32nd-octave temperaments: Difference between revisions

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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.

== Windrose ==

The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 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. Higher-dimensional versions of this are lower in badness than low-dimensional ones.

The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 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. Higher-dimensional versions of this are lower in badness than low-dimensional ones.

=== 7-limit ===

Subgroup:2.3.5.7

Subgroup: 2.3.5.7

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

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== Germanium ==

It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. 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]].

It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=612_270_494&limit=13 rym].

Subgroup: 2.3.5.7.11.13

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POTE Generator: 5.2381¢

== ~~Van Dijk~~ ==

== Dike ==

Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch ~~surname Van Dijk~~.

Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, ''dijk''.

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list:

Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808

Mapping: [{{monzo| 32 59 72 111 113 129 140 141 165 178 182 169}}, {{monzo|0 -18 5 -46 -5 -23 -20 -11 -44 -49 -51 -5}}]

POTE Generator: 17.2544¢

@@ Line 1: / Line 1: @@
 These are temperaments with period 1/32 of an octave.
+edo 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.
 == Windrose ==
-The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 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. Higher-dimensional versions of this are lower in badness than low-dimensional ones.
+The temperament is called windrose because there are 32 cardinal directions commonly assigned to a compass rose. It is defined as a 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. Higher-dimensional versions of this are lower in badness than low-dimensional ones.
 === 7-limit ===
-Subgroup:2.3.5.7
+Subgroup: 2.3.5.7
 Comma list: {{monzo|38, 9, -8, -12}}, {{monzo|15, -28, 32, -16}}
@@ Line 14: / Line 16: @@
 == Germanium ==
-It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. 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]].
+It is named after germanium, the 32nd element. Defined as 224 & 1376, this is the most just intonation-friendly 32nd-octave temperament. 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=612_270_494&limit=13 rym].
 Subgroup: 2.3.5.7.11.13
@@ Line 24: / Line 26: @@
 POTE Generator: 5.2381¢
-== Van Dijk ==
+== Dike ==
-Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch surname Van Dijk.
+Defined as a 2016dijk & 1600 temperament, since the warts on the val spell out the Dutch word for dike, ''dijk''.
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
-Comma list:
+Comma list: 4200/4199, 5916/5915, 7425/7424, 8991/8990, 33264/33263, 34452/34447, 59653/59644, 253487/253460, 930291/930248, 246938625/246907808
 Mapping: [{{monzo|	32	59	72	111	113	129 140 141 165 178 182 169}}, {{monzo|0	-18	5	-46	-5	-23	-20	-11	-44	-49	-51 -5}}]
 POTE Generator: 17.2544¢