5edo

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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, which equates 10/9 with 6/5, making it 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 also the optimal patent val for the no-5s trienstonic (or Zo) temperament, although this temperament is very inaccurate.

5edo is an example of an equipentatonic scale.

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 heptatonic chain-of-fifths notation, which is directly derived from the standard notation used in 12edo. The enharmonic unison is the minor 2nd, thus E and F are the same pitch.

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 (Sagittal sharp.png) and sagittal flat (Sagittal flat.png) respectively.

Sagittal notation

This notation uses the same sagittal sequence as EDOs 12, 19, and 26, and is a subset of the notations for EDOs 10, 15, 20, 25, 30, and 35b.

5-EDO Sagittal.svgSagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.

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 hexave. The circle of fifths: 1sn -- penta-4th -- penta-2nd -- penta-5th -- penta-3rd -- 1sn.

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. The circle of fifths: 1d -- 5d -- s3 -- s7 -- 4d -- 1d. When notating larger edos such as 8 or 13 this way, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid and 16/15 is a unisoid.

For note names, Kite often 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)

Approximation to JI

Selected 7-limit intervals

alt : Your browser has no SVG support.

5ed2-001.svg

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. It could be considered omniconsonant. 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 Blackwood comma, Pythagorean limma
5 27/25 [0 3 -2 133.238 Gugu Bug comma, large limma
5 16/15 [4 -1 -1 111.731 Gubi Dicot comma, classic chroma
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 comma
7 36/35 [2 2 -1 -1 48.770 Rugu Mint comma, septimal quartertone
7 49/48 [-4 -1 0 2 35.697 Zozo Semaphoresma, 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 comma
7 1728/1715 [6 3 -1 -3 13.074 Triru-agu Orwellisma
7 1029/1024 [-10 1 0 3 8.433 Latrizo Gamelisma
7 19683/19600 [-4 9 -2 -2 7.316 Labiruru Cataharry comma
7 5120/5103 [10 -6 1 -1 5.758 Saruyo Hemifamity comma
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 comma
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 comma, biome comma
13 676/675 [2 -3 -2 0 0 2 2.563 Bithogu Island comma, 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-TET scale MIDI:

Music

See also: Category:5edo tracks

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

Notes

  1. Ratios longer than 10 digits are presented by placeholders with informative hints.