3024, 4000/3993


POTE generator: ~14/11 = 416.718
[[Badness]]: 0.028307


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
== Gorgik ==
[[Subgroup]]: 2.3.5.7


EDOs: 72, 167, 239, 311, 1316c
[[Comma list]]: 2401/2400, 28672/28125


Badness: 0.0229
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


==13-limit==
: Mapping generators: ~2, ~8/7
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400


POTE generator: ~14/11 = 416.716
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


EDOs: 72, 311, 694, 1005c, 1699cd
[[Badness]]: 0.158384


Badness: 0.0209
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Newt=
Comma list: 176/175, 2401/2400, 2560/2541
Commas: 2401/2400, 33554432/33480783


POTE generator: ~49/40 = 351.113
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500


Wedgie: &lt;&lt;2 -57 -28 -95 -50 95||
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
Badness: 0.059260


Badness: 0.0419
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 176/175, 196/195, 364/363, 512/507
Commas: 2401/2400, 3025/3024, 19712/19683


POTE generator: ~49/40 = 351.115
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493


EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness: 0.0195
Badness: 0.032205


==13-limit==
== Fibo ==
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095
[[Subgroup]]: 2.3.5.7


POTE genertaor: ~49/40 = 351.117
[[Comma list]]: 2401/2400, 341796875/339738624


Map: [&lt;1 1 19 11 -10 -20|, &lt;0 2 -57 -28 46 81|]
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}


EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
: Mapping generators: ~2, ~125/96


Badness: 0.0138
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310


=Amicable=
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
Commas: 2401/2400, 1600000/1594323


POTE generator: ~21/20 = 84.880
Badness: 0.100511


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;20 52 31 36 -7 -74||
Comma list: 385/384, 1375/1372, 43923/43750


EDOs: 99, 212, 311, 410, 1131, 1541b
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


Badness: 0.0455
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318


=Septidiasemi=
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
Commas: 2401/2400, 2152828125/2147483648


POTE generator: ~15/14 = 119.297
Badness: 0.056514


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;26 -37 -12 -119 -92 76||
Comma list: 385/384, 625/624, 847/845, 1375/1372


EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}


Badness: 0.0441
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316


=Maviloid=
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
Commas: 2401/2400, 1224440064/1220703125


POTE generator: ~1296/875 = 678.810
Badness: 0.027429


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
== 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.


Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
[[Subgroup]]: 2.3.5.7


EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
[[Comma list]]: 2401/2400, 177147/175000


Badness: 0.0576
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


=Subneutral=
: Mapping generators: ~2, ~10/9
Commas: 2401/2400, 274877906944/274658203125


POTE generator: ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343


Map: [&lt;1 19 0 6}, &lt;0 -60 8 -11|]
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}


Wedgie: &lt;&lt;60 -8 11 -152 -151 48||
[[Badness]]: 0.125672


EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0458
Comma list: 243/242, 441/440, 43923/43750


=Osiris=
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
Commas: 2401/2400, 31381059609/31360000000


POTE generator: ~2800/2187 = 428.066
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345


Map: [&lt;1 13 33 21|, &lt;0 -32 -86 -51|]
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}


Wedgie: &lt;&lt;32 86 51 62 -9 -123||
Badness: 0.039962


EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0283
Comma list: 243/242, 351/350, 441/440, 847/845


=Gorgik=
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
Commas: 2401/2400, 28672/28125


POTE generator: ~8/7 = 227.512
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347


Map: [&lt;1 5 1 3|, &lt;0 -18 7 -1|]
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


Wedgie: &lt;&lt;18 -7 1 -53 -49 22||
Badness: 0.021849


EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.1584
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


==11-limit==
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
Commas: 176/175, 2401/2400, 2560/2541


POTE generator: ~8/7 = 227.500
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348


Map: [&lt;1 5 1 3 1|, &lt;0 -18 7 -1 13|]
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
Badness: 0.020295


Badness: 0.059
== Catafourth ==
{{See also| Sensipent family }}


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 196/195, 364/363, 512/507


POTE generator: ~8/7 = 227.493
[[Comma list]]: 2401/2400, 78732/78125


Map: [&lt;1 5 1 3 1 2|, &lt;0 -18 7 -1 13 9|]
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


EDOs: 21, 37, 58, 153bcef, 211bcdef
: Mapping generators: ~2, ~250/189


Badness: 0.0322
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235


=Fibo=
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
Commas: 2401/2400, 341796875/339738624


POTE generator: ~125/96 = 454.310
Badness: 0.079579


Map: [&lt;1 19 8 10|, &lt;0 -46 -15 -19|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;46 15 19 -83 -99 2||
Comma list: 243/242, 441/440, 78408/78125


EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}


Badness: 0.1005
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
Commas: 385/384, 1375/1372, 43923/43750


POTE generator: ~100/77 = 454.318
Badness: 0.036785


Map: [&lt;1 19 8 10 8|, &lt;0 -46 -15 -19 -12|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 37, 103, 140, 243e
Comma list: 243/242, 351/350, 441/440, 10985/10976


Badness: 0.0565
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
Commas: 385/384, 625/624, 847/845, 1375/1372


POTE generator: ~13/10 = 454.316
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}


Map: [&lt;1 19 8 10 8 9|, &lt;0 -46 -15 -19 -12 -14|]
Badness: 0.021694


EDOs: 37, 103, 140, 243e
== Cotritone ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0274
[[Comma list]]: 2401/2400, 390625/387072


=Mintone=
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
In addition to 2401/2400, mintone tempers out 177147/175000 = |-3 11 -5 -1&gt; 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.


==7-limit==
: Mappping generators: ~2, ~10/7
Commas: 2401/2400, 177147/175000


POTE generator: ~10/9 = 186.343
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385


Map: [&lt;1 5 9 7|, &lt;0 -22 -43 -27|]
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}


EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
[[Badness]]: 0.098322


Badness: 0.12567
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 385/384, 1375/1372, 4000/3993
Commas: 243/242, 441/440, 43923/43750


POTE generator: ~10/9 = 186.345
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}


Map: [&lt;1 5 9 7 12|, &lt;0 -22 -43 -27 -55|]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


EDOs: 58, 103, 161, 425b, 586b, 747bc
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}


Badness: 0.0400
Badness: 0.032225


==13-limit==
=== 13-limit ===
Commas: 243/242, 351/350, 441/440, 847/845
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 364/363, 385/384, 625/624
 
Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
 
Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}
 
Badness: 0.028683
 
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness 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]] ([[POTE]]): ~2 = 1\1, ~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
 
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
 
Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
 
Badness: 0.046181
 
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
 
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
 
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
 
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
 
: Mapping generators: ~2, ~3828125/2985984
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
 
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
 
[[Badness]] (Smith): 0.0597


POTE generator: ~10/9 = 186.347
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 9 7 12 11|, &lt;0 -22 -43 -27 -55 -47|]
Comma list: 2401/2400, 3025/3024, 766656/765625


EDOs: 58, 103, 161
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


Badness: 0.0218
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078


==17-limit==
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
Commas: 243/242, 351/350, 441/440, 561/560, 847/845


POTE generator: ~10/9 = 186.348
Badness (Smith): 0.0262


Map: [&lt;1 5 9 7 12 11 3|, &lt;0 -22 -43 -27 -55 -47 7|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 58, 103, 161, 264
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224


=Catafourth=
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
Commas: 2401/2400, 78732/78125


POTE generator: ~250/189 = 489.235
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069


Map: [&lt;1 13 17 13|, &lt;0 -28 -36 -25|]
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}


Wedgie: &lt;&lt;28 36 25 -8 -39 -43||
Badness (Smith): 0.0160


EDOs: 27, 76, 103, 130
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0796
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


==11-limit==
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
Commas: 243/242, 441/440, 78408/78125


POTE generator: ~250/189 = 489.252
Optimal tunings:  
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108


Map: [&lt;1 13 17 13 32|, &lt;0 -28 -36 -25 -70|]
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


EDOs: 103, 130, 233, 363, 493e, 856be
Badness (Smith): 0.0210


Badness: 0.0368
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41


==13-limit==
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Commas: 243/242, 351/350, 441/440, 10985/10976


POTE generator: ~65/49 = 489.256
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}


Map: [&lt;1 13 17 13 32 9|, &lt;0 -28 -36 -25 -70 -13|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111


EDOs: 103, 130, 233, 363
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


Badness: 0.0217
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
 
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 549755813888/533935546875
 
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
 
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
 
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
 
[[Badness]] (Sintel): 4.40
 
== Surmarvelpyth ==
''Surmarvelpyth'' 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
 
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}
 
: Mapping generators: ~2, ~675/448
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
 
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
 
[[Badness]]: 0.202249
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 820125/819896, 2097152/2096325
 
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
 
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}
 
Badness: 0.052308
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
 
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
 
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}
 
Badness: 0.032503
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=Cotritone=
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
Commas: 2401/2400, 390625/387072


POTE generator: ~7/5 = 583.3848
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}


Map: [&lt;1 -13 -4 -4|, &lt;0 30 13 14|]
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722


EDOs: 35, 37, 72, 109, 181, 253
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}


==11-limit==
Badness: 0.020995
Commas: 385/384, 1375/1372, 4000/3993


POTE generator: ~7/5 = 583.3872
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 -13 -4 -4 2|, &lt;0 30 13 14 3|]
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984


EDOs: 35, 37, 72, 109, 181, 253
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}


==13-limit==
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Commas: 169/168, 364/363, 385/384, 625/624


POTE generator: ~7/5 = 583.3866
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}


Map: [&lt;1 -13 -4 -4 2 -7|, &lt;0 30 13 14 3 22|]
Badness: 0.013771


EDOs: 37, 72, 109, 181f
== Notes ==


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Pages with mostly numerical content]]
[[Category:Breed]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"