5edo

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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	Zeta peak Zeta peak integer Zeta integral Zeta gap

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 2^1/5, 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)

Approximation to JI

Selected 7-limit intervals

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	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

Ear training

5edo ear-training exercises by Alex Ness available here:

• https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web

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:

• http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid

Music

Main article: Music in 5edo

See also: Category:5edo tracks

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