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


POTE generator: ~14/11 = 407.892
Optimal tunings:
* CTE: ~2 = 1200.000, ~325/256 = 407.691
* POTE: ~2 = 1200.000, ~325/256 = 407.736


Vals: {{Val list| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 50, 53, 103 }}


Badness: 0.100884
Badness (Smith): 0.024352


=== 13-limit ===
=== Coditonic ===
Comma list: 105/104, 126/125, 245/242, 1287/1280
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 6655/6561
 
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}


Mapping: [{{val|1 6 3 -4 -3 2}}, {{val|0 -13 -2 20 19 5}}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~242/189 = 407.528
* POTE: ~2 = 1200.000, ~242/189 = 407.567


POTE generator: ~14/11 = 407.887
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}


Vals: {{Val list| 3, 47, 50, 53d }}
Badness (Smith): 0.063876


Badness: 0.054997
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


[[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 family]]
[[Category:Pages with mostly numerical content]]
[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Ditonmic]]
[[Category:Ditonmic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Ditonmic_family"