32nd-octave temperaments: Difference between revisions
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These are temperaments with period 1/32 of an octave. | These are temperaments with period 1/32 of an octave. | ||
32edo is a wasteland as far as LCJI is concerned, but some of its multiples are good at harmonics, and thus can produce temperaments with period of 1/32 of an octave. | |||
== Windrose == | == Windrose == | ||
The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 608 & 1600 temperament. The maximal evenness pattern created inside the generator is a 12L 7s, if mapped to a keyboard, which has a 2 | The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 608 & 1600 temperament. The maximal evenness pattern created inside the generator is a 12L 7s, if mapped to a keyboard, which has a 2/3 step ratio and thus offers elegant microtempering that plays with the just noticeable difference. Higher-dimensional versions of this are lower in badness than low-dimensional ones. | ||
=== 7-limit === | === 7-limit === | ||
Subgroup:2.3.5.7 | Subgroup: 2.3.5.7 | ||
Comma list: {{monzo|38, 9, -8, -12}}, {{monzo|15, -28, 32, -16}} | Comma list: {{monzo|38, 9, -8, -12}}, {{monzo|15, -28, 32, -16}} | ||
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== Germanium == | == Germanium == | ||
It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. It tempers out [[3025/3024]], [[4096/4095]], [[4375/4374]] and [[9801/9800]] in the 13-limit, although if only these commas are taken, they make a rank 3 1/2-octave temperament called [ | It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. It tempers out [[3025/3024]], [[4096/4095]], [[4375/4374]] and [[9801/9800]] in the 13-limit, although if only these commas are taken, they make a rank 3 1/2-octave temperament called [http://x31eq.com/cgi-bin/rt.cgi?ets=612_270_494&limit=13 rym]. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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POTE Generator: 5.2381¢ | POTE Generator: 5.2381¢ | ||
== | == Dike == | ||
Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch | Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, ''dijk''. | ||
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37 | Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37 | ||
Comma list: | Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808 | ||
Mapping: [{{monzo| 32 59 72 111 113 129 140 141 165 178 182 169}}, {{monzo|0 -18 5 -46 -5 -23 -20 -11 -44 -49 -51 -5}}] | Mapping: [{{monzo| 32 59 72 111 113 129 140 141 165 178 182 169}}, {{monzo|0 -18 5 -46 -5 -23 -20 -11 -44 -49 -51 -5}}] | ||
POTE Generator: 17.2544¢ | POTE Generator: 17.2544¢ | ||
Revision as of 12:28, 5 June 2022
These are temperaments with period 1/32 of an octave.
32edo is a wasteland as far as LCJI is concerned, but some of its multiples are good at harmonics, and thus can produce temperaments with period of 1/32 of an octave.
Windrose
The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 608 & 1600 temperament. The maximal evenness pattern created inside the generator is a 12L 7s, if mapped to a keyboard, which has a 2/3 step ratio and thus offers elegant microtempering that plays with the just noticeable difference. Higher-dimensional versions of this are lower in badness than low-dimensional ones.
7-limit
Subgroup: 2.3.5.7
Comma list: [38, 9, -8, -12⟩, [15, -28, 32, -16⟩
Mapping: [⟨32 44 68 89], ⟨0 16 15 2]]
POTE Generator: 15.7517¢
Germanium
It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. It tempers out 3025/3024, 4096/4095, 4375/4374 and 9801/9800 in the 13-limit, although if only these commas are taken, they make a rank 3 1/2-octave temperament called rym.
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 256000/255789, 5942475/5940688
Mapping: [[32 51 75 89 110 118⟩, [0 -2 -5 6 5 3⟩]
POTE Generator: 5.2381¢
Dike
Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, dijk.
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808
Mapping: [[32 59 72 111 113 129 140 141 165 178 182 169⟩, [0 -18 5 -46 -5 -23 -20 -11 -44 -49 -51 -5⟩]
POTE Generator: 17.2544¢