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21edo supports {{w|tertian harmony}} with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The [[5/4]] major third is mapped to 400{{c}}, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to [[neogothic]] chords. In fact, [[6/5]] is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks [[consistency]] to the [[5-odd-limit]].
21edo supports {{w|tertian harmony}} with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The [[5/4]] major third is mapped to 400{{c}}, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to [[neogothic]] chords. In fact, [[6/5]] is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks [[consistency]] to the [[5-odd-limit]].
21edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations derive from and are inherited by [[84edo]], which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from [[63edo]], which is possibly a more sensible way to treat it.


In terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (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 terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (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.
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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|21|columns=11}}
{{Harmonics in equal|21|columns=11}}
{{Harmonics in equal|21|columns=11|start=12}}
{{Harmonics in equal|21|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 21edo (continued)}}


== Intervals ==
== Intervals ==
Inconsistent intervals are in ''italics''.
Note names are given with 7edo as the naturals and arrows (written as ^ and v) for inflections up/down by one edostep. Inconsistent intervals are in ''italics''.
{| class="wikitable right-1 right-2"
 
{| class="wikitable center-1 right-2 right-3"
|-
|-
! Steps
! #
! Name
! Cents
! Cents
! 43-odd-limit ratios*
! Approximate Ratios*
! Additional ratios<br>of 17, 19, and 27**
!Other ratios†
|-
|-
| 0
| 0
| C
| 0.00
| 0.00
| colspan="3" | 1/1
| [[1/1]]
|-
|-
| 1
| 1
| ^C
| 57.14
| 57.14
| 29/28, 30/29, 31/30, 32/31, 33/32
| [[21/20]]
| 28/27, 34/33, 39/38
|21/20
|-
|-
| 2
| 2
| vD
| 114.29
| 114.29
| 15/14, 16/15, 31/29, 33/31, 46/43
| [[14/13]], [[15/14]], [[16/15]]
| 17/16, 29/27
|14/13
|-
|-
| 3
| 3
| D
| 171.43
| 171.43
| 11/10, 32/29, 31/28, 43/39
| ''[[9/8]]'', ''[[13/12]]'', [[35/32]]
| 10/9, 19/17, 34/31
|''9/8, 13/12''
|-
|-
| 4
| 4
| ^D
| 228.57
| 228.57
| 8/7, 33/29
| [[8/7]], ''[[10/9]]''
| 17/15, 31/27, 38/33, 43/38
|
|-
|-
| 5
| 5
| vE
| 285.71
| 285.71
| 13/11, 33/28, 46/39
|''[[6/5]]'', [[7/6]]
| 19/16, 27/23, 32/27, 34/29
|''6/5,'' 7/6
|-
|-
| 6
| 6
| E
| 342.86
| 342.86
| 28/23, 39/32
| [[39/32]], [[128/105]], [[16/13]]
| 11/9, 17/14, 23/19, 38/31
|16/13
|-
|-
| 7
| 7
| ^E
| 400.00
| 400.00
| 29/23, 39/31
| [[5/4]], ''[[9/7]]''
| 19/15, 34/27, 43/34, 54/43
|5/4, ''9/7''
|-
|-
| 8
| 8
| vF
| 457.14
| 457.14
| 13/10, 30/23, 39/30, 43/33, 56/43
| [[13/10]], [[21/16]]
| 38/29
|21/16
|-
|-
| 9
| 9
| F
| 514.29
| 514.29
| 31/23, 39/29, 43/32, 58/43
| [[4/3]]
| 19/14, 23/17
|4/3
|-
|-
| 10
| 10
| ^F
| 571.43
| 571.43
| 32/23, 39/28, 46/33, 43/31, 60/43
| [[7/5]]
| 18/13, 38/27
|7/5
|-
|-
| 11
| 11
| vG
| 628.57
| 628.57
| 23/16, 56/39, 33/23, 43/30, 62/43
| [[10/7]]
| 13/9, 27/19
|10/7
|-
|-
| 12
| 12
| G
| 685.71
| 685.71
| 46/31, 58/39, 43/29, 64/43
| [[3/2]]
| 28/19, 34/23
|3/2
|-
|-
| 13
| 13
| ^G
| 742.86
| 742.86
| 20/13, 23/15, 60/39, 43/28, 66/43
| [[20/13]], [[32/21]]
| 29/19
|32/21
|-
|-
| 14
| 14
| vA
| 800.00
| 800.00
| 46/29, 62/39
| [[8/5]], ''[[14/9]]''
| 30/19, 27/17, 43/27, 68/43
|8/5, ''14/9''
|-
|-
| 15
| 15
| A
| 857.14
| 857.14
| 23/14, 64/39
| [[64/39]], [[105/64]], [[13/8]]
| 18/11, 28/17, 38/23, 31/19
|13/8
|-
|-
| 16
| 16
| ^A
| 914.29
| 914.29
| 22/13, 56/33, 39/23
|''[[5/3]]'', [[12/7]]
| 32/19, 27/16, 46/27, 29/17
|''5/3'', 12/7
|-
|-
| 17
| 17
| vB
| 971.43
| 971.43
| 7/4, 58/33
| [[7/4]], ''[[9/5]]''
| 30/17, 54/31, 33/19, 76/43
|
|-
|-
| 18
| 18
| B
| 1028.57
| 1028.57
| 20/11, 29/16, 56/31, 78/43
| ''[[16/9]]'', ''[[24/13]]'', [[64/35]]
| 9/5, 34/19, 31/17
|''16/9, 24/13''
|-
|-
| 19
| 19
| ^B
| 1085.71
| 1085.71
| 15/8, 28/15, 58/31, 62/33, 43/23
| [[13/7]], [[28/15]], [[15/8]]
| 32/17, 54/29
|13/7
|-
|-
| 20
| 20
| vC
| 1142.86
| 1142.86
| 29/15, 56/29, 31/16, 60/31, 64/33
| [[40/21]]
| 27/14, 33/17, 76/39
|40/21
|-
|-
| 21
| 21
| C
| 1200.00
| 1200.00
| colspan="3" | 2/1
| [[2/1]]
|}
|}
<nowiki/>*In the 2.15.7.33.39.23.29.31.43 subgroup
<nowiki/>*As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament
 
<nowiki>**</nowiki>Odd 27 by direct approximation
 
†All by patent val
 
Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val, so despite 9/8 and 10/9 being listed in the same row, 21edo is ''not'' a [[meantone]] system.


== Notation ==
== Notation ==
 
The following table gives a comparison of some notation systems for 21edo.
{| class="wikitable center-all right-3 right-5"
{| class="wikitable center-all right-3 right-5"
|-
|-
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! [[Cent]]s
! [[Cent]]s
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! [[5L 3s]] octotonic<br>notation
! [[5L 3s]] octatonic<br>notation
! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]
! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]
|+ style="font-size: 105%; white-space: nowrap;" | Notation systems for 21edo
|+
|-
|-
| 0
| 0
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| C (sus) down-four up-five
| C (sus) down-four up-five
|}
|}
== Approximation to JI ==
While 21edo does not approximate most low-limit just intervals well, it approximates a number of harmonics quite accurately. For example, 21edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable, though 19 being flat combined with 17 and 27 being sharp means that [[19/17]] and [[27/19]] are over 20 cents off. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations are also used by [[63edo]] and [[84edo]], which each cover many primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which can be derived from 63edo, and is possibly a more sensible way to treat it.
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | JI approximation of 21edo
|-
! Steps
! Cents
! Approximate ratios*
! Additional ratios<br>of 17, 19, and 27**
|-
| 0
| 0.00
| colspan="2" | [[1/1]]
|-
| 1
| 57.14
| [[29/28]], [[30/29]], [[31/30]], [[32/31]], [[33/32]]
| [[28/27]], [[34/33]], [[39/38]]
|-
| 2
| 114.29
| [[15/14]], [[16/15]], [[31/29]], [[33/31]], [[46/43]]
| [[17/16]], [[29/27]]
|-
| 3
| 171.43
| [[11/10]], [[32/29]], [[31/28]], [[43/39]]
| [[10/9]], [[19/17]], [[34/31]]
|-
| 4
| 228.57
| [[8/7]], [[33/29]]
| [[17/15]], [[31/27]], [[38/33]], [[43/38]]
|-
| 5
| 285.71
| [[13/11]], [[33/28]], [[46/39]]
| [[19/16]], [[27/23]], [[32/27]], [[34/29]]
|-
| 6
| 342.86
| [[28/23]], [[39/32]]
| [[11/9]], [[17/14]], [[23/19]], [[38/31]]
|-
| 7
| 400.00
| [[29/23]], [[39/31]]
| [[19/15]], [[34/27]], [[43/34]], [[54/43]]
|-
| 8
| 457.14
| [[13/10]], [[30/23]], [[39/30]], [[43/33]], [[56/43]]
| [[38/29]]
|-
| 9
| 514.29
| [[31/23]], [[39/29]], [[43/32]], [[58/43]]
| [[19/14]], [[23/17]]
|-
| 10
| 571.43
| [[32/23]], [[39/28]], [[46/33]], [[43/31]], [[60/43]]
| [[18/13]], [[38/27]]
|-
| 11
| 628.57
| [[23/16]], [[56/39]], [[33/23]], [[43/30]], [[62/43]]
| [[13/9]], [[27/19]]
|-
| 12
| 685.71
| [[46/31]], [[58/39]], [[43/29]], [[64/43]]
| [[28/19]], [[34/23]]
|-
| 13
| 742.86
| [[20/13]], [[23/15]], [[60/39]], [[43/28]], [[66/43]]
| [[29/19]]
|-
| 14
| 800.00
| [[46/29]], [[62/39]]
| [[30/19]], [[27/17]], [[43/27]], [[68/43]]
|-
| 15
| 857.14
| [[23/14]], [[64/39]]
| [[18/11]], [[28/17]], [[38/23]], [[31/19]]
|-
| 16
| 914.29
| [[22/13]], [[56/33]], [[39/23]]
| [[32/19]], [[27/16]], [[46/27]], [[29/17]]
|-
| 17
| 971.43
| [[7/4]], [[58/33]]
| [[30/17]], [[54/31]], [[33/19]], [[76/43]]
|-
| 18
| 1028.57
| [[20/11]], [[29/16]], [[56/31]], [[78/43]]
| [[9/5]], [[34/19]], [[31/17]]
|-
| 19
| 1085.71
| [[15/8]], [[28/15]], [[58/31]], [[62/33]], [[43/23]]
| [[32/17]], [[54/29]]
|-
| 20
| 1142.86
| [[29/15]], [[56/29]], [[31/16]], [[60/31]], [[64/33]]
| [[27/14]], [[33/17]], [[76/39]]
|-
| 21
| 1200.00
| colspan="2" | [[2/1]]
|}
<nowiki/>*43-odd-limit ratios of the 2.15.7.33.39.23.29.31.43 subgroup
<nowiki/>**Odd 27 by direct approximation
Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being expressable as 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val.
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 743: Line 845:


The subset 2 5 5 6 3 of 21edo is a good tuning for the [[magnetosphere scale]]{{idio}}.
The subset 2 5 5 6 3 of 21edo is a good tuning for the [[magnetosphere scale]]{{idio}}.
== Instruments ==
[[Lumatone mapping for 21edo|Lumatone mappings for 21edo]] are available.


== Music ==
== Music ==
Line 754: Line 859:
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/yLSIZxJnMh8 ''21edo waltz''] (2025)
* [https://www.youtube.com/shorts/yLSIZxJnMh8 ''21edo waltz''] (2025)
* [https://www.youtube.com/watch?v=1S4C-m_Dcno ''21edo improv''] (2026)
* [https://www.youtube.com/shorts/9a-nJ_Ml9z8 ''21edo groove''] (2026)


; [[Fabrizio Fiale]]
; [[Fabrizio Fiale]]
Line 779: Line 886:
; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [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)''] (2018)
* [https://www.youtube.com/watch?v=n0QA0ZQHPvk ''21edo Chacony, for two Harpsichords'']
* [https://www.youtube.com/watch?v=n0QA0ZQHPvk ''21edo Chacony, for two Harpsichords''] (2019)
* [https://www.youtube.com/watch?v=r0aKutu0gVg ''Twinkle Twinkle Little Star, with Shepard Effect''] (2023)
* [https://www.youtube.com/watch?v=r0aKutu0gVg ''Twinkle Twinkle Little Star, with Shepard Effect''] (2023)
* [https://www.youtube.com/watch?v=XlpAbSdy_sg ''Trio Sonata for Baroque Trio in 21 EDO''] (2026)


; [[Nick, The NRG]]
; [[Nick, The NRG]]
Line 803: Line 911:
; [[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)'']{{dead link}}, an album of xenharmonic Christmas covers, many are in 21 EDO


; [[Randy Wells]]
; [[Randy Wells]]
Line 817: Line 925:
== 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.
== See also ==
* [[Lumatone mapping for 21edo]]


[[Category:21edo| ]] <!-- main article -->
[[Category:21edo| ]] <!-- main article -->
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