32nd-octave temperaments
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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.
Temperaments discussed elsewhere include:
- Bezique, → Horwell temperaments
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 period 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]]
- mapping generators: ~4084101/4000000 = 1\32, ~48828125/46294416 = 90.749
Optimal tuning (CTE): ~48828125/46294416 = 90.749
Supporting ETs: 384bc, 608, 992, 1600, 2208, 2592
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 it should be noted that if only these commas are taken, they make a rank-3 1/2-octave temperament called rym. Thus germanium is a tempering of rym.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, [60 -15 -5 -10 1⟩
Mapping: [⟨32 1 -50 239 235], ⟨0 2 5 -6 -5]]
- mapping generators: ~134217728/131274675 = 1\32, ~77175/45056 = 932.260
Optimal tuning (CTE): ~77175/45056 = 932.260
Supporting ETs: 224, 704, 928, 1152, 1376, 1600
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 4375/4374, 5942475/5940688
Mapping: [⟨32 1 -50 239 235 193], ⟨0 2 5 -6 -5 -3]]
- mapping generators: ~1352/1323 = 1\32, ~245/143 = 932.263
Optimal tuning (CTE): ~245/143 = 932.263
Optimal ET sequence: 224, 1376, 1600
Embankment
Described as the 1600 & 2016 temperament. Due to 7-limit inconsistency of 2016edo, the temperament branches into polder, using 2016's patent val, and dam, using the 2016d val.
Subgroup: 2.3.5
Comma list: [-1591 160 576⟩
Mapping: [⟨32 5 87], ⟨0 18 -5]]
- mapping generators: ~[-348 35 126⟩ = 1\32, ~[627 -63 -227⟩ = 95.247
Optimal tuning (CTE): ~[627 -63 -227⟩ = 95.247
Supporting ETs: 416, 1184, 1600, 2016, 3616, 5216, ...
Polder
7/6 is reached in one generator.
Subgroup: 2.3.5.7
Comma list: [19 0 16 -20⟩, [90 -8 -20 -11⟩,
Mapping: [⟨32 5 87 100], ⟨0 18 -5 -4]]
- mapping generators: ~1352/1323 = 1\32, ~245/143 = 932.263
Optimal tuning (CTE): ~245/143 = 932.263
Supporting ETs: 416, 768b, 1184, 1600, 2016, 2784, ...
Dam
Due to complexity, dam is not a remarkably interesting temperament on its own, but in higher limits, its 37-limit extension dike is worth considering (see below).
Subgroup: 2.3.5.7
Comma list: [-54 3 20 1⟩, [-25 73 -4 -29⟩
Mapping: [⟨32 5 87 -27], ⟨0 18 -5 46]]
- mapping generators: ~2017815046875/1973822685184 = 1\32, ~16896102540283203125/15992037016835457024 = 95.247
Optimal tuning (CTE): ~16896102540283203125/15992037016835457024 = 95.247
Supporting ETs: 416d, 1600, 2016d, ...
Dike
37-limit
Defined as the 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, dijk. It is worth noting that in the 37-limit, 2016dijk val is better tuned than the patent val, being second only to 2016dhijk by error.
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]]
- mapping generators: ?~ 1\32, ~? = 17.2544