32nd-octave temperaments: Difference between revisions
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Fundamental quality improvement (Badness decreasing with dim isn't a fact specific to the temp. All temps are equally JI friendly. Mappings are composed by vals not monzos. Map-merge is a "the" not an "a") |
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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. | 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. | ||
{{Todo|inline=1| cleanup | expand | comment=Complete the data. Work out the data for intermediate subgroups. }} | |||
== Windrose == | == Windrose == | ||
The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as | The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as the 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]]. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
Comma list: {{monzo|38 | Comma list: {{monzo| 38 9 -8 -12 }}, {{monzo| 15 -28 32 -16 }} | ||
Mapping: [{{val|32 44 68 89}}, {{val|0 16 15 2}}] | Mapping: [{{val| 32 44 68 89 }}, {{val| 0 16 15 2 }}] | ||
POTE Generator: 15.7517¢ | POTE Generator: 15.7517¢ | ||
== Germanium == | == Germanium == | ||
It is named after germanium, the 32nd element | It is named after germanium, the 32nd element, defined as 224 & 1376. 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]]. | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 3025/3024, 4096/4095, 256000/255789, 5942475/5940688 | Comma list: 3025/3024, 4096/4095, 256000/255789, 5942475/5940688 | ||
Mapping: [{{ | Mapping: [{{val| 32 51 75 89 110 118 }}, {{val| 0 -2 -5 6 5 3 }}] | ||
POTE Generator: 5.2381¢ | POTE Generator: 5.2381¢ | ||
Vals: {{EDOs|224, 1376, 1600}} | Vals: {{EDOs| 224, 1376, 1600 }} | ||
== Dike == | == Dike == | ||
Defined as | Defined as the 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, ''dijk''. | ||
=== 37-limit === | === 37-limit === | ||
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 | ||
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Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808 | Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808 | ||
Mapping: [{{ | Mapping: [{{val| 32 59 72 111 113 129 140 141 165 178 182 169 }}, {{val| 0 -18 5 -46 -5 -23 -20 -11 -44 -49 -51 -5 }}] | ||
POTE Generator: 17.2544¢ | POTE Generator: 17.2544¢ | ||
Vals: {{EDOs|416dijk, 1600, 2016dijk}} | Vals: {{EDOs| 416dijk, 1600, 2016dijk }} | ||
[[Category:32edo]] | [[Category:32edo]] | ||
Revision as of 17:27, 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.
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Todo: cleanup , expand Complete the data. Work out the data for intermediate subgroups. |
Windrose
The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as the 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.
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. 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.
13-limit
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 the 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, dijk.
37-limit
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¢