32nd-octave temperaments: Difference between revisions

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.

 

* ''Bezique,'' → [[Horwell temperaments#Bezique|Horwell temperaments]]

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]]: {{monzo| 38 9 -8 -12 }}, {{monzo| 15 -28 32 -16 }}

[[Mapping]]: [{{val| 32 44 68 89 }}, {{val| 0 16 15 2 }}]

: mapping generators: ~4084101/4000000 = 1\32, ~48828125/46294416 = 90.749

 

 

 

Badness (Sintel): 79.660

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, {{monzo|60 -15 -5 -10 1}}

 

 

 

 

 

[[Subgroup]]: 2.3.5.7.11.13

 

[[Comma list]]: 3025/3024, 4096/4095, 4375/4374, 5942475/5940688

 

{{Mapping|legend=1| 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

 

 

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]]: {{monzo|-1591 160 576}}

: mapping generators: ~{{monzo|-348 35 126}} = 1\32, ~{{monzo|627 -63 -227}} = 95.247

[[Optimal tuning]] ([[CTE]]): ~{{monzo|627 -63 -227}} = 95.247

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=== 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.

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 }}]