5edo

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5edo
Prime factorization 5 (prime)
Step size 240¢
Fifth 3\5 = 720¢
Major 2nd 1\5 = 240¢
Minor 2nd 0\5 = 0¢
Augmented 1sn 1\5 = 240¢

5-edo divides the 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.)

Theory

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19
error absolute (¢) 0.0 +18.0 +93.7 -8.8 -71.3 +119.5 -105.0 -57.5
relative (%) 0 +8 +39 -4 -30 +50 -44 -24
nearest edomapping 5 3 2 4 2 4 0 1
fifthspan 0 +1 -1 -2 -1 -2 0 +2

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. 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.

Intervals

Steps 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

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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 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

5-EDO 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-edo scale MIDI:

Music

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

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