21edo

← 20edo 21edo 22edo →
Prime factorization	3 × 7
Step size	57.1429 ¢
Fifth	12\21 (685.714 ¢) (→ 4\7)
Semitones (A1:m2)	0:3 (0 ¢ : 171.4 ¢)
Consistency limit	3
Distinct consistency limit	3

21 equal divisions of the octave (abbreviated 21edo or 21ed2), also called 21-tone equal temperament (21tet) or 21 equal temperament (21et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 21 equal parts of about 57.1 ¢ each. Each step represents a frequency ratio of 2^1/21, or the 21st root of 2.

Theory

21edo contains three 7edo "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as an equalized "diatonic" scale, though non-mos options might also be preferable (such as omnidiatonic). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the chromatic semitone is equal to 0 cents (a fact characteristic of whitewood temperaments). So, another pair of accidentals (such as ups and downs) is usually used instead, though they might be "reskinned" as sharps and flats to aid melodic intuition.

21edo supports tertian harmony with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The 5/4 major third is mapped to 400 ¢, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to neogothic chords. In fact, 6/5 is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks consistency to the 5-odd-limit.

In terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (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.

Because 21edo is a Fibonacci edo, it contains an approximation to the logarithmic phi superfifth, which generates golden MOS scales 3L 2s, 5L 3s, and 8L 5s, with 21edo itself being an equalized version of 13L 8s.

Thanks to its sevenths, 21edo is an ideal tuning for its size for metallic harmony.

Odd harmonics

Approximation of odd harmonics in 21edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-16.2	+13.7	+2.6	+24.7	+20.1	+16.6	-2.6	+9.3	-11.8	-13.6	+0.3
Error	Relative (%)	-28.4	+24.0	+4.6	+43.2	+35.2	+29.1	-4.5	+16.3	-20.6	-23.9	+0.5
Steps (reduced)		33 (12)	49 (7)	59 (17)	67 (4)	73 (10)	78 (15)	82 (19)	86 (2)	89 (5)	92 (8)	95 (11)

Approximation of odd harmonics in 21edo (continued)
Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	+27.4	+8.4	-1.0	-2.2	+3.9	+16.3	-22.8	+0.4	+28.1	+2.8	-18.8
Error	Relative (%)	+47.9	+14.7	-1.8	-3.8	+6.8	+28.5	-39.9	+0.7	+49.1	+4.8	-32.9
Steps (reduced)		98 (14)	100 (16)	102 (18)	104 (20)	106 (1)	108 (3)	109 (4)	111 (6)	113 (8)	114 (9)	115 (10)

Intervals

Inconsistent intervals are in italics.

Steps	Cents	Approximate Ratios*
0	0.00	1/1
1	57.14	21/20
2	114.29	14/13, 15/14, 16/15
3	171.43	9/8, 13/12, 35/32
4	228.57	8/7, 10/9
5	285.71	6/5, 7/6
6	342.86	39/32, 128/105, 16/13
7	400.00	5/4, 9/7
8	457.14	13/10, 21/16
9	514.29	4/3
10	571.43	7/5
11	628.57	10/7
12	685.71	3/2
13	742.86	20/13, 32/21
14	800.00	8/5, 14/9
15	857.14	64/39, 105/64, 13/8
16	914.29	5/3, 12/7
17	971.43	7/4, 9/5
18	1028.57	16/9, 24/13, 64/35
19	1085.71	13/7, 28/15, 15/8
20	1142.86	40/21
21	1200.00	2/1

*As a 2.3.5.7.13-subgroup temperament

Notation

Notation systems for 21edo
Degree	Cents	Ups and downs notation			5L 3s octatonic notation	Extended-diatonic interval name
0	0.00	1	unison	C	C	Unison
1	57.14	^1 vv2	up unison, dud 2nd	^C vvD	C#	Subminor 2nd
2	114.29	^^1 v2	dup unison, down 2nd	^^C vD	Db	Minor 2nd
3	171.43	2	2nd	D	D	Submajor 2nd
4	228.57	^2 vv3	up 2nd, dud 3rd	^D vvE	D#	Supermajor 2nd
5	285.71	^^2 v3	dup 2nd, down 3rd	^^D vE	Eb	Subminor 3rd
6	342.86	3	3rd	E	E	Neutral 3rd
7	400.00	^3 vv4	up 3rd, dud 4th	^E vvF	E#/Fb	Major 3rd
8	457.14	^^3 v4	dup 3rd, down 4th	^^E vF	F	Third-fourth (naiadic)
9	514.29	4	4th	F	F#	Acute 4th
10	571.43	^4 vv5	up 4th, dud 5th	^F vvG	Gb	Narrow tritone
11	628.57	^^4 v5	dup 4th, down 5th	^^F vG	G	Wide tritone
12	685.71	5	5th	G	G#	Grave 5th
13	742.86	^5 vv6	up 5th, dud 6th	^G vvA	Hb	Fifth-sixth (cocytic)
14	800.00	^^5 v6	dup 5th, down 6th	^^G vA	H	Minor 6th
15	857.14	6	6th	A	H#/Ab	Neutral 6th
16	914.29	^6 vv7	up 6th, dud 7th	^A vvB	A	Supermajor 6th
17	971.43	^^6 v7	dup 6th, down 7th	^^A vB	A#	Subminor 7th
18	1028.57	7	7th	B	Bb	Supraminor 7th
19	1085.71	^7 vv8	up 7th, dud 8ve	^B vvC	B	Major 7th
20	1142.86	^^7 v8	dup 7th, down 8ve	^^B vC	B#/Cb	Supermajor 7th
21	1200.00	8	8ve	C	C	Octave

Sagittal notation

This notation uses the same sagittal sequence as 16-EDO, is a subset of the notation for 42b, and is a superset of the notation for 7-EDO.

Chords

Chord names

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. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

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

0-5-12 = C vE G = Cv = C down

0-7-12 = C ^E G = C^ = C up

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

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

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

0-6-12-17 = C E G vB = C,v7 = C add down-seven

0-5-12-18 = C vE G B = Cv,7 = C down add seven

0-5-12-17 = C vE G vB = Cv7 = C down-seven

For a more complete list, see Ups and downs notation#Chords and Chord Progressions.

Triadic harmony

One interesting feature of 21edo 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 inframinor, minor, neutral, major, and ultramajor 3rds respectively (or dud, down, perfect, up and dup). One can couple these with 21edo'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 harmonic series:

Steps	Cents	Ratio	Example in C	Written name	Spoken name
0-5-10	0-286-571	23:27:32	C vE vvG	Cv(vv5)	C down, dud five
0-4-11	0-229-629	7:8:10	C vvE vG	Cvv(v5)	C dud, down five
0-6-11	0-343-629	9:11:13	C E vG	C(v5)	C down-five
0-5-13	0-286-743	11:13:17	C vE ^G	Cv(^5)	C down up-five
0-8-13	0-457-743	13:17:20	C vF ^G	Cv4(^5)	C (sus) down-four up-five

Approximation to JI

While 21edo does not approximate most low-limit just intervals well, it approximates a number of harmonics quite accurately. For example, 21edo closely approximates the octave-reduced harmonics 7/4 (a subminor seventh), 15/8 (a major seventh), 23/16 (a wide tritone), 29/16 (a supraminor seventh), 31/16 (a supermajor seventh), 33/32 (a quartertone), 39/32 (a neutral third), and 43/32 (an acute fourth). The intervals 17/16, 19/16, 27/16 are approximated less accurately, but are still usable, though 19 being flat combined with 17 and 27 being sharp means that 19/17 and 27/19 are over 20 cents off. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as 10/9 being mapped wider than 9/8. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4 ¢. These approximations are also used by 63edo and 84edo, which each cover many primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which can be derived from 63edo, and is possibly a more sensible way to treat it.

JI approximation of 21edo
Steps	Cents	Approximate ratios*	Additional ratios of 17, 19, and 27**
0	0.00	1/1
1	57.14	29/28, 30/29, 31/30, 32/31, 33/32	28/27, 34/33, 39/38
2	114.29	15/14, 16/15, 31/29, 33/31, 46/43	17/16, 29/27
3	171.43	11/10, 32/29, 31/28, 43/39	10/9, 19/17, 34/31
4	228.57	8/7, 33/29	17/15, 31/27, 38/33, 43/38
5	285.71	13/11, 33/28, 46/39	19/16, 27/23, 32/27, 34/29
6	342.86	28/23, 39/32	11/9, 17/14, 23/19, 38/31
7	400.00	29/23, 39/31	19/15, 34/27, 43/34, 54/43
8	457.14	13/10, 30/23, 39/30, 43/33, 56/43	38/29
9	514.29	31/23, 39/29, 43/32, 58/43	19/14, 23/17
10	571.43	32/23, 39/28, 46/33, 43/31, 60/43	18/13, 38/27
11	628.57	23/16, 56/39, 33/23, 43/30, 62/43	13/9, 27/19
12	685.71	46/31, 58/39, 43/29, 64/43	28/19, 34/23
13	742.86	20/13, 23/15, 60/39, 43/28, 66/43	29/19
14	800.00	46/29, 62/39	30/19, 27/17, 43/27, 68/43
15	857.14	23/14, 64/39	18/11, 28/17, 38/23, 31/19
16	914.29	22/13, 56/33, 39/23	32/19, 27/16, 46/27, 29/17
17	971.43	7/4, 58/33	30/17, 54/31, 33/19, 76/43
18	1028.57	20/11, 29/16, 56/31, 78/43	9/5, 34/19, 31/17
19	1085.71	15/8, 28/15, 58/31, 62/33, 43/23	32/17, 54/29
20	1142.86	29/15, 56/29, 31/16, 60/31, 64/33	27/14, 33/17, 76/39
21	1200.00	2/1

*43-odd-limit ratios of the 2.15.7.33.39.23.29.31.43 subgroup

**Odd 27 by direct approximation

Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being expressable as 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val.

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 21edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 21edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/8, 16/15	2.554	4.5
7/4, 8/7	2.603	4.6
13/10, 20/13	2.929	5.1
13/11, 22/13	3.495	6.1
11/9, 18/11	4.551	8.0
15/14, 28/15	5.157	9.0
11/10, 20/11	6.424	11.2
13/9, 18/13	8.046	14.1
9/5, 10/9	10.975	19.2
7/5, 10/7	11.084	19.4
5/4, 8/5	13.686	24.0
13/7, 14/13	14.013	24.5
3/2, 4/3	16.241	28.4
13/8, 16/13	16.615	29.1
11/7, 14/11	17.508	30.6
7/6, 12/7	18.843	33.0
15/13, 26/15	19.170	33.5
11/8, 16/11	20.111	35.2
11/6, 12/11	20.792	36.4
9/7, 14/9	22.059	38.6
15/11, 22/15	22.665	39.7
13/12, 24/13	24.287	42.5
9/8, 16/9	24.661	43.2
5/3, 6/5	27.216	47.6

15-odd-limit intervals in 21edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
15/8, 16/15	2.554	4.5
7/4, 8/7	2.603	4.6
13/10, 20/13	2.929	5.1
13/11, 22/13	3.495	6.1
15/14, 28/15	5.157	9.0
11/10, 20/11	6.424	11.2
7/5, 10/7	11.084	19.4
5/4, 8/5	13.686	24.0
13/7, 14/13	14.013	24.5
3/2, 4/3	16.241	28.4
13/8, 16/13	16.615	29.1
11/7, 14/11	17.508	30.6
7/6, 12/7	18.843	33.0
15/13, 26/15	19.170	33.5
11/8, 16/11	20.111	35.2
15/11, 22/15	22.665	39.7
5/3, 6/5	29.927	52.4
9/8, 16/9	32.481	56.8
13/12, 24/13	32.856	57.5
9/7, 14/9	35.084	61.4
11/6, 12/11	36.351	63.6
9/5, 10/9	46.168	80.8
13/9, 18/13	49.097	85.9
11/9, 18/11	52.592	92.0

Regular temperament properties

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-, 11- and 13-limit gorgo, and 11- and 13-limit spartan. 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.

Uniform maps

13-limit uniform maps between 20.8 and 21.2
Min. size	Max. size	Wart notation	Map
20.6732	20.8381	21cdef	⟨21 33 48 58 72 77]
20.8381	20.8878	21cef	⟨21 33 48 59 72 77]
20.8878	20.9435	21ef	⟨21 33 49 59 72 77]
20.9435	20.9572	21e	⟨21 33 49 59 72 78]
20.9572	21.1361	21	⟨21 33 49 59 73 78]
21.1361	21.1943	21b	⟨21 34 49 59 73 78]
21.1943	21.2137	21bdd	⟨21 34 49 60 73 78]

Commas

21et tempers out the following commas. (Note: This assumes the val ⟨21 33 49 59 73 78].)

Prime Limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
3	2187/2048	[-11 7⟩	113.69	Lawa	Whtiewood comma, apotome, Pythagorean chroma
5	128/125	[7 0 -3⟩	41.06	Trigu	Augmented comma, diesis
5	(16 digits)	[-25 7 6⟩	31.57	Lala-tribiyo	Ampersand comma
5	(20 digits)	[32 -7 -9⟩	9.49	Sasa-tritrigu	Escapade comma
7	1029/1000	[-3 1 -3 3⟩	49.49	Trizogu	Keega
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Mint comma, septimal quartertone
7	(18 digits)	[-10 7 8 -7⟩	22.41	Lasepru-aquadbiyo	Blackjackisma
7	1029/1024	[-10 1 0 3⟩	8.43	Latrizo	Gamelisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma, septimal kleisma
7	16875/16807	[0 3 4 -5⟩	6.99	Quinru-aquadyo	Mirkwai comma
7	2401/2400	[-5 -1 -2 4⟩	0.72	Bizozogu	Breedsma
7	(30 digits)	[47 -7 -7 -7⟩	0.34	Trisa-seprugu	Akjaysma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry comma

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

Rank-2 temperaments

List of 21edo rank two temperaments by badness

Periods per 8ve	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	Hemiug
3	2\21	Augmented / august
3	3\21	Oodako
7	1\21	Whitewood

Scales

MOS scales

Since 21edo contains sub-edos of 3 and 7, it contains no heptatonic MOS scales (other than 7edo and a few very hard scales) 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-tone 3L 6s scale (related to Tcherepnin's scale in 12edo) is an excellent example.

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

21edo has the soft oneirotonic (5L 3s) MOS with generator 8\21; in addition to the naiadics that generate it, it has neutral thirds (instead of major thirds as in 13edo oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The 4-oneirosteps are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 16:23:30, 16:23:29 and their inversions.

Periods per octave	Generator	MOSes
1	2\21	1L 9s 10L 1s
1	4\21	5L 1s 5L 6s
1	5\21	4L 1s 4L 5s 4L 9s
1	8\21	3L 2s 5L 3s 8L 5s
3	2\21	3L 3s 3L 6s 9L 3s
3	3\21	3L 3s 6L 3s 6L 9s
7	1\21	7L 7s

List of useful MOS

August[6]: 5 2 5 2 5 2 (can use this like the augmented scale)
August[12]: 2 1 2 2 2 1 2 2 2 1 2 2 (can use this like the chromatic scale)
Oodako[6]: 3 4 3 4 3 4 (can use this like the whole tone scale)
Oodako[9]: 3 1 3 3 1 3 3 1 3 (optimised for no-fifths, no-fourths harmony, very xenharmonic)
Oodako[15]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
Slendric[5]: 4 4 4 4 5
Slendric[6]: 4 4 4 4 1 4
Slendric[11]: 1 3 1 3 1 3 1 3 1 3 1 (optimised for no-5s 17-limit harmony, very xenharmonic)
Whitewood[7]: 3 3 3 3 3 3 3 (identical to 7edo)
Whitewood[14]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1

Rank-3 scales

The rank-3 scale diasem (3 2 3 1 3 2 3 1 3 or 3 1 3 2 3 1 3 2 3 in 21edo) is the 21edo tempering of Zarlino diatonic with 1\21 comma steps added, resulting in two "major seconds" (171 ¢ and 228 ¢), two "minor thirds" (286 ¢ and 343 ¢) and two "fourths" (457 ¢ and 514 ¢). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).

Tetrachordal scales

While 21edo 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 21edo 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 up, up-2
4, 4, 1	(0)-229-457-(514)	C ^D ^^E F	Intense diatonic	C dup, up-2 & 6
5, 3, 1	(0)-286-457-(514)	C ^^D ^^E F	Archytas chromatic	C dup, dup-2
5, 2, 2	(0)-286-400-(514)	C ^^D ^E F	Weak chromatic	C up, dup 2 & 6
6, 2, 1	(0)-343-457-(514)	C ^3D ^^E F	Strong enharmonic	C dup, trup 2 & 6
7, 1, 1	(0)-400-457-(514)	C ^4D ^^E F	Pythagorean enharmonic	C dup, quadruple-up 2 & 6

∗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 21 EDO can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.