21edo

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Introduction

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.

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 Approximate Ratios *2 Approximate Ratios *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.

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 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

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.

Examples:

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 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

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.

Music