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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

degrees
size
in 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 / eighth
exactly 2/1

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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.

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, improvisations


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

    • 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.

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:
  • 0+1+3
  • 0+2+3
  • 0+1+3+4
  • 0+2+3+4

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 |.)

Comma
Value (cents)
Name
Second Name
Third Name
Monzo
256/243
90.225
Limma
Pythagorean Minor 2nd

| 8 -5 >
81/80
21.506
Syntonic Comma
Didymos Comma
Meantone Comma
| -4 4 -1 >
2889416/2882415
4.200
Vulture


| 24 -21 4 >
36/35
48.770
Septimal Quarter Tone


| 2 2 -1 -1 >
49/48
35.697
Slendro Diesis


| -4 -1 0 2 >
64/63
27.264
Septimal Comma
Archytas' Comma
Leipziger Komma
| 6 -2 0 -1 >
245/243
14.191
Sensamagic


| 0 -5 1 2 >
1728/1715
13.074
Orwellisma
Orwell Comma

| 6 3 -1 -3 >
1029/1024
8.433
Gamelisma


| -10 1 0 3 >
19683/19600
7.316
Cataharry


| -4 9 -2 -2 >
5120/5103
5.758
Hemifamity


| 10 -6 1 -1 >
1065875/1063543
3.792
Wadisma


| -26 -1 1 9 >
420175/419904
1.117
Wizma


| -6 -8 2 5 >
99/98
17.576
Mothwellsma


| -1 2 0 -2 1 >
896/891
9.688
Pentacircle


| 7 -4 0 1 -1 >
385/384
4.503
Keenanisma


| -7 -1 1 1 1 >
441/440
3.930
Werckisma


| -3 2 -1 2 -1 >
3025/3024
0.572
Lehmerisma


| -4 -3 2 -1 2 >
91/90
19.130
Superleap


| -1 -2 -1 1 0 1 >
676/675
2.563
Parizeksma


| 2 -3 -2 0 0 2 >
16/15
111.731
Diatonic semitone


| 4 -1 -1 >
14/13
128.298



| 1 0 0 1 0 -1 >
27/25
133.238
Large diatonic semit.


| 0 3 -2 >
11/10
165.004
Large neutral second


| -1 0 -1 0 1 >