# 5edo

 ← 4edo 5edo 6edo →
Prime factorization 5 (prime)
Step size 240¢
Fifth 3\5 (720¢)
(convergent)
Semitones (A1:m2) 1:0 (240¢ : 0¢)
Consistency limit 9
Distinct consistency limit 3
Special properties

5 equal divisions of the octave (abbreviated 5edo or 5ed2), also called 5-tone equal temperament (5tet) or 5 equal temperament (5et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5 equal parts of exactly 240 ¢ each. Each step represents a frequency ratio of 21/5, or the 5th root of 2.

5edo is notable for being the smallest edo containing xenharmonic intervals — 1edo, 2edo, 3edo, and 4edo are all subsets of 12edo.

## Theory

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 1, 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-odd-limit consistently, 3edo the 5-odd-limit, 4edo the 7-odd-limit and 5edo the 9-odd-limit, to represent the 11-odd-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). It is the optimal patent val for the no-5s trienstonic (or Zo) temperament, although this is a very inaccurate temperament.

### Prime harmonics

Approximation of prime harmonics in 5edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0 +18 +94 -9 -71 +119 -105 -58 +92 -70 +55
Relative (%) +0.0 +7.5 +39.0 -3.7 -29.7 +49.8 -43.7 -24.0 +38.2 -29.0 +22.9
Steps
(reduced)
5
(0)
8
(3)
12
(2)
14
(4)
17
(2)
19
(4)
20
(0)
21
(1)
23
(3)
24
(4)
25
(0)

### Subsets and supersets

5edo is the 3rd prime edo, after 2edo and 3edo and before 7edo. Multiples such as 10edo, 15edo, … up to 35edo, share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

## Intervals

Intervals of 5edo
Degree Cents Interval region Approximated JI intervals (error in ¢) Audio
3-limit 5-limit 7-limit Other
0 0 Unison (prime) 1/1 (just)
1 240 Supermajor second
Inframinor third
144/125 (-4.969)
125/108 (-13.076)
8/7 (+8.826)
7/6 (-26.871)
224/195 (-0.030)
2 480 Subfourth 4/3 (-18.045) 21/16 (+9.219) 33/25 (-0.686)
3 720 Superfifth 3/2 (+18.045) 32/21 (-9.219) 50/33 (+0.686)
4 960 Augmented sixth
Subminor seventh
216/125 (+13.076)
125/72 (+4.969)
12/7 (+26.871)
7/4 (-8.826)
195/112 (+0.030)
5 1200 Octave 2/1 (just)

## Notation

The usual notation system for 5edo is the chain-of-fifths notation, which is directly derived from the standard notation used in 12edo.

Notation of 5edo
Degree Cents Chain-of-fifths notation
Diatonic interval names Note names (on D)
0 0 Perfect unison (P1)
Minor second (m2)
Diminished third (d3)
D
Eb
Fb
1 240 Augmented unison (A1)
Major second (M2)
Minor third (m3)
Diminished fourth (d4)
D#
E
F
Gb
2 480 Augmented second (A2)
Major third (M3)
Perfect fourth (P4)
Diminished fifth (d5)
E#
F#
G
Ab
3 720 Augmented fourth (A4)
Perfect fifth (P5)
Minor sixth (m6)
Diminished seventh (d7)
G#
A
Bb
Cb
4 960 Augmented fifth (A5)
Major sixth (M6)
Minor seventh (m7)
Diminished octave (d8)
A#
B
C
Db
5 1200 Augmented sixth (A6)
Major seventh (M7)
Perfect octave (P8)
B#
C#
D

In 5edo:

• ups and downs notation is identical to circle-of-fifths notation;
• mixed sagittal notation is identical to circle-of-fifths notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp () and sagittal flat () respectively.

### Alternative notations

• via Reinhard's cents notation
• a four-line hybrid treble/bass staff.

Intervals can be named penta-2nd, penta-3rd, penta-4th, penta-5th and octave.

Kite Giedraitis has proposed pentatonic interval names that retain 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.

For note names, Kite omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode A619) or ⊧ (unicode 22A7) or 𐐆 (unicode 10406). Eef can also be used to notate 15edo.

## Solfege

Solfege of 5edo
Degree Cents Standard solfege
(movable do)
Uniform solfege
(1 vowel)
0 0 Do (P1) Da (P1)
1 240 Re (M2)
Me (m3)
Ra (M2)
Na (m3)
2 480 Mi (M3)
Fa (P4)
Ma (M3)
Fa (P4)
3 720 So (P5)
Le (m6)
Sa (P5)
Fla (m6)
4 960 La (M6)
Te (m7)
La (M6)
Tha (m7)
5 1200 Ti (M7)
Do (P8)
Da (P8)

## 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.

## Regular temperament properties

### Uniform maps

13-limit uniform maps between 4.5 and 5.5
Min. size Max. size Wart notation Map
4.5000 4.5221 5bcccddeefff 5 7 10 13 16 17]
4.5221 4.7292 5bcddeefff 5 7 11 13 16 17]
4.7292 4.7320 5bcddeef 5 7 11 13 16 18]
4.7320 4.7696 5cddeef 5 8 11 13 16 18]
4.7696 4.8088 5cddf 5 8 11 13 17 18]
4.8088 4.9528 5cf 5 8 11 14 17 18]
4.9528 4.9994 5f 5 8 12 14 17 18]
4.9994 5.0586 5 5 8 12 14 17 19]
5.0586 5.1650 5e 5 8 12 14 18 19]
5.1650 5.2696 5de 5 8 12 15 18 19]
5.2696 5.3477 5deff 5 8 12 15 18 20]
5.3477 5.3629 5deeeff 5 8 12 15 19 20]
5.3629 5.3835 5bbdeeeff 5 9 12 15 19 20]
5.3835 5.5000 5bbccdeeeff 5 9 13 15 19 20]

### Commas

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

Prime
Limit
Ratio[note 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: