19th-octave temperaments
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.
For graywood, see Syntonic–kleismic equivalence continuum#Graywood.
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
- 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
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 ¢
Badness (Sintel): 0.948