# 39edo

(Redirected from 39-edo)

39-EDO, 39-ED2 or 39-tET divides the octave in 39 equal parts of 30.76923 cents each one.

## Theory

If we take 22\39 as a fifth, 39edo can be used in mavila temperament, 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 superdiatonic LLLsLLLLs like a basical scale for notation and theory, suited in 16-ED2, and allied systems: 25-ED2 [1/3-tone 3;2]; 41-ED2 [1/5-tone 5;3]; and 57-ED2 [1/7-tone 7;4]. Hornbostel temperaments is included too with: 23-ED2 [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] & 62-ED2 [1/8-tone 8;3].

However, its 23\39 fifth, 5.737 cents sharp, is in much better tune than the mavila fifth which like all mavila fifths is very, very flat, in this case, 25 cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is 39 62 91 110 135].

A particular anecdote with this 39 divisions per 2/1 was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740?–1820), a little extract here.

As a superpyth system, 39edo is intermediate between 17edo and 22edo (39 being 17+22). While 17edo is superb for melody (as documented by George Secor), it doesn't 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 quarter-tone-sized, which results in a very strange-sounding 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 (the ideal diatonic semitone for melody being somewhere in between 60 and 80 cents, by Secor's estimates).

39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it doesn't 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).

## Intervals

 Armodue nomenclature 5;2 relation ‡ = Semisharp (1/5-tone up) b = Flat (3/5-tone down) # = Sharp (3/5-tone up) v = Semiflat (1/5-tone down)
Degree Cents Approximate Ratios
in 39d Val
Armodue
Notation
Ups and Downs Notation Nearest just interval
(Ratio, Cents, Error)
0 0.0000 1/1 1 P1 perfect unison D 1/1 0.0000 None
1 30.7692 81/80, 36/35 1‡ (9#) ^1 up unison,
downminor 2nd
^D,
vEb
57/56 30.6421 +0.1271
2 61.5385 28/27, 49/48, 33/32 2b m2 minor 2nd Eb 29/28 60.7513 +0.7872
3 92.3077 16/15, 25/24, 21/20 1# ^m2 upminor 2nd ^Eb 39/37 91.1386 +1.1691
4 123.0769 15/14 2v v~2 downmid 2nd ^^Eb 44/41 122.2555 +0.8214
5 153.8462 12/11, 11/10 2 ^~2 upmid 2nd vvE 35/32 155.1396 -1.2934
6 184.6154 10/9 2‡ vM2 downmajor 2nd vE 10/9 182.4037 +2.2117
7· 215.3846 8/7, 9/8 3b M2 major 2nd E 17/15 216.6867 -1.3021
8 246.1538 2# ^M2,
vm3
upmajor 2nd,
downminor 3rd
^E,
vF
15/13 247.7411 -1.5873
9 276.9231 7/6 3v m3 minor 3rd F 27/23 277.5907 -0.6676
10 307.6923 6/5 3 ^m3 upminor 3rd ^F 43/36 307.6077 +0.0846
11 338.4615 11/9 3‡ v~3 downmid 3rd ^^F 17/14 336.1295 +2.3320
12· 369.2308 27/22 4b ^~3 upmid 3rd vvF# 26/21 369.7468 -0.5160
13 400.0000 5/4 3# vM3 downmajor 3rd vF# 34/27 399.0904 +0.9096
14 430.7692 9/7, 14/11 4v (5b) M3 major 3rd F# 41/32 429.0624 +1.7068
15 461.5385 4 v4 down 4th vG 30/23 459.9944 +1.5441
16 492.3077 4/3 4‡ (5v) P4 perfect 4th G 85/64 491.2691 +1.0386
17· 523.0769 27/20 5 ^4 up 4th ^G 23/17 523.3189 -0.2420
18 553.8462 11/8 5‡ (4#) v~4 downmid 4th ^^G 11/8 551.3179 +2.5283
19 584.6154 7/5 6b ^~4,
^d5
upmid 4th,
updim 5th
vvG#,
^Ab
7/5 582.5122 +2.1032
20 615.3846 10/7 5# vA4,
v~5
downaug 4th,
downmid 5th
vG#,
^^Ab
10/7 617.4878 -2.1032
21 646.1538 16/11 6v ^~5 upmid 5th vvA 16/11 648.6821 -2.5283
22· 676.9231 40/27 6 v5 down 5th vA 34/23 676.6811 +0.2420
23 707.6923 3/2 6‡ P5 perfect 5th A 128/85 708.7309 -1.0386
24 738.4615 7b ^5 up 5th A^ 23/15 740.0056 -1.5441
25 769.2308 14/9, 11/7 6# m6 minor 6th Bb 64/41 770.9376 -1.7068
26 800.0000 8/5 7v ^m6 upminor 6th ^Bb 27/17 800.9096 -0.9096
27· 830.7692 44/27 7 v~6 downmid 6th ^^Bb 21/13 830.2532 +0.5160
28 861.5385 18/11 7‡ ^~6 upmid 6th vvB 28/17 863.8705 -2.3320
29 892.3077 5/3 8b vM6 downmajor 6th vB 72/43 892.3923 -0.0846
30 923.0769 12/7 7# M6 major 6th B 46/27 922.4093 +0.6676
31 953.8462 8v ^M6,
vm7
upmajor 6th,
downminor 7th
^B,
vC
26/15 952.2589 +1.5873
32· 984.6154 7/4, 16/9 8 m7 minor 7th C 30/17 983.3133 +1.3021
33 1015.3846 9/5 8‡ ^m7 upminor 7th ^C 9/5 1017.5963 -2.2117
34 1046.1538 11/6, 20/11 9b v~7 downmid 7th ^^C 64/35 1044.8604 +1.2934
35 1076.9231 28/15 8# ^~7 upmid 7th vvC# 41/22 1077.7445 -0.8214
36 1107.6923 15/8, 48/25, 40/21 9v (1b) vM7 downmajor 7th vC# 74/39 1108.8614 -1.1691
37 1138.4615 27/14, 49/48, 64/33 9 M7 major 7th C# 56/29 1139.2487 -0.7872
38 1169.2308 160/81, 35/18 9‡ (1v) v8 up-major 7th
down-8ve
^C#,
vD
112/57 1169.3579 -0.1271
39·· 1200.0000 2/1 1 P8 perfect 8ve D 2/1 1200.0000 None

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation #Chords and Chord Progressions.

## Just approximation

### Selected just intervals

prime 2 prime 3 prime 5 prime 7 prime 11
Error absolute (¢) 0.0 +5.7 +13.7 -15.0 +2.5
relative (%) 0.0 +18.6 +44.5 -48.7 +8.2

### Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 39et.

Note: the 39d val is used for lower error.

3-limit 5-limit 7-limit 11-limit
Octave stretch (¢) -1.81 -3.17 -3.78 -3.17
Error absolute (¢) 1.81 2.42 2.35 2.43
relative (%) 5.88 7.89 7.65 7.91

## Scales

14 14 11 - MOS of type 2L 1s

11 11 11 6 - MOS of type 3L 1s

10 10 10 9 - MOS of type 3L 1s

11 3 11 11 3 - MOS of type 3L 2s (Father pentatonic)

5 12 5 5 12 - MOS of type 2L 3s (Mavila pentatonic)

7 7 9 7 9 - MOS of type 2L 3s (Superpythagorean pentatonic)

8 8 8 8 7 - MOS of type 4L 1s (Bug pentatonic)

10 3 10 3 10 3 - MOS of type 3L 3s (Augmented hexatonic)

9 4 9 4 9 4 - MOS of type 3L 3s (Augmented hexatonic)

8 5 8 5 8 5 - MOS of type 3L 3s (Augmented hexatonic)

7 7 7 7 7 4 - MOS of type 5L 1s (Grumpy hexatonic)

5 5 7 5 5 5 7 - MOS of type 2L 5s (heptatonic Mavila Anti-Diatonic)

7 7 7 2 7 7 2 - MOS of type 5L 2s (heptatonic Superpythagorean diatonic)

5 5 5 5 5 5 5 4 - MOS of type 7L 1s (Grumpy octatonic)

5 5 5 2 5 5 5 5 2 - MOS of type 7L 2s (nonatonic Mavila Superdiatonic)

5 5 3 5 5 3 5 5 3 - MOS of type 6L 3s (unfair Augmented nonatonic)

5 4 4 5 4 4 5 4 4 - MOS of type 3L 6s (fair Augmented nonatonic)

4 4 4 4 4 4 4 4 4 3 - MOS of type 9L 1s (Grumpy decatonic)

3 3 5 3 3 3 5 3 3 3 5 - MOS of type 3L 8s (Anti-Sensi hendecatonic)

2 5 2 2 5 2 5 2 5 2 2 5 - MOS of type 5L 7s

3 3 3 4 3 3 3 4 3 3 3 4 - MOS of type 3L 9s

3 3 3 2 3 3 3 3 2 3 3 3 3 2 - MOS of type 11L 3s (Ketradektriatoh tetradecatonic)

3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 - MOS of type 9L 6s

3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 2 - MOS of type 7L 9s

2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 2 3 - MOS of type 5L 12s

2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 - MOS of type 3L 15s

3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 - MOS of type 10L 9s

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 - MOS of type 19L 1s

2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 - MOS of type 17L 5s

2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1 - MOS of type 16L 7s

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 - MOS of type 13L 13s

2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 - MOS of type 10L 19s

2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 - MOS of type 8L 23s

## 39edo and world music

39edo is a good candidate for a "universal tuning" in that it offers reasonable approximations of many different world music 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 would find 39edo an interesting possibility.

### Western

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 very pleasing.

Another option is to use a MODMOS, such as 7 6 3 7 6 7 3; this scale enables us to continue using pental rather than septimal thirds, but it has a false (wolf) fifth. When translating diatonic compositions into this scale, the wolf fifth can be avoided by introducing accidental notes when necessary. There are other MODMOS's 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's all have smaller-than-usual 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 Pachelbel's Canon in 39edo (using the 7 6 3 7 6 7 3 MODMOS), for example.

### Indian

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 5s MOS (where the generator is a perfect fifth).

### Arabic, Turkish, Persian

While 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)

whereas neither 17edo nor 24edo satisfy these properties.

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 would have loved augene.

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 mappings 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.

### Other

39edo offers a good approximation of pelog / mavila using the flat fifth as a generator.

It also offers many possible pentatonic scales, including the 2L+3S MOS (which is 9 7 7 9 7). Slendro can be approximated using this scale or using something like the quasi-equal 8 8 8 8 7. A more expressive pentatonic scale is the oneirotonic subset 9 6 9 9 6. Many Asian and African musical styles can thus be accommodated.

## Instruments (prototypes)

An illustrative image of a 39-ED2 keyboard

39-EDD fretboard visualization