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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:guest|guest]] and made on <tt>2013-03-04 09:13:43 UTC</tt>.<br>
: The original revision id was <tt>411783978</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
[[Lp tuning#L2%20tuning|L2 error]]: 5.046 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
[[Lp tuning#L2%20tuning|L2 error]]: 3.713 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
[[Lp tuning#L2%20tuning|L2 error]]: 0.6403 cents


=2.3.11=
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


==Neutraltet==
Badness: 0.165
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
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13


[[Lp tuning|POL2 generator]]: ~11/9 = 350.525
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


Gencom: [2 11/9; 243/242]
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 0 0 5|]


Map: [&lt;1 1 2|, &lt;0 2 5|]
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
[[Lp tuning#L2%20tuning|L2 error]]: 0.6756 cents


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]


=2.3.5.11=
Badness: 0.0564


==Porkypine==  
=== Daemotertiaschis ===
Commas: 55/54, 100/99
{{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.


[[Lp tuning|POL2 generator]]: ~10/9 = 164.078
Subgroup: 2.9.5.7.33.13.17


Gencom: [2 10/9; 55/54, 100/99]
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
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|]
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
[[Lp tuning#L2%20tuning|L2 error]]: 5.114 cents


==Mohaha==
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
Commas: 81/80, 121/120


[[Lp tuning|POL2 generator]]: ~11/9 = 348.094
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}


Gencom: [2 11/9; 81/80 121/120]
=== Baldy ===
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|]
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


Map: [&lt;1 1 0 2|, &lt;0 2 8 5|]
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.
EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd
[[Lp tuning#L2%20tuning|L2 error]]: 3.113 cents


===Music===
[[Subgroup]]: 2.9.5.7
[[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3|Mohaha10ping2]] by [[Billy Stiltner]]


=2.3.7.11=
[[Comma list]]: 225/224, 3125/3087


==Ennea==
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}
Commas: 41503/41472, 43923/43904


[[Lp tuning|POL2 generator]]: ~99/98 = 17.6258
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170


Gencom: [2 99/98; 41503/41472, 43923/43904]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
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|]
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd
[[Lp tuning#L2%20tuning|L2 error]]: 0.0857 cents


==Supra==  
==== 2.9.5.7.13 ====
Commas: 64/63, 99/98
{{See also|Chromatic pairs #Baldy}}


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.  


Gencom: [2 3/2; 64/63 99/98]
[[Subgroup]]: 2.9.5.7.13
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|]
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
EDOs: 5, 12, 17, 39c, 56c
[[Lp tuning#L2%20tuning|L2 error]]: 4.421 cents


==Skwares==
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
Commas: 99/98, 243/242


[[POTE tuning|POTE generator]]: ~9/7 = 425.244
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}


Gencom: [2 9/7; 99/98 243/242]
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
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|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
EDOs: 14, 17, 31, 48, 79, 127, 206bcd
[[Lp tuning#L2%20tuning|L2 error]]: 2.457 cents


==Hemif (11 limit)==
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents


Gencom: [2 11/9; 243/242 896/891]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
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|]
==== Baldanders ====
EDOs: 7, 17, 41, 58, 99d
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
[[Lp tuning#L2%20tuning|L2 error]]: 1.366 cents


=2.3.7.11.13=
Subgroup: 2.9.5.7.11


==Hemif (13 limit)==
Comma list: 100/99, 225/224, 245/242
Subgroup: 2.3.7.11.13
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|T2 error]]: 1.756 cents


==Parapyth (Rank 3)==
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
Commas: 352/351, 364/363


[[Lp tuning|POL2 tuning]]: &lt;1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


The gencom below gives [[Margo Schulter]]'s favored basis
Comma list: 100/99, 144/143, 225/224, 245/242
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|]
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
EDOs: 17, 41, 46, 58, 87, 104
[[Lp tuning#L2%20tuning|L2 error]]: 0.928 cents


==Leapfrog==
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
Commas: 169/168, 352/351, 364/363


[[Lp tuning|POL2 generator]]: ~3/2 = 704.745 cents
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}


Gencom: [2 3/2; 169/169 352/351 364/363]
== 2.3.25 subgroup ==
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|]
=== Shrub ===
EDOs: 17, 46, 63
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.
[[Lp tuning#L2%20tuning|L2 error]]: 1.847 cents


==Bleu==
Subgroup: 2.3.25
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
Edo join: 17 & 12


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
Comma list: [[2048/2025]]
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|]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
EDOs: [[xenharmonic/17edo|17]], [[xenharmonic/43edo|43]], [[xenharmonic/60edo|60c]]
[[Tp tuning#T2%20tuning|T2 error]]: 4.292 cents


=2.3.5.11=
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136


==Larry==  
==== 2.3.23.25.41 subgroup ====
Commas: 243/242, 4000/3993
''See also: [[Reversed meantone]]''
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
Edo join: 17 & 12


Gencom: [2 40/27; 243/242 4000/3993]
Comma list: 2048/2025, 576/575, 82/81
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|]
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
EDOs: 7, 58, 65, 137, 202
[[Lp tuning#L2%20tuning|L2 error]]: 0.6765 cents


=2.9.7.11=
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264


==Machine==  
===== Sburb =====
Commas: 64/63, 99/98
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
Subgroup: 2.3.7.23.25.41.59


Gencom: [2 8/7; 64/63 99/98]
Edo join: 17 & 12
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|]
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
EDOs: 5, 6, 11, 17, 28
[[Lp tuning#L2%20tuning|L2 error]]: 4.421 cents


==Apparatus==
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


Gencom: [2 77/72; 41503/41472 322102/321489]
== 2.9.5.11 subgroup ==
Gencom mapping: [&lt;1 5/2 0 3 5|, &lt;0 -19/2 0 -2 -16|]
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
[[Subgroup]]: 2.9.5.11.13
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
[[Lp tuning#L2%20tuning|L2 error]]: 0.1505 cents


==Mechanism==
[[Comma list]]: 45/44, 65/64, 81/80
Commas: 896/891, 26411/26244


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


Gencom: [2 9/7; 896/891 26411/26244]
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
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|]
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
EDOs: 8, 11, 30, 41, 52
[[Lp tuning#L2%20tuning|L2 error]]: 0.9531 cents


=2.9.15.7.11.13=
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


==Stacks (aka 2magic)==
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
Commas: 100/99, 105/104, 144/143, 196/195


[[Lp tuning|POL2 generator]]: ~9/7 = 438.977
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Music:
Gencom mapping: [&lt;1 5/2 5/2 5 2 7|, &lt;0 -5/2 -1/2 -6 4 -9|]
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


Sval map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 3 6 -4 9|]
== 2.9.7 subgroup ==
EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
=== Mabon ===
[[Lp tuning#L2%20tuning|L2 error]]: 3.772 cents
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.


===2.9.15.7.11 stacks===
Subgroup: 2.9.7
Commas: 100/99, 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 438.607
Comma basis: 44957696/43046721


Gencom: [2 9/7; 100/99 225/224 245/243]
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
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|]
Optimal tuning (CTE): ~729/448 = 870.792
EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
[[Lp tuning#L2%20tuning|L2 error]]: 2.741 cents


===2.9.15.7 stacks===
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
Commas: 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 439.296
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


Gencom: [2 9/7; 225/224 245/243]
Comma basis: 896/891, 1331/1296
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|]
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
[[Lp tuning#L2%20tuning|L2 error]]: 2.149 cents


=2.9.21=
Optimal tuning (CTE): ~16/11 = 870.966
==A-team==
Commas: 1029/1024


[[Lp tuning|POL2 generator]]: ~21/16 = 467.375
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}


Gencom: [2 21/16; 1029/1024]
== 2.9.7.11 subgroup ==
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3/2 0 -1/2|]
=== Apparatus ===
[[Subgroup]]: 2.9.7.11


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
[[Comma list]]: 41503/41472, 322102/321489
EDOs: 5, 13, 18, 41, 59, 77, 95
[[Lp tuning#L2%20tuning|L2 error]]: 0.6403


=2.3.7/5.11/5.13/5=
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
==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 generators: ~2, ~77/72


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
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 77/72; 41503/41472 322102/321489]
EDOs: 14, 29, 72, 101, 130, 159
[[Lp tuning#L2%20tuning|L2 error]]: 0.6274 cents


=2.3.7.11/5.13=
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
==Hypnosis==


Subgroup: 2.3.7.11/5.13
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
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]
[[Badness]]: 0.00263
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|]
=== Joan ===
EDOs: 17, 36, 118e, 125e, 161e, 197e
{{See also| Chromatic pairs #Joan }}
[[Lp tuning#L2%20tuning|L2 error]]: 1.317</pre></div>
 
<h4>Original HTML content:</h4>
Joan is related to [[casablanca]] as well as to [[orwell]].
<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:72:&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
 
[[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 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
 
== 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
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