39edo
| ← 38edo | 39edo | 40edo → |
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 30.769 ¢ each. Each step represents a frequency ratio of 21/39, or the 39th root of 2.
Theory
If we take 22\39 as a fifth, 39edo can be used in mavila temperament, 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 basical 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].
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 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 39edo, in some few ways, allied to 12et 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 39edo 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 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 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, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.
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
| 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 (%) | +19 | +44 | -49 | +37 | +8 | -32 | -37 | -41 | +33 | -30 | -42 | |
| Steps (reduced) |
62 (23) |
91 (13) |
109 (31) |
124 (7) |
135 (18) |
144 (27) |
152 (35) |
159 (3) |
166 (10) |
171 (15) |
176 (20) | |
Intervals
| Armodue nomenclature 5;2 relation |
|
| Degree | Cents | Approximate Ratios in 39d Val |
Armodue Notation |
Ups and Downs Notation | Nearest just interval (Ratio, Cents, Error) | ||||
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0000 | 1/1 | 1 | P1 | perfect unison | D | 1/1 | 0.0000 | None |
| 1 | 30.7692 | 81/80, 36/35, 50/49, 55/54, 56/55 | 1‡ (9#) | ^1 | up unison, downminor 2nd |
^D, vEb |
57/56 | 30.6421 | +0.1271 |
| 2 | 61.5385 | 28/27, 49/48, 33/32 | 2b | m2 | minor 2nd | Eb | 29/28 | 60.7513 | +0.7872 |
| 3 | 92.3077 | 16/15, 25/24, 21/20 | 1# | ^m2 | upminor 2nd | ^Eb | 39/37 | 91.1386 | +1.1691 |
| 4 | 123.0769 | 15/14 | 2v | v~2 | downmid 2nd | ^^Eb | 44/41 | 122.2555 | +0.8214 |
| 5 | 153.8462 | 12/11, 11/10 | 2 | ^~2 | upmid 2nd | vvE | 35/32 | 155.1396 | -1.2934 |
| 6 | 184.6154 | 10/9 | 2‡ | vM2 | downmajor 2nd | vE | 10/9 | 182.4037 | +2.2117 |
| 7· | 215.3846 | 9/8, 8/7 | 3b | M2 | major 2nd | E | 17/15 | 216.6867 | -1.3021 |
| 8 | 246.1538 | 81/70 | 2# | ^M2, vm3 |
upmajor 2nd, downminor 3rd |
^E, vF |
15/13 | 247.7411 | -1.5873 |
| 9 | 276.9231 | 7/6 | 3v | m3 | minor 3rd | F | 27/23 | 277.5907 | -0.6676 |
| 10 | 307.6923 | 6/5 | 3 | ^m3 | upminor 3rd | ^F | 43/36 | 307.6077 | +0.0846 |
| 11 | 338.4615 | 11/9 | 3‡ | v~3 | downmid 3rd | ^^F | 17/14 | 336.1295 | +2.3320 |
| 12· | 369.2308 | 27/22 | 4b | ^~3 | upmid 3rd | vvF# | 26/21 | 369.7468 | -0.5160 |
| 13 | 400.0000 | 5/4 | 3# | vM3 | downmajor 3rd | vF# | 34/27 | 399.0904 | +0.9096 |
| 14 | 430.7692 | 9/7, 14/11 | 4v (5b) | M3 | major 3rd | F# | 41/32 | 429.0624 | +1.7068 |
| 15 | 461.5385 | 35/27 | 4 | v4 | down 4th | vG | 30/23 | 459.9944 | +1.5441 |
| 16 | 492.3077 | 4/3 | 4‡ (5v) | P4 | perfect 4th | G | 85/64 | 491.2691 | +1.0386 |
| 17· | 523.0769 | 27/20 | 5 | ^4 | up 4th | ^G | 23/17 | 523.3189 | -0.2420 |
| 18 | 553.8462 | 11/8 | 5‡ (4#) | v~4 | downmid 4th | ^^G | 11/8 | 551.3179 | +2.5283 |
| 19 | 584.6154 | 7/5 | 6b | ^~4, ^d5 |
upmid 4th, updim 5th |
vvG#, ^Ab |
7/5 | 582.5122 | +2.1032 |
| 20 | 615.3846 | 10/7 | 5# | vA4, v~5 |
downaug 4th, downmid 5th |
vG#, ^^Ab |
10/7 | 617.4878 | -2.1032 |
| 21 | 646.1538 | 16/11 | 6v | ^~5 | upmid 5th | vvA | 16/11 | 648.6821 | -2.5283 |
| 22· | 676.9231 | 40/27 | 6 | v5 | down 5th | vA | 34/23 | 676.6811 | +0.2420 |
| 23 | 707.6923 | 3/2 | 6‡ | P5 | perfect 5th | A | 128/85 | 708.7309 | -1.0386 |
| 24 | 738.4615 | 54/35 | 7b | ^5 | up 5th | A^ | 23/15 | 740.0056 | -1.5441 |
| 25 | 769.2308 | 14/9, 11/7 | 6# | m6 | minor 6th | Bb | 64/41 | 770.9376 | -1.7068 |
| 26 | 800.0000 | 8/5 | 7v | ^m6 | upminor 6th | ^Bb | 27/17 | 800.9096 | -0.9096 |
| 27· | 830.7692 | 44/27 | 7 | v~6 | downmid 6th | ^^Bb | 21/13 | 830.2532 | +0.5160 |
| 28 | 861.5385 | 18/11 | 7‡ | ^~6 | upmid 6th | vvB | 28/17 | 863.8705 | -2.3320 |
| 29 | 892.3077 | 5/3 | 8b | vM6 | downmajor 6th | vB | 72/43 | 892.3923 | -0.0846 |
| 30 | 923.0769 | 12/7 | 7# | M6 | major 6th | B | 46/27 | 922.4093 | +0.6676 |
| 31 | 953.8462 | 140/81 | 8v | ^M6, vm7 |
upmajor 6th, downminor 7th |
^B, vC |
26/15 | 952.2589 | +1.5873 |
| 32· | 984.6154 | 16/9, 7/4 | 8 | m7 | minor 7th | C | 30/17 | 983.3133 | +1.3021 |
| 33 | 1015.3846 | 9/5 | 8‡ | ^m7 | upminor 7th | ^C | 9/5 | 1017.5963 | -2.2117 |
| 34 | 1046.1538 | 11/6, 20/11 | 9b | v~7 | downmid 7th | ^^C | 64/35 | 1044.8604 | +1.2934 |
| 35 | 1076.9231 | 28/15 | 8# | ^~7 | upmid 7th | vvC# | 41/22 | 1077.7445 | -0.8214 |
| 36 | 1107.6923 | 15/8, 48/25, 40/21 | 9v (1b) | vM7 | downmajor 7th | vC# | 74/39 | 1108.8614 | -1.1691 |
| 37 | 1138.4615 | 27/14, 96/49, 64/33 | 9 | M7 | major 7th | C# | 56/29 | 1139.2487 | -0.7872 |
| 38 | 1169.2308 | 160/81, 35/18, 49/25, 55/28, 108/55 | 9‡ (1v) | v8 | up-major 7th down-8ve |
^C#, vD |
112/57 | 1169.3579 | -0.1271 |
| 39·· | 1200.0000 | 2/1 | 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.
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
| Periods per octave |
Generator | Temperaments | MOS Scales |
|---|---|---|---|
| 1 | 1\39 | ||
| 1 | 2\39 | Unicorn | 1L 18s, 19L 1s |
| 1 | 4\39 | Negri | 1L 8s, 9L 1s, 10L 9s, 10L 19s |
| 1 | 5\39 | 1L 6s, 7L 1s, 8L 7s, 8L 15s, 8L 23s | |
| 1 | 7\39 | Machine | 1L 4s, 5L 1s, 6L 5s, 11L 6s, 11L 17s |
| 1 | 8\39 | Immunity | 4L 1s, 5L 4s, 5L 9s, 5L 14s, 5L 19s, 5L 24s, 5L 29s |
| 1 | 10\39 | Familia | 3L 1s, 4L 3s, 4L 7s, 4L 11s, 4L 15s, 4L 19s, 4L 23s, 4L 27s, 4L 31s |
| 1 | 11\39 | Amity | 3L 1s, 4L 3s, 7L 4s, 7L 11s, 7L 18s, 7L 25s |
| 1 | 14\39 | Hamity | 3L 2s, 3L 5s, 3L 8s, 11L 3s, 14L 11s |
| 1 | 16/39 | Superpyth (just over 1/5 comma) | 2L 3s, 5L 2s, 5L 7s, 5L 12s, 17L 5s |
| 1 | 17/39 | Mavila | 2L 3s, 2L 5s, 7L 2s, 7L 9s, 16L 7s |
| 1 | 19/39 | 2L_3s, 2L_5s, 2L_7s, 2L_9s, 2L_11s, 2L_13s etc ... 2L_35s | |
| 3 | 1\39 | ||
| 3 | 2\39 | 3L 3s, 3L 6s, 3L 9s, 3L 12s, 3L 15s, 18L 3s | |
| 3 | 3/39 | Augene | 3L 3s, 3L 6s, 3L 9s, 12L 3s, 12L 15s |
| 3 | 4\39 | Deflated | 3L 3s, 3L 6s, 9L 3s, 9L 12s, 9L 21s |
| 3 | 5/39 | Triforce | 3L 3s, 6L 3s, 9L 6s, 15L 9s |
| 3 | 6/39 | Terrain | 3L 3s, 6L 3s, 6L 9s, 6L 15, 6L 21s, 6L 27s |
| 13 | 1\39 | Tridecatonic | 13L_13s |
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
Lumatone mapping
See Lumatone mapping for 39edo