39edo: Difference between revisions

Line 5:

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]] (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.

Line 26:

! Approximate Ratios*

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

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

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

|-

| 0

Line 41:

| 30.77

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

| ^1, vm2

| ^1, vm2

| up unison, downminor 2nd

| up unison, downminor 2nd

| ^D, vEb

| ^D, vEb

| 57/56

| 30.64

Line 111:

| 246.15

| [[81/70]]

| ^M2, vm3

| ^M2, vm3

| upmajor 2nd, downminor 3rd

| upmajor 2nd, downminor 3rd

| ^E, vF

| ^E, vF

| 15/13

| 247.74

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| 584.62

| [[7/5]]

| vvA4, ^d5

| vvA4, ^d5

| dudaug 4th, updim 5th

| dudaug 4th, updim 5th

| vvG#, ^Ab

| vvG#, ^Ab

| 7/5

| 582.51

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| 615.38

| [[10/7]]

| vA4, ^^d5

| vA4, ^^d5

| downaug 4th, dupdim 5th

| downaug 4th, dupdim 5th

| vG#, ^^Ab

| vG#, ^^Ab

| 10/7

| 617.49

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| 953.85

| [[140/81]]

| ^M6, vm7

| ^M6, vm7

| upmajor 6th, downminor 7th

| upmajor 6th, downminor 7th

| ^B, vC

| ^B, vC

| 26/15

| 952.26

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| 1169.23

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

| ^M7, v8

| ^M7, v8

| upmajor 7th, down 8ve

| upmajor 7th, down 8ve

| ^C#, vD

| ^C#, vD

| 112/57

| 1169.36

Line 428:

| 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]].

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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" | [[Mapping]]

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

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

! colspan="2" | Tuning Error

|-

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=== Rank-2 temperaments ===

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

|+Table of temperaments by generator

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

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

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Line 836:

|-

| 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]]

|-

| 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]]

|-

| 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]]

|-

| 13

| 16\39 (1\39)

| 16\39 (1\39)

| 492.3 (30.8)

| 492.3 (30.8)

| [[Tridecatonic]]

| [[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 it is 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 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 very pleasing.

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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).

=== [[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:

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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]]).

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.

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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.

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.

== Instruments ==

=== Prototypes ===

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

@@ Line 5: / Line 5: @@
 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?–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]] (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 26: / Line 26: @@
 ! Approximate Ratios*
 ! colspan="3" | [[Ups and Downs Notation]]
-! colspan="3" | [[Nearest just interval|Nearest Just Interval]]<br>(Ratio, Cents, Error)
+! colspan="3" | [[Nearest just interval|Nearest Just Interval]]<br />(Ratio, Cents, Error)
 |-
 | 0
@@ Line 41: / Line 41: @@
 | 30.77
 | ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]
-| ^1,<br>vm2
+| ^1,<br />vm2
-| up unison, <br>downminor 2nd
+| up unison, <br />downminor 2nd
-| ^D, <br>vEb
+| ^D, <br />vEb
 | 57/56
 | 30.64
@@ Line 111: / Line 111: @@
 | 246.15
 | [[81/70]]
-| ^M2, <br>vm3
+| ^M2, <br />vm3
-| upmajor 2nd, <br>downminor 3rd
+| upmajor 2nd, <br />downminor 3rd
-| ^E, <br>vF
+| ^E, <br />vF
 | 15/13
 | 247.74
@@ Line 221: / Line 221: @@
 | 584.62
 | [[7/5]]
-| vvA4, <br>^d5
+| vvA4, <br />^d5
-| dudaug 4th, <br>updim 5th
+| dudaug 4th, <br />updim 5th
-| vvG#, <br>^Ab
+| vvG#, <br />^Ab
 | 7/5
 | 582.51
@@ Line 231: / Line 231: @@
 | 615.38
 | [[10/7]]
-| vA4, <br>^^d5
+| vA4, <br />^^d5
-| downaug 4th, <br>dupdim 5th
+| downaug 4th, <br />dupdim 5th
-| vG#, <br>^^Ab
+| vG#, <br />^^Ab
 | 10/7
 | 617.49
@@ Line 341: / Line 341: @@
 | 953.85
 | [[140/81]]
-| ^M6, <br>vm7
+| ^M6, <br />vm7
-| upmajor 6th, <br>downminor 7th
+| upmajor 6th, <br />downminor 7th
-| ^B, <br>vC
+| ^B, <br />vC
 | 26/15
 | 952.26
@@ Line 411: / Line 411: @@
 | 1169.23
 | ''[[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
+| ^C#, <br />vD
 | 112/57
 | 1169.36
@@ Line 428: / Line 428: @@
 | 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]].
@@ Line 696: / Line 696: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve Stretch (¢)
 ! colspan="2" | Tuning Error
 |-
@@ Line 736: / Line 737: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all right-3 left-4 left-5"
-|+Table of temperaments by generator
+|+ style="font-size: 105%;" | Table of temperaments by generator
 |-
-! Periods<br>per 8ve
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
@@ Line 835: / Line 836: @@
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 13
-| 16\39<br>(1\39)
+| 16\39<br />(1\39)
-| 492.3<br>(30.8)
+| 492.3<br />(30.8)
 | [[Tridecatonic]]
 | [[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 it is 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 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 very pleasing.
@@ Line 875: / Line 874: @@
 === 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, 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:
@@ Line 890: / Line 887: @@
 === 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 would have loved augene (→ [[Wikipedia: Coltrane changes]]).
 Tritone substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension.
@@ Line 900: / Line 896: @@
 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.
+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.
 == Instruments ==
 === Prototypes ===
 [[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]