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


[[Comma]]: 648/625
[[Comma list]]: 648/625


[[Mapping]]: [{{Val|4 0 3}}, {{Val|0 1 1}}]
{{Mapping|legend=1| 4 0 3 | 0 1 1 }}


[[POTE generator]]: ~3/2 = 699.507
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.507


{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}
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[[Comma list]]: 36/35, 50/49
[[Comma list]]: 36/35, 50/49


[[Mapping]]: [{{val|4 0 3 5}}, {{val|0 1 1 1}}]
{{Mapping|legend=1| 4 0 3 5 | 0 1 1 1 }}


{{Multival|legend=1|4 4 4 -3 -5 -2}}
{{Multival|legend=1|4 4 4 -3 -5 -2}}
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Comma list: 36/35, 50/49, 56/55
Comma list: 36/35, 50/49, 56/55


Mapping: [{{val| 4 0 3 5 14 }}, {{val| 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 tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
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Comma list: 36/35, 40/39, 50/49, 66/65
Comma list: 36/35, 40/39, 50/49, 66/65


Mapping: [{{val| 4 0 3 5 14 15 }}, {{val| 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 tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
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Comma list: 36/35, 45/44, 50/49
Comma list: 36/35, 45/44, 50/49


Mapping: [{{val| 4 0 3 5 -5 }}, {{val| 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 tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
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Comma list: 36/35, 50/49, 125/121
Comma list: 36/35, 50/49, 125/121


Mapping: [{{val| 4 1 4 6 6 }}, {{val| 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
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Comma list: 36/35, 50/49, 66/65, 125/121
Comma list: 36/35, 50/49, 66/65, 125/121


Mapping: [{{val| 4 1 4 6 6 7 }}, {{val| 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 tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
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== Hemidim ==
== Hemidim ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 648/625
[[Comma list]]: 49/48, 648/625


[[Mapping]]: [{{val|4 0 3 8}}, {{val|0 2 2 1}}]
{{Mapping|legend=1| 4 0 3 8 | 0 2 2 1 }}


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


[[POTE generator]]: ~7/6 = 252.555
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~7/6 = 252.555


{{Optimal ET sequence|legend=1| 4, 20c, 24, 52d, 76cdd }}
{{Optimal ET sequence|legend=1| 4, 20c, 24, 52d, 76cdd }}
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Comma list: 49/48, 77/75, 243/242
Comma list: 49/48, 77/75, 243/242


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


POTE generator: ~7/6 = 251.658
Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 251.658


{{Optimal ET sequence|legend=1| 4e, 20ce, 24, 76cdde }}
{{Optimal ET sequence|legend=1| 4e, 20ce, 24, 76cdde }}
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Comma list: 49/48, 66/65, 77/75, 243/242
Comma list: 49/48, 66/65, 77/75, 243/242


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


POTE generator: ~7/6 = 252.225
Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.225


{{Optimal ET sequence|legend=1| 4ef, 20cef, 24, 52de, 76cdde }}
{{Optimal ET sequence|legend=1| 4ef, 20cef, 24, 52de, 76cdde }}
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== Semidim ==
== Semidim ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 392/375
[[Comma list]]: 245/243, 392/375


[[Mapping]]: [{{val|8 0 6 -3}}, {{val|0 1 1 2}}]
{{Mapping|legend=1| 8 0 6 -3 | 0 1 1 2 }}


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


[[POTE generator]]: ~3/2 = 707.014
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\8, ~3/2 = 707.014


{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
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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}}]
Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}


POTE generator: ~3/2 = 706.645
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 706.645


{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
{{Optimal ET sequence|legend=1| 8d, 24, 32c, 56c }}
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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}}]
Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}


POTE generator: ~3/2 = 707.376
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 707.376


{{Optimal ET sequence|legend=1| 8d, 24, 32cf, 56cf }}
{{Optimal ET sequence|legend=1| 8d, 24, 32cf, 56cf }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dimipent family| ]] <!-- main article -->
[[Category:Dimipent family| ]] <!-- main article -->
[[Category:Diminished]]
[[Category:Diminished| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 11:15, 6 August 2023

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 list: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 699.507

Optimal ET sequence: 4, 8, 12

Badness: 0.047231

Diminished

Deutsch

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 ]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 699.523

Optimal ET sequence: 4, 8d, 12

Badness: 0.022401

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 56/55

Mapping: [⟨4 0 3 5 14], ⟨0 1 1 1 0]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109

Optimal ET sequence: 4, 8d, 12, 32cddee, 44cddeee

Badness: 0.022132

Scales: diminished12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 40/39, 50/49, 66/65

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 ET sequence: 4, 8d, 12f, 20cdef

Badness: 0.019509

Scales: diminished12

Demolished

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 50/49

Mapping: [⟨4 0 3 5 -5], ⟨0 1 1 1 3]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881

Optimal ET sequence: 12, 28, 40de

Badness: 0.026574

Cohedim

This temperament has been documented in Graham Breed's temperament finder as hemidim, the same name as 11-limit 4e & 24 and 13-limit 4ef & 24. For 11-limit 8bce & 12 temperament, cohedim arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 125/121

Mapping: [⟨4 1 4 6 6], ⟨0 2 2 2 3]]

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

Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679

Optimal ET sequence: 8bce, 12

Badness: 0.054965

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 50/49, 66/65, 125/121

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 ET sequence: 8bcef, 12f

Badness: 0.041707

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 ]]

Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.555

Optimal ET sequence: 4, 20c, 24, 52d, 76cdd

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]]

Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 251.658

Optimal ET sequence: 4e, 20ce, 24, 76cdde

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]]

Optimal tuning (POTE): ~6/5 = 1\4, ~7/6 = 252.225

Optimal ET sequence: 4ef, 20cef, 24, 52de, 76cdde

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 ]]

Optimal tuning (POTE): ~15/14 = 1\8, ~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]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 706.645

Optimal ET sequence: 8d, 24, 32c, 56c

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]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 707.376

Optimal ET sequence: 8d, 24, 32cf, 56cf

Badness: 0.030597

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