6edo

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Whole tone scale

6 equal divisions of the octave (6edo) is the tuning system derived by dividing the octave into 6 equal steps of 200 cents each, or the sixth root of 2. It is also known as the whole tone scale.

Theory

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)

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 2.5.7.9 subgroup temperament.

Related edos:

• Subsets: 2edo, 3edo
• Supersets: 12edo, 18edo, 24edo …
• Neighbours: 5edo, 7edo

Intervals

Steps	Cents	Interval			Approximate JI Ratios*
0	0	unison	P1	D	1/1
1	200	major 2nd	M2	E	8/7, 9/8, 10/9
2	400	major 3rd	M3	F#	5/4, 9/7
3	600	aug 4th, dim 5th	A4, d5	G#, Ab	7/5, 10/7
4	800	minor 6th	m6	Bb	8/5, 14/9
5	1000	minor 7th	m7	C	7/4, 9/5, 16/9
6	1200	perfect 8ve	P8	D	2/1

* based on treating 6edo as a 2.5.7.9 subgroup temperament; other approaches are possible.

Commas

6edo 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	Classic chromatic semitone
5	128/125	[7 0 -3⟩	41.06	Trigu	Diesis, augmented comma
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Small diesis, magic comma
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	Slendro diesis
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Tritonic diesis, jubilisma
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean
7	6144/6125	[11 1 -3 -2⟩	5.36	Sarurutrigu	Porwell
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

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) (Bandcamp)

Milan Guštar

• Dvandva (1987/2007)

Aaron Andrew Hunt

• "Prelude in 6ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)
• "Invention in 6ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)

NullPointerException Music

• "Exiting", from Edolian (2020)

Chris Vaisvil

• The Good Boundless (2011) (details)