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21edo

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 a basic "diatonic" scale, though maximum-variety-3 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 kind of accidental (such as ups and downs) is usually used instead, though it might be "reskinned" as sharps and flats to aid melodic intuition.

21edo supports tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.

In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics 7/4 (a subminor seventh), 17/16 (a semitone), 19/16 (a minor third), 23/16 (a tritone), and 29/16 (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals 16/15 and 27/16, if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.15.7.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.7.11/5.13/5.17/5 subgroup, which is possibly a more sensible way to treat it.

In terms of interval regions, 21edo 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.

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

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
	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

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
	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

Degree	Cents	Ups and downs notation			5L 3s octotonic notation	Extended-diatonic interval name	Approximate Ratios *1	Approximate Ratios *2	Approximate Ratios *3
0	0.00	1	unison	C	C	Unison	1/1	1/1	1/1
1	57.14	^1 vv2	up unison, dud 2nd	^C vvD	C#	Subminor 2nd	28/27, 30/29	35/34, 36/35	64/63
2	114.29	^^1 v2	dup unison, down 2nd	^^C vD	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, 11/10
4	228.57	^2 vv3	up 2nd, dud 3rd	^D vvE	D#	Supermajor 2nd	8/7	8/7	8/7, 10/9
5	285.71	^^2 v3	dup 2nd, down 3rd	^^D vE	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	400.00	^3 vv4	up 3rd, dud 4th	^E vvF	E#/Fb	Major 3rd	29/23	44/35	5/4, 9/7, 11/9, 14/11
8	457.14	^^3 v4	dup 3rd, down 4th	^^E vF	F	Third-fourth (naiadic)	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, dud 5th	^F vvG	Gb	Narrow tritone	32/23	18/13	7/5, 11/8
11	628.57	^^4 v5	dup 4th, down 5th	^^F vG	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, dud 6th	^G vvA	Hb	Fifth-sixth (cocytic)	23/15	17/11, 20/13, 26/17	20/13
14	800.00	^^5 v6	dup 5th, down 6th	^^G vA	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, dud 7th	^A vvB	A	Supermajor 6th	27/16, 46/27	17/10, 22/13	5/3, 12/7
17	971.43	^^6 v7	dup 6th, down 7th	^^A vB	A#	Subminor 7th	7/4	7/4	7/4, 9/5
18	1028.57	7	7th	B	Bb	Supraminor 7th	29/16, 9/5	9/5, 20/11	16/9, 20/11
19	1085.71	^7 vv8	up 7th, dud 8ve	^B vvC	B	Major 7th	15/8	17/9	15/8, 48/25
20	1142.86	^^7 v8	dup 7th, down 8ve	^^B vC	B#/Cb	Supermajor 7th	27/14, 29/15	35/18, 68/35	63/32
21	1200.00	8	8ve	C	C	Octave	2/1	2/1	2/1

∗1: based on treating 21edo as a 2.7.15.23.27.29 subgroup temperament

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

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

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Popular topics

• Just intonation – Tuning based on intervals with rational frequency ratios
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• Regular temperaments – (placeholder)
• Historical temperaments, such as Pythagorean tuning, Meantone temperaments, and Well temperaments