19th-octave temperaments: Difference between revisions
CompactStar (talk | contribs) I've discussed using this name for 19-comma in the XA discord. |
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{{ | {{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 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. | ||
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{{Mapping|legend=1|19 30 38|0 0 1}} | {{Mapping|legend=1|19 30 38|0 0 1}} | ||
[[Optimal tuning]] ([[CTE]]): ~2187/2048 = 1\19, ~5/4 = 386.314 | |||
[[Support]]ing [[ET]]s: {{EDOs|19, 38c, 57c, 76c, 95bcc }} | [[Support]]ing [[ET]]s: {{EDOs|19, 38c, 57c, 76c, 95bcc }} | ||
== Meanmag == | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: mapping generators: ~25/24, ~7 | : mapping generators: ~25/24, ~7 | ||
[[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396 | [[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396 | ||
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Badness: 0.066829 | Badness: 0.066829 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.045844 | Badness: 0.045844 | ||
== (5-limit) enneadecal == | |||
''For the 7-limit temperament, see [[Ragismic microtemperaments#Enneadecal]].'' | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 19073486328125/19042491875328 | |||
{{Mapping|legend=1|19 19 33|0 1 1}} | |||
[[Optimal tuning]] ([[CTE]]): ~648/625 = 1\19, ~3/2 = 701.990 | |||
[[Support]]ing [[ET]]s: {{EDOs|171, 19, 152, 323, 494, 665, 190, 133, 836, 817, 475, 114, 361, 209 }} | |||
== Undevigintone == | == Undevigintone == | ||
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Badness: 0.022933 | Badness: 0.022933 | ||
{{Navbox fractional-octave}} | |||
Latest revision as of 05:09, 12 June 2025
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.
Graywood
Subgroup: 2.3.5
Comma list: 1162261467/1073741824
Mapping: [⟨19 30 38], ⟨0 0 1]]
Optimal tuning (CTE): ~2187/2048 = 1\19, ~5/4 = 386.314
Supporting ETs: 19, 38c, 57c, 76c, 95bcc
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 tuning (POTE): ~25/24 = 1\19, ~8/7 = 238.396
Optimal ET sequence: 19, 38, 57, 76, 95bc
Badness: 0.077023
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 tuning (POTE): ~25/24 = 1\19, ~8/7 = 233.486
Optimal ET sequence: 19, 38, 57, 76
Badness: 0.066829
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 tuning (POTE): ~25/24 = 1\19, ~8/7 = 234.890
Optimal ET sequence: 19, 38, 57, 76
Badness: 0.045844
(5-limit) enneadecal
For the 7-limit temperament, see Ragismic microtemperaments#Enneadecal.
Subgroup: 2.3.5
Comma list: 19073486328125/19042491875328
Mapping: [⟨19 19 33], ⟨0 1 1]]
Optimal tuning (CTE): ~648/625 = 1\19, ~3/2 = 701.990
Supporting ETs: 171, 19, 152, 323, 494, 665, 190, 133, 836, 817, 475, 114, 361, 209
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 tuning (POTE): ~28/27 = 1\19, ~11/8 = 538.047
Badness: 0.036387
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 tuning (POTE): ~28/27 = 1\19, ~11/8 = 537.061
Badness: 0.022933