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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>
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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, ~
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"