19th-octave temperaments: Difference between revisions

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{{Infobox fractional-octave|19}}
{{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. 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.  
[[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 [[harmonic]]s [[7/1|7]], [[11/1|11]], and [[13/1|13]] mapped to an independent generator. Undevigintone has the same [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping as 19et with harmonic 11 mapped to an independent generator.  


See also [[enneadecal]] and [[superenneadecal]].  
See also [[enneadecal]] and [[superenneadecal]].  
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: mapping generators: ~25/24, ~7
: mapping generators: ~25/24, ~7


[[Optimal tuning]] ([[POTE]]): ~25/24 = 63.1579{{c}}, ~8/7 = 238.396{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~25/24 = 63.2931{{c}}, ~7/4 = 963.6625{{c}}
: [[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]] (Smith): 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 = 63.1579{{c}}, ~8/7 = 233.486{{c}}
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=0| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57 }}


Badness (Smith): 0.066829
Badness (Sintel): 2.21


==== 13-limit ====
==== 13-limit ====
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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 = 63.1579{{c}}, ~8/7 = 234.890{{c}}
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=0| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57, 76 }}


Badness (Smith): 0.045844
Badness (Sintel): 1.89


== Undevigintone ==
== Undevigintone ==
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: mapping generators: ~28/27, ~11
: mapping generators: ~28/27, ~11


[[Optimal tuning]] ([[POTE]]): ~28/27 = 63.1579{{c}}, ~11/8 = 538.047{{c}}
[[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]] (Smith): 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 = 63.1579{{c}}, ~11/8 = 537.061{{c}}
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=0| 19, 38df }}
{{Optimal ET sequence|legend=0| 19, 38df }}


Badness (Smith): 0.022933
Badness (Sintel): 0.948


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Revision as of 13:27, 23 March 2026

← 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. 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.

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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