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| C (sus) down-four up-five
| C (sus) down-four up-five
|}
|}
== Scales ==
=== MOS scales ===
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)|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 [[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"
|-
! Periods per octave
! Generator
! MOSes
|-
| 1
| 2\21
| [[1L 9s]] <br> [[10L 1s]]
|-
| 1
| 4\21
| [[5L 1s]]<br/>[[5L 6s]]
|-
| 1
| 5\21
| [[4L 1s]]<br/> [[4L 5s]]<br/> [[4L 9s]]
|-
| 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]]
|}
=== Rank-3 scales ===
The rank-3 scale [[diasem]] (3 2 3 1 3 2 3 1 3 or 3 1 3 2 3 1 3 2 3 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{c}} and 228{{c}}), two "minor thirds" (286{{c}} and 343{{c}}) and two "fourths" (457{{c}} and 514{{c}}). 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:
{| class="wikitable center-1 center-2"
|-
! [[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 &amp; 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 &amp; 6
|-
| 6, 2, 1
| (0)-343-457-(514)
| C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F
| Strong enharmonic
| C dup, trup 2 &amp; 6
|-
| 7, 1, 1
| (0)-400-457-(514)
| C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F
| Pythagorean enharmonic
| C dup, quadruple-up 2 &amp; 6
|}
&lowast;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 ([[Pelog21]]) sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.
Some modified versions of that Pelog-like scale, which vaguely resemble Japanese scales, include:
* 4 1 7 2 7
* 4 1 7 3 6
They sound best with with metallic and/or percussive timbres, such as the aperiodic timbres in [[Scale Workshop]].
The subset 2 5 5 6 3 of 21edo is a good tuning for the [[magnetosphere scale]]{{idio}}.


== Regular temperament properties ==
== Regular temperament properties ==
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|}
|}


== Books / literature ==
== Scales ==
* Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
=== MOS scales ===
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)|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 [[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"
|-
! Periods per octave
! Generator
! MOSes
|-
| 1
| 2\21
| [[1L 9s]] <br> [[10L 1s]]
|-
| 1
| 4\21
| [[5L 1s]]<br/>[[5L 6s]]
|-
| 1
| 5\21
| [[4L 1s]]<br/> [[4L 5s]]<br/> [[4L 9s]]
|-
| 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]]
|}
 
=== Rank-3 scales ===
The rank-3 scale [[diasem]] (3 2 3 1 3 2 3 1 3 or 3 1 3 2 3 1 3 2 3 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{c}} and 228{{c}}), two "minor thirds" (286{{c}} and 343{{c}}) and two "fourths" (457{{c}} and 514{{c}}). 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:
 
{| class="wikitable center-1 center-2"
|-
! [[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 &amp; 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 &amp; 6
|-
| 6, 2, 1
| (0)-343-457-(514)
| C ^<span style="font-size: 90%; vertical-align: super;">3</span>D ^^E F
| Strong enharmonic
| C dup, trup 2 &amp; 6
|-
| 7, 1, 1
| (0)-400-457-(514)
| C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F
| Pythagorean enharmonic
| C dup, quadruple-up 2 &amp; 6
|}
&lowast;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 ([[Pelog21]]) sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.
 
Some modified versions of that Pelog-like scale, which vaguely resemble Japanese scales, include:
* 4 1 7 2 7
* 4 1 7 3 6
They sound best with with metallic and/or percussive timbres, such as the aperiodic timbres in [[Scale Workshop]].
 
The subset 2 5 5 6 3 of 21edo is a good tuning for the [[magnetosphere scale]]{{idio}}.


== Music ==
== Music ==
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; [[User:Fitzgerald_Lee|Fitzgerald Lee]]
; [[User:Fitzgerald_Lee|Fitzgerald Lee]]
* [https://youtu.be/Nxn2FJWORIg ''Teetering Rag''] (2025)
* [https://youtu.be/Nxn2FJWORIg ''Teetering Rag''] (2025)
== Books / literature ==
* Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.


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