32edo: Difference between revisions

32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. It has unambiguous mappings for [[prime]]s up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.  

While even advocates of less-common [[edo]]s can struggle to find something about 32edo worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]], using its 9\32 generator of size 337.5 cents. [[Petr Pařízek]]'s preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune [[mohavila]], an 11-limit temperament which does not temper out sixix.

It also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family#Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the [[optimal patent val]] for the {{nowrap|15 & 32}} temperament, tempering out [[55/54]], 64/63, and [[245/242]].

The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[subgroup temperaments #Oceanfront|Oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[9/7]] and [[13/10]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker [[mode]]s. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.

=== Odd harmonics ===

{{Harmonics in equal|32}}

! colspan="3" | [[Ups and Downs Notation]]

! 13-limit Ratios

| 49/48, 50/49, 45/44

| 22/21, 25/24

| 16/15

| 12/11, 49/45

| 10/9, 39/35

| 8/7, 25/22

| 7/6, 64/55

| 6/5, 32/27

| 11/9, 39/32, 63/52

| 5/4, 26/21, 56/45, 96/77

| 14/11, 33/26, 80/63

| 13/10, 35/27, 64/49

| 4/3, 33/25, 160/121

| 27/20, 110/81

| 18/13, 11/8

| 7/5, 10/7, 99/70, 140/99

| 13/9, 16/11

| 40/27, 81/55

| 3/2, 50/33, 121/80

| 20/13, 54/35, 49/32

| 11/7, 52/33, 63/40

| 8/5, 21/13, 45/28, 77/48

| 18/11, 64/39, 104/63

| 5/3, 27/16

| 12/7, 55/32

| 7/4, 44/25

| 9/5, 70/39

| 11/6, 90/49

| 15/8

| 21/11, 48/25

| 96/49, 49/25, 88/45

| 2/1

=== Ups and downs notation ===

32edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

{{Sharpness-sharp5a}}

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

 

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.

This notation uses the same sagittal sequence as [[25edo#Sagittal notation|25-EDO]], and is a subset of the notation for [[64edo#Second-best fifth notation|64b]].

=== Zeta properties ===

Below is a plot of the [[Zeta]] function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.

[[File:plot32.png|alt=plot32.png|plot32.png]]

 

| zpi = 133

| steps = 32.0701500181780

| step size = 37.4179727665700

| tempered height = 4.471728

| pure height = 2.687487

| integral = 0.689412

| gap = 12.537826

| octave = 1197.37512853024

| consistent = 4

| distinct = 4

}}

[[Lumatone mapping for 32edo]]

; [[Brody Bigwood]]

* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)

* [https://www.youtube.com/watch?v=y2G6Fs2HMUs ''Canon on Twinkle Twinkle Little Star'', for organ] (2023) – ([https://www.youtube.com/watch?v=JWRGLa59ZwY for Baroque Oboe & Viola])

* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']

* [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 ''32 32 32 Nothing Less Will Do'']

* [https://www.youtube.com/watch?v=00kH3CqSgMY "Zinnia Riplet" (32-EDO)] (featured in [https://spectropolrecords.bandcamp.com/album/possible-worlds-vol-4 ''Possible Worlds Vol. 4''] of Spectropol Records)

* [https://soundcloud.com/overtoneshock/admins-hot-tub-32-edo ''Admin's Hot Tub'']

{{todo|add rank 2 temperaments table}}