19th-octave temperaments: Difference between revisions
Move graywood and 5-limit enneadecal to the equivalence continuum |
m Units & misc. cleanup |
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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 | ||
[[Optimal tuning]] ([[POTE]]): ~25/24 = | [[Optimal tuning]] ([[POTE]]): ~25/24 = 63.1579{{c}}, ~8/7 = 238.396{{c}} | ||
{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }} | {{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }} | ||
[[Badness]]: 0.077023 | [[Badness]] (Smith): 0.077023 | ||
=== 11-limit === | === 11-limit === | ||
| Line 26: | Line 25: | ||
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 = | Optimal tuning (POTE): ~25/24 = 63.1579{{c}}, ~8/7 = 233.486{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 38, 57, 76 }} | ||
Badness: 0.066829 | Badness (Smith): 0.066829 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 39: | Line 38: | ||
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 = | Optimal tuning (POTE): ~25/24 = 63.1579{{c}}, ~8/7 = 234.890{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 38, 57, 76 }} | ||
Badness: 0.045844 | Badness (Smith): 0.045844 | ||
== 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 = | [[Optimal tuning]] ([[POTE]]): ~28/27 = 63.1579{{c}}, ~11/8 = 538.047{{c}} | ||
{{Optimal ET sequence|legend=1| 19, 38d }} | {{Optimal ET sequence|legend=1| 19, 38d }} | ||
[[Badness]]: 0.036387 | [[Badness]] (Smith): 0.036387 | ||
=== 13-limit === | === 13-limit === | ||
| Line 67: | Line 65: | ||
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 = | Optimal tuning (POTE): ~28/27 = 63.1579{{c}}, ~11/8 = 537.061{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 38df }} | ||
Badness: 0.022933 | Badness (Smith): 0.022933 | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
Revision as of 13:09, 23 March 2026
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 tuning (POTE): ~25/24 = 63.1579 ¢, ~8/7 = 238.396 ¢
Optimal ET sequence: 19, 38, 57, 76, 95bc
Badness (Smith): 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 = 63.1579 ¢, ~8/7 = 233.486 ¢
Optimal ET sequence: 19, 38, 57, 76
Badness (Smith): 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 = 63.1579 ¢, ~8/7 = 234.890 ¢
Optimal ET sequence: 19, 38, 57, 76
Badness (Smith): 0.045844
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 = 63.1579 ¢, ~11/8 = 538.047 ¢
Badness (Smith): 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 = 63.1579 ¢, ~11/8 = 537.061 ¢
Badness (Smith): 0.022933