Ditonmic family: Difference between revisions

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This tempers out the ditonma, 1220703125/1207959552.
{{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 }}).


== Ditonic ==
== Ditonic ==
Subgroup: 2.3.5
Named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, 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.


[[Comma]]: 1220703125/1207959552
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val|1 6 3}}, {{val|0 -13 -2}}]
[[Comma list]]: 1220703125/1207959552


[[POTE generator]]: ~15625/12288 = 407.574
{{Mapping|legend=1| 1 6 3 | 0 -13 -2 }}


{{Val list|legend=1| 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
[[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 }}


[[Badness]]: 0.167086
{{Optimal ET sequence|legend=1| 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}


=== 7-limit ===
[[Badness]] (Smith): 0.167086
[[Comma list]]: 126/125, 8751645/8388608
 
== Coditone ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2125764/2100875


[[Mapping]]: [{{val|1 6 3 -4}}, {{val|0 -13 -2 20}}]
{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}


[[POTE generator]]: ~2625/2048 = 407.954
[[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 }}


{{Val list|legend=1| 3, 47, 50 }}
{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}


[[Badness]]: 0.242101
[[Badness]] (Smith): 0.064356


=== 11-limit ===
=== 11-limit ===
Comma list: 126/125, 245/242, 2079/2048
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 78408/78125
 
Mapping: {{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|legend=0| 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: {{mapping| 1 6 3 13 -3 2 | 0 -13 -2 -30 19 5 }}


Mapping: [{{val|1 6 3 -4 -3}}, {{val|0 -13 -2 20 19}}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~325/256 = 407.691
* POTE: ~2 = 1200.000, ~325/256 = 407.736


POTE generator: ~14/11 = 407.892
{{Optimal ET sequence|legend=0| 50, 53, 103 }}


Vals: {{Val list| 3, 47, 50, 53d }}
Badness (Smith): 0.024352


Badness: 0.100884
=== Coditonic ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 99/98, 176/175, 6655/6561
Comma list: 105/104, 126/125, 245/242, 1287/1280


Mapping: [{{val|1 6 3 -4 -3 2}}, {{val|0 -13 -2 20 19 5}}]
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}


POTE generator: ~14/11 = 407.887
Optimal tunings:
* CTE: ~2 = 1200.000, ~242/189 = 407.528
* POTE: ~2 = 1200.000, ~242/189 = 407.567


Vals: {{Val list| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}


Badness: 0.054997
Badness (Smith): 0.063876


== Coditone ==
==== 13-limit ====
<!-- Revised technical data for 11- and 13-limit by Xenllium on 10 June 2021. 11- and 13-limit coditone is for 50&53, not for 11-limit 3de&53 and 13-limit 3def&53. 11-limit 3de&53 and 13-limit 3def&53 is renamed "coditonic" by Xenllium on 10 June 2021. -->
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 225/224, 2125764/2100875
Comma list: 99/98, 176/175, 325/324, 847/845


[[Mapping]]: [{{val|1 6 3 13}}, {{val|0 -13 -2 -30}}]
Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}


[[POTE generator]]: ~1225/972 = 407.690
Optimal tunings:
* CTE: ~2 = 1200.000, ~242/189 = 407.514
* POTE: ~2 = 1200.000, ~33/26 = 407.541


{{Val list|legend=1| 3d, 50, 53, 103, 156 }}
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


[[Badness]]: 0.064356
Badness (Smith): 0.043989


=== 11-limit ===
== Diton ==
Comma list: 225/224, 385/384, 78408/78125
This extension is known as ''ditonic'' in [[Graham Breed]]'s temperament finder.


Mapping: [{{val|1 6 3 13 -3}}, {{val|0 -13 -2 -30 19}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~1225/972 = 407.741
[[Comma list]]: 126/125, 8751645/8388608


Vals: {{Val list| 50, 53, 103 }}
{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}


Badness: 0.044329
[[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 }}


==== 13-limit ====
{{Optimal ET sequence|legend=1| 3, 47, 50 }}
Comma list: 225/224, 351/350, 385/384, 847/845


Mapping: [{{val|1 6 3 13 -3 2}}, {{val|0 -13 -2 -30 19 5}}]
[[Badness]] (Smith): 0.242101


POTE generator: ~325/256 = 407.736
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 50, 53, 103 }}
Comma list: 126/125, 245/242, 2079/2048


Badness: 0.024352
Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}


=== Coditonic ===
Optimal tunings:
Comma list: 99/98, 176/175, 6655/6561
* CTE: ~2 = 1200.000, ~14/11 = 407.930
* POTE: ~2 = 1200.000, ~14/11 = 407.892


Mapping: [{{val|1 6 3 13 15}}, {{val|0 -13 -2 -30 -34}}]
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


POTE generator: ~242/189 = 407.567
Badness (Smith): 0.100884


Vals: {{Val list| 3de, 50e, 53 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.063876
Comma list: 105/104, 126/125, 245/242, 1287/1280


==== 13-limit ====
Mapping: {{mapping| 1 6 3 -4 -3 2 | 0 -13 -2 20 19 5 }}
Comma list: 99/98, 176/175, 325/324, 847/845


Mapping: [{{val|1 6 3 13 15 20}}, {{val|0 -13 -2 -30 -34 -48}}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~14/11 = 407.933
* POTE: ~2 = 1200.000, ~14/11 = 407.887


POTE generator: ~33/26 = 407.541
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Vals: {{Val list| 3def, 53 }}
Badness (Smith): 0.054997


Badness: 0.043989
== Notes ==


[[Category:Theory]]
[[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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