Ditonmic family: Difference between revisions
+description |
De-canonicalize septimal ditonic. Note where the true ditone is |
||
| Line 1: | Line 1: | ||
The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]] | 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 to the [[ditone]]. | 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 16: | Line 16: | ||
[[Badness]]: 0.167086 | [[Badness]]: 0.167086 | ||
== | == Diton == | ||
This extension is known as ''ditonic'' in [[Graham Breed]]'s temperament finder. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Revision as of 09:02, 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 tuning (POTE): ~2 = 1\1, ~15625/12288 = 407.574
Optimal ET sequence: 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc
Badness: 0.167086
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 tuning (POTE): ~2 = 1\1, ~2625/2048 = 407.954
Optimal ET sequence: 3, 47, 50
Badness: 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 tuning (POTE): ~2 = 1\1, ~14/11 = 407.892
Optimal ET sequence: 3, 47, 50, 53d
Badness: 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 tuning (POTE): ~2 = 1\1, ~14/11 = 407.887
Optimal ET sequence: 3, 47, 50, 53d
Badness: 0.054997
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 tuning (POTE): ~2 = 1\1, ~1225/972 = 407.690
Optimal ET sequence: 3d, 50, 53, 103, 156
Badness: 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 tuning (POTE): ~2 = 1\1, ~1225/972 = 407.741
Optimal ET sequence: 50, 53, 103
Badness: 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 tuning (POTE): ~2 = 1\1, ~325/256 = 407.736
Optimal ET sequence: 50, 53, 103
Badness: 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 tuning (POTE): ~2 = 1\1, ~242/189 = 407.567
Optimal ET sequence: 3de, 50e, 53
Badness: 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 tuning (POTE): ~2 = 1\1, ~33/26 = 407.541
Badness: 0.043989