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Subgroup 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>
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>2014-02-20 12:28:21 UTC</tt>.<br>
: The original revision id was <tt>490782324</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-thirteens subgroup temperaments]]
* [[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 =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  


[[Gencom]]: [2 8/7; 49/48]
=== Commatose ===
Gencom mapping: [&lt;1 2 0 3|, &lt;0 -2 0 -1|]
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]].


Map: [&lt;1 2 3|, &lt;0 -2 -1|]
[[Subgroup]]: 2.9.5.7
EDOs: 5, 14, 19, 24, 67c, 91c
[[Tp tuning#T2%20tuning|RMS error]]: 2.523 cents


==Archy==
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 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
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}


[[Lp tuning|POL2 generator]]: ~3/2 = 709.321
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765


Gencom: [2 3/2; 64/63]
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Gencom mapping: [&lt;1 1 0 4|, &lt;0 1 0 -2|]


Map: [&lt;1 2 2|, &lt;0 -1 2|]
[[Badness]]: 0.611
EDOs: 5, 12, 17, 22, 27, 137bc
[[Tp tuning#T2%20tuning|RMS error]]: 1.856 cents


==[[Slendric]]==  
==== 2.9.5.7.11 ====
Comma: 1029/1024
Subgroup: 2.9.5.7.11


[[Lp tuning|POL2 generator]]: ~8/7 = 233.688
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


Gencom: [2 8/7; 1029/1024]
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3 0 -1|]


Map: [&lt;1 1 3|, &lt;0 3 -1|]
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
[[Tp tuning#T2%20tuning|RMS error]]: 0.3202 cents


==Lie==
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
Comma: 177147/175616


[[Lp tuning|POL2 generator]]: ~81/56 = 633.525
Badness: 0.165


Map: [&lt;1 0 -3|, &lt;0 3 11|]
==== 2.9.5.7.11.13 ====
EDOs: 17, 36, 89, 125, 161, 358, 519b
Subgroup: 2.9.5.7.11.13
[[Tp tuning#T2%20tuning|RMS error]]: 0.3519 cents


=2.3.11=
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


==Neutraltet==
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
Neutraltet can be thought of as 2.3.11 version of either [[mohajira]] or [[maqamic]], and is the temperament optimizing the [[neutral tetrad]].


Comma: 243/242
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767


[[Lp tuning|POL2 generator]]: ~11/9 = 350.525
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


Gencom: [2 11/9; 243/242]
Badness: 0.0564
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 0 0 5|]


Map: [&lt;1 1 2|, &lt;0 2 5|]
=== Daemotertiaschis ===
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
{{See also|Schismatic family#Tertiaschis}}
[[Tp tuning#T2%20tuning|RMS error]]: 0.3021 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.


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
Subgroup: 2.9.5.7.33.13.17
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]


=2.3.5.11=
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


==Porkypine==
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
Commas: 55/54, 100/99


[[Lp tuning|POL2 generator]]: ~10/9 = 164.078
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


Gencom: [2 10/9; 55/54, 100/99]
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
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|]
=== Baldy ===
EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
{{See also|Schismatic family #Garibaldi}}
[[Tp tuning#T2%20tuning|RMS error]]: 2.287 cents
{{See also|No-threes subgroup temperaments #Frostburn}}


==Mohaha==
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: 81/80, 121/120


[[Lp tuning|POL2 generator]]: ~11/9 = 348.094
[[Subgroup]]: 2.9.5.7


Gencom: [2 11/9; 81/80 121/120]
[[Comma list]]: 225/224, 3125/3087
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|]
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}
EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd
[[Tp tuning#T2%20tuning|RMS error]]: 1.392 cents


===Music===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
[[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3|Mohaha10ping2]] by [[Billy Stiltner]]


=2.3.7.11=
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


==Ennea==
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
Commas: 41503/41472, 43923/43904


[[Lp tuning|POL2 generator]]: ~99/98 = 17.6258
==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}


Gencom: [2 99/98; 41503/41472, 43923/43904]
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.
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|]
[[Subgroup]]: 2.9.5.7.13
EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd
[[Tp tuning#T2%20tuning|RMS error]]: 0.0383 cents


==Supra==
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


Gencom: [2 3/2; 64/63 99/98]
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
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|]
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
EDOs: 5, 12, 17, 39c, 56c
[[Tp tuning#T2%20tuning|RMS error]]: 1.977 cents


==Skwares==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
Commas: 99/98, 243/242


[[POTE tuning|POTE generator]]: ~9/7 = 425.244
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}


Gencom: [2 9/7; 99/98 243/242]
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
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|]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
EDOs: 14, 17, 31, 48, 79, 127, 206bcd
[[Tp tuning#T2%20tuning|RMS error]]: 1.099 cents


==Hemif (11 limit)==  
==== Baldanders ====
Commas: 243/242, 896/891
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


POTE generator: ~11/9 = 351.535
Subgroup: 2.9.5.7.11


Gencom: [2 11/9; 243/242 896/891]
Comma list: 100/99, 225/224, 245/242
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 3 3 4 5 | 0 1 -4 -7 -9 }}
EDOs: 7, 17, 41, 58, 99d
[[Tp tuning#T2%20tuning|RMS error]]: 0.6108 cents


=2.3.7.11.13=
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743


==Suhajira==
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
Commas: 64/63, 78/77, 144/143


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


[[Tp tuning|POT2 generator]]: ~11/9 = 353.775
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


Map: [&lt;1 1 4 2 4|, &lt;0 2 -4 5 -1|]
Comma list: 100/99, 144/143, 225/224, 245/242
EDOs: 7, 10, 17, 44d, 61cd, 78cd
[[Tp tuning#T2%20tuning|RMS error]]: 1.953 cents


==Hemif==
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
Commas: 144/143, 243/242, 364/363
Related temperament: [[xenharmonic/Breedsmic temperaments|Hemififths]], namo


[[Tp tuning|POT2 generator]]: ~11/9 = 351.691
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
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|]
== 2.3.25 subgroup ==
EDOs: 7, 10, 17, 24, 41, 58
[[Tp tuning#T2%20tuning|RMS error]]: 0.7167 cents


==Skwares==
=== Shrub ===
Commas: 78/77, 99/98, 243/242
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.


[[Tp tuning|POT2 generator]]: ~9/7 = 424.457
Subgroup: 2.3.25


Map: [&lt;1 3 6 7 9|, &lt;0 -4 -9 -10 -15|]
Edo join: 17 & 12
EDOs: 17, 48e, 65de, 82c, 147ce
[[Tp tuning#T2%20tuning|RMS error]]: 1.769 cents


==Leapfrog==
Comma list: [[2048/2025]]
Commas: 169/168, 352/351, 364/363


[[Lp tuning|POL2 generator]]: ~3/2 = 704.745 cents
{{Mapping|legend=2| 1 1 7| 0 1 -4}}


Gencom: [2 3/2; 169/169 352/351 364/363]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
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|]
==== 2.3.23.25.41 subgroup ====
EDOs: 17, 46, 63
''See also: [[Reversed meantone]]''
[[Tp tuning#T2%20tuning|RMS error]]: 0.7541 cents


==Bleu==
Edo join: 17 & 12
Group: 2.3.7.11.13
Commas: 78/77, 99/98, 144/143
Sval name: 9&amp;17


[[Tp tuning|POL2 generator]]: ~13/12 = 139.990
Comma list: 2048/2025, 576/575, 82/81


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
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|]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
EDOs: [[xenharmonic/17edo|17]], [[xenharmonic/43edo|43]], [[xenharmonic/60edo|60c]]
[[Tp tuning#T2%20tuning|RMS error]]: 1.752 cents


==Parapyth (Rank 3)==  
===== Sburb =====
Commas: 352/351, 364/363
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


[[Lp tuning|POL2 tuning]]: &lt;1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
Subgroup: 2.3.7.23.25.41.59


The gencom below gives [[Margo Schulter]]'s favored basis
Edo join: 17 & 12
Gencom: [2 3/2 28/27; 352/351 364/363]
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|]
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
EDOs: 17, 41, 46, 58, 87, 104
[[Tp tuning#T2%20tuning|RMS error]]: 0.3789 cents


=2.3.5.11=
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}


==Larry==
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


Gencom: [2 40/27; 243/242 4000/3993]
[[Subgroup]]: 2.9.5.11.13
Gencom mapping: [&lt;1 5 12 0 12|, &lt;0 -6 -17 0 -15|]


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
[[Comma list]]: 45/44, 65/64, 81/80
EDOs: 7, 58, 65, 137, 202
[[Tp tuning#T2%20tuning|RMS error]]: 0.3025 cents


=2.9.7.11=
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


==Machine==
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]


Gencom: [2 8/7; 64/63 99/98]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
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|]
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
EDOs: 5, 6, 11, 17, 28
[[Tp tuning#T2%20tuning|RMS error]]: 1.977 cents


==Apparatus==
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


Gencom: [2 77/72; 41503/41472 322102/321489]
== 2.9.7 subgroup ==
Gencom mapping: [&lt;1 5/2 0 3 5|, &lt;0 -19/2 0 -2 -16|]
=== 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.


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
Subgroup: 2.9.7
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
[[Tp tuning#T2%20tuning|RMS error]]: 0.0673 cents


==Mechanism==
Comma basis: 44957696/43046721
Commas: 896/891, 26411/26244


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]


Gencom: [2 9/7; 896/891 26411/26244]
Optimal tuning (CTE): ~729/448 = 870.792
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|]
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
EDOs: 8, 11, 30, 41, 52
[[Tp tuning#T2%20tuning|RMS error]]: 0.4262 cents


=2.9.15.7.11.13=
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


==Stacks (aka 2magic)==
Comma basis: 896/891, 1331/1296
Commas: 100/99, 105/104, 144/143, 196/195


[[Lp tuning|POL2 generator]]: ~9/7 = 438.977
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Optimal tuning (CTE): ~16/11 = 870.966
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|]
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
[[Tp tuning#T2%20tuning|RMS error]]: 1.540 cents


===2.9.15.7.11 stacks===  
== 2.9.7.11 subgroup ==
Commas: 100/99, 225/224, 245/243
=== Apparatus ===
[[Subgroup]]: 2.9.7.11


[[Lp tuning|POL2 generator]]: ~9/7 = 438.607
[[Comma list]]: 41503/41472, 322102/321489


Gencom: [2 9/7; 100/99 225/224 245/243]
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
Gencom mapping: [&lt;1 5/2 5/2 5 2|, &lt;0 -5/2 -1/2 -6 4|]


Sval map: [&lt;1 0 2 -1 6|, &lt;0 5 3 6 -4|]
: mapping generators: ~2, ~77/72
EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
[[Tp tuning#T2%20tuning|RMS error]]: 1.226 cents


===2.9.15.7 stacks===
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
Commas: 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 439.296
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]


Gencom: [2 9/7; 225/224 245/243]
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
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|]
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
[[Tp tuning#T2%20tuning|RMS error]]: 1.074 cents


=2.9.21=
[[Badness]]: 0.00263
==A-team==
Commas: 1029/1024


[[Lp tuning|POL2 generator]]: ~21/16 = 467.375
=== Joan ===
{{See also| Chromatic pairs #Joan }}


Gencom: [2 21/16; 1029/1024]
Joan is related to [[casablanca]] as well as to [[orwell]].
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3/2 0 -1/2|]


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
[[Subgroup]]: 2.9.7.11
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=
[[Comma list]]: 99/98, 9317/9216
==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
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
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|]
: [[gencom]]: [2 11/8; 99/98 9317/9216]
EDOs: 14, 29, 72, 101, 130, 159
[[Tp tuning#T2%20tuning|RMS error]]: 0.2562 cents


=2.3.7.11/5.13=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
==Hypnosis==
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]
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
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|]
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
EDOs: 17, 36, 118e, 125e, 161e, 197e
 
[[Tp tuning#T2%20tuning|RMS error]]: 0.5379 cents</pre></div>
=== Machine ===
<h4>Original HTML content:</h4>
Machine is every other step of [[supra]], most interesting for its scale patterns.
<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:78:&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:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --
 
[[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: 2.9.15.7.11.13
 
Comma list: 100/99
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"