39edo: Difference between revisions
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== Theory == | == Theory == | ||
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]]. It also has a fine [[11/1|11]], and adding it to consideration | 39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]]. It also has a fine [[11/1|11]], and adding it to consideration one possible choice for 39et is the sharp-tending 39df [[val]] {{val| 39 62 91 '''110''' 135 '''145'''}}. | ||
As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(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 39et 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 because the [[diatonic semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|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. | As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(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 39et is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. 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 [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|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. | ||
Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], 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 [[7L 2s|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{{c}} flat. | Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], 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 [[7L 2s|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{{c}} flat. | ||
Revision as of 19:20, 29 May 2026
| ← 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 30.8 ¢ each. Each step represents a frequency ratio of 21/39, or the 39th root of 2.
Theory
39edo's perfect fifth is 5.8 ¢ sharp. Together with its best classical major third which is the familiar 400 ¢ of 12edo. We have two choices for a map for 7, but the sharp one works better with the 3 and 5. It also has a fine 11, and adding it to consideration one possible 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 39et is quasisuper. Alternatively, it can be seen as a hemifamity semaphore (that is, immunity) system in the patent val. 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 quartertone-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.
Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of mavila, 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).
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 (%) | +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) | |
| 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.
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
| Steps | Cents | Approximate ratios* | Ups and downs notation | Nearest just interval (Ratio, cents, error) | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 0.0 | 1/1 | P1 | perfect unison | D | 1/1 | 0.00 | None |
| 1 | 30.8 | 36/35, 50/49, 55/54, 56/55, 81/80 | ^1, vm2 |
up unison, downminor 2nd |
^D, vEb |
57/56 | 30.64 | +0.1271 |
| 2 | 61.5 | 28/27, 33/32, 49/48 | m2 | minor 2nd | Eb | 29/28 | 60.75 | +0.7872 |
| 3 | 92.3 | 16/15, 21/20, 25/24 | ^m2 | upminor 2nd | ^Eb | 39/37 | 91.14 | +1.1691 |
| 4 | 123.1 | 15/14 | ^^m2 | dupminor 2nd | ^^Eb | 44/41 | 122.26 | +0.8214 |
| 5 | 153.8 | 11/10, 12/11 | vvM2 | dudmajor 2nd | vvE | 35/32 | 155.14 | -1.2934 |
| 6 | 184.6 | 10/9 | vM2 | downmajor 2nd | vE | 10/9 | 182.40 | +2.2117 |
| 7 | 215.4 | 9/8, 8/7 | M2 | major 2nd | E | 17/15 | 216.69 | -1.3021 |
| 8 | 246.2 | 81/70 | ^M2, vm3 |
upmajor 2nd, downminor 3rd |
^E, vF |
15/13 | 247.74 | -1.5873 |
| 9 | 276.9 | 7/6 | m3 | minor 3rd | F | 27/23 | 277.59 | -0.6676 |
| 10 | 307.7 | 6/5 | ^m3 | upminor 3rd | ^F | 43/36 | 307.61 | +0.0846 |
| 11 | 338.5 | 11/9 | ^^m3 | dupminor 3rd | ^^F | 17/14 | 336.13 | +2.3320 |
| 12 | 369.2 | 27/22 | vvM3 | dudmajor 3rd | vvF# | 26/21 | 369.75 | -0.5160 |
| 13 | 400.0 | 5/4 | vM3 | downmajor 3rd | vF# | 34/27 | 399.09 | +0.9096 |
| 14 | 430.8 | 9/7, 14/11 | M3 | major 3rd | F# | 41/32 | 429.06 | +1.7068 |
| 15 | 461.5 | 35/27 | v4 | down 4th | vG | 30/23 | 459.99 | +1.5441 |
| 16 | 492.3 | 4/3 | P4 | perfect 4th | G | 85/64 | 491.27 | +1.0386 |
| 17 | 523.1 | 27/20 | ^4 | up 4th | ^G | 23/17 | 523.32 | -0.2420 |
| 18 | 553.8 | 11/8 | ^^4 | dup 4th | ^^G | 11/8 | 551.32 | +2.5283 |
| 19 | 584.6 | 7/5 | vvA4, ^d5 |
dudaug 4th, updim 5th |
vvG#, ^Ab |
7/5 | 582.51 | +2.1032 |
| 20 | 615.4 | 10/7 | vA4, ^^d5 |
downaug 4th, dupdim 5th |
vG#, ^^Ab |
10/7 | 617.49 | -2.1032 |
| 21 | 646.2 | 16/11 | vv5 | dud 5th | vvA | 16/11 | 648.68 | -2.5283 |
| 22 | 676.9 | 40/27 | v5 | down 5th | vA | 34/23 | 676.68 | +0.2420 |
| 23 | 707.7 | 3/2 | P5 | perfect 5th | A | 128/85 | 708.73 | -1.0386 |
| 24 | 738.5 | 54/35 | ^5 | up 5th | A^ | 23/15 | 740.01 | -1.5441 |
| 25 | 769.2 | 11/7, 14/9 | m6 | minor 6th | Bb | 64/41 | 770.94 | -1.7068 |
| 26 | 800.0 | 8/5 | ^m6 | upminor 6th | ^Bb | 27/17 | 800.91 | -0.9096 |
| 27 | 830.8 | 44/27 | ^^m6 | dupminor 6th | ^^Bb | 21/13 | 830.25 | +0.5160 |
| 28 | 861.5 | 18/11 | vvM6 | dudmajor 6th | vvB | 28/17 | 863.87 | -2.3320 |
| 29 | 892.3 | 5/3 | vM6 | downmajor 6th | vB | 72/43 | 892.39 | -0.0846 |
| 30 | 923.1 | 12/7 | M6 | major 6th | B | 46/27 | 922.41 | +0.6676 |
| 31 | 953.8 | 140/81 | ^M6, vm7 |
upmajor 6th, downminor 7th |
^B, vC |
26/15 | 952.26 | +1.5873 |
| 32 | 984.6 | 7/4, 16/9 | m7 | minor 7th | C | 30/17 | 983.31 | +1.3021 |
| 33 | 1015.4 | 9/5 | ^m7 | upminor 7th | ^C | 9/5 | 1017.60 | -2.2117 |
| 34 | 1046.2 | 11/6, 20/11 | ^^m7 | dupminor 7th | ^^C | 64/35 | 1044.86 | +1.2934 |
| 35 | 1076.9 | 28/15 | vvM7 | dudmajor 7th | vvC# | 41/22 | 1077.74 | -0.8214 |
| 36 | 1107.7 | 15/8, 40/21, 48/25 | vM7 | downmajor 7th | vC# | 74/39 | 1108.86 | -1.1691 |
| 37 | 1138.5 | 27/14, 96/49, 64/33 | M7 | major 7th | C# | 56/29 | 1139.25 | -0.7872 |
| 38 | 1169.2 | 35/18, 49/25, 55/28, 108/55, 160/81 | ^M7, v8 |
upmajor 7th, down 8ve |
^C#, vD |
112/57 | 1169.36 | -0.1271 |
| 39 | 1200.0 | 2/1 | P8 | perfect 8ve | D | 2/1 | 1200.00 | None |
* 11-limit in the 39d val, inconsistent intervals in italic
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See 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 | 
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| Flat symbol | |
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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.
| 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 |
| 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 |
| 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
| 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
- "Sinner's Finale" from Genshin Impact OST (2023) – covered by Bryan Deister (2025)
21st century
- 39edo (2023)
- 39edo jam (2025)
- 39edo improv (2025)
- Waltz in 39edo (2025)
- Tilt Your Head Down (2026)
- From Souvenirs of the Affliction (2025) – Bandcamp | YouTube
- "Resolute Prelude"
- "Residual Soliloquy"
- Romance On Other Planets (2021)