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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 19:09:37 UTC</tt>.<br>
: The original revision id was <tt>322353384</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]].
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]]).


=2.3.7=
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]].


==[[Semiphore and Godzilla|Semiphore]]==
This page uses an extended [[wart notation]] where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.
Comma: 49/48


POTE generator: ~8/7~7/6 = 250.385
== 2.3.… subgroups ==
=== Shrub ===
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  


Map: [&lt;1 2 3|, &lt;0 -2 -1|]
It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.
EDOs: 5, 19, 24


==Archy==  
==== 2.3.25 subgroup ====
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.
[[Subgroup]]: 2.3.25


Comma: 64/63
[[Comma list]]: 2048/2025


POTE generator: ~4/3 = 490.678
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


Map: [&lt;1 2 2|, &lt;0 -1 2|]
[[Optimal tuning]]s:  
EDOs: 5, 12, 17, 22, 27, 137bc
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


==[[Slendric]]==
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}
Comma: 1029/1024


POTE generator: ~8/7 = 233.688
[[Badness]] (Sintel): 0.234


Map: [&lt;1 1 3|, &lt;0 3 -1|]
===== 2.3.25.23 subgroup =====
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
Subgroup: 2.3.25.23
Badness: 0.00461


=2.3.11=
Comma list: 576/575, 736/729


==Neutraltet==
{{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }}
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 tunings:  
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}


POTE generator: ~11/9 = 350.525
{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}


Map: [&lt;1 1 2|, &lt;0 2 5|]
Badness (Sintel): 0.223
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
==== Sburb ====
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
Sburb is every other step of [[pajara]], extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a [[59/1|59th harmonic]] by setting the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]


=2.3.5.11=
Subgroup: 2.3.25.7


==Porcupine==
Comma list: 50/49, 64/63
Commas: 55/54, 100/99


POTE generator: 164.078
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}


Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -4|]
Optimal tunings:  
EDOs: 7, 15, 22, 37
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


==Mohaha==
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}
Commas: 81/80, 121/120


POTE generator: 348.094
Badness (Sintel): 0.253


Map: [&lt;1 1 0 2|, &lt;0 2 8 5|]
===== 2.3.25.7.23 subgroup =====
EDOs: 7, 10, 17, 24, 31
Subgroup: 2.3.25.7.23


=2.3.7.11=
Comma list: 50/49, 64/63, 162/161


==Ennea==
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}
Commas: 41503/41472, 43923/43904


POTE generator: ~99/98 = 17.6258
Optimal tunings:  
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


Sval map: [&lt;9 0 11 24|, &lt;0 2 2 1|]
{{Optimal ET sequence|legend=0| 5, 12, 17 }}
EDOs: 54, 63, 72, 135, 342, 477, 1089, 1566
Badness: 0.00426


==Supra==
Badness (Sintel): 0.419
Commas: 64/63, 99/98


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  


Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
[[Subgroup]]: 2.3.7.11/5.13
EDOs: 5, 12, 17, 39, 56
Badness: 0.00933


==Skwares==
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})
Commas: 99/98, 243/242


POTE generator: ~9/7 = 425.244
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
[[Optimal tuning]]s:  
EDOs: 14, 17, 31, 48, 79, 127, 206
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
Badness: 0.0107
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}


==Hemif==
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


Sval map: [&lt;1 1 -1 2|, &lt;0 2 13 5|]
=== Darian calendar ===
EDOs: 7, 17, 41, 58, 99
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].
Badness: 0.0109


=2.5.7.11=
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.


==Larry==
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
[[Subgroup]]: 2.3.35.11.19


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1 0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})
EDOs: 7, 58, 65, 137, 202
Badness: 0.0125


=2.9.7.11=
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


==Machine==
[[Optimal tuning]]s:
Commas: 64/63, 99/98
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


[[Smonzos and Svals|Sval map]]: [&lt;1 0 6 13|, &lt;0 1 -1 -3|]
=== Hydrothermal ===
EDOs: 5, 6, 11, 17, 28
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.
Badness: 0.00233


==Apparatus==
[[Subgroup]]: 2.3.7/5
Commas: 41503/41472, 322102/321489


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
: mapping generators: ~7/5, ~3
Badness: 0.00263


==Mechanism==
[[Optimal tuning]]s:
Commas: 896/891, 26411/26244
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}
: <nowiki/>* Wart for 7/5


[[Smonzos and Svals|Sval map]]: [&lt;1 5 5 2|, &lt;0 -5 -6 4|]
=== Argentic ===
EDOs: 8, 11, 30, 41, 52
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].  
Badness: 0.00439


=2.9.15.7.11.13=
[[Subgroup]]: 2.3.7/5


==Stacks (aka 2magic)==
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
Commas: 100/99, 105/104, 144/143, 196/195


POTE generator: ~9/7 = 438.977
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


Sval map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 3 6 -4 9|]
[[Optimal tuning]]s:  
EDOs: 11, 30, 41
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
Badness: 0.00602
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


===2.9.15.7.11 stacks===
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
Commas: 100/99, 225/224, 245/243


POTE generator: ~9/7 =438.443
[[Badness]] (Sintel): 0.119


Sval map: [&lt;1 0 2 -1 6|, &lt;0 5 3 6 -4|]
==== Edson ====
EDOs: 8, 11, 30, 41, 52
{{See also| Chromatic pairs #Edson }}
Badness: 0.00371


===2.9.15.7 stacks===
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  
Commas: 225/224, 245/243


POTE generator: ~9/7 = 438.850
[[Subgroup]]: 2.3.7/5.11/5.13/5


Sval map: [&lt;1 0 2 -1|, &lt;0 5 3 6|]
[[Comma list]]: [[196/195]] ({{monzo| 2 -1 2 0 -1 }}), [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[364/363]] ({{monzo| 2 -1 1 -2 1 }})
EDOs: 8, 11, 30, 41, 93, 134c, 175c
Badness: 0.00245


=2.9.21=
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
==A-team==
Commas: 1029/1024


POTE generator: ~21/16 = 467.756
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
[[Optimal tuning]]s:  
EDOs: 5, 13, 18, 41, 59, 254c, 313c, 372c, 472c, 490c
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
Badness: 0.000246
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}


</pre></div>
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 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:52:&amp;lt;img id=&amp;quot;wikitext@@
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}
 
Haumea is the add-7/5 extension of [[#Bridgetown|bridgetown]], which is in turn the add-11/5 extension of [[#Barbados|barbados]]. Other related temperamnets include [[schismatic family #Hemigari|hemigari]], [[the Archipelago #Parizekmic|parizekmic]], and [[aberschismic family #Namaka|namaka]].
 
[[Subgroup]]: 2.3.7/5.11/5.13/5
 
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})
 
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
 
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
: mapping generators: ~2, ~26/15
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}
 
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
 
=== Historical ===
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}
 
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
 
[[Subgroup]]: 2.3.7/5.11/5.13/5
 
[[Comma list]]: 364/363 ({{monzo| 2 -1 1 -2 1 }}), 441/440 ({{monzo| -3 2 2 -1 0 }}, 1001/1000 ({{monzo| -3 0 1 1 1 }})
 
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}
 
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
 
=== Direct breedsmic ===
This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].
 
[[Subgroup]]: 2.3.49/5
 
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})
 
{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}
 
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
: <nowiki/>* Wart for 49/5
 
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}
 
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].
 
[[Subgroup]]: 2.3.11.13/5
 
[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})
 
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}
 
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }}
: mapping generators: ~55/39, ~26/15
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}
 
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
 
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17
 
[[Comma list]]: 221/220 ({{monzo| -2 0 -1 1 1 }}), 243/242 ({{monzo| -1 5 -2 0 0 }}), 289/288 ({{monzo| -5 -2 0 0 2 }})
 
{{Mapping|legend=3| 2 0
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