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

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). 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)

JI approximation

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. 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

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

See also: Category:5edo tracks

Dmitry Bazhenov

• Hey, ule! (1st and 3rd sections) (Youtube) (2020)
• River flowing (Boris Tchaikovsky 5edo transcription) (2023, original piece from cycle Pentatonic composed in 1993)

Cenobyte

• 5 EDO Ukulele (2011)

Bryan Deister

• microtonal improvisation in 5edo (2023)

Francium

• something minimalistic idk (2022)
• Flight of the Bumblebee in 5edo (2023)

Andrew Heathwaite

• Pinta Penta in 5tET (2004) (alternative versions on SoundClick)

Hideya

• Like japanese curry (2020)
• Like a calm river flow (2022)

Aaron Andrew Hunt

• "Prelude in 5ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)
• "Invention in 5ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)

Aaron Krister Johnson

• 5tet funk^{[dead link]} (2004)

Vince Kaichan

• "Ether", from STAFFcirc vol. 7 (2021) (Bandcamp)

Brian McLaren

Various and sundry

Herman Miller

• Daybreak (2000)

NullPointerException Music

• Frontier (2020)
• "Winter Forest", from Edolian (2020)

Paul Rubenstein

Various, with electric guitars in 10edo and 15edo

X. J. Scott

• Sleeping Through It All (2004)

Bill Sethares

• 5-tet funk (2004)
• Pentacle (2004)

Sevish

• "Garden of Light", from Formless Shadows (2021)

Small Scale Revolution

• True Island : 5 Equal Divisions Of The Octave Ukulele (album) (2011)

Hans Straub

• Asîmchômsaia (2007)

Brian Wong

• Slendronica#1b (2009)

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