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]] ({{monzo|legend=1| -27 -2 13 }}, [[ratio]]: 1 220 703 125 / 1 207 959 552).


=Ditonic=
== Ditonic ==
Commas: 1220703125/1207959552
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.


POTE generator: ~15625/12288 = 407.574
[[Subgroup]]: 2.3.5


Map: [&lt;1 6 3|, &lt;0 -13 -2|]
[[Comma list]]: 1220703125/1207959552


EDOs: 47, 50, 53, 474c, 527c
{{Mapping|legend=1| 1 -7 1 | 0 13 2 }}
: mapping generators: ~2, ~24576/15625


Badness: 0.1671
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2971{{c}}, ~24576/15625 = 792.6223{{c}}
: [[error map]]: {{val| +0.297 +0.055 -0.772 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~24576/15625 = 792.4403{{c}}
: error map: {{val| 0.000 -0.231 -1.433 }}


==7-limit==
{{Optimal ET sequence|legend=1| 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
Commas: 126/125, 8751645/8388608


POTE generator: ~2625/2048 = 407.954
[[Badness]] (Sintel): 3.92


Map: [&lt;1 6 3 -4|, &lt;0 -13 -2 20|]
== Coditone ==
[[Subgroup]]: 2.3.5.7


EDOs: 3, 47, 50
[[Comma list]]: 225/224, 2125764/2100875


Badness: 0.2421
{{Mapping|legend=1| 1 -7 1 -17 | 0 13 2 30 }}


==11-limit==
[[Optimal tuning]]s:
Commas: 126/125, 245/242, 2079/2048
* [[WE]]: ~2 = 1200.4411{{c}}, ~1944/1225 = 792.6016{{c}}
: [[error map]]: {{val| +0.441 -1.223 -0.669 +1.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1944/1225 = 792.3270{{c}}
: error map: {{val| 0.000 -1.704 -1.660 +0.985 }}


POTE generator: ~14/11 = 407.892
{{Optimal ET sequence|legend=1| 50, 53, 103, 156 }}


Map: [&lt;1 6 3 -4 -3|, &lt;0 -13 -2 20 19|]
[[Badness]] (Sintel): 1.63


EDOs: 3, 47, 50, 53d
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1009
Comma list: 225/224, 385/384, 78408/78125


==13-limit==
{{Mapping|legend=0| 1 -7 1 -17 16 | 0 13 2 30 -19 }}
Commas: 105/104, 126/125, 245/242, 1287/1280


POTE generator: ~14/11 = 407.887
Optimal tunings:  
* WE: ~2 = 1200.8016{{c}}, ~198/125 = 792.7878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2692{{c}}


Map: [&lt;1 6 3 -4 -3 2|, &lt;0 -13 -2 20 19 5|]
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bcee }}


EDOs: 3, 47, 50, 53d
Badness (Sintel): 1.47


Badness: 0.0550
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Coditone=
Comma list: 225/224, 351/350, 385/384, 847/845
Commas: 225/224, 2125764/2100875


POTE generator: ~1225/972 = 407.690
{{Mapping|legend=0| 1 -7 1 -17 16 7 | 0 13 2 30 -19 -5 }}


Map: [&lt;1 6 3 13|, &lt;0 -13 -2 -30|]
Optimal tunings:  
* WE: ~2 = 1200.7372{{c}}, ~198/125 = 792.7511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2715{{c}}


EDOs: 50, 53, 103, 156
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bceef }}


==11-limit==
Badness (Sintel): 1.01
Commas: 99/98, 176/175, 6655/6561


POTE generator: ~242/189 = 407.567
=== Coditonic ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 3 13 15|, &lt;0 -13 -2 -30 -34|]
Comma list: 99/98, 176/175, 6655/6561


EDOs: 53
{{Mapping|legend=0| 1 -7 1 -17 -19 | 0 13 2 30 34 }}


Badness: 0.0639
Optimal tunings:  
* WE: ~2 = 1200.3578{{c}}, ~189/121 = 792.6693{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/121 = 792.4463{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}
Commas: 99/98, 176/175, 325/324, 847/845


POTE generator: ~33/26 = 407.541
Badness (Sintel): 2.11


Map: [&lt;1 6 3 13 15 20|, &lt;0 -13 -2 -30 -34 -48|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 53
Comma list: 99/98, 176/175, 325/324, 847/845


Badness: 0.0440
{{Mapping|legend=0| 1 -7 1 -17 -19 -28 | 0 13 2 30 34 48 }}


[[Category:Theory]]
Optimal tunings:
[[Category:Temperament family]]
* WE: ~2 = 1200.2797{{c}}, ~52/33 = 792.6439{{c}}
[[Category:Ditonmic]]
* CWE: ~2 = 1200.0000{{c}}, ~52/33 = 792.4682{{c}}
[[Category:Rank 2]]
 
{{Optimal ET sequence|legend=0| 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|legend=1| 1 -7 1 16 | 0 13 2 -20 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.9455{{c}}, ~2048/1323 = 793.3304{{c}}
: [[error map]]: {{val| +1.946 -2.278 +2.293 -4.305 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~2048/1323 = 792.0527{{c}}
: error map: {{val| 0.000 -5.270 -2.208 -9.879 }}
 
{{Optimal ET sequence|legend=1| 3, 47, 50 }}
 
[[Badness]] (Sintel): 6.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 245/242, 2079/2048
 
{{Mapping|legend=0| 1 -7 1 16 16 | 0 13 2 -20 -19 }}
 
Optimal tunings:
* WE: ~2 = 1201.6076{{c}}, ~11/7 = 793.1692{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1016{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -7 1 16 16 7 | 0 13 2 -20 -19 -5 }}
 
Optimal tunings:
* WE: ~2 = 1201.3885{{c}}, ~11/7 = 793.0294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1096{{c}}
 
{{Optimal ET sequence|legend=0| 3, 47, 50 }}
 
Badness (Sintel): 2.27
 
== References ==
 
[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Ditonic]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:33, 24 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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