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{{Infobox ET}}
{{Infobox ET}}
{{todo|add introduction}}
{{EDO intro|21}}


{{EDO intro|21}}
== Theory ==
== Theory ==
{{Harmonics in equal|steps=21|columns=14}}
{{Harmonics in equal|steps=21|columns=14}}


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 971.43¢ is only off in 21 EDO by 2.60 cents from just (968.83¢), which is better than any other EDO <26.
21edo provides both [[7edo]] 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 7edo or "equiheptatonic" scales, or as seven [[3edo]] ''augmented'' triads. The [[7/4]] at 971.43{{cent}} is only off in 21edo by 2.60{{cent}} from just (968.83{{cent}}), which is better than any other [[edo]] below 26.


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, 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.


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.
Of harmonics 3, 5, 7, 11, and 13, the only harmonic 21edo approximates with anything approaching a near-just flavor is the 7th harmonic. On the other hand, 21edo provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3{{cent}} or less), as well as a very reasonable approximation of the 27th harmonic (around 8{{cent}} sharp). As such, treating 21edo 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 21edo can be described as a ratio within the 29-odd-limit. 21edo 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.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-3 right-5"
{| class="wikitable center-all right-3 right-5"
|-
|-
! Degree
! [[Degree]]
! Cents
! [[Cent]]s
! colspan="3" | [[Ups and Downs Notation|Up/down notation]]
! colspan="3" | [[Ups and downs notation]]
! 5L3s Octotonic <br> Notation
! [[5L 3s]] octotonic <br> notation
! D.-R. Interval Types
! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]
! Approximate Ratios *1
! Approximate Ratios *1
! Approximate Ratios *2
! Approximate Ratios *2
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| ^^E <br> vF
| ^^E <br> vF
| F
| F
| Third-Fourth
| Third-fourth ([[naiadic]])
| 30/23
| 30/23
| 13/10, 17/13, 22/17
| 13/10, 17/13, 22/17
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| ^F <br> vvG
| ^F <br> vvG
| Gb
| Gb
| Narrow Tritone
| Narrow tritone
| 32/23
| 32/23
| 18/13
| 18/13
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| ^^F <br> vG
| ^^F <br> vG
| G
| G
| Wide Tritone
| Wide tritone
| 23/16
| 23/16
| 13/9
| 13/9
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| ^G <br> vvA
| ^G <br> vvA
| Hb
| Hb
| Fifth-Sixth
| Fifth-sixth ([[cocytic]])
| 23/15
| 23/15
| 17/11, 20/13, 26/17
| 17/11, 20/13, 26/17
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|}
|}


&lowast;1: based on treating 21 EDO as a 2.7.15.23.27.29 subgroup temperament
&lowast;1: based on treating 21edo as a 2.7.15.23.27.29 subgroup temperament


&lowast;2: based on treating 21 EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
&lowast;2: based on treating 21edo as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament


&lowast;3: based on treating 21 EDO as 13-limit laconic temperament
&lowast;3: based on treating 21edo as 13-limit laconic temperament


== Chord Names ==
== Chords ==
 
=== Chord names ===
Ups and downs can be used to name 21 EDO chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  
Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  


0-6-12 = C E G = C = C or C perfect
0-6-12 = C E G = C = C or C perfect
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0-5-12-17 = C vE G vB = Cv7 = C down-seven
0-5-12-17 = C vE G vB = Cv7 = C down-seven


For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].
 
== 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 inframinor, minor, neutral, major, and ultramajor 3rds respectively (or dud, down, perfect, up and dup). 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 harmonic series:
=== Triadic harmony ===
One interesting feature of 21edo 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 inframinor, minor, neutral, major, and ultramajor 3rds respectively (or dud, down, perfect, up and dup). One can couple these with 21edo'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 harmonic series:


{| class="wikitable center-1 center-2 center-3"
{| class="wikitable center-1 center-2 center-3"
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|}
|}


== Moment-of-Symmetry Scales ==
== Scales ==
 
=== MOS 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 21edo contains sub-edos of 3 and 7, it contains no heptatonic [[MOS scale]]s (other than 7edo and a few very [[Step ratio|hard]] scales) and a wealth of scales that repeat at a 1/3-octave period.


For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherepnin's scale in 12-TET) is an excellent example.
For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-tone [[3L 6s]] scale (related to Tcherepnin's scale in [[12edo]]) is an excellent example.


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 21edo generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7edo, 3edo, or a repetition of one of the other scales.


21 EDO 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 [[13 EDO]] oneirotonic), neogothic minor thirds, and Baroque diatonic semitones. The 4-oneirosteps are more tritone-like than fifth-like, unlike in 13 EDO, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 16:23:30, 16:23:29 and their inversions.
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 4-oneirosteps 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 16:23:30, 16:23:29 and their inversions.


{| class="wikitable"
{| class="wikitable"
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|}
|}


== Rank-3 Scales ==
=== Rank-3 scales ===
The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of Zarlino diatonic with 1\21 comma steps added, resulting in two "major seconds" (171c and 228c), two "minor thirds" (286c and 343c) and two "fourths" (457c and 514c). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{cent}} and 228{{cent}}), two "minor thirds" (286{{cent}} and 343{{cent}}) and two "fourths" (457{{cent}} and 514{{cent}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
 
== Tetrachordal Scales ==


While 21 EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21 EDO fourth is 9 steps, which can be divided into three parts in the following ways:
=== Tetrachordal scales ===
While 21edo lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21edo fourth is 9 steps, which can be divided into three parts in the following ways:


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Step Pattern
! [[Step pattern]]
! Cents
! [[Cents]]
! Example
! Example
! Name*
! Name*
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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 21 EDO 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 21 EDO can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.


== Other Scales ==
=== Other scales ===
The subset 2 3 7 2 7 of 21 EDO sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.
The subset 2 3 7 2 7 of 21edo sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.


The subset 2 5 5 6 3 of 21 EDO is a good tuning for the [[Aurora scale]].
The subset 2 5 5 6 3 of 21edo is a good tuning for the [[Aurora scale]].


== Regular temperament properties ==
== Regular temperament properties ==
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== Approaches ==
== Approaches ==
* [[21edo/Inthar's approach]]
* [[21edo/Inthar's approach]]


== Books / Literature ==
== Books / Literature ==
 
* Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.


== Music ==
== Music ==
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; [[Fabrizio Fiale]]
; [[Fabrizio Fiale]]
*[https://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE]
* [https://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE]
 
; [[Frédéric Gagné]]
* [https://youtu.be/tDjLcCictVQ?t=119 Tostarena: Ruins (21edo cover)], from [[XA Discord]]'s ''Xen Cover Project 2'' ([https://musescore.com/user/5995996/scores/8607089 score])
 
; [[Frédéric Gagné]], [[Ian Means]] and [[AraMax]]
* [https://www.youtube.com/watch?v=9rTbLQ9j1sE&t=135s Mirage Haze], from XA Discord's ''Deleted User EP'' ([https://musescore.com/user/5995996/scores/7852055 score])


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
*[https://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715 Anomalous Readings] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3 (MP3)]
* [https://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715 Anomalous Readings] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3 (MP3)]


; [[Inthar]]
; [[Inthar]]
*[[:File:The Angels' Library.mp3|The Angels' Library]] in the Sarnathian (23233233) mode of 21edo [[5L 3s]] ([[:File:The Angels' Library Score.pdf|score]])
* [[:File:The Angels' Library.mp3|The Angels' Library]] in the Sarnathian (23233233) mode of 21edo [[5L 3s]] ([[:File:The Angels' Library Score.pdf|score]])


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
*[http://soonlabel.com/xenharmonic/archives/2494 21-edo Trio for Organ] {{dead link}}
* [http://soonlabel.com/xenharmonic/archives/2494 21-edo Trio for Organ] {{dead link}}
*[http://soonlabel.com/xenharmonic/archives/2336 21-penny jingle] {{dead link}}
* [http://soonlabel.com/xenharmonic/archives/2336 21-penny jingle] {{dead link}}
* [https://www.youtube.com/watch?v=lpcqXD8tpXc Trio Sonata in 21edo for Organ (The Sewing Machine)]
* [https://www.youtube.com/watch?v=lpcqXD8tpXc Trio Sonata in 21edo for Organ (The Sewing Machine)]
* [https://www.youtube.com/watch?v=n0QA0ZQHPvk 21edo Chacony, for two Harpsichords]
* [https://www.youtube.com/watch?v=n0QA0ZQHPvk 21edo Chacony, for two Harpsichords]
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; [[Ron Sword]]
; [[Ron Sword]]
*[http://www.ronsword.com/sounds/21_improv.mp3 Short Clip of 21-edo Acoustic] {{dead link}}
* [http://www.ronsword.com/sounds/21_improv.mp3 Short Clip of 21-edo Acoustic] {{dead link}}
*[http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3 Open tuning Drone Improvisation in 21-edo] {{dead link}}
* [http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3 Open tuning Drone Improvisation in 21-edo] {{dead link}}


; [[Stephen Weigel]]
; [[Stephen Weigel]]
*[https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal Iridescent Wenge Fugue] (accepted to [https://www.seamusonline.org/ SEAMUS 2018] and [http://eabarndance.com/ Electroacoustic Barn Dance 2018])
* [https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal Iridescent Wenge Fugue] (accepted to [https://www.seamusonline.org/ SEAMUS 2018] and [http://eabarndance.com/ Electroacoustic Barn Dance 2018])
*[https://xenharmonicgod.bandcamp.com/album/weigel-family-christmas-xenharmonic-chocolate WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate)], an album of xenharmonic Christmas covers, many are in 21 EDO
* [https://xenharmonicgod.bandcamp.com/album/weigel-family-christmas-xenharmonic-chocolate WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate)], an album of xenharmonic Christmas covers, many are in 21 EDO


; [[Randy Wells]]
; [[Randy Wells]]
*[https://www.youtube.com/watch?v=sA5HfL3FjJU The Island Scene]
* [https://www.youtube.com/watch?v=sA5HfL3FjJU The Island Scene]


; [[Randy Winchester]]
; [[Randy Winchester]]
*[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3 Comets Over Flatland 15]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3 Comets Over Flatland 15]
*[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3 Comets Over Flatland 18]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3 Comets Over Flatland 18]


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/21edo"