19th-octave temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,152 edits
I've discussed using this name for 19-comma in the XA discord.
Adopt Template: Technical data page. Rework intro
 
(12 intermediate revisions by 8 users not shown)
Line 1: Line 1:
{{Fractional-octave navigation|19}}
{{Technical data page}}
[[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.  
{{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.  


== Graywood ==
For graywood, see [[Syntonic–kleismic equivalence continuum #Graywood]].
[[Subgroup]]: 2.3.5


[[Comma list]]: 1162261467/1073741824
See also [[ragismic microtemperaments #Enneadecal|enneadecal]] and [[sensamagic clan #Superenneadecal|superenneadecal]].


{{Mapping|legend=1|19 30 38|0 0 1}}
== Meanmag ==
 
[[Support]]ing [[ET]]s: {{EDOs|19, 38c, 57c, 76c, 95bcc }}
=== Meanmag ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 18: Line 17:


{{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 ===
Line 36: Line 36:
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


Line 49: Line 51:
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 ==
Line 61: Line 65:


{{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 ===
Line 77: Line 82:
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=0| 19, 38df }}


{{Optimal ET sequence|legend=1| 19, 38df }}
Badness (Sintel): 0.948


Badness: 0.022933
{{Navbox fractional-octave}}

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

View • Talk • EditFractional-octave temperaments 
← 14th • 15th • 16th • 17th • 18th • 19th-octave • 20th • 21st • 22nd • 23rd • 24th →
Retrieved from "https://en.xen.wiki/index.php?title=19th-octave_temperaments&oldid=236000"