Ditonmic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,475 edits
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "
Tags: Mobile edit Mobile web edit
 
(18 intermediate revisions by 7 users not shown)
Line 1: Line 1:
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 ==
Subgroup: 2.3.5
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.  


[[Comma]]: 1220703125/1207959552
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val|1 6 3}}, {{val|0 -13 -2}}]
[[Comma list]]: 1220703125/1207959552


[[POTE generator]]: ~15625/12288 = 407.574
{{Mapping|legend=1| 1 -7 1 | 0 13 2 }}
: mapping generators: ~2, ~24576/15625


{{Val list|legend=1| 3, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
[[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 }}


[[Badness]]: 0.167086
{{Optimal ET sequence|legend=1| 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}


=== 7-limit ===
[[Badness]] (Sintel): 3.92
[[Comma list]]: 126/125, 8751645/8388608
 
== Coditone ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2125764/2100875


[[Mapping]]: [{{val|1 6 3 -4}}, {{val|0 -13 -2 20}}]
{{Mapping|legend=1| 1 -7 1 -17 | 0 13 2 30 }}


[[POTE generator]]: ~2625/2048 = 407.954
[[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 }}


{{Val list|legend=1| 3, 47, 50 }}
{{Optimal ET sequence|legend=1| 50, 53, 103, 156 }}


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


=== 11-limit ===
=== 11-limit ===
Comma list: 126/125, 245/242, 2079/2048
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 78408/78125
 
{{Mapping|legend=0| 1 -7 1 -17 16 | 0 13 2 30 -19 }}
 
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|legend=0| 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: [{{val|1 6 3 -4 -3}}, {{val|0 -13 -2 20 19}}]
{{Mapping|legend=0| 1 -7 1 -17 16 7 | 0 13 2 30 -19 -5 }}


POTE generator: ~14/11 = 407.892
Optimal tunings:  
* WE: ~2 = 1200.7372{{c}}, ~198/125 = 792.7511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2715{{c}}


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


Badness: 0.100884
Badness (Sintel): 1.01


=== 13-limit ===
=== Coditonic ===
Comma list: 105/104, 126/125, 245/242, 1287/1280
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 6655/6561


Mapping: [{{val|1 6 3 -4 -3 2}}, {{val|0 -13 -2 20 19 5}}]
{{Mapping|legend=0| 1 -7 1 -17 -19 | 0 13 2 30 34 }}


POTE generator: ~14/11 = 407.887
Optimal tunings:  
* WE: ~2 = 1200.3578{{c}}, ~189/121 = 792.6693{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/121 = 792.4463{{c}}


Vals: {{Val list| 3, 47, 50, 53d }}
{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}


Badness: 0.054997
Badness (Sintel): 2.11


== Coditone ==
==== 13-limit ====
<!-- Revised technical data for 11- and 13-limit by Xenllium on 10 June 2021. 11- and 13-limit coditone is for 50&53, not for 11-limit 3de&53 and 13-limit 3def&53. 11-limit 3de&53 and 13-limit 3def&53 is renamed "coditonic" by Xenllium on 10 June 2021. -->
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 225/224, 2125764/2100875
Comma list: 99/98, 176/175, 325/324, 847/845


[[Mapping]]: [{{val|1 6 3 13}}, {{val|0 -13 -2 -30}}]
{{Mapping|legend=0| 1 -7 1 -17 -19 -28 | 0 13 2 30 34 48 }}


[[POTE generator]]: ~1225/972 = 407.690
Optimal tunings:  
* WE: ~2 = 1200.2797{{c}}, ~52/33 = 792.6439{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/33 = 792.4682{{c}}


{{Val list|legend=1| 3d, 50, 53, 103, 156 }}
{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}


[[Badness]]: 0.064356
Badness (Sintel): 1.82


=== 11-limit ===
== Diton ==
Comma list: 225/224, 385/384, 78408/78125
This low-accuracy temperament was considered the canonical extension of ditonic, and catalogued as so in [[Graham Breed]]'s Temperament Finder.


Mapping: [{{val|1 6 3 13 -3}}, {{val|0 -13 -2 -30 19}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~1225/972 = 407.741
[[Comma list]]: 126/125, 8751645/8388608


Vals: {{Val list| 50, 53, 103 }}
{{Mapping|legend=1| 1 -7 1 16 | 0 13 2 -20 }}


Badness: 0.044329
[[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 }}


==== 13-limit ====
{{Optimal ET sequence|legend=1| 3, 47, 50 }}
Comma list: 225/224, 351/350, 385/384, 847/845


Mapping: [{{val|1 6 3 13 -3 2}}, {{val|0 -13 -2 -30 19 5}}]
[[Badness]] (Sintel): 6.13


POTE generator: ~325/256 = 407.736
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 50, 53, 103 }}
Comma list: 126/125, 245/242, 2079/2048


Badness: 0.024352
{{Mapping|legend=0| 1 -7 1 16 16 | 0 13 2 -20 -19 }}


=== Coditonic ===
Optimal tunings:
Comma list: 99/98, 176/175, 6655/6561
* WE: ~2 = 1201.6076{{c}}, ~11/7 = 793.1692{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1016{{c}}


Mapping: [{{val|1 6 3 13 15}}, {{val|0 -13 -2 -30 -34}}]
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


POTE generator: ~242/189 = 407.567
Badness (Sintel): 3.34


Vals: {{Val list| 3de, 50e, 53 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.063876
Comma list: 105/104, 126/125, 245/242, 1287/1280


==== 13-limit ====
{{Mapping|legend=0| 1 -7 1 16 16 7 | 0 13 2 -20 -19 -5 }}
Comma list: 99/98, 176/175, 325/324, 847/845


Mapping: [{{val|1 6 3 13 15 20}}, {{val|0 -13 -2 -30 -34 -48}}]
Optimal tunings:  
* WE: ~2 = 1201.3885{{c}}, ~11/7 = 793.0294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1096{{c}}


POTE generator: ~33/26 = 407.541
{{Optimal ET sequence|legend=0| 3, 47, 50 }}


Vals: {{Val list| 3def, 53 }}
Badness (Sintel): 2.27


Badness: 0.043989
== References ==


[[Category:Theory]]
[[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"