# 39edo

 ← 38edo 39edo 40edo →
Prime factorization 3 × 13
Step size 30.7692¢
Fifth 23\39 (707.692¢)
Semitones (A1:m2) 5:2 (153.8¢ : 61.54¢)
Consistency limit 5
Distinct consistency limit 5

39 equal divisions of the octave (abbreviated 39edo or 39ed2), also called 39-tone equal temperament (39tet) or 39 equal temperament (39et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 39 equal parts of about 30.8 ¢ each. Each step represents a frequency ratio of 21/39, or the 39th root of 2.

## Theory

39edo's perfect fifth is 5.8 cents sharp. Together with its best classical major third which is the familiar 400 cents of 12edo, 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 39et, in some few ways, allied to 12et in supporting augene, and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine 11, and adding it to consideration we find that the equal temperament tempers out 99/98 and 121/120 also. This choice for 39et is the 39d val 39 62 91 110 135].

A particular anecdote with this system 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 = 17 + 22). 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 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 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 cents 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).

### Odd harmonics

Approximation of odd harmonics in 39edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +5.7 +13.7 -15.0 +11.5 +2.5 -9.8 -11.3 -12.6 +10.2 -9.2 -12.9
Relative (%) +18.6 +44.5 -48.7 +37.3 +8.2 -31.7 -36.9 -41.1 +33.1 -30.0 -41.9
Steps
(reduced)
62
(23)
91
(13)
109
(31)
124
(7)
135
(18)
144
(27)
152
(35)
159
(3)
166
(10)
171
(15)
176
(20)

### Subsets and supersets

Since 39 factors into 3 × 13, 39edo contains 3edo and 13edo as subsets.

## Intervals

Steps Cents Approximate Ratios* Ups and Downs Notation Nearest Just Interval
(Ratio, Cents, Error)
0 0.00 1/1 P1 perfect unison D 1/1 0.00 None
1 30.77 36/35, 50/49, 55/54, 56/55, 81/80 ^1,
vm2
up unison,
downminor 2nd
^D,
vEb
57/56 30.64 +0.1271
2 61.54 28/27, 33/32, 49/48 m2 minor 2nd Eb 29/28 60.75 +0.7872
3 92.31 16/15, 21/20, 25/24 ^m2 upminor 2nd ^Eb 39/37 91.14 +1.1691
4 123.08 15/14 ^^m2 dupminor 2nd ^^Eb 44/41 122.26 +0.8214
5 153.85 11/10, 12/11 vvM2 dudmajor 2nd vvE 35/32 155.14 -1.2934
6 184.62 10/9 vM2 downmajor 2nd vE 10/9 182.40 +2.2117
7 215.38 9/8, 8/7 M2 major 2nd E 17/15 216.69 -1.3021
8 246.15 81/70 ^M2,
vm3
upmajor 2nd,
downminor 3rd
^E,
vF
15/13 247.74 -1.5873
9 276.92 7/6 m3 minor 3rd F 27/23 277.59 -0.6676
10 307.69 6/5 ^m3 upminor 3rd ^F 43/36 307.61 +0.0846
11 338.46 11/9 ^^m3 dupminor 3rd ^^F 17/14 336.13 +2.3320
12 369.23 27/22 vvM3 dudmajor 3rd vvF# 26/21 369.75 -0.5160
13 400.00 5/4 vM3 downmajor 3rd vF# 34/27 399.09 +0.9096
14 430.77 9/7, 14/11 M3 major 3rd F# 41/32 429.06 +1.7068
15 461.54 35/27 v4 down 4th vG 30/23 459.99 +1.5441
16 492.31 4/3 P4 perfect 4th G 85/64 491.27 +1.0386
17 523.08 27/20 ^4 up 4th ^G 23/17 523.32 -0.2420
18 553.85 11/8 ^^4 dup 4th ^^G 11/8 551.32 +2.5283
19 584.62 7/5 vvA4,
^d5
dudaug 4th,
updim 5th
vvG#,
^Ab
7/5 582.51 +2.1032
20 615.38 10/7 vA4,
^^d5
downaug 4th,
dupdim 5th
vG#,
^^Ab
10/7 617.49 -2.1032
21 646.15 16/11 vv5 dud 5th vvA 16/11 648.68 -2.5283
22 676.92 40/27 v5 down 5th vA 34/23 676.68 +0.2420
23 707.69 3/2 P5 perfect 5th A 128/85 708.73 -1.0386
24 738.46 54/35 ^5 up 5th A^ 23/15 740.01 -1.5441
25 769.23 11/7, 14/9 m6 minor 6th Bb 64/41 770.94 -1.7068
26 800.00 8/5 ^m6 upminor 6th ^Bb 27/17 800.91 -0.9096
27 830.77 44/27 ^^m6 dupminor 6th ^^Bb 21/13 830.25 +0.5160
28 861.54 18/11 vvM6 dudmajor 6th vvB 28/17 863.87 -2.3320
29 892.31 5/3 vM6 downmajor 6th vB 72/43 892.39 -0.0846
30 923.08 12/7 M6 major 6th B 46/27 922.41 +0.6676
31 953.85 140/81 ^M6,
vm7
upmajor 6th,
downminor 7th
^B,
vC
26/15 952.26 +1.5873
32 984.62 7/4, 16/9 m7 minor 7th C 30/17 983.31 +1.3021
33 1015.38 9/5 ^m7 upminor 7th ^C 9/5 1017.60 -2.2117
34 1046.15 11/6, 20/11 ^^m7 dupminor 7th ^^C 64/35 1044.86 +1.2934
35 1076.92 28/15 vvM7 dudmajor 7th vvC# 41/22 1077.74 -0.8214
36 1107.69 15/8, 40/21, 48/25 vM7 downmajor 7th vC# 74/39 1108.86 -1.1691
37 1138.46 27/14, 96/49, 64/33 M7 major 7th C# 56/29 1139.25 -0.7872
38 1169.23 35/18, 49/25, 55/28, 108/55, 160/81 ^M7,
v8
upmajor 7th,
down 8ve
^C#,
vD
112/57 1169.36 -0.1271
39 1200.00 2/1 P8 perfect 8ve D 2/1 1200.00 None

* 11-limit in the 39d val, inconsistent intervals in italic

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

## Notation

### Ups and downs notation

Using Helmholtz–Ellis accidentals, 39edo can also be notated using ups and downs notation:

 Step Offset Sharp Symbol Flat Symbol 0 1 2 3 4 5 6 7 8 9 10 11 12

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.

### Armodue notation

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)
# Cents Armodue Notation Associated Ratios
0 0.0 1 1/1
1 30.8 1‡ (9#)
2 61.5 2b
3 92.3 1#
4 123.1 2v
5 153.8 2 11/10~12/11
6 184.6 2‡
7 · 215.4 3b 8/7
8 246.2 2#
9 276.9 3v
10 307.7 3 6/5~7/6
11 338.5 3‡
12 · 369.2 4b 5/4
13 400.0 3#
14 430.8 4v (5b)
15 461.5 4
16 492.3 4‡ (5v)
17 · 523.1 5 4/3~11/8
18 553.8 5‡ (4#)
19 584.6 6b 10/7
20 615.4 5# 7/5
21 646.2 6v
22 · 676.9 6 3/2~16/11
23 707.7 6‡
24 738.5 7b
25 769.2 6#
26 800.0 7v
27 · 830.8 7 8/5
28 861.5 7‡
29 892.3 8b 5/3~12/7
30 923.1 7#
31 953.8 8v
32 · 984.6 8 7/4
33 1015.4 8‡
34 1046.2 9b 11/6~20/11
35 1076.9 8#
36 1107.7 9v (1b)
37 1138.5 9
38 1169.2 9‡ (1v)
39 ·· 1200.0 1 2/1

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [62 -39 [39 62]] -1.81 1.81 5.88
2.3.5 128/125, 1594323/1562500 [39 62 91]] -3.17 2.42 7.89
2.3.5.7 64/63, 126/125, 2430/2401 [39 62 91 110]] (39d) -3.78 2.35 7.65
2.3.5.7.11 64/63, 99/98, 121/120, 126/125 [39 62 91 110 135]] (39d) -3.17 2.43 7.91

### Rank-2 temperaments

Table of temperaments by generator
Periods
per 8ve
Generator* Cents* Temperaments MOS Scales
1 1\39 30.8
1 2\39 61.5 Unicorn (39d) 1L 18s, 19L 1s
1 4\39 123.1 Negri (39c) 1L 8s, 9L 1s, 10L 9s, 10L 19s
1 5\39 153.8 1L 6s, 7L 1s, 8L 7s, 8L 15s, 8L 23s
1 7\39 215.4 Machine (39d) 1L 4s, 5L 1s, 6L 5s, 11L 6s, 11L 17s
1 8\39 246.2 Immunity (39) / immunized (39d) 4L 1s, 5L 4s, 5L 9s, 5L 14s, 5L 19s, 5L 24s, 5L 29s
1 10\39 307.7 Familia (39df) 3L 1s, 4L 3s, 4L 7s, 4L 11s, 4L 15s, 4L 19s, 4L 23s, 4L 27s, 4L 31s
1 11\39 338.5 Amity (39) / accord (39d) 3L 1s, 4L 3s, 7L 4s, 7L 11s, 7L 18s, 7L 25s
1 14\39 430.8 Hamity (39df) 3L 2s, 3L 5s, 3L 8s, 11L 3s, 14L 11s
1 16\39 492.3 Quasisuper (39d) 2L 3s, 5L 2s, 5L 7s, 5L 12s, 17L 5s
1 17\39 523.1 Mavila (39bc) 2L 3s, 2L 5s, 7L 2s, 7L 9s, 16L 7s
1 19\39 584.6 Pluto (39d) 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, 2L 13s etc. … 2L 35s
3 1\39 30.8
3 2\39 61.5 3L 3s, 3L 6s, 3L 9s, 3L 12s, 3L 15s, 18L 3s
3 6\39 184.6 Terrain / mirkat (39df) 3L 3s, 6L 3s, 6L 9s, 6L 15, 6L 21s, 6L 27s
3 8\39
(5\39)
246.2
(153.8)
Triforce (39) 3L 3s, 6L 3s, 9L 6s, 15L 9s
3 16\39
(3\39)
492.3
(92.3)
Augene (39d) 3L 3s, 3L 6s, 3L 9s, 12L 3s, 12L 15s
3 17\39
(4\39)
523.1
(123.0)
Deflated (39bd) 3L 3s, 3L 6s, 9L 3s, 9L 12s, 9L 21s
13 16\39
(1\39)
492.3
(30.8)
Tridecatonic 13L 13s

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

## 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, 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 6s) or 4 3 6 3 4 3 6 4 3 3. 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 (→ 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 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 39edo keyboard

39edo fretboard visualization

Randy Wells