39edo

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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 pions 7mus Armodue note ups anddowns

notation

Nearest Just

Interval

Cents value pions 7mus Error 11-limit Ratio Assuming

<39 62 91 110 135| Val

0 1 P1 perfect unison D 1/1 0 None 1/1
1 30.7692 32.6154 39.3846 (27.627616) 1‡ (9#) ^1 up unison D^ 57/56 30.6421 32.4806 39.2219 (27.38CF16) +0.1271
2 61.5385 65.2308 78.7692 (4E.C4EC16) 2b m2 minor 2nd Eb 29/28 60.7513 64.3964 77.76165 (4D.C2FB16) +0.7872
3 92.3077 97.84615 118.15385 (76.276216) 1# ^m2 upminor 2nd Eb^ 39/37 91.1386 96.6069 116.6574 (74.A84E16) +1.1691
4 123.0769 130.4615 157.5385 (9D.89D916) 2v v~2 downmid 2nd Eb^^ 44/41 122.2555 129.5909 156.4871 (9C.7CB216) +0.8214
5 153.84615 163.0769 196.9231 (C4.EC4F16 2 ^~2 upmid 2nd Evv 35/32 155.1396 164.448 198.5787 (C6.942716) -1.2934 12/11, 11/10
6 184.6154 195.6923 236.3077 (EC.4EC516) 2‡ vM2 downmajor 2nd Ev 10/9 182.4037 193.3479 233.47675 (E9.7A0C16) +2.2117 10/9
7· 215.3846 228.3077 275.6923 (113.B13B16) 3b M2 major 2nd E 17/15 216.6867 229.6879 277.359 (115.5BE516) -1.3021 8/7, 9/8
8 246.15385 260.9231 315.0769 (13B.13B116) 2# ^M2,

vm3

upmajor 2nd,

downminor 3rd

E^,

Fv

15/13 247.7411 262.6055 317.10855 (13D.1BCA16) -1.5873
9 276.9231 293.5385 354.4615 (162.762716) 3v m3 minor 3rd F 27/23 277.5907 294.2461 355.316 (163.50E816) -0.6676 7/6
10 307.6923 326.15385 393.84615 (189.D89E16) 3 ^m3 upminor 3rd F^ 43/36 307.6077 326.0642 393.7379 (189.BCE416) +0.0846 6/5
11 338.4615 358.7692 433.2308 (1B1.3B1416) 3‡ v~3 downmid 3rd F^^ 17/14 336.1295 356.2973 430.2458 (1AE.3EEA16) +2.332 11/9
12· 369.2308 391.3846 472.6154 (1D8.9D8A16) 4b ^~3 upmid 3rd F#vv 26/21 369.7468 391.9316 473.27585 (1D9.469E16) -0.516
13 400 424 512 (20016) 3# vM3 downmajor 3rd F#v 34/27 399.0904 423.0358 510.8357 (1FE.D5F216) +0.9096 5/4
14 430.7692 456.6154 551.3846 (227.627616) 4v (5b) M3 major 3rd F# 41/32 429.0624 454.80615 549.1999 (225.332B16) +1.7068 9/7, 14/11
15 461.5385 489.2308 590.7692 (24E.C4EC16) 4 v4 down 4th Gv 30/23 459.9944 487.594 588.7928 (24C.CAF416) +1.5441
16 492.3077 521.84615 630.15385 (276.276216) 4‡ (5v) P4 perfect 4th G 85/64 491.2691 520.7453 628.8245 (274.D31116) +1.0386 4/3
17· 523.0769 554.4615 669.5385 (29D.89D916) 5 ^4 up 4th G^ 23/17 523.3189 554.7181 669.8482 (29D.D92616) -0.242
18 553.8462 587.0769 708.9231 (2C4.EC4D16) 5‡ (4#) ^^4 double-up 4th G^^ 11/8 551.3179 584.397 705.687 (2C1.AFDD16) +2.5283 11/8
19 584.6154 619.6923 748.3077 (2EC.4EC516) 6b vvA4,

^d5

double-down aug 4th,

updim 5th

G#vv,

Ab^

7/5 582.5122 617.4629 745.6156 (2E9.9D9816) +2.1032 7/5
20 615.3846 652.3077 787.6923 (313.B13B16) 5# vA4,

^^d5

downaug 4th,

double-up dim 5th

G#v,

Ab^^

10/7 617.4878 654.5371 790.3844 (316.626816) -2.1032 10/7
21 646.1538 684.9231 827.0769 (33B.13B116) 6v vv5 double-down 5th Avv 16/11 648.6821 687.603 830.313 (33E.502316) -2.5283 16/11
22· 676.9231 717.5385 866.4615 (362.762716) 6 v5 down 5th Av 34/23 676.6811 717.2819 866.1518 (362.26DA16) +0.242
23 707.6923 750.15385 905.84615 (389.D89E16) 6‡ P5 perfect 5th A 128/85 708.7309 751.2547 907.1755 (38B.2CDE16) -1.0386 3/2
24 738.4615 782.7692 945.2308 (3B1.3B1416) 7b ^5 up 5th A^ 23/15 740.0056 784.406 947.2072 (3B3.350C16) -1.5441
25 769.2308 815.3846 984.6154 (3D8.9D8A16) 6# m6 minor 6th Bb 64/41 770.9376 817.19385 986.7991 (3DA.CCD416) -1.7068 14/9, 11/7
26 800 848 1024 (40016) 7v ^m6 upminor 6th Bb^ 27/17 800.9096 848.9642 1025.1643 (401.2A0E16) -0.9096 8/5
27· 830.7692 880.6154 1063.3846 (427.627616) 7 v~6 downmid 6th Bb^^ 21/13 830.2532 880.0784 1062.72415 (426.B96216) +0.516
28 861.5385 913.2308 1102.7692 (44E.C4EC16) 7‡ ^~6 upmid 6th Bvv 28/17 863.8705 915.7127 1105.7542 (451.C11616) -2.332 18/11
29 892.3077 945.84615 1142.15385 (476.276216) 8b vM6 downmajor 6th Bv 72/43 892.3923 945.9358 1142.2621 (476.431B16) -0.0846 5/3
30 923.0769 978.4615 1181.5385 (49D.89D916) 7# M6 major 6th B 46/27 922.4093 977.7539 1180.684 (49C.AF1816) +0.6676 12/7
31 953.8462 1011.0769 1220.9231 (4C4.EC4D16) 8v ^M6,

vm7

upmajor 6th,

downminor 7th

B^,

Cv

26/15 952.2589 1009.3945 1218.89145 (4C2.E45316) +1.5873
32· 984.6154 1043.6923 1260.3077 (4EC.4EC516) 8 m7 minor 7th C 30/17 983.3133 1042.3121 1258.641 (4EA.A41B16) +1.3021 7/4, 16/9
33 1015.3846 1076.3077 1299.6923 (513.B13B16) 8‡ ^m7 upminor 7th C^ 9/5 1017.5963 1079.6521 1302.52325 (516.85F316) -2.2117 9/5
34 1046.1538 1108.9231 1339.0769 (53B.13B116) 9b v~7 downmid 7th C^^ 64/35 1044.8604 1107.552 1337.4213 (539.6BD816) +1.2934 11/6, 20/11
35 1076.9231 1141.5385 1378.4615 (562.762716) 8# ^~7 upmid 7th C#vv 41/22 1077.7445 1142.4091 1379.5129 (563.8834D16) -0.8214
36 1107.6923 1174.15385 1417.84615 (589.D89E16) 9v (1b) vM7 downmajor 7th C#v 74/39 1108.8614 1175.3931 1419.3426 (58B.48A216) -1.1691
37 1138.4615 1206.7692 1457.2308 (5B1.3B1416) 9 M7 major 7th C# 56/29 1139.2487 1207.6036 1459.23835 (5A3.2C0516) -0.7872
38 1169.2308 1239.3846 1496.6154 (5D8.9D8A16) 9‡ (1v) v8 down-8ve Dv 112/57 1169.3579 1239.5194 1496.7781 (5C8.C73616) -0.1271
39··(or 0) 1200 1272 1536 (60016) 1 P8 perfect 8ve D 2/1 1200 1272 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):

TECLADO 39-EDD.PNG

An illustrative image of a 39-ED2 keyboard

Custom_700mm_5-str_Tricesanonaphonic_Guitar.png
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.