39 tone equal temperament

39-EDO, 39-ED2 or 39-tET divides the Octave (Ditave 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view 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]. 223-ED2, the best accuracy for Hornbostel temperament fits very good with Armodue like 1/29-tone 29;10 version. Note that 101, 131, 177 & 200 ED2s are tempered systems that Alexei Ogolevets (Ukraine, 1891 - 1967) was proposing in his List of Temperaments, in which the Armodue system fits very well in all these.

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).

39-EDO 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)

Degrees	Cents size	7mus	Armodue note	ups and downs notation			Nearest Just Interval	Cents value	7mus	Error	11-limit Ratio Assuming <39 62 91 110 135\| Val
0	0.0000	0	1	P1	perfect unison	D	1/1	0.0000	0	None	1/1
1	30.7692	39.3846 (27.6276₁₆)	1‡ (9#)	^1	up unison	D^	57/56	30.6421	39.2219 (27.38CF₁₆)	+0.1271
2	61.5385	78.7692 (4E.C4EC₁₆)	2b	m2	minor 2nd	Eb	29/28	60.7513	77.76165 (4D.C2FB₁₆)	+0.7872
3	92.3077	118.15385 (76.2762₁₆)	1#	^m2	upminor 2nd	Eb^	39/37	91.1386	116.6574 (74.A84E₁₆)	+1.1691
4	123.0769	157.5385 (9D.89D9₁₆)	2v	v~2	downmid 2nd	Eb^^	44/41	122.2555	156.4871 (9C.7CB2₁₆)	+0.8214
5	153.84615	196.9231 (C4.EC4F₁₆	2	^~2	upmid 2nd	Evv	35/32	155.1396	198.5787 (C6.9427₁₆)	-1.2934	12/11, 11/10
6	184.6154	236.3077 (EC.4EC5₁₆)	2‡	vM2	downmajor 2nd	Ev	10/9	182.4037	233.47675 (E9.7A0C₁₆)	+2.2117	10/9
7·	215.3846	275.6923 (113.B13B₁₆)	3b	M2	major 2nd	E	17/15	216.6867	277.359 (115.5BE5₁₆)	-1.3021	8/7, 9/8
8	246.15385	315.0769 (13B.13B1₁₆)	2#	^M2, vm3	upmajor 2nd, downminor 3rd	E^, Fv	15/13	247.7411	317.10855 (13D.1BCA₁₆)	-1.5873
9	276.9231	354.4615 (162.7627₁₆)	3v	m3	minor 3rd	F	27/23	277.5907	355.316 (163.50E8₁₆)	-0.6676	7/6
10	307.6923	393.84615 (189.D89E₁₆)	3	^m3	upminor 3rd	F^	43/36	307.6077	393.7379 (189.BCE4₁₆)	+0.0846	6/5
11	338.4615	433.2308 (1B1.3B14₁₆)	3‡	v~3	downmid 3rd	F^^	17/14	336.1295	430.2458 (1AE.3EEA₁₆)	+2.332	11/9
12·	369.2308	472.6154 (1D8.9D8A₁₆)	4b	^~3	upmid 3rd	F#vv	26/21	369.7468	473.27585 (1D9.469E₁₆)	-0.516
13	400	512 (200₁₆)	3#	vM3	downmajor 3rd	F#v	34/27	399.0904	510.8357 (1FE.D5F2₁₆)	+0.9096	5/4
14	430.7692	551.3846 (227.6276₁₆)	4v (5b)	M3	major 3rd	F#	41/32	429.0624	549.1999 (225.332B₁₆)	+1.7068	9/7, 14/11
15	461.5385	590.7692 (24E.C4EC₁₆)	4	v4	down 4th	Gv	30/23	459.9944	588.7928 (24C.CAF4₁₆)	+1.5441
16	492.3077	630.15385 (276.2762₁₆)	4‡ (5v)	P4	perfect 4th	G	85/64	491.2691	628.8245 (274.D311₁₆)	+1.0386	4/3
17·	523.0769	669.5385 (29D.89D9₁₆)	5	^4	up 4th	G^	23/17	523.3189	669.8482 (29D.D926₁₆)	-0.242
18	553.8462	708.9231 (2C4.EC4D₁₆)	5‡ (4#)	^^4	double-up 4th	G^^	11/8	551.3179	705.687 (2C1.AFDD₁₆)	+2.5283	11/8
19	584.6154	748.3077 (2EC.4EC5₁₆)	6b	vvA4, ^d5	double-down aug 4th, updim 5th	G#vv, Ab^	7/5	582.5122	745.6156 (2E9.9D98₁₆)	+2.1032	7/5
20	615.3846	787.6923 (313.B13B₁₆)	5#	vA4, ^^d5	downaug 4th, double-up dim 5th	G#v, Ab^^	10/7	617.4878	790.3844 (316.6268₁₆)	-2.1032	10/7
21	646.1538	827.0769 (33B.13B1₁₆)	6v	vv5	double-down 5th	Avv	16/11	648.6821	830.313 (33E.5023₁₆)	-2.5283	16/11
22·	676.9231	866.4615 (362.7627₁₆)	6	v5	down 5th	Av	34/23	676.6811	866.1518 (362.26DA₁₆)	+0.242
23	707.6923	905.84615 (389.D89E₁₆)	6‡	P5	perfect 5th	A	128/85	708.7309	907.1755 (38B.2CDE₁₆)	-1.0386	3/2
24	738.4615	945.2308 (3B1.3B14₁₆)	7b	^5	up 5th	A^	23/15	740.0056	947.2072 (3B3.350C₁₆)	-1.5441
25	769.2308	984.6154 (3D8.9D8A₁₆)	6#	m6	minor 6th	Bb	64/41	770.9376	986.7991 (3DA.CCD4₁₆)	-1.7068	14/9, 11/7
26	800	1024 (400₁₆)	7v	^m6	upminor 6th	Bb^	27/17	800.9096	1025.1643 (401.2A0E₁₆)	-0.9096	8/5
27·	830.7692	1063.3846 (427.6276₁₆)	7	v~6	downmid 6th	Bb^^	21/13	830.2532	1062.72415 (426.B962₁₆)	+0.516
28	861.5385	1102.7692 (44E.C4EC₁₆)	7‡	^~6	upmid 6th	Bvv	28/17	863.8705	1105.7542 (451.C116₁₆)	-2.332	18/11
29	892.3077	1142.15385 (476.2762₁₆)	8b	vM6	downmajor 6th	Bv	72/43	892.3923	1142.2621 (476.431B₁₆)	-0.0846	5/3
30	923.0769	1181.5385 (49D.89D9₁₆)	7#	M6	major 6th	B	46/27	922.4093	1180.684 (49C.AF18₁₆)	+0.6676	12/7
31	953.8462	1220.9231 (4C4.EC4D₁₆)	8v	^M6, vm7	upmajor 6th, downminor 7th	B^, Cv	26/15	952.2589	1218.89145 (4C2.E453₁₆)	+1.5873
32·	984.6154	1260.3077 (4EC.4EC5₁₆)	8	m7	minor 7th	C	30/17	983.3133	1258.641 (4EA.A41B₁₆)	+1.3021	7/4, 16/9
33	1015.3846	1299.6923 (513.B13B₁₆)	8‡	^m7	upminor 7th	C^	9/5	1017.5963	1302.52325 (516.85F3₁₆)	-2.2117	9/5
34	1046.1538	1339.0769 (53B.13B1₁₆)	9b	v~7	downmid 7th	C^^	64/35	1044.8604	1337.4213 (539.6BD8₁₆)	+1.2934	11/6, 20/11
35	1076.9231	1378.4615 (562.7627₁₆)	8#	^~7	upmid 7th	C#vv	41/22	1077.7445	1379.5129 (563.8834D₁₆)	-0.8214
36	1107.6923	1417.84615 (589.D89E₁₆)	9v (1b)	vM7	downmajor 7th	C#v	74/39	1108.8614	1419.3426 (58B.48A2₁₆)	-1.1691
37	1138.4615	1457.2308 (5B1.3B14₁₆)	9	M7	major 7th	C#	56/29	1139.2487	1459.23835 (5A3.2C05₁₆)	-0.7872
38	1169.2308	1496.6154 (5D8.9D8A₁₆)	9‡ (1v)	v8	down-8ve	Dv	112/57	1169.3579	1496.7781 (5C8.C736₁₆)	-0.1271
39··(or 0)	1200	1536 (600₁₆)	1	P8	perfect 8ve	D	2/1	1200	1536 (600₁₆)	None

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.

Instruments (prototypes):

An illustrative image of a 39-ED2 keyboard


39-EDD fretboard visualization

39 tone equal modes

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.