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{{Technical data page}}
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.
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.


=Hemififths=
Temperaments discussed elsewhere include:
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo|99edo]] and [[140edo|140edo]] providing good tunings, and [[239edo|239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie &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.
* ''[[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]]


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|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.
== Hemififths ==
{{Main| Hemififths }}


==5-limit==
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}}.
Comma: 858993459200/847288609443


POTE generator: ~655360/531441 = 351.476
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.


Map: [&lt;1 1 -5|, &lt;0 2 25|]
[[Subgroup]]: 2.3.5.7


EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
[[Comma list]]: 2401/2400, 5120/5103


Badness: 0.3728
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


==7-limit==
: mapping generators: ~2, ~49/40
Commas: 2401/2400, 5120/5103


7 and 9-limit minimax
[[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 }}


[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


Eigenvalues: 2, 5
[[Algebraic generator]]: (2 + sqrt(2))/2


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


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
[[Badness]] (Smith): 0.022243


EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0222
Comma list: 243/242, 441/440, 896/891


==11-limit==
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
Commas: 243/242, 441/440, 896/891


POTE generator: ~11/9 = 351.521
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


EDOs: 7, 17, 41, 58, 99
Badness (Smith): 0.023498


Badness: 0.0235
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 144/143, 196/195, 243/242, 364/363
Commas: 144/143, 196/195, 243/242, 364/363


POTE generator: ~11/9 = 351.573
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


EDOs: 7, 17, 41, 58, 99
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


Badness: 0.0191
Badness (Smith): 0.019090


=Semihemi=
=== Semihemi ===
Commas: 2401/2400, 3388/3375, 9801/9800
Subgroup: 2.3.5.7.11


POTE generator: ~49/40 = 351.505
Comma list: 2401/2400, 3388/3375, 5120/5103


Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}


EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
: mapping generators: ~99/70, ~400/231


Badness: 42.487
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047


==13-limit==
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
Commas: 352/351, 676/675, 847/845, 1716/1715


POTE generator: ~49/40 = 351.502
Badness (Smith): 0.042487


Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
Comma list: 352/351, 676/675, 847/845, 1716/1715


Badness: 0.0212
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


=Tertiaseptal=
Optimal tunings:
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 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


Commas: 2401/2400, 65625/65536
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}


POTE generator: ~256/245 = 77.191
Badness (Smith): 0.021188


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
=== 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.


EDOs: 15, 16, 31, 109, 140, 171
Subgroup: 2.3.5.7.11


Badness: 0.0130
Comma list: 2401/2400, 3025/3024, 5120/5103


==11-limit==
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
: Mapping generators: ~2, ~243/220


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378


EDOs: 15, 16, 31, 171, 202
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.0356
Badness (Smith): 0.040170


==13-limit==
==== 13-limit ====
Commas: 243/242, 441/440, 625/624, 3584/3575
Subgroup: 2.3.5.7.11.13


POTE generator: ~117/112 = 77.203
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Map: [&lt;1 3 2 3 7 1|, &lt;0 -22 5 -3 -55 42|]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


EDOs: 31, 140e, 171, 373ef, 544ef
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


Badness: 0.0369
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


==Tertia==
Badness (Smith): 0.031144
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
== Tertiaseptal ==
{{Main| Tertiaseptal }}


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


EDOs: 31, 109, 140, 171e, 311e
[[Subgroup]]: 2.3.5.7


Badness: 0.0302
[[Comma list]]: 2401/2400, 65625/65536


=Hemitert=
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
Commas: 2401/2400 3025/3024 65625/65536


POTE generator: ~45/44 = 38.596
: Mapping generators: ~2, ~256/245


Map: [&lt;1 3 2 3 6|, &lt;0 -44 10 -6 -79|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191


EDOs: 31, 280, 311, 342, 2021cde, 3731cde
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}


Badness: 0.0156
[[Badness]]: 0.012995


=Harry=
=== 11-limit ===
Commas: 2401/2400, 19683/19600
Subgroup: 2.3.5.7.11


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.
Comma list: 243/242, 441/440, 65625/65536


Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is &lt;&lt;12 34 20 30 ...||.
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}


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|130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227


[[POTE_tuning|POTE generator]]: ~21/20 = 83.156
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
Badness: 0.035576


Wedgie: &lt;&lt;12 34 20 26 -2 -49||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 14, 58, 72, 130, 202, 534, 938
Comma list: 243/242, 441/440, 625/624, 3584/3575


Badness: 0.0341
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
Commas: 243/242, 441/440, 4000/3993


[[POTE_tuning|POTE generator]]: ~21/20 = 83.167
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
Badness: 0.036876


EDOs: 14, 58, 72, 130, 202
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0159
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


==13-limit==
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
Commas: 243/242, 351/350, 441/440, 676/675


[[POTE_tuning|POTE generator]]: ~21/20 = 83.116
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}


EDOs: 14, 58, 72, 130, 462
Badness: 0.027398


Badness: 0.0130
=== Tertia ===
Subgroup:2.3.5.7.11


=Quasiorwell=
Comma list: 385/384, 1331/1323, 1375/1372
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.
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}


Commas: 2401/2400, 29360128/29296875
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173


POTE generator: ~1024/875 = 271.107
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
Badness: 0.030171


EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0358
Comma list: 352/351, 385/384, 625/624, 1331/1323


==11-limit==
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}


EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Badness: 0.028384


Badness: 0.0175
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162


EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}


Badness: 0.0179
Badness: 0.022416


=Decoid=
=== Tertiaseptia ===
Commas: 2401/2400, 67108864/66976875
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 231.099
Comma list: 2401/2400, 6250/6237, 65625/65536


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}


Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169


EDOs: 10, 120, 130, 270
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}


Badness: 0.0339
Badness: 0.056926


==11-limit==
==== 13-limit ====
Commas: 2401/2400, 5832/5825, 9801/9800
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 231.070
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}


EDOs: 130, 270, 670, 940, 1210
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168


Badness: 0.0187
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}


==13-limit==
Badness: 0.027474
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224


POTE generator: ~8/7 = 231.083
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197


EDOs: 130, 270, 940, 1480
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}


Badness: 0.0135
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


=Neominor=
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
Badness: 0.018773


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Weggie: &lt;&lt;6 41 22 51 18 -64||
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197


EDOs: 72, 161, 233, 305
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}


Badness: 0.0882
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
Badness: 0.017653


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


EDOs: 72, 161, 233, 305
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215


Badness: 0.0280
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
Commas: 169/168, 243/242, 364/363, 441/440


POTE generator: ~13/11 = 283.294
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
Badness: 0.015123


EDOs: 72, 161f, 233f
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.0269
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155


=Emmthird=
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167


POTE generator: ~2744/2187 = 392.988
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}


Map: [&lt;1 11 42 25|,  &lt;0 -14 -59 -33|]
Badness: 0.012181


Wedgie: &lt;&lt;14 59 33 61 13 -89||
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Badness: 0.0167
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}


=Quinmite=
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Commas: 2401/2400, 1959552/1953125


POTE generator: ~25/21 = 302.997
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
Badness: 0.012311


Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
==== 37-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37


EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Badness: 0.0373
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 }}


=Unthirds=
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
Commas: 2401/2400, 68359375/68024448


POTE generator: ~3969/3125 = 416.717
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
Badness: 0.010949


Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
==== 41-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41


EDOs: 72, 167, 239, 311, 694, 1005c
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930


Badness: 0.0753
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 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
Commas: 2401/2400, 3025/3024, 4000/3993


POTE generator: ~14/11 = 416.718
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
Badness: 0.009825


EDOs: 72, 167, 239, 311, 1316c
=== Hemitert ===
Subgroup: 2.3.5.7.11


Badness: 0.0229
Comma list: 2401/2400, 3025/3024, 65625/65536


==13-limit==
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400


POTE generator: ~14/11 = 416.716
: Mapping generators: ~2, ~45/44


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596


EDOs: 72, 311, 694, 1005c, 1699cd
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}


Badness: 0.0209
Badness: 0.015633


=Newt=
==== 13-limit ====
Commas: 2401/2400, 33554432/33480783
Subgroup: 2.3.5.7.11.13


POTE generator: ~49/40 = 351.113
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}


Wedgie: &lt;&lt;2 -57 -28 -95 -50 95||
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588


EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}


Badness: 0.0419
Badness: 0.033573


==11-limit==
==== 17-limit ====
Commas: 2401/2400, 3025/3024, 19712/19683
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~49/40 = 351.115
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}


EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589


Badness: 0.0195
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}


==13-limit==
Badness: 0.025298
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095


POTE genertaor: ~49/40 = 351.117
=== Semitert ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 19 11 -10 -20|, &lt;0 2 -57 -28 46 81|]
Comma list: 2401/2400, 9801/9800, 65625/65536


EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}


Badness: 0.0138
: Mapping generators: ~99/70, ~256/245


=Amicable=
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
Commas: 2401/2400, 1600000/1594323


POTE generator: ~21/20 = 84.880
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
Badness: 0.025790


Wedgie: &lt;&lt;20 52 31 36 -7 -74||
== 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.


EDOs: 99, 212, 311, 410, 1131, 1541b
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


Badness: 0.0455
[[Subgroup]]: 2.3.5.7


=Septidiasemi=
[[Comma list]]: 2401/2400, 29360128/29296875
Commas: 2401/2400, 2152828125/2147483648


POTE generator: ~15/14 = 119.297
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
: Mapping generators: ~2, ~875/512


Wedgie: &lt;&lt;26 -37 -12 -119 -92 76||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107


EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}


Badness: 0.0441
[[Badness]]: 0.035832


=Maviloid=
=== 11-limit ===
Commas: 2401/2400, 1224440064/1220703125
Subgroup: 2.3.5.7.11


POTE generator: ~1296/875 = 678.810
Comma list: 2401/2400, 3025/3024, 5632/5625


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}


Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111


EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}


Badness: 0.0576
Badness: 0.017540


=Subneutral=
=== 13-limit ===
Commas: 2401/2400, 274877906944/274658203125
Subgroup: 2.3.5.7.11.13


POTE generator: ~57344/46875 = 348.301
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


Map: [&lt;1 19 0 6}, &lt;0 -60 8 -11|]
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}


Wedgie: &lt;&lt;60 -8 11 -152 -151 48||
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107


EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.0458
Badness: 0.017921


=Osiris=
== Neominor ==
Commas: 2401/2400, 31381059609/31360000000
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.


POTE generator: ~2800/2187 = 428.066
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 13 33 21|, &lt;0 -32 -86 -51|]
[[Comma list]]: 2401/2400, 177147/175616


Wedgie: &lt;&lt;32 86 51 62 -9 -123||
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}


EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
: Mapping generators: ~2, ~189/160


Badness: 0.0283
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280


=Gorgik=
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Commas: 2401/2400, 28672/28125


POTE generator: ~8/7 = 227.512
[[Badness]]: 0.088221


Map: [&lt;1 5 1 3|, &lt;0 -18 7 -1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;18 -7 1 -53 -49 22||
Comma list: 243/242, 441/440, 35937/35840


EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}


Badness: 0.1584
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
Commas: 176/175, 2401/2400, 2560/2541


POTE generator: ~8/7 = 227.500
Badness: 0.027959


Map: [&lt;1 5 1 3 1|, &lt;0 -18 7 -1 13|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
Comma list: 169/168, 243/242, 364/363, 441/440


Badness: 0.059
Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
Commas: 176/175, 196/195, 364/363, 512/507


POTE generator: ~8/7 = 227.493
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}


Map: [&lt;1 5 1 3 1 2|, &lt;0 -18 7 -1 13 9|]
Badness: 0.026942


EDOs: 21, 37, 58, 153bcef, 211bcdef
== Emmthird ==
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Badness: 0.0322
[[Subgroup]]: 2.3.5.7


=Fibo=
[[Comma list]]: 2401/2400, 14348907/14336000
Commas: 2401/2400, 341796875/339738624


POTE generator: ~125/96 = 454.310
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}


Map: [&lt;1 19 8 10|, &lt;0 -46 -15 -19|]
: Mapping generators: ~2, ~2187/1372


Wedgie: &lt;&lt;46 15 19 -83 -99 2||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988


EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


Badness: 0.1005
[[Badness]]: 0.016736


==11-limit==
=== 11-limit ===
Commas: 385/384, 1375/1372, 43923/43750
Subgroup: 2.3.5.7.11


POTE generator: ~100/77 = 454.318
Comma list: 243/242, 441/440, 1792000/1771561


Map: [&lt;1 19 8 10 8|, &lt;0 -46 -15 -19 -12|]
Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}


EDOs: 37, 103, 140, 243e
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991


Badness: 0.0565
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


==13-limit==
Badness: 0.052358
Commas: 385/384, 625/624, 847/845, 1375/1372


POTE generator: ~13/10 = 454.316
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 19 8 10 8 9|, &lt;0 -46 -15 -19 -12 -14|]
Comma list: 243/242, 364/363, 441/440, 2200/2197


EDOs: 37, 103, 140, 243e
Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}


Badness: 0.0274
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989


=Mintone=
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
Commas: 2401/2400, 177147/175000


POTE generator: ~10/9 = 186.343
Badness: 0.026974


Map: [&lt;1 5 9 7|, &lt;0 -22 -43 -27|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Badness: 0.12567
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
Commas: 243/242, 441/440, 43923/43750


POTE generator: ~10/9 = 186.345
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}


Map: [&lt;1 5 9 7 12|, &lt;0 -22 -43 -27 -55|]
Badness: 0.023205


EDOs: 58, 103, 161, 425b, 586b, 747bc
== 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>.


Badness: 0.0400
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 2401/2400, 1959552/1953125
Commas: 243/242, 351/350, 441/440, 847/845


POTE generator: ~10/9 = 186.347
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}


Map: [&lt;1 5 9 7 12 11|, &lt;0 -22 -43 -27 -55 -47|]
: Mapping generators: ~2, ~42/25


EDOs: 58, 103, 161
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997


Badness: 0.0218
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}


=Catafourth=
[[Badness]]: 0.037322
Commas: 2400/2401, 78732/78125


POTE generator: ~250/189 = 489.235
== 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.


Map: [&lt;1 13 17 13|, &lt;0 -28 -36 -25|]
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).


Wedgie: &lt;&lt;[28 36 25 -8 -39 -43||
[[Subgroup]]: 2.3.5.7


EDOs: 27, 76, 103, 130
[[Comma list]]: 2401/2400, 68359375/68024448


Badness: 0.0796
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}


==11-limit==
: Mapping generators: ~2, ~6125/3888
Commas: 243/242, 441/440, 78408/78125


POTE generator: ~250/189 = 489.252
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717


Map: [&lt;1 13 17 13 32|, &lt;0 -28 -36 -25 -70|]
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}


EDOs: 103, 130, 233, 363, 493e, 856be
[[Badness]]: 0.075253


Badness: 0.0368
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 2401/2400, 3025/3024, 4000/3993
Commas: 243/242, 351/350, 441/440, 10985/10976


POTE generator: ~65/49 = 489.256
Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}


Map: [&lt;1 13 17 13 32 9|, &lt;0 -28 -36 -25 -70 -13|]
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718


EDOs: 103, 130, 233, 363
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}


Badness: 0.0217
Badness: 0.022926
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
 
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
 
Badness: 0.020888
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 33554432/33480783
 
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
 
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
 
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
 
[[Badness]]: 0.041878
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 19712/19683
 
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
 
Badness: 0.019461
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
 
Badness: 0.013830
 
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
 
== Septidiasemi ==
{{Main| Septidiasemi }}
 
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 2152828125/2147483648
 
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
 
: Mapping generators: ~2, ~28/15
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
 
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
 
[[Badness]]: 0.044115
 
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
 
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 939524096/935859375
 
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}
 
Badness: 0.090687
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}
 
Badness: 0.045773
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
 
Badness: 0.027322
 
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1224440064/1220703125
 
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
 
: Mapping generators: ~2, ~1296/875
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
 
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
 
[[Badness]]: 0.057632
 
== Subneutral ==
{{See also| Luna family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 274877906944/274658203125
 
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
 
: Mapping generators: ~2, ~57344/46875
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
 
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 
[[Badness]]: 0.045792
 
== Osiris ==
{{See also| Metric microtemperaments #Geb }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 31381059609/31360000000
 
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
 
: Mapping generators: ~2, ~2800/2187
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
 
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
 
[[Badness]]: 0.028307
 
== Gorgik ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 28672/28125
 
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
 
: Mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]]: 0.158384
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 2401/2400, 2560/2541
 
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
 
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
 
Badness: 0.059260
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 364/363, 512/507
 
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
 
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
 
Badness: 0.032205
 
== Fibo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 341796875/339738624
 
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
 
: Mapping generators: ~2, ~125/96
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness: 0.100511
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
 
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
 
Badness: 0.056514
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 847/845, 1375/1372
 
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
 
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
 
Badness: 0.027429
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175000
 
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
 
: Mapping generators: ~2, ~10/9
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
 
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
 
[[Badness]]: 0.125672
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 43923/43750
 
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
 
Badness: 0.039962
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 847/845
 
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
 
Badness: 0.021849
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
 
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
 
Badness: 0.020295
 
== Catafourth ==
{{See also| Sensipent family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 78732/78125
 
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
 
: Mapping generators: ~2, ~250/189
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
 
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
 
Badness: 0.079579
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 78408/78125
 
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
 
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
 
Badness: 0.036785
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 10985/10976
 
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
 
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
 
Badness: 0.021694
 
== Cotritone ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 390625/387072
 
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
 
: Mappping generators: ~2, ~10/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
 
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
 
[[Badness]]: 0.098322
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 4000/3993
 
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
 
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}
 
Badness: 0.032225
 
=== 13-limit ===
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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 766656/765625
 
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
 
Badness (Smith): 0.0262
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
 
Badness (Smith): 0.0160
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108
 
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
 
Badness (Smith): 0.0210
 
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41
 
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111
 
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
 
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
 
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 549755813888/533935546875
 
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
 
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
 
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
 
[[Badness]] (Sintel): 4.40
 
== Surmarvelpyth ==
''Surmarvelpyth'' 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
 
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
 
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
 
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness: 0.020995
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
 
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
 
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness: 0.013771
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
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