Subgroup temperaments: Difference between revisions

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* For no-twos, see [[Catalog of 3.5.7 subgroup rank two temperaments]].
* For no-twos, see [[Catalog of 3.5.7 subgroup rank two temperaments]].


Below are some other miscellaneous 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]].
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]].


== 2.9.5.7 subgroup ==
== 2.9.5.7 subgroup ==
=== Commatose ===
=== Commatose ===
Commatose is a fifthless 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 & 1789 temperament.  
Commatose is a fifthless 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 & 1789 temperament.  


Subgroup: 2.9.5.7
[[Subgroup]]: 2.9.5.7


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: {{val|1 9 6 13}}, {{val|0 -298 -188 -521}}
[[Mapping]]: [{{val| 1 9 6 13 }}, {{val| 0 -298 -188 -521 }}]


POTE Generator: 23.4763c ~ 531441/524288
[[Optimal tuning]] ([[POTE]]): ~531441/524288 = 23.4763


Vals: 460, 1329, 1789
{{Val list|legend=1| 460, 1329, 1789 }}


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


Mapping: {{val|1 9 6 13 16}}, {{val|0 -298 -188 -521 -641}}
Mapping: {{val| 1 9 6 13 16 }}, {{val| 0 -298 -188 -521 -641 }}


POTE Generator: 23.4766c ~ 531441/524288
Optimal tuning (POTE): ~531441/524288 = 23.4766


==== 2.9.5.7.11.13 ====
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13
Subgroup: 2.9.5.7.11.13


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


== 2.9.7.11 subgroup ==
== 2.9.7.11 subgroup ==
=== Machine ===
=== Machine ===
Subgroup: 2.9.7.11
[[Subgroup]]: 2.9.7.11


Commas: 64/63, 99/98
[[Comma list]]: 64/63, 99/98
 
[[Sval]] [[mapping]]: [{{val| 1 0 6 13 }}, {{val| 0 1 -1 -3 }}]


[[Gencom]]: [2 8/7; 64/63 99/98]
[[Gencom]]: [2 8/7; 64/63 99/98]


[[Gencom|Gencom mapping]]: [{{val|1 3/2 0 3 4}}, {{val|0 1/2 0 -1 -3}}]
[[Gencom]] [[mapping]]: [{{val| 1 3/2 0 3 4 }}, {{val| 0 1/2 0 -1 -3 }}]


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -1 -3}}]
[[Optimal tuning]] ([[POTE]]): ~8/7 = 214.384
 
[[Tp tuning|POL2 generator]]: ~8/7 = 214.384


{{Val list|legend=1| 5, 6, 11, 17, 28 }}
{{Val list|legend=1| 5, 6, 11, 17, 28 }}
Line 57: Line 57:


=== Mechanism ===
=== Mechanism ===
Subgroup: 2.9.7.11
[[Subgroup]]: 2.9.7.11


[[Comma list]]: 896/891, 26411/26244
[[Comma list]]: 896/891, 26411/26244
[[Sval]] [[mapping]]: [{{val| 1 5 5 2 }}, {{val| 0 -5 -6 4 }}]


[[Gencom]]: [2 9/7; 896/891 26411/26244]
[[Gencom]]: [2 9/7; 896/891 26411/26244]


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 5 2}}, {{val|0 -5/2 0 -6 4}}]
[[Gencom]] [[mapping]]: [{{val| 1 5/2 0 5 2 }}, {{val| 0 -5/2 0 -6 4 }}]


[[Mapping|Sval mapping]]: [{{val|1 5 5 2}}, {{val|0 -5 -6 4}}]
[[Optimal tuning]] ([[POTE]]): ~9/7 = 438.465
 
[[Tp tuning|POL2 generator]]: ~9/7 = 438.465


{{Val list|legend=1| 8, 11, 30, 41, 52 }}
{{Val list|legend=1| 8, 11, 30, 41, 52 }}
Line 74: Line 74:


=== Apparatus ===
=== Apparatus ===
Subgroup: 2.9.7.11
[[Subgroup]]: 2.9.7.11


[[Comma list]]: 41503/41472, 322102/321489
[[Comma list]]: 41503/41472, 322102/321489
[[Sval]] [[mapping]]: [{{val| 1 5 3 5 }}, {{val| 0 -19 -2 -16 }}]


[[Gencom]]: [2 77/72; 41503/41472 322102/321489]
[[Gencom]]: [2 77/72; 41503/41472 322102/321489]


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 3 5}}, {{val|0 -19/2 0 -2 -16}}]
[[Gencom]] [[mapping]]: [{{val| 1 5/2 0 3 5}}, {{val| 0 -19/2 0 -2 -16 }}]


[[Mapping|Sval mapping]]: [{{val|1 5 3 5}}, {{val|0 -19 -2 -16}}]
[[Optimal tuning]] ([[POTE]]): ~77/72 = 115.570
 
[[Tp tuning|POL2 generator]]: ~77/72 = 115.570


{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}
{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}
Line 91: Line 91:


== 2.9.15.7 subgroup ==
== 2.9.15.7 subgroup ==
=== Stacks (aka 2magic) ===
=== Stacks (aka 2magic) ===
Subgroup: 2.9.15.7
[[Subgroup]]: 2.9.15.7


[[Comma list]]: 225/224, 245/243
[[Comma list]]: 225/224, 245/243
[[Sval]] [[mapping]]: [{{val| 1 0 2 -1 }}, {{val| 0 5 3 6 }}]


[[Gencom]]: [2 9/7; 225/224 245/243]
[[Gencom]]: [2 9/7; 225/224 245/243]


[[Gencom|Gencom mapping]]: [{{val|1 5/2 5/2 5}}, {{val|0 -5/2 -1/2 -6}}]
[[Gencom]] [[mapping]]: [{{val| 1 5/2 5/2 5 }}, {{val| 0 -5/2 -1/2 -6 }}]


[[Mapping|Sval mapping]]: [{{val|1 0 2 -1}}, {{val|0 5 3 6}}]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~9/7 = 439.296
 
[[Tp tuning|POL2 generator]]: ~9/7 = 439.296


{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
Line 113: Line 112:


Comma list: 100/99, 225/224, 245/243
Comma list: 100/99, 225/224, 245/243
Sval mapping: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 3 6 -4 }}]


Gencom: [2 9/7; 100/99 225/224 245/243]
Gencom: [2 9/7; 100/99 225/224 245/243]


Gencom mapping: [{{val|1 5/2 5/2 5 2}}, {{val|0 -5/2 -1/2 -6 4}}]
Gencom mapping: [{{val| 1 5/2 5/2 5 2 }}, {{val| 0 -5/2 -1/2 -6 4 }}]
 
Sval mapping: [{{val|1 0 2 -1 6}}, {{val|0 5 3 6 -4}}]


POL2 generator: ~9/7 = 438.607
Optimal tuning (subgroup POTE): ~9/7 = 438.607


Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}
Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}
Line 130: Line 129:


Comma list: 100/99, 105/104, 144/143, 196/195
Comma list: 100/99, 105/104, 144/143, 196/195
Sval mapping: [{{val| 1 0 2 -1 6 -2 }}, {{val| 0 5 3 6 -4 9 }}]


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


Gencom mapping: [{{val|1 5/2 5/2 5 2 7}}, {{val|0 -5/2 -1/2 -6 4 -9}}]
Gencom mapping: [{{val| 1 5/2 5/2 5 2 7 }}, {{val| 0 -5/2 -1/2 -6 4 -9 }}]
 
Sval mapping: [{{val|1 0 2 -1 6 -2}}, {{val|0 5 3 6 -4 9}}]


POL2 generator: ~9/7 = 438.977
Optimal tuning (subgroup POTE): ~9/7 = 438.977


Vals: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
Optimal GPV sequence: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


RMS error: 1.540 cents
RMS error: 1.540 cents


== 2.9.21 subgroup ==
== 2.9.21 subgroup ==
=== A-team ===
[[Subgroup]]: 2.9.21


=== A-team ===
[[Comma list]]: 1029/1024
Subgroup: 2.9.21


[[Comma]]: 1029/1024
[[Sval]] [[mapping]]: [{{val| 1 2 4 }}, {{val| 0 3 1 }}]


[[Gencom]]: [2 21/16; 1029/1024]
[[Gencom]]: [2 21/16; 1029/1024]


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3/2 0 -1/2}}]
[[Gencom]] [[mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3/2 0 -1/2 }}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 4}}, {{val|0 3 1}}]


[[Tp tuning|POL2 generator]]: ~21/16 = 467.375
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/16 = 467.375


{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
Line 165: Line 163:
Subgroup: 2.9.5.21.11
Subgroup: 2.9.5.21.11


[[Commas]]: 81/80, 99/98, 385/384
Comma list: 81/80, 99/98, 385/384


[[Gencom]]: [2 21/16; 81/80 99/98 385/384]
Sval mapping: [{{val| 1 2 0 4 5 }}, {{val| 0 3 6 1 -4 }}]


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3 5}}, {{val|0 3/2 6 -1/2 -4}}]
Gencom: [2 21/16; 81/80 99/98 385/384]


[[Mapping|Sval mapping]]: [{{val|1 2 0 4 5}}, {{val|0 3 6 1 -4}}]
Gencom mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3/2 6 -1/2 -4 }}]


[[Tp tuning|POL2 generator]]: ~21/16 = 463.956
Optimal tuning (subgroup POTE): ~21/16 = 463.956


{{Val list|legend=1| 5, 13, 31}}
Optimal GPV sequence: {{Val list| 5, 13, 31}}


== Fractional subgroup temperaments ==
== Fractional subgroup temperaments ==
=== Historical ===
=== Historical ===
Subgroup: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Comma list]]: 364/363, 441/440, 1001/1000
[[Comma list]]: 364/363, 441/440, 1001/1000


[[Mapping|Sval mapping]]: [{{val|1 2 0 1 2}}, {{val|0 -6 7 2 -9}}]
[[Sval]] [[mapping]]: [{{val| 1 2 0 1 2 }}, {{val| 0 -6 7 2 -9 }}]


[[Tp tuning|POL2 generator]]: ~21/20 = 83.016
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016


{{Val list|legend=1| 14, 29, 72, 101, 130, 159 }}
{{Val list|legend=1| 14, 29, 72, 101, 130, 159 }}
Line 193: Line 190:


=== Hypnosis ===
=== Hypnosis ===
Subgroup: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7.11/5.13


[[Comma list]]: 169/168, 540/539, 729/728
[[Comma list]]: 169/168, 540/539, 729/728


[[Mapping|Sval mapping]]: [{{val|1 0 -3 8 0}}, {{val|0 3 11 -13 7}}]
[[Sval]] [[mapping]]: [{{val| 1 0 -3 8 0 }}, {{val| 0 3 11 -13 7 }}]


[[Tp tuning|POL2 generator]]: ~13/9 = 633.518
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518


{{Val list|legend=1| 17, 36, 118e, 125e, 161e, 197e }}
{{Val list|legend=1| 17, 36, 118e, 125e, 161e, 197e }}
Line 208: Line 205:


=== Oceanfront ===
=== Oceanfront ===
Subgroup: 2.3.7.13/5
[[Subgroup]]: 2.3.7.13/5


[[Comma list]]: 64/63, 91/90
[[Comma list]]: 64/63, 91/90


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


[[Tp tuning|POL2 generator]]: ~3/2 = 713.910
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910


{{Val list|legend=1| 5, 22, 27, 32, 37 }}
{{Val list|legend=1| 5, 22, 27, 32, 37 }}
Line 223: Line 220:


=== Marveltri ===
=== Marveltri ===
Subgroup: 2.5.9/7
[[Subgroup]]: 2.5.9/7


[[Comma list]]: 225/224
[[Comma list]]: 225/224


[[Mapping|Sval mapping]]: [{{val|1 2 1}}, {{val|0 1 -2}}]
[[Sval]] [[mapping]]: [{{val| 1 2 1 }}, {{val| 0 1 -2 }}]


[[Tp tuning|POL2 generator]]: ~5/4 = 383.638
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 383.638


{{Val list|legend=1| 3, 13, 16, 19, 22, 25 }}
{{Val list|legend=1| 3, 13, 16, 19, 22, 25 }}
Line 236: Line 233:


==== Sulis ====
==== Sulis ====
Subgroup: 2.5.9/7.11/9
[[Subgroup]]: 2.5.9/7.11/9


[[Comma list]]: 99/98, 176/175
[[Comma list]]: 99/98, 176/175


[[Mapping|Sval mapping]]: [{{val|1 2 1 -1}}, {{val|0 1 -2 4}}]
[[Sval]] [[mapping]]: [{{val| 1 2 1 -1 }}, {{val| 0 1 -2 4 }}]


[[Tp tuning|POL2 generator]]: ~5/4 = 386.558
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 386.558


{{Val list|legend=1| 3, …, 22, 25, 28, 31, 59 }}
{{Val list|legend=1| 3, …, 22, 25, 28, 31, 59 }}

Revision as of 18:26, 18 August 2022

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

For temperaments that omit various prime harmonics, see:

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.

2.9.5.7 subgroup

Commatose

Commatose is a fifthless 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 & 1789 temperament.

Subgroup: 2.9.5.7

Comma list: [28 -2 -19 8⟩, [9 -25 23 6⟩

Mapping: [⟨1 9 6 13], ⟨0 -298 -188 -521]]

Optimal tuning (POTE): ~531441/524288 = 23.4763

Template:Val list

2.9.5.7.11

Subgroup: 2.9.5.7.11

Comma list: [-7 7 -3 2 -4⟩, [17 0 -13 1 3⟩, [11 -2 -6 7 -3⟩

Mapping: ⟨1 9 6 13 16], ⟨0 -298 -188 -521 -641]

Optimal tuning (POTE): ~531441/524288 = 23.4766

2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

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

2.9.7.11 subgroup

Machine

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

Sval mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]

Gencom: [2 8/7; 64/63 99/98]

Gencom mapping: [⟨1 3/2 0 3 4], ⟨0 1/2 0 -1 -3]]

Optimal tuning (POTE): ~8/7 = 214.384

Template:Val list

RMS error: 1.977 cents

Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Sval mapping: [⟨1 5 5 2], ⟨0 -5 -6 4]]

Gencom: [2 9/7; 896/891 26411/26244]

Gencom mapping: [⟨1 5/2 0 5 2], ⟨0 -5/2 0 -6 4]]

Optimal tuning (POTE): ~9/7 = 438.465

Template:Val list

RMS error: 0.4262 cents

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Sval mapping: [⟨1 5 3 5], ⟨0 -19 -2 -16]]

Gencom: [2 77/72; 41503/41472 322102/321489]

Gencom mapping: [⟨1 5/2 0 3 5], ⟨0 -19/2 0 -2 -16]]

Optimal tuning (POTE): ~77/72 = 115.570

Template:Val list

RMS error: 0.0673 cents

2.9.15.7 subgroup

Stacks (aka 2magic)

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

Gencom: [2 9/7; 225/224 245/243]

Gencom mapping: [⟨1 5/2 5/2 5], ⟨0 -5/2 -1/2 -6]]

Optimal tuning (subgroup POTE): ~9/7 = 439.296

Template:Val list

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: [⟨1 0 2 -1 6], ⟨0 5 3 6 -4]]

Gencom: [2 9/7; 100/99 225/224 245/243]

Gencom mapping: [⟨1 5/2 5/2 5 2], ⟨0 -5/2 -1/2 -6 4]]

Optimal tuning (subgroup POTE): ~9/7 = 438.607

Vals: Template:Val list

RMS error: 1.226 cents

2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 3 6 -4 9]]

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

Gencom mapping: [⟨1 5/2 5/2 5 2 7], ⟨0 -5/2 -1/2 -6 4 -9]]

Optimal tuning (subgroup POTE): ~9/7 = 438.977

Optimal GPV sequence: Template:Val list

RMS error: 1.540 cents

2.9.21 subgroup

A-team

Subgroup: 2.9.21

Comma list: 1029/1024

Sval mapping: [⟨1 2 4], ⟨0 3 1]]

Gencom: [2 21/16; 1029/1024]

Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 0 -1/2]]

Optimal tuning (subgroup POTE): ~21/16 = 467.375

Template:Val list

RMS error: 0.3202 cents

2.9.5.21.11

Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: [⟨1 2 0 4 5], ⟨0 3 6 1 -4]]

Gencom: [2 21/16; 81/80 99/98 385/384]

Gencom mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]

Optimal tuning (subgroup POTE): ~21/16 = 463.956

Optimal GPV sequence: Template:Val list

Fractional subgroup temperaments

Historical

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 364/363, 441/440, 1001/1000

Sval mapping: [⟨1 2 0 1 2], ⟨0 -6 7 2 -9]]

Optimal tuning (subgroup POTE): ~21/20 = 83.016

Template:Val list

RMS error: 0.2562 cents

Hypnosis

Subgroup: 2.3.7.11/5.13

Comma list: 169/168, 540/539, 729/728

Sval mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]

Optimal tuning (subgroup POTE): ~13/9 = 633.518

Template:Val list

RMS error: 0.5379 cents

Related temperament: hypnos, tricot

Oceanfront

Subgroup: 2.3.7.13/5

Comma list: 64/63, 91/90

Sval mapping: [⟨1 0 6 -5], ⟨0 1 -2 4]]

Optimal tuning (subgroup POTE): ~3/2 = 713.910

Template:Val list

RMS error: 2.063 cents

Related temperament: superpyth, ultrapyth

Marveltri

Subgroup: 2.5.9/7

Comma list: 225/224

Sval mapping: [⟨1 2 1], ⟨0 1 -2]]

Optimal tuning (subgroup POTE): ~5/4 = 383.638

Template:Val list

RMS error: 0.4801 cents

Sulis

Subgroup: 2.5.9/7.11/9

Comma list: 99/98, 176/175

Sval mapping: [⟨1 2 1 -1], ⟨0 1 -2 4]]

Optimal tuning (subgroup POTE): ~5/4 = 386.558

Template:Val list

RMS error: 1.074 cents

Related temperament: minerva, würschmidt

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