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Breedsmic temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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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>2011-03-21 17:10:04 UTC</tt>.<br>
: The original revision id was <tt>212568090</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]]


Breedsmic temperaments are rank two temperaments tempering out the breedsma, 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)<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.


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.
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]]


=Hemififths=
== Hemififths ==
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| Hemififths }}


By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. [[99edo]] is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.


Commas: 2401/2400, 5120/5103
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.


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, 5120/5103


Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
Badness: 0.0222


==11-limit==
: mapping generators: ~2, ~49/40
Commas: 243/242, 441/440, 896/891


POTE generator: ~11/9 = 351.521
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}


Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]
[[Minimax tuning]]:
EDOs: 7, 17, 41, 58, 99
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
Badness: 0.0235
: {{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


==13-limit==
[[Algebraic generator]]: (2 + sqrt(2))/2
Commas: 144/143, 196/195, 243/242, 364/363


POTE generator: ~11/9 = 351.573
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}


Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]
[[Badness]] (Smith): 0.022243
EDOs: 7, 17, 41, 58, 99
Badness: 0.0191


=Tertiaseptal=
=== 11-limit ===
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.
Subgroup: 2.3.5.7.11


Commas: 2401/2400, 65625/65536
Comma list: 243/242, 441/440, 896/891


POTE generator: ~256/245 = 77.191
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]
Optimal tunings:  
EDOs: 15, 16, 31, 109, 140, 171
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
Badness: 0.0130
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206


==11-limit==
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
Commas: 243/242, 441/440, 65625/65536


POTE generator: ~256/245 = 77.227
Badness (Smith): 0.023498


Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]
==== 13-limit ====
EDOs: 15, 16, 31, 171, 202
Subgroup: 2.3.5.7.11.13
Badness: 0.0356


=Harry=
Comma list: 144/143, 196/195, 243/242, 364/363
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.
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


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 ...||.
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734


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


[[POTE tuning|POTE generator]]: ~21/20 = 83.156
Badness (Smith): 0.019090


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
=== Semihemi ===
Wedgie: &lt;&lt;12 34 20 26 -2 -49||
Subgroup: 2.3.5.7.11
EDOs: 14, 58, 72, 130, 202, 534, 938
Badness: 0.0341


==11-limit==
Comma list: 2401/2400, 3388/3375, 5120/5103
Commas: 243/242, 441/440, 4000/3993


[[POTE tuning|POTE generator]]: ~21/20 = 83.167
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
: mapping generators: ~99/70, ~400/231
EDOs: 14, 58, 72, 130, 202
Badness: 0.0159


==13-limit==
Optimal tunings:
Commas: 243/242, 351/350, 441/440, 676/675
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047


[[POTE tuning|POTE generator]]: ~21/20 = 83.116
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Badness (Smith): 0.042487
EDOs: 14, 58, 72, 130, 462
Badness: 0.0130


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


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.
Comma list: 352/351, 676/675, 847/845, 1716/1715


Commas: 2401/2400, 29360128/29296875
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


POTE generator: ~1024/875 = 271.107
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019


Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0358


==11-limit==
Badness (Smith): 0.021188
Commas: 2401/2400, 3025/3024, 5632/5625


POTE generator: ~90/77 = 271.111
=== 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.  


Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]
Subgroup: 2.3.5.7.11
EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Badness: 0.0175


==13-limit==
Comma list: 2401/2400, 3025/3024, 5120/5103
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095


POTE generator: ~90/77 = 271.107
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}


Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]
: Mapping generators: ~2, ~243/220
EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0179


=Decoid=
Optimal tunings:
Commas: 2401/2400, 67108864/66976875
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378


POTE generator: ~8/7 = 231.099
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
Badness (Smith): 0.040170
Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||
EDOs: 10, 120, 130, 270
Badness: 0.0339


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


POTE generator: ~8/7 = 231.070
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
EDOs: 130, 270, 670, 940, 1210
Badness: 0.0187


==13-limit==
Optimal tunings:
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470


POTE generator: ~8/7 = 231.083
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
Badness (Smith): 0.031144
EDOs: 130, 270, 940, 1480
 
Badness: 0.0135</pre></div>
== Tertiaseptal ==
<h4>Original HTML content:</h4>
{{Main| Tertiaseptal }}
<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;
 
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 65625/65536
 
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
 
: Mapping generators: ~2, ~256/245
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
 
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
 
[[Badness]]: 0.012995
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
 
Badness: 0.035576
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
 
Badness: 0.036876
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
 
Badness: 0.027398
 
=== Tertia ===
Subgroup:2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 1375/1372
 
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
 
Badness: 0.030171
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323
 
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
 
Badness: 0.028384
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
 
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
 
Optimal ET sequence: {{Optimal ET
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"