39edo: Difference between revisions

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== Theory ==

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 }}.

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 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].

A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|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). 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.

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.

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.

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.

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).

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 ===

=== Octave stretch ===

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.

There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used for this same purpose: 171zpi, 172zpi and 173zpi. The main zeta peak index page details all three tunings.

=== Subsets and supersets ===

Since 39 factors into {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.

== Intervals ==

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! Steps

! Cents

! Approximate ~~Ratios~~*

! Approximate ratios*

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

! colspan="3" | [[Ups and downs notation]]

! colspan="3" | [[Nearest just interval~~|Nearest Just Interval~~]] (Ratio, ~~Cents~~, ~~Error~~)

! colspan="3" | [[Nearest just interval]] (Ratio, cents, error)

|-

| 0

| 0.00

| 0.0

| [[1/1]]

| P1

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|-

| 1

| 30.77

| 30.8

| ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]

| ^1, vm2

| ^1, vm2

| up unison, downminor 2nd

| ^D, vEb

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|-

| 2

| 61.54

| 61.5

| [[28/27]], [[33/32]], ''[[49/48]]''

| m2

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|-

| 3

| 92.31

| 92.3

| ''[[16/15]]'', [[21/20]], ''[[25/24]]''

| ^m2

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|-

| 4

| 123.08

| 123.1

| [[15/14]]

| ^^m2

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|-

| 5

| 153.85

| 153.8

| [[11/10]], [[12/11]]

| vvM2

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|-

| 6

| 184.62

| 184.6

| [[10/9]]

| vM2

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|-

| 7

| 215.38

| 215.4

| [[9/8]], ''[[8/7]]''

| M2

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|-

| 8

| 246.15

| 246.2

| [[81/70]]

| ^M2, vm3

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|-

| 9

| 276.92

| 276.9

| [[7/6]]

| m3

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|-

| 10

| 307.69

| 307.7

| [[6/5]]

| ^m3

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|-

| 11

| 338.46

| 338.5

| [[11/9]]

| ^^m3

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|-

| 12

| 369.23

| 369.2

| [[27/22]]

| vvM3

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|-

| 13

| 400.00

| 400.0

| [[5/4]]

| vM3

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|-

| 14

| 430.77

| 430.8

| [[9/7]], [[14/11]]

| M3

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|-

| 15

| 461.54

| 461.5

| [[35/27]]

| v4

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|-

| 16

| 492.31

| 492.3

| [[4/3]]

| P4

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|-

| 17

| 523.08

| 523.1

| [[27/20]]

| ^4

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|-

| 18

| 553.85

| 553.8

| [[11/8]]

| ^^4

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|-

| 19

| 584.62

| 584.6

| [[7/5]]

| vvA4, ^d5

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|-

| 20

| 615.38

| 615.4

| [[10/7]]

| vA4, ^^d5

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|-

| 21

| 646.15

| 646.2

| [[16/11]]

| vv5

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|-

| 22

| 676.92

| 676.9

| [[40/27]]

| v5

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|-

| 23

| 707.69

| 707.7

| [[3/2]]

| P5

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|-

| 24

| 738.46

| 738.5

| [[54/35]]

| ^5

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|-

| 25

| 769.23

| 769.2

| [[11/7]], [[14/9]]

| m6

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|-

| 26

| 800.00

| 800.0

| [[8/5]]

| ^m6

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|-

| 27

| 830.77

| 830.8

| [[44/27]]

| ^^m6

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|-

| 28

| 861.54

| 861.5

| [[18/11]]

| vvM6

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|-

| 29

| 892.31

| 892.3

| [[5/3]]

| vM6

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|-

| 30

| 923.08

| 923.1

| [[12/7]]

| M6

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|-

| 31

| 953.85

| 953.8

| [[140/81]]

| ^M6, vm7

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|-

| 32

| 984.62

| 984.6

| ''[[7/4]]'', [[16/9]]

| m7

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|-

| 33

| 1015.38

| 1015.4

| [[9/5]]

| ^m7

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|-

| 34

| 1046.15

| 1046.2

| [[11/6]], [[20/11]]

| ^^m7

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|-

| 35

| 1076.92

| 1076.9

| [[28/15]]

| vvM7

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|-

| 36

| 1107.69

| 1107.7

| ''[[15/8]]'', [[40/21]], ''[[48/25]]''

| vM7

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|-

| 37

| 1138.46

| 1138.5

| [[27/14]], ''[[96/49]]'', [[64/33]]

| M7

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|-

| 38

| 1169.23

| 1169.2

| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]

| ^M7, v8

| ^M7, v8

| upmajor 7th, down 8ve

| upmajor 7th, down 8ve

| ^C#, vD

| 112/57

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|-

| 39

| 1200.00

| 1200.0

| [[2/1]]

| P8

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| None

|}

<nowiki>*~~</nowiki>~~ 11-limit in the 39d val, inconsistent intervals in ''italic''

<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~~]].

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 ==

=== Ups and downs notation ===

39edo can be notated with [[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).

Another notation uses [[Alternative symbols for ups and downs notation #Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].

==== Evo flavor ====

File:39-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

rect 120 80 240 106 [[33/32]]

default [[File:39-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:39-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

rect 120 80 240 106 [[33/32]]

default [[File:39-EDO_Revo_Sagittal.svg]]

</imagemap>

=== Armodue notation ===

; Armodue nomenclature 5;2 relation

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* '''v''' = Semiflat (1/5-tone down)

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

{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"

|-

! colspan="2" | #

! Cents

! Armodue~~ Notation~~

! Armodue notation

! Associated ~~Ratios~~

! Associated ratios

|-

| 0

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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|-

| 2.3

| {{~~monzo~~| 62 -39 }}

| {{Monzo| 62 -39 }}

| {{~~mapping~~| 39 62 }}

| {{Mapping| 39 62 }}

| -1.81

| −1.81

| 1.81

| 5.88

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| 2.3.5

| 128/125, 1594323/1562500

| {{~~mapping~~| 39 62 91 }}

| {{Mapping| 39 62 91 }}

| -3.17

| −3.17

| 2.42

| 7.89

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| 2.3.5.7

| 64/63, 126/125, 2430/2401

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

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

| -3.78

| −3.78

| 2.35

| 7.65

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| 2.3.5.7.11

| 64/63, 99/98, 121/120, 126/125

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

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

| -3.17

| −3.17

| 2.43

| 7.91

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=== Rank-2 temperaments ===

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

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

|+Table of temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! ~~Temperaments~~

! Temperament

! ~~MOS Scales~~

! Mos scales

|-

| 1

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| 61.5

| [[Unicorn]] (39d)

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

| [[1L 18s]], [[19L 1s]]

|-

| 1

| 4\39

| 123.1

| [[Negri]] (39)

| [[Negri]] (39c)

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

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

|-

| 1

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| 153.8

|

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

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

|-

| 1

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| 215.4

| [[Machine]] (39d)

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

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

|-

| 1

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| 246.2

| [[Immunity]] (39) / [[immunized]] (39d)

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

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

|-

| 1

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| 307.7

| [[Familia]] (39df)

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

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

|-

| 1

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| 338.5

| [[Amity]] (39) / [[accord]] (39d)

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

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

|-

| 1

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| 430.8

| [[Hamity]] (39df)

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

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

|-

| 1

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| 492.3

| [[Quasisuper]] (39d)

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

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

|-

| 1

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| 523.1

| [[Mavila]] (39bc)

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

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

|-

| 1

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| 584.6

| [[Pluto]] (39d)

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

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

|-

| 3

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| 61.5

|

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

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

|-

| 3

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| 184.6

| [[Terrain]] / [[mirkat]] (39df)

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

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

|-

| 3

| 8\39 (5\39)

| 8\39 (5\39)

| 246.2 (153.8)

| 246.2 (153.8)

| [[Triforce]] (39)

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

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

|-

| 3

| 16\39 (3\39)

| 16\39 (3\39)

| 492.3 (92.3)

| 492.3 (92.3)

| [[Augene]] (39d)

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

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

|-

| 3

| 17\39 (4\39)

| 17\39 (4\39)

| 523.1 (123.0)

| 523.1 (123.0)

| [[Deflated]] (39bd)

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

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

|-

| 13

| 16\39 (1\39)

| 16\39 (1\39)

| 492.3 (30.8)

| 492.3 (30.8)

| [[Tridecatonic]]

| [[13L 13s]]

| [[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

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

== 39edo and world music ==

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 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 pleasing.

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 [[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 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.

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.

The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|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.

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.

=== 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).

~~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 [[Arabic, Turkish, Persian music|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:

=== [[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)

Line 880:

Line 916:

=== 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.

~~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~~]]).

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 [[mapping]]s 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.

~~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~~.

=== 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.

~~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~~.

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.

~~=== Other ===~~

One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.

~~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.

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== 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|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 ===

[[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]

Line 904:

Line 961:

''39edo fretboard visualization''

==~~= Lumatone mapping =~~==

== Music ==

~~See~~ [[~~Lumatone mapping for~~ 39edo]]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)

* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)

* [https://www.youtube.com/shorts/4y11CWLIHNA ''Sinner's Finale - Genshin Impact (microtonal cover in 39edo)''] (2025)

~~== Music ==~~

; [[Randy Wells]]

* [https://www.youtube.com/watch?v=Q9wQV1J5eLE Romance On Other Planets] ~~by [[Randy Wells]]~~

* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)

~~[[Category:39edo]]~~

~~[[Category:Equal divisions of the octave|##]] ~~

~~[[Category:39-tone scales]]~~

~~[[Category:Modes]]~~

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|39}}
+{{ED intro}}
 == Theory ==
-edo'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 }}.
+edo'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 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].
+A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|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). 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.
+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.
-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] &amp; [[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.
+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] &amp; [[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.
-edo 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).
+edo 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 ===
 {{Harmonics in equal|39}}
+=== Octave stretch ===
+edo'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.
+There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used for this same purpose: 171zpi, 172zpi and 173zpi. The main zeta peak index page details all three tunings.
+=== Subsets and supersets ===
+Since 39 factors into {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.
 == Intervals ==
@@ Line 20: / Line 29: @@
 ! Steps
 ! Cents
-! Approximate Ratios*
+! Approximate ratios*
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
-! colspan="3" | [[Nearest just interval|Nearest Just Interval]]<br>(Ratio, Cents, Error)
+! colspan="3" | [[Nearest just interval]] <br>(Ratio, cents, error)
 |-
 | 0
-| 0.00
+| 0.0
 | [[1/1]]
 | P1
@@ Line 35: / Line 44: @@
 |-
 | 1
-| 30.77
+| 30.8
 | ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]
-| ^1,<br>vm2
+| ^1, <br>vm2
 | up unison, <br>downminor 2nd
 | ^D, <br>vEb
@@ Line 45: / Line 54: @@
 |-
 | 2
-| 61.54
+| 61.5
 | [[28/27]], [[33/32]], ''[[49/48]]''
 | m2
@@ Line 55: / Line 64: @@
 |-
 | 3
-| 92.31
+| 92.3
 | ''[[16/15]]'', [[21/20]], ''[[25/24]]''
 | ^m2
@@ Line 65: / Line 74: @@
 |-
 | 4
-| 123.08
+| 123.1
 | [[15/14]]
 | ^^m2
@@ Line 75: / Line 84: @@
 |-
 | 5
-| 153.85
+| 153.8
 | [[11/10]], [[12/11]]
 | vvM2
@@ Line 85: / Line 94: @@
 |-
 | 6
-| 184.62
+| 184.6
 | [[10/9]]
 | vM2
@@ Line 95: / Line 104: @@
 |-
 | 7
-| 215.38
+| 215.4
 | [[9/8]], ''[[8/7]]''
 | M2
@@ Line 105: / Line 114: @@
 |-
 | 8
-| 246.15
+| 246.2
 | [[81/70]]
 | ^M2, <br>vm3
@@ Line 115: / Line 124: @@
 |-
 | 9
-| 276.92
+| 276.9
 | [[7/6]]
 | m3
@@ Line 125: / Line 134: @@
 |-
 | 10
-| 307.69
+| 307.7
 | [[6/5]]
 | ^m3
@@ Line 135: / Line 144: @@
 |-
 | 11
-| 338.46
+| 338.5
 | [[11/9]]
 | ^^m3
@@ Line 145: / Line 154: @@
 |-
 | 12
-| 369.23
+| 369.2
 | [[27/22]]
 | vvM3
@@ Line 155: / Line 164: @@
 |-
 | 13
-| 400.00
+| 400.0
 | [[5/4]]
 | vM3
@@ Line 165: / Line 174: @@
 |-
 | 14
-| 430.77
+| 430.8
 | [[9/7]], [[14/11]]
 | M3
@@ Line 175: / Line 184: @@
 |-
 | 15
-| 461.54
+| 461.5
 | [[35/27]]
 | v4
@@ Line 185: / Line 194: @@
 |-
 | 16
-| 492.31
+| 492.3
 | [[4/3]]
 | P4
@@ Line 195: / Line 204: @@
 |-
 | 17
-| 523.08
+| 523.1
 | [[27/20]]
 | ^4
@@ Line 205: / Line 214: @@
 |-
 | 18
-| 553.85
+| 553.8
 | [[11/8]]
 | ^^4
@@ Line 215: / Line 224: @@
 |-
 | 19
-| 584.62
+| 584.6
 | [[7/5]]
 | vvA4, <br>^d5
@@ Line 225: / Line 234: @@
 |-
 | 20
-| 615.38
+| 615.4
 | [[10/7]]
 | vA4, <br>^^d5
@@ Line 235: / Line 244: @@
 |-
 | 21
-| 646.15
+| 646.2
 | [[16/11]]
 | vv5
@@ Line 245: / Line 254: @@
 |-
 | 22
-| 676.92
+| 676.9
 | [[40/27]]
 | v5
@@ Line 255: / Line 264: @@
 |-
 | 23
-| 707.69
+| 707.7
 | [[3/2]]
 | P5
@@ Line 265: / Line 274: @@
 |-
 | 24
-| 738.46
+| 738.5
 | [[54/35]]
 | ^5
@@ Line 275: / Line 284: @@
 |-
 | 25
-| 769.23
+| 769.2
 | [[11/7]], [[14/9]]
 | m6
@@ Line 285: / Line 294: @@
 |-
 | 26
-| 800.00
+| 800.0
 | [[8/5]]
 | ^m6
@@ Line 295: / Line 304: @@
 |-
 | 27
-| 830.77
+| 830.8
 | [[44/27]]
 | ^^m6
@@ Line 305: / Line 314: @@
 |-
 | 28
-| 861.54
+| 861.5
 | [[18/11]]
 | vvM6
@@ Line 315: / Line 324: @@
 |-
 | 29
-| 892.31
+| 892.3
 | [[5/3]]
 | vM6
@@ Line 325: / Line 334: @@
 |-
 | 30
-| 923.08
+| 923.1
 | [[12/7]]
 | M6
@@ Line 335: / Line 344: @@
 |-
 | 31
-| 953.85
+| 953.8
 | [[140/81]]
 | ^M6, <br>vm7
@@ Line 345: / Line 354: @@
 |-
 | 32
-| 984.62
+| 984.6
 | ''[[7/4]]'', [[16/9]]
 | m7
@@ Line 355: / Line 364: @@
 |-
 | 33
-| 1015.38
+| 1015.4
 | [[9/5]]
 | ^m7
@@ Line 365: / Line 374: @@
 |-
 | 34
-| 1046.15
+| 1046.2
 | [[11/6]], [[20/11]]
 | ^^m7
@@ Line 375: / Line 384: @@
 |-
 | 35
-| 1076.92
+| 1076.9
 | [[28/15]]
 | vvM7
@@ Line 385: / Line 394: @@
 |-
 | 36
-| 1107.69
+| 1107.7
 | ''[[15/8]]'', [[40/21]], ''[[48/25]]''
 | vM7
@@ Line 395: / Line 404: @@
 |-
 | 37
-| 1138.46
+| 1138.5
 | [[27/14]], ''[[96/49]]'', [[64/33]]
 | M7
@@ Line 405: / Line 414: @@
 |-
 | 38
-| 1169.23
+| 1169.2
 | ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]
-| ^M7,<br>v8
+| ^M7, <br>v8
-| upmajor 7th,<br>down 8ve
+| upmajor 7th, <br>down 8ve
 | ^C#, <br>vD
 | 112/57
@@ Line 415: / Line 424: @@
 |-
 | 39
-| 1200.00
+| 1200.0
 | [[2/1]]
 | P8
@@ Line 424: / Line 433: @@
 | None
 |}
-<nowiki>*</nowiki> 11-limit in the 39d val, inconsistent intervals in ''italic''
+<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]].
+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 ==
+=== Ups and downs notation ===
+edo can be notated with [[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).
+{{Sharpness-sharp5a}}
+Another notation uses [[Alternative symbols for ups and downs notation #Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp5}}
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].
+==== Evo flavor ====
+<imagemap>
+File:39-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[81/80]]
+rect 120 80 240 106 [[33/32]]
+default [[File:39-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:39-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[81/80]]
+rect 120 80 240 106 [[33/32]]
+default [[File:39-EDO_Revo_Sagittal.svg]]
+</imagemap>
 === Armodue notation ===
 ; Armodue nomenclature 5;2 relation
@@ Line 436: / Line 475: @@
 * '''v''' = Semiflat (1/5-tone down)
-{| class="wikitable center-all right-3 left-5"
+{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
 |-
 ! colspan="2" | #
 ! Cents
-! Armodue<br>Notation
+! Armodue notation
-! Associated Ratios
+! Associated ratios
 |-
 | 0
@@ Line 686: / Line 725: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal <br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 696: / Line 736: @@
 |-
 | 2.3
-| {{monzo| 62 -39 }}
+| {{Monzo| 62 -39 }}
-| {{mapping| 39 62 }}
+| {{Mapping| 39 62 }}
-| -1.81
+| −1.81
 | 1.81
 | 5.88
@@ Line 704: / Line 744: @@
 | 2.3.5
 | 128/125, 1594323/1562500
-| {{mapping| 39 62 91 }}
+| {{Mapping| 39 62 91 }}
-| -3.17
+| −3.17
 | 2.42
 | 7.89
@@ Line 711: / Line 751: @@
 | 2.3.5.7
 | 64/63, 126/125, 2430/2401
-| {{mapping| 39 62 91 110 }} (39d)
+| {{Mapping| 39 62 91 110 }} (39d)
-| -3.78
+| −3.78
 | 2.35
 | 7.65
@@ Line 718: / Line 758: @@
 | 2.3.5.7.11
 | 64/63, 99/98, 121/120, 126/125
-| {{mapping| 39 62 91 110 135 }} (39d)
+| {{Mapping| 39 62 91 110 135 }} (39d)
-| -3.17
+| −3.17
 | 2.43
 | 7.91
@@ Line 725: / Line 765: @@
 === Rank-2 temperaments ===
-{| class="wikitable center-all right-3 left-4 left-5"
+{| class="wikitable center-all left-4 left-5"
-|+Table of temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br>per 8ve
+! Periods <br />per 8ve
 ! Generator*
 ! Cents*
-! Temperaments
+! Temperament
-! MOS Scales
+! Mos scales
 |-
 | 1
@@ Line 744: / Line 784: @@
 | 61.5
 | [[Unicorn]] (39d)
-| [[1L 18s]], [[19L 1s]]
+| [[1L&nbsp;18s]], [[19L&nbsp;1s]]
 |-
 | 1
 | 4\39
 | 123.1
-| [[Negri]] (39)
+| [[Negri]] (39c)
-| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]
+| [[1L&nbsp;8s]], [[9L&nbsp;1s]], [[10L&nbsp;9s]], [[10L&nbsp;19s]]
 |-
 | 1
@@ Line 756: / Line 796: @@
 | 153.8
 |
-| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]
+| [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[8L&nbsp;7s]], [[8L&nbsp;15s]], [[8L&nbsp;23s]]
 |-
 | 1
@@ Line 762: / Line 802: @@
 | 215.4
 | [[Machine]] (39d)
-| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]
+| [[1L&nbsp;4s]], [[5L&nbsp;1s]], [[6L&nbsp;5s]], [[11L&nbsp;6s]], [[11L&nbsp;17s]]
 |-
 | 1
@@ Line 768: / Line 808: @@
 | 246.2
 | [[Immunity]] (39) / [[immunized]] (39d)
-| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]
+| [[4L&nbsp;1s]], [[5L&nbsp;4s]], [[5L&nbsp;9s]], [[5L&nbsp;14s]], [[5L&nbsp;19s]], [[5L&nbsp;24s]], [[5L&nbsp;29s]]
 |-
 | 1
@@ Line 774: / Line 814: @@
 | 307.7
 | [[Familia]] (39df)
-| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]
+| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], [[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], [[4L&nbsp;23s]], [[4L&nbsp;27s]], [[4L&nbsp;31s]]
 |-
 | 1
@@ Line 780: / Line 820: @@
 | 338.5
 | [[Amity]] (39) / [[accord]] (39d)
-| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]
+| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[7L&nbsp;4s]], [[7L&nbsp;11s]], [[7L&nbsp;18s]], [[7L&nbsp;25s]]
 |-
 | 1
@@ Line 786: / Line 826: @@
 | 430.8
 | [[Hamity]] (39df)
-| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]
+| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], [[11L&nbsp;3s]], [[14L&nbsp;11s]]
 |-
 | 1
@@ Line 792: / Line 832: @@
 | 492.3
 | [[Quasisuper]] (39d)
-| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]
+| [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]
 |-
 | 1
@@ Line 798: / Line 838: @@
 | 523.1
 | [[Mavila]] (39bc)
-| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[7L&nbsp;2s]], [[7L&nbsp;9s]], [[16L&nbsp;7s]]
 |-
 | 1
@@ Line 804: / Line 844: @@
 | 584.6
 | [[Pluto]] (39d)
-| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]] etc. … [[2L&nbsp;35s]]
 |-
 | 3
@@ Line 816: / Line 856: @@
 | 61.5
 |
-| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[3L&nbsp;12s]], [[3L&nbsp;15s]], [[18L&nbsp;3s]]
 |-
 | 3
@@ Line 822: / Line 862: @@
 | 184.6
 | [[Terrain]] / [[mirkat]] (39df)
-| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]
+| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[6L&nbsp;9s]], [[6L 15]], [[6L&nbsp;21s]], [[6L&nbsp;27s]]
 |-
 | 3
-| 8\39<br>(5\39)
+| 8\39 <br>(5\39)
-| 246.2<br>(153.8)
+| 246.2 <br>(153.8)
 | [[Triforce]] (39)
-| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]
+| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[9L&nbsp;6s]], [[15L&nbsp;9s]]
 |-
 | 3
-| 16\39<br>(3\39)
+| 16\39 <br>(3\39)
-| 492.3<br>(92.3)
+| 492.3 <br>(92.3)
 | [[Augene]] (39d)
-| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[12L&nbsp;3s]], [[12L&nbsp;15s]]
 |-
 | 3
-| 17\39<br>(4\39)
+| 17\39 <br>(4\39)
-| 523.1<br>(123.0)
+| 523.1 <br>(123.0)
 | [[Deflated]] (39bd)
-| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[9L&nbsp;3s]], [[9L&nbsp;12s]], [[9L&nbsp;21s]]
 |-
 | 13
-| 16\39<br>(1\39)
+| 16\39 <br>(1\39)
-| 492.3<br>(30.8)
+| 492.3 <br>(30.8)
 | [[Tridecatonic]]
-| [[13L 13s]]
+| [[13L&nbsp;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
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 == 39edo and world music ==
+edo 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.
-edo 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 ===
+edo 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.
-edo 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 [[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&nbsp;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.
-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.
+The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|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.
+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.
 === 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&nbsp;5s]] MOS (where the generator is a perfect fifth).
-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 [[Arabic, Turkish, Persian music|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:
-=== [[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)
@@ Line 880: / Line 916: @@
 === 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.
-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]]).
+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 [[mapping]]s 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.
-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.
+=== Other ===
+edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13.
-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.
+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.
-=== Other ===
+One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.
-edo 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.
+Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
 == 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|guitar.]]
+; Prime harmonics
+/1: string 2 open
+/1: string 5 fret 12 and string 7 fret 7
+/2: string 3 fret 9 and string 5 fret 4
+/4: string 1 fret 9 and string 3 fret 4
+/4: string 5 fret 8 and string 7 fret 3
+/8: string 2 fret 9 and string 4 fret 4
 === Prototypes ===
 [[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]
@@ Line 904: / Line 961: @@
 ''39edo fretboard visualization''
-=== Lumatone mapping ===
+== Music ==
-See [[Lumatone mapping for 39edo]]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
+* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
+* [https://www.youtube.com/shorts/4y11CWLIHNA ''Sinner's Finale - Genshin Impact (microtonal cover in 39edo)''] (2025)
-== Music ==
+; [[Randy Wells]]
-* [https://www.youtube.com/watch?v=Q9wQV1J5eLE Romance On Other Planets] by [[Randy Wells]]
+* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)
-[[Category:39edo]]
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
-[[Category:39-tone scales]]
-[[Category:Modes]]
 [[Category:Listen]]