5edo

Theory

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

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 father temperament.

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain 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 zeta integral edo, after 2EDO. It also is the smallest equal division representing the 9-limit 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 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.

Intervals

degrees	cents	Closest diatonic interval name	The "neighborhood" of just intervals
0	0	unison / prime	1/1
1	240	second, third	+8.826¢ from septimal second 8/7 -4.969¢ from diminished third 144/125 -13.076¢ from augmented second 125/108 -26.871¢ from septimal minor third 7/6
2	480	fourth	+9.219¢ from narrow fourth 21/16 -0.686¢ from smaller fourth 33/25 -18.045¢ from just fourth 4/3
3	720	fifth	+18.045¢ from just fifth 3/2 +0.686¢ from bigger fifth 50/33 -9.219¢ from wide fifth 32/21
4	960	sixth, seventh	26.871¢ from septimal major sixth 12/7 13.076¢ from diminished seventh 216/125 4.969¢ from augmented sixth 125/72 -8.826¢ from septimal seventh 7/4
5	1200	octave	2/1

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.

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

• By its cardinality, 5-edo is related to other pentatonic scales, and it is especially close in sound to many Indonesian slendros.
• 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.
• 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.

Cycles, Divisions

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

• Cycle of seconds: 0-1-2-3-4-0
• Cycle of fourths: 0-2-4-1-3-0
• Cycle of fifths: 0-3-1-4-2-0
• Cycle of sevenths: 0-4-3-2-1-0

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 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 |.)

Ratio	Monzo	cents	Color Name	Name	Second Name	Third Name
256/243	[8 -5⟩	90.225	Sawa	Limma	Pythagorean Minor 2nd
27/25	[0 3 -2⟩	133.238	Gugu	Large diatonic semit.
16/15	[4 -1 -1⟩	111.731	Gubi	Diatonic semitone
81/80	[-4 4 -1⟩	21.506	Gu	Syntonic Comma	Didymos Comma	Meantone Comma
2889416/2882415	[24 -21 4⟩	4.200	Sasa-quadyo	Vulture
36/35	[2 2 -1 -1⟩	48.770	Rugu	Septimal Quarter Tone
49/48	[-4 -1 0 2⟩	35.697	Zozo	Slendro Diesis
64/63	[6 -2 0 -1⟩	27.264	Ru	Septimal Comma	Archytas' Comma	Leipziger Komma
245/243	[0 -5 1 2⟩	14.191	Zozoyo	Sensamagic
1728/1715	[6 3 -1 -3⟩	13.074	Triru-agu	Orwellisma	Orwell Comma
1029/1024	[-10 1 0 3⟩	8.433	Latrizo	Gamelisma
19683/19600	[-4 9 -2 -2⟩	7.316	Labiruru	Cataharry
5120/5103	[10 -6 1 -1⟩	5.758	Saruyo	Hemifamity
1065875/1063543	[-26 -1 1 9⟩	3.792	Latritrizo-ayo	Wadisma
420175/419904	[-6 -8 2 5⟩	1.117	Quinzo-ayoyo	Wizma
11/10	[-1 0 -1 0 1⟩	165.004	Logu	Large neutral second
99/98	[-1 2 0 -2 1⟩	17.576	Loruru	Mothwellsma
896/891	[7 -4 0 1 -1⟩	9.688	Saluzo	Pentacircle
385/384	[-7 -1 1 1 1⟩	4.503	Lozoyo	Keenanisma
441/440	[-3 2 -1 2 -1⟩	3.930	Luzozogu	Werckisma
3025/3024	[-4 -3 2 -1 2⟩	0.572	Loloruyoyo	Lehmerisma
14/13	[1 0 0 1 0 -1⟩	128.298	Thuzo
91/90	[-1 -2 -1 1 0 1⟩	19.130	Thozogu	Superleap
676/675	[2 -3 -2 0 0 2⟩	2.563	Bithogu	Parizeksma

Ear Training

5edo ear-training exercises by Alex Ness available here:

• https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web

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

Music

• Herman Miller: Daybreak on Slendro Mountain (2000)
• Aaron Krister Johnson: 5tet funk (2004)
• Andrew Heathwaite: //Pinta Penta// (2004) play (rendered in 6 alternative pentatonics as well)
• Hans Straub: Asîmchômsaia play
• Brian Wong: Slendronica#1b play
• Brian McLaren: various and sundry
• Paul Rubenstein: various, with electric guitars in 10- and 15-edo
• X.J.Scott: Sleeping Through It All (2004)
• Bill Sethares: 5-tet funk (2004), Pentacle (2004)
• "Cenobyte" Ukulele http://www.youtube.com/watch?v=UKUCRnEJKKU
• "True Island" (album) by Small Scale Revolution (2011)
• Ralph Jarzombek: Micro12
• Prelude In 5ET by Aaron Andrew Hunt
• Invention In 5ET by Aaron Andrew Hunt

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