39edo: Difference between revisions

{{EDO intro}}

39edo's [[3/2|perfect fifth]] is 5.8 cents (¢) sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). 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]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. 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.  

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 cents flat.  

39edo's approximations of harmonics 3, 5, 7, and 11 can all be improved by slightly [[octave shrinking|compressing the octave]], using tunings such as [[62edt]] or [[101ed6]]. [[Ed255/128 #39ed255/128|39ed255/128]], a heavily compressed version of 39edo where the harmonics 13 and 17 are brought to tune at the cost of a worse 11, is also a possible choice.  

Since 39 factors into {{factorization|39}}, 39edo contains [[3edo]] and [[13edo]] as subsets.

! Approximate Ratios*

! colspan="3" | [[Ups and Downs Notation]]

! colspan="3" | [[Nearest just interval|Nearest Just Interval]]<br />(Ratio, Cents, Error)

| 0.00

| 30.77

| ^1,<br />vm2

| up unison, <br />downminor 2nd

| ^D, <br />vEb

| 61.54

| 92.31

| 123.08

| 153.85

| 184.62

| 215.38

| 246.15

| ^M2, <br />vm3

| upmajor 2nd, <br />downminor 3rd

| ^E, <br />vF

| 276.92

| 307.69

| 338.46

| 369.23

| 400.00

| 430.77

| 461.54

| 492.31

| 523.08

| 553.85

| 584.62

| vvA4, <br />^d5

| dudaug 4th, <br />updim 5th

| vvG#, <br />^Ab

| 615.38

| vA4, <br />^^d5

| downaug 4th, <br />dupdim 5th

| vG#, <br />^^Ab

| 646.15

| 676.92

| 707.69

| 738.46

| 769.23

| 800.00

| 830.77

| 861.54

| 892.31

| 923.08

| 953.85

| ^M6, <br />vm7

| upmajor 6th, <br />downminor 7th

| ^B, <br />vC

| 984.62

| 1015.38

| 1046.15

| 1076.92

| 1107.69

| 1138.46

| 1169.23

| ^M7,<br />v8

| upmajor 7th,<br />down 8ve

| ^C#, <br />vD

| 1200.00

<nowiki />* 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]].

Using [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals, 39edo can also be notated using [[ups and downs notation]]:  

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.

! Armodue Notation

! Associated Ratios

! rowspan="2" | Optimal<br />8ve stretch (¢)

| {{monzo| 62 -39 }}

| {{mapping| 39 62 }}

| &minus;1.81

| {{mapping| 39 62 91 }}

| &minus;3.17

| {{mapping| 39 62 91 110 }} (39d)

| &minus;3.78

| {{mapping| 39 62 91 110 135 }} (39d)

| &minus;3.17

{| class="wikitable center-all left-5"

! Periods<br />per 8ve

! MOS Scales

| [[1L 18s]], [[19L 1s]]

| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]

| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]

| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]

| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]

| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]

| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]

| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]

| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]

| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]

| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]

| 8\39<br />(5\39)

| 246.2<br />(153.8)

| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]

| 16\39<br />(3\39)

| 492.3<br />(92.3)

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]

| 17\39<br />(4\39)

| 523.1<br />(123.0)

| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]

| 16\39<br />(1\39)

| 492.3<br />(30.8)

| [[13L 13s]]

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

39edo is a good candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[Approaches to Musical Tuning|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.

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 [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|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 [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the '''7 6 3 7 6 7 3''' MODMOS), for example.

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).

While [[Arabic, Turkish, Persian|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:

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 (&rarr; [[Wikipedia: Coltrane changes]]).

39edo offers approximations of [[pelog]] and [[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{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

=== Lumatone mapping ===

See [[Lumatone mapping for 39edo]]