21edo

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21edo

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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

Theory

Approximation of odd harmonics in 21edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29
Error	Absolute (¢)	-16.2	+13.7	+2.6	+24.7	+20.1	+16.6	-2.6	+9.3	-11.8	-13.6	+0.3	+27.4	+8.4	-1.0
	Relative (%)	-28.4	+24.0	+4.6	+43.2	+35.2	+29.1	-4.5	+16.3	-20.6	-23.9	+0.5	+47.9	+14.7	-1.8
Steps (reduced)		33 (12)	49 (7)	59 (17)	67 (4)	73 (10)	78 (15)	82 (19)	86 (2)	89 (5)	92 (8)	95 (11)	98 (14)	100 (16)	102 (18)

21 EDO provides both 7 EDO as a subset and the familiar 400-cent major third, while also giving some higher-limit JI possibilities. 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 971.43¢ is only off in 21 EDO by 2.60 cents from just (968.83¢), which is better than any other EDO <26.

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.

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 JI interpretation 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.

Intervals

Degree	Cents	Up/down notation			5L3s Octotonic Notation	D.-R. Interval Types	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
4	228.57	^2 vv3	up 2nd, dud 3rd	^D vvE	D#	Supermajor 2nd	8/7	8/7	8/7, 10/9, 11/10
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	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	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, 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, 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 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

Ups and downs can be used to name 21 EDO 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 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 ultraminor, minor, neutral, major, and ultramajor 3rds respectively (or dud, down, perfect, up and dup). 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 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

Moment-of-Symmetry Scales

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 Tcherepnin'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.

21 EDO 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 13 EDO oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The 4-oneirosteps are more tritone-like than fifth-like, unlike in 13 EDO, 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

Rank-3 scales

The rank-3 scale diasem (323132313 or 313231323 in 21edo) is the 21edo tempering of Zarlino diatonic with 1\21 comma steps added, resulting in two "major seconds" (171c and 228c), two "minor thirds" (286c and 343c) and two "fourths" (457c and 514c). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).

Tetrachordal Scales

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 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, triple-up 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.

As a regular temperament

The patent val for 21 EDO 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.

Rank two temperaments

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

Commas

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

Prime Limit	Ratio	Monzo	Cents	Color name	Name(s)
3	2187/2048	[-11 7⟩	113.69	Lawa	Apotome
5	128/125	[7 0 -3⟩	41.06	Trigu	Diesis, Augmented Comma
5	(16 digits)	[-25 7 6⟩	31.57	Lala-tribiyo	Ampersand, Ampersand's 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	Septimal Quarter Tone
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	Septimal Kleisma, Marvel Comma
7	16875/16807	[0 3 4 -5⟩	6.99	Quinru-aquadyo	Mirkwai
7	2401/2400	[-5 -1 -2 4⟩	0.72	Bizozogu	Breedsma
7	(30 digits)	[47 -7 -7 -7⟩	0.34	Trisa-seprugu	Akjaysma, 5\7 Octave Comma
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

Approaches

• 21edo/Inthar's approach

Books / Literature

Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.

Music

• Iridescent Wenge Fugue by Stephen Weigel (accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018)
• WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate), an album of xenharmonic Christmas covers played by Stephen Weigel, many are in 21 EDO
• 21-edo Trio for Organ by Claudi Meneghin
• 21-penny jingle by Claudi Meneghin
• Short Clip of 21-edo Acoustic ^{[dead link]} by Ron Sword
• Open tuning Drone Improvisation in 21-edo ^{[dead link]} by Ron Sword
• Anomalous Readings (MP3) by Andrew Heathwaite
• Comets Over Flatland 15 by Randy Winchester
• Comets Over Flatland 18 by Randy Winchester
• L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE by Fabrizio Fulvio Fausto Fiale
• The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo 5L 3s (score)

See also

• Lumatone mapping for 21edo