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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>2012-04-18 19:09:16 UTC</tt>.<br>
: The original revision id was <tt>322353290</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]]
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]]).


==[[Semiphore and Godzilla|Semiphore]]==
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


POTE 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]].  


Map: [&lt;1 2 3|, &lt;0 -2 -1|]
=== Commatose ===
EDOs: 5, 19, 24
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]].


==Archy==
[[Subgroup]]: 2.9.5.7
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
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


POTE generator: ~4/3 = 490.678
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}


Map: [&lt;1 2 2|, &lt;0 -1 2|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
EDOs: 5, 12, 17, 22, 27, 137bc


==[[Slendric]]==
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Comma: 1029/1024


POTE generator: ~8/7 = 233.688
[[Badness]]: 0.611


Map: [&lt;1 1 3|, &lt;0 3 -1|]
==== 2.9.5.7.11 ====
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
Subgroup: 2.9.5.7.11
Badness: 0.00461


=2.3.11=
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


==Neutraltet==
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
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, ~531441/524288 = 23.4767


POTE generator: ~11/9 = 350.525
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


Map: [&lt;1 1 2|, &lt;0 2 5|]
Badness: 0.165
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
==== 2.9.5.7.11.13 ====
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
Subgroup: 2.9.5.7.11.13
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]


=2.3.5.11=
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


==Porcupine==
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
Commas: 55/54, 100/99


POTE generator: 164.078
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767


Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -4|]
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
EDOs: 7, 15, 22, 37


==Mohaha==
Badness: 0.0564
Commas: 81/80, 121/120


POTE generator: 348.094
=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
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.


Map: [&lt;1 1 0 2|, &lt;0 2 8 5|]
Subgroup: 2.9.5.7.33.13.17
EDOs: 7, 10, 17, 24, 31


=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


POTE generator: ~99/98 = 17.6258
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


Sval map: [&lt;9 0 11 24|, &lt;0 2 2 1|]
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
EDOs: 54, 63, 72, 135, 342, 477, 1089, 1566
Badness: 0.00426


==Supra==  
=== Baldy ===
Commas: 64/63, 99/98
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


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


Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
[[Subgroup]]: 2.9.5.7
EDOs: 5, 12, 17, 39, 56
Badness: 0.00933


==Skwares==
[[Comma list]]: 225/224, 3125/3087
Commas: 99/98, 243/242


POTE generator: ~9/7 = 425.244
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
EDOs: 14, 17, 31, 48, 79, 127, 206
Badness: 0.0107


==Hemif==
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]


Sval map: [&lt;1 1 -1 2|, &lt;0 2 13 5|]
==== 2.9.5.7.13 ====
EDOs: 7, 17, 41, 58, 99
{{See also|Chromatic pairs #Baldy}}
Badness: 0.0109


=2.5.7.11=
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.  


==Larry==
[[Subgroup]]: 2.9.5.7.13
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
EDOs: 7, 58, 65, 137, 202
Badness: 0.0125


=2.9.7.11=
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}


==Machine==
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090


[[Smonzos and Svals|Sval map]]: [&lt;1 0 6 13|, &lt;0 1 -1 -3|]
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
EDOs: 5, 6, 11, 17, 28
Badness: 0.00233


==Apparatus==
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
==== Baldanders ====
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
Badness: 0.00263


==Mechanism==
Subgroup: 2.9.5.7.11
Commas: 896/891, 26411/26244


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
Comma list: 100/99, 225/224, 245/242


[[Smonzos and Svals|Sval map]]: [&lt;1 5 5 2|, &lt;0 -5 -6 4|]
{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
EDOs: 8, 11, 30, 41, 52
Badness: 0.00439


=2.9.15.7.11.13=
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743


==Stacks (aka 2magic)==
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
Commas: 100/99, 105/104, 144/143, 196/195


POTE generator: ~9/7 = 438.977
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]


Sval map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 3 6 -4 9|]
===== 2.9.5.7.11.13 =====
EDOs: 11, 30, 41
Subgroup: 2.9.5.7.11.13
Badness: 0.00602


===2.9.15.7.11 stacks===
Comma list: 100/99, 144/143, 225/224, 245/242
Commas: 100/99, 225/224, 245/243


POTE generator: ~9/7 =438.443
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}


Sval map: [&lt;1 0 2 -1 6|, &lt;0 5 3 6 -4|]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
EDOs: 8, 11, 30, 41, 52
Badness: 0.00371


===2.9.15.7 stacks===
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
Commas: 225/224, 245/243


POTE generator: ~9/7 = 438.850
== 2.3.25 subgroup ==


Sval map: [&lt;1 0 2 -1|, &lt;0 5 3 6|]
=== Shrub ===
EDOs: 8, 11, 30, 41, 93, 134c, 175c
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.
Badness: 0.00245


=2.9.21=
Subgroup: 2.3.25
==A-team==
Commas: 1029/1024


POTE generator: ~21/16 = 467.756
Edo join: 17 & 12


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
Comma list: [[2048/2025]]
EDOs: 5, 13, 18, 41, 59, 254c, 313c, 372c, 472c, 490c
Badness: 0.000246


{{Mapping|legend=2| 1 1 7| 0 1 -4}}


</pre></div>
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
<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:52:&amp;lt;img id=&amp;quot;wikitext@
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
 
Edo join: 17 & 12
 
Comma list: 2048/2025, 576/575, 82/81
 
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
 
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
 
Subgroup: 2.3.7.23.25.41.59
 
Edo join: 17 & 12
 
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
 
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
 
== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}
 
[[Subgroup]]: 2.9.5.11.13
 
[[Comma list]]: 45/44, 65/64, 81/80
 
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
 
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
 
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
 
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
 
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)
 
== 2.9.7 subgroup ==
=== 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.
 
Subgroup: 2.9.7
 
Comma basis: 44957696/43046721
 
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
 
Optimal tuning (CTE): ~729/448 = 870.792
 
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
 
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11
 
Comma basis: 896/891, 1331/1296
 
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
 
Optimal tuning (CTE): ~16/11 = 870.966
 
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
 
== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 41503/41472, 322102/321489
 
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
 
: mapping generators: ~2, ~77/72
 
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
 
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
 
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
 
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
 
[[Badness]]: 0.00263
 
=== Joan ===
{{See also| Chromatic pairs #Joan }}
 
Joan is related to [[casablanca]] as well as to [[orwell]].
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 99/98, 9317/9216
 
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}
 
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
 
: [[gencom]]: [2 11/8; 99/98 9317/9216]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
 
{{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
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"