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Ditonmic family: Difference between revisions

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== 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.  
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 splits [[~]][[5/2]] in two for a generator, which happens to be an interval very close in size to the [[octave complement]] of the ditone (i.e. the Pythagorean major third, [[81/64]]). It can be described as 3 & 53, and is part of the [[schismic–Mercator equivalence continuum]] with equivalence number ''n'' = 13/2. Its [[ploidacot]] is eta-13-cot. Note that the ditone itself is 52 generator steps away.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 9: Line 9:
[[Comma list]]: 1220703125/1207959552
[[Comma list]]: 1220703125/1207959552


{{Mapping|legend=1| 1 6 3 | 0 -13 -2 }}
{{Mapping|legend=1| 1 -7 1 | 0 13 2 }}
: mapping generators: ~2, ~15625/12288
: mapping generators: ~2, ~24576/15625


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~15625/12288 = 407.534{{c}}
* [[WE]]: ~2 = 1200.2971{{c}}, ~24576/15625 = 792.6223{{c}}
: [[error map]]: {{val| 0.000 +0.099 -1.382 }}
: [[error map]]: {{val| +0.297 +0.055 -0.772 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~15625/12288 = 407.574{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~24576/15625 = 792.4403{{c}}
: error map: {{val| 0.000 -0.416 -1.462 }}
: error map: {{val| 0.000 -0.231 -1.433 }}


{{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]] (Smith): 0.167086
[[Badness]] (Sintel): 3.92


== Coditone ==
== Coditone ==
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[[Comma list]]: 225/224, 2125764/2100875
[[Comma list]]: 225/224, 2125764/2100875


{{Mapping|legend=1| 1 6 3 13 | 0 -13 -2 -30 }}
{{Mapping|legend=1| 1 -7 1 -17 | 0 13 2 30 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~1225/972 = 407.642{{c}}
* [[WE]]: ~2 = 1200.4411{{c}}, ~1944/1225 = 792.6016{{c}}
: [[error map]]: {{val| 0.000 -1.298 -1.597 +1.921 }}
: [[error map]]: {{val| +0.441 -1.223 -0.669 +1.722 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~1225/972 = 407.690{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1944/1225 = 792.3270{{c}}
: error map: {{val| 0.000 -1.921 -1.693 +0.483 }}
: error map: {{val| 0.000 -1.704 -1.660 +0.985 }}


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


[[Badness]] (Smith): 0.064356
[[Badness]] (Sintel): 1.63


=== 11-limit ===
=== 11-limit ===
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Comma list: 225/224, 385/384, 78408/78125
Comma list: 225/224, 385/384, 78408/78125


Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}
{{Mapping|legend=0| 1 -7 1 -17 16 | 0 13 2 30 -19 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~1225/972 = 407.688{{c}}
* WE: ~2 = 1200.8016{{c}}, ~198/125 = 792.7878{{c}}
* POTE: ~2 = 1200.000{{c}}, ~1225/972 = 407.741{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2692{{c}}


{{Optimal ET sequence|legend=0| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bcee }}


Badness (Smith): 0.044329
Badness (Sintel): 1.47


==== 13-limit ====
==== 13-limit ====
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Comma list: 225/224, 351/350, 385/384, 847/845
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|legend=0| 1 -7 1 -17 16 7 | 0 13 2 30 -19 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~325/256 = 407.691{{c}}
* WE: ~2 = 1200.7372{{c}}, ~198/125 = 792.7511{{c}}
* POTE: ~2 = 1200.000{{c}}, ~325/256 = 407.736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2715{{c}}


{{Optimal ET sequence|legend=0| 50, 53, 103 }}
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bceef }}


Badness (Smith): 0.024352
Badness (Sintel): 1.01


=== Coditonic ===
=== Coditonic ===
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Comma list: 99/98, 176/175, 6655/6561
Comma list: 99/98, 176/175, 6655/6561


Mapping: {{mapping| 1 6 3 13 15 | 0 -13 -2 -30 -34 }}
{{Mapping|legend=0| 1 -7 1 -17 -19 | 0 13 2 30 34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~242/189 = 407.528{{c}}
* WE: ~2 = 1200.3578{{c}}, ~189/121 = 792.6693{{c}}
* POTE: ~2 = 1200.000{{c}}, ~242/189 = 407.567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/121 = 792.4463{{c}}


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


Badness (Smith): 0.063876
Badness (Sintel): 2.11


==== 13-limit ====
==== 13-limit ====
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Comma list: 99/98, 176/175, 325/324, 847/845
Comma list: 99/98, 176/175, 325/324, 847/845


Mapping: {{mapping| 1 6 3 13 15 20 | 0 -13 -2 -30 -34 -48 }}
{{Mapping|legend=0| 1 -7 1 -17 -19 -28 | 0 13 2 30 34 48 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~242/189 = 407.514{{c}}
* WE: ~2 = 1200.2797{{c}}, ~52/33 = 792.6439{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/26 = 407.541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/33 = 792.4682{{c}}


{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


Badness (Smith): 0.043989
Badness (Sintel): 1.82


== Diton ==
== Diton ==
This extension is known as ''ditonic'' in [[Graham Breed]]'s temperament finder.  
This low-accuracy temperament was considered the canonical extension of ditonic, and catalogued as so in [[Graham Breed]]'s Temperament Finder.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 126/125, 8751645/8388608
[[Comma list]]: 126/125, 8751645/8388608


{{Mapping|legend=1| 1 6 3 -4 | 0 -13 -2 20 }}
{{Mapping|legend=1| 1 -7 1 16 | 0 13 2 -20 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~2625/2048 = 407.922{{c}}
* [[WE]]: ~2 = 1201.9455{{c}}, ~2048/1323 = 793.3304{{c}}
: [[error map]]: {{val| 0.000 -4.939 -2.157 -10.388 }}
: [[error map]]: {{val| +1.946 -2.278 +2.293 -4.305 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~2625/2048 = 407.954{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~2048/1323 = 792.0527{{c}}
: error map: {{val| 0.000 -5.353 -2.221 -9.751 }}
: error map: {{val| 0.000 -5.270 -2.208 -9.879 }}


{{Optimal ET sequence|legend=1| 3, 47, 50 }}
{{Optimal ET sequence|legend=1| 3, 47, 50 }}


[[Badness]] (Smith): 0.242101
[[Badness]] (Sintel): 6.13


=== 11-limit ===
=== 11-limit ===
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Comma list: 126/125, 245/242, 2079/2048
Comma list: 126/125, 245/242, 2079/2048


Mapping: {{mapping| 1 6 3 -4 -3 | 0 -13 -2 20 19 }}
{{Mapping|legend=0| 1 -7 1 16 16 | 0 13 2 -20 -19 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 407.930{{c}}
* WE: ~2 = 1201.6076{{c}}, ~11/7 = 793.1692{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/11 = 407.892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1016{{c}}


{{Optimal ET sequence|legend=0| 3, 47, 50 }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness (Smith): 0.100884
Badness (Sintel): 3.34


=== 13-limit ===
=== 13-limit ===
Line 138: Line 138:
Comma list: 105/104, 126/125, 245/242, 1287/1280
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|legend=0| 1 -7 1 16 16 7 | 0 13 2 -20 -19 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 407.933{{c}}
* WE: ~2 = 1201.3885{{c}}, ~11/7 = 793.0294{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/11 = 407.887{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1096{{c}}


{{Optimal ET sequence|legend=0| 3, 47, 50 }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness (Smith): 0.054997
Badness (Sintel): 2.27


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Ditonmic_family"