Subgroup temperaments: Difference between revisions

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== Suhajira ==
== Suhajira ==
Subgroup: 2.3.7.11.13
[[Comma list]]: 64/63, 78/77, 144/143
[[Comma list]]: 64/63, 78/77, 144/143


Line 255: Line 257:


== Hemif ==
== Hemif ==
Subgroup: 2.3.7.11.13
[[Comma list]]: 144/143, 243/242, 364/363
[[Comma list]]: 144/143, 243/242, 364/363


Line 272: Line 276:


== Skwares ==
== Skwares ==
Subgroup: 2.3.7.11.13
[[Comma list]]: 78/77, 99/98, 243/242
[[Comma list]]: 78/77, 99/98, 243/242


Line 288: Line 294:
== Leapfrog ==
== Leapfrog ==
{{see also|Gentle region}}
{{see also|Gentle region}}
Subgroup: 2.3.7.11.13


[[Comma list]]: 169/168, 352/351, 364/363
[[Comma list]]: 169/168, 352/351, 364/363
Line 331: Line 339:


== Parapyth (Rank 3) ==
== Parapyth (Rank 3) ==
Subgroup: 2.3.7.11.13
[[Comma list]]: 352/351, 364/363
[[Comma list]]: 352/351, 364/363


Line 337: Line 347:
[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
[[Gencom]]: [2 3/2 28/27; 352/351 364/363]


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


[[Mapping|Sval mapping]]: [<1 0 0 7 12|, <0 1 0 -4 -7|, <0 0 1 1 1|]
[[Mapping|Sval mapping]]: [{{val|1 0 0 7 12}}, {{val|0 1 0 -4 -7}}, {{val|0 0 1 1 1}}]


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
Line 351: Line 361:
== Neutral ==
== 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]].
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]].
Subgroup: 2.3.11


[[Comma]]: 243/242
[[Comma]]: 243/242
Line 356: Line 368:
[[Gencom]]: [2 11/9; 243/242]
[[Gencom]]: [2 11/9; 243/242]


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


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


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


{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}


[[Tp tuning#T2 tuning|RMS error]]: 0.3021 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents


[[neutraltet7|Seven note albitonic scale]]
[[neutraltet7|Seven note albitonic scale]]
Line 372: Line 384:
[[neutraltet17|Seventeen note mega chromatic scale]]
[[neutraltet17|Seventeen note mega chromatic scale]]


= 2.9.7.11 subgroup =
= 2.5.7 subgroup =
 
== Didacus ==
Subgroup: 2.5.7
 
[[Comma]]: 3136/3125
 
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
 
[[Gencom]]: [2 28/25; 3136/3125]
 
[[Gencom|Gencom map]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
 
[[Tp tuning|POL2 generator]]: ~28/25 = 93.772


==Machine==
{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Commas: 64/63, 99/98


[[POTE_tuning|POTE generator]]: ~17/15 = 214.384
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


Gencom: [2 8/7; 64/63 99/98]
==Llywelyn==
Subgroup: 2.5.7


Gencom mapping: [<1 3/2 0 3 4|, <0 1/2 0 -1 -3|]
[[Comma]]: 4194304/4117715


[[Smonzos_and_Svals|Sval map]]: [<1 0 6 13|, <0 1 -1 -3|]
[[Gencom]]: [2 8/7; 4194304/4117715]


EDOs: 5, 6, 11, 17, 28
[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]


==Apparatus==
[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
Commas: 41503/41472, 322102/321489


[[POTE_tuning|POTE generator]]: ~31/29 = 115.570
{{Val list|legend=1| 16, 37 }}


Gencom: [2 77/72; 41503/41472 322102/321489]
[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents


Gencom mapping: [<1 5/2 0 3 5|, <0 -19/2 0 -2 -16|]
==Rainy==
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Smonzos_and_Svals|Sval map]]: [<1 5 3 5|, <0 -19 -2 -16|]
Subgroup: 2.5.7


EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
[[Comma]]: [[2100875/2097152]]


[[Tp_tuning#T2 tuning|RMS error]]: 0.0673 cents
[[Gencom]]: [2 256/245; 2100875/2097152]


==Mechanism==
[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
Commas: 896/891, 26411/26244


[[POTE_tuning|POTE generator]]: ~9/7 = 438.465
[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]


Gencom: [2 9/7; 896/891 26411/26244]
[[Tp tuning|POL2 generator]]: ~256/245 = 77.205


Gencom mapping: [<1 5/2 0 5 2|, <0 -5/2 0 -6 4|]
{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


[[Smonzos_and_Svals|Sval map]]: [<1 5 5 2|, <0 -5 -6 4|]
[[Tp tuning #T2 tuning|RMS error]]: 0.058596 cents


EDOs: 8, 11, 30, 41, 52
= 2.9.7.11 subgroup =


[[Tp_tuning#T2 tuning|RMS error]]: 0.4262 cents
== Machine ==
Subgroup: 2.9.7.11


= 2.9.15.7.11.13 subgroup =
Commas: 64/63, 99/98


==Stacks (aka 2magic)==
[[Gencom]]: [2 8/7; 64/63 99/98]
Commas: 100/99, 105/104, 144/143, 196/195


[[Lp_tuning|POL2 generator]]: ~9/7 = 438.977
[[Gencom|Gencom mapping]]: [{{val|1 3/2 0 3 4}}, {{val|0 1/2 0 -1 -3}}]


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -1 -3}}]
 
[[Tp tuning|POL2 generator]]: ~8/7 = 214.384


Gencom mapping: [<1 5/2 5/2 5 2 7|, <0 -5/2 -1/2 -6 4 -9|]
{{Val list|legend=1| 5, 6, 11, 17, 28 }}


Sval map: [<1 0 2 -1 6 -2|, <0 5 3 6 -4 9|]
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents


EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
== Apparatus ==
Subgroup: 2.9.7.11


[[Tp_tuning#T2 tuning|RMS error]]: 1.540 cents
[[Comma list]]: 41503/41472, 322102/321489


===2.9.15.7.11 stacks===
[[Gencom]]: [2 77/72; 41503/41472 322102/321489]
Commas: 100/99, 225/224, 245/243


[[Lp_tuning|POL2 generator]]: ~9/7 = 438.607
[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 3 5}}, {{val|0 -19/2 0 -2 -16}}]


Gencom: [2 9/7; 100/99 225/224 245/243]
[[Mapping|Sval mapping]]: [{{val|1 5 3 5}}, {{val|0 -19 -2 -16}}]


Gencom mapping: [<1 5/2 5/2 5 2|, <0 -5/2 -1/2 -6 4|]
[[Tp tuning|POL2 generator]]: ~77/72 = 115.570


Sval map: [<1 0 2 -1 6|, <0 5 3 6 -4|]
{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}


EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents


[[Tp_tuning#T2 tuning|RMS error]]: 1.226 cents
== Mechanism ==
Subgroup: 2.9.7.11


===2.9.15.7 stacks===
[[Comma list]]: 896/891, 26411/26244
Commas: 225/224, 245/243


[[Lp_tuning|POL2 generator]]: ~9/7 = 439.296
[[Gencom]]: [2 9/7; 896/891 26411/26244]


Gencom: [2 9/7; 225/224 245/243]
[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 5 2}}, {{val|0 -5/2 0 -6 4}}]


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


Sval map: [<1 0 2 -1|, <0 5 3 6|]
[[Tp tuning|POL2 generator]]: ~9/7 = 438.465


EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
{{Val list|legend=1| 8, 11, 30, 41, 52 }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


= 2.9.21 subgroup =
= 2.9.15.7 subgroup =


==A-team==
== Stacks (aka 2magic) ==
Commas: 1029/1024
Subgroup: 2.9.15.7


[[Lp_tuning|POL2 generator]]: ~17/13 = 467.375
[[Comma list]]: 225/224, 245/243


Gencom: [2 21/16; 1029/1024]
[[Gencom]]: [2 9/7; 225/224 245/243]


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


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


EDOs: 5, 13, 18, 41, 59, 77, 95
[[Tp tuning|POL2 generator]]: ~9/7 = 439.296


[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents
{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}


= 2.3.7/5.11/5.13/5 subgroup =
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


==Historical==
=== 2.9.15.7.11 stacks ===
Subgroup: 2.3.7/5.11/5.13/5
Subgroup: 2.9.15.7.11


Commas: 364/363, 441/440, 1001/1000
Comma list: 100/99, 225/224, 245/243


[[Lp_tuning|POL2 generator]]: ~21/20 = 83.016
Gencom: [2 9/7; 100/99 225/224 245/243]


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
Gencom mapping: [{{val|1 5/2 5/2 5 2}}, {{val|0 -5/2 -1/2 -6 4}}]


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


Sval map: [<1 2 0 1 2|, <0 -6 7 2 -9|]
POL2 generator: ~9/7 = 438.607


EDOs: 14, 29, 72, 101, 130, 159
Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.2562 cents
RMS error: 1.226 cents


= 2.3.7.11/5.13 subgroup =
=== 2.9.15.7.11.13 stacks ===
Subgroup: 2.9.15.7.11.13


==Hypnosis==
Comma list: 100/99, 105/104, 144/143, 196/195
Subgroup: 2.3.7.11/5.13


Commas: 169/168, 540/539, 729/728
Gencom: [2 9/7; 100/99 105/104 144/143 196/195]


Related temperament: [[Swetismic_temperaments#Hypnos-13-limit|hypnos]], [[Hemifamity_temperaments#Tricot-13-limit|tricot]]
Gencom mapping: [{{val|1 5/2 5/2 5 2 7}}, {{val|0 -5/2 -1/2 -6 4 -9}}]


[[Gencom]]: [2 13/9; 169/168, 540/539, 729/728]
Sval map: [{{val|1 0 2 -1 6 -2}}, {{val|0 5 3 6 -4 9}}]


Gencom map: [<1 0 -4 -3 4 0|, <0 3 13/2 11 -13/2 7|]
POL2 generator: ~9/7 = 438.977


Sval map: [<1 0 -3 8 0|, <0 3 11 -13 7|]
Vals: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


EDOs: 17, 36, 118e, 125e, 161e, 197e
RMS error: 1.540 cents


[[Tp_tuning#T2 tuning|RMS error]]: 0.5379 cents
= 2.9.21 subgroup =


= 2.5.7 subgroup =
== A-team ==
Subgroup: 2.9.21


==Didacus==
[[Comma]]: 1029/1024
Subgroup: 2.5.7


Commas: 3136/3125
[[Gencom]]: [2 21/16; 1029/1024]


Related temperaments: [[Chromatic_pairs#Roulette|roulette]], [[Gamelismic clan#Hemithirds|hemithirds]]
[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3/2 0 -1/2}}]


[[Gencom]]: [2 28/25; 3136/3125]
[[Mapping|Sval mapping]]: [{{val|1 2 4}}, {{val|0 3 1}}]


Gencom map: [<1 0 2 2|, <0 0 2 5|]
[[Tp tuning|POL2 generator]]: ~17/13 = 467.375


Sval map: [<1 2 2|, <0 2 5|]
{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}


EDOs: 6, 19, 25, 31, 37, 99, 130, 161, 353
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


[[Tp_tuning#T2 tuning|RMS error]]: 0.2138 cents
= 2.3.7/5.11/5.13/5 subgroup =


==Llywelyn==
== Historical ==
Subgroup: 2.5.7
Subgroup: 2.3.7/5.11/5.13/5


Commas: 4194304/4117715
Commas: 364/363, 441/440, 1001/1000


[[Gencom]]: [2 8/7; 4194304/4117715]
[[Lp_tuning|POL2 generator]]: ~21/20 = 83.016


Gencom map: [<1 0 1 3|, <0 0 7 -1|]
[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]


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


EDOs: 16, 37
Sval map: [<1 2 0 1 2|, <0 -6 7 2 -9|]


[[Tp_tuning#T2 tuning|RMS error]]: 0.5391 cents
EDOs: 14, 29, 72, 101, 130, 159


==Rainy==
[[Tp_tuning#T2 tuning|RMS error]]: 0.2562 cents


Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
= 2.3.7.11/5.13 subgroup =


Subgroup: 2.5.7
== Hypnosis ==
Subgroup: 2.3.7.11/5.13


Commas: [[2100875/2097152]]
Commas: 169/168, 540/539, 729/728


[[Gencom]]: [2 256/245; 2100875/2097152]
Related temperament: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]


Gencom map: [{{Val| 1 0 2 3 }}, {{Val| 0 0 5 -3}}]
[[Gencom]]: [2 13/9; 169/168, 540/539, 729/728]


Sval map: [{{Val| 1 2 3 }}, {{Val| 0 5 -3}}]
Gencom map: [<1 0 -4 -3 4 0|, <0 3 13/2 11 -13/2 7|]


POTE generator: 77.2047 cents
Sval map: [<1 0 -3 8 0|, <0 3 11 -13 7|]


EDOs: 31, 47, 62, 78, 93, 109, 140, 171, 202, 233
EDOs: 17, 36, 118e, 125e, 161e, 197e


[[Tp_tuning#T2 tuning|RMS error]]: 0.058596 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.5379 cents


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 05:38, 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

Subgroup: 2.3.7.11.13

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

Subgroup: 2.3.7.11.13

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

Subgroup: 2.3.7.11.13

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

Subgroup: 2.3.7.11.13

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)

Subgroup: 2.3.7.11.13

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.

Subgroup: 2.3.11

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

2.5.7 subgroup

Didacus

Subgroup: 2.5.7

Comma: 3136/3125

Related temperaments: roulette, hemithirds

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

Gencom map: [⟨1 0 2 2], ⟨0 0 2 5]]

Sval mapping: [⟨1 2 2], ⟨0 2 5]]

POL2 generator: ~28/25 = 93.772

Template:Val list

RMS error: 0.2138 cents

Llywelyn

Subgroup: 2.5.7

Comma: 4194304/4117715

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

Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]

Sval mapping: [⟨1 1 3], ⟨0 7 -1]]

POL2 generator: ~8/7 = 226.910

Template:Val list

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

Comma: 2100875/2097152

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

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

Sval mapping: [⟨1 2 3], ⟨0 5 -3]]

POL2 generator: ~256/245 = 77.205

Template:Val list

RMS error: 0.058596 cents

2.9.7.11 subgroup

Machine

Subgroup: 2.9.7.11

Commas: 64/63, 99/98

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

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

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

POL2 generator: ~8/7 = 214.384

Template:Val list

RMS error: 1.977 cents

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

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

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

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

POL2 generator: ~77/72 = 115.570

Template:Val list

RMS error: 0.0673 cents

Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

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

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

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

POL2 generator: ~9/7 = 438.465

Template:Val list

RMS error: 0.4262 cents

2.9.15.7 subgroup

Stacks (aka 2magic)

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

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

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

Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

POL2 generator: ~9/7 = 439.296

Template:Val list

RMS error: 1.074 cents

2.9.15.7.11 stacks

Subgroup: 2.9.15.7.11

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

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

POL2 generator: ~9/7 = 438.607

Vals: Template:Val list

RMS error: 1.226 cents

2.9.15.7.11.13 stacks

Subgroup: 2.9.15.7.11.13

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

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]]

POL2 generator: ~9/7 = 438.977

Vals: Template:Val list

RMS error: 1.540 cents

2.9.21 subgroup

A-team

Subgroup: 2.9.21

Comma: 1029/1024

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

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

Sval mapping: [⟨1 2 4], ⟨0 3 1]]

POL2 generator: ~17/13 = 467.375

Template:Val list

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

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