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Diminished family: Difference between revisions

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→Dimipent: Precision (DKW)
+CTE tunings. Review generators and optimal ET sequences. Move DKW tuning to the main article.
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== Dimipent ==
== Dimipent ==
{{Main| Diminished }}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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{{Mapping|legend=1| 4 0 3 | 0 1 1 }}
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.507
* [[CTE]]: ~6/5 = 1\4, ~3/2 = 696.9833 (~25/24 = 96.9833)
* [[DKW theory|DKW]]: ~6/5 = 1\4, ~3/2 = 690.289
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.507 (~25/24 = 99.507)


{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
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== Diminished ==
== Diminished ==
<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>
<div style="float: right">[[:de:Verminderte Temperaturen|Deutsch]]</div>
{{Main| Diminished }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Multival|legend=1|4 4 4 -3 -5 -2}}
{{Multival|legend=1|4 4 4 -3 -5 -2}}


[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523
[[Optimal tuning]]s:
* [[CTE]]: ~6/5 = 1\4, ~3/2 = 691.9545 (~21/20 = 91.9545)
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.523 (~21/20 = 99.523)


{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
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Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
Optimal tunings:
* CTE: ~6/5 = 1\4, ~3/2 = 691.9546 (~21/20 = 91.9546)
* POTE: ~6/5 = 1\4, ~3/2 = 709.109 (~15/14 = 109.109)


{{Optimal ET sequence|legend=1| 4, 8d, 12, 32cddee, 44cddeee }}
Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12, 32cddee, 44cddeee }}


Badness: 0.022132
Badness: 0.022132
Scales: [[diminished12]]


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}
Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
Optimal tunings:
* CTE: ~6/5 = 1\4, ~3/2 = 691.9544 (~21/20 = 91.9544)
* POTE: ~6/5 = 1\4, ~3/2 = 713.773 (~15/14 = 113.773)


{{Optimal ET sequence|legend=1| 4, 8d, 12f, 20cdef }}
Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12f, 20cdef }}


Badness: 0.019509
Badness: 0.019509
Scales: [[diminished12]]


=== Demolished ===
=== Demolished ===
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Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
Optimal tunings:
* CTE: ~6/5 = 1\4, ~3/2 = 687.7473 (~21/20 = 87.7473)
* POTE: ~6/5 = 1\4, ~3/2 = 689.881 (~21/20 = 89.881)


{{Optimal ET sequence|legend=1| 12, 28, 40de }}
Optimal ET sequence: {{Optimal ET sequence| 12, 28, 40de }}


Badness: 0.026574
Badness: 0.026574
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Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}


Mapping generators: ~6/5, ~11/7
: mapping generators: ~6/5, ~11/7


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679
Optimal tunings:
* CTE: ~6/5 = 1\4, ~11/7 = 793.4233 (~12/11 = 106.5767)
* POTE: ~6/5 = 1\4, ~11/7 = 798.4212 (~12/11 = 101.679)


{{Optimal ET sequence|legend=1| 8bce, 12 }}
Optimal ET sequence: {{Optimal ET sequence| 8bce, 12 }}


Badness: 0.054965
Badness: 0.054965
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Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
Optimal tunings:
* CTE: ~6/5 = 1\4, ~11/7 = 791.3751 (~12/11 = 108.6249)
* POTE: ~6/5 = 1\4, ~11/7 = 797.701 (~12/11 = 102.299)


{{Optimal ET sequence|legend=1| 8bcef, 12f }}
Optimal ET sequence: {{Optimal ET sequence| 8bcef, 12f }}


Badness: 0.041707
Badness: 0.041707
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{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}
: mapping generators: ~6/5, ~7/4


{{Multival|legend=1| 8 8 4 -6 -16 -13 }}
{{Multival|legend=1| 8 8 4 -6 -16 -13 }}


[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~7/6 = 252.555
[[Optimal tuning]]s:
* [[CTE]]: ~6/5 = 1\4, ~7/4 = 948.2575 (~36/35 = 48.2575)
* [[POTE]]: ~6/5 = 1\4, ~7/4 = 947.445 (~36/35 = 47.445)


{{Optimal ET sequence|legend=1| 4, 20c, 24, 52d, 76cdd }}
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}


[[Badness]]: 0.086378
[[Badness]]: 0.086378
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Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5}}
Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5}}


Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 251.658
Optimal tunings:
* CTE: ~6/5 = 1\4, ~7/4 = 949.8722 (~36/35 = 49.8722)
* POTE: ~6/5 = 1\4, ~7/4 = 948.342 (~36/35 = 48.342)


{{Optimal ET sequence|legend=1| 4e, 20ce, 24, 76cdde }}
Optimal ET sequence: {{Optimal ET sequence| 4e, 20ce, 24 }}


Badness: 0.056576
Badness: 0.056576
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Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}


Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.225
Optimal tunings:
* CTE: ~6/5 = 1\4, ~7/4 = 949.3664 (~36/35 = 49.3664)
* POTE: ~6/5 = 1\4, ~7/4 = 947.775 (~36/35 = 47.775)


{{Optimal ET sequence|legend=1| 4ef, 20cef, 24, 52de, 76cdde }}
Optimal ET sequence: {{Optimal ET sequence| 4ef, 24 }}


Badness: 0.039030
Badness: 0.039030
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{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}
: mapping generators: ~15/14, ~3


{{Multival|legend=1| 8 8 16 -6 3 15 }}
{{Multival|legend=1| 8 8 16 -6 3 15 }}


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\8, ~3/2 = 707.014
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 1\8, ~3/2 = 702.7651 (~36/35 = 47.2349)
* [[POTE]]: ~15/14 = 1\8, ~3/2 = 707.014 (~36/35 = 42.986)


{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
{{Optimal ET sequence|legend=1| 8d, 24, 32c }}


[[Badness]]: 0.107523
[[Badness]]: 0.107523
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Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 706.645
Optimal tunings:
* CTE: ~12/11 = 1\8, ~3/2 = 702.6622 (~36/35 = 47.3378)
* POTE: ~12/11 = 1\8, ~3/2 = 706.645 (~36/35 = 43.355)


{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32c }}


Badness: 0.047598
Badness: 0.047598
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Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 707.376
Optimal tunings:
* CTE: ~12/11 = 1\8, ~3/2 = 701.9520 (~36/35 = 48.0480)
* POTE: ~12/11 = 1\8, ~3/2 = 707.376 (~36/35 = 42.624)


{{Optimal ET sequence|legend=1| 8d, 24, 32cf, 56cf }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32cf }}


Badness: 0.030597
Badness: 0.030597
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