**39-EDO, 39-ED2** or **39-tET** divides the Octave (Ditave 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of [[xenharmonic/7L 2s|Superdiatonic]] LLLsLLLLs like a basical scale for notation and theory, suited in [[xenharmonic/16edo|16-ED2]], and allied systems: [[xenharmonic/25edo|25-ED2]] [1/3-tone 3;2]; [[xenharmonic/41edo|41-ED2]] [1/5-tone 5;3]; and [[xenharmonic/57edo|57]] ED2 [1/7-tone 7;4]. **Hornbostel Temperaments** is included too with: [[xenharmonic/23edo|23-ED2]] [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] & [[xenharmonic/62edo|62-ED2]] [1/8-tone 8;3]. [[223edo|223-ED2]], the best accuracy for Hornbostel temperament fits very good with Armodue like 1/29-tone 29;10 version. Note that [[101edo|101]], [[131edo|131]], [[177edo|177]] & [[200edo|200]] ED2s are tempered systems that [[http://www.h-pi.com/eop-ogolevets.html|Alexei Ogolevets]] (Ukraine, 1891 - 1967) was proposing in his List of Temperaments, in which the Armodue system fits very well in all these.
== Theory ==
However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is <39 62 91 110 135|.
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 have two choices for a tuning of [[7/1|7]], but the sharp one blends better with the sharp 3 and 5 as their errors cancel with their ratios, thus yielding significantly higher average accuracy than the [[patent val]] in the [[7-odd-limit|7-]] and [[9-odd-limit]]. It also has a fine [[11/1|11]], and adding it to consideration the best choice for 39et is the sharp-tending 39df val {{val| 39 62 91 '''110''' 135 '''145''' }}.
A particular anecdote with this 39 divisions per 2/1 was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740? - 1820), a little extract [[http://ml.oxfordjournals.org/content/76/2/291.extract.jpg|here]].
As a superpyth system, 39edo is intermediate between 17edo and 22edo (39 being 17+22). While 17edo is superb for melody (as documented by George Secor), it doesn't 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 (the ideal diatonic semitone for melody being somewhere in between 60 and 80 cents, 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 39d 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 assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.
39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it doesn't 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|columns=11}}
{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}
=== As a tuning of other temperaments ===
39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[128/125|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 [[12edo|12et]] in [[support]]ing [[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]].
==__**39-EDO Intervals**__==
Alternatively, the patent val tempers out [[49/48]] to yield [[semaphore]], and provides a reasonable tuning of [[triforce]] beyond [[15edo]], and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports [[negri]] and 5-limit [[Syntonic–chromatic_equivalence_continuum#Sixix_(5-limit)|sixix]].
|| **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) ||
||~ **Degrees** ||~ **Armodue note** ||~ **Cents size** ||~ **[[xenharmonic/Nearest just interval|Nearest Just I]]nterval** ||~ **Cents value** ||~ **Error** ||~ 11-limit Ratio Assuming
If we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]] through the 39bc val, 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).
Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.
As 39edo is a rare case where a non-patent val does significantly better than the patent val, we provide two tables, for those who look for the most accurate temperament available and for those who would like to explore the potential utilities in this edo.
==**__39 tone equal [[xenharmonic/modes|modes]]__:**==
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! colspan="3" | [[Ups and downs notation]]
|-
| 0
| 0.0
| P1
| perfect unison
| D
|-
| 1
| 30.8
| ^1, <br>vm2
| up unison, <br>downminor 2nd
| ^D, <br>vEb
|-
| 2
| 61.5
| m2
| minor 2nd
| Eb
|-
| 3
| 92.3
| ^m2
| upminor 2nd
| ^Eb
|-
| 4
| 123.1
| ^^m2
| dupminor 2nd
| ^^Eb
|-
| 5
| 153.8
| vvM2
| dudmajor 2nd
| vvE
|-
| 6
| 184.6
| vM2
| downmajor 2nd
| vE
|-
| 7
| 215.4
| M2
| major 2nd
| E
|-
| 8
| 246.2
| ^M2, <br>vm3
| upmajor 2nd, <br>downminor 3rd
| ^E, <br>vF
|-
| 9
| 276.9
| m3
| minor 3rd
| F
|-
| 10
| 307.7
| ^m3
| upminor 3rd
| ^F
|-
| 11
| 338.5
| ^^m3
| dupminor 3rd
| ^^F
|-
| 12
| 369.2
| vvM3
| dudmajor 3rd
| vvF#
|-
| 13
| 400.0
| vM3
| downmajor 3rd
| vF#
|-
| 14
| 430.8
| M3
| major 3rd
| F#
|-
| 15
| 461.5
| v4
| down 4th
| vG
|-
| 16
| 492.3
| P4
| perfect 4th
| G
|-
| 17
| 523.1
| ^4
| up 4th
| ^G
|-
| 18
| 553.8
| ^^4
| dup 4th
| ^^G
|-
| 19
| 584.6
| vvA4, <br>^d5
| dudaug 4th, <br>updim 5th
| vvG#, <br>^Ab
|-
| 20
| 615.4
| vA4, <br>^^d5
| downaug 4th, <br>dupdim 5th
| vG#, <br>^^Ab
|-
| 21
| 646.2
| vv5
| dud 5th
| vvA
|-
| 22
| 676.9
| v5
| down 5th
| vA
|-
| 23
| 707.7
| P5
| perfect 5th
| A
|-
| 24
| 738.5
| ^5
| up 5th
| A^
|-
| 25
| 769.2
| m6
| minor 6th
| Bb
|-
| 26
| 800.0
| ^m6
| upminor 6th
| ^Bb
|-
| 27
| 830.8
| ^^m6
| dupminor 6th
| ^^Bb
|-
| 28
| 861.5
| vvM6
| dudmajor 6th
| vvB
|-
| 29
| 892.3
| vM6
| downmajor 6th
| vB
|-
| 30
| 923.1
| M6
| major 6th
| B
|-
| 31
| 953.8
| ^M6, <br>vm7
| upmajor 6th, <br>downminor 7th
| ^B, <br>vC
|-
| 32
| 984.6
| m7
| minor 7th
| C
|-
| 33
| 1015.4
| ^m7
| upminor 7th
| ^C
|-
| 34
| 1046.2
| ^^m7
| dupminor 7th
| ^^C
|-
| 35
| 1076.9
| vvM7
| dudmajor 7th
| vvC#
|-
| 36
| 1107.7
| vM7
| downmajor 7th
| vC#
|-
| 37
| 1138.5
| M7
| major 7th
| C#
|-
| 38
| 1169.2
| ^M7, <br>v8
| upmajor 7th, <br>down 8ve
| ^C#, <br>vD
|-
| 39
| 1200.0
| P8
| perfect 8ve
| D
|}
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.
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Kite's ups and downs notation #Chords and chord progressions]].
===Western:===
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}
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.
=== Kite's ups and downs notation ===
39edo can also be notated with [[Kite's ups and downs notation|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).
{{Ups and downs sharpness}}
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, the wolf fifth can be avoided by introducing accidental notes when necessary. 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.
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].
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.
==== 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>
===**Indian:**===
==== 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>
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 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 sytems. 39edo is a Level 2 system because:
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal <br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 62 -39 }}
| {{Mapping| 39 62 }}
| −1.81
| 1.81
| 5.88
|-
| 2.3.5
| 128/125, 1594323/1562500
| {{Mapping| 39 62 91 }}
| −3.17
| 2.42
| 7.89
|-
| 2.3.5.7
| 64/63, 126/125, 2430/2401
| {{Mapping| 39 62 91 110 }} (39d)
| −3.78
| 2.35
| 7.65
|-
| 2.3.5.7.11
| 64/63, 99/98, 121/120, 126/125
| {{Mapping| 39 62 91 110 135 }} (39d)
| −3.17
| 2.43
| 7.91
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-4 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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 [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|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 [[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.
=== 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 [[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.
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.
=== 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 [[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:
* It has two types of "neutral" seconds (154 and 185 cents)
* It has two types of "neutral" seconds (154 and 185 cents)
Line 149:
Line 1,230:
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.
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:**===
=== 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 [[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.
=== 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{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
'''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
The harmonic seventh ("barbershop seventh") tetrad is reasonably well approximated in 39edo, and some temperaments (augene in particular) are liberally supplied with them. John Coltrane [[https://en.wikipedia.org/wiki/Coltrane_changes|would have loved augene]].
7/4: string 5 fret 8 and string 7 fret 3
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.
11/8: string 2 fret 9 and string 4 fret 4
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 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 accomodated.</pre></div>
<strong>39-EDO, 39-ED2</strong> or <strong>39-tET</strong> divides the Octave (Ditave 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%202s">Superdiatonic</a> LLLsLLLLs like a basical scale for notation and theory, suited in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16edo">16-ED2</a>, and allied systems: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/25edo">25-ED2</a> [1/3-tone 3;2]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41-ED2</a> [1/5-tone 5;3]; and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/57edo">57</a> ED2 [1/7-tone 7;4]. <strong>Hornbostel Temperaments</strong> is included too with: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/23edo">23-ED2</a> [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] &amp; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/62edo">62-ED2</a> [1/8-tone 8;3]. <a class="wiki_link" href="/223edo">223-ED2</a>, the best accuracy for Hornbostel temperament fits very good with Armodue like 1/29-tone 29;10 version. Note that <a class="wiki_link" href="/101edo">101</a>, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/177edo">177</a> &amp; <a class="wiki_link" href="/200edo">200</a> ED2s are tempered systems that <a class="wiki_link_ext" href="http://www.h-pi.com/eop-ogolevets.html" rel="nofollow">Alexei Ogolevets</a> (Ukraine, 1891 - 1967) was proposing in his List of Temperaments, in which the Armodue system fits very well in all these.<br />
However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is &lt;39 62 91 110 135|.<br />
A particular anecdote with this 39 divisions per 2/1 was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740? - 1820), a little extract <a class="wiki_link_ext" href="http://ml.oxfordjournals.org/content/76/2/291.extract.jpg" rel="nofollow">here</a>.<br />
<br />
As a superpyth system, 39edo is intermediate between 17edo and 22edo (39 being 17+22). While 17edo is superb for melody (as documented by George Secor), it doesn't 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 &quot;diatonic semitone&quot; 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 (the ideal diatonic semitone for melody being somewhere in between 60 and 80 cents, by Secor's estimates).<br />
<br />
39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it doesn't do as good of a job at approximating JI as some other systems do. Because it can also approximate mavila as well as &quot;anti-mavila&quot; (oneirotonic), the latter of which it inherits from <a class="wiki_link" href="/13edo">13edo</a>, this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).<br />
* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)
* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)
; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "Resolute Prelude"
** "Residual Soliloquy"
<table class="wiki_table">
; [[Randy Wells]]
<tr>
* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x39 tone equal temperament-39edo and world music:"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong><u>39edo and world music:</u></strong></h2>
<br />
39edo is a good candidate for a &quot;universal tuning&quot; 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x39 tone equal temperament-39edo and world music:-Western:"></a><!-- ws:end:WikiTextHeadingRule:10 -->Western:</h3>
<br />
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: <strong>7 7 2 7 7 7 2</strong>). Because 39edo is a superpyth rather than a meantone system, this means that the harmonic quality of its diatonic scale will differ somewhat, since &quot;minor&quot; and &quot;major&quot; 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.<br />
<br />
Another option is to use a MODMOS, such as <strong>7 6 3 7 6 7 3</strong>; 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, the wolf fifth can be avoided by introducing accidental notes when necessary. 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 <em>many</em> different ways, acquiring a distinctly different but still harmonious character each time.<br />
<br />
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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x39 tone equal temperament-39edo and world music:-Indian:"></a><!-- ws:end:WikiTextHeadingRule:12 --><strong>Indian:</strong></h3>
<br />
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).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x39 tone equal temperament-39edo and world music:-Arabic, Turkish, Persian:"></a><!-- ws:end:WikiTextHeadingRule:14 --><strong><a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">Arabic, Turkish, Persian</a>:</strong></h3>
<br />
While middle-eastern music is commonly approximated using 24edo, 39edo offers a potentially better alternative. 17edo and 24edo both satisfy the &quot;Level 1&quot; requirements for maqam tuning sytems. 39edo is a Level 2 system because:<br />
<br />
<ul><li>It has two types of &quot;neutral&quot; seconds (154 and 185 cents)</li><li>It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents)</li></ul><br />
whereas neither 17edo nor 24edo satisfy these properties.<br />
<br />
39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a &quot;major-like&quot; wide neutral third and a wide &quot;neutral&quot; second approaching 10/9), will likely be especially well suited to 39edo.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x39 tone equal temperament-39edo and world music:-Blues / Jazz / African-American:"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong>Blues / Jazz / African-American:</strong></h3>
<br />
The harmonic seventh (&quot;barbershop seventh&quot;) tetrad is reasonably well approximated in 39edo, and some temperaments (augene in particular) are liberally supplied with them. John Coltrane <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Coltrane_changes" rel="nofollow">would have loved augene</a>.<br />
<br />
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 <em>resolution</em> rather than a suspension.<br />
<br />
Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a &quot;blue major third&quot; 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x39 tone equal temperament-39edo and world music:-Other:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Other:</h3>
<br />
39edo offers a good approximation of pelog / mavila using the flat fifth as a generator.<br />
<br />
It also offers <em>many</em> possible pentatonic scales, including the 2L+3S MOS (which is <strong>9 7 7 9 7</strong>). Slendro can be approximated using this scale or using something like the quasi-equal <strong>8 8 8 8 7</strong>. A more expressive pentatonic scale is the oneirotonic subset <strong>9 6 9 9 6</strong>. Many Asian and African musical styles can thus be accomodated.</body></html></pre></div>
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.
39edo's perfect fifth is 5.8 cents sharp, together with its best classical major third which is the familiar 400 cents of 12edo. We have two choices for a tuning of 7, but the sharp one blends better with the sharp 3 and 5 as their errors cancel with their ratios, thus yielding significantly higher average accuracy than the patent val in the 7- and 9-odd-limit. It also has a fine 11, and adding it to consideration the best 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 39d 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 assuming that the MOS diatonic is used, because the diatonic semitone is quartertone-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.
39edo, 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 supportingaugene, 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.
Alternatively, the patent val tempers out 49/48 to yield semaphore, and provides a reasonable tuning of triforce beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp. The 39c val supports negri and 5-limit sixix.
If we take 22\39 as a fifth, 39edo can be used as a tuning of mavila through the 39bc val, 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).
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
As 39edo is a rare case where a non-patent val does significantly better than the patent val, we provide two tables, for those who look for the most accurate temperament available and for those who would like to explore the potential utilities in this edo.
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
Flat symbol
Sagittal notation
This notation uses the same sagittal sequence as 46edo.
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.
