32edo: Difference between revisions

Since 32 factors into primes as 2⁵, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.

=== 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]].

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

What follows is a comparison of compressed-octave 32edo tunings.

* Step size: 37.500{{c}}, octave size: 1200.0{{c}}

Pure-octaves 32edo approximates all harmonics up to 16 within 15.5{{c}}.

; [[WE|32et, 13-limit WE tuning]]

* Step size: 37.481{{c}}, octave size: 1199.4{{c}}

Compressing the octave of 32edo by around half a cent results in improved primes 3, 7 and 11, but worse primes 5 and 13. This approximates all harmonics up to 16 within 18.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}

; [[WE|32et, 11-limit WE tuning]]

Compressing the octave of 32edo by around 1.5{{c}} results in improved primes 3, 7 and 11, but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 16.4{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.

* Step size: 37.431{{c}}, octave size: 1197.8{{c}}

Compressing the octave of 32edo by around 2{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 18.6{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.

{{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}

{{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}

; [[zpi|133zpi]]  

Compressing the octave of 32edo by around NNN{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 17.4{{c}}. The tuning 133zpi does this.

Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute 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.

[[Lumatone mapping for 32edo]]

* [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'']

* [https://billystiltner.bandcamp.com/album/1332 ''1332'']

* [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'']