Subgroup temperaments: Difference between revisions

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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]].
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]].


=2.3.7=
= 2.3.5.11 subgroup =


==[[Semaphore_and_Godzilla|Semaphore]]==
== Porkypine ==
Comma: 49/48
Subgroup: 2.3.5.11


[[Lp_tuning|POL2 generator]]: ~15/13= 250.385
[[Comma list]]: 55/54, 100/99


[[Gencom]]: [2 8/7; 49/48]
[[Gencom]]: [2 10/9; 55/54, 100/99]


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


Map: [<1 2 3|, <0 -2 -1|]
[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -3 -5 -4}}]


EDOs: 5, 14, 19, 24, 67c, 91c
[[Tp tuning|POL2 generator]]: ~11/10 = 164.078


[[Tp_tuning#T2 tuning|RMS error]]: 2.523 cents
{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd ]}


==Archy==
[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
Archy (properly pronounced "arky", after the Greek theorist Archytas) 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
== Mohaha ==
Subgroup: 2.3.5.11


[[Lp_tuning|POL2 generator]]: ~3/2 = 709.321
[[Comma list]]: 81/80, 121/120


Gencom: [2 3/2; 64/63]
[[Gencom]]: [2 11/9; 81/80 121/120]


Gencom mapping: [<1 1 0 4|, <0 1 0 -2|]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]


Map: [<1 2 2|, <0 -1 2|]
[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]


EDOs: 5, 12, 17, 22, 27, 137bc
[[Tp tuning|POL2 generator]]: ~11/9 = 348.094


[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}


==[[Slendric]]==
[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
Comma: 1029/1024


[[Lp_tuning|POL2 generator]]: ~8/7 = 233.688
===Music===
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


Gencom: [2 8/7; 1029/1024]
== Larry ==
Subgroup: 2.3.5.11


Gencom mapping: [<1 1 0 3|, <0 3 0 -1|]
[[Comma list]]: 243/242, 4000/3993


Map: [<1 1 3|, <0 3 -1|]
Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]


EDOs: 5, 21, 26, 31, 36, 77, 113, 190
[[Gencom]]: [2 40/27; 243/242 4000/3993]


[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents
[[Gencom|Gencom mapping]]: [<1 5 12 0 12|, <0 -6 -17 0 -15|]


==Lee==
[[Mapping|Sval mapping]]: [<1 5 12 12|, <0 -6 -17 -15|]
Comma: 177147/175616


[[Lp_tuning|POL2 generator]]: ~13/9 = 633.525
[[Tp tuning|POL2 generator]]: ~40/27 = 683.166


Map: [<1 0 -3|, <0 3 11|]
{{Val list|legend=1| 7, 58, 65, 137, 202 }}


EDOs: 17, 36, 89, 125, 161, 358, 519b
[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents


[[Tp_tuning#T2 tuning|RMS error]]: 0.3519 cents
= 2.3.7 subgroup =


==Skwares==
==[[Semaphore and Godzilla|Semaphore]]==
Comma: 19683/19208
Subgroup: 2.3.7


[[Lp_tuning|POL2 generator]]: ~23/18 = 425.365 cents
[[Comma]]: 49/48


Map: [<1 3 6|, <0 -4 -9|]
[[Gencom]]: [2 8/7; 49/48]


EDOs: 14, 17, 31, 48, 79, 189b, 268bc, 347bc
[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]


[[Tp_tuning#T2 tuning|RMS error]]: 1.149 cents
[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]


=2.3.11=
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385


==Neutral==
{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira (further down on this page) and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Comma: 243/242
[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents


[[Lp_tuning|POL2 generator]]: ~11/9 = 350.525
== Archy ==
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.


Gencom: [2 11/9; 243/242]
Subgroup: 2.3.7


Gencom mapping: [<1 1 0 0 2|, <0 2 0 0 5|]
[[Comma]]: 64/63


Map: [<1 1 2|, <0 2 5|]
[[Gencom]]: [2 3/2; 64/63]


EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]


[[Tp_tuning#T2 tuning|RMS error]]: 0.3021 cents
[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]


[[neutraltet7|Seven note albitonic scale]]
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321


[[neutraltet10|Ten note chromatic scale]]
{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}


[[neutraltet17|Seventeen note mega chromatic scale]]
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents


=2.3.5.11=
==[[Slendric]]==
Subgroup: 2.3.7


==Porkypine==
[[Comma]]: 1029/1024
Commas: 55/54, 100/99


[[Lp_tuning|POL2 generator]]: ~11/10 = 164.078
[[Gencom]]: [2 8/7; 1029/1024]


Gencom: [2 10/9; 55/54, 100/99]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]


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


Map: [<1 2 3 4|, <0 -3 -5 -4|]
[[Tp tuning|POL2 generator]]: ~8/7 = 233.688


EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}


[[Tp_tuning#T2 tuning|RMS error]]: 2.287 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


==Mohaha==
== Lee ==
Commas: 81/80, 121/120
Subgroup: 2.3.7


[[Lp_tuning|POL2 generator]]: ~11/9 = 348.094
[[Comma]]: 177147/175616


Gencom: [2 11/9; 81/80 121/120]
[[Gencom]]: [2 81/56; 177147/175616]


Gencom mapping: [<1 1 0 0 2|, <0 2 8 0 5|]
[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]


Map: [<1 1 0 2|, <0 2 8 5|]
[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]


EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd
[[Tp tuning|POL2 generator]]: ~81/56 = 633.525


[[Tp_tuning#T2 tuning|RMS error]]: 1.392 cents
{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}


===Music===
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


==Larry==
== Skwares ==
Commas: 243/242, 4000/3993
Subgroup: 2.3.7


Related temperaments: [[Gravity_family#Gravity|gravity]], [[Gravity_family#Harry|harry]]
[[Comma]]: 19683/19208


POTE generator: ~40/27 = 683.166
[[Gencom]]: [2 9/7; 19683/19208]


Gencom: [2 40/27; 243/242 4000/3993]
[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]


Gencom mapping: [<1 5 12 0 12|, <0 -6 -17 0 -15|]
[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]


Map: [<1 5 12 12|, <0 -6 -17 -15|]
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365 cents


EDOs: 7, 58, 65, 137, 202
{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.3025 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents


= 2.3.7.11 =
= 2.3.7.11 subgroup =


== Ennea ==
== Ennea ==
Subgroup: 2.3.7.11
[[Comma list]]: 41503/41472, 43923/43904
[[Comma list]]: 41503/41472, 43923/43904
[[Lp_tuning|POL2 generator]]: ~99/98 = 17.6258


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
Line 162: Line 161:
[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]


[[Mapping|Sval map]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
 
[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258


{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.0383 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents


== Supra ==
== Supra ==
Subgroup: 2.3.7.11
[[Comma list]]: 64/63, 99/98
[[Comma list]]: 64/63, 99/98
[[POTE_tuning|POTE generator]]: ~3/2 = 707.192


[[Gencom]]: [2 3/2; 64/63 99/98]
[[Gencom]]: [2 3/2; 64/63 99/98]
Line 177: Line 178:
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]


[[Mapping|Sval map]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192


{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
{{Val list|legend=1| 5, 12, 17, 39c, 56c }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents


== Skwares ==
== Skwares ==
Subgroup: 2.3.7.11
[[Comma list]]: 99/98, 243/242
[[Comma list]]: 99/98, 243/242
[[POTE_tuning|POTE generator]]: ~23/18 = 425.244


[[Gencom]]: [2 9/7; 99/98 243/242]
[[Gencom]]: [2 9/7; 99/98 243/242]
Line 192: Line 195:
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]


[[Mapping|Sval map]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
 
[[Tp tuning|POL2 generator]]: ~23/18 = 425.244


{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.099 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents


== Hemif ==
== Hemif ==
Subgroup: 2.3.7.11
[[Comma list]]: 243/242, 896/891
[[Comma list]]: 243/242, 896/891
[[POTE generator]]: ~11/9 = 351.535


[[Gencom]]: [2 11/9; 243/242 896/891]
[[Gencom]]: [2 11/9; 243/242 896/891]
Line 207: Line 212:
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]


[[Mapping|Sval map]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535


{{Val list|legend=1| 7, 17, 41, 58, 99d }}
{{Val list|legend=1| 7, 17, 41, 58, 99d }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.6108 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents


== Bleu ==
== Bleu ==
Subgroup: 2.3.7.11
[[Comma list]]: 99/98, 864/847
[[Comma list]]: 99/98, 864/847
[[POTE generator]]: ~12/11 = 140.005


[[Gencom]]: [2 12/11; 99/98 864/847]
[[Gencom]]: [2 12/11; 99/98 864/847]
Line 222: Line 229:
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]


[[Mapping|Sval Map]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
 
[[Tp tuning|POL2 generator]]: ~12/11 = 140.005


{{Val list| 9, 17, 43, 60c }}
{{Val list| 9, 17, 43, 60c }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.829 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents


=2.3.7.11.13=
= 2.3.7.11.13 subgroup =


==Suhajira==
== Suhajira ==
Commas: 64/63, 78/77, 144/143
[[Comma list]]: 64/63, 78/77, 144/143


[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
[[Gencom]]: [2 11/9; 64/63 78/77 144/143]


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


[[Tp_tuning|POT2 generator]]: ~11/9 = 353.775
[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]


Map: [<1 1 4 2 4|, <0 2 -4 5 -1|]
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775


EDOs: 7, 10, 17, 44d, 61cd, 78cd
{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.953 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents


==Hemif==
== Hemif ==
Commas: 144/143, 243/242, 364/363
[[Comma list]]: 144/143, 243/242, 364/363


Related temperament: [[Breedsmic_temperaments|Hemififths]], namo
Related temperament: [[Breedsmic_temperaments|Hemififths]], namo


[[Tp_tuning|POT2 generator]]: ~11/9 = 351.691
[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
 
{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents


Gencom map: [<1 1 0 -1 2 4|, <0 2 0 13 5 -1|]
== Skwares ==
[[Comma list]]: 78/77, 99/98, 243/242


Map: [<1 1 -1 2 4|, <0 2 13 5 -1|]
[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]


EDOs: 7, 10, 17, 24, 41, 58
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}, {{val|0 -4 0 -9 -10 -15}}]


[[Tp_tuning#T2 tuning|RMS error]]: 0.7167 cents
[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}, {{val|0 -4 -9 -10 -15}}]


==Skwares==
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
Commas: 78/77, 99/98, 243/242


[[Tp_tuning|POT2 generator]]: ~23/18 = 424.457
{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}


Map: [<1 3 6 7 9|, <0 -4 -9 -10 -15|]
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents


EDOs: 17, 48e, 65de, 82c, 147ce
== Leapfrog ==
{{see also|Gentle region}}


[[Tp_tuning#T2 tuning|RMS error]]: 1.769 cents
[[Comma list]]: 169/168, 352/351, 364/363


==Leapfrog==
[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
:''See also: [[Gentle region]]''
Commas: 169/168, 352/351, 364/363


[[Lp_tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]


Gencom: [2 3/2; 169/169 352/351 364/363]
[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]


Gencom mapping: [<1 1 0 -6 -3 -1|, <0 1 0 15 11 8|]
[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents


Sval map: [<1 0 -21 -14 -9|, <0 1 15 11 8|]
{{Val list|legend=1| 17, 46, 63 }}


EDOs: 17, 46, 63
[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents


[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
===Music===
===Music===
*Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[User:IlL|IlL]] (in progress):
*Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[User:IlL|IlL]] (in progress):
**I. Prelude
**I. Prelude
Line 299: Line 313:
**[[:File:Locrian Suite Gigue.mp3|VII. Gigue]]
**[[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


==Bleu==
== Bleu ==
Group: 2.3.7.11.13
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 78/77, 99/98, 144/143
 
[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
 
[[Tp tuning|POL2 generator]]: ~13/12 = 139.990
 
{{Val list|legend=1| 17, 43, 60c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
 
== Parapyth (Rank 3) ==
[[Comma list]]: 352/351, 364/363
 
The gencom below gives [[Margo Schulter]]'s favored basis
 
[[Gencom]]: [2 3/2 28/27; 352/351 364/363]


Commas: 78/77, 99/98, 144/143
[[Gencom|Gencom mapping]]: [<1 1 0 1 4 6|, <0 1 0 3 -1 -4|, <0 0 0 1 1 1|]


Sval name: 9&17
[[Mapping|Sval mapping]]: [<1 0 0 7 12|, <0 1 0 -4 -7|, <0 0 1 1 1|]


[[Tp_tuning|POL2 generator]]: ~13/12 = 139.990
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}


Gencom map: [<1 1 0 2 3 3|, <0 5 0 7 4 6|]
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


Map: [<1 1 2 3 3|, <0 5 7 4 6|]
= 2.3.11 subgroup =


EDOs: [[17edo|17]], [[43edo|43]], [[60edo|60c]]
== Neutral ==
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


[[Tp_tuning#T2 tuning|RMS error]]: 1.752 cents
[[Comma]]: 243/242


==Parapyth (Rank 3)==
[[Gencom]]: [2 11/9; 243/242]
Commas: 352/351, 364/363


[[Lp_tuning|POL2 tuning]]: <1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
[[Gencom|Gencom mapping]]: [<1 1 0 0 2|, <0 2 0 0 5|]


The gencom below gives [[Margo Schulter]]'s favored basis
[[Mapping|Sval mapping]]: [<1 1 2|, <0 2 5|]


Gencom: [2 3/2 28/27; 352/351 364/363]
[[Lp tuning|POL2 generator]]: ~11/9 = 350.525


Gencom mapping: [<1 1 0 1 4 6|, <0 1 0 3 -1 -4|, <0 0 0 1 1 1|]
{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}


Sval map: [<1 0 0 7 12|, <0 1 0 -4 -7|, <0 0 1 1 1|]
[[Tp tuning#T2 tuning|RMS error]]: 0.3021 cents


EDOs: 17, 41, 46, 58, 87, 104
[[neutraltet7|Seven note albitonic scale]]


[[Tp_tuning#T2 tuning|RMS error]]: 0.3789 cents
[[neutraltet10|Ten note chromatic scale]]
 
[[neutraltet17|Seventeen note mega chromatic scale]]


=2.9.7.11=
= 2.9.7.11 subgroup =


==Machine==
==Machine==
Line 382: Line 419:
[[Tp_tuning#T2 tuning|RMS error]]: 0.4262 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.4262 cents


=2.9.15.7.11.13=
= 2.9.15.7.11.13 subgroup =


==Stacks (aka 2magic)==
==Stacks (aka 2magic)==
Line 429: Line 466:
[[Tp_tuning#T2 tuning|RMS error]]: 1.074 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.074 cents


=2.9.21=
= 2.9.21 subgroup =


==A-team==
==A-team==
Line 446: Line 483:
[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents


=2.3.7/5.11/5.13/5=
= 2.3.7/5.11/5.13/5 subgroup =


==Historical==
==Historical==
Line 465: Line 502:
[[Tp_tuning#T2 tuning|RMS error]]: 0.2562 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.2562 cents


=2.3.7.11/5.13=
= 2.3.7.11/5.13 subgroup =


==Hypnosis==
==Hypnosis==
Line 484: Line 521:
[[Tp_tuning#T2 tuning|RMS error]]: 0.5379 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.5379 cents


=2.5.7=
= 2.5.7 subgroup =


==Didacus==
==Didacus==

Revision as of 05:02, 15 May 2021

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

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.

2.3.5.11 subgroup

Porkypine

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Gencom: [2 10/9; 55/54, 100/99]

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

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

POL2 generator: ~11/10 = 164.078

{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd ]}

RMS error: 2.287 cents

Mohaha

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Gencom: [2 11/9; 81/80 121/120]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

POL2 generator: ~11/9 = 348.094

Template:Val list

RMS error: 1.392 cents

Music

Mohaha10ping2 by Billy Stiltner

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Related temperaments: gravity, harry

Gencom: [2 40/27; 243/242 4000/3993]

Gencom mapping: [<1 5 12 0 12|, <0 -6 -17 0 -15|]

Sval mapping: [<1 5 12 12|, <0 -6 -17 -15|]

POL2 generator: ~40/27 = 683.166

Template:Val list

RMS error: 0.3025 cents

2.3.7 subgroup

Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

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

Sval mapping: [⟨1 2 3], ⟨0 -2 -1]]

POL2 generator: ~7/6 = 250.385

Template:Val list

RMS error: 2.523 cents

Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

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

Sval mapping: [⟨1 2 2], ⟨0 -1 2]]

POL2 generator: ~3/2 = 709.321

Template:Val list

RMS error: 1.856 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

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

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

POL2 generator: ~8/7 = 233.688

Template:Val list

RMS error: 0.3202 cents

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]

Sval mapping: [⟨1 0 -3], ⟨0 3 11]]

POL2 generator: ~81/56 = 633.525

Template:Val list

RMS error: 0.3519 cents

Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]

Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]

POL2 generator: ~9/7 = 425.365 cents

Template:Val list

RMS error: 1.149 cents

2.3.7.11 subgroup

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]

Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

Template:Val list

RMS error: 0.0383 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

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

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

Template:Val list

RMS error: 1.977 cents

Skwares

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]

Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]

POL2 generator: ~23/18 = 425.244

Template:Val list

RMS error: 1.099 cents

Hemif

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

POL2 generator: ~11/9 = 351.535

Template:Val list

RMS error: 0.6108 cents

Bleu

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

POL2 generator: ~12/11 = 140.005

Template:Val list

RMS error: 1.829 cents

2.3.7.11.13 subgroup

Suhajira

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

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

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

POL2 generator: ~11/9 = 353.775

Template:Val list

RMS error: 1.953 cents

Hemif

Comma list: 144/143, 243/242, 364/363

Related temperament: Hemififths, namo

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

Template:Val list

RMS error: 0.7167 cents

Skwares

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [{{val|1 3 0 6 7 9}, ⟨0 -4 0 -9 -10 -15]]

Sval mapping: [{{val|1 3 6 7 9}, ⟨0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

Template:Val list

RMS error: 1.769 cents

Leapfrog

See also: Gentle region

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]

Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

Template:Val list

RMS error: 0.7541 cents

Music

Bleu

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 13/12; 78/77 99/98 144/143]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

POL2 generator: ~13/12 = 139.990

Template:Val list

RMS error: 1.752 cents

Parapyth (Rank 3)

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

Gencom mapping: [<1 1 0 1 4 6|, <0 1 0 3 -1 -4|, <0 0 0 1 1 1|]

Sval mapping: [<1 0 0 7 12|, <0 1 0 -4 -7|, <0 0 1 1 1|]

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

Template:Val list

RMS error: 0.3789 cents

2.3.11 subgroup

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Comma: 243/242

Gencom: [2 11/9; 243/242]

Gencom mapping: [<1 1 0 0 2|, <0 2 0 0 5|]

Sval mapping: [<1 1 2|, <0 2 5|]

POL2 generator: ~11/9 = 350.525

Template:Val list

RMS error: 0.3021 cents

Seven note albitonic scale

Ten note chromatic scale

Seventeen note mega chromatic scale

2.9.7.11 subgroup

Machine

Commas: 64/63, 99/98

POTE generator: ~17/15 = 214.384

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

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

Sval map: [<1 0 6 13|, <0 1 -1 -3|]

EDOs: 5, 6, 11, 17, 28

RMS error: 1.977 cents

Apparatus

Commas: 41503/41472, 322102/321489

POTE generator: ~31/29 = 115.570

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

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

Sval map: [<1 5 3 5|, <0 -19 -2 -16|]

EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623

RMS error: 0.0673 cents

Mechanism

Commas: 896/891, 26411/26244

POTE generator: ~9/7 = 438.465

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

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

Sval map: [<1 5 5 2|, <0 -5 -6 4|]

EDOs: 8, 11, 30, 41, 52

RMS error: 0.4262 cents

2.9.15.7.11.13 subgroup

Stacks (aka 2magic)

Commas: 100/99, 105/104, 144/143, 196/195

POL2 generator: ~9/7 = 438.977

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|]

Sval map: [<1 0 2 -1 6 -2|, <0 5 3 6 -4 9|]

EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef

RMS error: 1.540 cents

2.9.15.7.11 stacks

Commas: 100/99, 225/224, 245/243

POL2 generator: ~9/7 = 438.607

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|]

Sval map: [<1 0 2 -1 6|, <0 5 3 6 -4|]

EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,

RMS error: 1.226 cents

2.9.15.7 stacks

Commas: 225/224, 245/243

POL2 generator: ~9/7 = 439.296

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

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

Sval map: [<1 0 2 -1|, <0 5 3 6|]

EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c

RMS error: 1.074 cents

2.9.21 subgroup

A-team

Commas: 1029/1024

POL2 generator: ~17/13 = 467.375

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

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

Sval map: [<1 2 4|, <0 3 1|]

EDOs: 5, 13, 18, 41, 59, 77, 95

RMS error: 0.3202 cents

2.3.7/5.11/5.13/5 subgroup

Historical

Subgroup: 2.3.7/5.11/5.13/5

Commas: 364/363, 441/440, 1001/1000

POL2 generator: ~21/20 = 83.016

Gencom: [2 21/20; 364/363 441/440 1001/1000]

Gencom map: [<1 2 -3/4 -3/4 1/4 5/4|, <0 -6 0 7 2 -9|]

Sval map: [<1 2 0 1 2|, <0 -6 7 2 -9|]

EDOs: 14, 29, 72, 101, 130, 159

RMS error: 0.2562 cents

2.3.7.11/5.13 subgroup

Hypnosis

Subgroup: 2.3.7.11/5.13

Commas: 169/168, 540/539, 729/728

Related temperament: hypnos, tricot

Gencom: [2 13/9; 169/168, 540/539, 729/728]

Gencom map: [<1 0 -4 -3 4 0|, <0 3 13/2 11 -13/2 7|]

Sval map: [<1 0 -3 8 0|, <0 3 11 -13 7|]

EDOs: 17, 36, 118e, 125e, 161e, 197e

RMS error: 0.5379 cents

2.5.7 subgroup

Didacus

Subgroup: 2.5.7

Commas: 3136/3125

Related temperaments: roulette, hemithirds

Gencom: [2 28/25; 3136/3125]

Gencom map: [<1 0 2 2|, <0 0 2 5|]

Sval map: [<1 2 2|, <0 2 5|]

EDOs: 6, 19, 25, 31, 37, 99, 130, 161, 353

RMS error: 0.2138 cents

Llywelyn

Subgroup: 2.5.7

Commas: 4194304/4117715

Gencom: [2 8/7; 4194304/4117715]

Gencom map: [<1 0 1 3|, <0 0 7 -1|]

Sval map: [<1 1 3|, <0 7 -1|]

EDOs: 16, 37

RMS error: 0.5391 cents

Rainy

Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal.

Subgroup: 2.5.7

Commas: 2100875/2097152

Gencom: [2 256/245; 2100875/2097152]

Gencom map: [⟨1 0 2 3], ⟨0 0 5 -3]]

Sval map: [⟨1 2 3], ⟨0 5 -3]]

POTE generator: 77.2047 cents

EDOs: 31, 47, 62, 78, 93, 109, 140, 171, 202, 233

RMS error: 0.058596 cents

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