21edo: Difference between revisions

| de = 21-EDO

| ja = 21平均律

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 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]].

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.

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]].

{{Harmonics in equal|21|columns=11}}

{{Harmonics in equal|21|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 21edo (continued)}}

{| class="wikitable center-1 right-2 right-3"

| 21

<nowiki/>*As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament

The following table gives a comparison of some notation systems for 21edo.

|-

| 114.29

|-

|-

| 228.57

| up 2nd, <br> dud 3rd

| 285.71

| dup 2nd, <br> down 3rd

| 342.86

| up 3rd, <br> dud 4th

| 457.14

| dup 3rd, <br> down 4th

| up 4th, <br> dud 5th

| dup 4th, <br> down 5th

| up 5th, <br> dud 6th

| Fifth-sixth ([[cocytic]])

| 800.00

|-

| 1142.86

| dup 7th, <br> down 8ve

=== 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]].

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

0-6-12-17 = C E G vB = C,v7 = C add 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:

== 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.

|+ style="white-space: nowrap;" | JI approximation of 21edo

! Additional ratios<br>of 17, 19, and 27**

| colspan="2" | [[1/1]]

| 1

|-

| 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]]

|-

| 228.57

| [[17/15]], [[31/27]], [[38/33]], [[43/38]]

| 5

| 285.71

| [[13/11]], [[33/28]], [[46/39]]

| 342.86

| [[11/9]], [[17/14]], [[23/19]], [[38/31]]

| 400.00

| [[19/15]], [[34/27]], [[43/34]], [[54/43]]

|-

| [[13/10]], [[30/23]], [[39/30]], [[43/33]], [[56/43]]

|-

| [[31/23]], [[39/29]], [[43/32]], [[58/43]]

| [[19/14]], [[23/17]]

|-

| [[18/13]], [[38/27]]

| [[23/16]], [[56/39]], [[33/23]], [[43/30]], [[62/43]]

| [[13/9]], [[27/19]]

| [[46/31]], [[58/39]], [[43/29]], [[64/43]]

| [[28/19]], [[34/23]]

| [[20/13]], [[23/15]], [[60/39]], [[43/28]], [[66/43]]

| [[29/19]]

|-

|-

|-

|-

| [[7/4]], [[58/33]]

| [[30/17]], [[54/31]], [[33/19]], [[76/43]]

|-

| 1028.57

|-

| 1085.71

| [[15/8]], [[28/15]], [[58/31]], [[62/33]], [[43/23]]

| [[32/17]], [[54/29]]

| 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

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}}

=== Uniform maps ===

{{Uniform map|edo=21}}