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==Introduction==
==Theory==


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.
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 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.


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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.
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.


== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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! | Cents
! | Cents
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]]
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]]
! | 5L3s Octotonic Notation
! | 5L3s Octotonic  
Notation
! | D.-R. Interval Types
! | D.-R. Interval Types
! | Approximate Ratios *1
! | Approximate Ratios *1
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double-down 2nd
double-down 2nd
| | C^
| | ^C


Dvv
vvD
| style="text-align:center;" | C#
| style="text-align:center;" | C#
| style="text-align:center;" | Subminor 2nd
| style="text-align:center;" | Subminor 2nd
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down 2nd
down 2nd
| | C^^
| | ^^C


Dv
vD
| style="text-align:center;" | Db
| style="text-align:center;" | Db
| style="text-align:center;" | Minor 2nd
| style="text-align:center;" | Minor 2nd
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double-down 3rd
double-down 3rd
| | D^
| | ^D


Evv
vvE
| style="text-align:center;" | D#
| style="text-align:center;" | D#
| style="text-align:center;" | Supermajor 2nd
| style="text-align:center;" | Supermajor 2nd
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down 3rd
down 3rd
| | D^^
| | ^^D


Ev
vE
| style="text-align:center;" | Eb
| style="text-align:center;" | Eb
| style="text-align:center;" | Subminor 3rd
| style="text-align:center;" | Subminor 3rd
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double-down 4th
double-down 4th
| | E^
| | ^E


Fvv
vvF
| style="text-align:center;" | E#/Fb
| style="text-align:center;" | E#/Fb
| style="text-align:center;" | Major 3rd
| style="text-align:center;" | Major 3rd
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down 4th
down 4th
| | E^^
| | ^^E


Fv
vF
| style="text-align:center;" | F
| style="text-align:center;" | F
| style="text-align:center;" | Third-Fourth
| style="text-align:center;" | Third-Fourth
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double-down 5th
double-down 5th
| | F^
| | ^F


Gvv
vvG
| style="text-align:center;" | Gb
| style="text-align:center;" | Gb
| style="text-align:center;" | Narrow Tritone
| style="text-align:center;" | Narrow Tritone
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down 5th
down 5th
| | F^^
| | ^^F


Gv
vG
| style="text-align:center;" | G
| style="text-align:center;" | G
| style="text-align:center;" | Wide Tritone
| style="text-align:center;" | Wide Tritone
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double-down 6th
double-down 6th
| | G^
| | ^G


Avv
vvA
| style="text-align:center;" | Hb
| style="text-align:center;" | Hb
| style="text-align:center;" | Fifth-Sixth
| style="text-align:center;" | Fifth-Sixth
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down 6th
down 6th
| | G^^
| | ^^G


Av
vA
| style="text-align:center;" | H
| style="text-align:center;" | H
| style="text-align:center;" | Minor 6th
| style="text-align:center;" | Minor 6th
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double-down 7th
double-down 7th
| | A^
| | ^A


Bvv
vvB
| style="text-align:center;" | A
| style="text-align:center;" | A
| style="text-align:center;" | Supermajor 6th
| style="text-align:center;" | Supermajor 6th
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down 7th
down 7th
| | A^^
| | ^^A


Bv
vB
| style="text-align:center;" | A#
| style="text-align:center;" | A#
| style="text-align:center;" | Subminor 7th
| style="text-align:center;" | Subminor 7th
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double-down 8ve
double-down 8ve
| | B^
| | ^B


Cvv
vvC
| style="text-align:center;" | B
| style="text-align:center;" | B
| style="text-align:center;" | Major 7th
| style="text-align:center;" | Major 7th
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down 8ve
down 8ve
| | B^^
| | ^^B


Cv
vC
| style="text-align:center;" | B#/Cb
| style="text-align:center;" | B#/Cb
| style="text-align:center;" | Supermajor 7th
| style="text-align:center;" | Supermajor 7th
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0-6-12 = C E G = C = C or C perfect
0-6-12 = C E G = C = C or C perfect


0-5-12 = C Ev G = C(v3) = C down-three
0-5-12 = C vE G = C(v3) = C down-three


0-7-12 = C E^ G = C(^3) = C up-three
0-7-12 = C ^E G = C(^3) = C up-three


0-6-11 = C E Gv = C(v5) = C down-five
0-6-11 = C E vG = C(v5) = C down-five


0-7-13 = C E^ G^ = C(^3,^5) = C up-three up-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
0-6-12-18 = C E G B = C7 = C seven


0-6-12-17 = C E G Bv = C(v7) = C down-seven
0-6-12-17 = C E G vB = C,v7 = C add down-seven


0-5-12-18 = C Ev G B = C7(v3) = C seven down-three
0-5-12-18 = C vE G B = C7(v3) = C seven down-three


0-5-12-17 = C Ev G Bv = C.v7 = C dot down seven
0-5-12-17 = C vE G vB = Cv7 = C down-seven


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


==Triadic Harmony==
==Triadic Harmony==
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| style="text-align:center;" | 0-286-571
| style="text-align:center;" | 0-286-571
| style="text-align:center;" | 23:27:32
| style="text-align:center;" | 23:27:32
| | C Ev Gvv
| | C vE vvG
| | C.v(vv5)
| | Cv(vv5)
| | C dot down, double-down five
| | C down, double-down five
|-
|-
| style="text-align:center;" | 0-4-11
| style="text-align:center;" | 0-4-11
| style="text-align:center;" | 0-229-629
| style="text-align:center;" | 0-229-629
| style="text-align:center;" | 7:8:10
| style="text-align:center;" | 7:8:10
| | C Evv Gv
| | C vvE vG
| | C.vv(v5)
| | Cvv(v5)
| | C dot double-down, down five
| | C double-down, down five
|-
|-
| style="text-align:center;" | 0-6-11
| style="text-align:center;" | 0-6-11
| style="text-align:center;" | 0-343-629
| style="text-align:center;" | 0-343-629
| style="text-align:center;" | 9:11:13
| style="text-align:center;" | 9:11:13
| | C E Gv
| | C E vG
| | C(v5)
| | C(v5)
| | C down-five
| | C down-five
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| style="text-align:center;" | 0-286-743
| style="text-align:center;" | 0-286-743
| style="text-align:center;" | 11:13:17
| style="text-align:center;" | 11:13:17
| | C Ev G^
| | C vE ^G
| | C.v(^5)
| | Cv(^5)
| | C dot down up-five
| | C down up-five
|-
|-
| style="text-align:center;" | 0-8-13
| style="text-align:center;" | 0-8-13
| style="text-align:center;" | 0-457-743
| style="text-align:center;" | 0-457-743
| style="text-align:center;" | 13:17:20
| style="text-align:center;" | 13:17:20
| | C Fv G^
| | C vF ^G
| | C.v4(^5)
| | Cv4(^5)
| | C (sus) down-four up-five
| | C (sus) down-four up-five
|-
|-
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| style="text-align:center;" | 0-286-857
| style="text-align:center;" | 0-286-857
| style="text-align:center;" | 11:13:18
| style="text-align:center;" | 11:13:18
| | C Ev A
| | C vE A
| | A(v5)
| | A(v5)
| | (inversion of 9:11:13)
| | (inversion of 9:11:13)
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| style="text-align:center;" | 4, 3, 2
| style="text-align:center;" | 4, 3, 2
| style="text-align:center;" | (0)-229-400-(514)
| style="text-align:center;" | (0)-229-400-(514)
| | C D^ E^ F
| | C ^D ^E F
| style="text-align:center;" | Soft diatonic
| style="text-align:center;" | Soft diatonic
| | C upperfect up-2
| | C upperfect up-2
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| style="text-align:center;" | 4, 4, 1
| style="text-align:center;" | 4, 4, 1
| style="text-align:center;" | (0)-229-457-(514)
| style="text-align:center;" | (0)-229-457-(514)
| | C D^ E^^ F
| | C ^D ^^E F
| style="text-align:center;" | Intense diatonic
| style="text-align:center;" | Intense diatonic
| | C up-2 & 6, double-up-3 & 7
| | C up-2 & 6, double-up-3 & 7
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| style="text-align:center;" | 5, 3, 1
| style="text-align:center;" | 5, 3, 1
| style="text-align:center;" | (0)-286-457-(514)
| style="text-align:center;" | (0)-286-457-(514)
| | C D^^ E^^ F
| | C ^^D ^^E F
| style="text-align:center;" | Archytas chromatic
| style="text-align:center;" | Archytas chromatic
| | C double-up-2, 3, 6 and 7
| | C double-up-2, 3, 6 and 7
Line 491: Line 492:
| style="text-align:center;" | 5, 2, 2
| style="text-align:center;" | 5, 2, 2
| style="text-align:center;" | (0)-286-400-(514)
| style="text-align:center;" | (0)-286-400-(514)
| | C D^^ E^ F
| | C ^D ^E F
| style="text-align:center;" | Weak chromatic
| style="text-align:center;" | Weak chromatic
| | C double-up 2 & 6, up-3 & 7
| | C double-up 2 & 6, up-3 & 7
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| style="text-align:center;" | 6, 2, 1
| style="text-align:center;" | 6, 2, 1
| style="text-align:center;" | (0)-343-457-(514)
| style="text-align:center;" | (0)-343-457-(514)
| | C D^<span style="font-size: 90%; vertical-align: super;">3</span> E^^ F
| | C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F
| style="text-align:center;" | Strong enharmonic
| style="text-align:center;" | Strong enharmonic
| | C triple-up 2 &amp; 6, double-up 3 &amp; 7
| | C triple-up 2 &amp; 6, double-up 3 &amp; 7
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| style="text-align:center;" | 7, 1, 1
| style="text-align:center;" | 7, 1, 1
| style="text-align:center;" | (0)-400-457-(514)
| style="text-align:center;" | (0)-400-457-(514)
| | C D^<span style="font-size: 90%; vertical-align: super;">4</span> E^^ F
| | C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F
| style="text-align:center;" | Pythagorean enharmonic
| style="text-align:center;" | Pythagorean enharmonic
| | C quadruple-up 2 &amp; 6, double-up 3 &amp; 7
| | C quadruple-up 2 &amp; 6, double-up 3 &amp; 7
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==Commas==
==Commas==
21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)
21 EDO tempers out the following 13-limit [[commas]]. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Comma
! | [[Ratio]]
! | Monzo
! | [[Monzo]]
! | Value (Cents)
! | [[Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 1
! | Name 2
! | Name 2
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| |<nowiki> | -11 7 </nowiki>&gt;
| |<nowiki> | -11 7 </nowiki>&gt;
| style="text-align:right;" | 113.69
| style="text-align:right;" | 113.69
| style="text-align:center;" |Lawa
| style="text-align:center;" | Apotome
| style="text-align:center;" | Apotome
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 7 0 -3 </nowiki>&gt;
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:right;" | 41.06
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | Augmented Comma
Line 587: Line 591:
| |<nowiki> | -25 7 6 </nowiki>&gt;
| |<nowiki> | -25 7 6 </nowiki>&gt;
| style="text-align:right;" | 31.57
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | Ampersand's Comma
| |  
| |  
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| |<nowiki> | 32 -7 -9 </nowiki>&gt;
| |<nowiki> | 32 -7 -9 </nowiki>&gt;
| style="text-align:right;" | 9.49
| style="text-align:right;" | 9.49
| style="text-align:center;" |Sasa-tritrigu
| style="text-align:center;" | Escapade Comma
| style="text-align:center;" | Escapade Comma
| |  
| |  
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| |<nowiki> | -3 1 -3 3 </nowiki>&gt;
| |<nowiki> | -3 1 -3 3 </nowiki>&gt;
| style="text-align:right;" | 49.49
| style="text-align:right;" | 49.49
| style="text-align:center;" |Trizogu
| style="text-align:center;" | Keega
| style="text-align:center;" | Keega
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| style="text-align:right;" | 48.77
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | Septimal Quarter Tone
| |  
| |  
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| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| style="text-align:right;" | 22.41
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| style="text-align:right;" | 8.43
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| style="text-align:right;" | 7.71
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | Marvel Comma
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| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:right;" | 6.99
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 635: Line 647:
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| style="text-align:right;" | 0.72
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | Breedsma
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 641: Line 654:
| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| style="text-align:right;" | 0.34
| style="text-align:right;" | 0.34
| style="text-align:center;" |Trisa-seprugu
| style="text-align:center;" | Akjaysma
| style="text-align:center;" | Akjaysma
| style="text-align:center;" | 5\7 Octave Comma
| style="text-align:center;" | 5\7 Octave Comma
Line 647: Line 661:
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| style="text-align:right;" | 17.58
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| style="text-align:right;" | 9.86
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| style="text-align:right;" | 3.03
| style="text-align:right;" | 3.03
| style="text-align:center;" |Triluyo
| style="text-align:center;" | Wizardharry
| style="text-align:center;" | Wizardharry
| style="text-align:center;" |  
| style="text-align:center;" |  
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