Ditonmic family: Difference between revisions

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m Spell 1\1 in decimal, note the badness metric being used, and misc. cleanup
+CTE tunings and error maps
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{{Mapping|legend=1| 1 6 3 | 0 -13 -2 }}
{{Mapping|legend=1| 1 6 3 | 0 -13 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~15625/12288 = 407.574
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~15625/12288 = 407.534
: [[error map]]: {{val| 0.000 +0.099 -1.382 }}
* [[POTE]]: ~2 = 1200.000, ~15625/12288 = 407.574
: error map: {{val| 0.000 -0.416 -1.462 }}


{{Optimal ET sequence|legend=1| 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
{{Optimal ET sequence|legend=1| 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}


[[Badness]] (Smith): 0.167086
[[Badness]] (Smith): 0.167086
Line 25: Line 29:
{{Multival|legend=1| 13 2 30 -27 11 64 }}
{{Multival|legend=1| 13 2 30 -27 11 64 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~1225/972 = 407.690
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~1225/972 = 407.642
: [[error map]]: {{val| 0.000 -1.298 -1.597 +1.921 }}
* [[POTE]]: ~2 = 1200.000, ~1225/972 = 407.690
: error map: {{val| 0.000 -1.921 -1.693 +0.483 }}


{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}
{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}
Line 38: Line 46:
Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}
Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}


Optimal tuning (POTE): ~2 = 1200.000, ~1225/972 = 407.741
Optimal tunings:
* CTE: ~2 = 1200.000, ~1225/972 = 407.688
* POTE: ~2 = 1200.000, ~1225/972 = 407.741


{{Optimal ET sequence|legend=0| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103 }}
Line 51: Line 61:
Mapping: {{mapping| 1 6 3 13 -3 2 | 0 -13 -2 -30 19 5 }}
Mapping: {{mapping| 1 6 3 13 -3 2 | 0 -13 -2 -30 19 5 }}


Optimal tuning (POTE): ~2 = 1200.000, ~325/256 = 407.736
Optimal tunings:
* CTE: ~2 = 1200.000, ~325/256 = 407.691
* POTE: ~2 = 1200.000, ~325/256 = 407.736


{{Optimal ET sequence|legend=0| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103 }}
Line 64: Line 76:
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}


Optimal tuning (POTE): ~2 = 1200.000, ~242/189 = 407.567
Optimal tunings:
* CTE: ~2 = 1200.000, ~242/189 = 407.528
* POTE: ~2 = 1200.000, ~242/189 = 407.567


{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}
Line 77: Line 91:
Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}
Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}


Optimal tuning (POTE): ~2 = 1200.000, ~33/26 = 407.541
Optimal tunings:
* CTE: ~2 = 1200.000, ~242/189 = 407.514
* POTE: ~2 = 1200.000, ~33/26 = 407.541


{{Optimal ET sequence|legend=0| 3def, 53 }}
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


Badness (Smith): 0.043989
Badness (Smith): 0.043989
Line 94: Line 110:
{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}
{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~2625/2048 = 407.954
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 407.922
: [[error map]]: {{val| 0.000 -4.939 -2.157 -10.388 }}
* [[POTE]]: ~2 = 1200.000, ~2625/2048 = 407.954
: error map: {{val| 0.000 -5.353 -2.221 -9.751 }}


{{Optimal ET sequence|legend=1| 3, 47, 50 }}
{{Optimal ET sequence|legend=1| 3, 47, 50 }}
Line 107: Line 127:
Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}
Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}


Optimal tuning (POTE): ~2 = 1200.000, ~14/11 = 407.892
Optimal tunings:
* CTE: ~2 = 1200.000, ~14/11 = 407.930
* POTE: ~2 = 1200.000, ~14/11 = 407.892


{{Optimal ET sequence|legend=0| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness (Smith): 0.100884
Badness (Smith): 0.100884
Line 120: Line 142:
Mapping: {{mapping| 1 6 3 -4 -3 2 | 0 -13 -2 20 19 5 }}
Mapping: {{mapping| 1 6 3 -4 -3 2 | 0 -13 -2 20 19 5 }}


Optimal tuning (POTE): ~2 = 1200.000, ~14/11 = 407.887
Optimal tunings:
* CTE: ~2 = 1200.000, ~14/11 = 407.933
* POTE: ~2 = 1200.000, ~14/11 = 407.887


{{Optimal ET sequence|legend=0| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness (Smith): 0.054997
Badness (Smith): 0.054997

Revision as of 09:18, 13 March 2025

The ditonmic family of temperaments tempers out the ditonma (ratio: 1220703125/1207959552, monzo: [-27 -2 13⟩).

Ditonic

Named by Petr Pařízek in 2011[1], ditonic can be described as the 50 & 53 temperament. It splits ~8/5 in two for a generator, which happens to be an interval very close in size to the ditone, ~81/64. Note that the ditone itself is 52 generator steps away.

Subgroup: 2.3.5

Comma list: 1220703125/1207959552

Mapping: [⟨1 6 3], ⟨0 -13 -2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15625/12288 = 407.534
error map: ⟨0.000 +0.099 -1.382]
  • POTE: ~2 = 1200.000, ~15625/12288 = 407.574
error map: ⟨0.000 -0.416 -1.462]

Optimal ET sequence: 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc

Badness (Smith): 0.167086

Coditone

Subgroup: 2.3.5.7

Comma list: 225/224, 2125764/2100875

Mapping: [⟨1 6 3 13], ⟨0 -13 -2 -30]]

Wedgie: ⟨⟨ 13 2 30 -27 11 64 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~1225/972 = 407.642
error map: ⟨0.000 -1.298 -1.597 +1.921]
  • POTE: ~2 = 1200.000, ~1225/972 = 407.690
error map: ⟨0.000 -1.921 -1.693 +0.483]

Optimal ET sequence: 3d, 50, 53, 103, 156

Badness (Smith): 0.064356

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 78408/78125

Mapping: [⟨1 6 3 13 -3], ⟨0 -13 -2 -30 19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~1225/972 = 407.688
  • POTE: ~2 = 1200.000, ~1225/972 = 407.741

Optimal ET sequence: 50, 53, 103

Badness (Smith): 0.044329

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 385/384, 847/845

Mapping: [⟨1 6 3 13 -3 2], ⟨0 -13 -2 -30 19 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~325/256 = 407.691
  • POTE: ~2 = 1200.000, ~325/256 = 407.736

Optimal ET sequence: 50, 53, 103

Badness (Smith): 0.024352

Coditonic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 6655/6561

Mapping: [⟨1 6 3 13 15], ⟨0 -13 -2 -30 -34]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~242/189 = 407.528
  • POTE: ~2 = 1200.000, ~242/189 = 407.567

Optimal ET sequence: 3de, 50e, 53

Badness (Smith): 0.063876

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 325/324, 847/845

Mapping: [⟨1 6 3 13 15 20], ⟨0 -13 -2 -30 -34 -48]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~242/189 = 407.514
  • POTE: ~2 = 1200.000, ~33/26 = 407.541

Optimal ET sequence: 3def, 50eff, 53

Badness (Smith): 0.043989

Diton

This extension is known as ditonic in Graham Breed's temperament finder.

Subgroup: 2.3.5.7

Comma list: 126/125, 8751645/8388608

Mapping: [⟨1 6 3 -4], ⟨0 -13 -2 20]]

Wedgie: ⟨⟨ 13 2 -20 -27 -68 -52 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~2625/2048 = 407.922
error map: ⟨0.000 -4.939 -2.157 -10.388]
  • POTE: ~2 = 1200.000, ~2625/2048 = 407.954
error map: ⟨0.000 -5.353 -2.221 -9.751]

Optimal ET sequence: 3, 47, 50

Badness (Smith): 0.242101

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 2079/2048

Mapping: [⟨1 6 3 -4 -3], ⟨0 -13 -2 20 19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 407.930
  • POTE: ~2 = 1200.000, ~14/11 = 407.892

Optimal ET sequence: 3, 47, 50

Badness (Smith): 0.100884

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 1287/1280

Mapping: [⟨1 6 3 -4 -3 2], ⟨0 -13 -2 20 19 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 407.933
  • POTE: ~2 = 1200.000, ~14/11 = 407.887

Optimal ET sequence: 3, 47, 50

Badness (Smith): 0.054997

Notes

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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