Diminished family: Difference between revisions

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The '''dimipent family''' tempers out the major diesis aka diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12EDO.  
The '''dimipent family''' tempers out the major diesis aka diminished comma, [[648/625]], the amount by which four [[6/5]] minor thirds exceed an [[octave]], and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.  


== Dimipent ==
== Dimipent ==
Line 7: Line 6:
[[Comma]]: 648/625
[[Comma]]: 648/625


[[POTE tuning|POTE generator]]: ~3/2 = 699.507
[[Mapping]]: [{{Val|4 0 3}}, {{Val|0 1 1}}]


[[Map]]: [<4 0 3|, <0 1 1|]
[[POTE generator]]: ~3/2 = 699.507


{{Vals|legend=1| 4, 8, 12 }}
{{Val list|legend=1| 4, 8, 12 }}


[[Badness]]: 0.047231
[[Badness]]: 0.047231
Line 22: Line 21:
[[Comma list]]: 36/35, 50/49
[[Comma list]]: 36/35, 50/49


[[POTE tuning|POTE generator]]: ~3/2 = 699.523
[[Mapping]]: [{{val|4 0 3 5}}, {{val|0 1 1 1}}]


[[Map]]: [<4 0 3 5|, <0 1 1 1|]
{{Multival|legend=1|4 4 4 -3 -5 -2}}


Wedgie: <<4 4 4 -3 -5 -2||
[[POTE generator]]: ~3/2 = 699.523


{{Vals|legend=1| 4, 8d, 12 }}
{{Val list|legend=1| 4, 8d, 12 }}


[[Badness]]: 0.022401
[[Badness]]: 0.022401
Line 37: Line 36:
[[Comma list]]: 49/48, 648/625
[[Comma list]]: 49/48, 648/625


[[POTE tuning|POTE generator]]: ~8/7 = 252.555
[[Mapping]]: [{{val|4 0 3 8}}, {{val|0 2 2 1}}]


[[Map]]: [<4 0 3 8|, <0 2 2 1|]
{{Multival|legend=1|8 8 4 -6 -16 -13}}


Wedgie: <<8 8 4 -6 -16 -13||
[[POTE generator]]: ~7/6 = 252.555


{{Vals|legend=1| 24, 52d, 76cd }}
{{Val list|legend=1| 4, 20c, 24, 52d, 76cdd }}


[[Badness]]: 0.0864
[[Badness]]: 0.086378


=== 11-limit ===
=== 11-limit ===
Line 52: Line 51:
Comma list: 49/48, 77/75, 243/242
Comma list: 49/48, 77/75, 243/242


POTE generator: ~8/7 = 251.658
Mapping: [{{val|4 0 3 8 -2}}, {{val|0 2 2 1 5}}]


Map: [<4 0 3 8 -2|, <0 2 2 1 5|]
POTE generator: ~7/6 = 251.658


{{Vals|legend=1| 24, 76cde }}
Vals: {{Val list| 4e, 20ce, 24, 76cdde }}


Badness: 0.0566
Badness: 0.056576


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 66/65, 77/75, 648/625
Comma list: 49/48, 66/65, 77/75, 243/242


POTE generator: ~8/7 = 252.225
Mapping: [{{val|4 0 3 8 -2 -1}}, {{val|0 2 2 1 5 5}}]


Map: [<4 0 3 8 -2 -1|, <0 2 2 1 5 5|]
POTE generator: ~7/6 = 252.225


{{Vals|legend=1| 24, 52de, 76cde }}
Vals: {{Val list| 4ef, 20cef, 24, 52de, 76cdde }}


Badness: 0.039
Badness: 0.039030


== Semidim ==
== Semidim ==
Line 78: Line 77:
[[Comma list]]: 245/243, 392/375
[[Comma list]]: 245/243, 392/375


[[POTE tuning|POTE generator]]: ~3/2 = 707.014
[[Mapping]]: [{{val|8 0 6 -3}}, {{val|0 1 1 2}}]


[[Map]]: [<8 0 6 -3|, <0 1 1 2|]
{{Multival|legend=1|8 8 16 -6 3 15}}


Wedgie: <<8 8 16 -6 3 15||
[[POTE generator]]: ~3/2 = 707.014


{{Vals|legend=1| 24, 32c, 56c }}
{{Vals|legend=1| 8d, 24, 32c, 56c }}


[[Badness]]: 0.1075
[[Badness]]: 0.107523


=== 11-limit ===
=== 11-limit ===
Line 92: Line 91:


Comma list: 56/55, 77/75, 245/243
Comma list: 56/55, 77/75, 245/243
Mapping: [{{val|8 0 6 -3 15}}, {{val|0 1 1 2 1}}]


POTE generator: ~3/2 = 706.645
POTE generator: ~3/2 = 706.645


Map: [<8 0 6 -3 15|, <0 1 1 2 1|]
Vals: {{Val list| 8d, 24, 32c, 56c }}


{{Vals|legend=1| 24, 32c, 56c }}
Badness: 0.047598
 
Badness: 0.0476


=== 13-limit ===
=== 13-limit ===
Line 105: Line 104:


Comma list: 56/55, 66/65, 77/75, 507/500
Comma list: 56/55, 66/65, 77/75, 507/500
Mapping: [{{val|8 0 6 -3 15 17}}, {{val|0 1 1 2 1 1}}]


POTE generator: ~3/2 = 707.376
POTE generator: ~3/2 = 707.376


Map: [<8 0 6 -3 15 17|, <0 1 1 2 1 1|]
Vals: {{Val list| 8d, 24, 32cf, 56cf }}


{{Vals|legend=1| 24, 32cf, 56cf }}
Badness: 0.030597
 
Badness: 0.0306


[[Category:Theory]]
[[Category:Theory]]
Line 119: Line 118:
[[Category:Diminished]]
[[Category:Diminished]]
[[Category:Rank-2]]
[[Category:Rank-2]]
[[Category:Todo:review]]

Revision as of 22:37, 7 June 2021

The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12EDO.

Dimipent

Subgroup: 2.3.5

Comma: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

POTE generator: ~3/2 = 699.507

Template:Val list

Badness: 0.047231

Diminished

See also: Jubilismic clan § Diminished

Subgroup: 2.3.5.7

Comma list: 36/35, 50/49

Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]

Wedgie: ⟨⟨ 4 4 4 -3 -5 -2 ]]

POTE generator: ~3/2 = 699.523

Template:Val list

Badness: 0.022401

Hemidim

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping: [⟨4 0 3 8], ⟨0 2 2 1]]

Wedgie: ⟨⟨ 8 8 4 -6 -16 -13 ]]

POTE generator: ~7/6 = 252.555

Template:Val list

Badness: 0.086378

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2], ⟨0 2 2 1 5]]

POTE generator: ~7/6 = 251.658

Vals: Template:Val list

Badness: 0.056576

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2 -1], ⟨0 2 2 1 5 5]]

POTE generator: ~7/6 = 252.225

Vals: Template:Val list

Badness: 0.039030

Semidim

Subgroup: 2.3.5.7

Comma list: 245/243, 392/375

Mapping: [⟨8 0 6 -3], ⟨0 1 1 2]]

Wedgie: ⟨⟨ 8 8 16 -6 3 15 ]]

POTE generator: ~3/2 = 707.014

Optimal ET sequence: 8d, 24, 32c, 56c

Badness: 0.107523

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 245/243

Mapping: [⟨8 0 6 -3 15], ⟨0 1 1 2 1]]

POTE generator: ~3/2 = 706.645

Vals: Template:Val list

Badness: 0.047598

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 507/500

Mapping: [⟨8 0 6 -3 15 17], ⟨0 1 1 2 1 1]]

POTE generator: ~3/2 = 707.376

Vals: Template:Val list

Badness: 0.030597

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