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21 equal divisions of the octave


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:

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


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:


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:


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

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

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

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




Books / Literature:

Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
external image ron1.jpgexternal image 21tetsm.JPG
21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)

Compositions/Listening:


Iridescent Wenge Fugue by Stephen Weigel
21-edo Trio for Organ, by Claudi Meneghin
21-penny jingle, by Claudi Meneghin
Short Clip of 21-edo Acoustic by Ron Sword
Open tuning Drone Improvisation in 21-edo by Ron Sword
Anomalous Readings play 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