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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]]
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