21edo: Difference between revisions
Add 13-limit uniform maps |
Add 2 tracks by me, include alternate interseptimal names, todo:add introduction no longer needed, use lowercase "edo" (as usual on the wiki), more internal links, misc. edits |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{EDO intro|21}} | ||
== Theory == | == Theory == | ||
{{Harmonics in equal|steps=21|columns=14}} | {{Harmonics in equal|steps=21|columns=14}} | ||
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, | 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 | 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]] | ||
! | ! [[Cent]]s | ||
! colspan="3" | [[Ups and | ! colspan="3" | [[Ups and downs notation]] | ||
! | ! [[5L 3s]] octotonic <br> notation | ||
! | ! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]] | ||
! Approximate Ratios *1 | ! Approximate Ratios *1 | ||
! Approximate Ratios *2 | ! Approximate Ratios *2 | ||
| Line 125: | Line 123: | ||
| ^^E <br> vF | | ^^E <br> vF | ||
| F | | F | ||
| Third- | | Third-fourth ([[naiadic]]) | ||
| 30/23 | | 30/23 | ||
| 13/10, 17/13, 22/17 | | 13/10, 17/13, 22/17 | ||
| Line 147: | Line 145: | ||
| ^F <br> vvG | | ^F <br> vvG | ||
| Gb | | Gb | ||
| Narrow | | Narrow tritone | ||
| 32/23 | | 32/23 | ||
| 18/13 | | 18/13 | ||
| Line 158: | Line 156: | ||
| ^^F <br> vG | | ^^F <br> vG | ||
| G | | G | ||
| Wide | | Wide tritone | ||
| 23/16 | | 23/16 | ||
| 13/9 | | 13/9 | ||
| Line 180: | Line 178: | ||
| ^G <br> vvA | | ^G <br> vvA | ||
| Hb | | Hb | ||
| Fifth- | | Fifth-sixth ([[cocytic]]) | ||
| 23/15 | | 23/15 | ||
| 17/11, 20/13, 26/17 | | 17/11, 20/13, 26/17 | ||
| Line 274: | Line 272: | ||
|} | |} | ||
∗1: based on treating | ∗1: based on treating 21edo as a 2.7.15.23.27.29 subgroup temperament | ||
∗2: based on treating | ∗2: based on treating 21edo as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament | ||
∗3: based on treating | ∗3: based on treating 21edo as 13-limit laconic temperament | ||
== Chord | == Chords == | ||
=== Chord names === | |||
Ups and downs can be used to name | 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 | ||
| Line 302: | Line 300: | ||
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 | For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]]. | ||
One interesting feature of | === 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" | ||
| Line 353: | Line 350: | ||
|} | |} | ||
== | == Scales == | ||
=== MOS scales === | |||
Since | 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- | 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 | 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. | ||
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" | ||
| Line 398: | Line 395: | ||
|} | |} | ||
== Rank-3 | === 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" ( | 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). | ||
While | === 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 | ! [[Step pattern]] | ||
! Cents | ! [[Cents]] | ||
! Example | ! Example | ||
! Name* | ! Name* | ||
| Line 459: | Line 455: | ||
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 | === Other scales === | ||
The subset 2 3 7 2 7 of | 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 | The subset 2 5 5 6 3 of 21edo is a good tuning for the [[Aurora scale]]. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 640: | Line 636: | ||
== 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] | ||
| Line 670: | Line 670: | ||
; [[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 == | ||
Revision as of 04:13, 12 September 2023
| ← 20edo | 21edo | 22edo → |
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -16.2 | +13.7 | +2.6 | +24.7 | +20.1 | +16.6 | -2.6 | +9.3 | -11.8 | -13.6 | +0.3 | +27.4 | +8.4 | -1.0 |
| Relative (%) | -28.4 | +24.0 | +4.6 | +43.2 | +35.2 | +29.1 | -4.5 | +16.3 | -20.6 | -23.9 | +0.5 | +47.9 | +14.7 | -1.8 | |
| Steps (reduced) |
33 (12) |
49 (7) |
59 (17) |
67 (4) |
73 (10) |
78 (15) |
82 (19) |
86 (2) |
89 (5) |
92 (8) |
95 (11) |
98 (14) |
100 (16) |
102 (18) | |
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 ¢ is only off in 21edo by 2.60 ¢ from just (968.83 ¢), which is better than any other edo below 26.
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 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 ¢ or less), as well as a very reasonable approximation of the 27th harmonic (around 8 ¢ 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
| 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 |
| 4 | 228.57 | ^2 vv3 |
up 2nd, dud 3rd |
^D vvE |
D# | Supermajor 2nd | 8/7 | 8/7 | 8/7, 10/9, 11/10 |
| 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, 20/11 |
| 18 | 1028.57 | 7 | 7th | B | Bb | Supraminor 7th | 29/16, 9/5 | 9/5, 20/11 | 16/9 |
| 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
Chords
Chord names
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
0-5-12 = C vE G = Cv = C down
0-7-12 = C ^E G = C^ = C up
0-6-11 = C E vG = C(v5) = C down-five
0-7-13 = C ^E ^G = C^(^5) = C up up-five
0-6-12-18 = C E G B = C7 = C seven
0-6-12-17 = C E G vB = C,v7 = C add down-seven
0-5-12-18 = C vE G B = Cv,7 = C down add 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.
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:
| Steps | Cents | Ratio | Example in C | Written name | Spoken name |
|---|---|---|---|---|---|
| 0-5-10 | 0-286-571 | 23:27:32 | C vE vvG | Cv(vv5) | C down, dud five |
| 0-4-11 | 0-229-629 | 7:8:10 | C vvE vG | Cvv(v5) | C dud, down five |
| 0-6-11 | 0-343-629 | 9:11:13 | C E vG | C(v5) | C down-five |
| 0-5-13 | 0-286-743 | 11:13:17 | C vE ^G | Cv(^5) | C down up-five |
| 0-8-13 | 0-457-743 | 13:17:20 | C vF ^G | Cv4(^5) | C (sus) down-four up-five |
Scales
MOS scales
Since 21edo contains sub-edos of 3 and 7, it contains no heptatonic MOS scales (other than 7edo and a few very 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-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 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.
21edo has the soft oneirotonic (5L 3s) MOS with generator 8\21; in addition to the naiadics 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.
| Periods per octave | Generator | MOSes |
|---|---|---|
| 1 | 2\21 | 1L 9s 10L 1s |
| 1 | 4\21 | 5L 1s 5L 6s |
| 1 | 5\21 | 4L 1s 4L 5s 4L 9s |
| 1 | 8\21 | 3L 2s 5L 3s 8L 5s |
| 3 | 2\21 | 3L 3s 3L 6s 9L 3s |
| 3 | 3\21 | 3L 3s 6L 3s 6L 9s |
| 7 | 1\21 | 7L 7s |
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" (171 ¢ and 228 ¢), two "minor thirds" (286 ¢ and 343 ¢) and two "fourths" (457 ¢ and 514 ¢). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
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:
| Step pattern | Cents | Example | Name* | Ups/downs name |
|---|---|---|---|---|
| 3, 3, 3 | (0)-171-343-(514) | C D E F | Equable diatonic | C perfect |
| 4, 3, 2 | (0)-229-400-(514) | C ^D ^E F | Soft diatonic | C up, up-2 |
| 4, 4, 1 | (0)-229-457-(514) | C ^D ^^E F | Intense diatonic | C dup, up-2 & 6 |
| 5, 3, 1 | (0)-286-457-(514) | C ^^D ^^E F | Archytas chromatic | C dup, dup-2 |
| 5, 2, 2 | (0)-286-400-(514) | C ^^D ^E F | Weak chromatic | C up, dup 2 & 6 |
| 6, 2, 1 | (0)-343-457-(514) | C ^3D ^^E F | Strong enharmonic | C dup, trup 2 & 6 |
| 7, 1, 1 | (0)-400-457-(514) | C ^4D ^^E F | Pythagorean enharmonic | C dup, quadruple-up 2 & 6 |
∗These names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
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
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 21edo is a good tuning for the Aurora scale.
Regular temperament properties
The patent val for 21 EDO 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.
Uniform maps
| Min. size | Max. size | Wart notation | Map |
|---|---|---|---|
| 20.6732 | 20.8381 | 21cdef | ⟨21 33 48 58 72 77] |
| 20.8381 | 20.8878 | 21cef | ⟨21 33 48 59 72 77] |
| 20.8878 | 20.9435 | 21ef | ⟨21 33 49 59 72 77] |
| 20.9435 | 20.9572 | 21e | ⟨21 33 49 59 72 78] |
| 20.9572 | 21.1361 | 21 | ⟨21 33 49 59 73 78] |
| 21.1361 | 21.1943 | 21b | ⟨21 34 49 59 73 78] |
| 21.1943 | 21.2137 | 21bdd | ⟨21 34 49 60 73 78] |
Commas
21edo tempers out the following commas. (Note: This assumes the val ⟨21 33 49 59 73 78].)
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 3 | 2187/2048 | [-11 7⟩ | 113.69 | Lawa | Apotome |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis, Augmented Comma |
| 5 | (16 digits) | [-25 7 6⟩ | 31.57 | Lala-tribiyo | Ampersand, Ampersand's Comma |
| 5 | (20 digits) | [32 -7 -9⟩ | 9.49 | Sasa-tritrigu | Escapade Comma |
| 7 | 1029/1000 | [-3 1 -3 3⟩ | 49.49 | Trizogu | Keega |
| 7 | 36/35 | [2 2 -1 -1⟩ | 48.77 | Rugu | Septimal Quarter Tone |
| 7 | (18 digits) | [-10 7 8 -7⟩ | 22.41 | Lasepru-aquadbiyo | Blackjackisma |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.43 | Latrizo | Gamelisma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Septimal Kleisma, Marvel Comma |
| 7 | 16875/16807 | [0 3 4 -5⟩ | 6.99 | Quinru-aquadyo | Mirkwai |
| 7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Bizozogu | Breedsma |
| 7 | (30 digits) | [47 -7 -7 -7⟩ | 0.34 | Trisa-seprugu | Akjaysma, 5\7 Octave Comma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 11 | 4000/3993 | [5 -1 3 0 -3⟩ | 3.03 | Triluyo | Wizardharry |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Rank-2 temperaments
List of 21edo rank two temperaments by badness
| Periods per octave | Generator | Temperaments |
|---|---|---|
| 1 | 1\21 | Escapade |
| 1 | 2\21 | Miracle |
| 1 | 4\21 | Slendric/Gorgo/Gidorah |
| 1 | 5\21 | Subklei |
| 1 | 8\21 | Tridec |
| 1 | 10\21 | Triton |
| 3 | 1\21 | |
| 3 | 2\21 | Augmented/August |
| 3 | 3\21 | Oodako |
| 7 | 1\21 | Whitewood |
Approaches
Books / Literature
- Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
Music
- Tostarena: Ruins (21edo cover), from XA Discord's Xen Cover Project 2 (score)
- Mirage Haze, from XA Discord's Deleted User EP (score)
- The Angels' Library in the Sarnathian (23233233) mode of 21edo 5L 3s (score)
- 21-edo Trio for Organ [dead link]
- 21-penny jingle [dead link]
- Trio Sonata in 21edo for Organ (The Sewing Machine)
- 21edo Chacony, for two Harpsichords
- Iridescent Wenge Fugue (accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018)
- WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate), an album of xenharmonic Christmas covers, many are in 21 EDO