5edo: Difference between revisions

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{{interwiki

__FORCETOC__{{interwiki

| de =

| en = 5edo

| es = 5 EDO

| ja = 5平均律

}}~~__FORCETOC__~~

}}

~~-----~~

~~'''5-edo''' divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of two.~~

== Theory ==

5-edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[EDO|edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)

'''5-edo''' divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of two. 5-edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[EDO|edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)

~~There~~ is a ~~lot of near~~-~~equipentatonic world music~~, ~~just google "gyil" or "amadinda" or "slendro"~~.

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].

~~=== Listen~~ to the ~~sound~~ of ~~the 5~~-~~edo scale ===~~

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

~~For any musician~~, ~~there~~ is ~~no substitute for~~ the ~~experience of a particular xenharmonic sound~~. ~~The user going by~~ the ~~name Hyacinth on Wikipedia~~ and ~~Wikimedia Commons has many xenharmonic MIDI's~~ and ~~has graciously copylefted them! This~~ is ~~his 5~~-~~edo~~ scale ~~MIDI:~~

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and 5edo the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

* http://commons.~~wikimedia.org/wiki/File:~~5-~~tet_scale_on_C~~.~~mid~~

=== Intervals ~~in 5-edo =~~==

== Intervals ==

{| class="wikitable"

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| style="text-align:center;" | 5

| style="text-align:center;" | 1200

| style="text-align:center;" | octave ~~/ eighth~~

| style="text-align:center;" | octave

| exactly 2/1

|}

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[[:File:5ed2-001.svg|5ed2-001.svg]]

== Notation ==

* via Reinhard's cents notation

* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C

* a four-line hybrid treble/bass staff.

[[KiteGiedraitis|Kite Giedraitis]] has proposed a pentatonic notation that retains the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. When notating larger edos such as 8 or 13, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid.

== Observations ==

=== Related scales ===

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Line 92:

* For the same reason there are many "circle sisters":

** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.

~~=== As a temperament ===~~

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and 5edo the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

=== Cycles, Divisions ===

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* Cycle of sevenths: 0-4-3-2-1-0

== Commas ~~Tempered~~ ==

=== Harmony ===

5-EDO tempers out the following commas. (Note: This assumes the val < 5 8 12 14 17 19 |.)

5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

In contrast to other EDOs, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10-EDO]]).

Important chords:

* 0+1+3

* 0+2+3

* 0+1+3+4

* 0+2+3+4

=== Melody ===

Smallest EDO that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

=== Chord or scale? ===

Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.

== Commas ==

5-EDO tempers out the following [[commas]]. (Note: This assumes the val < 5 8 12 14 17 19 |.)

{| class="wikitable"

|-

! [[~~Comma~~]]

! [[Ratio]]

! [[Monzo]]

! [[cent]]s

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|}

== 5-edo ~~in Musicmaking ==~~

== Ear Training ==

5edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web here].

For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:

* http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid

==~~= Compositions, Improvisations =~~==

== Music ==

* [http://www.io.com/%7Ehmiller/ Herman Miller]: ''[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 Daybreak on Slendro Mountain]'' (2000)

* Aaron K. Johnson: ''[http://www.akjmusic.com/audio/5tet_funk.mp3 5tet funk]'' (2004)

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* Ralph Jarzombek: [http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 Micro12]

There is ~~a lot of 5edo~~ world music, search for "gyil" or "amadinda" or "slendro".

There is much 5-edo (or nearly so) world music, just search for "gyil" or "amadinda" or "slendro".

~~=== Ear Training ===~~

~~5edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web here].~~

~~=== Notation ===~~

* via Reinhard's cents notation

* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C

* a four-line hybrid treble/bass staff.

~~=== Harmony ===~~

5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

~~In contrast to other EDOs, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10-EDO]]).~~

~~Important chords:~~

* 0+1+3

* 0+2+3

* 0+1+3+4

* 0+2+3+4

~~=== Melody ===~~

Smallest EDO that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

~~=== Chord or scale? ===~~

Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.

[[Category:5-tone]]

@@ Line 1: / Line 1: @@
-{{interwiki
+__FORCETOC__{{interwiki
 | de =
 | en = 5edo
 | es = 5 EDO
 | ja = 5平均律
-}}__FORCETOC__
+}}
------
-'''5-edo''' divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of two.
 == Theory ==
--edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[EDO|edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)
+'''5-edo''' divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of two. 5-edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[EDO|edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)
-There is a lot of near-equipentatonic world music, just google "gyil" or "amadinda" or "slendro".
+If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].
-=== Listen to the sound of the 5-edo scale ===
+Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
-For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:
+Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and 5edo the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
-* http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid
-=== Intervals in 5-edo ===
+== Intervals ==
 {| class="wikitable"
@@ Line 77: / Line 71: @@
 | style="text-align:center;" | 5
 | style="text-align:center;" | 1200
-| style="text-align:center;" | octave / eighth
+| style="text-align:center;" | octave
 | exactly 2/1
 |}
@@ Line 84: / Line 78: @@
 [[:File:5ed2-001.svg|5ed2-001.svg]]
+== Notation ==
+* via Reinhard's cents notation
+* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C
+* a four-line hybrid treble/bass staff.
+[[KiteGiedraitis|Kite Giedraitis]] has proposed a pentatonic notation that retains the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. When notating larger edos such as 8 or 13, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid.
+== Observations ==
 === Related scales ===
@@ Line 90: / Line 92: @@
 * For the same reason there are many "circle sisters":
 ** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.
-=== As a temperament ===
-If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].
-Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
-Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and 5edo the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
 === Cycles, Divisions ===
@@ Line 107: / Line 101: @@
 * Cycle of sevenths: 0-4-3-2-1-0
-== Commas Tempered ==
+=== Harmony ===
--EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)
+edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.
+In contrast to other EDOs, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10-EDO]]).
+Important chords:
+* 0+1+3
+* 0+2+3
+* 0+1+3+4
+* 0+2+3+4
+=== Melody ===
+Smallest EDO that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.
+=== Chord or scale? ===
+Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.
+== Commas ==
+-EDO tempers out the following [[commas]]. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)
 {| class="wikitable"
 |-
-! [[Comma]]
+! [[Ratio]]
 ! [[Monzo]]
 ! [[cent]]s
@@ Line 313: / Line 324: @@
 |}
-== 5-edo in Musicmaking ==
+== Ear Training ==
+edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web here].
+For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:
+* http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid
-=== Compositions, Improvisations ===
+== Music ==
 * [http://www.io.com/%7Ehmiller/ Herman Miller]: ''[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 Daybreak on Slendro Mountain]'' (2000)
 * Aaron K. Johnson: ''[http://www.akjmusic.com/audio/5tet_funk.mp3 5tet funk]'' (2004)
@@ Line 327: / Line 343: @@
 * Ralph Jarzombek: [http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 Micro12]
-There is a lot of 5edo world music, search for "gyil" or "amadinda" or "slendro".
+There is much 5-edo (or nearly so) world music, just search for "gyil" or "amadinda" or "slendro".
-=== Ear Training ===
-edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web here].
-=== Notation ===
-* via Reinhard's cents notation
-* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C
-* a four-line hybrid treble/bass staff.
-=== Harmony ===
-edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.
-In contrast to other EDOs, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10-EDO]]).
-Important chords:
-* 0+1+3
-* 0+2+3
-* 0+1+3+4
-* 0+2+3+4
-=== Melody ===
-Smallest EDO that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.
-=== Chord or scale? ===
-Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.
 [[Category:5-tone]]