39edo: Difference between revisions

39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]]. It also has a fine [[11/1|11]], and adding it to consideration the choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the [[diatonic semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.  

39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].  

{| class="wikitable center-all right-2 left-3 right-9 right-10"

! Steps

! Cents

| [[1/1]]

| ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]

| [[28/27]], [[33/32]], ''[[49/48]]''

| ''[[16/15]]'', [[21/20]], ''[[25/24]]''

| [[9/8]], ''[[8/7]]''

| ''[[7/4]]'', [[16/9]]

| ''[[15/8]]'', [[40/21]], ''[[48/25]]''

| [[27/14]], ''[[96/49]]'', [[64/33]]

| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]

| [[2/1]]

=== Ups and downs notation ===

39edo 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).

39edo is a [[zeta valley edo]] and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|101ed6]] or [[zpi|173zpi]].

39edo can be usefully mapped onto the val 39dfgijk. The [[Tenney-Euclidean]] tuning of this regular temperament is 30.67475 cents per step, which is closely approximated by [[62edt]] and [[173zpi]].

* ''Tilt Your Head Down - 39edo '' (2026) [https://www.youtube.com/shorts/Tf812DJzoEA <nowiki>[short]</nowiki>], [https://www.youtube.com/watch?v=Vzife15uUU4 <nowiki>[full song]</nowiki>]