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Schismatic family: Difference between revisions

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== Bischismic ==
== Bischismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 32805/32768
[[Comma list]]: 3136/3125, 32805/32768
Line 1,020: Line 1,020:
{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}
{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}


[[POTE generator]]: ~3/2 = 701.592
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~3/2 = 701.592


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,037: Line 1,037:
Mapping: [{{val| 2 0 30 69 102 }}, {{val| 0 1 -8 -20 -30 }}]
Mapping: [{{val| 2 0 30 69 102 }}, {{val| 0 1 -8 -20 -30 }}]


POTE generator: ~3/2 = 701.612
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.612


Optimal GPV sequence: {{Val list| 12, 106de, 118, 130, 248 }}
Optimal GPV sequence: {{Val list| 12, 106de, 118, 130, 248 }}
Line 1,050: Line 1,050:
Mapping: [{{val| 2 0 30 69 102 -75 }}, {{val| 0 1 -8 -20 -30 26 }}]
Mapping: [{{val| 2 0 30 69 102 -75 }}, {{val| 0 1 -8 -20 -30 26 }}]


POTE generator: ~3/2 = 701.590
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.590


Optimal GPV sequence: {{Val list| 12, 106def, 118, 130, 248, 378 }}
Optimal GPV sequence: {{Val list| 12, 106def, 118, 130, 248, 378 }}
Line 1,063: Line 1,063:
Mapping: [{{val| 2 0 30 69 102 -75 5 }}, {{val| 0 1 -8 -20 -30 26 1 }}]
Mapping: [{{val| 2 0 30 69 102 -75 5 }}, {{val| 0 1 -8 -20 -30 26 1 }}]


POTE generator: ~3/2 = 701.600
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.600


Optimal GPV sequence: {{Val list| 12, 106def, 118, 130, 248g }}
Optimal GPV sequence: {{Val list| 12, 106def, 118, 130, 248g }}
Line 1,076: Line 1,076:
Mapping: [{{val| 2 0 30 69 102 131 }}, {{val| 0 1 -8 -20 -30 -39 }}]
Mapping: [{{val| 2 0 30 69 102 131 }}, {{val| 0 1 -8 -20 -30 -39 }}]


POTE generator: ~3/2 = 701.565
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.565


Optimal GPV sequence: {{Val list| 12f, 106deff, 118f, 130 }}
Optimal GPV sequence: {{Val list| 12f, 106deff, 118f, 130 }}
Line 1,089: Line 1,089:
Mapping: [{{val| 2 0 30 69 102 131 5 }}, {{val| 0 1 -8 -20 -30 -39 1 }}]
Mapping: [{{val| 2 0 30 69 102 131 5 }}, {{val| 0 1 -8 -20 -30 -39 1 }}]


POTE generator: ~3/2 = 701.595
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.595


Optimal GPV sequence: {{Val list| 12f, 106deff, 118f, 130, 248fg, 378fgg }}
Optimal GPV sequence: {{Val list| 12f, 106deff, 118f, 130, 248fg, 378fgg }}
Line 1,096: Line 1,096:


== Kleischismic ==
== Kleischismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 1500625/1492992
[[Comma list]]: 32805/32768, 1500625/1492992
Line 1,106: Line 1,106:
{{Multival|legend=1| 4 -32 38 -60 49 178 }}
{{Multival|legend=1| 4 -32 38 -60 49 178 }}


[[POTE generator]]: ~36/35 = 50.920
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~36/35 = 50.920


{{Val list|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
{{Val list|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
Line 1,119: Line 1,119:
Mapping: [{{val| 2 1 22 -15 8 }}, {{val| 0 2 -16 19 -1 }}]
Mapping: [{{val| 2 1 22 -15 8 }}, {{val| 0 2 -16 19 -1 }}]


POTE generator: ~36/35 = 50.918
Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.918


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212, 330e, 542de }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212, 330e, 542de }}
Line 1,132: Line 1,132:
Mapping: [{{val| 2 1 22 -15 8 15 }}, {{val| 0 2 -16 19 -1 -7 }}]
Mapping: [{{val| 2 1 22 -15 8 15 }}, {{val| 0 2 -16 19 -1 -7 }}]


POTE generator: ~36/35 = 50.938
Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.938


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212f }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212f }}
Line 1,145: Line 1,145:
Mapping: [{{val| 2 1 22 -15 8 15 6 }}, {{val| 0 2 -16 19 -1 -7 2 }}]
Mapping: [{{val| 2 1 22 -15 8 15 6 }}, {{val| 0 2 -16 19 -1 -7 2 }}]


POTE generator: ~36/35 = 50.942
Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.942


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212fg }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212fg }}
Line 1,158: Line 1,158:
Mapping: [{{val| 2 1 22 -15 8 -36 }}, {{val| 0 2 -16 19 -1 40 }}]
Mapping: [{{val| 2 1 22 -15 8 -36 }}, {{val| 0 2 -16 19 -1 40 }}]


POTE generator: ~36/35 = 50.951
Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.951


Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212 }}
Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212 }}
Line 1,171: Line 1,171:
Mapping: [{{val| 2 1 22 -15 8 -36 6 }}, {{val| 0 2 -16 19 -1 40 2 }}]
Mapping: [{{val| 2 1 22 -15 8 -36 6 }}, {{val| 0 2 -16 19 -1 40 2 }}]


POTE generator: ~36/35 = 50.948
Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.948


Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212g }}
Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212g }}
Line 1,178: Line 1,178:


== Squirrel ==
== Squirrel ==
The squirrel temperament (29&36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
The squirrel temperament (29 & 36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 32805/32768
[[Comma list]]: 686/675, 32805/32768
Line 1,188: Line 1,188:
{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}
{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}


[[POTE generator]]: ~160/147 = 166.140
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 166.140


{{Val list|legend=1| 29, 36, 65 }}
{{Val list|legend=1| 29, 36, 65 }}
Line 1,201: Line 1,201:
Mapping: [{{val| 1 2 -1 1 0 }}, {{val| 0 -3 24 13 25 }}]
Mapping: [{{val| 1 2 -1 1 0 }}, {{val| 0 -3 24 13 25 }}]


POTE generator: ~11/10 = 166.097
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097


Optimal GPV sequence: {{Val list| 29, 36, 65 }}
Optimal GPV sequence: {{Val list| 29, 36, 65 }}
Line 1,214: Line 1,214:
Mapping: [{{val| 1 2 -1 1 0 3 }}, {{val| 0 -3 24 13 25 5 }}]
Mapping: [{{val| 1 2 -1 1 0 3 }}, {{val| 0 -3 24 13 25 5 }}]


POTE generator: ~11/10 = 166.054
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054


Optimal GPV sequence: {{Val list| 29, 36, 65f, 94df, 159df }}
Optimal GPV sequence: {{Val list| 29, 36, 65f, 94df, 159df }}
Line 1,221: Line 1,221:


== Tertiaschis ==
== Tertiaschis ==
The ''tertiaschis'' temperament (94&159) has a ~11/10 generator, sharing the same 2.3.5.11 with [[#Squirrel]], but tempers out 1071785/1062882 for prime 7.  
The ''tertiaschis'' temperament (94&159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071785/1062882 for prime 7.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 1071875/1062882
[[Comma list]]: 32805/32768, 1071875/1062882
Line 1,231: Line 1,231:
{{Multival|legend=1| 3 -24 52 -45 74 188 }}
{{Multival|legend=1| 3 -24 52 -45 74 188 }}


[[POTE generator]]: ~192/175 = 166.019
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 166.019


{{Val list|legend=1| 65, 94, 159, 253, 412cd }}
{{Val list|legend=1| 65, 94, 159, 253, 412cd }}
Line 1,244: Line 1,244:
Mapping: [{{val| 1 2 -1 10 0}}, {{val| 0 -3 24 -52 25}}]
Mapping: [{{val| 1 2 -1 10 0}}, {{val| 0 -3 24 -52 25}}]


POTE generator: ~11/10 = 166.017
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017


Optimal GPV sequence: {{Val list| 65, 94, 159, 253, 412cd }}
Optimal GPV sequence: {{Val list| 65, 94, 159, 253, 412cd }}
Line 1,257: Line 1,257:
Mapping: [{{val| 1 2 -1 10 0 12}}, {{val| 0 -3 24 -52 25 -60}}]
Mapping: [{{val| 1 2 -1 10 0 12}}, {{val| 0 -3 24 -52 25 -60}}]


POTE generator: ~11/10 = 166.016
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016


Optimal GPV sequence: {{Val list| 65f, 94, 159, 253, 412cdf, 665ccdef }}
Optimal GPV sequence: {{Val list| 65f, 94, 159, 253, 412cdf, 665ccdef }}
Line 1,270: Line 1,270:
Mapping: [{{val| 1 2 -1 10 0 12 -2}}, {{val| 0 -3 24 -52 25 -60 44}}]
Mapping: [{{val| 1 2 -1 10 0 12 -2}}, {{val| 0 -3 24 -52 25 -60 44}}]


POTE generator: ~11/10 = 166.012
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012


Optimal GPV sequence: {{Val list| 65f, 94, 159, 253 }}
Optimal GPV sequence: {{Val list| 65f, 94, 159, 253 }}
Line 1,277: Line 1,277:


== Countertertiaschis ==
== Countertertiaschis ==
The ''countertertiaschis'' temperament (159&224) has a ~11/10 generator, sharing the same 2.3.5.11 with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.  
The ''countertertiaschis'' temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 244140625/243045684
[[Comma list]]: 32805/32768, 244140625/243045684
Line 1,285: Line 1,285:
[[Mapping]]: [{{val| 1 2 -1 -12 }}, {{val| 0 -3 24 107 }}]
[[Mapping]]: [{{val| 1 2 -1 -12 }}, {{val| 0 -3 24 107 }}]


[[POTE generator]]: ~625/567 = 166.0621
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/567 = 166.0621


{{Val list|legend=1| 65d, 159, 224, 383, 607 }}
{{Val list|legend=1| 65d, 159, 224, 383, 607 }}
Line 1,298: Line 1,298:
Mapping: [{{val| 1 2 -1 -12 0 }}, {{val| 0 -3 24 107 25 }}]
Mapping: [{{val| 1 2 -1 -12 0 }}, {{val| 0 -3 24 107 25 }}]


POTE generator: ~11/10 = 166.0628
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628


Optimal GPV sequence: {{Val list| 65d, 159, 224, 383, 607 }}
Optimal GPV sequence: {{Val list| 65d, 159, 224, 383, 607 }}
Line 1,311: Line 1,311:
Mapping: [{{val| 1 2 -1 -12 0 -10 }}, {{val| 0 -3 24 107 25 99 }}]
Mapping: [{{val| 1 2 -1 -12 0 -10 }}, {{val| 0 -3 24 107 25 99 }}]


POTE generator: ~11/10 = 166.0628
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628


Optimal GPV sequence: {{Val list| 65d, 159, 224, 383, 607 }}
Optimal GPV sequence: {{Val list| 65d, 159, 224, 383, 607 }}
Line 1,320: Line 1,320:
{{See also| Stearnsmic clan #Pogo }}
{{See also| Stearnsmic clan #Pogo }}


The pogo temperament (94&130) splits the period in two to address the difference between [[#Tertiaschis]] and [[#Countertertiaschis]]. The schismic tempering of the fifth is just about right for the stearnsma.  
The pogo temperament (94 & 130) splits the period in two to address the difference between [[#Tertiaschis]] and [[#Countertertiaschis]]. The schismic tempering of the fifth is just about right for tempering out the stearnsma.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 118098/117649
[[Comma list]]: 32805/32768, 118098/117649
Line 1,332: Line 1,332:
{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
{{Multival|legend=1| 6 -48 10 -90 -1 158 }}


[[POTE generator]]: ~9/7 = 433.901
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901


{{Val list|legend=1| 36, 94, 130, 224, 354 }}
{{Val list|legend=1| 36, 94, 130, 224, 354 }}
Line 1,345: Line 1,345:
Mapping: [{{val| 2 1 22 2 25 }}, {{val| 0 3 -24 5 -25 }}]
Mapping: [{{val| 2 1 22 2 25 }}, {{val| 0 3 -24 5 -25 }}]


Mapping generators: ~99/70, ~9/7
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
 
POTE generator: ~9/7 = 433.911


Optimal GPV sequence: {{Val list| 36, 94, 130, 224, 354, 578 }}
Optimal GPV sequence: {{Val list| 36, 94, 130, 224, 354, 578 }}
Line 1,360: Line 1,358:
Mapping: [{{val| 2 1 22 2 25 -2 }}, {{val| 0 3 -24 5 -25 13 }}]
Mapping: [{{val| 2 1 22 2 25 -2 }}, {{val| 0 3 -24 5 -25 13 }}]


Mapping generators: ~99/70, ~9/7
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
 
POTE generator: ~9/7 = 433.911


Optimal GPV sequence: {{Val list| 36, 94, 130, 224, 354, 578 }}
Optimal GPV sequence: {{Val list| 36, 94, 130, 224, 354, 578 }}
Line 1,369: Line 1,365:


== Term ==
== Term ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 250047/250000
[[Comma list]]: 32805/32768, 250047/250000
Line 1,379: Line 1,375:
{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}
{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}


[[POTE generator]]: ~3/2 = 701.742
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~3/2 = 701.742


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzos (unchanged intervals): 2, 6/5
* [[7-odd-limit]] [[eigenmonzo basis]]: 2.5/3
* [[9-odd-limit]] eigenmonzos (unchanged intervals): 2, 9/7
* [[9-odd-limit]] eigenmonzo basis: 2.9/7


{{Val list|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
{{Val list|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
Line 1,390: Line 1,386:


=== Terminal ===
=== Terminal ===
The ''terminal'' temperament (12&159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.  
The ''terminal'' temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,398: Line 1,394:
Mapping: [{{val| 3 0 45 94 134 }}, {{val| 0 1 -8 -18 -26 }}]
Mapping: [{{val| 3 0 45 94 134 }}, {{val| 0 1 -8 -18 -26 }}]


POTE generator: ~3/2 = 701.824
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824


Optimal GPV sequence: {{Val list| 12, 147de, 159, 330 }}
Optimal GPV sequence: {{Val list| 12, 147de, 159, 330 }}
Line 1,411: Line 1,407:
Mapping: [{{val| 3 0 45 94 134 168 }}, {{val| 0 1 -8 -18 -26 -33 }}]
Mapping: [{{val| 3 0 45 94 134 168 }}, {{val| 0 1 -8 -18 -26 -33 }}]


POTE generator: ~3/2 = 701.821
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821


Optimal GPV sequence: {{Val list| 12f, 147def, 159, 330 }}
Optimal GPV sequence: {{Val list| 12f, 147def, 159, 330 }}
Line 1,424: Line 1,420:
Mapping: [{{val| 3 0 45 94 134 168 -2 }}, {{val| 0 1 -8 -18 -26 -33 3 }}]
Mapping: [{{val| 3 0 45 94 134 168 -2 }}, {{val| 0 1 -8 -18 -26 -33 3 }}]


POTE generator: ~3/2 = 701.810
Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810


Optimal GPV sequence: {{Val list| 12f, 147def, 159, 171, 330 }}
Optimal GPV sequence: {{Val list| 12f, 147def, 159, 171, 330 }}
Line 1,437: Line 1,433:
Mapping: [{{val| 3 0 45 94 -137 }}, {{val| 0 1 -8 -18 31 }}]
Mapping: [{{val| 3 0 45 94 -137 }}, {{val| 0 1 -8 -18 31 }}]


POTE generator: ~3/2 = 701.685
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685


Optimal GPV sequence: {{Val list| 12e, 159e, 171, 183, 354, 537, 891de }}
Optimal GPV sequence: {{Val list| 12e, 159e, 171, 183, 354, 537, 891de }}
Line 1,450: Line 1,446:
Mapping: [{{val| 3 0 45 94 -137 -103 }}, {{val| 0 1 -8 -18 31 24 }}]
Mapping: [{{val| 3 0 45 94 -137 -103 }}, {{val| 0 1 -8 -18 31 24 }}]


POTE generator: ~3/2 = 701.689
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689


Optimal GPV sequence: {{Val list| 171, 183, 354, 891de, 1245dee, 1599ddee }}
Optimal GPV sequence: {{Val list| 171, 183, 354, 891de, 1245dee, 1599ddee }}
Line 1,463: Line 1,459:
Mapping: [{{val| 3 0 45 94 -137 -103 -2 }}, {{val| 0 1 -8 -18 31 24 3 }}]
Mapping: [{{val| 3 0 45 94 -137 -103 -2 }}, {{val| 0 1 -8 -18 31 24 3 }}]


POTE generator: ~3/2 = 701.688
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688


Optimal GPV sequence: {{Val list| 171, 183, 354, 891de, 1245dee, 1599ddee }}
Optimal GPV sequence: {{Val list| 171, 183, 354, 891de, 1245dee, 1599ddee }}
Line 1,470: Line 1,466:


=== Semiterm ===
=== Semiterm ===
The ''semiterm'' temperament (12&342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
The ''semiterm'' temperament (12 & 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,480: Line 1,476:
Mapping generators: ~55/49, ~3
Mapping generators: ~55/49, ~3


POTE generator: ~3/2 = 701.7460
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460


Optimal GPV sequence: {{Val list| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
Optimal GPV sequence: {{Val list| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
Line 1,493: Line 1,489:
Mapping: [{{val| 6 0 90 188 287 355 }}, {{val| 0 1 -8 -18 -28 -35 }}]
Mapping: [{{val| 6 0 90 188 287 355 }}, {{val| 0 1 -8 -18 -28 -35 }}]


POTE tuning: ~3/2 = 701.7256
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256


Optimal GPV sequence: {{Val list| 12f, 330eff, 342f, 696f }} *
Optimal GPV sequence: {{Val list| 12f, 330eff, 342f, 696f }} *
Line 1,510: Line 1,506:
Mapping generators: ~63/50, ~693/400
Mapping generators: ~63/50, ~693/400


POTE generator: ~12/11 = 150.872
Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)


Optimal GPV sequence: {{Val list| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
Optimal GPV sequence: {{Val list| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
Line 1,523: Line 1,519:
Mapping: [{{val| 3 0 45 94 8 42 }}, {{val| 0 2 -16 -36 1 -13 }}]
Mapping: [{{val| 3 0 45 94 8 42 }}, {{val| 0 2 -16 -36 1 -13 }}]


POTE generator: ~12/11 = 150.873
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)


Optimal GPV sequence: {{Val list| 24d, 159, 183, 342f }}
Optimal GPV sequence: {{Val list| 24d, 159, 183, 342f }}
Line 1,536: Line 1,532:
Mapping: [{{val| 3 0 45 94 8 42 -2 }}, {{val| 0 2 -16 -36 1 -13 6 }}]
Mapping: [{{val| 3 0 45 94 8 42 -2 }}, {{val| 0 2 -16 -36 1 -13 6 }}]


POTE generator: ~12/11 = 150.867
Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)


Optimal GPV sequence: {{Val list| 24d, 159, 183, 342f, 525f, 867ff }}
Optimal GPV sequence: {{Val list| 24d, 159, 183, 342f, 525f, 867ff }}
Line 1,551: Line 1,547:
Mapping generators: ~1536/1225, ~34300/19683
Mapping generators: ~1536/1225, ~34300/19683


[[CTE generator]]: ~34300/19683 = 950.9654
[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654


{{Val list|legend=1| 24, 135, 159, 612ccdd }}
{{Val list|legend=1| 24, 135, 159, 612ccdd }}
Line 1,564: Line 1,560:
Mapping: [{{val| 3 0 45 -32 8 }}, {{val| 0 2 -16 17 1 }}]
Mapping: [{{val| 3 0 45 -32 8 }}, {{val| 0 2 -16 17 1 }}]


POTE generator: ~121/70 = 950.9629
Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9629


Optimal GPV sequence: {{Val list| 24, 135, 159 }}
Optimal GPV sequence: {{Val list| 24, 135, 159 }}
Line 1,571: Line 1,567:


== Sesquiquartififths ==
== Sesquiquartififths ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 32805/32768
[[Comma list]]: 2401/2400, 32805/32768
Line 1,581: Line 1,577:
{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}
{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}


[[POTE generator]]: ~448/405 = 175.434
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 175.434


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,600: Line 1,596:
Mapping generators: ~2, ~256/231
Mapping generators: ~2, ~256/231


POTE generator: ~256/231 = 175.406
Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406


Optimal GPV sequence: {{Val list| 41, 89, 130, 301e, 431e }}
Optimal GPV sequence: {{Val list| 41, 89, 130, 301e, 431e }}
Line 1,613: Line 1,609:
Mapping: [{{val| 1 1 7 5 2 -2 }}, {{val| 0 4 -32 -15 10 39 }}]
Mapping: [{{val| 1 1 7 5 2 -2 }}, {{val| 0 4 -32 -15 10 39 }}]


POTE generator: ~72/65 = 175.409
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409


Optimal GPV sequence: {{Val list| 41, 89, 130, 301e, 431e }}
Optimal GPV sequence: {{Val list| 41, 89, 130, 301e, 431e }}
Line 1,626: Line 1,622:
Mapping: [{{val| 2 2 14 10 23 }}, {{val| 0 4 -32 -15 -55 }}]
Mapping: [{{val| 2 2 14 10 23 }}, {{val| 0 4 -32 -15 -55 }}]


POTE generator: ~448/405 = 175.435
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435


Optimal GPV sequence: {{Val list| 82e, 130, 212, 342, 1156, 1498, 1840d }}
Optimal GPV sequence: {{Val list| 82e, 130, 212, 342, 1156, 1498, 1840d }}
Line 1,633: Line 1,629:


== Quintilipyth ==
== Quintilipyth ==
The ''quintilipyth'' temperament (12&253, formerly ''[[40ed10 #Regular temperaments|quintilischis]]'' temperament) slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832, quinruyoyo) in the 7-limit.
The ''quintilipyth'' temperament (12 & 253, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 9765625/9680832
[[Comma list]]: 32805/32768, 9765625/9680832


[[Mapping]]: [{{val|1 2 -1 -4}}, {{val|0 -5 40 82}}]
[[Mapping]]: [{{val| 1 2 -1 -4 }}, {{val| 0 -5 40 82 }}]


{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}
{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}


[[POTE generator]]: ~625/588 = 99.625
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/588 = 99.625


{{Val list|legend=1| 12, 253, 265 }}
{{Val list|legend=1| 12, 253, 265 }}
Line 1,656: Line 1,652:
Mapping: [{{val|1 2 -1 -4 -7}}, {{val|0 -5 40 82 126}}]
Mapping: [{{val|1 2 -1 -4 -7}}, {{val|0 -5 40 82 126}}]


POTE generator: ~35/33 = 99.616
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616


Optimal GPV sequence: {{Val list| 12, 253, 265, 518c, 783cc }}
Optimal GPV sequence: {{Val list| 12, 253, 265, 518c, 783cc }}
Line 1,669: Line 1,665:
Mapping: [{{val|1 2 -1 -4 -7 -9}}, {{val|0 -5 40 82 126 153}}]
Mapping: [{{val|1 2 -1 -4 -7 -9}}, {{val|0 -5 40 82 126 153}}]


POTE generator: ~35/33 = 99.612
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612


Optimal GPV sequence: {{Val list| 12f, 253, 518c, 771cc }}
Optimal GPV sequence: {{Val list| 12f, 253, 518c, 771cc }}
Line 1,680: Line 1,676:
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619


Mapping: [{{val|1 2 -1 -4 -7 -9 5}}, {{val|0 -5 40 82 126 153 -11}}]
Mapping: [{{val| 1 2 -1 -4 -7 -9 5 }}, {{val| 0 -5 40 82 126 153 -11 }}]


POTE generator: ~18/17 = 99.612
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612


Optimal GPV sequence: {{Val list| 12f, 253, 518c, 771cc }}
Optimal GPV sequence: {{Val list| 12f, 253, 518c, 771cc }}
Line 1,693: Line 1,689:
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971


Mapping: [{{val|1 2 -1 -4 -7 -9 5 4}}, {{val|0 -5 40 82 126 153 -11 3}}]
Mapping: [{{val| 1 2 -1 -4 -7 -9 5 4 }}, {{val| 0 -5 40 82 126 153 -11 3 }}]


POTE generator: ~18/17 = 99.615
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615


Optimal GPV sequence: {{Val list| 12f, 253, 265, 518ch }}
Optimal GPV sequence: {{Val list| 12f, 253, 265, 518ch }}
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