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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.


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 2.9/5.11/5.13/5.17/5.7 subgroup, which is possibly a more sensible way to treat it.
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.


== Intervals ==
== Intervals ==
Line 30: Line 30:
! Cents
! Cents
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]]
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]]
! 5L3s Octatonic <br> Notation
! 5L3s Octotonic <br> Notation
! D.-R. Interval Types
! D.-R. Interval Types
! Approximate Ratios *1 *2 *3
! Approximate Ratios *1
! Approximate Ratios *2
! Approximate Ratios *3
|-
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | 0.00
| style="text-align:center;" | 0.00
| style="text-align:right;" | 1
| style="text-align:right;" | 1
0
| style="text-align:center;" | unison
| style="text-align:center;" | 1sn
| | C
| | C
| style="text-align:center;" | C
| style="text-align:center;" | C
| style="text-align:center;" | Unison
| 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;" | 1/1
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:center;" | 57.14
| style="text-align:center;" | 57.14
| style="text-align:right;" | ^1,
| style="text-align:right;" | ^1


vv2
vv2
 
| style="text-align:center;" | up unison,
^0,
 
vv1
| style="text-align:center;" | up 1sn,


double-down 2nd
double-down 2nd
 
| | ^C
up 1sn,
 
double-down 1st
| | ^C,


vvD
vvD
| style="text-align:center;" | C#
| style="text-align:center;" | C#
| style="text-align:center;" | Subminor 2nd
| style="text-align:center;" | Subminor 2nd
Subminor 1st
| style="text-align:center;" | 28/27, 30/29
| style="text-align:center;" | 28/27, 30/29
35/34, 36/35
| style="text-align:center;" | 35/34, 36/35
 
| style="text-align:center;" | 64/63
64/63
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:center;" | 114.29
| style="text-align:center;" | 114.29
| style="text-align:right;" | ^^1,
| style="text-align:right;" | ^^1


v2
v2
 
| style="text-align:center;" | double-up unison,
^^0,
 
v1
| style="text-align:center;" | double-up 1sn,


down 2nd
down 2nd
 
| | ^^C
double-up 1sn,
 
down 1st
| | ^^C,


vD
vD
| style="text-align:center;" | Db
| style="text-align:center;" | Db
| style="text-align:center;" | Minor 2nd
| style="text-align:center;" | Minor 2nd
Minor 1st
| style="text-align:center;" | 16/15, 15/14, 29/27
| style="text-align:center;" | 16/15, 15/14, 29/27
18/17
| style="text-align:center;" | 18/17
 
| style="text-align:center;" | 16/15, 25/24
16/15, 25/24
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:center;" | 171.43
| style="text-align:center;" | 171.43
| style="text-align:right;" | 2
| style="text-align:right;" | 2
1
| style="text-align:center;" | 2nd
| style="text-align:center;" | 2nd
1st
| | D
| | D
| style="text-align:center;" | D
| style="text-align:center;" | D
| style="text-align:center;" | Submajor 2nd
| style="text-align:center;" | Submajor 2nd
Submajor 1st
| style="text-align:center;" | 10/9, 32/29
| style="text-align:center;" | 10/9, 32/29
10/9, 11/10
| style="text-align:center;" | 10/9,11/10
 
| style="text-align:center;" | 9/8
9/8
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:center;" | 228.57
| style="text-align:center;" | 228.57
| style="text-align:right;" | ^2,
| style="text-align:right;" | ^2


vv3
vv3
^1,
vv2
| style="text-align:center;" | up 2nd,
| style="text-align:center;" | up 2nd,


double-down 3rd
double-down 3rd
 
| | ^D
up 1st,
 
double-down 2nd
| | ^D,


vvE
vvE
| style="text-align:center;" | D#
| style="text-align:center;" | D#
| style="text-align:center;" | Supermajor 2nd
| style="text-align:center;" | Supermajor 2nd
Supermajor 1st
| style="text-align:center;" | 8/7
| style="text-align:center;" | 8/7
8/7, 10/9, 11/10
| style="text-align:center;" | 8/7
| style="text-align:center;" | 8/7, 10/9, 11/10
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:center;" | 285.71
| style="text-align:center;" | 285.71
| style="text-align:right;" | ^^2,
| style="text-align:right;" | ^^2


v3
v3
^^1,
v2
| style="text-align:center;" | double-up 2nd,
| style="text-align:center;" | double-up 2nd,


down 3rd
down 3rd
 
| | ^^D
double-up 1st,
 
down 2nd
| | ^^D,


vE
vE
| style="text-align:center;" | Eb
| style="text-align:center;" | Eb
| style="text-align:center;" | Subminor 3rd
| style="text-align:center;" | Subminor 3rd
Subminor 2nd
| style="text-align:center;" | 27/23, 32/27
| style="text-align:center;" | 27/23, 32/27
13/11, 20/17
| style="text-align:center;" | 13/11, 20/17
 
| style="text-align:center;" | 6/5, 7/6
6/5, 7/6
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:center;" | 342.86
| style="text-align:center;" | 342.86
| style="text-align:right;" | 3
| style="text-align:right;" | 3
2
| style="text-align:center;" | 3rd
| style="text-align:center;" | 3rd
2nd
| | E
| | E
| style="text-align:center;" | E
| style="text-align:center;" | E
| style="text-align:center;" | Neutral 3rd
| style="text-align:center;" | Neutral 3rd
Neutral 2nd
| style="text-align:center;" | 28/23
| style="text-align:center;" | 28/23
11/9
| style="text-align:center;" | 11/9
 
| style="text-align:center;" | 16/13
16/13
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:center;" | 400.00
| style="text-align:center;" | 400.00
| style="text-align:right;" | ^3,
| style="text-align:right;" | ^3


vv4
vv4
^2,
vv3
| style="text-align:center;" | up 3rd,
| style="text-align:center;" | up 3rd,


double-down 4th
double-down 4th
 
| | ^E
up 2nd,
 
double-down 3rd
| | ^E,


vvF
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
Major 2nd
| style="text-align:center;" | 29/23
| style="text-align:center;" | 29/23
44/35
| style="text-align:center;" | 44/35
 
| style="text-align:center;" | 5/4, 9/7, 11/9, 14/11
5/4, 9/7, 11/9, 14/11
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:center;" | 457.14
| style="text-align:center;" | 457.14
| style="text-align:right;" | ^^3,
| style="text-align:right;" | ^^3


v4
v4
^^2,
v3
| style="text-align:center;" | double-up 3rd,
| style="text-align:center;" | double-up 3rd,


down 4th
down 4th
double-up 3rd,
down 3rd
| | ^^E
| | ^^E


Line 228: Line 167:
| style="text-align:center;" | F
| style="text-align:center;" | F
| style="text-align:center;" | Third-Fourth
| style="text-align:center;" | Third-Fourth
Second-Third
| style="text-align:center;" | 30/23
| style="text-align:center;" | 30/23
13/10, 17/13, 22/17
| style="text-align:center;" | 13/10, 17/13, 22/17
 
| style="text-align:center;" | 13/10
13/10
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:center;" | 514.29
| style="text-align:center;" | 514.29
| style="text-align:right;" | 4
| style="text-align:right;" | 4
3
| style="text-align:center;" | 4th
| style="text-align:center;" | 4th
3rd
| | F
| | F
| style="text-align:center;" | F#
| style="text-align:center;" | F#
| style="text-align:center;" | Acute 4th
| style="text-align:center;" | Acute 4th
Acute 3rd
| style="text-align:center;" | 161/120, 256/189
| style="text-align:center;" | 161/120, 256/189
35/26
| style="text-align:center;" | 35/26
 
| style="text-align:center;" | 4/3, 18/13
4/3, 18/13
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:center;" | 571.43
| style="text-align:center;" | 571.43
| style="text-align:right;" | ^4,
| style="text-align:right;" | ^4


vv5
vv5
^3,
vv4
| style="text-align:center;" | up 4th,
| style="text-align:center;" | up 4th,


double-down 5th
double-down 5th
 
| | ^F
up 3rd,
 
double-down 4th
| | ^F,


vvG
vvG
Line 271: Line 196:
| style="text-align:center;" | Narrow Tritone
| style="text-align:center;" | Narrow Tritone
| style="text-align:center;" | 32/23
| style="text-align:center;" | 32/23
18/13
| style="text-align:center;" | 18/13
 
| style="text-align:center;" | 7/5, 11/8
7/5, 11/8
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:center;" | 628.57
| style="text-align:center;" | 628.57
| style="text-align:right;" | ^^4,
| style="text-align:right;" | ^^4


v5
v5
^^3,
v4
| style="text-align:center;" | double-up 4th,
| style="text-align:center;" | double-up 4th,


down 5th
down 5th
 
| | ^^F
double-up 3rd,
 
down 4th
| | ^^F,


vG
vG
Line 297: Line 213:
| style="text-align:center;" | Wide Tritone
| style="text-align:center;" | Wide Tritone
| style="text-align:center;" | 23/16
| style="text-align:center;" | 23/16
13/9
| style="text-align:center;" | 13/9
 
| style="text-align:center;" | 10/7, 16/11
10/7, 16/11
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:center;" | 685.71
| style="text-align:center;" | 685.71
| style="text-align:right;" | 5
| style="text-align:right;" | 5
4
| style="text-align:center;" | 5th
| style="text-align:center;" | 5th
4th
| | G
| | G
| style="text-align:center;" | G#
| style="text-align:center;" | G#
| style="text-align:center;" | Grave 5th
| style="text-align:center;" | Grave 5th
Grave 4th
| style="text-align:center;" | 189/128, 240/161
| style="text-align:center;" | 189/128, 240/161
52/35
| style="text-align:center;" | 52/35
 
| style="text-align:center;" | 3/2, 13/9
3/2, 13/9
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:center;" | 742.86
| style="text-align:center;" | 742.86
| style="text-align:right;" | ^5,
| style="text-align:right;" | ^5


vv6
vv6
^4,
vv5
| style="text-align:center;" | up 5th,
| style="text-align:center;" | up 5th,


double-down 6th
double-down 6th
 
| | ^G
up 4th,
 
double-down 5th
| | ^G,


vvA
vvA
| style="text-align:center;" | Hb
| style="text-align:center;" | Hb
| style="text-align:center;" | Fifth-Sixth
| style="text-align:center;" | Fifth-Sixth
Fourth-Fifth
| style="text-align:center;" | 23/15
| style="text-align:center;" | 23/15
17/11, 20/13, 26/17
| style="text-align:center;" | 17/11, 20/13, 26/17
 
| style="text-align:center;" | 20/13
20/13
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:center;" | 800.00
| style="text-align:center;" | 800.00
| style="text-align:right;" | ^^5,
| style="text-align:right;" | ^^5


v6
v6
^^4,
v5
| style="text-align:center;" | double-up 5th,
| style="text-align:center;" | double-up 5th,


down 6th
down 6th
 
| | ^^G
double-up 4th,
 
down 5th
| | ^^G,


vA
vA
| style="text-align:center;" | H
| style="text-align:center;" | H
| style="text-align:center;" | Minor 6th
| style="text-align:center;" | Minor 6th
Minor 5th
| style="text-align:center;" | 46/29
| style="text-align:center;" | 46/29
35/22
| style="text-align:center;" | 35/22
 
| style="text-align:center;" | 8/5, 11/7, 14/9, 18/11
8/5, 11/7, 14/9, 18/11
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:center;" | 857.14
| style="text-align:center;" | 857.14
| style="text-align:right;" | 6
| style="text-align:right;" | 6
5
| style="text-align:center;" | 6th
| style="text-align:center;" | 6th
5th
| | A
| | A
| style="text-align:center;" | H#/Ab
| style="text-align:center;" | H#/Ab
| style="text-align:center;" | Neutral 6th
| style="text-align:center;" | Neutral 6th
Neutral 5th
| style="text-align:center;" | 23/14
| style="text-align:center;" | 23/14
18/11
| style="text-align:center;" | 18/11
 
| style="text-align:center;" | 13/8
13/8
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:center;" | 914.29
| style="text-align:center;" | 914.29
| style="text-align:right;" | ^6,
| style="text-align:right;" | ^6


vv7
vv7
^5,
vv6
| style="text-align:center;" | up 6th,
| style="text-align:center;" | up 6th,


double-down 7th
double-down 7th
 
| | ^A
up 5th,
 
double-down 6th
| | ^A,


vvB
vvB
| style="text-align:center;" | A
| style="text-align:center;" | A
| style="text-align:center;" | Supermajor 6th
| style="text-align:center;" | Supermajor 6th
Supermajor 5th
| style="text-align:center;" | 27/16, 46/27
| style="text-align:center;" | 27/16, 46/27
17/10, 22/13
| style="text-align:center;" | 17/10, 22/13
 
| style="text-align:center;" | 5/3, 12/7
5/3, 12/7
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
| style="text-align:center;" | 971.43
| style="text-align:center;" | 971.43
| style="text-align:right;" | ^^6,
| style="text-align:right;" | ^^6


v7
v7
^^5,
v6
| style="text-align:center;" | double-up 6th,
| style="text-align:center;" | double-up 6th,


down 7th
down 7th
 
| | ^^A
double-up 7th,
 
down 6th
| | ^^A,


vB
vB
| style="text-align:center;" | A#
| style="text-align:center;" | A#
| style="text-align:center;" | Subminor 7th,
| style="text-align:center;" | Subminor 7th
Subminor 6th
| style="text-align:center;" | 7/4
| style="text-align:center;" | 7/4
7/4, 9/5, 20/11
| style="text-align:center;" | 7/4
| style="text-align:center;" | 7/4, 9/5, 20/11
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
| style="text-align:center;" | 1028.57
| style="text-align:center;" | 1028.57
| style="text-align:right;" | 7
| style="text-align:right;" | 7
6
| style="text-align:center;" | 7th
| style="text-align:center;" | 7th
6th
| | B
| | B
| style="text-align:center;" | Bb
| style="text-align:center;" | Bb
| style="text-align:center;" | Supraminor 7th
| style="text-align:center;" | Supraminor 7th
Supraminor 6th
| style="text-align:center;" | 29/16, 9/5
| style="text-align:center;" | 29/16, 9/5
9/5, 20/11
| style="text-align:center;" | 9/5, 20/11
 
| style="text-align:center;" | 16/9
16/9
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 19
| style="text-align:center;" | 1085.71
| style="text-align:center;" | 1085.71
| style="text-align:right;" | ^7,
| style="text-align:right;" | ^7


vv8
vv8
^8,
vv7
| style="text-align:center;" | up 7th,
| style="text-align:center;" | up 7th,


double-down 8ve
double-down 8ve
 
| | ^B
up 6th,
 
double-down 7th
| | ^B,


vvC
vvC
| style="text-align:center;" | B
| style="text-align:center;" | B
| style="text-align:center;" | Major 7th
| style="text-align:center;" | Major 7th
Major 6th
| style="text-align:center;" | 15/8
| style="text-align:center;" | 15/8
17/9
| style="text-align:center;" | 17/9
 
| style="text-align:center;" | 15/8, 48/25
15/8, 48/25
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:center;" | 1142.86
| style="text-align:center;" | 1142.86
| style="text-align:right;" | ^^7,
| style="text-align:right;" | ^^7


v8
v8
^^6,
v7
| style="text-align:center;" | double-up 7th,
| style="text-align:center;" | double-up 7th,


down 8ve
down 8ve
 
| | ^^B
double-up 6th,
 
down 7th
| | ^^B,


vC
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
Supermajor 6th
| style="text-align:center;" | 27/14, 29/15
| style="text-align:center;" | 27/14, 29/15
35/18, 68/35
| style="text-align:center;" | 35/18, 68/35
 
| style="text-align:center;" | 63/32
63/32
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:center;" | 1200.00
| style="text-align:center;" | 1200.00
| style="text-align:right;" | 8
| style="text-align:right;" | 8
7
| style="text-align:center;" | 8ve
| style="text-align:center;" | 8ve
7th
| | C
| | C
| style="text-align:center;" | C
| style="text-align:center;" | C
| style="text-align:center;" | Octave
| style="text-align:center;" | Octave
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
|}
|}
Line 520: Line 365:
&lowast;1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
&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 2.9/5.11/5.13/5.17/5.7 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
&lowast;3: based on treating 21-EDO as 13-limit laconic temperament
Retrieved from "https://en.xen.wiki/w/21edo"