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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>2012-10-24 23:42:29 UTC</tt>.<br>
: The original revision id was <tt>376252958</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]].
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]].


==[[Semaphore and Godzilla|Semaphore]]==
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


[[Lp tuning|POL2 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.  


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


Map: [&lt;1 2 3|, &lt;0 -2 -1|]
==== 2.3.25 subgroup ====
EDOs: 5, 14, 19, 24, 67c, 91c
[[Subgroup]]: 2.3.25
[[Lp tuning#L2 tuning|L2 error]]: 5.046 cents


==Archy==
[[Comma list]]: 2048/2025
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=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


[[Lp tuning|POL2 generator]]: ~3/2 = 709.321
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


Gencom: [2 3/2; 64/63]
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}
Gencom mapping: [&lt;1 1 0 4|, &lt;0 1 0 -2|]


Map: [&lt;1 2 2|, &lt;0 -1 2|]
[[Badness]] (Sintel): 0.234
EDOs: 5, 12, 17, 22, 27, 137bc
[[Lp tuning#L2 tuning|L2 error]]: 3.713 cents


==[[Slendric]]==  
===== 2.3.25.23 subgroup =====
Comma: 1029/1024
Subgroup: 2.3.25.23


[[Lp tuning|POL2 generator]]: ~8/7 = 233.688
Comma list: 576/575, 736/729


Gencom: [2 8/7; 1029/1024]
{{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }}
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3 0 -1|]


Map: [&lt;1 1 3|, &lt;0 3 -1|]
Optimal tunings:  
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}
[[Lp tuning#L2 tuning|L2 error]]: 0.6403 cents
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}


=2.3.11=
{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}


==Neutraltet==
Badness (Sintel): 0.223
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
==== Sburb ====
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.


[[Lp tuning|POL2 generator]]: ~11/9 = 350.525
Subgroup: 2.3.25.7


Gencom: [2 11/9; 243/242]
Comma list: 50/49, 64/63
Gencom mapping: [&lt;1 1 0 0 2|, &lt;0 2 0 0 5|]


Map: [&lt;1 1 2|, &lt;0 2 5|]
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}
EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
[[Lp tuning#L2 tuning|L2 error]]: 0.6756 cents


[[xenharmonic/neutraltet7|Seven note albitonic scale]]
Optimal tunings:
[[xenharmonic/neutraltet10|Ten note chromatic scale]]
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
[[xenharmonic/neutraltet17|Seventeen note mega chromatic scale]]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


=2.3.5.11=
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}


==Porkypine==
Badness (Sintel): 0.253
Commas: 55/54, 100/99


[[Lp tuning|POL2 generator]]: ~10/9 = 164.078
===== 2.3.25.7.23 subgroup =====
Subgroup: 2.3.25.7.23


Gencom: [2 10/9; 55/54, 100/99]
Comma list: 50/49, 64/63, 162/161
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 0 11 6 -5 | 0 1 -4 -2 6 }}
EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
[[Lp tuning#L2 tuning|L2 error]]: 5.114 cents


==Mohaha==
Optimal tunings:
Commas: 81/80, 121/120
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


[[Lp tuning|POL2 generator]]: ~11/9 = 348.094
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


Gencom: [2 11/9; 81/80 121/120]
Badness (Sintel): 0.419
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|]
=== Hypnosis ===
EDOs: 7, 10, 17, 24, 31, 69d, 100d, 131bd
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  
[[Lp tuning#L2 tuning|L2 error]]: 3.113 cents


===Music===
[[Subgroup]]: 2.3.7.11/5.13
[[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3|Mohaha10ping2]] by [[Billy Stiltner]]


=2.3.7.11=
[[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 }})


==Ennea==
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
Commas: 41503/41472, 43923/43904
: mapping generators: ~2, ~13/9


[[Lp tuning|POL2 generator]]: ~99/98 = 17.6258
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}


Gencom: [2 99/98; 41503/41472, 43923/43904]
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
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|]
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd
[[Lp tuning#L2 tuning|L2 error]]: 0.0857 cents


==Supra==  
=== Darian calendar ===
Commas: 64/63, 99/98
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]].


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


Gencom: [2 3/2; 64/63 99/98]
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.
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|]
[[Subgroup]]: 2.3.35.11.19
EDOs: 5, 12, 17, 39c, 56c
[[Lp tuning#L2 tuning|L2 error]]: 4.421 cents


==Skwares==
[[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 }})
Commas: 99/98, 243/242


[[POTE tuning|POTE generator]]: ~9/7 = 425.244
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


Gencom: [2 9/7; 99/98 243/242]
[[Optimal tuning]]s:  
Gencom mapping: [&lt;1 3 0 6 7|, &lt;0 -4 0 -9 -10]]
* [[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}}


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}
EDOs: 14, 17, 31, 48, 79, 127, 206bcd
[[Lp tuning#L2 tuning|L2 error]]: 2.457 cents


==Hemif (11 limit)==  
=== Hydrothermal ===
Commas: 243/242, 896/891
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.


POTE generator: ~11/9 = 351.535
[[Subgroup]]: 2.3.7/5


Gencom: [2 11/9; 243/242 896/891]
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})
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=3| 2 0 1 | 0 1 0 }}
EDOs: 7, 17, 41, 58, 99d
: mapping generators: ~7/5, ~3
[[Lp tuning#L2 tuning|L2 error]]: 1.366 cents


=2.3.7.11.13=
[[Optimal tuning]]s:
* [[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}}


==Hemif (13 limit)==
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}
Subgroup: 2.3.7.11.13
: <nowiki/>* Wart for 7/5
Commas: 144/143, 243/242, 364/363
Related temperament: [[xenharmonic/Breedsmic temperaments|Hemififths]], namo


[[Tp tuning|POT2 generator]]: ~11/9 = 351.691
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].  


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
[[Subgroup]]: 2.3.7/5
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|]
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
EDOs: 7, 10, 17, 24, 41, 58
[[Tp tuning#T2 tuning|T2 error]]: 1.756 cents


==Parapyth (Rank 3)==
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
Commas: 352/351, 364/363
: mapping generators: ~2, ~3


[[Lp tuning|POL2 tuning]]: &lt;1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


Gencom: [2 3 7; 352/351 364/363]
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
Gencom mapping: [&lt;1 0 0 7 12|, &lt;0 1 0 -4 -7|, &lt;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|]
[[Badness]] (Sintel): 0.119
EDOs: 17, 41, 46, 58, 87, 104
[[Lp tuning#L2 tuning|L2 error]]: 0.928 cents


==Leapfrog==
==== Edson ====
Commas: 169/168, 352/351, 364/363
{{See also| Chromatic pairs #Edson }}


[[Lp tuning|POL2 generator]]: ~3/2 = 704.745 cents
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  


Gencom: [2 3/2; 169/169 352/351 364/363]
[[Subgroup]]: 2.3.7/5.11/5.13/5
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|]
[[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: 17, 46, 63
[[Lp tuning#L2 tuning|L2 error]]: 1.847 cents


==Bleu==
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
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
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
[[Optimal tuning]]s:  
Gencom map: [&lt;1 1 0 2 3 3|, &lt;0 5 0 7 4 6|]
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


Map: [&lt;1 1 2 3 3|, &lt;0 5 7 4 6|]
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
EDOs: [[xenharmonic/17edo|17]], [[xenharmonic/43edo|43]], [[xenharmonic/60edo|60c]]
[[Tp tuning#T2 tuning|T2 error]]: 4.292 cents


=2.5.7.11=
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


==Larry==  
==== Haumea ====
Commas: 243/242, 4000/3993
{{See also| Chromatic pairs #Haumea }}
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
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]].  


Gencom: [2 40/27; 243/242 4000/3993]
[[Subgroup]]: 2.3.7/5.11/5.13/5
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]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})
EDOs: 7, 58, 65, 137, 202
[[Lp tuning#L2 tuning|L2 error]]: 0.6765 cents


=2.9.7.11=
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


==Machine==
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
Commas: 64/63, 99/98
: mapping generators: ~2, ~26/15


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


Gencom: [2 8/7; 64/63 99/98]
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}
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|]
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
EDOs: 5, 6, 11, 17, 28
[[Lp tuning#L2 tuning|L2 error]]: 4.421 cents


==Apparatus==  
=== Historical ===
Commas: 41503/41472, 322102/321489
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
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.


Gencom: [2 77/72; 41503/41472 322102/321489]
[[Subgroup]]: 2.3.7/5.11/5.13/5
Gencom mapping: [&lt;1 5/2 0 3 5|, &lt;0 -19/2 0 -2 -16|]


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


==Mechanism==
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
Commas: 896/891, 26411/26244
: mapping generators: ~2, ~21/20


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


Gencom: [2 9/7; 896/891 26411/26244]
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}
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|]
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
EDOs: 8, 11, 30, 41, 52
[[Lp tuning#L2 tuning|L2 error]]: 0.9531 cents


=2.9.15.7.11.13=
=== 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]].  


==Stacks (aka 2magic)==
[[Subgroup]]: 2.3.49/5
Commas: 100/99, 105/104, 144/143, 196/195


[[Lp tuning|POL2 generator]]: ~9/7 = 438.977
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
Gencom mapping: [&lt;1 5/2 5/2 5 2 7|, &lt;0 -5/2 -1/2 -6 4 -9|]
: mapping generators: ~2, ~49/40


Sval map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 3 6 -4 9|]
[[Optimal tuning]]:
EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
[[Lp tuning#L2 tuning|L2 error]]: 3.772 cents
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}


===2.9.15.7.11 stacks===
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
Commas: 100/99, 225/224, 245/243
: <nowiki/>* Wart for 49/5


[[Lp tuning|POL2 generator]]: ~9/7 = 438.607
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


Gencom: [2 9/7; 100/99 225/224 245/243]
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].
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|]
[[Subgroup]]: 2.3.11.13/5
EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
[[Lp tuning#L2 tuning|L2 error]]: 2.741 cents


===2.9.15.7 stacks===
[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})
Commas: 225/224, 245/243


[[Lp tuning|POL2 generator]]: ~9/7 = 439.296
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}


Gencom: [2 9/7; 225/224 245/243]
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }}
Gencom mapping: [&lt;1 5/2 5/2 5|, &lt;0 -5/2 -1/2 -6|]
: mapping generators: ~55/39, ~26/15


Sval map: [&lt;1 0 2 -1|, &lt;0 5 3 6|]
[[Optimal tuning]]s:  
EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
[[Lp tuning#L2 tuning|L2 error]]: 2.149 cents
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}


=2.9.21=
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}
==A-team==
Commas: 1029/1024


[[Lp tuning|POL2 generator]]: ~21/16 = 467.375
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


Gencom: [2 21/16; 1029/1024]
==== Pakkanian hemipyth ====
Gencom mapping: [&lt;1 1 0 3|, &lt;0 3/2 0 -1/2|]
[[Subgroup]]: 2.3.11.13/5.17


Sval map: [&lt;1 2 4|, &lt;0 3 1|]
[[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 }})
EDOs: 5, 13, 18, 41, 59, 77, 95
[[Lp tuning#L2 tuning|L2 error]]: 0.6403


=2.3.7/5.11/5.13/5=
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}
==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
[[Optimal tuning]]s:
* [[Tp tuning|subgroup WE]]: ~17/12 = 600.1781{{c}}, ~26/15 = 950.7656{{c}} (~15/13 = 249.2344{{c}})
* [[Tp tuning|subgroup CWE]]: ~17/12 = 600.0000{{c}}, ~26/15 = 950.6011{{c}} (~15/13 = 249.3989{{c}})


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
Gencom map: [&lt;1 2 -3/4 -3/4 1/4 5/4|, &lt;0 -6 0 7 2 -9|]
: <nowiki/>* Wart for 13/5


Sval map: [&lt;1 2 0 1 2|, &lt;0 -6 7 2 -9|]
=== Bridgetown ===
EDOs: 14, 29, 72, 101, 130, 159
{{See also| Chromatic pairs #Bridgetown }}
[[Lp tuning#L2 tuning|L2 error]]: 0.6274 cents


=2.3.7.11/5.13=
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of [[#Haumea|haumea]] and the add-11/5 [[extension]] of [[#Barbados|barbados]].  
==Hypnosis==


Subgroup: 2.3.7.11/5.13
[[Subgroup]]: 2.3.11/5.13/5
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]
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})
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|]
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}
EDOs: 17, 36, 118e, 125e, 161e, 197e
 
[[Lp tuning#L2 tuning|L2 error]]: 1.317</pre></div>
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
<h4>Original HTML content:</h4>
: mapping generators: ~2, ~26/15
<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:
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}
 
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
: <nowiki/>* Wart for 11/5
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
 
=== Seventeen-cot ===
Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026.
 
[[Subgroup]]: 2.3.11/5.13/5
 
[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})
 
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}}
: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}}
: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }}
 
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161… }}
 
[[Badness]] (Sintel): 0.080
 
=== Blackweed ===
Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out [[256/243]] alike but in the 2.3.11/7 subgroup. [[20edo]] is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.
 
[[Subgroup]]: 2.3.11/7
 
[[Comma list]]: 256/243 ({{monzo| 8 -5 0 }})
 
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
: error map: {{val| 0.000 +18.045 +2.475 }}
 
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
 
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}
 
Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain [[retraction]] of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
 
[[Subgroup]]: 2.3.11/7.13/7
 
[[Comma list]]: 352/351 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }})
 
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}
 
{{Mapping|legend=5| 1 0 0 -19/3 2/3 17/3 | 0 1 0 11/3 -1/3 -10/3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}
 
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
=== Fendo ===
Fendo tempers out [[40/39]] in the 2.3.13/5 subgroup. This equates [[4/3]] with the tendo third [[13/10]]. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with [[father family #Father|father]], as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by [[User: VectorGraphics|Vector]] in 2025 as a contraction of ''fourth'' and ''tendo third''.
 
In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.
 
[[5edo]] and [[13edo]] make good fendo tunings.
 
[[Subgroup]]: 2.3.13/5
 
[[Comma list]]: 40/39 ({
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