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 tuning of [[7/1|7]]. The sharp one yields [[superpyth]] temperament, while the flat (patent) one yields [[semaphore]] (and also [[hemifamity]]) temperament.

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 39d is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. 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 assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 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.  

{{Harmonics in equal|39|columns=11}}

{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}

=== As a tuning of other temperaments ===

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. Alternatively, the patent val tempers out 49/48 to yield semaphore. 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]].  

Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.

{| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"

! rowspan="2" | Steps

! rowspan="2" | Cents

! rowspan="2" | Ratios of the<br>[[2.3.5.11 subgroup]]

! colspan="3" rowspan="2" | [[Ups and downs notation]]

| colspan=3 | [[1/1]]

| ''[[28/27]]'', [[64/63]]

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

| [[33/32]]

| ''[[21/20]]'', [[36/35]]

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

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

| ''[[50/49]]''

| [[9/8]]

| ''[[8/7]]''

| [[16/9]]

| ''[[7/4]]''

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

| ''[[49/25]]''

| [[64/33]]

| [[35/18]], ''[[40/21]]''

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

| [[108/55]], [[160/81]]

| [[63/32]], ''[[27/14]]''

| ''[[35/18]]'', [[49/25]]

| colspan=3 | [[2/1]]

=== Stein–Zimmermann–Gould notation ===

 

=== Kite's ups and downs notation ===

39edo can also be notated with [[Kite's ups and downs notation|Kite's 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).

{{Ups and downs sharpness}}

 

=== Interval mappings ===

{{Q-odd-limit intervals}}

{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}

<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

 

 

Some might consider 39edo a candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[approaches to musical tuning|traditions]]; it is one of the simplest edos that can make this claim. Because of this, composers wishing to combine multiple world music traditions (for example, [[gamelan]] with [[maqam]] singing) within one unified framework might find 39edo an interesting possibility.

* [https://www.youtube.com/shorts/1T_xrZpUslQ ''39edo improv''] (2025)

* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)

* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)

 

* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]