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| Augmented 1sn = 1\5 = 240¢
| Augmented 1sn = 1\5 = 240¢
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'''5-edo''' divides the octave into 5 equal parts, making its smallest interval exactly 240 [[cent]]s, 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]] containing xenharmonic intervals - 1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].
'''5 equal divisions of the octave''' (or '''5edo''') is the [[tuning system]] derived by dividing the [[octave]] into 5 equal steps of 240 [[cent]]s each, or the fifth root of two. 5edo is the third [[prime EDO|prime edo]], after [[2edo]] and [[3edo]]. Most importantly, 5edo is the smallest [[edo]] containing xenharmonic intervals — 1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].


== Theory ==
== Theory ==
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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]].
If 5edo is regarded as a temperament, which is to say as 5-TET, 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.
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]]. Nevertheless, because the comma tempered out for this EDO's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second EDO to demonstrate 3-to-2 [[telicity]].
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]]. Nevertheless, because the comma tempered out for this edo's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second edo to demonstrate 3-to-2 [[telicity]].


== Intervals ==
== Intervals ==
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=== Related scales ===
=== Related scales ===


* By its cardinality, 5-edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.
* By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.
* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.
* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.
* For the same reason there are many "circle sisters":
* 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.
** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.
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=== Cycles, Divisions ===
=== Cycles, Divisions ===


5 is a prime number so 5-edo contains no sub-edos. Only simple cycles:
5 is a prime number so 5edo contains no sub-edos. Only simple cycles:


* Cycle of seconds: 0-1-2-3-4-0
* Cycle of seconds: 0-1-2-3-4-0
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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.
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]]).
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]]).


Important chords:
Important chords:
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=== Melody ===
=== 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.
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? ===
=== Chord or scale? ===
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== Commas ==
== Commas ==


5-EDO [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.  
5edo [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
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== Music ==
== Music ==


* [http://www.io.com/%7Ehmiller/ Herman Miller]: ''[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 Daybreak on Slendro Mountain]'' (2000)
{| class="wikitable sortable"
* [[Aaron Krister Johnson]]: ''[http://www.akjmusic.com/audio/5tet_funk.mp3 5tet funk]'' (2004)
!Title
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=1519939 Andrew Heathwaite: //Pinta Penta// (2004)] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3 play] (rendered in 6 alternative pentatonics as well)
!Composer
* [[Hans Straub]]: [http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia Asîmchômsaia] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3 play]
!Year
* [[Brian Wong]]: [http://bwong.ca/template1.php?sub=3 Slendronica#1b] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg play]
!Genre
* [[Brian McLaren]]: various and sundry<li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>X.J.Scott: ''Sleeping Through It All'' (2004)
!Additional links
* [[Bill Sethares]]: ''5-tet funk'' (2004), ''Pentacle'' (2004)
|-
* "Cenobyte" Ukulele http://www.youtube.com/watch?v=UKUCRnEJKKU
|[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 ''Daybreak'']
* "[http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele True Island]" (album) by Small Scale Revolution (2011)
|[[Herman Miller]]
* [[Ralph Jarzombek]]: [http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 Micro12]
|2000
* [https://soundcloud.com/uz1kt3k/prelude-in-5et Prelude In 5ET] by [[Aaron Andrew Hunt]]
|Folk (Indonesian)
* [https://soundcloud.com/uz1kt3k/invention-in-5et Invention In 5ET] by [[Aaron Andrew Hunt]]
|
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (first and third parts in 5-edo)
|-
*[https://youtu.be/Xh2EUwg34pk "Winter Forest"] by NullPointerException Music (from his 1-EDO to 31-EDO album "Edolian")
|[https://www.soundclick.com/music/songInfo.cfm?songID=1519939 ''Pinta Penta in 5tET'']
|[[Andrew Heathwaite]]
|2004
|Electronic
|[https://www.soundclick.com/artist/default.cfm?bandID=122613&content=songs Alternative versions (SoundClick)]
|-
|[http://www.akjmusic.com/audio/5tet_funk.mp3 ''5tet funk'']{{dead link}}
|[[Aaron Krister Johnson]]
|2004
|Funk
|
|-
|[https://www.soundclick.com/music/songInfo.cfm?songID=1522036 ''Sleeping Through It All'']
|[[X. J. Scott]]
|2004
|Electronic
|
|-
|''5-tet funk''
|[[Bill Sethares]]
|2004
|Funk
|
|-
|''Pentacle''
|[[Bill Sethares]]
|2004
|(?)
|
|-
|[http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia ''Asîmchômsaia'']
|[[Hans Straub]]
|2005 (?)
|Folk (Japanese)
|
|-
|[https://soundcloud.com/brian-wong/slendronica-1b ''Slendronica#1b'']
|[[Brian Wong]]
|2009 (?)
|Folk (Indonesian)
|
|-
|[https://www.youtube.com/watch?v=UKUCRnEJKKU ''Random Explorations''] (for ukulele)
|Cenobyte
|2011
|Folk
|
|-
|[https://www.jamendo.com/album/104474/true-island-5-equal-divisions-of-the-octave-ukulele ''True Island : 5 Equal Divisions Of The Octave Ukulele'' (album)]
|[https://www.jamendo.com/artist/374790/small-scale-revolution Small Scale Revolution]
|2011
|Folk
|
|-
|[http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 ''Micro12'']
|[http://webzoom.freewebs.com/ralphjarzombek Ralph Jarzombek]
|2010 (?)
|Electronic
|
|-
|[https://soundcloud.com/uz1kt3k/prelude-in-5et ''Prelude in 5ET'']
|[[Aaron Andrew Hunt]]
|2015
|Neobaroque
|
|-
|[https://soundcloud.com/uz1kt3k/invention-in-5et ''Invention in 5ET'']
|[[Aaron Andrew Hunt]]
|2015
|Neobaroque
|
|-
|[https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!'' (1st and 3rd sections)]
|Dmitriy Bazhenov
|2020
|Ambiance
|
|-
|data-sort-value="Winter Forest (from Edolian)"|[https://www.youtube.com/watch?v=Xh2EUwg34pk "Winter Forest"] (from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''])
|NullPointerException Music
|2020
|Classical
|
|-
|data-sort-value="Ether (from STAFFcirc vol. 7)"|"[https://soundcloud.com/sexytoadsandfrogsfriendcircle/5-vince-kaichan-ether Ether]" (from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''])
|Vince Kaichan
|2021
|Electronic
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]
|}
 
* [[Brian McLaren]]: various and sundry
* [[Paul Rubenstein]]: various, with electric guitars in 10- and 15edo


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


[[Category:5-tone]]
[[Category:5-tone]]
Retrieved from "https://en.xen.wiki/w/5edo"