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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.

DegreeCentsNotation

5L3sOctotonicNotationD.-R. IntervalTypesApproximateRatios *1ApproximateRatios *2ApproximateRatios *3vv2

double-down 2nd

Dvv

v2

down 2nd

Dv

vv3

double-down 3rd

Evv

v3

down 3rd

Ev

vv4

double-down 4th

Fvv

v4

down 4th

Fv

vv5

double-down 5th

Gvv

v5

down 5th

Gv

vv6

double-down 6th

Avv

v6

down 6th

Av

vv7

double-down 7th

Bvv

v7

down 7th

Bv

vv8

double-down 8ve

Cvv

v8

down 8ve

Cv

*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:

StepsCentsRatioExample in CWritten nameSpoken name## 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: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 badnessper octave

## 13-limit Commas

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)

Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.Books / Literature:21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)Compositions/Listening:Iridescent Wenge Fugue by Stephen Weigel, accepted to SEAMUS 2018 and Electroacoustic Barn Dance 201821-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