21 equal divisions of the octave[edit]

Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo augmented triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO <26.

21-EDO as a temperament:[edit]

In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit laconic temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

Degree	Cents	Up/down Notation			5L3s Octotonic Notation	D.-R. Interval Types	Approximate *Ratios 1**	*ApproximateRatios 2**	*ApproximateRatios 3**
0	0	1	unison	C	C	Unison	1/1	1/1	1/1
1	57.14	^1 vv2	up unison, double-down 2nd	C^ Dvv	C#	Subminor 2nd	28/27, 30/29	35/34, 36/35	64/63
2	114.29	^^1 v2	double-up unison, down 2nd	C^^ Dv	Db	Minor 2nd	16/15, 15/14, 29/27	18/17	16/15, 25/24
3	171.43	2	2nd	D	D	Submajor 2nd	10/9, 32/29	10/9,11/10	9/8
4	228.57	^2 vv3	up 2nd, double-down 3rd	D^ Evv	D#	Supermajor 2nd	8/7	8/7	8/7, 10/9, 11/10
5	285.71	^^2 v3	double-up 2nd, down 3rd	D^^ Ev	Eb	Subminor 3rd	27/23, 32/27	13/11, 20/17	6/5, 7/6
6	342.86	3	3rd	E	E	Neutral 3rd	28/23	11/9	16/13
7	400	^3 vv4	up 3rd, double-down 4th	E^ Fvv	E#/Fb	Major 3rd	29/23	44/35	5/4, 9/7, 11/9, 14/11
8	457.14	^^3 v4	double-up 3rd, down 4th	E^^ Fv	F	Third-Fourth	30/23	13/10, 17/13, 22/17	13/10
9	514.29	4	4th	F	F#	Acute 4th	161/120, 256/189	35/26	4/3, 18/13
10	571.43	^4 vv5	up 4th, double-down 5th	F^ Gvv	Gb	Narrow Tritone	32/23	18/13	7/5, 11/8
11	628.57	^^4 v5	double-up 4th, down 5th	F^^ Gv	G	Wide Tritone	23/16	13/9	10/7, 16/11
12	685.71	5	5th	G	G#	Grave 5th	189/128, 240/161	52/35	3/2, 13/9
13	742.86	^5 vv6	up 5th, double-down 6th	G^ Avv	Hb	Fifth-Sixth	23/15	17/11, 20/13, 26/17	20/13
14	800	^^5 v6	double-up 5th, down 6th	G^^ Av	H	Minor 6th	46/29	35/22	8/5, 11/7, 14/9, 18/11
15	857.14	6	6th	A	H#/Ab	Neutral 6th	23/14	18/11	13/8
16	914.29	^6 vv7	up 6th, double-down 7th	A^ Bvv	A	Supermajor 6th	27/16, 46/27	17/10, 22/13	5/3, 12/7
17	971.43	^^6 v7	double-up 6th, down 7th	A^^ Bv	A#	Subminor 7th	7/4	7/4	7/4, 9/5, 20/11
18	1028.57	7	7th	B	Bb	Supraminor 7th	29/16, 9/5	9/5, 20/11	16/9
19	1085.71	^7 vv8	up 7th, double-down 8ve	B^ Cvv	B	Major 7th	15/8	17/9	15/8, 48/25
20	1142.86	^^7 v8	double-up 7th, down 8ve	B^^ Cv	B#/Cb	Supermajor 7th	27/14, 29/15	35/18, 68/35	63/32
21	1200	8	8ve	C	C	Octave	2/1	2/1	2/1

1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament

2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament

3: based on treating 21-EDO as 13-limit laconic temperament

Chord Names[edit]

Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.

0-6-12 = C E G = C = C or C perfect

0-5-12 = C Ev G = C(v3) = C down-three

0-7-12 = C E^ G = C(^3) = C up-three

0-6-11 = C E Gv = C(v5) = C down-five

0-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five

0-6-12-18 = C E G B = C7 = C seven

0-6-12-17 = C E G Bv = C(v7) = C down-seven

0-5-12-18 = C Ev G B = C7(v3) = C seven down-three

0-5-12-17 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

21-tone scales:

augment6

augment9

augment12

Triadic Harmony in 21-EDO:[edit]

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:

Steps	Cents	Ratio	Example in C	Written name	Spoken name
0-5-10	0-286-571	23:27:32	C Ev Gvv	C.v(vv5)	C dot down, double-down five
0-4-11	0-229-629	7:8:10	C Evv Gv	C.vv(v5)	C dot double-down, down five
0-6-11	0-343-629	9:11:13	C E Gv	C(v5)	C down-five
0-5-13	0-286-743	11:13:17	C Ev G^	C.v(^5)	C dot down up-five
0-8-13	0-457-743	13:17:20	C Fv G^	C.v4(^5)	C (sus) down-four up-five
0-5-15	0-286-857	11:13:18	C Ev A	A(v5)	(inversion of 9:11:13)

Moment-of-Symmetry Scales in 21-EDO:[edit]

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.

For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.

For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

Tetrachordal Scales in 21-EDO[edit]

While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:

Step Pattern	Cents	Example	Name*	Ups/downs name
3, 3, 3	(0)-171-343-(514)	C D E F	Equable diatonic	C perfect
4, 3, 2	(0)-229-400-(514)	C D^ E^ F	Soft diatonic	C upperfect up-2
4, 4, 1	(0)-229-457-(514)	C D^ E^^ F	Intense diatonic	C up-2 & 6, double-up-3 & 7
5, 3, 1	(0)-286-457-(514)	C D^^ E^^ F	Archytas chromatic	C double-up-2, 3, 6 and 7
5, 2, 2	(0)-286-400-(514)	C D^^ E^ F	Weak chromatic	C double-up 2 & 6, up-3 & 7
6, 2, 1	(0)-343-457-(514)	C D^3 E^^ F	Strong enharmonic	C triple-up 2 & 6, double-up 3 & 7
7, 1, 1	(0)-400-457-(514)	C D^4 E^^ F	Pythagorean enharmonic	C quadruple-up 2 & 6, double-up 3 & 7

these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.

Rank two temperaments[edit]

List of 21edo rank two temperaments by badness

Periods per octave	Generator	Temperaments
1	1\21	Escapade
1	2\21	Miracle
1	4\21	Slendric/Gorgo/Gidorah
1	5\21	Subklei
1	8\21	Tridec
1	10\21	Triton
3	1\21
3	2\21	Augmented/August
3	3\21	Oodako
7	1\21	Whitewood

13-limit Commas[edit]

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)

Comma	Monzo	Value (Cents)	Name 1	Name 2
2187/2048	\| -11 7 >	113.69	Apotome
128/125	\| 7 0 -3 >	41.06	Diesis	Augmented Comma
	\| -25 7 6 >	31.57	Ampersand's Comma
	\| 32 -7 -9 >	9.49	Escapade Comma
1029/1000	\| -3 1 -3 3 >	49.49	Keega
36/35	\| 2 2 -1 -1 >	48.77	Septimal Quarter Tone
	\| -10 7 8 -7 >	22.41	Blackjackisma
1029/1024	\| -10 1 0 3 >	8.43	Gamelisma
225/224	\| -5 2 2 -1 >	7.71	Septimal Kleisma	Marvel Comma
16875/16807	\| 0 3 4 -5 >	6.99	Mirkwai
2401/2400	\| -5 -1 -2 4 >	0.72	Breedsma
	\| 47 -7 -7 -7 >	0.34	Akjaysma	5\7 Octave Comma
99/98	\| -1 2 0 -2 1 >	17.58	Mothwellsma
176/175	\| 4 0 -2 -1 1 >	9.86	Valinorsma
4000/3993	\| 5 -1 3 0 -3 >	3.03	Wizardharry