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
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[[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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 ===
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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: {{mapping| 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 ==

Revision as of 08:21, 21 September 2026

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 (monzo: [-27 -2 13⟩, ratio: 1 220 703 125 / 1 207 959 552).

Ditonic

Named by Petr Pařízek in 2011[1], 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

Comma list: 1220703125/1207959552

Mapping: [⟨1 -7 1], ⟨0 13 2]]

mapping generators: ~2, ~24576/15625

Optimal tunings:

  • WE: ~2 = 1200.2971 ¢, ~24576/15625 = 792.6223 ¢
error map: ⟨+0.297 +0.055 -0.772]
  • CWE: ~2 = 1200.0000 ¢, ~24576/15625 = 792.4403 ¢
error map: ⟨0.000 -0.231 -1.433]

Optimal ET sequence: 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc

Badness (Sintel): 3.92

Coditone

Subgroup: 2.3.5.7

Comma list: 225/224, 2125764/2100875

Mapping: [⟨1 -7 1 -17], ⟨0 13 2 30]]

Optimal tunings:

  • WE: ~2 = 1200.4411 ¢, ~1944/1225 = 792.6016 ¢
error map: ⟨+0.441 -1.223 -0.669 +1.722]
  • CWE: ~2 = 1200.0000 ¢, ~1944/1225 = 792.3270 ¢
error map: ⟨0.000 -1.704 -1.660 +0.985]

Optimal ET sequence: 50, 53, 103, 156

Badness (Sintel): 1.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 78408/78125

Mapping: [⟨1 -7 1 -17 16], ⟨0 13 2 30 -19]]

Optimal tunings:

  • WE: ~2 = 1200.8016 ¢, ~198/125 = 792.7878 ¢
  • CWE: ~2 = 1200.0000 ¢, ~198/125 = 792.2692 ¢

Optimal ET sequence: 50, 53, 103, 259be, 362bcee

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 385/384, 847/845

Mapping: [⟨1 -7 1 -17 16 7], ⟨0 13 2 30 -19 -5]]

Optimal tunings:

  • WE: ~2 = 1200.7372 ¢, ~198/125 = 792.7511 ¢
  • CWE: ~2 = 1200.0000 ¢, ~198/125 = 792.2715 ¢

Optimal ET sequence: 50, 53, 103, 259be, 362bceef

Badness (Sintel): 1.01

Coditonic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 1 -17 -19], ⟨0 13 2 30 34]]

Optimal tunings:

  • WE: ~2 = 1200.3578 ¢, ~189/121 = 792.6693 ¢
  • CWE: ~2 = 1200.0000 ¢, ~189/121 = 792.4463 ¢

Optimal ET sequence: 3de, 50e, 53

Badness (Sintel): 2.11

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 325/324, 847/845

Mapping: [⟨1 -7 1 -17 -19 -28], ⟨0 13 2 30 34 48]]

Optimal tunings:

  • WE: ~2 = 1200.2797 ¢, ~52/33 = 792.6439 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/33 = 792.4682 ¢

Optimal ET sequence: 3def, 50eff, 53

Badness (Sintel): 1.82

Diton

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

Comma list: 126/125, 8751645/8388608

Mapping: [⟨1 -7 1 16], ⟨0 13 2 -20]]

Optimal tunings:

  • WE: ~2 = 1201.9455 ¢, ~2048/1323 = 793.3304 ¢
error map: ⟨+1.946 -2.278 +2.293 -4.305]
  • CWE: ~2 = 1200.000 ¢, ~2048/1323 = 792.0527 ¢
error map: ⟨0.000 -5.270 -2.208 -9.879]

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 6.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 2079/2048

Mapping: [⟨1 -7 1 16 16], ⟨0 13 2 -20 -19]]

Optimal tunings:

  • WE: ~2 = 1201.6076 ¢, ~11/7 = 793.1692 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 792.1016 ¢

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 3.34

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 1 16 16 7], ⟨0 13 2 -20 -19 -5]]

Optimal tunings:

  • WE: ~2 = 1201.3885 ¢, ~11/7 = 793.0294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 792.1096 ¢

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 2.27

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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