Breedsmic temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-01-13 21:51:57 UTC</tt>.<br>
: The original revision id was <tt>537227354</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


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.


Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 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.
Temperaments discussed elsewhere include:
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


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


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


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


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


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


Map: [&lt;1 1 -5|, &lt;0 2 25|]
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
EDOs: 41, 58, 99, 239, 338, 915b, 1253bc
Badness: 0.3728


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


7 and 9-limit minimax
[[Optimal tuning]]s:
[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
Eigenvalues: 2, 5
: [[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 }}


Algebraic generator: (2 + sqrt(2))/2
[[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


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
[[Algebraic generator]]: (2 + sqrt(2))/2
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
Badness: 0.0222


==11-limit==
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}
Commas: 243/242, 441/440, 896/891


POTE generator: ~11/9 = 351.521
[[Badness]] (Smith): 0.022243


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
=== 11-limit ===
EDOs: 7, 17, 41, 58, 99
Subgroup: 2.3.5.7.11
Badness: 0.0235


==13-limit==
Comma list: 243/242, 441/440, 896/891
Commas: 144/143, 196/195, 243/242, 364/363


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


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
Optimal tunings:  
EDOs: 7, 17, 41, 58, 99
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
Badness: 0.0191
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


=Semihemi=
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
Commas: 2401/2400, 3388/3375, 9801/9800


POTE generator: ~49/40 = 351.505
Badness (Smith): 0.023498


Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]
==== 13-limit ====
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
Subgroup: 2.3.5.7.11.13
Badness: 42.487


==13-limit==
Comma list: 144/143, 196/195, 243/242, 364/363
Commas: 352/351, 676/675, 847/845, 1716/1715


POTE generator: ~49/40 = 351.502
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]
Optimal tunings:  
EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
Badness: 0.0212
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


=Tertiaseptal=
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
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]] 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
Badness (Smith): 0.019090


POTE generator: ~256/245 = 77.191
=== Semihemi ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
Comma list: 2401/2400, 3388/3375, 5120/5103
EDOs: 15, 16, 31, 109, 140, 171
Badness: 0.0130


==11-limit==
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
: mapping generators: ~99/70, ~400/231


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
Optimal tunings:  
EDOs: 15, 16, 31, 171, 202
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
Badness: 0.0356
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047


==13-limit==
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
Commas: 243/242, 441/440, 625/624, 3584/3575


POTE generator: ~117/112 = 77.203
Badness (Smith): 0.042487


Map: [&lt;1 3 2 3 7 1|, &lt;0 -22 5 -3 -55 42|]
==== 13-limit ====
EDOs: 31, 140e, 171, 373ef, 544ef
Subgroup: 2.3.5.7.11.13
Badness: 0.0369


==Tertia==
Comma list: 352/351, 676/675, 847/845, 1716/1715
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


Map: [&lt;1 3 2 3 5|, &lt;0 -22 5 -3 -24|]
Optimal tunings:  
EDOs: 31, 109, 140, 171e, 311e
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
Badness: 0.0302
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


=Hemitert=
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
Commas: 2401/2400 3025/3024 65625/65536


POTE generator: ~45/44 = 38.596
Badness (Smith): 0.021188


Map: [&lt;1 3 2 3 6|, &lt;0 -44 10 -6 -79|]
=== Quadrafifths ===
EDOs: 31, 280, 311, 342, 2021cde, 3731cde
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.  
Badness: 0.0156


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


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: 2401/2400, 3025/3024, 5120/5103


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 1 -5 -1 8 | 0 4 50 26 -31 }}


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
: Mapping generators: ~2, ~243/220


[[POTE tuning|POTE generator]]: ~21/20 = 83.156
Optimal tunings:
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
Wedgie: &lt;&lt;12 34 20 26 -2 -49||
EDOs: 14, 58, 72, 130, 202, 534, 938
Badness: 0.0341


==11-limit==
Badness (Smith): 0.040170
Commas: 243/242, 441/440, 4000/3993


[[POTE tuning|POTE generator]]: ~21/20 = 83.167
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
EDOs: 14, 58, 72, 130, 202
Badness: 0.0159


==13-limit==
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
Commas: 243/242, 351/350, 441/440, 676/675


[[POTE tuning|POTE generator]]: ~21/20 = 83.116
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
EDOs: 14, 58, 72, 130, 462
Badness: 0.0130


=Quasiorwell=
Badness (Smith): 0.031144
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.
== Tertiaseptal ==
{{Main| Tertiaseptal }}


Commas: 2401/2400, 29360128/29296875
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.


POTE generator: ~1024/875 = 271.107
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
[[Comma list]]: 2401/2400, 65625/65536
EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0358


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


POTE generator: ~90/77 = 271.111
: Mapping generators: ~2, ~256/245


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Badness: 0.0175


==13-limit==
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
[[Badness]]: 0.012995


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
=== 11-limit ===
EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Subgroup: 2.3.5.7.11
Badness: 0.0179


=Decoid=
Comma list: 243/242, 441/440, 65625/65536
Commas: 2401/2400, 67108864/66976875


POTE generator: ~8/7 = 231.099
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
EDOs: 10, 120, 130, 270
Badness: 0.0339


==11-limit==
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
Commas: 2401/2400, 5832/5825, 9801/9800


POTE generator: ~8/7 = 231.070
Badness: 0.035576


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
==== 13-limit ====
EDOs: 130, 270, 670, 940, 1210
Subgroup: 2.3.5.7.11.13
Badness: 0.0187


==13-limit==
Comma list: 243/242, 441/440, 625/624, 3584/3575
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224


POTE generator: ~8/7 = 231.083
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
EDOs: 130, 270, 940, 1480
Badness: 0.0135


=Neominor=
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
Badness: 0.036876


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
==== 17-limit ====
Weggie: &lt;&lt;6 41 22 51 18 -64||
Subgroup: 2.3.5.7.11.13.17
EDOs: 72, 161, 233, 305
Badness: 0.0882


==11-limit==
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
EDOs: 72, 161, 233, 305
Badness: 0.0280


==13-limit==
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
Commas: 169/168, 243/242, 364/363, 441/440


POTE generator: ~13/11 = 283.294
Badness: 0.027398


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
=== Tertia ===
EDOs: 72, 161f, 233f
Subgroup:2.3.5.7.11
Badness: 0.0269


=Emmthird=
Comma list: 385/384, 1331/1323, 1375/1372
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}


POTE generator: ~2744/2187 = 392.988
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173


Map: [&lt;1 11 42 25|, &lt;0 -14 -59 -33|]
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
Wedgie: &lt;&lt;14 59 33 61 13 -89||
EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Badness: 0.0167


=Quinmite=
Badness: 0.030171
Commas: 2401/2400, 1959552/1953125


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


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
Comma list: 352/351, 385/384, 625/624, 1331/1323
Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Badness: 0.0373


=Unthirds=
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
Commas: 2401/2400, 68359375/68024448


POTE generator: ~3969/3125 = 416.717
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
EDOs: 72, 167, 239, 311, 694, 1005c
Badness: 0.0753


==11-limit==
Badness: 0.028384
Commas: 2401/2400, 3025/3024, 4000/3993


POTE generator: ~14/11 = 416.718
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
EDOs: 72, 167, 239, 311, 1316c
Badness: 0.0229


==13-limit==
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400


POTE generator: ~14/11 = 416.716
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
EDOs: 72, 311, 694, 1005c, 1699cd
Badness: 0.0209


=Newt=
Badness: 0.022416
Commas: 2401/2400, 33554432/33480783


POTE generator: ~49/40 = 351.113
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
Comma list: 2401/2400, 6250/6237, 65625/65536
Wedgie: &lt;&lt;2 -57 -28 -95 -50 95||
EDOs: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bc
Badness: 0.0419


==11-limit==
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
Commas: 2401/2400, 3025/3024, 19712/19683


POTE generator: ~49/40 = 351.115
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Badness: 0.0195


==13-limit==
Badness: 0.056926
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: 625/624, 2080/2079, 2200/2197, 2401/2400
EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
Badness: 0.0138


=Amicable=
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
Commas: 2401/2400, 1600000/1594323


POTE generator: ~21/20 = 84.880
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
Wedgie: &lt;&lt;20 52 31 36 -7 -74||
EDOs: 99, 212, 311, 410, 1131, 1541b
Badness: 0.0455


=Septidiasemi=
Badness: 0.027474
Commas: 2401/2400, 2152828125/2147483648


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


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Wedgie: &lt;&lt;26 -37 -12 -119 -92 76||
EDOs: 10, 151, 161, 171, 3581bcd, 3752bcd, 3923bcd, 4094bcd, 4265bcd, 4436bcd, 4607bcd
Badness: 0.0441


=Maviloid=
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
Commas: 2401/2400, 1224440064/1220703125


POTE generator: ~1296/875 = 678.810
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
Badness: 0.0576


=Subneutral=
Badness: 0.018773
Commas: 2401/2400, 274877906944/274658203125


POTE generator: ~57344/46875 = 348.301
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 19 0 6}, &lt;0 -60 8 -11|]
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Wedgie: &lt;&lt;60 -8 11 -152 -151 48||
EDOs: 31, 348, 379, 410, 441, 1354, 1795, 2236
Badness: 0.0458


=Osiris=
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
Commas: 2401/2400, 31381059609/31360000000


POTE generator: ~2800/2187 = 428.066
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169


Map: [&lt;1 13 33 21|, &lt;0 -32 -86 -51|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Wedgie: &lt;&lt;32 86 51 62 -9 -123||
EDOs: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955d
Badness: 0.0283


=Gorgik=
Badness: 0.017653
Commas: 2401/2400, 28672/28125


POTE generator: ~8/7 = 227.512
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 5 1 3|, &lt;0 -18 7 -1|]
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
Wedgie: &lt;&lt;18 -7 1 -53 -49 22||
EDOs: 21, 37, 58, 153bc, 211bcd, 269bcd
Badness: 0.1584


==11-limit==
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
Commas: 176/175, 2401/2400, 2560/2541


POTE generator: ~8/7 = 227.500
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168


Map: [&lt;1 5 1 3 1|, &lt;0 -18 7 -1 13|]
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
EDOs: 21, 37, 58, 153bce, 211bcde, 269bcde
Badness: 0.059


==13-limit==
Badness: 0.015123
Commas: 176/175, 196/195, 364/363, 512/507


POTE generator: ~8/7 = 227.493
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Map: [&lt;1 5 1 3 1 2|, &lt;0 -18 7 -1 13 9|]
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
EDOs: 21, 37, 58, 153bcef, 211bcdef
Badness: 0.0322


=Fibo=
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
Commas: 2401/2400, 341796875/339738624


POTE generator: ~125/96 = 454.310
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167


Map: [&lt;1 19 8 10|, &lt;0 -46 -15 -19|]
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
Wedgie: &lt;&lt;46 15 19 -83 -99 2||
EDOs: 37, 103, 140, 243, 383, 1009cd, 1392cd
Badness: 0.1005


==11-limit==
Badness: 0.012181
Commas: 385/384, 1375/1372, 43923/43750


POTE generator: ~100/77 = 454.318
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Map: [&lt;1 19 8 10 8|, &lt;0 -46 -15 -19 -12|]
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
EDOs: 37, 103, 140, 243e
Badness: 0.0565


==13-limit==
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
Commas: 385/384, 625/624, 847/845, 1375/1372


POTE generator: ~13/10 = 454.316
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


Map: [&lt;1 19 8 10 8 9|, &lt;0 -46 -15 -19 -12 -14|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
EDOs: 37, 103, 140, 243e
Badness: 0.0274


=Mintone=
Badness: 0.012311
Commas: 2401/2400, 177147/175000


POTE generator: ~10/9 = 186.343
==== 37-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37


Map: [&lt;1 5 9 7|, &lt;0 -22 -43 -27|]
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
EDOs: 45, 58, 103, 161, 586b, 747bc, 908bc
Badness: 0.12567


==11-limit==
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 }}
Commas: 243/242, 441/440, 43923/43750


POTE generator: ~10/9 = 186.345
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170


Map: [&lt;1 5 9 7 12|, &lt;0 -22 -43 -27 -55|]
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
EDOs: 58, 103, 161, 425b, 586b, 747bc
Badness: 0.0400


==13-limit==
Badness: 0.010949
Commas: 243/242, 351/350, 441/440, 847/845


POTE generator: ~10/9 = 186.347
==== 41-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41


Map: [&lt;1 5 9 7 12 11|, &lt;0 -22 -43 -27 -55 -47|]
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
EDOs: 58, 103, 161
 
Badness: 0.0218</pre></div>
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 }}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Breedsmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:92:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;a href="#Hemififths"&gt;Hemififths&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;!-- ws:
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, ~

Latest revision as of 20:57, 18 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 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.

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)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.

Temperaments discussed elsewhere include:

Hemififths

Main article: Hemififths

Hemififths may be described as the 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(1/25), giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14(1/13), 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[clarification needed].

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

Subgroup: 2.3.5.7

Comma list: 2401/2400, 5120/5103

Mapping: [⟨1 1 -5 -1], ⟨0 2 25 13]]

mapping generators: ~2, ~49/40

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~49/40 = 351.4464
error map: ⟨0.0000 +0.9379 -0.1531 -0.0224]
  • POTE: ~2 = 1200.0000, ~49/40 = 351.4774
error map: ⟨0.0000 +0.9999 +0.6221 +0.0307]

Minimax tuning:

[[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: (2 + sqrt(2))/2

Optimal ET sequence: 41, 58, 99, 239, 338

Badness (Smith): 0.022243

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~11/9 = 351.4289
  • POTE: ~2 = 1200.0000, ~11/9 = 351.5206

Optimal ET sequence: 17c, 41, 58, 99e

Badness (Smith): 0.023498

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 364/363

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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 41, 157, 198, 437f, 635bcff

Badness (Smith): 0.031144

Tertiaseptal

Main article: 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 & 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: [⟨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: 31, 109, 140, 171

Badness: 0.012995

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227

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: [⟨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: 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: [⟨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: 31, 109eg, 140e, 171

Badness: 0.027398

Tertia

Subgroup:2.3.5.7.11

Comma list: 385/384, 1331/1323, 1375/1372

Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]

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

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: [⟨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: 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: [⟨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: 31, 109g, 140, 311e, 451ee

Badness: 0.022416

Tertiaseptia

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 65625/65536

Mapping: [⟨1 3 2 3 -4], ⟨0 -22 5 -3 116]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169

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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 140, 171, 311

Badness: 0.009825

Hemitert

Subgroup: 2.3.5.7.11

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

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: 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: [⟨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: 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: [⟨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: 31, 280, 311, 653f, 964f

Badness: 0.025298

Semitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 65625/65536

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: 62e, 140, 202, 342

Badness: 0.025790

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = [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 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7's, or 3841/38, 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: [⟨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: 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: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]

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

Optimal ET sequence: 31, 208, 239, 270

Badness: 0.017540

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

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: 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: [⟨1 3 12 8], ⟨0 -6 -41 -22]]

Mapping generators: ~2, ~189/160

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

Optimal ET sequence: 72, 161, 233, 305

Badness: 0.088221

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 35937/35840

Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]

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

Optimal ET sequence: 72, 161, 233, 305

Badness: 0.027959

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 364/363, 441/440

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: 72, 161f, 233f

Badness: 0.026942

Emmthird

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 14348907/14336000

Mapping: [⟨1 11 42 25], ⟨0 -14 -59 -33]]

Mapping generators: ~2, ~2187/1372

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

Optimal ET sequence: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d

Badness: 0.016736

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 11 42 25 27], ⟨0 -14 -59 -33 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991

Optimal ET sequence: 58, 113, 171

Badness: 0.052358

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: [⟨1 11 42 25 27 38], ⟨0 -14 -59 -33 -35 -51]]

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

Optimal ET sequence: 58, 113, 171

Badness: 0.026974

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197

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: 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[1][2].

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [⟨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: 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 72 & 311 temperament, the 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 (undecimal 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: [⟨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: 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: [⟨1 29 33 25 25], ⟨0 -42 -47 -34 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718

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: [⟨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: 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: [⟨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: 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: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

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

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: [⟨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: 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: [⟨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: 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb

Septidiasemi

Main article: 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: [⟨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: 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd

Badness: 0.044115

Sedia

The sedia temperament (10&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: [⟨1 25 -31 -8 62], ⟨0 -26 37 12 -65]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279

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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 157, 171, 1012, 1183, 1354, 1525, 1696

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨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: 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: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]

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

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: [⟨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: 21, 37, 58, 153bcef, 211bccdeeff

Badness: 0.032205

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [⟨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: 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: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]

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

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: [⟨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: 37, 66b, 103, 140, 243e

Badness: 0.027429

Mintone

In addition to 2401/2400, mintone tempers out 177147/175000 = [-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 & 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: [⟨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: 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: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]

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

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: [⟨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: 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: [⟨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: 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: [⟨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: 27, 76, 103, 130

Badness: 0.079579

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 78408/78125

Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]

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

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: [⟨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: 103, 130, 233, 363

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [⟨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: 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: [⟨1 17 9 10 5], ⟨0 -30 -13 -14 -3]]

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

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: [⟨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: 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: [⟨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: 31, 117c, 148bc, 179bc

Badness: 0.110820

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1815/1792

Mapping: [⟨1 1 9 6 2], ⟨0 2 -23 -11 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639

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 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 Scottish town, where a 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, [24 13 -18 -1⟩

Mapping: [⟨1 -25 -16 -13], ⟨0 74 51 44]]

Mapping generators: ~2, ~3828125/2985984

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
error map: ⟨0.0000 -0.0270 +0.1502 -0.1120]
  • CWE: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
error map: ⟨0.0000 -0.0205 +0.1547 -0.1081]

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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, [93 -32 -17 -1⟩

Mapping: [⟨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: 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: [⟨1 43 -74 -25 36], ⟨0 -70 129 47 -55]]

Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720

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: [⟨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: 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: [⟨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: 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: [⟨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: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.013771

Notes

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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