21edo: Difference between revisions

| de = 21-EDO

{{Infobox ET}}

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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]].

 

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]].

Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].

=== 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)}}

! [[Degree]]

! [[Cent]]s

! colspan="3" | [[Ups and downs notation]]

! [[5L 3s]] octatonic<br>notation

! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]

| Third-fourth ([[naiadic]])

| Narrow tritone

| Wide tritone

| Fifth-sixth ([[cocytic]])

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]].

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).  

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:

== 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"

! Steps

! Cents

! Approximate ratios*

| 0

| colspan="2" | [[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]]

| 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]]

| [[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]]

| 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]]

21et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 21 33 49 59 73 78 }}.)

| Whtiewood comma, apotome, Pythagorean chroma

| Augmented comma, diesis

| [[Ampersand comma]]

| [[Escapade comma]]

| Mint comma, septimal quartertone

| Marvel comma, septimal kleisma

| Mirkwai comma

| [[Akjaysma]]

| Wizardharry comma

=== Rank-2 temperaments ===

| [[Escapade]]

 

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.

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"

 

* Slendric[11]: 1 3 1 3 1 3 1 3 1 3 1 (optimised for no-5s [[17-limit]] harmony, very xenharmonic)

 

 

 

| C ^<span style="font-size: 90%; vertical-align: super;">4</span>D ^^E F

 

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.

 

Some [[modmos]] of the [[miracle]] temperament are available in 21edo:

 

 

 

 

 

[[Lumatone mapping for 21edo|Lumatone mappings for 21edo]] are available.

; [[User:Fitzgerald_Lee|Fitzgerald Lee]]

* [https://youtu.be/Nxn2FJWORIg ''Teetering Rag''] (2025)

== Books / literature ==

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