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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


POTE generator: ~11/9 = 351.573
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
Badness: 0.017921


EDOs: 7ccd, 17c, 41, 58, 99ef
== Neominor ==
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.


Badness: 0.0191
[[Subgroup]]: 2.3.5.7


== Semihemi ==
[[Comma list]]: 2401/2400, 177147/175616
Commas: 2401/2400, 3388/3375, 9801/9800


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


Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]
: Mapping generators: ~2, ~189/160


EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280


Badness: 0.042487
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
 
[[Badness]]: 0.088221
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 35937/35840
 
Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276
 
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
 
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


POTE generator: ~49/40 = 351.502
Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}


Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294


EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}


Badness: 0.0212
Badness: 0.026942


=Tertiaseptal=
== Emmthird ==
Aside from the breedsma, [[tertiaseptal]] tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 65625/65536
[[Subgroup]]: 2.3.5.7


POTE generator: ~256/245 = 77.191
[[Comma list]]: 2401/2400, 14348907/14336000


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}


EDOs: 15, 16, 31, 109, 140, 171
: Mapping generators: ~2, ~2187/1372


Badness: 0.0130
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988


==11-limit==
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
[[Badness]]: 0.016736


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 15, 16, 31, 171, 202
Comma list: 243/242, 441/440, 1792000/1771561


Badness: 0.0356
Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
 
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
 
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
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
 
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
 
Badness: 0.023205
 
== 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>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1959552/1953125
 
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
 
: Mapping generators: ~2, ~42/25
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
 
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
 
[[Badness]]: 0.037322
 
== 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.
 
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 68359375/68024448
 
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
 
: Mapping generators: ~2, ~6125/3888
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
 
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
 
[[Badness]]: 0.075253
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 4000/3993
 
Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


POTE generator: ~68/65 = 77.201
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}


Map: [&lt;1 3 2 3 7 1 1|, &lt;0 -22 5 -3 -55 42 48|]
Badness: 0.022926


EDOs: 31, 140e, 171
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0274
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


==Tertia==
Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716


Map: [&lt;1 3 2 3 5|, &lt;0 -22 5 -3 -24|]
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}


EDOs: 31, 109, 140, 171e, 311e
Badness: 0.020888


Badness: 0.0302
== Newt ==
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.


== Hemitert ==
[[Subgroup]]: 2.3.5.7
Commas: 2401/2400, 3025/3024, 65625/65536


POTE generator: ~45/44 = 38.596
[[Comma list]]: 2401/2400, 33554432/33480783


Map: [&lt;1 3 2 3 6|, &lt;0 -44 10 -6 -79|]
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}


EDOs: 31, 280, 311, 342, 2021cde, 3731cde
: mapping generators: ~2, ~49/40


Badness: 0.0156
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113


=Harry=
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
{{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.
[[Badness]]: 0.041878


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 ...||.
=== 11-limit ===
Subgroup: 2.3.5.7.11


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.
Comma list: 2401/2400, 3025/3024, 19712/19683


Commas: 2401/2400, 19683/19600
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}


[[POTE_tuning|POTE generator]]: ~21/20 = 83.156
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}


Wedgie: &lt;&lt;12 34 20 26 -2 -49||
Badness: 0.019461


EDOs: 14c, 58, 72, 130, 202, 534, 938
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0341
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095


==11-limit==
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
Commas: 243/242, 441/440, 4000/3993


[[POTE_tuning|POTE generator]]: ~21/20 = 83.167
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}


EDOs: 14c, 58, 72, 130, 202
Badness: 0.013830


Badness: 0.0159
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19


==13-limit==
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
Commas: 243/242, 351/350, 441/440, 676/675


POTE generator: ~21/20 = 83.116
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117


EDOs: 58, 72, 130, 462
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}


Badness: 0.0130
== Septidiasemi ==
{{Main| Septidiasemi }}


==17-limit==
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
Commas: 221/220, 243/242, 289/288, 351/350, 441/440


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, 2152828125/2147483648


EDOs: 58, 72, 130, 202g
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


Badness: 0.0127
: Mapping generators: ~2, ~28/15


=Quasiorwell=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
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| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}


Commas: 2401/2400, 29360128/29296875
[[Badness]]: 0.044115


POTE generator: ~1024/875 = 271.107
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
Subgroup: 2.3.5.7.11


EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Comma list: 243/242, 441/440, 939524096/935859375


Badness: 0.0358
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
Badness: 0.090687


EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0175
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


==13-limit==
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}


EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.045773


Badness: 0.0179
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=Decoid=
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
{{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]].
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}


Commas: 2401/2400, 67108864/66976875
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


POTE generator: ~8/7 = 231.099
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
Badness: 0.027322


Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}


EDOs: 10, 120, 130, 270
[[Subgroup]]: 2.3.5.7


Badness: 0.0339
[[Comma list]]: 2401/2400, 1224440064/1220703125


==11-limit==
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
Commas: 2401/2400, 5832/5825, 9801/9800


POTE generator: ~8/7 = 231.070
: Mapping generators: ~2, ~1296/875


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810


EDOs: 130, 270, 670, 940, 1210
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


Badness: 0.0187
[[Badness]]: 0.057632


==13-limit==
== Subneutral ==
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224
{{See also| Luna family }}


POTE generator: ~8/7 = 231.083
[[Subgroup]]: 2.3.5.7


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
[[Comma list]]: 2401/2400, 274877906944/274658203125


EDOs: 130, 270, 940, 1480
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


Badness: 0.0135
: Mapping generators: ~2, ~57344/46875


=Neominor=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
[[Badness]]: 0.045792


Weggie: &lt;&lt;6 41 22 51 18 -64||
== Osiris ==
{{See also| Metric microtemperaments #Geb }}


EDOs: 72, 161, 233, 305
[[Subgroup]]: 2.3.5.7


Badness: 0.0882
[[Comma list]]: 2401/2400, 31381059609/31360000000


==11-limit==
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
: Mapping generators: ~2, ~2800/2187


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066


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


Badness: 0.0280
[[Badness]]: 0.028307


==13-limit==
== Gorgik ==
Commas: 169/168, 243/242, 364/363, 441/440
[[Subgroup]]: 2.3.5.7


POTE generator: ~13/11 = 283.294
[[Comma list]]: 2401/2400, 28672/28125


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


EDOs: 72, 161f, 233f
: Mapping generators: ~2, ~8/7


Badness: 0.0269
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512


=Emmthird=
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
[[Badness]]: 0.158384


POTE generator: ~2744/2187 = 392.988
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 11 42 25|, &lt;0 -14 -59 -33|]
Comma list: 176/175, 2401/2400, 2560/2541


Wedgie: &lt;&lt;14 59 33 61 13 -89||
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500


Badness: 0.0167
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


=Quinmite=
Badness: 0.059260
Commas: 2401/2400, 1959552/1953125


POTE generator: ~25/21 = 302.997
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
Comma list: 176/175, 196/195, 364/363, 512/507


Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}


EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493


Badness: 0.0373
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}


=Unthirds=
Badness: 0.032205
Commas: 2401/2400, 68359375/68024448


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


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
[[Comma list]]: 2401/2400, 341796875/339738624


Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}


EDOs: 72, 167, 239, 311, 694, 1005c
: Mapping generators: ~2, ~125/96


Badness: 0.0753
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310


==11-limit==
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
Commas: 2401/2400, 3025/3024, 4000/3993


POTE generator: ~14/11 = 416.718
Badness: 0.100511


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 72, 167, 239, 311, 1316c
Comma list: 385/384, 1375/1372, 43923/43750


Badness: 0.0229
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400


POTE generator: ~14/11 = 416.716
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
Badness: 0.056514


EDOs: 72, 311, 694, 1005c, 1699cd
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0209
Comma list: 385/384, 625/624, 847/845, 1375/1372


=Newt=
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
Commas: 2401/2400, 33554432/33480783


POTE generator: ~49/40 = 351.113
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


Wedgie: &lt;&lt;2 -57 -28 -95 -50 95||
Badness: 0.027429


EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
== 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.


Badness: 0.0419
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 2401/2400, 177147/175000
Commas: 2401/2400, 3025/3024, 19712/19683


POTE generator: ~49/40 = 351.115
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
: Mapping generators: ~2, ~10/9


EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343


Badness: 0.0195
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}


==13-limit==
[[Badness]]: 0.125672
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095


POTE genertaor: ~49/40 = 351.117
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 19 11 -10 -20|, &lt;0 2 -57 -28 46 81|]
Comma list: 243/242, 441/440, 43923/43750


EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}


Badness: 0.0138
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345


=Amicable=
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
{{see also| Amity family }}


Commas: 2401/2400, 1600000/1594323
Badness: 0.039962


POTE generator: ~21/20 = 84.880
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
Comma list: 243/242, 351/350, 441/440, 847/845


Wedgie: &lt;&lt;20 52 31 36 -7 -74||
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}


EDOs: 99, 212, 311, 410, 1131, 1541b
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347


Badness: 0.0455
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


=Septidiasemi=
Badness: 0.021849
Commas: 2401/2400, 2152828125/2147483648


POTE generator: ~15/14 = 119.297
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Wedgie: &lt;&lt;26 -37 -12 -119 -92 76||
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}


EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348


Badness: 0.0441
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


=Maviloid=
Badness: 0.020295
{{see also| Ragismic microtemperaments #Parakleismic }}


Commas: 2401/2400, 1224440064/1220703125
== Catafourth ==
{{See also| Sensipent family }}


POTE generator: ~1296/875 = 678.810
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
[[Comma list]]: 2401/2400, 78732/78125


Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
: Mapping generators: ~2, ~250/189


Badness: 0.0576
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235


=Subneutral=
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
Commas: 2401/2400, 274877906944/274658203125


POTE generator: ~57344/46875 = 348.301
Badness: 0.079579


Map: [&lt;1 19 0 6}, &lt;0 -60 8 -11|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;60 -8 11 -152 -151 48||
Comma list: 243/242, 441/440, 78408/78125


EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}


Badness: 0.0458
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252


=Osiris=
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
Commas: 2401/2400, 31381059609/31360000000


POTE generator: ~2800/2187 = 428.066
Badness: 0.036785


Map: [&lt;1 13 33 21|, &lt;0 -32 -86 -51|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;32 86 51 62 -9 -123||
Comma list: 243/242, 351/350, 441/440, 10985/10976


EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}


Badness: 0.0283
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256


=Gorgik=
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
Commas: 2401/2400, 28672/28125


POTE generator: ~8/7 = 227.512
Badness: 0.021694


Map: [&lt;1 5 1 3|, &lt;0 -18 7 -1|]
== Cotritone ==
[[Subgroup]]: 2.3.5.7


Wedgie: &lt;&lt;18 -7 1 -53 -49 22||
[[Comma list]]: 2401/2400, 390625/387072


EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}


Badness: 0.1584
: Mappping generators: ~2, ~10/7


==11-limit==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
Commas: 176/175, 2401/2400, 2560/2541


POTE generator: ~8/7 = 227.500
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}


Map: [&lt;1 5 1 3 1|, &lt;0 -18 7 -1 13|]
[[Badness]]: 0.098322


EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.059
Comma list: 385/384, 1375/1372, 4000/3993


==13-limit==
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
Commas: 176/175, 196/195, 364/363, 512/507


POTE generator: ~8/7 = 227.493
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


Map: [&lt;1 5 1 3 1 2|, &lt;0 -18 7 -1 13 9|]
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}


EDOs: 21, 37, 58, 153bcef, 211bcdef
Badness: 0.032225


Badness: 0.0322
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Fibo=
Comma list: 169/168, 364/363, 385/384, 625/624
Commas: 2401/2400, 341796875/339738624


POTE generator: ~125/96 = 454.310
Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}


Map: [&lt;1 19 8 10|, &lt;0 -46 -15 -19|]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387


Wedgie: &lt;&lt;46 15 19 -83 -99 2||
Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}


EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
Badness: 0.028683


Badness: 0.1005
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 385/384, 1375/1372, 43923/43750


POTE generator: ~100/77 = 454.318
[[Comma list]]: 2401/2400, 3645/3584


Map: [&lt;1 19 8 10 8|, &lt;0 -46 -15 -19 -12|]
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}


EDOs: 37, 103, 140, 243e
: Mapping generators: ~2, ~49/40


Badness: 0.0565
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603


==13-limit==
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
Commas: 385/384, 625/624, 847/845, 1375/1372


POTE generator: ~13/10 = 454.316
[[Badness]]: 0.110820


Map: [&lt;1 19 8 10 8 9|, &lt;0 -46 -15 -19 -12 -14|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 37, 103, 140, 243e
Comma list: 243/242, 441/440, 1815/1792


Badness: 0.0274
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


=Mintone=
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
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==
Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
Commas: 2401/2400, 177147/175000


POTE generator: ~10/9 = 186.343
Badness: 0.046181


Map: [&lt;1 5 9 7|, &lt;0 -22 -43 -27|]
== 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.


EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
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.


Badness: 0.12567
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.


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 243/242, 441/440, 43923/43750


POTE generator: ~10/9 = 186.345
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


Map: [&lt;1 5 9 7 12|, &lt;0 -22 -43 -27 -55|]
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}


EDOs: 58, 103, 161, 425b, 586b, 747bc
: Mapping generators: ~2, ~3828125/2985984


Badness: 0.0400
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
Commas: 243/242, 351/350, 441/440, 847/845


POTE generator: ~10/9 = 186.347
[[Badness]] (Smith): 0.0597


Map: [&lt;1 5 9 7 12 11|, &lt;0 -22 -43 -27 -55 -47|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 58, 103, 161
Comma list: 2401/2400, 3025/3024, 766656/765625


Badness: 0.0218
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


==17-limit==
Optimal tunings:
Commas: 243/242, 351/350, 441/440, 561/560, 847/845
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078


POTE generator: ~10/9 = 186.348
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


Map: [&lt;1 5 9 7 12 11 3|, &lt;0 -22 -43 -27 -55 -47 7|]
Badness (Smith): 0.0262


EDOs: 58, 103, 161, 264
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Catafourth=
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
{{see also| Sensipent family }}


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


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"