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Subgroup temperaments: Difference between revisions

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=== Commatose ===
=== Commatose ===
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]].
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
[[Subgroup]]: 2.9.5.7
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[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}
{{Mapping|legend=3| 1 -289 -182 -508 | 0 298 188 521 }}
: mapping generators: ~2, ~1048576/531441


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
[[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 }}
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}


[[Badness]]: 0.611
[[Badness]] (Sintel): 30.9


==== 2.9.5.7.11 ====
==== 2.9.5.7.11 ====
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Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
{{Mapping|legend=2| 1 -289 -182 -508 -625 | 0 298 188 521 641 }}


Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
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=1| 460, 869e, 1329, 1789, 3118 }}
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}


Badness: 0.165
Badness (Sintel): 8.67


==== 2.9.5.7.11.13 ====
==== 2.9.5.7.11.13 ====
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Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
{{Mapping|legend=2| 0 -289 -182 -508 -625 -312 | 0 298 188 521 641 322 }}


Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
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=1| 460, 869e, 1329, 1789, 3118 }}
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}


Badness: 0.0564
Badness (Sintel): 3.30


=== Daemotertiaschis ===
=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
{{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.
 
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
Subgroup: 2.9.5.7.33.13.17
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Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
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|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}
: mapping generators: ~2, ~33/20


Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
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}}


[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459* }}
 
Badness (Sintel): 0.439


=== Baldy ===
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also| Chromatic pairs #Baldy }}
{{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.
Baldy is every other step of [[schismatic family #Garibaldi|garibaldi]]. It is also the add-9 extension of [[no-threes subgroup temperaments #Frostburn|frostburn]]. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
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[[Comma list]]: 225/224, 3125/3087
[[Comma list]]: 225/224, 3125/3087


{{Mapping|legend=3| 1 3 3 4 | 0 1 -4 -7 }}
{{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }}
 
: mapping generators: ~2, ~9
[[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]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}}


==== 2.9.5.7.13 ====
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }}
{{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 & 35 temperament.  
[[Badness]] (Sintel): 0.274


[[Subgroup]]: 2.9.5.7.13
==== 2.9.5.7.13 subgroup ====
Subgroup: 2.9.5.7.13


[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
Comma list: 225/224, 325/324, 640/637


{{Mapping|legend=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


{{Mapping|legend=5| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
{{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}


: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
Optimal tunings:  
* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }}


{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
Badness (Sintel): 0.386


[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
RMS error: 0.5999 cents
 
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]


==== Baldanders ====
==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones.


Subgroup: 2.9.5.7.11
Subgroup: 2.9.5.7.11
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Comma list: 100/99, 225/224, 245/242
Comma list: 100/99, 225/224, 245/242


{{Mapping|legend=3| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
Optimal tunings:
* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
Badness (Sintel): 0.389


===== 2.9.5.7.11.13 =====
===== 2.9.5.7.11.13 =====
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Comma list: 100/99, 144/143, 225/224, 245/242
Comma list: 100/99, 144/143, 225/224, 245/242


{{Mapping|legend=3| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
{{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.6385{{c}}, ~9/8 = 204.3526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }}


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
Badness (Sintel): 0.653


=== Glacial ===
=== Glacial ===
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Music:
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


=== Mabon ===
=== Mabon ===
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