# 5edo

(Redirected from 5-edo)

## 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 exactly 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 exactly 2/1

## 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.

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:

## Music

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