19th-octave temperaments

Fractional-octave temperaments

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← 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 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 ¢

Optimal ET sequence: 19, 38d

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 ¢

Optimal ET sequence: 19, 38df

Badness (Smith): 0.022933