39edo

Theory

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].

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	Armodue Notation	Ups and Downs Notation			Nearest Just Interval	Cents	Error	11-limit Ratio in ⟨39 62 91 110 135] (39d) Val
0	0.0000	1	P1	perfect unison	D	1/1	0.0000	None	1/1
1	30.7692	1‡ (9#)	^1	up unison, downminor 2nd	^D, vEb	57/56	30.6421	+0.1271
2	61.5385	2b	m2	minor 2nd	Eb	29/28	60.7513	+0.7872
3	92.3077	1#	^m2	upminor 2nd	^Eb	39/37	91.1386	+1.1691
4	123.0769	2v	v~2	downmid 2nd	^^Eb	44/41	122.2555	+0.8214
5	153.8462	2	^~2	upmid 2nd	vvE	35/32	155.1396	-1.2934	12/11, 11/10
6	184.6154	2‡	vM2	downmajor 2nd	vE	10/9	182.4037	+2.2117	10/9
7·	215.3846	3b	M2	major 2nd	E	17/15	216.6867	-1.3021	8/7, 9/8
8	246.1538	2#	^M2, vm3	upmajor 2nd, downminor 3rd	^E, vF	15/13	247.7411	-1.5873
9	276.9231	3v	m3	minor 3rd	F	27/23	277.5907	-0.6676	7/6
10	307.6923	3	^m3	upminor 3rd	^F	43/36	307.6077	+0.0846	6/5
11	338.4615	3‡	v~3	downmid 3rd	^^F	17/14	336.1295	+2.3320	11/9
12·	369.2308	4b	^~3	upmid 3rd	vvF#	26/21	369.7468	-0.5160
13	400.0000	3#	vM3	downmajor 3rd	vF#	34/27	399.0904	+0.9096	5/4
14	430.7692	4v (5b)	M3	major 3rd	F#	41/32	429.0624	+1.7068	9/7, 14/11
15	461.5385	4	v4	down 4th	vG	30/23	459.9944	+1.5441
16	492.3077	4‡ (5v)	P4	perfect 4th	G	85/64	491.2691	+1.0386	4/3
17·	523.0769	5	^4	up 4th	^G	23/17	523.3189	-0.242
18	553.8462	5‡ (4#)	v~4	downmid 4th	^^G	11/8	551.3179	+2.5283	11/8
19	584.6154	6b	^~4, ^d5	upmid 4th, updim 5th	vvG#, ^Ab	7/5	582.5122	+2.1032	7/5
20	615.3846	5#	vA4, v~5	downaug 4th, downmid 5th	vG#, ^^Ab	10/7	617.4878	-2.1032	10/7
21	646.1538	6v	^~5	upmid 5th	vvA	16/11	648.6821	-2.5283	16/11
22·	676.9231	6	v5	down 5th	vA	34/23	676.6811	+0.242
23	707.6923	6‡	P5	perfect 5th	A	128/85	708.7309	-1.0386	3/2
24	738.4615	7b	^5	up 5th	A^	23/15	740.0056	-1.5441
25	769.2308	6#	m6	minor 6th	Bb	64/41	770.9376	-1.7068	14/9, 11/7
26	800.0000	7v	^m6	upminor 6th	^Bb	27/17	800.9096	-0.9096	8/5
27·	830.7692	7	v~6	downmid 6th	^^Bb	21/13	830.2532	+0.516
28	861.5385	7‡	^~6	upmid 6th	vvB	28/17	863.8705	-2.332	18/11
29	892.3077	8b	vM6	downmajor 6th	vB	72/43	892.3923	-0.0846	5/3
30	923.0769	7#	M6	major 6th	B	46/27	922.4093	+0.6676	12/7
31	953.8462	8v	^M6, vm7	upmajor 6th, downminor 7th	^B, vC	26/15	952.2589	+1.5873
32·	984.6154	8	m7	minor 7th	C	30/17	983.3133	+1.3021	7/4, 16/9
33	1015.3846	8‡	^m7	upminor 7th	^C	9/5	1017.5963	-2.2117	9/5
34	1046.1538	9b	v~7	downmid 7th	^^C	64/35	1044.8604	+1.2934	11/6, 20/11
35	1076.9231	8#	^~7	upmid 7th	vvC#	41/22	1077.7445	-0.8214
36	1107.6923	9v (1b)	vM7	downmajor 7th	vC#	74/39	1108.8614	-1.1691
37	1138.4615	9	M7	major 7th	C#	56/29	1139.2487	-0.7872
38	1169.2308	9‡ (1v)	v8	down-8ve	vD	112/57	1169.3579	-0.1271
39··	1200.0000	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.

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