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Schismatic family: Difference between revisions

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* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6
: [[Eigenmonzo basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
: Eigenmonzos (unchanged intervals): 2, 9/7
: Eigenmonzo basis: 2.7/3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


POTE generator: ~3/2 = 702.157
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: Eigenmonzos (unchanged intervals): 2, 9/7
: Eigenmonzo basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 1 0 15 25 -33 -28 }}, {{val| 0 1 -8 -14 23 20 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 }}, {{val| 0 1 -8 -14 23 20 }}]


POTE generator: ~3/2 = 702.113
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
: Eigenmonzos (unchanged intervals): 2, 14/13
: Eigenmonzo basis: 2.13/7


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 1 0 15 25 -33 -28 -7 }}, {{val| 0 1 -8 -14 23 20 7 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 -7 }}, {{val| 0 1 -8 -14 23 20 7 }}]


POTE generator: ~3/2 = 702.092
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092


Optimal GPV sequence: {{Val list| 41, 53, 94g }}
Optimal GPV sequence: {{Val list| 41, 53, 94g }}
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Mapping: [{{val| 1 0 15 25 -33 -28 -7 9 }}, {{val| 0 1 -8 -14 23 20 7 -3 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 -7 9 }}, {{val| 0 1 -8 -14 23 20 7 -3 }}]


POTE generator: ~3/2 = 702.079
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079


Optimal GPV sequence: {{Val list| 41, 53, 94g }}
Optimal GPV sequence: {{Val list| 41, 53, 94g }}
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Mapping: [{{val| 1 0 15 25 -33 -28 77 }}, {{val| 0 1 -8 -14 23 20 -46 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 77 }}, {{val| 0 1 -8 -14 23 20 -46 }}]


POTE generator: ~3/2 = 702.097
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097


Optimal GPV sequence: {{Val list| 41g, 53, 94, 241ce, 335cde }}
Optimal GPV sequence: {{Val list| 41g, 53, 94, 241ce, 335cde }}
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Mapping: [{{val| 1 0 15 25 -33 -28 77 9 }}, {{val| 0 1 -8 -14 23 20 -46 -3 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 77 9 }}, {{val| 0 1 -8 -14 23 20 -46 -3 }}]


POTE generator: ~3/2 = 702.098
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098


Optimal GPV sequence: {{Val list| 41g, 53, 94, 241ceh, 335cdehh }}
Optimal GPV sequence: {{Val list| 41g, 53, 94, 241ceh, 335cdehh }}
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Mapping: [{{val| 1 0 15 25 -33 -28 -72 }}, {{val| 0 1 -8 -14 23 20 48 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 -72 }}, {{val| 0 1 -8 -14 23 20 48 }}]


POTE generator: ~3/2 = 702.144
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144


Optimal GPV sequence: {{Val list| 41, 53g, 94 }}
Optimal GPV sequence: {{Val list| 41, 53g, 94 }}
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Mapping: [{{val| 1 0 15 25 -33 -28 -72 9 }}, {{val| 0 1 -8 -14 23 20 48 -3 }}]
Mapping: [{{val| 1 0 15 25 -33 -28 -72 9 }}, {{val| 0 1 -8 -14 23 20 48 -3 }}]


POTE generator: ~3/2 = 702.135
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135


Optimal GPV sequence: {{Val list| 41, 53g, 94 }}
Optimal GPV sequence: {{Val list| 41, 53g, 94 }}
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


POTE generator: ~3/2 = 702.321
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: Eigenmonzos (unchanged intervals): 2, 11/9
: Eigenmonzo basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 239: Line 239:
Mapping: [{{val| 1 0 15 25 32 37 }}, {{val| 0 1 -8 -14 -18 -21 }}]
Mapping: [{{val| 1 0 15 25 32 37 }}, {{val| 0 1 -8 -14 -18 -21 }}]


POTE generator: ~3/2 = 702.559
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: Eigenmonzos (unchanged intervals): 2, 13/9
: Eigenmonzo basis: 2.13/9


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 1 0 15 25 32 37 -7 }}, {{val| 0 1 -8 -14 -18 -21 7 }}]
Mapping: [{{val| 1 0 15 25 32 37 -7 }}, {{val| 0 1 -8 -14 -18 -21 7 }}]


POTE generator: ~3/2 = 702.312
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312


Optimal GPV sequence: {{Val list| 12f, 29, 41 }}
Optimal GPV sequence: {{Val list| 12f, 29, 41 }}
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Mapping: [{{val| 1 0 15 25 32 37 -7 9 }}, {{val| 0 1 -8 -14 -18 -21 7 -3 }}]
Mapping: [{{val| 1 0 15 25 32 37 -7 9 }}, {{val| 0 1 -8 -14 -18 -21 7 -3 }}]


POTE generator: ~3/2 = 702.357
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357


Optimal GPV sequence: {{Val list| 12f, 29, 41 }}
Optimal GPV sequence: {{Val list| 12f, 29, 41 }}
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Mapping: [{{val| 1 0 15 25 32 37 58 }}, {{val| 0 1 -8 -14 -18 -21 -34 }}]
Mapping: [{{val| 1 0 15 25 32 37 58 }}, {{val| 0 1 -8 -14 -18 -21 -34 }}]


POTE generator: ~3/2 = 702.725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725


Optimal GPV sequence: {{Val list| 12fg, 29g, 41, 70cd, 111cd }}
Optimal GPV sequence: {{Val list| 12fg, 29g, 41, 70cd, 111cd }}
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Mapping: [{{val| 1 0 15 25 32 37 58 9 }}, {{val| 0 1 -8 -14 -18 -21 -34 -3 }}]
Mapping: [{{val| 1 0 15 25 32 37 58 9 }}, {{val| 0 1 -8 -14 -18 -21 -34 -3 }}]


POTE generator: ~3/2 = 702.753
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753


Optimal GPV sequence: {{Val list| 12fg, 29g, 41, 70cd, 111cdh, 181ccddh }}
Optimal GPV sequence: {{Val list| 12fg, 29g, 41, 70cd, 111cdh, 181ccddh }}
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Mapping: [{{val| 1 0 15 25 32 37 12 }}, {{val| 0 1 -8 -14 -18 -21 -5 }}]
Mapping: [{{val| 1 0 15 25 32 37 12 }}, {{val| 0 1 -8 -14 -18 -21 -5 }}]


POTE generator: ~3/2 = 702.717
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717


Optimal GPV sequence: {{Val list| 12f, 29g, 41g, 70cdgg }}
Optimal GPV sequence: {{Val list| 12f, 29g, 41g, 70cdgg }}
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Mapping: [{{val| 1 0 15 25 32 37 12 9 }}, {{val| 0 1 -8 -14 -18 -21 -5 -3 }}]
Mapping: [{{val| 1 0 15 25 32 37 12 9 }}, {{val| 0 1 -8 -14 -18 -21 -5 -3 }}]


POTE generator: ~3/2 = 702.716
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716


Optimal GPV sequence: {{Val list| 12f, 29g, 41g, 70cdgg }}
Optimal GPV sequence: {{Val list| 12f, 29g, 41g, 70cdgg }}
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


POTE generator: ~3/2 = 701.725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: Eigenmonzos (unchanged intervals): 2, 11/9
: Eigenmonzo basis: 2.11/9


Optimal GPV sequence: {{Val list| 12, 41e, 53, 118d, 171de }}
Optimal GPV sequence: {{Val list| 12, 41e, 53, 118d, 171de }}
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Mapping: [{{val| 1 0 15 25 51 56 }}, {{val| 0 1 -8 -14 -30 -33 }}
Mapping: [{{val| 1 0 15 25 51 56 }}, {{val| 0 1 -8 -14 -30 -33 }}


POTE generator: ~3/2 = 701.747
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: Eigenmonzos (unchanged intervals): 2, 11/9
: Eigenmonzo basis: 2.11/9


Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 118d, 171de }}
Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 118d, 171de }}
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Mapping: [{{val| 1 0 15 25 51 56 -7 }}, {{val| 0 1 -8 -14 -30 -33 7 }}
Mapping: [{{val| 1 0 15 25 51 56 -7 }}, {{val| 0 1 -8 -14 -30 -33 7 }}


POTE generator: ~3/2 = 701.680
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680


Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 65d, 118dg }}
Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 65d, 118dg }}
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Mapping: [{{val| 1 0 15 25 51 56 -7 9 }}, {{val| 0 1 -8 -14 -30 -33 7 -3 }}
Mapping: [{{val| 1 0 15 25 51 56 -7 9 }}, {{val| 0 1 -8 -14 -30 -33 7 -3 }}


POTE generator: ~3/2 = 701.705
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705


Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 65d, 118dg }}
Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 65d, 118dg }}
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Mapping generators: ~2, ~110/63
Mapping generators: ~2, ~110/63


POTE generator: ~63/55 = 248.918
Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)


Optimal GPV sequence: {{Val list| 29, 53, 82e, 135e, 188ce }}
Optimal GPV sequence: {{Val list| 29, 53, 82e, 135e, 188ce }}
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Mapping generators: ~2, ~26/15
Mapping generators: ~2, ~26/15


POTE generator: ~15/13 = 248.918
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)


Optimal GPV sequence: {{Val list| 29, 53, 82e, 135ef, 188cef }}
Optimal GPV sequence: {{Val list| 29, 53, 82e, 135ef, 188cef }}
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=== Karadeniz ===
=== Karadeniz ===
{{see also| Turkish maqam music temperaments #Karadeniz temperament }}
{{See also| Turkish maqam music temperaments #Karadeniz temperament }}


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 434: Line 434:
Mapping: [{{val| 1 1 7 11 2 }}, {{val| 0 2 -16 -28 5 }}]
Mapping: [{{val| 1 1 7 11 2 }}, {{val| 0 2 -16 -28 5 }}]


POTE generator: ~11/9 = 350.994
Mapping generators: ~2, ~11/9
 
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994


Optimal GPV sequence: {{Val list| 41, 106, 147 }}
Optimal GPV sequence: {{Val list| 41, 106, 147 }}
Line 447: Line 449:
Mapping: [{{val| 1 1 7 11 2 -8 }}, {{val| 0 2 -16 -28 5 40 }}]
Mapping: [{{val| 1 1 7 11 2 -8 }}, {{val| 0 2 -16 -28 5 40 }}]


POTE generator: ~11/9 = 351.014
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014


Optimal GPV sequence: {{Val list| 41, 106, 147 }}
Optimal GPV sequence: {{Val list| 41, 106, 147 }}
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Mapping generators: ~2, ~11/10
Mapping generators: ~2, ~11/10


POTE generator: ~11/10 = 165.974
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974


Optimal GPV sequence: {{Val list| 29, 65d, 94, 441cde, 535cde, 629cde }}
Optimal GPV sequence: {{Val list| 29, 65d, 94, 441cde, 535cde, 629cde }}
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Mapping generators: ~2, ~11/10
Mapping generators: ~2, ~11/10


POTE generator: ~11/10 = 165.963
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963


Optimal GPV sequence: {{Val list| 29, 65d, 94 }}
Optimal GPV sequence: {{Val list| 29, 65d, 94 }}
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== Schism ==
== Schism ==
{{see also| Archytas clan #Schism }}
{{See also| Archytas clan #Schism }}


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 701.556
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
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Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


POTE generator ~3/2 = 702.136
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136


Optimal GPV sequence: {{Val list| 12, 29de, 41de }}
Optimal GPV sequence: {{Val list| 12, 29de, 41de }}
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== Pontiac ==
== Pontiac ==
{{main| Pontiac }}
{{Main| Pontiac }}


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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