Ditonmic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m Cleanup
Tags: Mobile edit Mobile web edit
 
(8 intermediate revisions by 3 users not shown)
Line 1: Line 1:
The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]], 1220703125/1207959552 = {{monzo| -27 -2 13 }}.
{{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 ==
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.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 8: Line 11:
{{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


== Septimal ditonic ==
== Coditone ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 8751645/8388608
[[Comma list]]: 225/224, 2125764/2100875


{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}
{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}


{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~2625/2048 = 407.954
{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}


{{Optimal ET sequence|legend=1| 3, 47, 50 }}
[[Badness]] (Smith): 0.064356
 
[[Badness]]: 0.242101


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 245/242, 2079/2048
Comma list: 225/224, 385/384, 78408/78125


Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}
Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.892
Optimal tunings:
* CTE: ~2 = 1200.000, ~1225/972 = 407.688
* POTE: ~2 = 1200.000, ~1225/972 = 407.741


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


Badness: 0.100884
Badness (Smith): 0.044329


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 126/125, 245/242, 1287/1280
Comma list: 225/224, 351/350, 385/384, 847/845


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


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.887
Optimal tunings:
* CTE: ~2 = 1200.000, ~325/256 = 407.691
* POTE: ~2 = 1200.000, ~325/256 = 407.736


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


Badness: 0.054997
Badness (Smith): 0.024352


== Coditone ==
=== Coditonic ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 2125764/2100875
Comma list: 99/98, 176/175, 6655/6561


{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}
Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}


{{Multival|legend=1| 13 2 30 -27 11 64 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~242/189 = 407.528
* POTE: ~2 = 1200.000, ~242/189 = 407.567


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1225/972 = 407.690
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}


{{Optimal ET sequence|legend=1| 3d, 50, 53, 103, 156 }}
Badness (Smith): 0.063876


[[Badness]]: 0.064356
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 99/98, 176/175, 325/324, 847/845
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 78408/78125
Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}


Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}
Optimal tunings:  
* CTE: ~2 = 1200.000, ~242/189 = 407.514
* POTE: ~2 = 1200.000, ~33/26 = 407.541


Optimal tuning (POTE): ~2 = 1\1, ~1225/972 = 407.741
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


Optimal ET sequence: {{optimal ET sequence| 50, 53, 103 }}
Badness (Smith): 0.043989


Badness: 0.044329
== Diton ==
This extension is known as ''ditonic'' in [[Graham Breed]]'s temperament finder.  


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 385/384, 847/845
[[Comma list]]: 126/125, 8751645/8388608


Mapping: {{mapping| 1 6 3 13 -3 2 | 0 -13 -2 -30 19 5 }}
{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~325/256 = 407.736
[[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: {{optimal ET sequence| 50, 53, 103 }}
{{Optimal ET sequence|legend=1| 3, 47, 50 }}


Badness: 0.024352
[[Badness]] (Smith): 0.242101


=== Coditonic ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 6655/6561
Comma list: 126/125, 245/242, 2079/2048


Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}
Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}


Optimal tuning (POTE): ~2 = 1\1, ~242/189 = 407.567
Optimal tunings:
* CTE: ~2 = 1200.000, ~14/11 = 407.930
* POTE: ~2 = 1200.000, ~14/11 = 407.892


Optimal ET sequence: {{optimal ET sequence| 3de, 50e, 53 }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness: 0.063876
Badness (Smith): 0.100884


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 176/175, 325/324, 847/845
Comma list: 105/104, 126/125, 245/242, 1287/1280
 
Mapping: {{mapping| 1 6 3 -4 -3 2 | 0 -13 -2 20 19 5 }}


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


Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 407.541
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Optimal ET sequence: {{optimal ET sequence| 3def, 53 }}
Badness (Smith): 0.054997


Badness: 0.043989
== 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
Retrieved from "https://en.xen.wiki/index.php?title=Ditonmic_family&oldid=200842"