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Subgroup temperaments: Difference between revisions

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== 2.9.… subgroups ==
== 2.9.… subgroups ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
=== Commatose ===
Commatose is a 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]].
[[Subgroup]]: 2.9.5.7
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
{{Mapping|legend=3| 1 -289 -182 -508 | 0 298 188 521 }}
: mapping generators: ~2, ~1048576/531441
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9727{{c}}, ~1048576/531441 = 1176.4969{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1048576/531441 = 1176.5236{{c}}
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
[[Badness]] (Sintel): 30.9
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
{{Mapping|legend=2| 1 -289 -182 -508 -625 | 0 298 188 521 641 }}
Optimal tunings:
* WE: ~2 = 1199.9833{{c}}, ~531441/524288 = 1176.5070{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~531441/524288 = 1176.5234{{c}}
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}
Badness (Sintel): 8.67
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
{{Mapping|legend=2| 0 -289 -182 -508 -625 -312 | 0 298 188 521 641 322 }}
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~7056/3575 = 23.4767{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7056/3575 = 23.4767{{c}}
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}
Badness (Sintel): 3.30
=== 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 (hence the name – daemo- + tertiaschis), 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 }}
: mapping generators: ~2, ~33/20
Optimal tunings:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}}
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459* }}
Badness (Sintel): 0.439


=== Baldy ===
=== Baldy ===
Line 613: Line 545:
Badness (Sintel): 0.653
Badness (Sintel): 0.653


=== Glacial ===
=== Daemotertiaschis ===
{{See also| Chromatic pairs #Glacial }}
{{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 (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.


[[Subgroup]]: 2.9.5.11.13
Subgroup: 2.9.5.7.33.13.17


[[Comma list]]: 45/44, 65/64, 81/80
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}
: mapping generators: ~2, ~33/20


{{Mapping|legend=5| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
Optimal tunings:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}}


: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459* }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
[[Badness]] (Sintel): 0.439


{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
=== A-team ===
{{See also| Dual-fifth temperaments #Dual-3 A-Team }}


[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
A-team is every other step of [[gamelismic clan #Mothra|mothra]].  


Music:
[[Subgroup]]: 2.9.5.21
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


=== Mabon ===
[[Comma list]]: 81/80 ({{monzo| -4 2 -1 0 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})
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
{{Mapping|legend=3| 1 2 0 4 | 0 3 6 1 }}


Comma basis: 44957696/43046721
{{Mapping|legend=5| 1 1 0 3 | 0 3/2 6 -1/2 }}
: mapping generators: ~2, ~21/16


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.9303{{c}}, ~21/16 = 464.7465{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 464.5029{{c}}


Optimal tuning (CTE): ~729/448 = 870.792
{{Optimal ET sequence|legend=1| 13, 18, 31 }}


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
[[Badness]] (Sintel): 0.177


==== 2.9.7.11 subgroup ====
===== 2.9.5.21.11 subgroup =====
Subgroup: 2.9.7.11
Subgroup: 2.9.5.21.11


Comma basis: 896/891, 1331/1296
Comma list: 81/80 ({{monzo| -4 2 -1 0 }}), 99/98 ({{monzo| -1 2 0 -2 1 }}), 385/384 ({{monzo| -7 -1 1 1 1 }})


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
{{Mapping|legend=2| 1 2 0 4 5 | 0 3 6 1 -4 }}


Optimal tuning (CTE): ~16/11 = 870.966
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
Optimal tunings:
* Subgroup WE: ~2 = 1201.3979{{c}}, ~21/16 = 464.6020{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 464.1242{{c}}


=== Apparatus ===
{{Optimal ET sequence|legend=0| 13, 31, 44, 75b }}
[[Subgroup]]: 2.9.7.11


[[Comma list]]: 41503/41472, 322102/321489
Badness (Sintel): 0.213


{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


: mapping generators: ~2, ~77/72
[[Subgroup]]: 2.9.5.11.13


{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
[[Comma list]]: 45/44, 65/64, 81/80


: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
{{Mapping|legend=5| 1 0 -4 0 -6 10 | 0 1/2 2 0 3 -2 }}
: mapping generators: ~2, ~9


{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.6916{{c}}, ~9/8 = 186.1446{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 186.1740{{c}}


[[Badness]]: 0.00263
{{Optimal ET sequence|legend=1| 6, 13, 45bf }}


=== Joan ===
[[Badness]] (Sintel): 0.281
{{See also| Chromatic pairs #Joan }}


Joan is related to [[casablanca]] as well as to [[orwell]].  
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


[[Subgroup]]: 2.9.7.11
Music:
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


[[Comma list]]: 99/98, 9317/9216
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[Magic family #Magic|magic]].


{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}
[[Subgroup]]: 2.9.15.7


{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
[[Comma list]]: 225/224 ({{monzo| -5 0 2 -1 }}), 245/243 ({{monzo| 0 -3 1 2 }})


: [[gencom]]: [2 11/8; 99/98 9317/9216]
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
{{Mapping|legend=5| 1 0 2 -1 | 0 5/2 1/2 6 }}
: mapping generators: ~2, ~14/9


{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1201.0786{{c}}, ~14/9 = 761.3878{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 760.9151{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 71, 112* }}


=== Machine ===
<nowiki/>* Wart for 15
Machine is every other step of [[supra]], most interesting for its scale patterns.


[[Subgroup]]: 2.9.7.11
[[Badness]] (Sintel): 0.0884


[[Comma list]]: 64/63, 99/98
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11


: sval mapping generators: ~2, ~9
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 }}), 225/224 ({{monzo| -5 0 2 -1 0 }}), 245/243 ({{monzo| 0 -3 1 2 0 }})


{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
{{Mapping|legend=2| 1 0 2 -1 6 | 0 5 3 6 -4 }}


: [[gencom]]: [2 8/7; 64/63 99/98]
{{Mapping|legend=4| 1 0 2 -1 6 | 0 5/2 1/2 6 -4 }}


[[Optimal tuning]]s:
Optimal tunings:  
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* Subgroup WE: ~2 = 1200.1372{{c}}, ~14/9 = 761.4799{{c}}
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 761.4017{{c}}


{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
{{Optimal ET sequence|legend=0| 8, 11, 30, 41, 52, 93, 145 }}


[[Badness]]: 0.00233
Badness (Sintel): 0.166


=== Penta a.k.a. mechanism ===
RMS error: 1.226 cents
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.  


[[Subgroup]]: 2.9.7.11
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


[[Comma list]]: 896/891, 26411/26244
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 0 }}), 105/104 ({{monzo| -3 0 1 1 0 -1 }}), 144/143 ({{monzo| 4 1 0 0 -1 -1 }}), 196/195 ({{monzo| 2 0 -1 2 0 -1 }})


{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
{{Mapping|legend=2| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


: sval mapping generators: ~2, ~14/9
{{Mapping|legend=4| 1 0 2 -1 6 -2 | 0 5/2 1/2 6 -4 9 }}


{{Mapping|legend=5| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
Optimal tunings:
* Subgroup WE: ~2 = 1200.0331{{c}}, ~14/9 = 760.8754{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 760.8568{{c}}


: [[gencom]]: [2 9/7; 896/891 26411/26244]
{{Optimal ET sequence|legend=0| 11, 30, 41 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
Badness (Sintel): 0.257


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
RMS error: 1.540 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
=== B-team ===
B-team may be described as 23 & 41 and is every other step of [[gamelismic clan #Rodan|rodan]].


[[Badness]]: 0.00439
[[Subgroup]]: 2.9.15.21


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
[[Comma list]]: 245/243 ({{monzo| 0 -4 1 2 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})


=== Novisept ===
{{Mapping|legend=3| 1 2 0 4 | 0 3 10 1 }}
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
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.2146{{c}}, ~21/16 = 468.9174{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 468.8518{{c}}


[[Comma list]]: 729/728, 442/441, 833/832
{{Optimal ET sequence|legend=1| 18, 23, 41, 64, 151 }}


{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Badness]] (Sintel): 0.155


[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
==== 2.9.15.21.33 subgroup ====
Subgroup: 2.9.15.21.33


Badness (Dirichlet): 0.142
Comma list: 245/243 ({{monzo| 0 -4 1 2 0 }}), 385/384 ({{monzo| -7 -2 1 1 1 }}), 441/440 ({{monzo| -3 1 -1 2 -1 }})


=== Demon ===
{{Mapping|legend=2| 1 2 0 4 7 | 0 3 10 1 -5 }}
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
Optimal tunings:
* Subgroup WE: ~2 = 1200.0553{{c}}, ~21/16 = 468.9391{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 468.9189{{c}}


[[Comma list]]: [[1331/1296]]
{{Optimal ET sequence|legend=0| 18, 23, 41, 64, 87, 151, 238b }}


{{Mapping|legend=2|1 1 2|0 3 2}}
Badness (Sintel): 0.160


[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
=== Mabon ===
Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], and named so by {{u|Dummy index}} in 2024.  


{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
[[Subgroup]]: 2.9.7


=== Genius ===
[[Comma basis]]: 44957696/43046721


Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
{{Mapping|legend=3| 1 1 -3 | 0 3 8 }}
: mapping generators: ~2, ~729/448


[[Subgroup]]: 2.9.11
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0000{{c}}, ~729/448 = 870.4003{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~729/448 = 871.1047{{c}}


[[Comma list]]: [[131769/131072]]
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62, 73b, 157bb }}


{{Mapping|legend=2|1 1 4|0 4 -1}}
[[Badness]] (Sintel): 4.08


[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
Comma basis: 896/891, 1331/1296


=== Stacks (a.k.a. 2magic) ===
{{Mapping|legend=2| 1 1 -3 2 | 0 3 8 2 }}
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
Optimal tunings:  
* WE: ~2 = 1200.1182{{c}}, ~18/11 = 870.9904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 870.9234{{c}}


[[Comma list]]: 225/224, 245/243
{{Optimal ET sequence|legend=0| 7d, 11, 40, 51, 62 }}


{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
Badness (Sintel): 4.08


: sval mapping generators: ~2, ~14/9
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.


{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
It was named by [[Mike Battaglia]] in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96074.html Yahoo! Tuning Group | ''Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament'']</ref>


: [[gencom]]: [2 9/7; 225/224 245/243]
[[Subgroup]]: 2.9.7.11


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
[[Comma list]]: 64/63, 99/98


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
{{Mapping|legend=5| 1 0 0 6 13 | 0 1/2 0 -1 -3 }}
: mapping generators: ~2, ~9


==== 2.9.15.7.11 ====
[[Optimal tuning]]s:
Subgroup: 2.9.15.7.11
* [[WE]]: ~2 = 1197.2650{{c}}, ~9/8 = 213.8957{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 214.9961{{c}}


Comma list: 100/99, 225/224, 245/243
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}


Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
[[Badness]] (Sintel): 0.176


{{Mapping|legend=4| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
=== Apparatus ===
Apparatus is the restriction of [[gamelismic clan #Revelation|revelation]] to the 2.9.7.11 subgroup, and every fourth step of [[breedsmic temperaments #Quasiorwell|quasiorwell]], without prime 5.


: gencom: [2 9/7; 100/99 225/224 245/243]
[[Subgroup]]: 2.9.7.11


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
[[Comma list]]: 41503/41472, 322102/321489


Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
{{Mapping|legend=3| 1 -14 1 -11 | 0 19 2 16 }}


RMS error: 1.226 cents
{{Mapping|legend=5| 1 -7 0 1 -11 | 0 19/2 0 2 16 }}
: mapping generators: ~2, ~144/77


==== 2.9.15.7.11.13 ====
[[Optimal tuning]]s:
Subgroup: 2.9.15.7.11.13
* [[WE]]: ~2 = 1200.0088{{c}}, ~144/77 = 1084.4377{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/77 = 1084.4304{{c}}


Comma list: 100/99, 105/104, 144/143, 196/195
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}


Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
[[Badness]] (Sintel): 0.199


{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.


: gencom: [2 9/7; 100/99 105/104 144/143 196/195]
[[Subgroup]]: 2.9.7.11


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
[[Comma list]]: 896/891, 26411/26244


Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}


RMS error: 1.540 cents
{{Mapping|legend=5| 1 0 0 -1 6 | 0 5/2 0 6 -4 }}
: mapping generators: ~2, ~14/9


=== A-team ===
[[Optimal tuning]]s:
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].  
* [[WE]]: ~2 = 1199.4843{{c}}, ~14/9 = 761.2079{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 761.4979{{c}}


[[Subgroup]]: 2.9.21
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}


[[Comma list]]: 1029/1024
[[Badness]] (Sintel): 0.331


{{Mapping|legend=3| 1 2 4 | 0 3 1 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


: sval mapping generators: ~2, ~21/16
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}
=== Joan ===
{{See also| Chromatic pairs #Joan }}


: [[gencom]]: [2 21/16; 1029/1024]
Joan is the restriction of [[starling temperaments #Casablanca|casablanca]] to the 2.9.7.11 subgroup, as well as every other step of [[orwell]], without prime 5.


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375
[[Subgroup]]: 2.9.7.11


{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
[[Comma list]]: 99/98, 9317/9216


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}


==== 2.9.5.21 ====
{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''
: mapping generators: ~2, ~11/8


Subgroup: 2.9.5.21
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8640{{c}}, ~11/8 = 543.0623{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 542.8446{{c}}


[[Comma]] list: 81/80, 1029/1024
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd }}


Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}
[[Badness]] (Sintel): 0.407


Mapping generators: ~2, ~21/16
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents


Optimal ([[Lp tuning|POL2]]) generator: 464.3865
=== Novisept ===
Named by {{u|Pailiaq}} in 2026, novisept is generated by a one-cent-flat [[9/7]], such that stacking 5 of them gives [[7/4]]. It can be formed by doubling both generator and period of [[gizzard]].


{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
[[Subgroup]]: 2.9.7.13.17


===== 2.9.5.21.11 =====
[[Comma list]]: 442/441, 729/728, 833/832
Subgroup: 2.9.5.21.11


Comma list: 81/80, 99/98, 385/384
{{Mapping|legend=3| 1 1 1 -1 3 | 0 6 5 13 3 }}
: mapping generators: ~2, ~9/7


Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3044{{c}}, ~9/7 = 433.9023{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 433.8360{{c}}


{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}
{{Optimal ET sequence|legend=1| 11, 25f, 36, 47, 83, 130, 213g, 296g }}


: gencom: [2 21/16; 81/80 99/98 385/384]
[[Badness]] (Sintel): 0.142


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
=== Genius ===
Genius tempers out the [[Alpharabian comma]] in the 2.9.11 subgroup; it is therefore every other step of [[no-fives subgroup temperaments #Alphaxenean|alphaxenean]]. It was named by {{u|Akselai}} in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.


{{Optimal ET sequence|legend=1| 5, 13, 31 }}
[[Subgroup]]: 2.9.11


==== B-team ====
[[Comma list]]: 131769/131072
B-team (23 & 41) is every other step of [[rodan]].


Subgroup: 2.9.15.21.33
{{Mapping|legend=2| 1 1 4 | 0 4 -1 }}
: mapping generators: ~2, ~16/11


Comma list: 245/243, 385/384, 441/440
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3181{{c}}, ~16/11 = 650.8508{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 650.7321{{c}}


Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
{{Optimal ET sequence|legend=1| 11, 24, 59, 83, 142, 225, 367e, 592be, 959beee, 1326bbeeee }}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
[[Badness]] (Sintel): 0.198


{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
=== Demon ===
Demon is a temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s 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 {{u|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 the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament.


== 4.3.… subgroups ==
[[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup.  
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
[[Subgroup]]: 2.9.11


[[Comma list]]: 15625/15552
[[Comma list]]: 1331/1296


{{Mapping|legend=3| 1 3 3 | 0 -6 -5 }}
{{Mapping|legend=3| 1 1 2 | 0 3 2 }}
: mapping generators: ~2, ~18/11


: Mapping generators: ~4, ~5/3
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.1151{{c}}, ~18/11 = 869.4724{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/11 = 869.3196{{c}}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
{{Optimal ET sequence|legend=1| 4, 7, 11, 18, 29, 76e }}


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
[[Badness]] (Sintel): 0.332


== 4.3.… subgroups ==
=== Tetrameantone ===
=== Tetrameantone ===
{{Main| Tetrameantone }}
{{Main| Tetrameantone }}
Named by [[Budjarn Lambeth]] in 2024, tetrameantone is the double-octave restriction of [[meantone family #Meantone|meantone]].


[[Subgroup]]: 4.3.5
[[Subgroup]]: 4.3.5
Line 929: Line 912:
[[Comma list]]: 81/80
[[Comma list]]: 81/80


{{Mapping|legend=3| 1 1 2 | 0 -1 -4 }}
{{Mapping|legend=3| 1 0 -2 | 0 1 4 }}
: mapping generators: ~4, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2402.7812{{c}}, ~3 = 1898.4361{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~3 = 1896.6512{{c}}
 
[[Optimal ET sequence]]: {{EDs|equave=4| 5, 14, 19, 43, 62, 81, 262b, 343b }}
 
[[Badness]] (Sintel): 0.0866


: Mapping generators: ~4, ~4/3
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
Named by {{u|CompactStar}} in 2023, tetrahanson is the double-octave restriction of [[kleismic family #Kleismic a.k.a. hanson|hanson]].  


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
[[Subgroup]]: 4.3.5
 
[[Comma list]]: 15625/15552
 
{{Mapping|legend=3| 1 -3 -2 | 0 6 5 }}
: mapping generators: ~4, ~12/5
 
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2400.3318{{c}}, ~12/5 = 1517.2163{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~12/5 = 1517.0308{{c}}
 
[[Optimal ET sequence]]: {{EDs|equave=4| 8, 11, 19, 68, 87, 106, 511, 617, 723, 829, 935c, 1764c }}
 
[[Badness]] (Sintel): 0.155


=== Tetramagic ===
=== Tetramagic ===
Named by [[Budjarn Lambeth]] in 2024, tetramagic is the double-octave restriction of [[magic family #Magic|magic]].


[[Subgroup]]: 4.3.5
[[Subgroup]]: 4.3.5
Line 944: Line 951:


{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
: mapping generators: ~4, ~5/4


: Mapping generators: ~4, ~5/4
[[Optimal tuning]]s:  
* [[WE]]: ~4 = 2402.4898{{c}}, ~5/4 = 380.4527{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~5/4 = 380.2194{{c}}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
[[Optimal ET sequence]]: {{EDs|equave=4| 6, 13, 19, 63, 82, 101, 120, 221c, 341c }}


[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
[[Badness]] (Sintel): 0.459


=== Blacktetra ===
=== Blacktetra ===
Named by [[Budjarn Lambeth]] in 2024, blacktetra is the double-octave restriction of [[blackwood family #Blackwood|blackwood]].


[[Subgroup]]: 4.3.5
[[Subgroup]]: 4.3.5
Line 957: Line 968:
[[Comma list]]: 256/243
[[Comma list]]: 256/243


{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}
{{Mapping|legend=3| 5 4 0 | 0 0 1 }}
: mapping generators: ~4, ~5


: Mapping generators: ~4, ~16/15
[[Optimal tuning]]s:  
* [[WE]]: ~4/3 = 477.7018{{c}}, ~5/4 = 397.6806{{c}} (~16/15 = 80.0212{{c}})
* [[CWE]]: ~4/3 = 480.0000{{c}}, ~5/4 = 395.1255{{c}} (~16/15 = 84.7745{{c}})


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
[[Optimal ET sequence]]: {{EDs|equave=4| 5, 20, 25, 30 }}


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
[[Badness]] (Sintel): 0.748


== 4.6.… subgroups ==
== 4.6.… subgroups ==
Line 2,006: Line 2,020:
: <nowiki />* wart for 5/2
: <nowiki />* wart for 5/2


= Related temperament collections =
== See also ==
* [[Dual-fifth temperaments]]
* [[Dual-fifth temperaments]]
* [[Irrational-subgroup temperaments]]
* [[Irrational-subgroup temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments
== References ==
<references/>


[[Category:Subgroup temperaments| ]] <!-- main article -->
[[Category:Subgroup temperaments| ]] <!-- main article -->
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