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5 Equal Divisions of the Octave: Theory | 5-edo in Musicmaking

## 5 Equal Divisions of the Octave: Theory

## "Equal Pentatonic"

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.)

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

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

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

## Intervals in 5-edo

in cents

interval name

-4.969¢ from diminished third 144/125

-13.076¢ from augmented second 125/108

-26.871¢ from septimal minor third 7/6

-0.686¢ from smaller fourth 33/25

-18.045¢ from just fourth 4/3

+0.686¢ from bigger fifth 50/33

-9.219¢ from wide fifth 32/21

13.076¢ from diminished seventh 216/125

4.969¢ from augmented sixth 125/72

-8.826¢ from septimal seventh 7/4

## Related scales

## 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 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.

## 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

## 5-edo in Musicmaking

Compositions, improvisationsDaybreak on Slendro Mountain(2000)5tet funk(2004)Sleeping Through It All(2004)5-tet funk(2004),Pentacle(2004)There is a lot of 5edo world music, search for "gyil" or "amadinda" or "slendro".

## Ear Training

5edo ear-training exercises by Alex Ness available here.## Notation

## 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.Important chords:

## Melody

Smallest edo which can be used for melodies in a "standard" way. Relatively large step of 240 c 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 Tempered

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