Diminished family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 10: Line 10:
[[POTE generator]]: ~3/2 = 699.507
[[POTE generator]]: ~3/2 = 699.507


{{Val list|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}


[[Badness]]: 0.047231
[[Badness]]: 0.047231
Line 27: Line 27:
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523


{{Val list|legend=1| 4, 8d, 12 }}
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}


[[Badness]]: 0.022401
[[Badness]]: 0.022401
Line 40: Line 40:
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109


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


Badness: 0.022132
Badness: 0.022132
Line 55: Line 55:
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773


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


Badness: 0.019509
Badness: 0.019509
Line 70: Line 70:
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881


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


Badness: 0.026574
Badness: 0.026574
Line 87: Line 87:
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679


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


Badness: 0.054965
Badness: 0.054965
Line 100: Line 100:
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299


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


Badness: 0.041707
Badness: 0.041707
Line 115: Line 115:
[[POTE generator]]: ~7/6 = 252.555
[[POTE generator]]: ~7/6 = 252.555


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


[[Badness]]: 0.086378
[[Badness]]: 0.086378
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POTE generator: ~7/6 = 251.658
POTE generator: ~7/6 = 251.658


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


Badness: 0.056576
Badness: 0.056576
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POTE generator: ~7/6 = 252.225
POTE generator: ~7/6 = 252.225


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


Badness: 0.039030
Badness: 0.039030
Line 169: Line 169:
POTE generator: ~3/2 = 706.645
POTE generator: ~3/2 = 706.645


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


Badness: 0.047598
Badness: 0.047598
Line 182: Line 182:
POTE generator: ~3/2 = 707.376
POTE generator: ~3/2 = 707.376


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


Badness: 0.030597
Badness: 0.030597

Revision as of 16:07, 7 May 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: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

POTE generator: ~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 ]]

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~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 ]]

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

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]]

POTE generator: ~3/2 = 707.376

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

Badness: 0.030597

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