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 2^1/39, 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 have two choices for a tuning of 7, but the sharp one blends better with the sharp 3 and 5 as their errors cancel with their ratios, thus yielding significantly higher average accuracy than the patent val in the 7- and 9-odd-limit. It also has a fine 11, and adding it to consideration the best choice for 39et is the sharp-tending 39df val ⟨39 62 91 110 135 145].

As a superpyth system, 39edo is intermediate between 17edo and 22edo (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. 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.

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)

Approximation of odd harmonics in 39edo (continued)

Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	-3.4	-13.6	-14.2	-6.6	+8.3	-1.3	-5.2	-4.0	+1.7	+11.6	-5.6
	Relative (%)	-11.0	-44.1	-46.1	-21.4	+26.9	-4.2	-16.9	-13.1	+5.5	+37.6	-18.2
Steps (reduced)		181 (25)	185 (29)	189 (33)	193 (37)	197 (2)	200 (5)	203 (8)	206 (11)	209 (14)	212 (17)	214 (19)

As a tuning of other temperaments

39edo, with its 400 ¢ major third, 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. 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 12et in supporting 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.

Alternatively, the patent val tempers out 49/48 to yield semaphore, and provides a reasonable tuning of triforce beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports negri and 5-limit sixix.

If we take 22\39 as a fifth, 39edo can be used as a tuning of mavila through the 39bc val, 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 ¢ 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).

Subsets and supersets

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

Intervals

As 39edo is a rare case where a non-patent val does significantly better than the patent val, we provide two tables, for those who look for the most accurate temperament available and for those who would like to explore the potential utilities in this edo.

Table of intervals, 39df val

#	Cents	Approximate ratios*
0	0.0	1/1
1	30.8	36/35, 50/49, 55/54, 56/55, 81/80
2	61.5	22/21, 28/27, 33/32, 49/48
3	92.3	16/15, 21/20, 25/24
4	123.1	14/13, 15/14
5	153.8	11/10, 12/11, 13/12
6	184.6	10/9
7	215.4	9/8, 8/7
8	246.2	15/13
9	276.9	7/6
10	307.7	6/5
11	338.5	11/9, 16/13
12	369.2	26/21, 27/22
13	400.0	5/4
14	430.8	9/7, 14/11
15	461.5	13/10
16	492.3	4/3
17	523.1	27/20
18	553.8	11/8, 18/13, 15/11
19	584.6	7/5
20	615.4	10/7
21	646.2	13/9, 16/11, 22/15
22	676.9	40/27
23	707.7	3/2
24	738.5	20/13
25	769.2	11/7, 14/9
26	800.0	8/5
27	830.8	21/13, 44/27
28	861.5	18/11, 13/8
29	892.3	5/3
30	923.1	12/7
31	953.8	26/15
32	984.6	16/9, 7/4
33	1015.4	9/5
34	1046.2	11/6, 20/11, 24/13
35	1076.9	13/7, 28/15
36	1107.7	15/8, 40/21, 48/25
37	1138.5	21/11, 27/14, 64/33, 96/49
38	1169.2	35/18, 49/25, 108/55, 160/81
39	1200.0	2/1

* As a 13-limit temperament

Table of intervals, various vals

#	Cents	Ratios of the 2.3.11 subgroup	Intervals of 5 and 7
			39c val	39 val	39d val
0	0.0	1/1
1	30.8		28/27, 50/49, 64/63	28/27, 64/63, 81/80	36/35, 50/49, 81/80
2	61.5	33/32, 256/243		21/20, 36/35	22/21, 28/27, 49/48
3	92.3		21/20, 22/21, 36/35	16/15, 22/21, 25/24	16/15, 21/20, 25/24
4	123.1		15/14, 16/15		15/14
5	153.8	12/11	10/9	11/10, 15/14	11/10
6	184.6		11/10	10/9	10/9
7	215.4	9/8			8/7
8	246.2		7/6, 8/7	7/6, 8/7	81/70
9	276.9	32/27			7/6
10	307.7			6/5	6/5
11	338.5	11/9	6/5
12	369.2	27/22	5/4
13	400.0		14/11	5/4, 14/11	5/4
14	430.8	81/64		35/27	9/7, 14/11
15	461.5		9/7, 21/16	9/7, 21/16	35/27
16	492.3	4/3
17	523.1		15/11	27/20	27/20
18	553.8	11/8	27/20	7/5, 15/11	15/11
19	584.6		7/5		7/5
20	615.4		10/7		10/7
21	646.2	16/11	40/27	10/7, 22/15	22/15
22	676.9		22/15	40/27	40/27
23	707.7	3/2
24	738.5		14/9, 32/21	14/9, 32/21	54/35
25	769.2	128/81		54/35	11/7, 14/9
26	800.0		11/7	8/5, 11/7	8/5
27	830.8	44/27	8/5
28	861.5	18/11	5/3
29	892.3			5/3	5/3
30	923.1	27/16			12/7
31	953.8		7/4, 12/7	7/4, 12/7	140/81
32	984.6	16/9			7/4
33	1015.4		20/11	9/5	9/5
34	1046.2	11/6	9/5	20/11, 28/15	20/11
35	1076.9		15/8, 28/15		28/15
36	1107.7		21/11, 35/18, 40/21	15/8, 21/11, 48/25	15/8, 40/21, 48/25
37	1138.5	64/33, 243/128		35/18, 40/21	27/14, 96/49
38	1169.2		27/14, 49/25, 63/32	27/14, 63/32, 160/81	35/18, 49/25, 160/81
39	1200.0	2/1

Proposed interval names and solfèges

Table of proposed interval names and solfèges

#	Cents	Ups and downs notation
0	0.0	P1	perfect unison	D
1	30.8	^1, vm2	up unison, downminor 2nd	^D, vEb
2	61.5	m2	minor 2nd	Eb
3	92.3	^m2	upminor 2nd	^Eb
4	123.1	^^m2	dupminor 2nd	^^Eb
5	153.8	vvM2	dudmajor 2nd	vvE
6	184.6	vM2	downmajor 2nd	vE
7	215.4	M2	major 2nd	E
8	246.2	^M2, vm3	upmajor 2nd, downminor 3rd	^E, vF
9	276.9	m3	minor 3rd	F
10	307.7	^m3	upminor 3rd	^F
11	338.5	^^m3	dupminor 3rd	^^F
12	369.2	vvM3	dudmajor 3rd	vvF#
13	400.0	vM3	downmajor 3rd	vF#
14	430.8	M3	major 3rd	F#
15	461.5	v4	down 4th	vG
16	492.3	P4	perfect 4th	G
17	523.1	^4	up 4th	^G
18	553.8	^^4	dup 4th	^^G
19	584.6	vvA4, ^d5	dudaug 4th, updim 5th	vvG#, ^Ab
20	615.4	vA4, ^^d5	downaug 4th, dupdim 5th	vG#, ^^Ab
21	646.2	vv5	dud 5th	vvA
22	676.9	v5	down 5th	vA
23	707.7	P5	perfect 5th	A
24	738.5	^5	up 5th	A^
25	769.2	m6	minor 6th	Bb
26	800.0	^m6	upminor 6th	^Bb
27	830.8	^^m6	dupminor 6th	^^Bb
28	861.5	vvM6	dudmajor 6th	vvB
29	892.3	vM6	downmajor 6th	vB
30	923.1	M6	major 6th	B
31	953.8	^M6, vm7	upmajor 6th, downminor 7th	^B, vC
32	984.6	m7	minor 7th	C
33	1015.4	^m7	upminor 7th	^C
34	1046.2	^^m7	dupminor 7th	^^C
35	1076.9	vvM7	dudmajor 7th	vvC#
36	1107.7	vM7	downmajor 7th	vC#
37	1138.5	M7	major 7th	C#
38	1169.2	^M7, v8	upmajor 7th, down 8ve	^C#, vD
39	1200.0	P8	perfect 8ve	D

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Kite's ups and downs notation #Chords and chord progressions.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol													
Flat symbol													

Kite's ups and downs notation

39edo can also be notated with 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).

Step offset	0	1	2	3	4	5	6	7	8	9	10	11
Sharp symbol
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as 46edo.

Evo flavor

Revo flavor

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

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 39edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 39edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/13, 26/15	1.587	5.2
7/5, 10/7	2.103	6.8
9/5, 10/9	2.212	7.2
11/8, 16/11	2.528	8.2
11/6, 12/11	3.209	10.4
15/14, 28/15	3.634	11.8
9/7, 14/9	4.315	14.0
13/7, 14/13	5.221	17.0
3/2, 4/3	5.737	18.6
13/10, 20/13	7.325	23.8
5/3, 6/5	7.949	25.8
11/9, 18/11	8.946	29.1
13/9, 18/13	9.536	31.0
13/8, 16/13	9.758	31.7
7/6, 12/7	10.052	32.7
11/10, 20/11	11.158	36.3
15/8, 16/15	11.346	36.9
9/8, 16/9	11.475	37.3
13/11, 22/13	12.287	39.9
11/7, 14/11	13.261	43.1
5/4, 8/5	13.686	44.5
15/11, 22/15	13.874	45.1
7/4, 8/7	14.980	48.7
13/12, 24/13	15.273	49.6

15-odd-limit intervals in 39edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
9/5, 10/9	2.212	7.2
11/8, 16/11	2.528	8.2
11/6, 12/11	3.209	10.4
13/7, 14/13	5.221	17.0
3/2, 4/3	5.737	18.6
5/3, 6/5	7.949	25.8
11/9, 18/11	8.946	29.1
13/8, 16/13	9.758	31.7
11/10, 20/11	11.158	36.3
9/8, 16/9	11.475	37.3
13/11, 22/13	12.287	39.9
5/4, 8/5	13.686	44.5
7/4, 8/7	14.980	48.7
13/12, 24/13	15.496	50.4
15/11, 22/15	16.895	54.9
11/7, 14/11	17.508	56.9
15/8, 16/15	19.424	63.1
7/6, 12/7	20.717	67.3
13/9, 18/13	21.233	69.0
13/10, 20/13	23.445	76.2
9/7, 14/9	26.454	86.0
7/5, 10/7	28.666	93.2
15/13, 26/15	29.182	94.8
15/14, 28/15	34.403	111.8

15-odd-limit intervals by 39df val mapping

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/13, 26/15	1.587	5.2
7/5, 10/7	2.103	6.8
9/5, 10/9	2.212	7.2
11/8, 16/11	2.528	8.2
11/6, 12/11	3.209	10.4
15/14, 28/15	3.634	11.8
9/7, 14/9	4.315	14.0
13/7, 14/13	5.221	17.0
3/2, 4/3	5.737	18.6
13/10, 20/13	7.325	23.8
5/3, 6/5	7.949	25.8
11/9, 18/11	8.946	29.1
13/9, 18/13	9.536	31.0
7/6, 12/7	10.052	32.7
11/10, 20/11	11.158	36.3
9/8, 16/9	11.475	37.3
11/7, 14/11	13.261	43.1
5/4, 8/5	13.686	44.5
13/12, 24/13	15.273	49.6
7/4, 8/7	15.789	51.3
15/11, 22/15	16.895	54.9
13/11, 22/13	18.483	60.1
15/8, 16/15	19.424	63.1
13/8, 16/13	21.011	68.3

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 rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Temperament	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 distinct

Octave stretch or compression

39edo is a zeta valley edo and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly compressing the octave, to get a tuning like 101ed6 or 173zpi.

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.

39edo and world music

Some might consider 39edo a 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 might 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 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 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 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, Iranian

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 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 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 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^{[clarification needed]} and African ^{[clarification needed]} musical styles can thus be accommodated.

Scales

• Quasisuper[7] MOS scale: 7 7 2 7 7 7 2
• Quasisuper[7] pental modmos: 7 6 3 7 6 7 3
• 3L 6s modmos: 7 3 3 3 7 3 3 7 3
• Extended quasisuper: 4 3 6 3 4 3 6 4 3 3
• 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

Instruments

Lumatone mapping

See Lumatone mapping for 39edo

Skip fretting

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

Prime harmonics

1/1: string 2 open

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

An illustrative image of a 39edo keyboard

39edo fretboard visualization

Music

Modern renderings

HOYO-MiX

• "Sinner's Finale" from Genshin Impact OST (2023) – covered by Bryan Deister (2025)

21st century

Bryan Deister

• 39edo (2023)
• 39edo jam (2025)
• 39edo improv (2025)
• Waltz in 39edo (2025)
• Tilt Your Head Down (2026)

groundfault

• From Souvenirs of the Affliction (2025) – Bandcamp | YouTube
  • "Resolute Prelude"
  • "Residual Soliloquy"

Randy Wells

• Romance On Other Planets (2021)