Ditonmic family: Difference between revisions

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{{Technical data page}}
The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]] ([[ratio]]: 1220703125/1207959552, {{monzo|legend=1| -27 -2 13 }}).
The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]] ([[ratio]]: 1220703125/1207959552, {{monzo|legend=1| -27 -2 13 }}).


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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 = 1\1, ~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]]: 0.167086
[[Badness]] (Smith): 0.167086


== Coditone ==
== Coditone ==
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{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}
{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}


{{Multival|legend=1| 13 2 30 -27 11 64 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~1225/972 = 407.642
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1225/972 = 407.690
: [[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 }}


[[Badness]]: 0.064356
[[Badness]] (Smith): 0.064356


=== 11-limit ===
=== 11-limit ===
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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 = 1\1, ~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: {{optimal ET sequence| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103 }}


Badness: 0.044329
Badness (Smith): 0.044329


==== 13-limit ====
==== 13-limit ====
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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 = 1\1, ~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: {{optimal ET sequence| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103 }}


Badness: 0.024352
Badness (Smith): 0.024352


=== Coditonic ===
=== Coditonic ===
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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 = 1\1, ~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: {{optimal ET sequence| 3de, 50e, 53 }}
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}


Badness: 0.063876
Badness (Smith): 0.063876


==== 13-limit ====
==== 13-limit ====
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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 = 1\1, ~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: {{optimal ET sequence| 3def, 53 }}
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


Badness: 0.043989
Badness (Smith): 0.043989


== Diton ==
== Diton ==
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{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}
{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}


{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 407.922
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2625/2048 = 407.954
: [[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 }}


[[Badness]]: 0.242101
[[Badness]] (Smith): 0.242101


=== 11-limit ===
=== 11-limit ===
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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 = 1\1, ~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: {{optimal ET sequence| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness: 0.100884
Badness (Smith): 0.100884


=== 13-limit ===
=== 13-limit ===
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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 = 1\1, ~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: {{optimal ET sequence| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness: 0.054997
Badness (Smith): 0.054997


== Notes ==
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Ditonmic]]
[[Category:Ditonmic]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:29, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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]]

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]]

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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