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21edo
Theory
Notes: Excellent odd harmonics 7, 15, 23, 29, 31, 33, 39, 43, all derived from 84edo; less accurate but still usable 17, 19, 27; also note 3*21 subgroup from 63edo
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 "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.
21edo supports 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 ¢, 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.
21edo closely approximates the octave-reduced harmonics 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. 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 ¢. These approximations derive from and are inherited by 84edo, which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from 63edo, which is possibly a more sensible way to treat it.
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.
Because 21edo is a 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.
Thanks to its sevenths, 21edo is an ideal tuning for its size for metallic harmony.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -16.2 | +13.7 | +2.6 | +24.7 | +20.1 | +16.6 | -2.6 | +9.3 | -11.8 | -13.6 | +0.3 |
| Relative (%) | -28.4 | +24.0 | +4.6 | +43.2 | +35.2 | +29.1 | -4.5 | +16.3 | -20.6 | -23.9 | +0.5 | |
| Steps (reduced) |
33 (12) |
49 (7) |
59 (17) |
67 (4) |
73 (10) |
78 (15) |
82 (19) |
86 (2) |
89 (5) |
92 (8) |
95 (11) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +27.4 | +8.4 | -1.0 | -2.2 | +3.9 | +16.3 | -22.8 | +0.4 | +28.1 | +2.8 | -18.8 |
| Relative (%) | +47.9 | +14.7 | -1.8 | -3.8 | +6.8 | +28.5 | -39.9 | +0.7 | +49.1 | +4.8 | -32.9 | |
| Steps (reduced) |
98 (14) |
100 (16) |
102 (18) |
104 (20) |
106 (1) |
108 (3) |
109 (4) |
111 (6) |
113 (8) |
114 (9) |
115 (10) | |
Intervals
Inconsistent intervals are in italics.
| Steps | Cents | 43-odd-limit ratios* | Additional ratios |
|---|---|---|---|
| 0 | 0.00 | ||
| 1 | 57.14 | ||
| 2 | 114.29 | ||
| 3 | 171.43 | ||
| 4 | 228.57 | ||
| 5 | 285.71 | ||
| 6 | 342.86 | ||
| 7 | 400.00 | ||
| 8 | 457.14 | ||
| 9 | 514.29 | ||
| 10 | 571.43 | ||
| 11 | 628.57 | ||
| 12 | 685.71 | ||
| 13 | 742.86 | ||
| 14 | 800.00 | ||
| 15 | 857.14 | ||
| 16 | 914.29 | ||
| 17 | 971.43 | ||
| 18 | 1028.57 | ||
| 19 | 1085.71 | ||
| 20 | 1142.86 | ||
| 21 | 1200.00 |
*In the 2.15.7.33.39.23.29.31.43 subgroup
| Degree | Cents | Ups and downs notation | 5L 3s octotonic notation |
Extended-diatonic interval name |
||
|---|---|---|---|---|---|---|
| 0 | 0.00 | 1 | unison | C | C | Unison |
| 1 | 57.14 | ^1 vv2 | up unison, dud 2nd |
^C vvD |
C# | Subminor 2nd |
| 2 | 114.29 | ^^1 v2 |
dup unison, down 2nd |
^^C vD |
Db | Minor 2nd |
| 3 | 171.43 | 2 | 2nd | D | D | Submajor 2nd |
| 4 | 228.57 | ^2 vv3 |
up 2nd, dud 3rd |
^D vvE |
D# | Supermajor 2nd |
| 5 | 285.71 | ^^2 v3 |
dup 2nd, down 3rd |
^^D vE |
Eb | Subminor 3rd |
| 6 | 342.86 | 3 | 3rd | E | E | Neutral 3rd |
| 7 | 400.00 | ^3 vv4 |
up 3rd, dud 4th |
^E vvF |
E#/Fb | Major 3rd |
| 8 | 457.14 | ^^3 v4 |
dup 3rd, down 4th |
^^E vF |
F | Third-fourth (naiadic) |
| 9 | 514.29 | 4 | 4th | F | F# | Acute 4th |
| 10 | 571.43 | ^4 vv5 |
up 4th, dud 5th |
^F vvG |
Gb | Narrow tritone |
| 11 | 628.57 | ^^4 v5 |
dup 4th, down 5th |
^^F vG |
G | Wide tritone |
| 12 | 685.71 | 5 | 5th | G | G# | Grave 5th |
| 13 | 742.86 | ^5 vv6 |
up 5th, dud 6th |
^G vvA |
Hb | Fifth-sixth (cocytic) |
| 14 | 800.00 | ^^5 v6 |
dup 5th, down 6th |
^^G vA |
H | Minor 6th |
| 15 | 857.14 | 6 | 6th | A | H#/Ab | Neutral 6th |
| 16 | 914.29 | ^6 vv7 |
up 6th, dud 7th |
^A vvB |
A | Supermajor 6th |
| 17 | 971.43 | ^^6 v7 |
dup 6th, down 7th |
^^A vB |
A# | Subminor 7th |
| 18 | 1028.57 | 7 | 7th | B | Bb | Supraminor 7th |
| 19 | 1085.71 | ^7 vv8 |
up 7th, dud 8ve |
^B vvC |
B | Major 7th |
| 20 | 1142.86 | ^^7 v8 |
dup 7th, down 8ve |
^^B vC |
B#/Cb | Supermajor 7th |
| 21 | 1200.00 | 8 | 8ve | C | C | Octave |
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