Smate family: Difference between revisions

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


[[Comma list]]: 2048/1875
[[Comma list]]: 2048/1875


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


[[POTE generator]]: ~5/4 = 420.855
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 420.855


{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}
{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }}
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== Septimal smate ==
== Septimal smate ==
{{see also| Mint temperaments #Smate }}
{{See also| Mint temperaments #Smate }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2048/1875
[[Comma list]]: 36/35, 2048/1875


[[Mapping]]: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}


{{Multival|legend=1| 4 -1 9 -11 3 24 }}
{{Multival|legend=1| 4 -1 9 -11 3 24 }}


[[POTE generator]]: ~5/4 = 422.275
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 422.275


{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37cd }}
{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }}


[[Badness]]: 0.077871
[[Badness]]: 0.077871
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Comma list: 36/35, 56/55, 243/242
Comma list: 36/35, 56/55, 243/242


Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}


POTE generator: ~5/4 = 422.217
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217


{{Optimal ET sequence|legend=1| 14, 17c, 37cde }}
{{Optimal ET sequence|legend=1| 3de, 14, 17c, 37ccddee }}


Badness: 0.042518
Badness: 0.042518
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Comma list: 26/25, 36/35, 56/55, 243/242  
Comma list: 26/25, 36/35, 56/55, 243/242  


Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }}


POTE generator: ~5/4 = 423.020
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020


{{Optimal ET sequence|legend=1| 14, 17c }}
{{Optimal ET sequence|legend=1| 3de, 14, 17c }}


Badness: 0.036836
Badness: 0.036836


== Hemismate ==
== Hemismate ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 256/245, 392/375  
[[Comma list]]: 256/245, 392/375  


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


{{Multival|legend=1| 8 -2 1 -22 -21 8 }}
{{Multival|legend=1| 8 -2 1 -22 -21 8 }}


[[POTE generator]]: ~8/7 = 210.452
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 210.452


{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcd }}
{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcd }}
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Comma list: 56/55, 77/75, 256/245
Comma list: 56/55, 77/75, 256/245


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


POTE generator: ~8/7 = 210.481
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.481


{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcde }}
{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcde }}
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Comma list: 26/25, 56/55, 77/75, 256/245
Comma list: 26/25, 56/55, 77/75, 256/245


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


POTE generators: ~8/7 = 210.974
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.974


{{Optimal ET sequence|legend=1| 6, 11, 17c }}
{{Optimal ET sequence|legend=1| 6, 11, 17c }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Smate]]
[[Category:Smate family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 10:48, 6 August 2023

The smate family of temperaments tempers out 2048/1875, the smate comma, resulting in equation of four just major thirds (5/4) with the just perfect eleventh (8/3). It therefore requires an extremely sharp tuning of the just major third. 17edo and 20edo provide it and make for good tunings.

Smate

Subgroup: 2.3.5

Comma list: 2048/1875

Mapping: [⟨1 3 2], ⟨0 -4 1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 420.855

Optimal ET sequence: 3, 11, 14, 17c, 20c, 37c

Badness: 0.178624

Septimal smate

See also: Mint temperaments § Smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

Wedgie: ⟨⟨ 4 -1 9 -11 3 24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.275

Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd

Badness: 0.077871

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 243/242

Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217

Optimal ET sequence: 3de, 14, 17c, 37ccddee

Badness: 0.042518

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 243/242

Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020

Optimal ET sequence: 3de, 14, 17c

Badness: 0.036836

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]

Wedgie: ⟨⟨ 8 -2 1 -22 -21 8 ]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.452

Optimal ET sequence: 6, 11, 17c, 40bcd

Badness: 0.154301

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 256/245

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.481

Optimal ET sequence: 6, 11, 17c, 40bcde

Badness: 0.065528

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 77/75, 256/245

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.974

Optimal ET sequence: 6, 11, 17c

Badness: 0.050472

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