21edo: Difference between revisions
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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. | 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. | ||
== 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 | ! 5L3s Octatonic <br> Notation | ||
! D.-R. Interval Types | ! D.-R. Interval Types | ||
! Approximate Ratios *1 | ! Approximate Ratios *1 *2 *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 | ||
| style="text-align:center;" | | 0 | ||
| 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 | | 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 | |||
^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 | |||
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 | |||
^^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 | |||
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 | |||
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;" | 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 | |||
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 | |||
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 | |||
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 167: | Line 228: | ||
| 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 | |||
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 | |||
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 196: | Line 271: | ||
| 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 | |||
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 213: | Line 297: | ||
| 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 | |||
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 | |||
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 | |||
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 | |||
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 | |||
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 | |||
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;" | 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 | |||
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 | |||
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 | |||
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 | ||
|} | |} | ||
| Line 365: | Line 520: | ||
∗1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament | ∗1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament | ||
∗2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5. | ∗2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.7 subgroup temperament | ||
∗3: based on treating 21-EDO as 13-limit laconic temperament | ∗3: based on treating 21-EDO as 13-limit laconic temperament | ||