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Subgroup temperaments: Difference between revisions

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m →2.3.13/5 subgroup: More specific todo category (not a perfect match, but a good spot to be found by someone who can help)
 
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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>
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-09-23 16:44:41 UTC</tt>.<br>
: The original revision id was <tt>593183188</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]]
Below are some subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].


=2.3.7=
For temperaments that omit various prime harmonics, see:
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).


==[[Semaphore and Godzilla|Semaphore]]==
Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].
Comma: 49/48


[[Lp tuning|POL2 generator]]: ~8/7~7/6 = 250.385
= Composite subgroup temperaments =
== 2.3.35 subgroup ==
=== Darian calendar ===
Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]].


[[Gencom]]: [2 8/7; 49/48]
==== 2.3.11.19 subgroup ====
Gencom mapping: [&lt;1 2 0 3|, &lt;0 -2 0 -1|]
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


Map: [&lt;1 2 3|, &lt;0 -2 -1|]
[[Subgroup]]: 2.3.11.19
EDOs: 5, 14, 19, 24, 67c, 91c
[[Tp tuning#T2%20tuning|RMS error]]: 2.523 cents


==Archy==
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}
Archy can be thought of as "no-5's [[dominant]]" or "no-5's [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.


Comma: 64/63
: sval mapping generators: ~6291456/5285401, ~25289/24576


[[Lp tuning|POL2 generator]]: ~3/2 = 709.321
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257


Gencom: [2 3/2; 64/63]
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
Gencom mapping: [&lt;1 1 0 4|, &lt;0 1 0 -2|]


Map: [&lt;1 2 2|, &lt;0 -1 2|]
==== 2.3.35.11.19 subgroup ====
EDOs: 5, 12, 17, 22, 27, 137bc
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
[[Tp tuning#T2%20tuning|RMS error]]: 1.856 cents


==[[Slendric]]==
Subgroup: 2.3.35.11.19
Comma: 1029/1024


[[Lp tuning|POL2 generator]]: ~8/7 = 233.688
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}


Gencom: [2 8/7; 1029/1024]
: sval mapping generators: ~2240/1881, ~36/35
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3 0 -1|]


Map: [&lt;1 1 3|, &lt;0 3 -1|]
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
[[Tp tuning#T2%20tuning|RMS error]]: 0.3202 cents


==Lee==
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
Comma: 177147/175616


[[Lp tuning|POL2 generator]]: ~81/56 = 633.525
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  


Map: [&lt;1 0 -3|, &lt;0 3 11|]
=== Commatose ===
EDOs: 17, 36, 89, 125, 161, 358, 519b
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].
[[Tp tuning#T2%20tuning|RMS error]]: 0.3519 cents


==Skwares==
[[Subgroup]]: 2.9.5.7
Comma: 19683/19208


[[Lp tuning|POL2 generator]]: ~9/7 = 425.365 cents
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


Map: [&lt;1 3 6|, &lt;0 -4 -9|]
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}
EDOs: 14, 17, 31, 48, 79, 189b, 268bc, 347bc
[[Tp tuning#T2%20tuning|RMS error]]: 1.149 cents


=2.3.11=
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765


==Neuter==
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Neuter can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira (further down on this page) and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Comma: 243/242
[[Badness]]: 0.611


[[Lp tuning|POL2 generator]]: ~11/9 = 350.525
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11


Gencom: [2 11/9; 243/242]
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 0 0 5|]


Map: [&lt;1 1 2|, &lt;0 2 5|]
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
[[Tp tuning#T2%20tuning|RMS error]]: 0.3021 cents


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]


=2.3.5.11=
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


==Porkypine==
Badness: 0.165
Commas: 55/54, 100/99


[[Lp tuning|POL2 generator]]: ~10/9 = 164.078
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13


Gencom: [2 10/9; 55/54, 100/99]
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
Gencom mapping: [&lt;1 2 3 0 4|, &lt;0 -3 -5 0 -4|]


Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -4|]
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
[[Tp tuning#T2%20tuning|RMS error]]: 2.287 cents


==Mohaha==
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
Commas: 81/80, 121/120


[[Lp tuning|POL2 generator]]: ~11/9 = 348.094
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


Gencom: [2 11/9; 81/80 121/120]
Badness: 0.0564
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|]


Map: [&lt;1 1 0 2|, &lt;0 2 8 5|]
=== Daemotertiaschis ===
EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd
{{See also|Schismatic family#Tertiaschis}}
[[Tp tuning#T2%20tuning|RMS error]]: 1.392 cents
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.


===Music===
Subgroup: 2.9.5.7.33.13.17
[[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3|Mohaha10ping2]] by [[Billy Stiltner]]


=2.3.7.11=
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


==Ennea==
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
Commas: 41503/41472, 43923/43904


[[Lp tuning|POL2 generator]]: ~99/98 = 17.6258
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


Gencom: [2 99/98; 41503/41472, 43923/43904]
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
Gencom mapping: [&lt;1 14/9 0 25/9 31/9|, &lt;0 2 0 2 1|]


Sval map: [&lt;9 0 11 24|, &lt;0 2 2 1|]
=== Baldy ===
EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd
{{See also|Schismatic family #Garibaldi}}
[[Tp tuning#T2%20tuning|RMS error]]: 0.0383 cents
{{See also|No-threes subgroup temperaments #Frostburn}}


==Supra==
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
[[Subgroup]]: 2.9.5.7


Gencom: [2 3/2; 64/63 99/98]
[[Comma list]]: 225/224, 3125/3087
Gencom mapping: [&lt;1 1 0 4 7|, &lt;0 1 0 -2 -6|]


Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}
EDOs: 5, 12, 17, 39c, 56c
[[Tp tuning#T2%20tuning|RMS error]]: 1.977 cents


==Skwares==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
Commas: 99/98, 243/242


[[POTE tuning|POTE generator]]: ~9/7 = 425.244
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


Gencom: [2 9/7; 99/98 243/242]
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
Gencom mapping: [&lt;1 3 0 6 7|, &lt;0 -4 0 -9 -10]]


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
==== 2.9.5.7.13 ====
EDOs: 14, 17, 31, 48, 79, 127, 206bcd
{{See also|Chromatic pairs #Baldy}}
[[Tp tuning#T2%20tuning|RMS error]]: 1.099 cents


==Hemif==
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
[[Subgroup]]: 2.9.5.7.13


Gencom: [2 11/9; 243/242 896/891]
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
Gencom mapping: [&lt;1 1 0 -1 2|, &lt;0 2 0 13 5|]


Sval map: [&lt;1 1 -1 2|, &lt;0 2 13 5|]
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
EDOs: 7, 17, 41, 58, 99d
[[Tp tuning#T2%20tuning|RMS error]]: 0.6108 cents


==Bleu==
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
Commas: 99/98, 864/847


POTE generator: ~12/11 = 140.005
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]


Map: [&lt;1 1 2 3|, &lt;0 5 7 4|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
EDOs: 9, 17, 43, 60c
RMS error: 1.829 cents


=2.3.7.11.13=
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}


==Suhajira==
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
Commas: 64/63, 78/77, 144/143


Gencom: [2 11/9; 64/63 78/77 144/143]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
Gencom map: [&lt;1 1 0 4 2 4|, &lt;0 2 0 -4 5 -1|]


[[Tp tuning|POT2 generator]]: ~11/9 = 353.775
==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


Map: [&lt;1 1 4 2 4|, &lt;0 2 -4 5 -1|]
Subgroup: 2.9.5.7.11
EDOs: 7, 10, 17, 44d, 61cd, 78cd
[[Tp tuning#T2%20tuning|RMS error]]: 1.953 cents


==Hemif==
Comma list: 100/99, 225/224, 245/242
Commas: 144/143, 243/242, 364/363
Related temperament: [[xenharmonic/Breedsmic temperaments|Hemififths]], namo


[[Tp tuning|POT2 generator]]: ~11/9 = 351.691
{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
Gencom map: [&lt;1 1 0 -1 2 4|, &lt;0 2 0 13 5 -1|]


Map: [&lt;1 1 -1 2 4|, &lt;0 2 13 5 -1|]
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
EDOs: 7, 10, 17, 24, 41, 58
[[Tp tuning#T2%20tuning|RMS error]]: 0.7167 cents


==Skwares==
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
Commas: 78/77, 99/98, 243/242


[[Tp tuning|POT2 generator]]: ~9/7 = 424.457
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


Map: [&lt;1 3 6 7 9|, &lt;0 -4 -9 -10 -15|]
Comma list: 100/99, 144/143, 225/224, 245/242
EDOs: 17, 48e, 65de, 82c, 147ce
[[Tp tuning#T2%20tuning|RMS error]]: 1.769 cents


==Leapfrog==
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
Commas: 169/168, 352/351, 364/363


[[Lp tuning|POL2 generator]]: ~3/2 = 704.745 cents
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414


Gencom: [2 3/2; 169/169 352/351 364/363]
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
Gencom mapping: [&lt;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 8|]


Sval map: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 8|]
== 2.3.25 subgroup ==
EDOs: 17, 46, 63
[[Tp tuning#T2%20tuning|RMS error]]: 0.7541 cents


==Bleu==  
=== Shrub ===
Group: 2.3.7.11.13
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Commas: 78/77, 99/98, 144/143
Sval name: 9&amp;17


[[Tp tuning|POL2 generator]]: ~13/12 = 139.990
Subgroup: 2.3.25


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
Edo join: 17 & 12
Gencom map: [&lt;1 1 0 2 3 3|, &lt;0 5 0 7 4 6|]


Map: [&lt;1 1 2 3 3|, &lt;0 5 7 4 6|]
Comma list: [[2048/2025]]
EDOs: [[xenharmonic/17edo|17]], [[xenharmonic/43edo|43]], [[xenharmonic/60edo|60c]]
[[Tp tuning#T2%20tuning|RMS error]]: 1.752 cents


==Parapyth (Rank 3)==
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Commas: 352/351, 364/363


[[Lp tuning|POL2 tuning]]: &lt;1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136


The gencom below gives [[Margo Schulter]]'s favored basis
==== 2.3.23.25.41 subgroup ====
Gencom: [2 3/2 28/27; 352/351 364/363]
''See also: [[Reversed meantone]]''
Gencom mapping: [&lt;1 1 0 1 4 6|, &lt;0 1 0 3 -1 -4|, &lt;0 0 0 1 1 1|]


Sval map: [&lt;1 0 0 7 12|, &lt;0 1 0 -4 -7|, &lt;0 0 1 1 1|]
Edo join: 17 & 12
EDOs: 17, 41, 46, 58, 87, 104
[[Tp tuning#T2%20tuning|RMS error]]: 0.3789 cents


=2.3.5.11=
Comma list: 2048/2025, 576/575, 82/81


==Larry==
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264


Gencom: [2 40/27; 243/242 4000/3993]
===== Sburb =====
Gencom mapping: [&lt;1 5 12 0 12|, &lt;0 -6 -17 0 -15|]
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
Subgroup: 2.3.7.23.25.41.59
EDOs: 7, 58, 65, 137, 202
[[Tp tuning#T2%20tuning|RMS error]]: 0.3025 cents


=2.9.7.11=
Edo join: 17 & 12


==Machine==
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}


Gencom: [2 8/7; 64/63 99/98]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
Gencom mapping: [&lt;1 3/2 0 3 4|, &lt;0 1/2 0 -1 -3|]


[[Smonzos and Svals|Sval map]]: [&lt;1 0 6 13|, &lt;0 1 -1 -3|]
== 2.9.5.11 subgroup ==
EDOs: 5, 6, 11, 17, 28
=== Glacial ===
[[Tp tuning#T2%20tuning|RMS error]]: 1.977 cents
{{See also| Chromatic pairs #Glacial }}


==Apparatus==
[[Subgroup]]: 2.9.5.11.13
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
[[Comma list]]: 45/44, 65/64, 81/80


Gencom: [2 77/72; 41503/41472 322102/321489]
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
Gencom mapping: [&lt;1 5/2 0 3 5|, &lt;0 -19/2 0 -2 -16|]


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
[[Tp tuning#T2%20tuning|RMS error]]: 0.0673 cents


==Mechanism==
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
Commas: 896/891, 26411/26244


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


Gencom: [2 9/7; 896/891 26411/26244]
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
Gencom mapping: [&lt;1 5/2 0 5 2|, &lt;0 -5/2 0 -6 4|]


[[Smonzos and Svals|Sval map]]: [&lt;1 5 5 2|, &lt;0 -5 -6 4|]
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
EDOs: 8, 11, 30, 41, 52
[[Tp tuning#T2%20tuning|RMS error]]: 0.4262 cents


=2.9.15.7.11.13=
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


==Stacks (aka 2magic)==  
== 2.9.7 subgroup ==
Commas: 100/99, 105/104, 144/143, 196/195
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


[[Lp tuning|POL2 generator]]: ~9/7 = 438.977
Subgroup: 2.9.7


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Comma basis: 44957696/43046721
Gencom mapping: [&lt;1 5/2 5/2 5 2 7|, &lt;0 -5/2 -1/2 -6 4 -9|]


Sval map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 3 6 -4 9|]
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
[[Tp tuning#T2%20tuning|RMS error]]: 1.540 cents


===2.9.15.7.11 stacks===
Optimal tuning (CTE): ~729/448 = 870.792
Commas: 100/99, 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 438.607
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...


Gencom: [2 9/7; 100/99 225/224 245/243]
==== 2.9.7.11 subgroup ====
Gencom mapping: [&lt;1 5/2 5/2 5 2|, &lt;0 -5/2 -1/2 -6 4|]
Subgroup: 2.9.7.11


Sval map: [&lt;1 0 2 -1 6|, &lt;0 5 3 6 -4|]
Comma basis: 896/891, 1331/1296
EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
[[Tp tuning#T2%20tuning|RMS error]]: 1.226 cents


===2.9.15.7 stacks===
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
Commas: 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 439.296
Optimal tuning (CTE): ~16/11 = 870.966


Gencom: [2 9/7; 225/224 245/243]
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
Gencom mapping: [&lt;1 5/2 5/2 5|, &lt;0 -5/2 -1/2 -6|]


Sval map: [&lt;1 0 2 -1|, &lt;0 5 3 6|]
== 2.9.7.11 subgroup ==
EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
=== Apparatus ===
[[Tp tuning#T2%20tuning|RMS error]]: 1.074 cents
[[Subgroup]]: 2.9.7.11


=2.9.21=
[[Comma list]]: 41503/41472, 322102/321489
==A-team==
Commas: 1029/1024


[[Lp tuning|POL2 generator]]: ~21/16 = 467.375
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}


Gencom: [2 21/16; 1029/1024]
: mapping generators: ~2, ~77/72
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3/2 0 -1/2|]


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
EDOs: 5, 13, 18, 41, 59, 77, 95
[[Tp tuning#T2%20tuning|RMS error]]: 0.3202 cents


=2.3.7/5.11/5.13/5=
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
==Historical==
Subgroup: 2.3.7/5.11/5.13/5
Commas: 364/363, 441/440, 1001/1000


[[Lp tuning|POL2 generator]]: ~21/20 = 83.016
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
Gencom map: [&lt;1 2 -3/4 -3/4 1/4 5/4|, &lt;0 -6 0 7 2 -9|]


Sval map: [&lt;1 2 0 1 2|, &lt;0 -6 7 2 -9|]
[[Badness]]: 0.00263
EDOs: 14, 29, 72, 101, 130, 159
[[Tp tuning#T2%20tuning|RMS error]]: 0.2562 cents


=2.3.7.11/5.13=  
=== Joan ===
==Hypnosis==  
{{See also| Chromatic pairs #Joan }}
Subgroup: 2.3.7.11/5.13
Commas: 169/168, 540/539, 729/728
Related temperament: [[Swetismic temperaments#Hypnos-13-limit|hypnos]], [[Hemifamity temperaments#Tricot-13-limit|tricot]]


[[Gencom]]: [2 13/9; 169/168, 540/539, 729/728]
Joan is related to [[casablanca]] as well as to [[orwell]].
Gencom map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 13/2 11 -13/2 7|]


Sval map: [&lt;1 0 -3 8 0|, &lt;0 3 11 -13 7|]
[[Subgroup]]: 2.9.7.11
EDOs: 17, 36, 118e, 125e, 161e, 197e
[[Tp tuning#T2%20tuning|RMS error]]: 0.5379 cents


=2.5.7=
[[Comma list]]: 99/98, 9317/9216


==Didacus==
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}
Subgroup: 2.5.7
Commas: 3136/3125
Related temperaments: [[Chromatic+pairs#Roulette]]


[[Gencom]]: [2 28/25 3136/3125]
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
Gencom map: [&lt;1 0 2 2|, &lt;0 0 2 5|]


Sval map: [&lt;1 2 2|, &lt;0 2 5|]
: [[gencom]]: [2 11/8; 99/98 9317/9216]
EDOs: 6, 19, 25, 31, 37, 99, 130, 161, 353
 
[[Tp tuning#T2%20tuning|RMS error]]: 0.2138 cents</pre></div>
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
<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;Subgroup temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:86:&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:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;a href="#x2.3.7"&gt;2.3.7&lt;/a&gt
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
 
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 64/63, 99/98
 
{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}
 
: sval mapping generators: ~2, ~9
 
{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
 
: [[gencom]]: [2 8/7; 64/63 99/98]
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843
 
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
 
[[Badness]]: 0.00233
 
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 896/891, 26411/26244
 
{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}
 
: sval mapping generators: ~2, ~14/9
 
{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
 
: [[gencom]]: [2 9/7; 896/891 26411/26244]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
 
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
 
[[Badness]]: 0.00439
 
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
 
== 2.9.7.13.17 subgroup ==
 
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].
 
[[Subgroup]]: 2.9.7.13.17
 
[[Comma list]]: 729/728, 442/441, 833/832
 
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
 
Badness (Dirichlet): 0.142
 
== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.
 
[[Subgroup]]: 2.9.11
 
[[Comma list]]: [[1331/1296]]
 
{{Mapping|legend=2|1 1 2|0 3 2}}
 
[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
 
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
 
=== Genius ===
 
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
 
[[Subgroup]]: 2.9.11
 
[[Comma list]]: [[131769/131072]]
 
{{Mapping|legend=2|1 1 4|0 4 -1}}
 
[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
 
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
 
== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].
 
[[Subgroup]]: 2.9.15.7
 
[[Comma list]]: 225/224, 245/243
 
{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}
 
: sval mapping generators: ~2, ~14/9
 
{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
 
: [[gencom]]: [2 9/7; 225/224 245/243]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
 
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
 
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11
 
Comma list: 100/99, 225/224, 245/243
 
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
 
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
 
: gencom: [2 9/7; 100/99 225/224 245/243]
 
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
 
Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
 
RMS error: 1.226 cents
 
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"