6edo

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6edo

7edo →

Prime factorization	2 × 3
Step size	200 ¢
Fifth	4\6 (800 ¢) (→ 2\3)
Semitones (A1:m2)	4:-2 (800 ¢ : -400 ¢)
Dual sharp fifth	4\6 (800 ¢) (→ 2\3)
Dual flat fifth	3\6 (600 ¢) (→ 1\2)
Dual major 2nd	1\6 (200 ¢) (convergent)
Consistency limit	7
Distinct consistency limit	3
Special properties	Highly composite

English Wikipedia has an article on:

Whole tone scale

6 equal divisions of the octave (abbreviated 6edo or 6ed2), also called 6-tone equal temperament (6tet) or 6 equal temperament (6et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 6 equal parts of exactly 200 ¢ each. Each step represents a frequency ratio of 2^1/6, or the 6th root of 2. It is also known as the whole tone scale.

Theory

As a subset of 12edo, 6edo can be notated on a five-line staff with standard notation. It is the first edo that is not a zeta peak, has lower consistency than the one that precedes it, and the highest edo that has no single period mode of symmetry scales other than using the single step as a generator. This means it is relatively poor for its size at creating traditional tonal music, with 5edo and 7edo both having much better representations of the third harmonic, but has still seen more use than most edos other than 12, since it can be played on any 12-tone instrument.

While 6edo does not well approximate the 3rd harmonic, it does contain a good approximation of the 9th harmonic. Therefore, 6edo can be treated as a 2.9.5.7-subgroup temperament.

Odd harmonics

Approximation of odd harmonics in 6edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+98.0	+13.7	+31.2	-3.9	+48.7	-40.5	-88.3	+95.0	-97.5	-70.8	-28.3
	Relative (%)	+49.0	+6.8	+15.6	-2.0	+24.3	-20.3	-44.1	+47.5	-48.8	-35.4	-14.1
Steps (reduced)		10 (4)	14 (2)	17 (5)	19 (1)	21 (3)	22 (4)	23 (5)	25 (1)	25 (1)	26 (2)	27 (3)

Subsets and supersets

• Subsets: 2edo and 3edo
• Supersets: 12edo, 18edo, 24edo …

Intervals

Intervals of 6edo

Degree	Cents	Interval region	Approximated JI intervals				Audio
			3-limit	5-limit	7-limit	Other
0	0	Unison (prime)	1/1 (just)
1	200	Major second	9/8 (−3.910)	10/9 (+17.596)	28/25 (+3.802) 8/7 (−31.174)	19/17 (+7.442) 55/49 (+0.020) 64/57 (−0.532) 17/15 (−16.687)
2	400	Major third	81/64 (−7.820)	5/4 (+13.686)	63/50 (−0.108) 9/7 (−35.084)	34/27 (+0.910)
3	600	Tritone			7/5 (+17.488) 10/7 (−17.488)	24/17 (+3.000) 99/70 (−0.088) 17/12 (−3.000)
4	800	Minor sixth	128/81 (+7.820)	8/5 (−13.686)	14/9 (+35.084) 100/63 (+0.108)	27/17 (−0.910)
5	1000	Minor seventh	16/9 (+3.910)	9/5 (−17.596)	7/4 (+31.174) 25/14 (−3.802)	30/17 (+16.687) 57/32 (+0.532) 98/55 (−0.020) 34/19 (−7.442)
6	1200	Octave	2/1 (just)

Notation

Notation of 6edo

Degree	Cents	12edo subset notation
		Diatonic interval names	Note names (on D)
0	0	Perfect unison (P1)	D
1	200	Major second (M2) Diminished third (d3)	E Fb
2	400	Major third (M3) Diminished fourth (d4)	F# Gb
3	600	Augmented fourth (A4) Diminished fifth (d5)	G# Ab
4	800	Augmented fifth (A5) Minor sixth (m6)	A# Bb
5	1000	Augmented sixth (A6) Minor seventh (m7)	B# C
6	1200	Perfect octave (P8)	D

In 6edo:

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

Sagittal notation

This notation is a subset of the notations for EDOs 12, 18, 24, 36, 48, 60, 72, and 84.

Evo flavor

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

Revo flavor

Solfege

Solfege of 6edo

Degree	Cents	12edo subset standard solfege (movable do)	12edo subset uniform solfege (2–3 vowels)
0	0	Do	Da
1	200	Re	Ra
2	400	Mi	Ma (M3) Fo (d4)
3	600	Fi (A4) Se (d5)	Pa (A4) Sha (d5)
4	800	Si (A5) Le (m6)	Su (A5) Fla (m6)
5	1000	Li (A6) Te (m7)	Lu (A6) Tha (m7)
6	1200	Do	Da

Regular temperament properties

Commas

6et tempers out the following commas. This assumes val ⟨6 10 14 17 21 22].

Prime limit	Ratio	Monzo	Cents	Color name	Name(s)
3	32/27	[5 -3⟩	294.13	Wa	Pythagorean minor third
5	25/24	[-3 -1 2⟩	70.67	Yoyo	Dicot comma, classic chroma
5	128/125	[7 0 -3⟩	41.06	Trigu	Augmented comma, diesis
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Magic comma, small diesis
5	(12 digits)	[17 1 -8⟩	11.45	Saquadbigu	Würschmidt comma
5	(30 digits)	[-44 -3 21⟩	6.72	Trila-septriyo	Mutt comma
7	49/48	[-4 -1 0 2⟩	35.70	Zozo	Semaphoresma, slendro diesis
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma, tritonic diesis
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean comma
7	6144/6125	[11 1 -3 -2⟩	5.36	Sarurutrigu	Porwell comma
7	2401/2400	[-5 -1 -2 4⟩	0.72	Bizozogu	Breedsma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
13	13/12	[-2 -1 0 0 0 1⟩	138.57	tho 2nd	Tridecimal neutral second

Music

Bryan Deister

• 6edo improvisation (2024)

Chimeratio

• "Bowser breaks into Arnold Schoenberg's house and steals six of the twelve Tone Crystals (every other one), activating The 666666-Year-Curse Mechanism", from STAFFcirc vol. 7 (2021) – SoundCloud | Bandcamp

Milan Guštar

• Dvandva (1987/2007)

Aaron Andrew Hunt

• From The Equal-Tempered Keyboard (1999–2022)
  • "Prelude in 6ET" – Bandcamp | SoundCloud^{[dead link]}
  • "Invention in 6ET" – Bandcamp | SoundCloud^{[dead link]}

NullPointerException Music

• "Exiting", from Edolian (2020)

Phanomium

• Heximal (2024)

Chris Vaisvil

• The Good Boundless (2011) – blog | play