32nd-octave temperaments: Difference between revisions
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These are temperaments with period 1/32 of an octave. | {{Technical data page}} | ||
{{Infobox fractional-octave|32}} | |||
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#Bezique|Horwell temperaments]] | |||
== Windrose == | == Windrose == | ||
{{Mathematical interest}} | |||
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]]. | 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]]. | ||
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[[Support|Supporting]] [[ET|ETs]]: {{EDOs|384bc, 608, 992, 1600, 2208, 2592}} | [[Support|Supporting]] [[ET|ETs]]: {{EDOs|384bc, 608, 992, 1600, 2208, 2592}} | ||
Badness (Sintel): 79.660 | |||
== Germanium == | == Germanium == | ||
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== Embankment == | == 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. -1591 160 576 | 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|legend=1| 32 5 87 | 0 18 -5 }} | |||
: mapping generators: ~{{monzo|-348 35 126}} = 1\32, ~{{monzo|627 -63 -227}} = 95.247 | |||
[[Optimal tuning]] ([[CTE]]): ~{{monzo|627 -63 -227}} = 95.247 | |||
[[Support]]ing [[ET]]s: {{EDOs|416, 1184, 1600, 2016, 3616, 5216}}, ... | |||
=== Polder === | |||
[[7/6]] is reached in one generator. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: {{monzo|19 0 16 -20}}, {{monzo|90 -8 -20 -11}}, | |||
{{Mapping|legend=1| 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 | |||
[[Support]]ing [[ET]]s: {{EDOs|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: {{monzo|-54 3 20 1}}, {{monzo|-25 73 -4 -29}} | |||
{{mapping|legend=1|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: {{EDOs|416d, 1600, 2016d}}, ... | |||
=== Dike === | === Dike === | ||
==== 37-limit ==== | ==== 37-limit ==== | ||
Defined as the 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, ''dijk''. | 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 | Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37 | ||
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{{Optimal ET sequence|legend=1| 416dijk, 1600, 2016dijk }} | {{Optimal ET sequence|legend=1| 416dijk, 1600, 2016dijk }} | ||
{{Navbox fractional-octave}} | |||