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* [[Just intonation]] – Tuning based on [[interval]]s with {{W|rational number}} [[frequency ratio]]s | * [[Just intonation]] – Tuning based on [[interval]]s with {{W|rational number}} [[frequency ratio]]s | ||
* [[EDO|Equal divisions of the octave]] and other [[equal-step tuning]]s | * [[EDO|Equal divisions of the octave]] and other [[equal-step tuning]]s | ||
* [[MOS scale|Moment of Symmetry (MOS) scales]] – Scales with at most two distinct sizes (e.g. {{w|major and minor}}) for each interval class, [[MOS scale#Equivalent definitions and generalizations| | * [[MOS scale|Moment of Symmetry (MOS) scales]] – Scales with at most two distinct sizes (e.g. {{w|major and minor}}) for each interval class, among [[MOS scale#Equivalent definitions and generalizations|many other things]] | ||
* [[Regular temperaments]] – | * [[Regular temperaments]] – Tuning systems that appear the same everywhere, great for free modulation; [[equal temperament]]s are a basic example | ||
* [[Historical temperaments]], such as [[Pythagorean tuning]], [[meantone]] temperaments, and [[well temperament]]s | * [[Historical temperaments]], such as [[Pythagorean tuning]], [[meantone]] temperaments, and [[well temperament]]s | ||
Revision as of 05:44, 13 January 2026
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21edo
Theory
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 a basic "diatonic" scale, though maximum-variety-3 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 kind of accidental (such as ups and downs) is usually used instead, though it might be "reskinned" as sharps and flats to aid melodic intuition.
21edo supports tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.
In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics 7/4 (a subminor seventh), 17/16 (a semitone), 19/16 (a minor third), 23/16 (a tritone), and 29/16 (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals 16/15 and 27/16, if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.15.7.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.7.11/5.13/5.17/5 subgroup, which is possibly a more sensible way to treat it.
In terms of interval regions, 21edo 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.
Because 21edo is a Fibonacci edo, it contains an approximation to the logarithmic phi superfifth, which generates golden MOS scales 8L 5s, 5L 3s, and 3L 2s.
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
| Degree | Cents | Ups and downs notation | 5L 3s octotonic notation |
Extended-diatonic interval name |
Approximate Ratios *1 | Approximate Ratios *2 | Approximate Ratios *3 | ||
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 1 | unison | C | C | Unison | 1/1 | 1/1 | 1/1 |
| 1 | 57.14 | ^1 vv2 | up unison, dud 2nd |
^C vvD |
C# | Subminor 2nd | 28/27, 30/29 | 35/34, 36/35 | 64/63 |
| 2 | 114.29 | ^^1 v2 |
dup unison, down 2nd |
^^C vD |
Db | Minor 2nd | 16/15, 15/14, 29/27 | 18/17 | 16/15, 25/24 |
| 3 | 171.43 | 2 | 2nd | D | D | Submajor 2nd | 10/9, 32/29 | 10/9,11/10 | 9/8, 11/10 |
| 4 | 228.57 | ^2 vv3 |
up 2nd, dud 3rd |
^D vvE |
D# | Supermajor 2nd | 8/7 | 8/7 | 8/7, 10/9 |
| 5 | 285.71 | ^^2 v3 |
dup 2nd, down 3rd |
^^D vE |
Eb | Subminor 3rd | 27/23, 32/27 | 13/11, 20/17 | 6/5, 7/6 |
| 6 | 342.86 | 3 | 3rd | E | E | Neutral 3rd | 28/23 | 11/9 | 16/13 |
| 7 | 400.00 | ^3 vv4 |
up 3rd, dud 4th |
^E vvF |
E#/Fb | Major 3rd | 29/23 | 44/35 | 5/4, 9/7, 11/9, 14/11 |
| 8 | 457.14 | ^^3 v4 |
dup 3rd, down 4th |
^^E vF |
F | Third-fourth (naiadic) | 30/23 | 13/10, 17/13, 22/17 | 13/10 |
| 9 | 514.29 | 4 | 4th | F | F# | Acute 4th | 161/120, 256/189 | 35/26 | 4/3, 18/13 |
| 10 | 571.43 | ^4 vv5 |
up 4th, dud 5th |
^F vvG |
Gb | Narrow tritone | 32/23 | 18/13 | 7/5, 11/8 |
| 11 | 628.57 | ^^4 v5 |
dup 4th, down 5th |
^^F vG |
G | Wide tritone | 23/16 | 13/9 | 10/7, 16/11 |
| 12 | 685.71 | 5 | 5th | G | G# | Grave 5th | 189/128, 240/161 | 52/35 | 3/2, 13/9 |
| 13 | 742.86 | ^5 vv6 |
up 5th, dud 6th |
^G vvA |
Hb | Fifth-sixth (cocytic) | 23/15 | 17/11, 20/13, 26/17 | 20/13 |
| 14 | 800.00 | ^^5 v6 |
dup 5th, down 6th |
^^G vA |
H | Minor 6th | 46/29 | 35/22 | 8/5, 11/7, 14/9, 18/11 |
| 15 | 857.14 | 6 | 6th | A | H#/Ab | Neutral 6th | 23/14 | 18/11 | 13/8 |
| 16 | 914.29 | ^6 vv7 |
up 6th, dud 7th |
^A vvB |
A | Supermajor 6th | 27/16, 46/27 | 17/10, 22/13 | 5/3, 12/7 |
| 17 | 971.43 | ^^6 v7 |
dup 6th, down 7th |
^^A vB |
A# | Subminor 7th | 7/4 | 7/4 | 7/4, 9/5 |
| 18 | 1028.57 | 7 | 7th | B | Bb | Supraminor 7th | 29/16, 9/5 | 9/5, 20/11 | 16/9, 20/11 |
| 19 | 1085.71 | ^7 vv8 |
up 7th, dud 8ve |
^B vvC |
B | Major 7th | 15/8 | 17/9 | 15/8, 48/25 |
| 20 | 1142.86 | ^^7 v8 |
dup 7th, down 8ve |
^^B vC |
B#/Cb | Supermajor 7th | 27/14, 29/15 | 35/18, 68/35 | 63/32 |
| 21 | 1200.00 | 8 | 8ve | C | C | Octave | 2/1 | 2/1 | 2/1 |
∗1: based on treating 21edo as a 2.7.15.23.27.29 subgroup temperament
∗2: based on treating 21edo as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
∗3: based on treating 21edo as 13-limit laconic temperament
Main page
Welcome to the Xenharmonic Wiki!
The Xenharmonic Wiki is an open resource dedicated to musical tuning systems, focusing on xenharmonic music while also documenting historical tunings and tuning practices from world traditions. It covers the theory and practical applications of these systems.
For a lengthier introduction, see Xenharmonic Wiki: Introduction.
If you are new to musical tuning
- Why use alternative tunings?
- What are microtonal and xenharmonic music?
- Listen to alternatively tuned music, in case you're wondering what it all sounds like.
- Discover approaches to musical tuning
- Explore links to xenharmonic websites
- Browse the library of published works about microtonal/xenharmonic music
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Popular topics
- Just intonation – Tuning based on intervals with rational number frequency ratios
- Equal divisions of the octave and other equal-step tunings
- Moment of Symmetry (MOS) scales – Scales with at most two distinct sizes (e.g. major and minor) for each interval class, among many other things
- Regular temperaments – Tuning systems that appear the same everywhere, great for free modulation; equal temperaments are a basic example
- Historical temperaments, such as Pythagorean tuning, meantone temperaments, and well temperaments
Practical xenharmonics
Contributing to the Xenharmonic Wiki
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