Ditonmic family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
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This tempers out the ditonma, 1220703125/1207959552.
The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]], 1220703125/1207959552 = {{monzo| -27 -2 13 }}.


== Ditonic ==
== Ditonic ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: 1220703125/1207959552
[[Comma list]]: 1220703125/1207959552


[[Mapping]]: [{{val|1 6 3}}, {{val|0 -13 -2}}]
{{Mapping|legend=1| 1 6 3 | 0 -13 -2 }}


[[POTE generator]]: ~15625/12288 = 407.574
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15625/12288 = 407.574


{{Optimal ET sequence|legend=1| 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
{{Optimal ET sequence|legend=1| 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
Line 14: Line 14:
[[Badness]]: 0.167086
[[Badness]]: 0.167086


=== 7-limit ===
== Septimal ditonic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 8751645/8388608
[[Comma list]]: 126/125, 8751645/8388608


[[Mapping]]: [{{val|1 6 3 -4}}, {{val|0 -13 -2 20}}]
{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}


{{Multival|legend=1|13 2 -20 -27 -68 -52}}
{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}


[[POTE generator]]: ~2625/2048 = 407.954
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2625/2048 = 407.954


{{Optimal ET sequence|legend=1| 3, 47, 50 }}
{{Optimal ET sequence|legend=1| 3, 47, 50 }}
Line 34: Line 34:
Comma list: 126/125, 245/242, 2079/2048
Comma list: 126/125, 245/242, 2079/2048


Mapping: [{{val|1 6 3 -4 -3}}, {{val|0 -13 -2 20 19}}]
Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}


POTE generator: ~14/11 = 407.892
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.892


{{Optimal ET sequence|legend=1| 3, 47, 50, 53d }}
Optimal ET sequence: {{optimal ET sequence| 3, 47, 50, 53d }}


Badness: 0.100884
Badness: 0.100884
Line 47: Line 47:
Comma list: 105/104, 126/125, 245/242, 1287/1280
Comma list: 105/104, 126/125, 245/242, 1287/1280


Mapping: [{{val|1 6 3 -4 -3 2}}, {{val|0 -13 -2 20 19 5}}]
Mapping: {{mapping| 1 6 3 -4 -3 2 | 0 -13 -2 20 19 5 }}


POTE generator: ~14/11 = 407.887
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.887


{{Optimal ET sequence|legend=1| 3, 47, 50, 53d }}
Optimal ET sequence: {{optimal ET sequence| 3, 47, 50, 53d }}


Badness: 0.054997
Badness: 0.054997


== Coditone ==
== Coditone ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 2125764/2100875
[[Comma list]]: 225/224, 2125764/2100875


[[Mapping]]: [{{val|1 6 3 13}}, {{val|0 -13 -2 -30}}]
{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}


{{Multival|legend=1|13 2 30 -27 11 64}}
{{Multival|legend=1| 13 2 30 -27 11 64 }}


[[POTE generator]]: ~1225/972 = 407.690
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1225/972 = 407.690


{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}
{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}
Line 75: Line 75:
Comma list: 225/224, 385/384, 78408/78125
Comma list: 225/224, 385/384, 78408/78125


Mapping: [{{val|1 6 3 13 -3}}, {{val|0 -13 -2 -30 19}}]
Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}


POTE generator: ~1225/972 = 407.741
Optimal tuning (POTE): ~2 = 1\1, ~1225/972 = 407.741


{{Optimal ET sequence|legend=1| 50, 53, 103 }}
Optimal ET sequence: {{optimal ET sequence| 50, 53, 103 }}


Badness: 0.044329
Badness: 0.044329
Line 88: Line 88:
Comma list: 225/224, 351/350, 385/384, 847/845
Comma list: 225/224, 351/350, 385/384, 847/845


Mapping: [{{val|1 6 3 13 -3 2}}, {{val|0 -13 -2 -30 19 5}}]
Mapping: {{mapping| 1 6 3 13 -3 2 | 0 -13 -2 -30 19 5 }}


POTE generator: ~325/256 = 407.736
Optimal tuning (POTE): ~2 = 1\1, ~325/256 = 407.736


{{Optimal ET sequence|legend=1| 50, 53, 103 }}
Optimal ET sequence: {{optimal ET sequence| 50, 53, 103 }}


Badness: 0.024352
Badness: 0.024352
Line 101: Line 101:
Comma list: 99/98, 176/175, 6655/6561
Comma list: 99/98, 176/175, 6655/6561


Mapping: [{{val|1 6 3 13 15}}, {{val|0 -13 -2 -30 -34}}]
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}


POTE generator: ~242/189 = 407.567
Optimal tuning (POTE): ~2 = 1\1, ~242/189 = 407.567


{{Optimal ET sequence|legend=1| 3de, 50e, 53 }}
Optimal ET sequence: {{optimal ET sequence| 3de, 50e, 53 }}


Badness: 0.063876
Badness: 0.063876
Line 114: Line 114:
Comma list: 99/98, 176/175, 325/324, 847/845
Comma list: 99/98, 176/175, 325/324, 847/845


Mapping: [{{val|1 6 3 13 15 20}}, {{val|0 -13 -2 -30 -34 -48}}]
Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}


POTE generator: ~33/26 = 407.541
Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 407.541


{{Optimal ET sequence|legend=1| 3def, 53 }}
Optimal ET sequence: {{optimal ET sequence| 3def, 53 }}


Badness: 0.043989
Badness: 0.043989

Revision as of 13:14, 17 July 2024

The ditonmic family of temperaments tempers out the ditonma, 1220703125/1207959552 = [-27 -2 13⟩.

Ditonic

Subgroup: 2.3.5

Comma list: 1220703125/1207959552

Mapping: [⟨1 6 3], ⟨0 -13 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~15625/12288 = 407.574

Optimal ET sequence: 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc

Badness: 0.167086

Septimal ditonic

Subgroup: 2.3.5.7

Comma list: 126/125, 8751645/8388608

Mapping: [⟨1 6 3 -4], ⟨0 -13 -2 20]]

Wedgie: ⟨⟨ 13 2 -20 -27 -68 -52 ]]

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 407.954

Optimal ET sequence: 3, 47, 50

Badness: 0.242101

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 2079/2048

Mapping: [⟨1 6 3 -4 -3], ⟨0 -13 -2 20 19]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.892

Optimal ET sequence: 3, 47, 50, 53d

Badness: 0.100884

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 1287/1280

Mapping: [⟨1 6 3 -4 -3 2], ⟨0 -13 -2 20 19 5]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.887

Optimal ET sequence: 3, 47, 50, 53d

Badness: 0.054997

Coditone

Subgroup: 2.3.5.7

Comma list: 225/224, 2125764/2100875

Mapping: [⟨1 6 3 13], ⟨0 -13 -2 -30]]

Wedgie: ⟨⟨ 13 2 30 -27 11 64 ]]

Optimal tuning (POTE): ~2 = 1\1, ~1225/972 = 407.690

Optimal ET sequence: 3d, 50, 53, 103, 156

Badness: 0.064356

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 78408/78125

Mapping: [⟨1 6 3 13 -3], ⟨0 -13 -2 -30 19]]

Optimal tuning (POTE): ~2 = 1\1, ~1225/972 = 407.741

Optimal ET sequence: 50, 53, 103

Badness: 0.044329

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 385/384, 847/845

Mapping: [⟨1 6 3 13 -3 2], ⟨0 -13 -2 -30 19 5]]

Optimal tuning (POTE): ~2 = 1\1, ~325/256 = 407.736

Optimal ET sequence: 50, 53, 103

Badness: 0.024352

Coditonic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 6655/6561

Mapping: [⟨1 6 3 13 15], ⟨0 -13 -2 -30 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~242/189 = 407.567

Optimal ET sequence: 3de, 50e, 53

Badness: 0.063876

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 325/324, 847/845

Mapping: [⟨1 6 3 13 15 20], ⟨0 -13 -2 -30 -34 -48]]

Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 407.541

Optimal ET sequence: 3def, 53

Badness: 0.043989

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