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{{interwiki
{{interwiki
| de =  
| de = 21-EDO
| en = 21edo
| en = 21edo
| es =  
| es =  
| ja = 21平均律
| ja = 21平均律
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 * 7
{{ED intro}}
| Step size = 57.143
 
| Fifth type = 12\21 = 685.714¢ = [[7-edo]]
== Theory ==
| Major 2nd = 3\21 = 171¢
21edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as an equalized "[[5L 2s|diatonic]]" scale, though non-mos options might also be preferable (such as [[omnidiatonic]]). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the [[chromatic semitone]] is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another pair of accidentals (such as ups and downs) is usually used instead, though they might be "reskinned" as sharps and flats to aid melodic intuition.  
| Minor 2nd = 3\21 = 171¢
 
| Augmented 1sn = 0\21 = 0¢
21edo supports {{w|tertian harmony}} with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The [[5/4]] major third is mapped to 400{{c}}, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to [[neogothic]] chords. In fact, [[6/5]] is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks [[consistency]] to the [[5-odd-limit]].
}}
 
==Theory==
In terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (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.
{| class="wikitable"
! colspan="2" |
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |Error
!absolute (¢)
|0
| -16.24
|13.7
|2.6
|20.1
|16.6
|9.3
| -11.8
|-
![[Relative error|relative]] (%)
|0
| -28
|24
|5
|35
|29
|16
| -21
|-
! colspan="2" |[[nearest edomapping]]
|21
|12
|7
|17
|10
|15
|2
|5
|}
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 968.826 cents is only off in 21-tone by 2.6 cents, 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.
Because 21edo is a {{W|Fibonacci sequence|Fibonacci}} edo, it contains an approximation to the [[logarithmic phi]] superfifth, which generates golden MOS scales [[3L 2s]], [[5L 3s]], and [[8L 5s]], with 21edo itself being an equalized version of [[13L 8s]].


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 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.
Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].


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_Family|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.
=== Odd harmonics ===
{{Harmonics in equal|21|columns=11}}
{{Harmonics in equal|21|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 21edo (continued)}}


== Intervals ==
== Intervals ==
{| class="wikitable"
Note names are given with 7edo as the naturals and arrows (written as ^ and v) for inflections up/down by one edostep. Inconsistent intervals are in ''italics''.
 
{| class="wikitable center-1 right-2 right-3"
|-
|-
! Degree
! #
! Name
! Cents
! Cents
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]]
! Approximate Ratios*
! 5L3s Octotonic <br> Notation
|-
! D.-R. Interval Types
| 0
! Approximate Ratios *1
| C
! Approximate Ratios *2
| 0.00
! Approximate Ratios *3
| [[1/1]]
|-
| 1
| ^C
| 57.14
| [[21/20]]
|-
| 2
| vD
| 114.29
| [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| D
| 171.43
| ''[[9/8]]'', ''[[13/12]]'', [[35/32]]
|-
| 4
| ^D
| 228.57
| [[8/7]], ''[[10/9]]''
|-
| 5
| vE
| 285.71
|''[[6/5]]'', [[7/6]]
|-
| 6
| E
| 342.86
| [[39/32]], [[128/105]], [[16/13]]
|-
| 7
| ^E
| 400.00
| [[5/4]], ''[[9/7]]''
|-
| 8
| vF
| 457.14
| [[13/10]], [[21/16]]
|-
| 9
| F
| 514.29
| [[4/3]]
|-
| 10
| ^F
| 571.43
| [[7/5]]
|-
| 11
| vG
| 628.57
| [[10/7]]
|-
| 12
| G
| 685.71
| [[3/2]]
|-
| 13
| ^G
| 742.86
| [[20/13]], [[32/21]]
|-
|-
| style="text-align:center;" | 0
| 14
| style="text-align:center;" | 0.00
| vA
| style="text-align:right;" | 1
| 800.00
| style="text-align:center;" | unison
| [[8/5]], ''[[14/9]]''
| | C
| style="text-align:center;" | C
| style="text-align:center;" | Unison
| style="text-align:center;" | 1/1
| style="text-align:center;" | 1/1
| style="text-align:center;" | 1/1
|-
|-
| style="text-align:center;" | 1
| 15
| style="text-align:center;" | 57.14
| A
| style="text-align:right;" | ^1
| 857.14
| [[64/39]], [[105/64]], [[13/8]]
|-
| 16
| ^A
| 914.29
|''[[5/3]]'', [[12/7]]
|-
| 17
| vB
| 971.43
| [[7/4]], ''[[9/5]]''
|-
| 18
| B
| 1028.57
| ''[[16/9]]'', ''[[24/13]]'', [[64/35]]
|-
| 19
| ^B
| 1085.71
| [[13/7]], [[28/15]], [[15/8]]
|-
| 20
| vC
| 1142.86
| [[40/21]]
|-
| 21
| C
| 1200.00
| [[2/1]]
|}
<nowiki/>*As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament


vv2
== Notation ==
| style="text-align:center;" | up unison,
The following table gives a comparison of some notation systems for 21edo.
 
{| class="wikitable center-all right-3 right-5"
double-down 2nd
|-
| | ^C
! [[Degree]]
 
! [[Cent]]s
vvD
! colspan="3" | [[Ups and downs notation]]
| style="text-align:center;" | C#
! [[5L 3s]] octatonic<br>notation
| style="text-align:center;" | Subminor 2nd
! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]
| style="text-align:center;" | 28/27, 30/29
|+
| style="text-align:center;" | 35/34, 36/35
|-
| style="text-align:center;" | 64/63
| 0
| 0.00
| 1
| unison
| C
| C
| Unison
|-
| 1
| 57.14
| ^1  vv2
| up unison, <br> dud 2nd
| ^C <br> vvD
| C#
| Subminor 2nd
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 114.29
| 114.29
| style="text-align:right;" | ^^1
| ^^1 <br> v2
 
| dup unison, <br> down 2nd
v2
| ^^C <br> vD
| style="text-align:center;" | double-up unison,
| Db
 
| Minor 2nd
down 2nd
| | ^^C
 
vD
| style="text-align:center;" | Db
| style="text-align:center;" | Minor 2nd
| style="text-align:center;" | 16/15, 15/14, 29/27
| style="text-align:center;" | 18/17
| style="text-align:center;" | 16/15, 25/24
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 171.43
| 171.43
| style="text-align:right;" | 2
| 2
| style="text-align:center;" | 2nd
| 2nd
| | D
| D
| style="text-align:center;" | D
| D
| style="text-align:center;" | Submajor 2nd
| Submajor 2nd
| style="text-align:center;" | 10/9, 32/29
| style="text-align:center;" | 10/9,11/10
| style="text-align:center;" | 9/8
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 228.57
| 228.57
| style="text-align:right;" | ^2
| ^2 <br> vv3
 
| up 2nd, <br> dud 3rd
vv3
| ^D <br> vvE
| style="text-align:center;" | up 2nd,
| D#
 
| Supermajor 2nd
double-down 3rd
| | ^D
 
vvE
| style="text-align:center;" | D#
| style="text-align:center;" | Supermajor 2nd
| style="text-align:center;" | 8/7
| style="text-align:center;" | 8/7
| style="text-align:center;" | 8/7, 10/9, 11/10
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 285.71
| 285.71
| style="text-align:right;" | ^^2
| ^^2 <br> v3
 
| dup 2nd, <br> down 3rd
v3
| ^^D <br> vE
| style="text-align:center;" | double-up 2nd,
| Eb
 
| Subminor 3rd
down 3rd
| | ^^D
 
vE
| style="text-align:center;" | Eb
| style="text-align:center;" | Subminor 3rd
| style="text-align:center;" | 27/23, 32/27
| style="text-align:center;" | 13/11, 20/17
| style="text-align:center;" | 6/5, 7/6
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 342.86
| 342.86
| style="text-align:right;" | 3
| 3
| style="text-align:center;" | 3rd
| 3rd
| | E
| E
| style="text-align:center;" | E
| E
| style="text-align:center;" | Neutral 3rd
| Neutral 3rd
| style="text-align:center;" | 28/23
| style="text-align:center;" | 11/9
| style="text-align:center;" | 16/13
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 400.00
| 400.00
| style="text-align:right;" | ^3
| ^3 <br> vv4
 
| up 3rd, <br> dud 4th
vv4
| ^E <br> vvF
| style="text-align:center;" | up 3rd,
| E#/Fb
 
| Major 3rd
double-down 4th
| | ^E
 
vvF
| style="text-align:center;" | E#/Fb
| style="text-align:center;" | Major 3rd
| style="text-align:center;" | 29/23
| style="text-align:center;" | 44/35
| style="text-align:center;" | 5/4, 9/7, 11/9, 14/11
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 457.14
| 457.14
| style="text-align:right;" | ^^3
| ^^3 <br> v4
 
| dup 3rd, <br> down 4th
v4
| ^^E <br> vF
| style="text-align:center;" | double-up 3rd,
| F
 
| Third-fourth ([[naiadic]])
down 4th
| | ^^E
 
vF
| style="text-align:center;" | F
| style="text-align:center;" | Third-Fourth
| style="text-align:center;" | 30/23
| style="text-align:center;" | 13/10, 17/13, 22/17
| style="text-align:center;" | 13/10
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 514.29
| 514.29
| style="text-align:right;" | 4
| 4
| style="text-align:center;" | 4th
| 4th
| | F
| F
| style="text-align:center;" | F#
| F#
| style="text-align:center;" | Acute 4th
| Acute 4th
| style="text-align:center;" | 161/120, 256/189
| style="text-align:center;" | 35/26
| style="text-align:center;" | 4/3, 18/13
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 571.43
| 571.43
| style="text-align:right;" | ^4
| ^4 <br> vv5
 
| up 4th, <br> dud 5th
vv5
| ^F <br> vvG
| style="text-align:center;" | up 4th,
| Gb
 
| Narrow tritone
double-down 5th
| | ^F
 
vvG
| style="text-align:center;" | Gb
| style="text-align:center;" | Narrow Tritone
| style="text-align:center;" | 32/23
| style="text-align:center;" | 18/13
| style="text-align:center;" | 7/5, 11/8
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 628.57
| 628.57
| style="text-align:right;" | ^^4
| ^^4 <br> v5
 
| dup 4th, <br> down 5th
v5
| ^^F <br> vG
| style="text-align:center;" | double-up 4th,
| G
 
| Wide tritone
down 5th
| | ^^F
 
vG
| style="text-align:center;" | G
| style="text-align:center;" | Wide Tritone
| style="text-align:center;" | 23/16
| style="text-align:center;" | 13/9
| style="text-align:center;" | 10/7, 16/11
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 685.71
| 685.71
| style="text-align:right;" | 5
| 5
| style="text-align:center;" | 5th
| 5th
| | G
| G
| style="text-align:center;" | G#
| G#
| style="text-align:center;" | Grave 5th
| Grave 5th
| style="text-align:center;" | 189/128, 240/161
| style="text-align:center;" | 52/35
| style="text-align:center;" | 3/2, 13/9
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 742.86
| 742.86
| style="text-align:right;" | ^5
| ^5 <br> vv6
 
| up 5th, <br> dud 6th
vv6
| ^G <br> vvA
| style="text-align:center;" | up 5th,
| Hb
 
| Fifth-sixth ([[cocytic]])
double-down 6th
| | ^G
 
vvA
| style="text-align:center;" | Hb
| style="text-align:center;" | Fifth-Sixth
| style="text-align:center;" | 23/15
| style="text-align:center;" | 17/11, 20/13, 26/17
| style="text-align:center;" | 20/13
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 800.00
| 800.00
| style="text-align:right;" | ^^5
| ^^5 <br> v6
 
| dup 5th, <br> down 6th
v6
| ^^G <br> vA
| style="text-align:center;" | double-up 5th,
| H
 
| Minor 6th
down 6th
| | ^^G
 
vA
| style="text-align:center;" | H
| style="text-align:center;" | Minor 6th
| style="text-align:center;" | 46/29
| style="text-align:center;" | 35/22
| style="text-align:center;" | 8/5, 11/7, 14/9, 18/11
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 857.14
| 857.14
| style="text-align:right;" | 6
| 6
| style="text-align:center;" | 6th
| 6th
| | A
| A
| style="text-align:center;" | H#/Ab
| H#/Ab
| style="text-align:center;" | Neutral 6th
| Neutral 6th
| style="text-align:center;" | 23/14
| style="text-align:center;" | 18/11
| style="text-align:center;" | 13/8
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 914.29
| 914.29
| style="text-align:right;" | ^6
| ^6 <br> vv7
 
| up 6th, <br> dud 7th
vv7
| ^A <br> vvB
| style="text-align:center;" | up 6th,
| A
 
| Supermajor 6th
double-down 7th
| | ^A
 
vvB
| style="text-align:center;" | A
| style="text-align:center;" | Supermajor 6th
| style="text-align:center;" | 27/16, 46/27
| style="text-align:center;" | 17/10, 22/13
| style="text-align:center;" | 5/3, 12/7
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 971.43
| 971.43
| style="text-align:right;" | ^^6
| ^^6 <br> v7
 
| dup 6th, <br> down 7th
v7
| ^^A <br> vB
| style="text-align:center;" | double-up 6th,
| A#
 
| Subminor 7th
down 7th
| | ^^A
 
vB
| style="text-align:center;" | A#
| style="text-align:center;" | Subminor 7th
| style="text-align:center;" | 7/4
| style="text-align:center;" | 7/4
| style="text-align:center;" | 7/4, 9/5, 20/11
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 1028.57
| 1028.57
| style="text-align:right;" | 7
| 7
| style="text-align:center;" | 7th
| 7th
| | B
| B
| style="text-align:center;" | Bb
| Bb
| style="text-align:center;" | Supraminor 7th
| Supraminor 7th
| style="text-align:center;" | 29/16, 9/5
| style="text-align:center;" | 9/5, 20/11
| style="text-align:center;" | 16/9
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 1085.71
| 1085.71
| style="text-align:right;" | ^7
| ^7 <br> vv8
 
| up 7th, <br> dud 8ve
vv8
| ^B <br> vvC
| style="text-align:center;" | up 7th,
| B
 
| Major 7th
double-down 8ve
| | ^B
 
vvC
| style="text-align:center;" | B
| style="text-align:center;" | Major 7th
| style="text-align:center;" | 15/8
| style="text-align:center;" | 17/9
| style="text-align:center;" | 15/8, 48/25
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 1142.86
| 1142.86
| style="text-align:right;" | ^^7
| ^^7 <br> v8
 
| dup 7th, <br> down 8ve
v8
| ^^B <br> vC
| style="text-align:center;" | double-up 7th,
| B#/Cb
 
| Supermajor 7th
down 8ve
| | ^^B
 
vC
| style="text-align:center;" | B#/Cb
| style="text-align:center;" | Supermajor 7th
| style="text-align:center;" | 27/14, 29/15
| style="text-align:center;" | 35/18, 68/35
| style="text-align:center;" | 63/32
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 1200.00
| 1200.00
| style="text-align:right;" | 8
| 8
| style="text-align:center;" | 8ve
| 8ve
| | C
| C
| style="text-align:center;" | C
| C
| style="text-align:center;" | Octave
| Octave
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
|}
|}


&lowast;1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
=== Sagittal notation ===
 
This notation uses the same sagittal sequence as [[16edo#Sagittal notation|16-EDO]], is a subset of the notation for [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
&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
{{Sagittal chart|}}
 
==Chord Names==


== Chords ==
=== 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. 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).  
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. 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).  


Line 431: Line 360:
0-5-12-17 = C vE G vB = Cv7 = C down-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|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].


==Triadic Harmony==
=== Triadic harmony ===
One interesting feature of 21edo 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 inframinor, minor, neutral, major, and ultramajor 3rds respectively (or dud, down, perfect, up and dup). One can couple these with 21edo'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:


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:
{| 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, 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
|}


{| class="wikitable"
== Approximation to JI ==
While 21edo does not approximate most low-limit just intervals well, it approximates a number of harmonics quite accurately. For example, 21edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable, though 19 being flat combined with 17 and 27 being sharp means that [[19/17]] and [[27/19]] are over 20 cents off. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations are also used by [[63edo]] and [[84edo]], which each cover many primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which can be derived from 63edo, and is possibly a more sensible way to treat it.
 
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | JI approximation of 21edo
|-
! Steps
! Cents
! Approximate ratios*
! Additional ratios<br>of 17, 19, and 27**
|-
| 0
| 0.00
| colspan="2" | [[1/1]]
|-
| 1
| 57.14
| [[29/28]], [[30/29]], [[31/30]], [[32/31]], [[33/32]]
| [[28/27]], [[34/33]], [[39/38]]
|-
| 2
| 114.29
| [[15/14]], [[16/15]], [[31/29]], [[33/31]], [[46/43]]
| [[17/16]], [[29/27]]
|-
| 3
| 171.43
| [[11/10]], [[32/29]], [[31/28]], [[43/39]]
| [[10/9]], [[19/17]], [[34/31]]
|-
| 4
| 228.57
| [[8/7]], [[33/29]]
| [[17/15]], [[31/27]], [[38/33]], [[43/38]]
|-
| 5
| 285.71
| [[13/11]], [[33/28]], [[46/39]]
| [[19/16]], [[27/23]], [[32/27]], [[34/29]]
|-
| 6
| 342.86
| [[28/23]], [[39/32]]
| [[11/9]], [[17/14]], [[23/19]], [[38/31]]
|-
| 7
| 400.00
| [[29/23]], [[39/31]]
| [[19/15]], [[34/27]], [[43/34]], [[54/43]]
|-
| 8
| 457.14
| [[13/10]], [[30/23]], [[39/30]], [[43/33]], [[56/43]]
| [[38/29]]
|-
| 9
| 514.29
| [[31/23]], [[39/29]], [[43/32]], [[58/43]]
| [[19/14]], [[23/17]]
|-
| 10
| 571.43
| [[32/23]], [[39/28]], [[46/33]], [[43/31]], [[60/43]]
| [[18/13]], [[38/27]]
|-
| 11
| 628.57
| [[23/16]], [[56/39]], [[33/23]], [[43/30]], [[62/43]]
| [[13/9]], [[27/19]]
|-
| 12
| 685.71
| [[46/31]], [[58/39]], [[43/29]], [[64/43]]
| [[28/19]], [[34/23]]
|-
| 13
| 742.86
| [[20/13]], [[23/15]], [[60/39]], [[43/28]], [[66/43]]
| [[29/19]]
|-
| 14
| 800.00
| [[46/29]], [[62/39]]
| [[30/19]], [[27/17]], [[43/27]], [[68/43]]
|-
| 15
| 857.14
| [[23/14]], [[64/39]]
| [[18/11]], [[28/17]], [[38/23]], [[31/19]]
|-
|-
! | Steps
| 16
! | Cents
| 914.29
! | Ratio
| [[22/13]], [[56/33]], [[39/23]]
! | Example in C
| [[32/19]], [[27/16]], [[46/27]], [[29/17]]
! | Written name
! | Spoken name
|-
|-
| style="text-align:center;" | 0-5-10
| 17
| style="text-align:center;" | 0-286-571
| 971.43
| style="text-align:center;" | 23:27:32
| [[7/4]], [[58/33]]
| | C vE vvG
| [[30/17]], [[54/31]], [[33/19]], [[76/43]]
| | Cv(vv5)
| | C down, double-down five
|-
|-
| style="text-align:center;" | 0-4-11
| 18
| style="text-align:center;" | 0-229-629
| 1028.57
| style="text-align:center;" | 7:8:10
| [[20/11]], [[29/16]], [[56/31]], [[78/43]]
| | C vvE vG
| [[9/5]], [[34/19]], [[31/17]]
| | Cvv(v5)
| | C double-down, down five
|-
|-
| style="text-align:center;" | 0-6-11
| 19
| style="text-align:center;" | 0-343-629
| 1085.71
| style="text-align:center;" | 9:11:13
| [[15/8]], [[28/15]], [[58/31]], [[62/33]], [[43/23]]
| | C E vG
| [[32/17]], [[54/29]]
| | C(v5)
| | C down-five
|-
|-
| style="text-align:center;" | 0-5-13
| 20
| style="text-align:center;" | 0-286-743
| 1142.86
| style="text-align:center;" | 11:13:17
| [[29/15]], [[56/29]], [[31/16]], [[60/31]], [[64/33]]
| | C vE ^G
| [[27/14]], [[33/17]], [[76/39]]
| | Cv(^5)
| | C down up-five
|-
|-
| style="text-align:center;" | 0-8-13
| 21
| style="text-align:center;" | 0-457-743
| 1200.00
| style="text-align:center;" | 13:17:20
| colspan="2" | [[2/1]]
| | C vF ^G
| | Cv4(^5)
| | C (sus) down-four up-five
|}
|}
<nowiki/>*43-odd-limit ratios of the 2.15.7.33.39.23.29.31.43 subgroup


==Moment-of-Symmetry Scales==
<nowiki/>**Odd 27 by direct approximation


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.
Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being expressable as 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val.


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.
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals}}


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.
== Regular temperament properties ==
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-, 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.


Examples:
=== Uniform maps ===
*[[augment6]]
{{Uniform map|edo=21}}
*[[augment9]]
*[[augment12]]


21edo also has the [[oneirotonic]] (5L 3s) MOS with generator 8\21; it supports the 13&21 temperament on the 2.5.9.11.13.17 subgroup (approximating 4:5:9:11:13:17).
=== Commas ===
21et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 21 33 49 59 73 78 }}.)


==Tetrachordal Scales==
{| class="commatable wikitable center-all left-3 right-4 left-6"
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:
|-
 
! [[Harmonic limit|Prime<br>Limit]]
{| class="wikitable"
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[2187/2048]]
| {{monzo| -11 7 }}
| 113.69
| Lawa
| Whtiewood comma, apotome, Pythagorean chroma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quartertone
|-
| 7
| <abbr title="854296875/843308032">(18 digits)</abbr>
| {{monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
|-
! | Step Pattern
| 7
! | Cents
| [[1029/1024]]
! | Example
| {{monzo| -10 1 0 3 }}
! | Name*
| 8.43
! | Ups/downs name
| Latrizo
| Gamelisma
|-
|-
| style="text-align:center;" | 3, 3, 3
| 7
| style="text-align:center;" | (0)-171-343-(514)
| [[225/224]]
| | C D E F
| {{monzo| -5 2 2 -1 }}
| style="text-align:center;" | Equable diatonic
| 7.71
| | C perfect
| Ruyoyo
| Marvel comma, septimal kleisma
|-
|-
| style="text-align:center;" | 4, 3, 2
| 7
| style="text-align:center;" | (0)-229-400-(514)
| [[16875/16807]]
| | C ^D ^E F
| {{monzo| 0 3 4 -5 }}
| style="text-align:center;" | Soft diatonic
| 6.99
| | C upperfect up-2
| Quinru-aquadyo
| Mirkwai comma
|-
|-
| style="text-align:center;" | 4, 4, 1
| 7
| style="text-align:center;" | (0)-229-457-(514)
| [[2401/2400]]
| | C ^D ^^E F
| {{monzo| -5 -1 -2 4 }}
| style="text-align:center;" | Intense diatonic
| 0.72
| | C up-2 &amp; 6, double-up-3 &amp; 7
| Bizozogu
| Breedsma
|-
|-
| style="text-align:center;" | 5, 3, 1
| 7
| style="text-align:center;" | (0)-286-457-(514)
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| | C ^^D ^^E F
| {{monzo| 47 -7 -7 -7 }}
| style="text-align:center;" | Archytas chromatic
| 0.34
| | C double-up-2, 3, 6 and 7
| Trisa-seprugu
| [[Akjaysma]]
|-
|-
| style="text-align:center;" | 5, 2, 2
| 11
| style="text-align:center;" | (0)-286-400-(514)
| [[99/98]]
| | C ^D ^E F
| {{monzo| -1 2 0 -2 1 }}
| style="text-align:center;" | Weak chromatic
| 17.58
| | C double-up 2 &amp; 6, up-3 &amp; 7
| Loruru
| Mothwellsma
|-
|-
| style="text-align:center;" | 6, 2, 1
| 11
| style="text-align:center;" | (0)-343-457-(514)
| [[176/175]]
| | C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F
| {{monzo| 4 0 -2 -1 1 }}
| style="text-align:center;" | Strong enharmonic
| 9.86
| | C triple-up 2 &amp; 6, double-up 3 &amp; 7
| Lorugugu
| Valinorsma
|-
|-
| style="text-align:center;" | 7, 1, 1
| 11
| style="text-align:center;" | (0)-400-457-(514)
| [[4000/3993]]
| | C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F
| {{monzo| 5 -1 3 0 -3 }}
| style="text-align:center;" | Pythagorean enharmonic
| 3.03
| | C quadruple-up 2 &amp; 6, double-up 3 &amp; 7
| Triluyo
| Wizardharry comma
|}
|}
&lowast;These names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
<references/>


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-2 temperaments ===
 
* [[List of 21edo rank two temperaments by badness]]
==Rank two temperaments==
[[List of 21edo rank two temperaments by badness]]


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Periods per octave
! Periods<br>per 8ve
! | Generator
! Generator
! | Temperaments
! Temperaments
|-
|-
| | 1
| 1
| | 1\21
| 1\21
| | [[Escapade_family#Escapade|Escapade]]
| [[Escapade]]
|-
|-
| | 1
| 1
| | 2\21
| 2\21
| | [[Gamelismic_clan#Miracle|Miracle]]
| [[Miracle]]
|-
|-
| | 1
| 1
| | 4\21
| 4\21
| | [[Slendric]]/[[Gamelismic_clan#Gorgo|Gorgo]]/[[Gamelismic_clan#Gidorah|Gidorah]]
| [[Slendric]] / [[gorgo]] / [[gidorah]]
|-
|-
| | 1
| 1
| | 5\21
| 5\21
| | [[Mint_temperaments#Subklei|Subklei]]
| [[Subklei]]
|-
|-
| | 1
| 1
| | 8\21
| 8\21
| | [[Chromatic_pairs#Tridec|Tridec]]
| [[Tridec]]
|-
|-
| | 1
| 1
| | 10\21
| 10\21
| | [[Marvel_temperaments#Triton|Triton]]
| [[Triton]]
|-
|-
| | 3
| 3
| | 1\21
| 1\21
| |
| [[Hemiug]]
|-
|-
| | 3
| 3
| | 2\21
| 2\21
| | [[Augmented_family|Augmented]]/[[August]]
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
|-
| | 3
| 3
| | 3\21
| 3\21
| | [[Oodako]]
| [[Oodako]]
|-
|-
| | 7
| 7
| | 1\21
| 1\21
| | [[Apotome_family|Whitewood]]
| [[Whitewood]]
|}
|}


==Commas==
== Scales ==
21 EDO tempers out the following 13-limit [[commas]]. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)
=== MOS scales ===
Since 21edo contains sub-edos of 3 and 7, it contains no heptatonic [[MOS scale]]s (other than 7edo and a few very [[Step ratio|hard]] scales) 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)|augmented]] temperament) yields the most harmonically-efficient scales. The 9-tone [[3L 6s]] scale (related to Tcherepnin's scale in [[12edo]]) is an excellent example.
 
For scales with a full-octave period, only 6 degrees of 21edo generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7edo, 3edo, or a repetition of one of the other scales.
 
21edo 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 [[13edo]] oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The 4-oneirosteps are more tritone-like than fifth-like, unlike in 13edo, 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.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | [[Ratio]]
! Periods per octave
! | [[Monzo]]
! Generator
! | [[Cents]]
! MOSes
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
|-
|-
| style="text-align:center;" | 2187/2048
| 1
| |<nowiki> | -11 7 </nowiki>&gt;
| 2\21
| style="text-align:right;" | 113.69
| [[1L 9s]] <br> [[10L 1s]]
| style="text-align:center;" |Lawa
| style="text-align:center;" | Apotome
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 128/125
| 1
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| 4\21
| style="text-align:right;" | 41.06
| [[5L 1s]]<br/>[[5L 6s]]
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
|-
|-
| style="text-align:center;" |  
| 1
| |<nowiki> | -25 7 6 </nowiki>&gt;
| 5\21
| style="text-align:right;" | 31.57
| [[4L 1s]]<br/> [[4L 5s]]<br/> [[4L 9s]]
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| |
|-
|-
| style="text-align:center;" |  
| 1
| |<nowiki> | 32 -7 -9 </nowiki>&gt;
| 8\21
| style="text-align:right;" | 9.49
| [[3L 2s]]<br/> [[5L 3s]]<br/> [[8L 5s]]
| style="text-align:center;" |Sasa-tritrigu
| style="text-align:center;" | Escapade Comma
| |
|-
|-
| style="text-align:center;" | 1029/1000
| 3
| |<nowiki> | -3 1 -3 3 </nowiki>&gt;
| 2\21
| style="text-align:right;" | 49.49
| [[3L 3s]]<br/> [[3L 6s]]<br/> [[9L 3s]]
| style="text-align:center;" |Trizogu
| style="text-align:center;" | Keega
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 36/35
| 3
| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| 3\21
| style="text-align:right;" | 48.77
| [[3L 3s]]<br/> [[6L 3s]]<br/>[[6L 9s]]
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| |
|-
|-
| style="text-align:center;" |
| 7
| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| 1\21
| style="text-align:right;" | 22.41
| [[7L 7s]]
| style="text-align:center;" |Lasepru-aquadbiyo
|}
| style="text-align:center;" | Blackjackisma
 
| style="text-align:center;" |
==== List of useful MOS ====
* [[August]][6]: 5 2 5 2 5 2 (can use this like the augmented scale)
* August[12]: 2 1 2 2 2 1 2 2 2 1 2 2 (can use this like the chromatic scale)
* [[Oodako]][6]: 3 4 3 4 3 4 (can use this like the whole tone scale)
* Oodako[9]: 3 1 3 3 1 3 3 1 3 (optimised for no-fifths, no-fourths harmony, very [[xenharmonic]])
* Oodako[15]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
* [[Slendric]][5]: 4 4 4 4 5
* Slendric[6]: 4 4 4 4 1 4
* Slendric[11]: 1 3 1 3 1 3 1 3 1 3 1 (optimised for no-5s [[17-limit]] harmony, very xenharmonic)
* [[Whitewood]][7]: 3 3 3 3 3 3 3 (identical to [[7edo]])
* Whitewood[14]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1
 
=== Rank-3 scales ===
The rank-3 scale [[diasem]] (3 2 3 1 3 2 3 1 3 or 3 1 3 2 3 1 3 2 3 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{c}} and 228{{c}}), two "minor thirds" (286{{c}} and 343{{c}}) and two "fourths" (457{{c}} and 514{{c}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
 
=== Tetrachordal scales ===
While 21edo 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 21edo fourth is 9 steps, which can be divided into three parts in the following ways:
 
{| class="wikitable center-1 center-2"
|-
|-
| style="text-align:center;" | 1029/1024
! [[Step pattern]]
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
! [[Cents]]
| style="text-align:right;" | 8.43
! Example
| style="text-align:center;" |Latrizo
! Name*
| style="text-align:center;" | Gamelisma
! Ups/downs name
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 225/224
| 3, 3, 3
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| (0)-171-343-(514)
| style="text-align:right;" | 7.71
| C D E F
| style="text-align:center;" |Ruyoyo
| Equable diatonic
| style="text-align:center;" | Septimal Kleisma
| C perfect
| style="text-align:center;" | Marvel Comma
|-
|-
| style="text-align:center;" | 16875/16807
| 4, 3, 2
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| (0)-229-400-(514)
| style="text-align:right;" | 6.99
| C ^D ^E F
| style="text-align:center;" |Quinru-aquadyo
| Soft diatonic
| style="text-align:center;" | Mirkwai
| C up, up-2
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 2401/2400
| 4, 4, 1
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| (0)-229-457-(514)
| style="text-align:right;" | 0.72
| C ^D ^^E F
| style="text-align:center;" |Bizozogu
| Intense diatonic
| style="text-align:center;" | Breedsma
| C dup, up-2 &amp; 6
| style="text-align:center;" |
|-
|-
| style="text-align:center;" |
| 5, 3, 1
| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| (0)-286-457-(514)
| style="text-align:right;" | 0.34
| C ^^D ^^E F
| style="text-align:center;" |Trisa-seprugu
| Archytas chromatic
| style="text-align:center;" | Akjaysma
| C dup, dup-2
| style="text-align:center;" | 5\7 Octave Comma
|-
|-
| style="text-align:center;" | 99/98
| 5, 2, 2
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| (0)-286-400-(514)
| style="text-align:right;" | 17.58
| C ^^D ^E F
| style="text-align:center;" |Loruru
| Weak chromatic
| style="text-align:center;" | Mothwellsma
| C up, dup 2 &amp; 6
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 176/175
| 6, 2, 1
| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| (0)-343-457-(514)
| style="text-align:right;" | 9.86
| C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F
| style="text-align:center;" |Lorugugu
| Strong enharmonic
| style="text-align:center;" | Valinorsma
| C dup, trup 2 &amp; 6
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 4000/3993
| 7, 1, 1
| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| (0)-400-457-(514)
| style="text-align:right;" | 3.03
| C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F
| style="text-align:center;" |Triluyo
| Pythagorean enharmonic
| style="text-align:center;" | Wizardharry
| C dup, quadruple-up 2 &amp; 6
| style="text-align:center;" |
|}
|}
&lowast;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.
=== Other scales ===
Some [[modmos]] of the [[miracle]] temperament are available in 21edo:
* Modmos of miracle[8]: 2 5 2 3 3 1 3 2
* Modmos of miracle[11]: 2 3 1 1 2 3 2 1 1 3 2
The subset 2 3 7 2 7 of 21edo ([[Pelog21]]) sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.
Some modified versions of that Pelog-like scale, which vaguely resemble Japanese scales, include:
* 4 1 7 2 7
* 4 1 7 3 6
They sound best with with metallic and/or percussive timbres, such as the aperiodic timbres in [[Scale Workshop]].
The subset 4 5 3 5 4 of 21edo is a kooky pseudo-[[equipentatonic]] scale.
The subset 2 5 5 6 3 of 21edo is a good tuning for the [[magnetosphere scale]]{{idio}}.
== Instruments ==
[[Lumatone mapping for 21edo|Lumatone mappings for 21edo]] are available.
== Music ==
; [[Abnormality]]
* [https://www.youtube.com/watch?v=EcHBY0S024s ''DEUS EX 5L1s''] (2025)
; [[Beheld]]
* [https://www.youtube.com/watch?v=xTcevFGxB_Q ''Huge vibe'']
* [https://www.youtube.com/watch?v=FW8AMFdkDw4 ''Hearty vibe''] (2024)
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/yLSIZxJnMh8 ''21edo waltz''] (2025)
* [https://www.youtube.com/watch?v=1S4C-m_Dcno ''21edo improv''] (2026)
* [https://www.youtube.com/shorts/9a-nJ_Ml9z8 ''21edo groove''] (2026)
; [[Fabrizio Fiale]]
* [https://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco ''L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE'']
; [[Francium]]
* "Gordon Guide" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/3O8rX7wHSYFV2AXj6yVc5u Spotify] | [https://francium223.bandcamp.com/track/gordon-guide Bandcamp] | [https://www.youtube.com/watch?v=pOJhks06uLI YouTube] – in Gorgo[11], 21edo tuning
* [https://www.youtube.com/watch?v=llr_vws6alY ''Gumballs & Party Dancing''] (2025)
; [[Frédéric Gagné]]
* [https://youtu.be/tDjLcCictVQ?t=119 ''Tostarena: Ruins (21edo cover)''], from [[XA Discord]]'s ''Xen Cover Project 2'' ([https://musescore.com/user/5995996/scores/8607089 score])
; [[Frédéric Gagné]], [[Ian Means]] and [[User:AraMax|AraMax]]
* [https://www.youtube.com/watch?v=9rTbLQ9j1sE&t=135s ''Mirage Haze''], from XA Discord's ''Deleted User EP'' ([https://musescore.com/user/5995996/scores/7852055 score])


==Books / Literature==
; [[Andrew Heathwaite]]
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
* [https://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715 ''Anomalous Readings''] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3 (MP3)]


==Music==
; [[Inthar]]
*''[https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal Iridescent Wenge Fugue]'' by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] (accepted to [https://www.seamusonline.org/ SEAMUS 2018] and [http://eabarndance.com/ Electroacoustic Barn Dance 2018])
* ''[[:File:The Angels' Library.mp3|The Angels' Library]] in the Sarnathian (23233233) mode of 21edo [[5L 3s]]'' ([[:File:The Angels' Library Score.pdf|score]])
*[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 21edo
 
*''[http://soonlabel.com/xenharmonic/archives/2494 21-edo Trio for Organ]'' by [[Claudi Meneghin]]
; [[Budjarn Lambeth]]
*''[http://soonlabel.com/xenharmonic/archives/2336 21-penny jingle]'' by Claudi Meneghin
* [https://www.youtube.com/watch?v=tIwfO1lZWAc ''The Bells In The Rain (uses a scale designed for metallic timbres)''] (2025)
*''[http://www.ronsword.com/sounds/21_improv.mp3 Short Clip of 21-edo Acoustic]'' {{dead link}} by [[Ron_Sword|Ron Sword]]
 
*''[http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3 Open tuning Drone Improvisation in 21-edo]'' {{dead link}} by Ron Sword
; [[Claudi Meneghin]]
*''[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715 Anomalous Readings]'' [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3 (MP3)] by [[Andrew Heathwaite]]
* [http://soonlabel.com/xenharmonic/archives/2336 ''21-penny jingle''] {{dead link}}
*''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3 Comets Over Flatland 15]'' by [[Randy Winchester]]
* [https://www.youtube.com/watch?v=lpcqXD8tpXc ''Trio Sonata in 21edo for Organ (The Sewing Machine)''] (2018)
*''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3 Comets Over Flatland 18]'' by Randy Winchester
* [https://www.youtube.com/watch?v=n0QA0ZQHPvk ''21edo Chacony, for two Harpsichords''] (2019)
*''[http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE]'' by [[Fabrizio_Fiale|Fabrizio Fulvio Fausto Fiale]]
* [https://www.youtube.com/watch?v=r0aKutu0gVg ''Twinkle Twinkle Little Star, with Shepard Effect''] (2023)
* [https://www.youtube.com/watch?v=XlpAbSdy_sg ''Trio Sonata for Baroque Trio in 21 EDO''] (2026)
 
; [[Nick, The NRG]]
* [https://www.youtube.com/watch?v=LHyrWBH57sw ''Moonlight Shanty'']
 
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=9RCuWizgTbg ''Edolian - Twenty-One''] (2020)
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=oGsc0TcjE2k ''3 Canon Exercises in 21EDO the 3 modes of August9 / Messiaen Mode 3''] (2024)
 
; [[Relyt R]]
* from ''Xuixo'' (2023)
** ''Silicon Burning'' [https://relytr.bandcamp.com/track/silicon-burning-21-edo Bandcamp] | [https://open.spotify.com/track/4MxWvoBpGAbBVvvD3uZhDp Spotify]
** ''10 Megakelvin'' [https://relytr.bandcamp.com/track/10-megakelvin-21-edo Bandcamp] | [https://open.spotify.com/track/7D8gGwgdctEyoAxDqfgLEI Spotify]
 
; [[Ron Sword]]
* [http://www.ronsword.com/sounds/21_improv.mp3 ''Short Clip of 21-edo Acoustic''] {{dead link}}
* [http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3 ''Open tuning Drone Improvisation in 21-edo''] {{dead link}}
 
; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal ''Iridescent Wenge Fugue''] (accepted to [https://www.seamusonline.org/ SEAMUS 2018] and [http://eabarndance.com/ Electroacoustic Barn Dance 2018])
* [https://xenharmonicgod.bandcamp.com/album/weigel-family-christmas-xenharmonic-chocolate ''WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate)'']{{dead link}}, an album of xenharmonic Christmas covers, many are in 21 EDO
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=sA5HfL3FjJU ''The Island Scene'']
 
; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3 ''Comets Over Flatland 15'']
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3 ''Comets Over Flatland 18'']
 
; [[User:Fitzgerald_Lee|Fitzgerald Lee]]
* [https://youtu.be/Nxn2FJWORIg ''Teetering Rag''] (2025)
 
== Books / literature ==
* Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.


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