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 {{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]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic [[semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

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

39edo 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 ===

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=== 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 pleasing.

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

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

Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that xenharmonic to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the '''7 6 3 7 6 7 3''' MODMOS), for example.

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 ===

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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 might 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.

[[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.

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

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

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 {{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]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic semitone is quartertone-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.
+As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the diatonic [[semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.
 Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] &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.
-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 ===
@@ Line 870: / Line 870: @@
 === 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 pleasing.
 Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.
@@ Line 876: / Line 876: @@
 The MOS and the MODMOS's all have smaller-than-usual semitones, which makes them more effective for melody than their counterparts in 12edo or meantone systems.
-Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that xenharmonic to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the '''7 6 3 7 6 7 3''' MODMOS), for example.
+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 ===
@@ 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 might 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.
+[[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.
@@ Line 901: / Line 901: @@
 edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator.
-It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive pentatonic scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
+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 ==