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 cents sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.

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

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

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding 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)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding [[diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by [[Secor]]'s estimates.

Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25 cents flat.

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== 39edo and world music ==

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.

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.

=== 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 ~~very~~ 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 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'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.

Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The MOS and the MODMOS's all have smaller-than-usual semitones, which makes them more effective for melody than their counterparts in 12edo or meantone systems.

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=== 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, ~~Persian]]~~ ===

=== 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:

While [[Arabic, Turkish, Persian|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:

* It has two types of "neutral" seconds (154 and 185 cents)

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=== 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 ~~would~~ have loved augene (→ [[Wikipedia: Coltrane changes]]).

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.

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 ~~a good approximation~~ of pelog / mavila using the flat fifth as a generator.

39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator.

It also offers ''many'' possible pentatonic scales, including the [[2L~~ ~~3s]] MOS (which is '''9 7 7 9 7'''). Slendro can be approximated using this scale or using something like the quasi-equal '''8 8 8 8 7'''. A more expressive pentatonic scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian and African musical styles can thus be accommodated.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive pentatonic scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Instruments ==

@@ Line 3: / Line 3: @@
 == 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 cents sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.
-A particular anecdote with this system was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740?&ndash;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 [[Charles Clagget]] (Ireland, 1740?&ndash;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]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding 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)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding [[diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by [[Secor]]'s estimates.
 Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] &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.
@@ Line 867: / Line 867: @@
 == 39edo and world music ==
-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.
+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.
 === 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 very 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 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'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.
+Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.
 The MOS and the MODMOS's all have smaller-than-usual semitones, which makes them more effective for melody than their counterparts in 12edo or meantone systems.
@@ Line 879: / Line 879: @@
 === 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, Persian]] ===
+=== 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:
+While [[Arabic, Turkish, Persian|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:
 * It has two types of "neutral" seconds (154 and 185 cents)
@@ Line 892: / Line 892: @@
 === 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 would have loved augene (&rarr; [[Wikipedia: Coltrane changes]]).
+The harmonic seventh ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (&rarr; [[Wikipedia: Coltrane changes]]).
 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.
+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 ===
-edo offers a good approximation of pelog / mavila using the flat fifth as a generator.
+edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator.
-It also offers ''many'' possible pentatonic scales, including the [[2L&nbsp;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.
+It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive pentatonic scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
 == Instruments ==