# 5edo

(Redirected from 5 EDO)
 Prime factorization 5 (prime) Step size 240.000¢ Fifth 3\5 (720¢) Major 2nd 1\5 (240¢) Semitones (A1:m2) 1:0 (240¢ : 0¢) Consistency limit 9

5 equal divisions of the octave (or 5edo) is the tuning system derived by dividing the octave into 5 equal steps of 240 cents each, or the fifth root of two. 5edo is the third 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

Approximation of prime harmonics in 5edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0 +18 +94 -9 -71 +119 -105 -58 +92 -70
relative (%) +0 +8 +39 -4 -30 +50 -44 -24 +38 -29
Steps
(reduced)
5
(0)
8
(3)
12
(2)
14
(4)
17
(2)
19
(4)
20
(0)
21
(1)
23
(3)
24
(4)
A chromatic 5edo scale on C.

If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that 16/15 is tempered out. This means in 5tet 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-odd-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. 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 — that is, when not counting the comparatively trivial 1edo.

In addition, considering 5edo as a no-5's temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next edo doing it better being 17), and is the optimal patent val for the no-5s trienstonic (or Zo) temperament.

## Intervals

Steps Cents Closest diatonic
interval name
The "neighborhood" of just intervals Audio
0 0 unison / prime 1/1
1 240 second, third +8.826¢ from septimal second 8/7
-0.030¢ from 224/195
-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
+0.030¢ from 195/112
-8.826¢ from septimal seventh 7/4
5 1200 octave 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, 5edo 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 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":
• 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 5edo 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 10edo).

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

5edo tempers out the following commas. This assumes the val 5 8 12 14 17 19].

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 256/243 [8 -5 90.225 Sawa Limma, Pythagorean diatonic semitone
5 27/25 [0 3 -2 133.238 Gugu Large limma
5 16/15 [4 -1 -1 111.731 Gubi Classic diatonic semitone
5 81/80 [-4 4 -1 21.506 Gu Syntonic comma, Didymus comma, meantone comma
5 (22 digits) [24 -21 4 4.200 Sasa-quadyo Vulture
7 36/35 [2 2 -1 -1 48.770 Rugu Septimal quarter tone
7 49/48 [-4 -1 0 2 35.697 Zozo Slendro diesis
7 64/63 [6 -2 0 -1 27.264 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 245/243 [0 -5 1 2 14.191 Zozoyo Sensamagic
7 1728/1715 [6 3 -1 -3 13.074 Triru-agu Orwellisma, Orwell comma
7 1029/1024 [-10 1 0 3 8.433 Latrizo Gamelisma
7 19683/19600 [-4 9 -2 -2 7.316 Labiruru Cataharry
7 5120/5103 [10 -6 1 -1 5.758 Saruyo Hemifamity
7 (18 digits) [-26 -1 1 9 3.792 Latritrizo-ayo Wadisma
7 (12 digits) [-6 -8 2 5 1.117 Quinzo-ayoyo Wizma
11 11/10 [-1 0 -1 0 1 165.004 Logu Large undecimal neutral 2nd
11 99/98 [-1 2 0 -2 1 17.576 Loruru Mothwellsma
11 896/891 [7 -4 0 1 -1 9.688 Saluzo Pentacircle
11 385/384 [-7 -1 1 1 1 4.503 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.930 Luzozogu Werckisma
11 3025/3024 [-4 -3 2 -1 2 0.572 Loloruyoyo Lehmerisma
13 14/13 [1 0 0 1 0 -1 128.298 Thuzo Tridecimal 2/3-tone, trienthird
13 91/90 [-1 -2 -1 1 0 1 19.130 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2 2.563 Bithogu Island comma, parizeksma
1. Ratios longer than 10 digits are presented by placeholders with informative hints

## 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-TET scale MIDI:

## Music

Daybreak Herman Miller 2000 Folk (Indonesian)
Pinta Penta in 5tET Andrew Heathwaite 2004 Electronic Alternative versions (SoundClick)
Sleeping Through It All X. J. Scott 2004 Electronic
5-tet funk Bill Sethares 2004 Funk
Pentacle Bill Sethares 2004 (?)
Asîmchômsaia Hans Straub 2005 (?) Folk (Japanese)
Slendronica#1b Brian Wong 2009 (?) Folk (Indonesian)
Random Explorations (for ukulele) Cenobyte 2011 Folk
True Island : 5 Equal Divisions Of The Octave Ukulele (album) Small Scale Revolution 2011 Folk
Micro12 Ralph Jarzombek 2010 (?) Electronic
Prelude in 5ET Aaron Andrew Hunt 2015 Neobaroque
Invention in 5ET Aaron Andrew Hunt 2015 Neobaroque
Hey, ule! (1st and 3rd sections) Dmitry Bazhenov 2020 Ambiance
"Winter Forest" (from Edolian) NullPointerException Music 2020 Classical
"Ether" (from STAFFcirc vol. 7) Vince Kaichan 2021 Electronic Album (Bandcamp)

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