32nd-octave temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
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
Badness (Sintel): 79.660
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
Optimal ET sequence: 416dijk, 1600, 2016dijk