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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 of 17, 19, and 27** |
Other ratios† |
|---|---|---|---|---|
| 0 | 0.00 | 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 | |
| 16 | 914.29 | 22/13, 56/33, 39/23 | 32/19, 27/16, 46/27, 29/17 | |
| 17 | 971.43 | 7/4, 58/33 | 30/17, 54/31, 33/19, 76/43 | |
| 18 | 1028.57 | 20/11, 29/16, 56/31, 78/43 | 9/5, 34/19, 31/17 | |
| 19 | 1085.71 | 15/8, 28/15, 58/31, 62/33, 43/23 | 32/17, 54/29 | |
| 20 | 1142.86 | 29/15, 56/29, 31/16, 60/31, 64/33 | 27/14, 33/17, 76/39 | |
| 21 | 1200.00 | 2/1 |
*In the 2.15.7.33.39.23.29.31.43 subgroup
**Odd 27 by direct approximation (Note: Fractions like 27/15 = 9/5 are included.)
†All by patent val
| 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 |
56edo
Theory
56edo shares its near perfect quality of classical major third with 28edo, which it doubles, while also adding a superpythagorean 5th that is a convergent towards the bronze metallic mean, following 17edo and preceding 185edo. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third 5/4 and the Pythagorean major third 81/64. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in 224edo, which is the quadruple of 56edo.
56edo has unambiguous approximations to prime harmonics up to 19, and possibly up to 29. However harmonic 3 is quite sharp, leading harmonic 9 to be even more so, and causing intervals like 10/9, 9/7, and 13/9 to be inconsistent. Therefore, 56edo is not very popular compared to edos like 53 and 58.
One step of 56edo is the closest direct approximation to the syntonic comma, 81/80, with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by patent val mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) Barium temperament realizes this proximity through regular temperament theory, and is supported by notable edos like 224edo, 1848edo, and 2520edo, which is a highly composite edo.
56edo can be used to tune hemithirds, superkleismic, sycamore and keen temperaments, and using ⟨56 89 130 158] (56d) as the equal temperament val, for pajara. It provides the optimal patent val for 7-, 11- and 13-limit sycamore, and the 11-limit 56d val is close to the POTE tuning for undecimal pajara.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +5.19 | -0.60 | -4.54 | +5.82 | -4.81 | +2.19 | +2.49 | -6.85 | -1.01 | -9.32 |
| Relative (%) | +0.0 | +24.2 | -2.8 | -21.2 | +27.2 | -22.5 | +10.2 | +11.6 | -31.9 | -4.7 | -43.5 | |
| Steps (reduced) |
56 (0) |
89 (33) |
130 (18) |
157 (45) |
194 (26) |
207 (39) |
229 (5) |
238 (14) |
253 (29) |
272 (48) |
277 (53) | |
Subsets and supersets
Since 56 factors into 23 × 7, 56edo has subset edos 2, 4, 7, 8, 14, 28.
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