21edo: Difference between revisions

== Theory ==

 

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.

 

{| class="wikitable center-all right-3 right-5"

! colspan="3" | [[Ups and Downs Notation|Up/down notation]]

| 0

| 0.00

| 1

| unison

| C

| C

| Unison

| 1/1

| 1/1

| 1/1

| 1

| 57.14

| ^1 <br> vv2

| up unison, <br> double-down 2nd

| ^C <br> vvD

| C#

| Subminor 2nd

| 28/27, 30/29

| 35/34, 36/35

| 64/63

| 2

| 114.29

| ^^1 <br> v2

| double-up unison, <br> down 2nd

| ^^C <br> 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 <br> vv3

| up 2nd, <br> double-down 3rd

| ^D <br> vvE

| D#

| Supermajor 2nd

| 8/7

| 8/7

| 8/7, 10/9, 11/10

| 5

| 285.71

| ^^2 <br> v3

| double-up 2nd, <br> down 3rd

| ^^D <br> 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 <br> vv4

| up 3rd, <br> double-down 4th

| ^E <br> vvF

| E#/Fb

| Major 3rd

| 29/23

| 44/35

| 5/4, 9/7, 11/9, 14/11

| 8

| 457.14

| ^^3 <br> v4

| double-up 3rd, <br> down 4th

| ^^E <br> 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 <br> vv5

| up 4th, <br> double-down 5th

| ^F <br> vvG

| Gb

| Narrow Tritone

| 32/23

| 18/13

| 7/5, 11/8

| 11

| 628.57

| ^^4 <br> v5

| double-up 4th, <br> down 5th

| ^^F <br> 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 <br> vv6

| up 5th, <br> double-down 6th

| ^G <br> vvA

| Hb

| Fifth-Sixth

| 23/15

| 17/11, 20/13, 26/17

| 20/13

| 14

| 800.00

| ^^5 <br> v6

| double-up 5th, <br> down 6th

| ^^G <br> 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 <br> vv7

| up 6th, <br> double-down 7th

| ^A <br> vvB

| A

| Supermajor 6th

| 27/16, 46/27

| 17/10, 22/13

| 5/3, 12/7

| 17

| 971.43

| ^^6 <br> v7

| double-up 6th, <br> down 7th

| ^^A <br> 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 <br> vv8

| up 7th, <br> double-down 8ve

| ^B <br> vvC

| B

| Major 7th

| 15/8

| 17/9

| 15/8, 48/25

| 20

| 1142.86

| ^^7 <br> v8

| double-up 7th, <br> down 8ve

| ^^B <br> 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

&lowast;1: based on treating 21 EDO as a 2.7.15.23.27.29 subgroup temperament

&lowast;2: based on treating 21 EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament

&lowast;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).  

== 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 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:

{| class="wikitable center-1 center-2 center-3"

! 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, double-down five

| 0-4-11

| 0-229-629

| 7:8:10

| C vvE vG

| Cvv(v5)

| C double-down, 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 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 [[Step ratio|soft]] [[oneirotonic]] (5L 3s) MOS with generator 8\21; in addition to the [[naiadic]]s that generate it, it has neutral thirds (instead of major thirds as in [[13 EDO]] oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The oneirofifths (4-step intervals) 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 9:10:11:13 and 16:23:30.

! Periods per octave

! Generator

! MOSes

| 1

| 2\21

| [[1L 9s]] <br> [[10L 1s]]

| 1

| 4\21

| [[5L 1s]]<br/>[[5L 6s]]

| 1

| 5\21

| [[4L 1s]]<br/> [[4L 5s]]<br/> [[4L 9s]]

| 1

| 8\21

| [[3L 2s]]<br/> [[5L 3s]]<br/> [[8L 5s]]

| 3

| 2\21

| [[3L 3s]]<br/> [[3L 6s]]<br/> [[9L 3s]]

| 3

| 3\21

| [[3L 3s]]<br/> [[6L 3s]]<br/>[[6L 9s]]

| 7

| 1\21

| [[7L 7s]]

== Tetrachordal Scales ==

 

{| class="wikitable center-1 center-2"

! 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 &amp; 6, double-up-3 &amp; 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 &amp; 6, up-3 &amp; 7

| 6, 2, 1

| (0)-343-457-(514)

| C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F

| Strong enharmonic

| C triple-up 2 &amp; 6, double-up 3 &amp; 7

| 7, 1, 1

| (0)-400-457-(514)

| C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F

| Pythagorean enharmonic

| C quadruple-up 2 &amp; 6, double-up 3 &amp; 7

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.

 

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.

! Periods per octave

! Generator

! Temperaments

| 1

| 1\21

| [[Escapade_family#Escapade|Escapade]]

| 1

| 2\21

| [[Gamelismic_clan#Miracle|Miracle]]

| 1

| 4\21

| [[Slendric]]/[[Gamelismic_clan#Gorgo|Gorgo]]/[[Gamelismic_clan#Gidorah|Gidorah]]

| 1

| 5\21

| [[Mint_temperaments#Subklei|Subklei]]

| 1

| 8\21

| [[Chromatic_pairs#Tridec|Tridec]]

| 1

| 10\21

| [[Marvel_temperaments#Triton|Triton]]

| 3

| 1\21

|  

| 3

| 2\21

| [[Augmented_family|Augmented]]/[[August]]

| 3

| 3\21

| [[Oodako]]

| 7

| 1\21

| [[Apotome_family|Whitewood]]

== Books / Literature ==

 

== Music ==

 

*[https://xenharmonicgod.bandcamp.com/album/weigel-family-christmas-xenharmonic-chocolate WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate)], an album of xenharmonic Christmas covers played by Stephen Weigel, many are in 21 EDO