19th-octave temperaments: Difference between revisions

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[[19edo]] has excellent 5-limit accuracy, but its quality of higher-limit approximation can be improved. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families. Meanmag has the same 5-limit mapping as 19et with harmonics 7, 11, and 13 mapped to an independent generator. Undevigintone has the same 2.3.5.7.13 subgroup mapping as 19et with harmonic 11 mapped to an independent generator.  
{{Technical data page}}
{{Infobox fractional-octave|19}}
[[19edo]] has excellent [[5-limit]] accuracy, but its quality of higher-limit approximation can be improved. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families.  


See also [[enneadecal]] and [[superenneadecal]].  
* Graywood has the same 3-limit mapping as 19et, but harmonic [[5/1|5]] is mapped to an independent generator.
* Meanmag has the same 5-limit mapping as 19et, but harmonic 7 is mapped to an independent generator.
* Undevigintone has the same 7-limit mapping as 19et, but harmonic 11 is mapped to an independent generator.
 
For graywood, see [[Syntonic–kleismic equivalence continuum #Graywood]].
 
See also [[ragismic microtemperaments #Enneadecal|enneadecal]] and [[sensamagic clan #Superenneadecal|superenneadecal]].  


== Meanmag ==
== Meanmag ==
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{{Mapping|legend=1| 19 30 44 0 | 0 0 0 1 }}
{{Mapping|legend=1| 19 30 44 0 | 0 0 0 1 }}
: mapping generators: ~25/24, ~7
: mapping generators: ~25/24, ~7


{{Multival|legend=1| 0 0 19 0 30 44 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~25/24 = 63.2931{{c}}, ~7/4 = 963.6625{{c}}
[[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396
: [[error map]]: {{val| +2.569 -3.162 -1.417 -0.026 }}
* [[CWE]]: ~25/24 = 63.1579{{c}}, ~7/4 = 963.4030{{c}}
: error map: {{val| 0.000 -7.218 -7.366 -5.423 }}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }}
{{Optimal ET sequence|legend=1| 19, 57, 76, 171bbccdd }}


[[Badness]]: 0.077023
[[Badness]] (Sintel): 1.95


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 19 30 44 0 119 | 0 0 0 1 -1 }}
Mapping: {{mapping| 19 30 44 0 119 | 0 0 0 1 -1 }}


Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 233.486
Optimal tunings:
* WE: ~25/24 = 63.2535{{c}}, ~7/4 = 967.9769{{c}}
* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 966.6112{{c}}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57 }}


Badness: 0.066829
Badness (Sintel): 2.21


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Mapping: {{mapping| 19 30 44 0 119 17 | 0 0 0 1 -1 1 }}
Mapping: {{mapping| 19 30 44 0 119 17 | 0 0 0 1 -1 1 }}


Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 234.890
Optimal tunings:
* WE: ~25/24 = 63.2422{{c}}, ~7/4 = 966.3987{{c}}
* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 965.3984{{c}}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57, 76 }}


Badness: 0.045844
Badness (Sintel): 1.89


== Undevigintone ==
== Undevigintone ==
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{{Mapping|legend=1| 19 30 44 53 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 19 30 44 53 0 | 0 0 0 0 1 }}
: mapping generators: ~28/27, ~11
: mapping generators: ~28/27, ~11


[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\19, ~11/8 = 538.047
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 63.3591{{c}}, ~11/8 = 539.7611{{c}}
* [[CWE]]: ~28/27 = 63.1579{{c}}, ~11/8 = 540.6837{{c}}


{{Optimal ET sequence|legend=1| 19, 38d }}
{{Optimal ET sequence|legend=1| 19, 38d }}


[[Badness]]: 0.036387
[[Badness]] (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~28/27 = 1\19, ~11/8 = 537.061
Optimal tunings:
* WE: ~28/27 = 63.3741{{c}}, ~11/8 = 538.8996{{c}}
* CWE: ~28/27 = 63.1579{{c}}, ~11/8 = 539.4216{{c}}


{{Optimal ET sequence|legend=1| 19, 38df }}
{{Optimal ET sequence|legend=0| 19, 38df }}


Badness: 0.022933
Badness (Sintel): 0.948


[[Category:Temperament collections]]
{{Navbox fractional-octave}}
[[Category:Fractional-octave temperaments]]
[[Category:19edo]]
[[Category:Rank 2]]

Latest revision as of 13:27, 17 August 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.

← 18th-octave • 19th-octave • 20th-octave →

19edo has excellent 5-limit accuracy, but its quality of higher-limit approximation can be improved. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families.

  • Graywood has the same 3-limit mapping as 19et, but harmonic 5 is mapped to an independent generator.
  • Meanmag has the same 5-limit mapping as 19et, but harmonic 7 is mapped to an independent generator.
  • Undevigintone has the same 7-limit mapping as 19et, but harmonic 11 is mapped to an independent generator.

For graywood, see Syntonic–kleismic equivalence continuum #Graywood.

See also enneadecal and superenneadecal.

Meanmag

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3072

Mapping: [⟨19 30 44 0], ⟨0 0 0 1]]

mapping generators: ~25/24, ~7

Optimal tunings:

  • WE: ~25/24 = 63.2931 ¢, ~7/4 = 963.6625 ¢
error map: ⟨+2.569 -3.162 -1.417 -0.026]
  • CWE: ~25/24 = 63.1579 ¢, ~7/4 = 963.4030 ¢
error map: ⟨0.000 -7.218 -7.366 -5.423]

Optimal ET sequence: 19, 57, 76, 171bbccdd

Badness (Sintel): 1.95

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 625/616

Mapping: [⟨19 30 44 0 119], ⟨0 0 0 1 -1]]

Optimal tunings:

  • WE: ~25/24 = 63.2535 ¢, ~7/4 = 967.9769 ¢
  • CWE: ~25/24 = 63.1579 ¢, ~7/4 = 966.6112 ¢

Optimal ET sequence: 19, 38, 57

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 625/616

Mapping: [⟨19 30 44 0 119 17], ⟨0 0 0 1 -1 1]]

Optimal tunings:

  • WE: ~25/24 = 63.2422 ¢, ~7/4 = 966.3987 ¢
  • CWE: ~25/24 = 63.1579 ¢, ~7/4 = 965.3984 ¢

Optimal ET sequence: 19, 38, 57, 76

Badness (Sintel): 1.89

Undevigintone

Subgroup: 2.3.5.7.11

Comma list: 49/48, 81/80, 126/125

Mapping: [⟨19 30 44 53 0], ⟨0 0 0 0 1]]

mapping generators: ~28/27, ~11

Optimal tunings:

  • WE: ~28/27 = 63.3591 ¢, ~11/8 = 539.7611 ¢
  • CWE: ~28/27 = 63.1579 ¢, ~11/8 = 540.6837 ¢

Optimal ET sequence: 19, 38d

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 81/80, 126/125

Mapping: [⟨19 30 44 53 0 70], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~28/27 = 63.3741 ¢, ~11/8 = 538.8996 ¢
  • CWE: ~28/27 = 63.1579 ¢, ~11/8 = 539.4216 ¢

Optimal ET sequence: 19, 38df

Badness (Sintel): 0.948

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