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==Theory==
==Theory==
{{Primes in edo|21|columns=9}}
{| class="wikitable center-all"
! colspan="2" | <!-- empty cell -->
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | Error
! absolute (¢)
| 0.0
| -16.2
| +13.7
| +2.6
| +20.1
| +16.6
| +9.3
| -11.8
|-
! [[Relative error|relative]] (%)
| 0
| -28
| +24
| +5
| +35
| +29
| +16
| -21
|-
! colspan="2" | [[nearest edomapping]]
| 21
| 12
| 7
| 17
| 10
| 15
| 2
| 5
|}


21-edo provides both 7-edo as a subset and the familiar 400-cent major third, while also giving some higher-limit JI possibilities. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.
21-edo provides both 7-edo as a subset and the familiar 400-cent major third, while also giving some higher-limit JI possibilities. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.
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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 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.
In temperament terms, 21-EDO can be treated as a 13-limit temperament, but 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.
 
The [[patent val]] for 21edo tempers out [[128/125]] and [[2187/2000]] in the [[5-limit]], and supplies the [[optimal patent val]] for the 5-limit [[laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-, 11- and 13-limit [[gorgo]], and 11- and 13-limit spartan. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.


== Intervals ==
== Intervals ==
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==Triadic Harmony==
==Triadic Harmony==


One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing ultraminor, minor, neutral, major, and ultramajor 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:
One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:


{| class="wikitable"
{| class="wikitable"
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==Moment-of-Symmetry Scales==
==Moment-of-Symmetry Scales==
Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.
Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.


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For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.
For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.


21edo has the [[Step ratio|soft]] [[oneirotonic]] (5L 3s) MOS with generator 8\21; in addition to the [[naiadic]]s that generate it, it has neutral thirds (instead of major thirds as in [[13edo]] oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The oneirofifths (4-step intervals) are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 9:10:11:13 and 16:23:30.
Examples:
*[[augment6]]
*[[augment9]]
*[[augment12]]


{| class="wikitable"
21edo has the [[oneirotonic]] (5L 3s) MOS with generator 8\21; it supports the 13&21 temperament on the 2.9/5.11/5.13/5 subgroup (approximating 5:9:11:13).
|-
! | Periods per octave
! | Generator
! | MOSes
|-
| | 1
| | 2\21
| | [[10L 1s]]
|-
| | 1
| | 4\21
| | [[5L 1s]]<br/>[[5L 6s]]
|-
| | 1
| | 5\21
| | [[4L 1s]]<br/> [[4L 5s]]<br/> [[4L 9s]] <br/> Pathological 4L 13s
|-
| | 1
| | 8\21
| | [[3L 2s]]<br/> [[5L 3s]]<br/> [[8L 5s]]
|-
| | 3
| | 2\21
| | [[3L 3s]]<br/> [[3L 6s]]<br/> [[9L 3s]]
|-
| | 3
| | 3\21
| | [[3L 3s]]<br/> [[6L 3s]]<br/>[[6L 9s]]
|-
| | 7
| | 1\21
| | [[7L 7s]]
|}


==Tetrachordal Scales==
==Tetrachordal Scales==
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The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.
The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.


== As a regular temperament ==
==Rank two temperaments==
The [[patent val]] for 21edo tempers out [[128/125]] and [[2187/2000]] in the [[5-limit]], and supplies the [[optimal patent val]] for the 5-limit [[laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-, 11- and 13-limit [[gorgo]], and 11- and 13-limit spartan. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
 
=== Rank two temperaments ===
[[List of 21edo rank two temperaments by badness]]
[[List of 21edo rank two temperaments by badness]]


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|}
|}


=== Commas ===
==Commas==
21 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 21 33 49 59 73 78 }}.)
21 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 21 33 49 59 73 78 }}.)


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