39edo: Difference between revisions

{{ED intro}}

 

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.  

Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.  

39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

39edo is a reasonable tuning of [[triforce]] beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp.

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

 

 

{| 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="2" | Intervals of 7

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

! Patent val

! 39d val

| 0.0

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

| 30.8

| [[55/54]], [[81/80]]

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

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

| ^1, <br>vm2

| 61.5

| [[33/32]]

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

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

| 92.3

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

| [[21/20]]

| 123.1

| 153.8

| [[11/10]], [[12/11]]

| ''[[15/14]]''

|

| 184.6

|

| 7

| 215.4

| [[9/8]]

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

| 246.2

| 276.9

| 307.7

|

| 338.5

|

| 12

| 369.2

|

| 400.0

| ''[[14/11]]''

| 430.8

| 461.5

| 492.3

|

| 17

| 523.1

|

| 553.8

| ''[[7/5]]''

| 584.6

| 615.4

| 646.2

| ''[[10/7]]''

| 22

| 676.9

|

| 707.7

|

| 738.5

| 769.2

|

| ''[[54/35]]''

| [[11/7]], [[14/9]]

| 800.0

| ''[[11/7]]''

| 27

| 830.8

|

| 861.5

|

| 892.3

|

| 923.1

| 953.8

| 32

| 984.6

| [[16/9]]

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

| 1015.4

|

| 1046.2

| ''[[28/15]]''

| 1076.9

| 1107.7

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

| [[40/21]]

| 1138.5

| [[64/33]]

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

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

| 1169.2

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

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

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

| ^M7, <br>v8

| upmajor 7th, <br>down 8ve

| 39

| 1200.0

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

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

! rowspan="2" | [[Comma list]]

! rowspan="2" | Optimal <br>8ve stretch (¢)

! colspan="2" | Tuning error

| {{Monzo| 62 -39 }}

| {{Mapping| 39 62 }}

| −1.81

| {{Mapping| 39 62 91 }}

| −3.17

| {{Mapping| 39 62 91 110 }} (39d)

| −3.78

| {{Mapping| 39 62 91 110 135 }} (39d)

| −3.17

{| class="wikitable center-all left-4 left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods <br />per 8ve

! Temperament

! Mos scales

| [[1L&nbsp;18s]], [[19L&nbsp;1s]]

| [[Negri]] (39c)

| [[1L&nbsp;8s]], [[9L&nbsp;1s]], [[10L&nbsp;9s]], [[10L&nbsp;19s]]

| [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[8L&nbsp;7s]], [[8L&nbsp;15s]], [[8L&nbsp;23s]]

| [[1L&nbsp;4s]], [[5L&nbsp;1s]], [[6L&nbsp;5s]], [[11L&nbsp;6s]], [[11L&nbsp;17s]]

| [[4L&nbsp;1s]], [[5L&nbsp;4s]], [[5L&nbsp;9s]], [[5L&nbsp;14s]], [[5L&nbsp;19s]], [[5L&nbsp;24s]], [[5L&nbsp;29s]]

| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], [[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], [[4L&nbsp;23s]], [[4L&nbsp;27s]], [[4L&nbsp;31s]]

| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[7L&nbsp;4s]], [[7L&nbsp;11s]], [[7L&nbsp;18s]], [[7L&nbsp;25s]]

| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], [[11L&nbsp;3s]], [[14L&nbsp;11s]]

| [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]

| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[7L&nbsp;2s]], [[7L&nbsp;9s]], [[16L&nbsp;7s]]

| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]] etc. … [[2L&nbsp;35s]]

| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[3L&nbsp;12s]], [[3L&nbsp;15s]], [[18L&nbsp;3s]]

| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[6L&nbsp;9s]], [[6L 15]], [[6L&nbsp;21s]], [[6L&nbsp;27s]]

| 8\39 <br>(5\39)

| 246.2 <br>(153.8)

| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[9L&nbsp;6s]], [[15L&nbsp;9s]]

| 16\39 <br>(3\39)

| 492.3 <br>(92.3)

| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[12L&nbsp;3s]], [[12L&nbsp;15s]]

| 17\39 <br>(4\39)

| 523.1 <br>(123.0)

| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[9L&nbsp;3s]], [[9L&nbsp;12s]], [[9L&nbsp;21s]]

| 16\39 <br>(1\39)

| 492.3 <br>(30.8)

| [[13L&nbsp;13s]]

<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

 

 

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.

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.

Another option is to use a [[modmos]], such as 7 6 3 7 6 7 3; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either 7 3 3 3 7 3 3 7 3 (a [[modmos]] of type [[3L&nbsp;6s]]) or 4 3 6 3 4 3 6 4 3 3. There are other modmos scales that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|semitones]], which makes them more effective for melody than their counterparts in 12edo or meantone systems.

Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that [[xenharmonic]] to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the 7 6 3 7 6 7 3 modmos), for example.

=== Arabic, Turkish, Iranian ===

While [[Arabic, Turkish, Persian music|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:

39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] mos (which is 9 7 7 9 7). [[Slendro]] can be approximated using that scale or using something like the [[quasi-equal]] 8 8 8 8 7 or 8 8 7 8 8.  

One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.  

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

* Quasisuper[22] MOS scale (resembles [[Indian]] [[sruti]]): 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 2

* Slendro approximations: 9 7 7 9 7 or 8 8 8 8 7 or 8 8 7 8 8

* An expressive [[oneirotonic]] subset: 9 6 9 9 6

* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''

== Music ==

 

=== 21st century ===

* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)

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

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

 

; [[groundfault]]

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

* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)