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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>2011-09-27 08:37:45 UTC</tt>.<br>
: The original revision id was <tt>258599756</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">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:
==[[Slendric]]==
* [[No-elevens subgroup temperaments]]
Comma: 1029/1024
* [[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]]).


POTE generator: ~8/7 = 233.688
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]].


Map: [&lt;1 1 3|, &lt;0 3 -1|]
= Composite subgroup temperaments =
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
== 2.3.35 subgroup ==
Badness: 0.00461
=== 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]].


=2.5.7.11=
==== 2.3.11.19 subgroup ====
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


==Larry==
[[Subgroup]]: 2.3.11.19
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
: sval mapping generators: ~6291456/5285401, ~25289/24576
EDOs: 7, 58, 65, 137, 202
Badness: 0.0125


=2.3.7.11=
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257


==Ennea==
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
Commas: 41503/41472, 43923/43904


POTE generator: ~99/98 = 17.6258
==== 2.3.35.11.19 subgroup ====
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.


Sval map: [&lt;9 0 11 24|, &lt;0 2 2 1|]
Subgroup: 2.3.35.11.19
EDOs: 54, 63, 72, 135, 342, 477, 1089, 1566
Badness: 0.00426


==Supra==
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
: sval mapping generators: ~2240/1881, ~36/35


Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
EDOs: 5, 12, 17, 39, 56
Badness: 0.00933


==Skwares==
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
Commas: 99/98, 243/242


POTE generator: ~9/7 = 425.244
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
=== Commatose ===
EDOs: 14, 17, 31, 48, 79, 127, 206
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]].
Badness: 0.0107


==Hemif==
[[Subgroup]]: 2.9.5.7
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


Sval map: [&lt;1 1 -1 2|, &lt;0 2 13 5|]
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}
EDOs: 7, 17, 41, 58, 99
Badness: 0.0109


=2.9.7.11=
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765


==Machine==
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
[[Badness]]: 0.611


[[Smonzos and Svals|Sval map]]: [&lt;1 0 6 13|, &lt;0 1 -1 -3|]
==== 2.9.5.7.11 ====
EDOs: 5, 6, 11, 17, 28
Subgroup: 2.9.5.7.11
Badness: 0.00233


==Apparatus==
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
Badness: 0.00263


==Mechanism==
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
Commas: 896/891, 26411/26244


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
Badness: 0.165


[[Smonzos and Svals|Sval map]]: [&lt;1 5 5 2|, &lt;0 -5 -6 4|]
==== 2.9.5.7.11.13 ====
EDOs: 8, 11, 30, 41, 52
Subgroup: 2.9.5.7.11.13
Badness: 0.00439</pre></div>
 
<h4>Original HTML content:</h4>
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
<div style="width:100%; max-height:400pt; overflow:auto; background-color
 
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
 
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
 
Badness: 0.0564
 
=== 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.
 
Subgroup: 2.9.5.7.33.13.17
 
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
 
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
 
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
 
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
 
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}
 
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.
 
[[Subgroup]]: 2.9.5.7
 
[[Comma list]]: 225/224, 3125/3087
 
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
 
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
 
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
 
==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}
 
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.
 
[[Subgroup]]: 2.9.5.7.13
 
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
 
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
 
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
 
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
 
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
 
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
 
==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
 
Subgroup: 2.9.5.7.11
 
Comma list: 100/99, 225/224, 245/242
 
{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
 
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
 
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
 
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13
 
Comma list: 100/99, 144/143, 225/224, 245/242
 
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
 
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
 
== 2.3.25 subgroup ==
 
=== Shrub ===
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.
 
Subgroup: 2.3.25
 
Edo join: 17 & 12
 
Comma list: [[2048/2025]]
 
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
 
==== 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,
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