Ditonmic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
+description
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "
Tags: Mobile edit Mobile web edit
 
(13 intermediate revisions by 4 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]] ({{monzo|legend=1| -27 -2 13 }}, [[ratio]]: 1 220 703 125 / 1 207 959 552).


== 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 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 8: 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, ~24576/15625


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15625/12288 = 407.574
[[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 }}


{{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]] (Sintel): 3.92


== 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 -7 1 -17 | 0 13 2 30 }}


{{Multival|legend=1| 13 2 -20 -27 -68 -52 }}
[[Optimal tuning]]s:
* [[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2625/2048 = 407.954
{{Optimal ET sequence|legend=1| 50, 53, 103, 156 }}
 
{{Optimal ET sequence|legend=1| 3, 47, 50 }}


[[Badness]]: 0.242101
[[Badness]] (Sintel): 1.63


=== 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|legend=0| 1 -7 1 -17 16 | 0 13 2 30 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.892
Optimal tunings:
* WE: ~2 = 1200.8016{{c}}, ~198/125 = 792.7878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2692{{c}}


Optimal ET sequence: {{optimal ET sequence| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bcee }}


Badness: 0.100884
Badness (Sintel): 1.47


=== 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|legend=0| 1 -7 1 -17 16 7 | 0 13 2 30 -19 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 407.887
Optimal tunings:
* WE: ~2 = 1200.7372{{c}}, ~198/125 = 792.7511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2715{{c}}


Optimal ET sequence: {{optimal ET sequence| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bceef }}


Badness: 0.054997
Badness (Sintel): 1.01


== 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|legend=0| 1 -7 1 -17 -19 | 0 13 2 30 34 }}


{{Multival|legend=1| 13 2 30 -27 11 64 }}
Optimal tunings:
* WE: ~2 = 1200.3578{{c}}, ~189/121 = 792.6693{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/121 = 792.4463{{c}}


[[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 (Sintel): 2.11


[[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|legend=0| 1 -7 1 -17 -19 -28 | 0 13 2 30 34 48 }}


Mapping: {{mapping| 1 6 3 13 -3 | 0 -13 -2 -30 19 }}
Optimal tunings:  
* WE: ~2 = 1200.2797{{c}}, ~52/33 = 792.6439{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/33 = 792.4682{{c}}


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 (Sintel): 1.82


Badness: 0.044329
== Diton ==
This low-accuracy temperament was considered the canonical extension of ditonic, and catalogued as so 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 -7 1 16 | 0 13 2 -20 }}


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


Badness: 0.024352
[[Badness]] (Sintel): 6.13


=== 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|legend=0| 1 -7 1 16 16 | 0 13 2 -20 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~242/189 = 407.567
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: {{optimal ET sequence| 3de, 50e, 53 }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness: 0.063876
Badness (Sintel): 3.34


==== 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 13 15 20 | 0 -13 -2 -30 -34 -48 }}
{{Mapping|legend=0| 1 -7 1 16 16 7 | 0 13 2 -20 -19 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 407.541
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: {{optimal ET sequence| 3def, 53 }}
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Badness: 0.043989
Badness (Sintel): 2.27


== Notes ==
== References ==


[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Ditonic]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Ditonmic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Ditonmic]]
[[Category:Rank 2]]

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
Retrieved from "https://en.xen.wiki/index.php?title=Ditonmic_family&oldid=237756"