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== Theory ==
== Theory ==
{{Harmonics in equal|steps=21|columns=14}}
21edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as an equalized "[[5L 2s|diatonic]]" scale, though non-mos options might also be preferable (such as [[omnidiatonic]]). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the [[chromatic semitone]] is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another pair of accidentals (such as ups and downs) is usually used instead, though they might be "reskinned" as sharps and flats to aid melodic intuition.


21edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as a basic "diatonic" scale, though maximum-variety-3 options might also be preferable (such as omnidiatonic). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the chromatic semitone is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another kind of accidental (such as ups and downs) is usually used instead, though it might be "reskinned" as sharps and flats to aid melodic intuition.  
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 tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.  
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 just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but 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.
Because 21edo is a {{W|Fibonacci sequence|Fibonacci}} edo, it contains an approximation to the [[logarithmic phi]] superfifth, which generates golden MOS scales [[3L 2s]], [[5L 3s]], and [[8L 5s]], with 21edo itself being an equalized version of [[13L 8s]].


In terms of interval regions, 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.
Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].


Because 21edo is a Fibonacci edo, it contains an approximation to the [[logarithmic phi]] superfifth, which generates golden MOS scales 8L 5s, 5L 3s, and 3L 2s.
=== Odd harmonics ===
{{Harmonics in equal|21|columns=11}}
{{Harmonics in equal|21|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 21edo (continued)}}
 
== Intervals ==
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''.


Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].
{| class="wikitable center-1 right-2 right-3"
|-
! #
! Name
! Cents
! Approximate Ratios*
|-
| 0
| C
| 0.00
| [[1/1]]
|-
| 1
| ^C
| 57.14
| [[21/20]]
|-
| 2
| vD
| 114.29
| [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| D
| 171.43
| ''[[9/8]]'', ''[[13/12]]'', [[35/32]]
|-
| 4
| ^D
| 228.57
| [[8/7]], ''[[10/9]]''
|-
| 5
| vE
| 285.71
|''[[6/5]]'', [[7/6]]
|-
| 6
| E
| 342.86
| [[39/32]], [[128/105]], [[16/13]]
|-
| 7
| ^E
| 400.00
| [[5/4]], ''[[9/7]]''
|-
| 8
| vF
| 457.14
| [[13/10]], [[21/16]]
|-
| 9
| F
| 514.29
| [[4/3]]
|-
| 10
| ^F
| 571.43
| [[7/5]]
|-
| 11
| vG
| 628.57
| [[10/7]]
|-
| 12
| G
| 685.71
| [[3/2]]
|-
| 13
| ^G
| 742.86
| [[20/13]], [[32/21]]
|-
| 14
| vA
| 800.00
| [[8/5]], ''[[14/9]]''
|-
| 15
| A
| 857.14
| [[64/39]], [[105/64]], [[13/8]]
|-
| 16
| ^A
| 914.29
|''[[5/3]]'', [[12/7]]
|-
| 17
| vB
| 971.43
| [[7/4]], ''[[9/5]]''
|-
| 18
| B
| 1028.57
| ''[[16/9]]'', ''[[24/13]]'', [[64/35]]
|-
| 19
| ^B
| 1085.71
| [[13/7]], [[28/15]], [[15/8]]
|-
| 20
| vC
| 1142.86
| [[40/21]]
|-
| 21
| C
| 1200.00
| [[2/1]]
|}
<nowiki/>*As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament


== Intervals ==
== 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"
|-
|-
Line 29: Line 152:
! [[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]]
! Approximate Ratios *1
|+
! Approximate Ratios *2
! Approximate Ratios *3
|-
|-
| 0
| 0
Line 42: Line 163:
| C
| C
| Unison
| Unison
| 1/1
| 1/1
| 1/1
|-
|-
| 1
| 1
Line 53: Line 171:
| C#
| C#
| Subminor 2nd
| Subminor 2nd
| 28/27, 30/29
| 35/34, 36/35
| 64/63
|-
|-
| 2
| 2
Line 64: Line 179:
| Db
| Db
| Minor 2nd
| Minor 2nd
| 16/15, 15/14, 29/27
| 18/17
| 16/15, 25/24
|-
|-
| 3
| 3
Line 75: Line 187:
| D
| D
| Submajor 2nd
| Submajor 2nd
| 10/9, 32/29
| 10/9,11/10
| 9/8
|-
|-
| 4
| 4
Line 86: Line 195:
| D#
| D#
| Supermajor 2nd
| Supermajor 2nd
| 8/7
| 8/7
| 8/7, 10/9, 11/10
|-
|-
| 5
| 5
Line 97: Line 203:
| Eb
| Eb
| Subminor 3rd
| Subminor 3rd
| 27/23, 32/27
| 13/11, 20/17
| 6/5, 7/6
|-
|-
| 6
| 6
Line 108: Line 211:
| E
| E
| Neutral 3rd
| Neutral 3rd
| 28/23
| 11/9
| 16/13
|-
|-
| 7
| 7
Line 119: Line 219:
| E#/Fb
| E#/Fb
| Major 3rd
| Major 3rd
| 29/23
| 44/35
| 5/4, 9/7, 11/9, 14/11
|-
|-
| 8
| 8
Line 130: Line 227:
| F
| F
| Third-fourth ([[naiadic]])
| Third-fourth ([[naiadic]])
| 30/23
| 13/10, 17/13, 22/17
| 13/10
|-
|-
| 9
| 9
Line 141: Line 235:
| F#
| F#
| Acute 4th
| Acute 4th
| 161/120, 256/189
| 35/26
| 4/3, 18/13
|-
|-
| 10
| 10
Line 152: Line 243:
| Gb
| Gb
| Narrow tritone
| Narrow tritone
| 32/23
| 18/13
| 7/5, 11/8
|-
|-
| 11
| 11
Line 163: Line 251:
| G
| G
| Wide tritone
| Wide tritone
| 23/16
| 13/9
| 10/7, 16/11
|-
|-
| 12
| 12
Line 174: Line 259:
| G#
| G#
| Grave 5th
| Grave 5th
| 189/128, 240/161
| 52/35
| 3/2, 13/9
|-
|-
| 13
| 13
Line 185: Line 267:
| Hb
| Hb
| Fifth-sixth ([[cocytic]])
| Fifth-sixth ([[cocytic]])
| 23/15
| 17/11, 20/13, 26/17
| 20/13
|-
|-
| 14
| 14
Line 196: Line 275:
| H
| H
| Minor 6th
| Minor 6th
| 46/29
| 35/22
| 8/5, 11/7, 14/9, 18/11
|-
|-
| 15
| 15
Line 207: Line 283:
| H#/Ab
| H#/Ab
| Neutral 6th
| Neutral 6th
| 23/14
| 18/11
| 13/8
|-
|-
| 16
| 16
Line 218: Line 291:
| A
| A
| Supermajor 6th
| Supermajor 6th
| 27/16, 46/27
| 17/10, 22/13
| 5/3, 12/7
|-
|-
| 17
| 17
Line 229: Line 299:
| A#
| A#
| Subminor 7th
| Subminor 7th
| 7/4
| 7/4
| 7/4, 9/5, 20/11
|-
|-
| 18
| 18
Line 240: Line 307:
| Bb
| Bb
| Supraminor 7th
| Supraminor 7th
| 29/16, 9/5
| 9/5, 20/11
| 16/9
|-
|-
| 19
| 19
Line 251: Line 315:
| B
| B
| Major 7th
| Major 7th
| 15/8
| 17/9
| 15/8, 48/25
|-
|-
| 20
| 20
Line 262: Line 323:
| B#/Cb
| B#/Cb
| Supermajor 7th
| Supermajor 7th
| 27/14, 29/15
| 35/18, 68/35
| 63/32
|-
|-
| 21
| 21
Line 273: Line 331:
| C
| C
| Octave
| Octave
| 2/1
| 2/1
| 2/1
|}
|}


&lowast;1: based on treating 21edo as a 2.7.15.23.27.29 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 21edo as 13-limit laconic temperament
== Notation ==
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[16edo#Sagittal notation|16-EDO]], is a subset of the notation for [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
This notation uses the same sagittal sequence as [[16edo#Sagittal notation|16-EDO]], is a subset of the notation for [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].


<imagemap>
{{Sagittal chart|}}
File:21-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 343 0 503 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 343 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:21-EDO_Sagittal.svg]]
</imagemap>


== Chords ==
== Chords ==
Line 368: Line 409:
| 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 599: Line 768:
* Slendric[11]: 1 3 1 3 1 3 1 3 1 3 1 (optimised for no-5s [[17-limit]] harmony, very xenharmonic)
* Slendric[11]: 1 3 1 3 1 3 1 3 1 3 1 (optimised for no-5s [[17-limit]] harmony, very xenharmonic)
* [[Whitewood]][7]: 3 3 3 3 3 3 3 (identical to [[7edo]])
* [[Whitewood]][7]: 3 3 3 3 3 3 3 (identical to [[7edo]])
* [[Whitewood]][14]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1
* Whitewood[14]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1


=== Rank-3 scales ===
=== Rank-3 scales ===
Line 662: Line 831:


=== Other scales ===
=== Other scales ===
Some [[modmos]] of the [[miracle]] temperament are available in 21edo:
* Modmos of miracle[8]: 2 5 2 3 3 1 3 2
* Modmos of miracle[11]: 2 3 1 1 2 3 2 1 1 3 2
The subset 2 3 7 2 7 of 21edo ([[Pelog21]]) sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.
The subset 2 3 7 2 7 of 21edo ([[Pelog21]]) sounds similar to the ''Pelog lima'' mode of the [[Pelog]] scale.


Line 668: Line 841:
* 4 1 7 3 6
* 4 1 7 3 6
They sound best with with metallic and/or percussive timbres, such as the aperiodic timbres in [[Scale Workshop]].
They sound best with with metallic and/or percussive timbres, such as the aperiodic timbres in [[Scale Workshop]].
The subset 4 5 3 5 4 of 21edo is a kooky pseudo-[[equipentatonic]] scale.


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 681: 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 706: 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 730: 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 744: 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 -->
Retrieved from "https://en.xen.wiki/w/21edo"