5edo: Difference between revisions

=5 Equal Divisions of the Octave: Theory=

==="Equal Pentatonic"===

5-edo divides the 1200-[[cent|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|2edo]] and [[3edo|3edo]]. Most importantly, 5-edo is the smallest [[EDO|edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)

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

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

==Intervals in 5-edo==

! | degrees

! | size

 

in [[cent|cents]]

! | Closest diatonic

 

interval name

! | The "neighborhood" of just intervals

| | exactly 1/1

| | +8.826¢ from septimal second [[8/7|8/7]]

| | +9.219¢ from narrow fourth [[21/16|21/16]]

| | +18.045¢ from just fifth [[3/2|3/2]]

| | 26.871¢ from septimal major sixth [[12/7|12/7]]

| | exactly 2/1

==Related scales==

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

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|9-limit]] [[consistent|consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo|4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit|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|2edo]] represents the [[3-limit|3-limit]] consistently, [[3edo|3edo]] the [[5-limit|5-limit]], [[4edo|4edo]] the [[7-limit|7-limit]] and [[5edo|5edo]] the [[9-limit|9-limit]], to represent the [[11-limit|11-limit]] consistently with a [[Patent_val|patent val]] requires going all the way to [[22edo|22edo]].

==Cycles, Divisions==

Cycle of seconds: 0-1-2-3-4-0

Cycle of fourths: 0-2-4-1-3-0

 

 

=5-edo in Musicmaking=

 

=='''Compositions''', improvisations==

<ul><ul><li>[http://www.io.com/%7Ehmiller/ Herman Miller]: ''[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 Daybreak on Slendro Mountain]'' (2000)</li><li>Aaron K. Johnson: ''[http://www.akjmusic.com/audio/5tet_funk.mp3 5tet funk]'' (2004)</li><li>[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)</li><li>[[Hans_Straub|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]</li><li>[[Brian_Wong|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]</li><li>Brian McLaren: various and sundry</li><li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>X.J.Scott: ''Sleeping Through It All'' (2004)</li><li>Bill Sethares: ''5-tet funk'' (2004), ''Pentacle'' (2004)</li><li>"Cenobyte" Ukulele [http://www.youtube.com/watch?v=UKUCRnEJKKU  http://www.youtube.com/watch?v=UKUCRnEJKKU]</li><li>"[http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele True Island]" (album) by Small Scale Revolution (2011)</li><li>Ralph Jarzombek: [http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 Micro12]</li></ul></ul>

==Ear Training==

==Notation==

<ul><ul><li>via Reinhard's cents notation</li><li>naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C</li><li>a four-line hybrid treble/bass staff.</li></ul></ul>

==Harmony==

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 10-EDO).

 

==Melody==

==Chord or scale?==

==Commas Tempered==

! | Comma

! | Value (cents)

! | Name

! | Second Name

! | Third Name

! | Monzo

| | Limma

| | Pythagorean Minor 2nd

| |  

| | | 8 -5 &gt;

| | Syntonic Comma

| | Didymos Comma

| | Meantone Comma

| | | -4 4 -1 &gt;

| | Vulture

| |  

| |  

| | | 24 -21 4 &gt;

| | Septimal Quarter Tone

| |  

| |  

| | | 2 2 -1 -1 &gt;

| | Slendro Diesis

| |  

| |  

| | | -4 -1 0 2 &gt;

| | Septimal Comma

| | Archytas' Comma

| | Leipziger Komma

| | | 6 -2 0 -1 &gt;

| | Sensamagic

| |  

| |  

| | | 0 -5 1 2 &gt;

| | Orwellisma

| | Orwell Comma

| |  

| | | 6 3 -1 -3 &gt;

| | Gamelisma

| |  

| |  

| | | -10 1 0 3 &gt;

| | Cataharry

| |  

| |  

| | | -4 9 -2 -2 &gt;

| | Hemifamity

| |  

| |  

| | | 10 -6 1 -1 &gt;

| | Wadisma

| |  

| |  

| | | -26 -1 1 9 &gt;

| | Wizma

| |  

| |  

| | | -6 -8 2 5 &gt;

| | Mothwellsma

| |  

| |  

| | | -1 2 0 -2 1 &gt;

| | Pentacircle

| |  

| |  

| | | 7 -4 0 1 -1 &gt;

| | Keenanisma

| |  

| |  

| | | -7 -1 1 1 1 &gt;

| | Werckisma

| |  

| |  

| | | -3 2 -1 2 -1 &gt;

| | Lehmerisma

| |  

| |  

| | | -4 -3 2 -1 2 &gt;

| | Superleap

| |  

| |  

| | | -1 -2 -1 1 0 1 &gt;

| | Parizeksma

| |  

| |  

| | | 2 -3 -2 0 0 2 &gt;

| | Diatonic semitone

| |  

| |  

| | | 4 -1 -1 &gt;

| |  

| |  

| |  

| | | 1 0 0 1 0 -1 &gt;

| | Large diatonic semit.

| |  

| |  

| | | 0 3 -2 &gt;

| | Large neutral second

| |  

| |  

| | | -1 0 -1 0 1 &gt;