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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]] [[regular temperament|temperaments]] [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -5 -1 -2 4 }}, [[ratio]]: 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>2012-04-10 21:51:04 UTC</tt>.<br>
: The original revision id was <tt>319294640</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:
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
* ''[[Neptune (temperament)|Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Maviloid]]'' (+1224440064/1220703125) → [[Parakleismic family #Maviloid|Parakleismic family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood 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.
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  


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


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.
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 {{nowrap| 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. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  


Commas: 2401/2400, 5120/5103
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) 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.


7 and 9-limit minimax
[[Subgroup]]: 2.3.5.7
[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]
Eigenvalues: 2, 5


Algebraic generator: (2 + sqrt(2))/2
[[Comma list]]: 2401/2400, 65625/65536


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
{{Mapping|legend=1| 1 -19 7 0 | 0 22 -5 3 }}
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
: mapping generators: ~2, ~245/128
Badness: 0.0222


==11-limit==
[[Optimal tuning]]s:
Commas: 243/242, 441/440, 896/891
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979{{c}})
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899{{c}})
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}


POTE generator: ~11/9 = 351.521
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
[[Badness]] (Sintel): 0.329
EDOs: 7, 17, 41, 58, 99
Badness: 0.0235


==13-limit==  
=== 11-limit ===
Commas: 144/143, 196/195, 243/242, 364/363
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 351.573
Comma list: 243/242, 441/440, 65625/65536


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
Mapping: {{mapping| 1 -19 7 0 -48 | 0 22 -5 3 55 }}
EDOs: 7, 17, 41, 58, 99
Badness: 0.0191


=Semihemi=
Optimal tunings:
Commas: 2401/2400, 3388/3375, 9801/9800
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})


POTE generator: ~49/40 = 351.505
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]
Badness (Sintel): 1.18
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd
Badness: 42.487


==13-limit==
==== 13-limit ====
Commas: 352/351, 676/675, 847/845, 1716/1715
Subgroup: 2.3.5.7.11.13


POTE generator: ~49/40 = 351.502
Comma list: 243/242, 441/440, 625/624, 3584/3575


Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]
Mapping: {{mapping| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}
EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf
Badness: 0.0212


=Tertiaseptal=
Optimal tunings:
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] 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.
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


Commas: 2401/2400, 65625/65536
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}


POTE generator: ~256/245 = 77.191
Badness (Sintel): 1.52


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
==== 17-limit ====
EDOs: 15, 16, 31, 109, 140, 171
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0130


==11-limit==
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
Mapping: {{mapping| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
Optimal tunings:  
EDOs: 15, 16, 31, 171, 202
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
Badness: 0.0356
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})


==Tertia==
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}
Commas: 385/384, 1331/1323, 1375/1372


POTE generator: ~22/21 = 77.173
Badness (Sintel): 1.40


Map: [&lt;1 3 2 3 5|, &lt;0 -22 5 -3 -24|]
=== Tertia ===
EDOs: 31, 109, 140, 171e, 311e
Subgroup:2.3.5.7.11
Badness: 0.0302


=Hemitert=
Comma list: 385/384, 1331/1323, 1375/1372
Commas: 2401/2400 3025/3024 65625/65536


POTE generator: ~45/44 = 38.596
Mapping: {{mapping| 1 -19 7 0 -19 | 0 22 -5 3 24 }}


Map: [&lt;1 3 2 3 6|, &lt;0 -44 10 -6 -79|]
Optimal tunings:  
EDOs: 31, 280, 311, 342, 2021cde, 3731cde
* WE: ~2 = 1200.2336{{c}}, ~21/11 = 1123.0454{{c}} (~22/21 = 77.1882{{c}})
Badness: 0.0156
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8311{{c}} (~22/21 = 77.1689{{c}})


=Harry=
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}
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.
Badness (Sintel): 0.997


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


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
Comma list: 352/351, 385/384, 625/624, 1331/1323


[[POTE tuning|POTE generator]]: ~21/20 = 83.156
Mapping: {{mapping| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
Optimal tunings:  
Wedgie: &lt;&lt;12 34 20 26 -2 -49||
* WE: ~2 = 1200.1395{{c}}, ~21/11 = 1122.9727{{c}} (~22/21 = 77.1669{{c}})
EDOs: 14, 58, 72, 130, 202, 534, 938
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8426{{c}} (~22/21 = 77.1574{{c}})
Badness: 0.0341


==11-limit==
{{Optimal ET sequence|legend=0| 31, 78f, 109, 140 }}
Commas: 243/242, 441/440, 4000/3993


[[POTE tuning|POTE generator]]: ~21/20 = 83.167
Badness (Sintel): 1.17


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
==== 17-limit ====
EDOs: 14, 58, 72, 130, 202
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0159


==13-limit==
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Commas: 243/242, 351/350, 441/440, 676/675


[[POTE tuning|POTE generator]]: ~21/20 = 83.116
Mapping: {{mapping| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Optimal tunings:  
EDOs: 14, 58, 72, 130, 462
* WE: ~2 = 1200.1655{{c}}, ~21/11 = 1122.9926{{c}} (~22/21 = 77.1729{{c}})
Badness: 0.0130
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8376{{c}} (~22/21 = 77.1624{{c}})


=Quasiorwell=
{{Optimal ET sequence|legend=0| 31, 78fg, 109g, 140 }}
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.
Badness (Sintel): 1.14


Commas: 2401/2400, 29360128/29296875
=== Tertiaseptia ===
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.


POTE generator: ~1024/875 = 271.107
Subgroup: 2.3.5.7.11


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
Comma list: 2401/2400, 6250/6237, 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: {{mapping| 1 -19 7 0 112 | 0 22 -5 3 -116 }}
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
Optimal tunings:  
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Badness: 0.0175


==13-limit==
Badness (Sintel): 1.88
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0179


=Decoid=
Mapping: {{mapping| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}
Commas: 2401/2400, 67108864/66976875


POTE generator: ~8/7 = 231.099
Optimal tunings:  
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}
Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
EDOs: 10, 120, 130, 270
Badness: 0.0339


==11-limit==
Badness (Sintel): 1.14
Commas: 2401/2400, 5832/5825, 9801/9800


POTE generator: ~8/7 = 231.070
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
EDOs: 130, 270, 670, 940, 1210
Badness: 0.0187


==13-limit==
Mapping: {{mapping| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224


POTE generator: ~8/7 = 231.083
Optimal tunings:  
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
EDOs: 130, 270, 940, 1480
Badness: 0.0135


=Neominor=
Badness (Sintel): 0.956
Commas: 2401/2400, 177147/175616


POTE generator: ~189/160 = 283.280
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23


Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197
Weggie: &lt;&lt;6 41 22 51 18 -64||
EDOs: 72, 161, 233, 305
Badness: 0.0882


==11-limit==
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}
Commas: 243/242, 441/440, 35937/35840


POTE: ~33/28 = 283.276
Optimal tunings:  
* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})


Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}
EDOs: 72, 161, 233, 305
Badness: 0.0280


==13-limit==
Badness (Sintel): 0.944
Commas: 169/168, 243/242, 364/363, 441/440


POTE generator: ~13/11 = 283.294
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23.29


Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155
EDOs: 72, 161f, 233f
Badness: 0.0269


=Emmthird=
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Commas: 2401/2400, 14348907/14336000
Optimal tunings:  
* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})


POTE generator: ~2744/2187 = 392.988
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}


Map: [&lt;1 11 42 25|,  &lt;0 -14 -59 -33|]
Badness (Sintel): 0.858
Wedgie: &lt;&lt;14 59 33 61 13 -89||
EDOs: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d
Badness: 0.0167


=Quinmite=
=== Hemitert ===
Commas: 2401/2400, 1959552/1953125
Subgroup: 2.3.5.7.11


POTE generator: ~25/21 = 302.997
Comma list: 2401/2400, 3025/3024, 65625/65536


Map: [&lt;1 27 24 20|, &lt;0 -34 -29 -23|]
Mapping: {{mapping| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
Wedgie: &lt;&lt;34 29 23 -33 -59 -28||
: mapping generators: ~2, ~88/45
EDOs: 95, 99, 202, 301, 400, 701, 1001c, 1802c, 2903c
Badness: 0.0373


=Unthirds=
Optimal tunings:
Commas: 2401/2400, 68359375/68024448
* WE: ~2 = 1200.1008{{c}}, ~88/45 = 1161.5020{{c}} (~45/44 = 38.5988{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4053{{c}} (~45/44 = 38.5947{{c}})


POTE generator: ~3969/3125 = 416.717
{{Optimal ET sequence|legend=0| 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee }}


Map: [&lt;1 29 33 25|, &lt;0 -42 -47 -34|]
Badness (Sintel): 0.517
Wedgie: &lt;&lt;42 47 34 -23 -64 -53||
EDOs: 72, 167, 239, 311, 694, 1005c
Badness: 0.0753


==11-limit==
==== 13-limit ====
Commas: 2401/2400, 3025/3024, 4000/3993
Subgroup: 2.3.5.7.11.13


POTE generator: ~14/11 = 416.718
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Map: [&lt;1 29 33 25 25|, &lt;0 -42 -47 -34 -33|]
Mapping: {{mapping| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}
EDOs: 72, 167, 239, 311, 1316c
Badness: 0.0229


==13-limit==
Optimal tunings:
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400
* WE: ~2 = 1199.9822{{c}}, ~88/45 = 1161.3952{{c}} (~45/44 = 38.5871{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4123{{c}} (~45/44 = 38.5877{{c}})


POTE generator: ~14/11 = 416.716
{{Optimal ET sequence|legend=0| 31, 280, 311 }}


Map: [&lt;1 29 33 25 25 99|, &lt;0 -42 -47 -34 -33 -146|]
Badness (Sintel): 1.39
EDOs: 72, 311, 694, 1005c, 1699cd
Badness: 0.0209


=Newt=
==== 17-limit ====
Commas: 2401/2400, 33554432/33480783
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~49/40 = 351.113
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Map: [&lt;1 1 19 11|, &lt;0 2 -57 -28|]
Mapping: {{mapping| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}
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==
Optimal tunings:
Commas: 2401/2400, 3025/3024, 19712/19683
* WE: ~2 = 1200.0042{{c}}, ~88/45 = 1161.4149{{c}} (~45/44 = 38.5893{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4109{{c}} (~45/44 = 38.5891{{c}})


POTE generator: ~49/40 = 351.115
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f }}


Map: [&lt;1 1 19 11 -10|, &lt;0 2 -57 -28 46|]
Badness (Sintel): 1.29
EDOs: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b
Badness: 0.0195


==13-limit==
=== Semitert ===
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Subgroup: 2.3.5.7.11


POTE genertaor: ~49/40 = 351.117
Comma list: 2401/2400, 9801/9800, 65625/65536


Map: [&lt;1 1 19 11 -10 -20|, &lt;0 2 -57 -28 46 81|]
Mapping: {{mapping| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
EDOs: 41, 229, 270, 581, 851, 2283b, 3134b
: mapping generators: ~99/70, ~693/512
Badness: 0.0138


=Amicable=
Optimal tunings:
Commas: 2401/2400, 1600000/1594323
* WE: ~99/70 = 600.0548{{c}}, ~693/512 = 522.8547{{c}} (~256/245 = 77.2002{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~693/512 = 522.8069{{c}} (~256/245 = 77.1931{{c}})


POTE generator: ~21/20 = 84.880
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}


Map: [&lt;1 3 6 5|, &lt;0 -20 -52 -31|]
Badness (Sintel): 0.853
Wedgie: &lt;&lt;20 52 31 36 -7 -74||
EDOs: 99, 212, 311, 410, 1131, 1541b
Badness: 0.0455


=Septidiasemi=
== Emmthird ==
Commas: 2401/2400, 2152828125/2147483648
Emmthird tempers out the [[scheme comma]] and may be described as the {{nowrap| 58 & 171 }} temperament. The generator for emmthird is flatter than [[81/64]] by a lee comma, [[177147/175616]], and sharper than [[5/4]] by the hemimage comma, [[10976/10935]]. The [[ploidacot]] for this temperament is delta-14-cot.


POTE generator: ~15/14 = 119.297
The [[11-limit]] version, which tempers out [[243/242]] and [[441/440]], has much lower accuracy and is [[support]]ed by much fewer equal temperaments.  


Map: [&lt;1 25 -31 -8|, &lt;0 -26 37 12|]
[[Subgroup]]: 2.3.5.7
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=
[[Comma list]]: 2401/2400, 14348907/14336000
Commas: 2401/2400, 1224440064/1220703125


POTE generator: ~1296/875 = 678.810
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~2744/2187


Map: [&lt;1 31 34 26|, &lt;0 -52 -56 -41|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;52 56 41 -32 -81 -62||
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
EDOs: 76, 99, 274, 373, 472, 571, 1043, 1614
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
Badness: 0.0576
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


[[Badness]] (Sintel): 0.424


</pre></div>
=== 11-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11
<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:64:&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:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;a href="#Hemififths"&gt;Hemififths&lt;/a&gt;&lt
 
Comma list: 243/242, 441/440, 1792000/1771561
 
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}
 
Optimal tunings:
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.73
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
 
Optimal tunings:
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.11
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.18
 
== Hemififths ==
{{Main| 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}}.
 
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|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7412{{c}}
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