39edo: Difference between revisions

A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|Charles Clagget}} (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here].

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.  

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).

{{Harmonics in equal|39}}

=== Octave stretch ===

 

=== Subsets and supersets ===

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

! Steps

! Approximate ratios*

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

! colspan="3" | [[Nearest just interval]] <br>(Ratio, cents, error)

| 0.0

| 0.00

| 30.8

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

| ^1, <br>vm2

| 30.64

| 61.5

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

| 60.75

| 92.3

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

| 91.14

| 123.1

| ^^m2

| dupminor 2nd

| 122.26

| 153.8

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

| vvM2

| dudmajor 2nd

| 155.14

| 184.6

| 182.40

| 7

| 215.4

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

| 216.69

| 246.2

| 247.74

| 276.9

| 277.59

| 307.7

| 307.61

| 338.5

| ^^m3

| dupminor 3rd

| 336.13

| 12

| 369.2

| vvM3

| dudmajor 3rd

| 369.75

| 400.0

| 399.09

| 430.8

| 429.06

| 461.5

| 459.99

| 492.3

| 491.27

| 17

| 523.1

| 523.32

| 553.8

| ^^4

| dup 4th

| 551.32

| 584.6

| vvA4, <br>^d5

| dudaug 4th, <br>updim 5th

| 582.51

| 615.4

| vA4, <br>^^d5

| downaug 4th, <br>dupdim 5th

| 617.49

| 646.2

| vv5

| dud 5th

| 648.68

| 22

| 676.9

| 676.68

| 707.7

| 708.73

| 738.5

| 740.01

| 769.2

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

| 770.94

| 800.0

| 800.91

| 27

| 830.8

| ^^m6

| dupminor 6th

| 830.25

| 861.5

| vvM6

| dudmajor 6th

| 863.87

| 892.3

| 892.39

| 923.1

| 922.41

| 953.8

| 952.26

| 32

| 984.6

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

| 983.31

| 1015.4

| 1017.60

| 1046.2

| ^^m7

| dupminor 7th

| 1044.86

| 1076.9

| vvM7

| dudmajor 7th

| 1077.74

| 1107.7

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

| 1108.86

| 1138.5

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

| 1139.25

| 1169.2

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

| ^M7, <br>v8

| upmajor 7th, <br>down 8ve

| 1169.36

| 39

| 1200.0

| 1200.00

File:39-EDO_Revo_Sagittal.svg

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 120 80 240 106 [[33/32]]

default [[File:39-EDO_Revo_Sagittal.svg]]

 

{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"

! colspan="2" | #

! Cents

| 1

| [[1/1]]

|-

| 30.8

|  

| 2

| 4‡ (5v)

|  

| 523.1

| 18

| 769.2

| 6#

| 892.3

| 5/3~12/7

|-

| 984.6

| 7/4

| 8#

| 1200.0

| 2/1

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

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

! rowspan="2" | [[Mapping]]

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

! colspan="2" | Tuning error

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

| 2.3

| {{Mapping| 39 62 }}

| −1.81

=== Rank-2 temperaments ===

39edo is a good 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.

39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic mos: 7 7 2 7 7 7 2). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.

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.

A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L&nbsp;5s]] MOS (where the generator is a perfect fifth).

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:

* It has two types of "neutral" seconds (154 and 185 cents)

* It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents)

39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a "major-like" wide neutral third and a wide "neutral" second approaching 10/9), will likely be especially well suited to 39edo.

=== Blues / Jazz / African-American ===

The [[harmonic seventh]] ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (→ [[Wikipedia: Coltrane changes]]).

[[Tritone]] substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension.

Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a "blue major third" can be identified as either of the two neutral thirds. There are two possible [[mapping]]s for [[7/4]] which are about equal in closeness. The sharp mapping is the normal one because it works better with the [[5/4]] and [[3/2]], but using the flat one instead (as an accidental) allows for another type of blue note.

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.

See [[Lumatone mapping for 39edo]]

'''Skip fretting system 39 2 5''' is a [[skip-fretting]] system for [[39edo]]. All examples on this page are for 7-string [[Guitar|guitar.]]

2/1: string 5 fret 12 and string 7 fret 7

3/2: string 3 fret 9 and string 5 fret 4

5/4: string 1 fret 9 and string 3 fret 4

7/4: string 5 fret 8 and string 7 fret 3  

11/8: string 2 fret 9 and string 4 fret 4

=== Prototypes ===

[[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]

''An illustrative image of a 39edo keyboard''

[[File:Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|alt=Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|826x203px|Custom_700mm_5-str_Tricesanonaphonic_Guitar.png]]

''39edo fretboard visualization''

* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)

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

* [https://www.youtube.com/shorts/4y11CWLIHNA ''Sinner's Finale - Genshin Impact (microtonal cover in 39edo)''] (2025)

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

[[Category:Listen]]