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Breedsmic temperaments: Difference between revisions

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


It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.


=Hemififths=
Temperaments discussed elsewhere include:
{{main|Hemififths}}
* ''[[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 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 ==
{{Main| Hemififths }}


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.
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}}.


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


POTE generator: ~655360/531441 = 351.476
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 -5|, &lt;0 2 25|]
[[Comma list]]: 2401/2400, 5120/5103


EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


Badness: 0.3728
: mapping generators: ~2, ~49/40


==7-limit==
[[Optimal tuning]]s:
Commas: 2401/2400, 5120/5103
* [[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 and 9-limit minimax
[[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


[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]
[[Algebraic generator]]: (2 + sqrt(2))/2


Eigenvalues: 2, 5
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}


Algebraic generator: (2 + sqrt(2))/2
[[Badness]] (Smith): 0.022243


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
Comma list: 243/242, 441/440, 896/891


Badness: 0.0222
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


==11-limit==
Optimal tunings:
Commas: 243/242, 441/440, 896/891
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


POTE generator: ~11/9 = 351.521
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
Badness (Smith): 0.023498


EDOs: 7ccd, 17c, 41, 58, 99e
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0235
Comma list: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Smith): 0.019090
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Smith): 0.042487
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Smith): 0.021188
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 
: Mapping generators: ~2, ~243/220
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Smith): 0.040170
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Smith): 0.031144
 
== Tertiaseptal ==
{{Main| Tertiaseptal }}
 
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 65625/65536
 
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
 
: Mapping generators: ~2, ~256/245
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
 
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
 
[[Badness]]: 0.012995
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
 
Badness: 0.035576
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
 
Badness: 0.036876
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
 
Badness: 0.027398
 
=== Tertia ===
Subgroup:2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 1375/1372
 
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
 
Badness: 0.030171
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323
 
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
 
Badness: 0.028384
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
 
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
 
Badness: 0.022416
 
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 6250/6237, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
 
Badness: 0.056926
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
 
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
 
Badness: 0.027474
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.018773
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
 
Badness: 0.017653
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
 
Badness: 0.015123
 
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
 
Badness: 0.012181
 
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.012311
 
==== 37-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.010949
 
==== 41-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
 
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.009825
 
=== Hemitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
 
: Mapping generators: ~2, ~45/44
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
 
Badness: 0.015633
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
 
Badness: 0.033573
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
 
Badness: 0.025298
 
=== Semitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 65625/65536
 
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
 
: Mapping generators: ~99/70, ~256/245
 
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
 
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
 
Badness: 0.025790
 
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 29360128/29296875
 
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
 
: Mapping generators: ~2, ~875/512
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
 
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
 
[[Badness]]: 0.035832
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5632/5625
 
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
 
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
 
Badness: 0.017540


=== 13-limit ===
=== 13-limit ===
Commas: 144/143, 196/195, 243/242, 364/363
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
 
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
 
Badness: 0.017921
 
== Neominor ==
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175616
 
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
 
: Mapping generators: ~2, ~189/160


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


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


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


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


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


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


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


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


Badness: 0.042487
Badness: 0.027959


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.0356
Badness: 0.052358


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


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


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


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


Badness: 0.0369
Badness: 0.026974


=== 17-limit ===
=== 17-limit ===
Commas: 243/242, 375/374, 441/440, 625/624, 3584/3575
Subgroup: 2.3.5.7.11.13.17
 
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
 
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
 
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


POTE generator: ~68/65 = 77.201
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}


Map: [&lt;1 3 2 3 7 1 1|, &lt;0 -22 5 -3 -55 42 48|]
== Septidiasemi ==
{{Main| Septidiasemi }}


EDOs: 31, 140e, 171
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.


Badness: 0.0274
[[Subgroup]]: 2.3.5.7


==Tertia==
[[Comma list]]: 2401/2400, 2152828125/2147483648
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}


Map: [&lt;1 3 2 3 5|, &lt;0 -22 5 -3 -24|]
: Mapping generators: ~2, ~28/15


EDOs: 31, 109, 140, 171e, 311e
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297


Badness: 0.0302
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}


== Hemitert ==
[[Badness]]: 0.044115
Commas: 2401/2400, 3025/3024, 65625/65536


POTE generator: ~45/44 = 38.596
=== 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 3 2 3 6|, &lt;0 -44 10 -6 -79|]
Subgroup: 2.3.5.7.11


EDOs: 31, 280, 311, 342, 2021cde, 3731cde
Comma list: 243/242, 441/440, 939524096/935859375


Badness: 0.0156
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}


=Harry=
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
{{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.
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}


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 ...||.
Badness: 0.090687


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


Commas: 2401/2400, 19683/19600
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


[[POTE_tuning|POTE generator]]: ~21/20 = 83.156
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


Wedgie: &lt;&lt;12 34 20 26 -2 -49||
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}


EDOs: 14c, 58, 72, 130, 202, 534, 938
Badness: 0.045773


Badness: 0.0341
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==11-limit==
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Commas: 243/242, 441/440, 4000/3993


[[POTE_tuning|POTE generator]]: ~21/20 = 83.167
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


EDOs: 14c, 58, 72, 130, 202
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Badness: 0.0159
Badness: 0.027322


==13-limit==
== Maviloid ==
Commas: 243/242, 351/350, 441/440, 676/675
{{See also| Ragismic microtemperaments #Parakleismic }}


POTE generator: ~21/20 = 83.116
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
[[Comma list]]: 2401/2400, 1224440064/1220703125


EDOs: 58, 72, 130, 462
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}


Badness: 0.0130
: Mapping generators: ~2, ~1296/875


==17-limit==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
Commas: 221/220, 243/242, 289/288, 351/350, 441/440


POTE generator: ~21/20 = 83.168
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


Map: [&lt;2 4 7 7 9 11 9|, &lt;0 -6 -17 -10 -15 -26 -6|]
[[Badness]]: 0.057632


EDOs: 58, 72, 130, 202g
== Subneutral ==
{{See also| Luna family }}


Badness: 0.0127
[[Subgroup]]: 2.3.5.7


=Quasiorwell=
[[Comma list]]: 2401/2400, 274877906944/274658203125
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|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


Commas: 2401/2400, 29360128/29296875
: Mapping generators: ~2, ~57344/46875


POTE generator: ~1024/875 = 271.107
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}


EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
[[Badness]]: 0.045792


Badness: 0.0358
== Osiris ==
{{See also| Metric microtemperaments #Geb }}


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
[[Comma list]]: 2401/2400, 31381059609/31360000000


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}


EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
: Mapping generators: ~2, ~2800/2187


Badness: 0.0175
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066


==13-limit==
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
[[Badness]]: 0.028307


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
== Gorgik ==
[[Subgroup]]: 2.3.5.7


EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
[[Comma list]]: 2401/2400, 28672/28125


Badness: 0.0179
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


=Decoid=
: Mapping generators: ~2, ~8/7
{{see also|Qintosec family #Decoid}}


In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of [[10edo]]. Either 8/7 or 16/15 can be used its generator. It may be described as the 10&amp;270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512


Commas: 2401/2400, 67108864/66976875
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


POTE generator: ~8/7 = 231.099
[[Badness]]: 0.158384


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
Comma list: 176/175, 2401/2400, 2560/2541


EDOs: 10, 120, 130, 270
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


Badness: 0.0339
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
Commas: 2401/2400, 5832/5825, 9801/9800


POTE generator: ~8/7 = 231.070
Badness: 0.059260


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 130, 270, 670, 940, 1210
Comma list: 176/175, 196/195, 364/363, 512/507


Badness: 0.0187
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224


POTE generator: ~8/7 = 231.083
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
Badness: 0.032205


EDOs: 130, 270, 940, 1480
== Fibo ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0135
[[Comma list]]: 2401/2400, 341796875/339738624


=Neominor=
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
: Mapping generators: ~2, ~125/96


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310


Weggie: &lt;&lt;6 41 22 51 18 -64||
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}


EDOs: 72, 161, 233, 305
Badness: 0.100511


Badness: 0.0882
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 385/384, 1375/1372, 43923/43750
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318


EDOs: 72, 161, 233, 305
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


Badness: 0.0280
Badness: 0.056514


==13-limit==
=== 13-limit ===
Commas: 169/168, 243/242, 364/363, 441/440
Subgroup: 2.3.5.7.11.13


POTE generator: ~13/11 = 283.294
Comma list: 385/384, 625/624, 847/845, 1375/1372


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}


EDOs: 72, 161f, 233f
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316


Badness: 0.0269
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}


=Emmthird=
Badness: 0.027429
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
== 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.


POTE generator: ~2744/2187 = 392.988
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 11 42 25|, &lt;0 -14 -59 -33|]
[[Comma list]]: 2401/2400, 177147/175000


Wedgie: &lt;&lt;14 59 33 61 13 -89||
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
: Mapping generators: ~2, ~10/9


Badness: 0.0167
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343


=Quinmite=
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
Commas: 2401/2400, 1959552/1953125


POTE generator: ~25/21 = 302.997
[[Badness]]: 0.125672


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
Comma list: 243/242, 441/440, 43923/43750


EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}


Badness: 0.0373
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345


=Unthirds=
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
Commas: 2401/2400, 68359375/68024448


POTE generator: ~3969/3125 = 416.717
Badness: 0.039962


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
Comma list: 243/242, 351/350, 441/440, 847/845


EDOs: 72, 167, 239, 311, 694, 1005c
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}


Badness: 0.0753
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
Commas: 2401/2400, 3025/3024, 4000/3993


POTE generator: ~14/11 = 416.718
Badness: 0.021849


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 72, 167, 239, 311, 1316c
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Badness: 0.0229
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400


POTE generator: ~14/11 = 416.716
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
Badness: 0.020295


EDOs: 72, 311, 694, 1005c, 1699cd
== Catafourth ==
{{See also| Sensipent family }}


Badness: 0.0209
[[Subgroup]]: 2.3.5.7


=Newt=
[[Comma list]]: 2401/2400, 78732/78125
Commas: 2401/2400, 33554432/33480783


POTE generator: ~49/40 = 351.113
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
: Mapping generators: ~2, ~250/189


Wedgie: &lt;&lt;2 -57 -28 -95 -50 95||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235


EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}


Badness: 0.0419
Badness: 0.079579


==11-limit==
=== 11-limit ===
Commas: 2401/2400, 3025/3024, 19712/19683
Subgroup: 2.3.5.7.11


POTE generator: ~49/40 = 351.115
Comma list: 243/242, 441/440, 78408/78125


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}


EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252


Badness: 0.0195
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}


==13-limit==
Badness: 0.036785
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095


POTE genertaor: ~49/40 = 351.117
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 19 11 -10 -20|, &lt;0 2 -57 -28 46 81|]
Comma list: 243/242, 351/350, 441/440, 10985/10976


EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}


Badness: 0.0138
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256


=Amicable=
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
Commas: 2401/2400, 1600000/1594323


POTE generator: ~21/20 = 84.880
Badness: 0.021694


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
== Cotritone ==
[[Subgroup]]: 2.3.5.7


Wedgie: &lt;&lt;20 52 31 36 -7 -74||
[[Comma list]]: 2401/2400, 390625/387072


EDOs: 99, 212, 311, 410, 1131, 1541b
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}


Badness: 0.0455
: Mappping generators: ~2, ~10/7


=Septidiasemi=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
Commas: 2401/2400, 2152828125/2147483648


POTE generator: ~15/14 = 119.297
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
[[Badness]]: 0.098322


Wedgie: &lt;&lt;26 -37 -12 -119 -92 76||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
Comma list: 385/384, 1375/1372, 4000/3993


Badness: 0.0441
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}


=Maviloid=
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Commas: 2401/2400, 1224440064/1220703125


POTE generator: ~1296/875 = 678.810
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
Badness: 0.032225


Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
Comma list: 169/168, 364/363, 385/384, 625/624


Badness: 0.0576
Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}


=Subneutral=
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Commas: 2401/2400, 274877906944/274658203125


POTE generator: ~57344/46875 = 348.301
Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}


Map: [&lt;1 19 0 6}, &lt;0 -60 8 -11|]
Badness: 0.028683


Wedgie: &lt;&lt;60 -8 11 -152 -151 48||
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''


EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
[[Subgroup]]: 2.3.5.7


Badness: 0.0458
[[Comma list]]: 2401/2400, 3645/3584


=Osiris=
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
Commas: 2401/2400, 31381059609/31360000000


POTE generator: ~2800/2187 = 428.066
: Mapping generators: ~2, ~49/40


Map: [&lt;1 13 33 21|, &lt;0 -32 -86 -51|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603


Wedgie: &lt;&lt;32 86 51 62 -9 -123||
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}


EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
[[Badness]]: 0.110820


Badness: 0.0283
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Gorgik=
Comma list: 243/242, 441/440, 1815/1792
Commas: 2401/2400, 28672/28125


POTE generator: ~8/7 = 227.512
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


Map: [&lt;1 5 1 3|, &lt;0 -18 7 -1|]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639


Wedgie: &lt;&lt;18 -7 1 -53 -49 22||
Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}


EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
Badness: 0.046181


Badness: 0.1584
== 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.  


==11-limit==
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.
Commas: 176/175, 2401/2400, 2560/2541


POTE generator: ~8/7 = 227.500
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.


Map: [&lt;1 5 1 3 1|, &lt;0 -18 7 -1 13|]
[[Subgroup]]: 2.3.5.7


EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


Badness: 0.059
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}


==13-limit==
: Mapping generators: ~2, ~3828125/2985984
Commas: 176/175, 196/195, 364/363, 512/507


POTE generator: ~8/7 = 227.493
[[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 }}


Map: [&lt;1 5 1 3 1 2|, &lt;0 -18 7 -1 13 9|]
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}


EDOs: 21, 37, 58, 153bcef, 211bcdef
[[Badness]] (Smith): 0.0597


Badness: 0.0322
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Fibo=
Comma list: 2401/2400, 3025/3024, 766656/765625
Commas: 2401/2400, 341796875/339738624


POTE generator: ~125/96 = 454.310
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


Map: [&lt;1 19 8 10|, &lt;0 -46 -15 -19|]
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078


Wedgie: &lt;&lt;46 15 19 -83 -99 2||
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
Badness (Smith): 0.0262


Badness: 0.1005
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
Commas: 385/384, 1375/1372, 43923/43750


POTE generator: ~100/77 = 454.318
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}


Map: [&lt;1 19 8 10 8|, &lt;0 -46 -15 -19 -12|]
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069


EDOs: 37, 103, 140, 243e
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}


Badness: 0.0565
Badness (Smith): 0.0160


==13-limit==
=== 17-limit ===
Commas: 385/384, 625/624, 847/845, 1375/1372
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~13/10 = 454.316
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


Map: [&lt;1 19 8 10 8 9|, &lt;0 -46 -15 -19 -12 -14|]
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}


EDOs: 37, 103, 140, 243e
Optimal tunings:  
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108


Badness: 0.0274
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


=Mintone=
Badness (Smith): 0.0210
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==
=== 2.3.5.7.11.13.17.41 subgroup ===
Commas: 2401/2400, 177147/175000
Subgroup: 2.3.5.7.11.13.17.41


POTE generator: ~10/9 = 186.343
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


Map: [&lt;1 5 9 7|, &lt;0 -22 -43 -27|]
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}


EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
Optimal tunings:  
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111


Badness: 0.12567
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


==11-limit==
== Hemigoldis ==
Commas: 243/242, 441/440, 43923/43750
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''


POTE generator: ~10/9 = 186.345
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.


Map: [&lt;1 5 9 7 12|, &lt;0 -22 -43 -27 -55|]
[[Subgroup]]: 2.3.5.7


EDOs: 58, 103, 161, 425b, 586b, 747bc
[[Comma list]]: 2401/2400, 549755813888/533935546875


Badness: 0.0400
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}


==13-limit==
: mapping generators: ~2, ~7/4
Commas: 243/242, 351/350, 441/440, 847/845


POTE generator: ~10/9 = 186.347
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690


Map: [&lt;1 5 9 7 12 11|, &lt;0 -22 -43 -27 -55 -47|]
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}


EDOs: 58, 103, 161
[[Badness]] (Sintel): 4.40


Badness: 0.0218
== 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.


==17-limit==
[[Subgroup]]: 2.3.5.7
Commas: 243/242, 351/350, 441/440, 561/560, 847/845


POTE generator: ~10/9 = 186.348
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}


Map: [&lt;1 5 9 7 12 11 3|, &lt;0 -22 -43 -27 -55 -47 7|]
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}


EDOs: 58, 103, 161, 264
: Mapping generators: ~2, ~675/448


=Catafourth=
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
Commas: 2401/2400, 78732/78125


POTE generator: ~250/189 = 489.235
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}


Map: [&lt;1 13 17 13|, &lt;0 -28 -36 -25|]
[[Badness]]: 0.202249


Wedgie: &lt;&lt;28 36 25 -8 -39 -43||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 27, 76, 103, 130
Comma list: 2401/2400, 820125/819896, 2097152/2096325


Badness: 0.0796
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}


==11-limit==
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
Commas: 243/242, 441/440, 78408/78125


POTE generator: ~250/189 = 489.252
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}


Map: [&lt;1 13 17 13 32|, &lt;0 -28 -36 -25 -70|]
Badness: 0.052308


EDOs: 103, 130, 233, 363, 493e, 856be
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0368
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167


==13-limit==
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
Commas: 243/242, 351/350, 441/440, 10985/10976


POTE generator: ~65/49 = 489.256
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723


Map: [&lt;1 13 17 13 32 9|, &lt;0 -28 -36 -25 -70 -13|]
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}


EDOs: 103, 130, 233, 363
Badness: 0.032503


Badness: 0.0217
=== 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]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"