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.  

32edo provides a good 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. Sixix temperament is generated by an approximate [[6/5]], which is tuned very sharp so that 5 of them octave-reduce to [[4/3]]. [[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. The 9\32 generator can also be used for the much more inaccurate [[mohavila]] temperament which treats the 675-cent fifth from [[16edo]] as 3/2 (basically it's [[mavila]]'s response to neutral third temperaments).

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 [[kangaroo]] temperament, tempering out [[55/54]], 64/63, and [[245/242]]. This is generated by a flat [[16/11]] such that 3 of them reduce to [[3/2]] and 6 of them reduce to [[8/7]]. Its 15-note scale could potentially be used as a [[detempering]] of [[15edo]].

The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The diatonic [[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]]. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675{{c}} (inherited from [[16edo]]), but it is much more inaccurate than the sharp fifth, so 32edo cannot truly be considered a [[dual-fifth]] system.

{{Harmonics in equal|32}}

Since 32 is a power of two and factors as 2⁵, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.

! 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

| [[Slender]] (32)

| 7/5

| [[Progress]] (32cf)

| 14\33<br>(1\32)

| 487.5<br>(37.5)

| 4/3<br>(36/35)

| 14\33<br>(1\32)

| 487.5<br>(37.5)

| 4/3<br>(45/44)

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct