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 }}.

A particular anecdote with this system was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here].

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] (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 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 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).

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

| 61.54

| 92.31

| 123.08

| 153.85

| 184.62

| 215.38

| 246.15

| 276.92

| 307.69

| 338.46

| 369.23

| 400.00

| 430.77

| 461.54

| 492.31

| 523.08

| 553.85

| 584.62

| 615.38

| 646.15

| 676.92

| 707.69

| 738.46

| 769.23

| 800.00

| 830.77

| 861.54

| 892.31

| 923.08

| 953.85

| 984.62

| 1015.38

| 1046.15

| 1076.92

| 1107.69

| 1138.46

| 1169.23

| ^M7,<br>v8

| upmajor 7th,<br>down 8ve

| 1200.00

<nowiki>*</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" | [[Comma list|Comma List]]

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

! colspan="2" | Tuning Error

| {{monzo| 62 -39 }}

| {{mapping| 39 62 }}

| -1.81

| {{mapping| 39 62 91 }}

| -3.17

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

| -3.78

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

| -3.17

{| class="wikitable center-all right-3 left-4 left-5"

|+Table of temperaments by generator

! Periods<br>per 8ve

! Temperaments

! MOS Scales

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

| [[Negri]] (39)

| [[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>*</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 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.

 

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 [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).

 

=== [[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:

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.

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.

=== Lumatone mapping ===

See [[Lumatone mapping for 39edo]]