6edo

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.

6edo 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
Error	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

Intervals of 6edo
Degree	Cents	Interval region	Approximated JI intervals* (error in ¢)				Audio
Degree	Cents	Interval region	3-limit	5-limit	7-limit	Other	Audio
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)

* based on treating 6edo as a subset of 12edo, itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible (e.g. for 6edo as a 2.5.7.9 subgroup temperament, ignore the "Other" column).

Notation

Notation of 6edo
Degree	Cents	12edo subset notation
Degree	Cents	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.

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

Uniform maps

13-limit uniform maps between 5.5 and 6.5
Min. size	Max. size	Wart notation	Map
5.5000	5.5212	6bcdddeeeffff	⟨6 9 13 15 19 20]
5.5212	5.5399	6bcdeeeffff	⟨6 9 13 16 19 20]
5.5399	5.6368	6bcdeeeff	⟨6 9 13 16 19 21]
5.6368	5.8101	6bcdeff	⟨6 9 13 16 20 21]
5.8101	5.8141	6bcde	⟨6 9 13 16 20 22]
5.8141	5.8774	6bde	⟨6 9 14 16 20 22]
5.8774	5.9258	6be	⟨6 9 14 17 20 22]
5.9258	5.9938	6b	⟨6 9 14 17 21 22]
5.9938	6.0804	6	⟨6 10 14 17 21 22]
6.0804	6.2149	6f	⟨6 10 14 17 21 23]
6.2149	6.2336	6eef	⟨6 10 14 17 22 23]
6.2336	6.2448	6ddeef	⟨6 10 14 18 22 23]
6.2448	6.3506	6ccddeef	⟨6 10 15 18 22 23]
6.3506	6.5000	6ccddeefff	⟨6 10 15 18 22 24]

Commas

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

Prime Limit	Ratio^{[note 1]}	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

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

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