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This is a page where I will draft edits before making them on the actual page. This may possibly include drafting a new page to be created. If you have something to add to any of them, or any concerns, please suggest them on the talk page. If a template is set to debug, make sure to remove that setting when editing the target page.
This is a page where I will draft edits before making them on the actual page. This may possibly include drafting a new page to be created. If you have something to add to any of them, or any concerns, please suggest them on the talk page. If a template is set to debug, make sure to remove that setting when editing the target page.


= 21edo =
= Pajara =
== Theory ==
There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.
''Notes: Excellent odd harmonics 7, 15, 23, 29, 31, 33, 39, 43, all derived from 84edo, decent 17, 19, 27''


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.  
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.  


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.
{| class="wikitable center-1 right-2 right-4"
 
|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
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 2.15.7.23.27.29 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 2.9/5.7.11/5.13/5.17/5 subgroup, which is possibly a more sensible way to treat it.
|-
 
! rowspan="2" | #
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.
! colspan="2" | Period 0
 
! colspan="2" | Period 1
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.
 
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}}
 
== Intervals ==
{| class="wikitable center-all right-3 right-5"
|-
|-
! [[Degree]]
! Cents*
! [[Cent]]s
! Approximate ratios
! colspan="3" | [[Ups and downs notation]]
! Cents*
! [[5L 3s]] octotonic<br>notation
! Approximate ratios
! [[Extended-diatonic interval names|Extended-diatonic <br> interval name]]
! Approximate Ratios *1
! Approximate Ratios *2
! Approximate Ratios *3
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1
| '''1/1'''
| unison
| 600.0
| C
| 7/5, 10/7
| C
| Unison
| 1/1
| 1/1
| 1/1
|-
|-
| 1
| 1
| 57.14
| 707.2
| ^1  vv2
| '''3/2'''
| up unison, <br> dud 2nd
| 107.2
| ^C <br> vvD
| 15/14, 16/15, 21/20
| C#
| Subminor 2nd
| 28/27, 30/29
| 35/34, 36/35
| 64/63
|-
|-
| 2
| 2
| 114.29
| 214.4
| ^^1 <br> v2
| '''8/7''', '''9/8'''
| dup unison, <br> down 2nd
| 814.4
| ^^C <br> vD
| '''8/5'''
| Db
| Minor 2nd
| 16/15, 15/14, 29/27
| 18/17
| 16/15, 25/24
|-
|-
| 3
| 3
| 171.43
| 921.5
| 2
| 12/7
| 2nd
| 321.5
| D
| 6/5
| D
| Submajor 2nd
| 10/9, 32/29
| 10/9,11/10
| 9/8, 11/10
|-
|-
| 4
| 4
| 228.57
| 428.7
| ^2 <br> vv3
| 9/7, 14/11
| up 2nd, <br> dud 3rd
| 1028.7
| ^D <br> vvE
| 9/5, 20/11
| D#
| Supermajor 2nd
| 8/7
| 8/7
| 8/7, 10/9
|-
|-
| 5
| 5
| 285.71
| 1135.9
| ^^2 <br> v3
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| dup 2nd, <br> down 3rd
| 535.9
| ^^D <br> vE
| 15/11, 27/20
| Eb
| Subminor 3rd
| 27/23, 32/27
| 13/11, 20/17
| 6/5, 7/6
|-
|-
| 6
| 6
| 342.86
| 643.1
| 3
| '''16/11'''
| 3rd
| 43.1
| E
| 45/44, 56/55, 81/80
| E
|}
| Neutral 3rd
 
| 28/23
{| class="wikitable center-1 right-2 right-4"
| 11/9
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
| 16/13
|-
|-
| 7
! rowspan="2" | #
| 400.00
! colspan="2" | Period 0
| ^3 <br> vv4
! colspan="2" | Period 1
| up 3rd, <br> dud 4th
| ^E <br> vvF
| E#/Fb
| Major 3rd
| 29/23
| 44/35
| 5/4, 9/7, 11/9, 14/11
|-
|-
| 8
! Cents*
| 457.14
! Approximate ratios
| ^^3 <br> v4
! Cents*
| dup 3rd, <br> down 4th
! Approximate ratios
| ^^E <br> vF
| F
| Third-fourth ([[naiadic]])
| 30/23
| 13/10, 17/13, 22/17
| 13/10
|-
|-
| 9
| 0
| 514.29
| 0.0
| 4
| '''1/1'''
| 4th
| 600.0
| F
| 7/5, 10/7
| F#
| Acute 4th
| 161/120, 256/189
| 35/26
| 4/3, 18/13
|-
|-
| 10
| 1
| 571.43
| 709.6
| ^4 <br> vv5
| '''3/2'''
| up 4th, <br> dud 5th
| 109.6
| ^F <br> vvG
| 15/14, 16/15, 21/20
| Gb
| Narrow tritone
| 32/23
| 18/13
| 7/5, 11/8
|-
|-
| 11
| 2
| 628.57
| 219.1
| ^^4 <br> v5
| '''8/7''', '''9/8'''
| dup 4th, <br> down 5th
| 819.1
| ^^F <br> vG
| '''8/5'''
| G
| Wide tritone
| 23/16
| 13/9
| 10/7, 16/11
|-
|-
| 12
| 3
| 685.71
| 928.7
| 5
| 12/7
| 5th
| 328.7
| G
| 6/5, 11/9
| G#
| Grave 5th
| 189/128, 240/161
| 52/35
| 3/2, 13/9
|-
|-
| 13
| 4
| 742.86
| 438.2
| ^5 <br> vv6
| 9/7
| up 5th, <br> dud 6th
| 1038.2
| ^G <br> vvA
| 9/5, 11/6
| Hb
| Fifth-sixth ([[cocytic]])
| 23/15
| 17/11, 20/13, 26/17
| 20/13
|-
|-
| 14
| 5
| 800.00
| 1147.8
| ^^5 <br> v6
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| dup 5th, <br> down 6th
| 547.8
| ^^G <br> vA
| '''11/8''', 27/20
| H
| Minor 6th
| 46/29
| 35/22
| 8/5, 11/7, 14/9, 18/11
|-
|-
| 15
| 857.14
| 6
| 6
| 6th
| 657.3
| A
| 22/15
| H#/Ab
| 57.3
| Neutral 6th
| 22/21, 33/32, 81/80
| 23/14
| 18/11
| 13/8
|-
| 16
| 914.29
| ^6 <br> vv7
| up 6th, <br> dud 7th
| ^A <br> vvB
| A
| Supermajor 6th
| 27/16, 46/27
| 17/10, 22/13
| 5/3, 12/7
|-
| 17
| 971.43
| ^^6 <br> v7
| dup 6th, <br> down 7th
| ^^A <br> vB
| A#
| Subminor 7th
| 7/4
| 7/4
| 7/4, 9/5
|-
| 18
| 1028.57
| 7
| 7th
| B
| Bb
| Supraminor 7th
| 29/16, 9/5
| 9/5, 20/11
| 16/9, 20/11
|-
| 19
| 1085.71
| ^7 <br> vv8
| up 7th, <br> dud 8ve
| ^B <br> vvC
| B
| Major 7th
| 15/8
| 17/9
| 15/8, 48/25
|-
| 20
| 1142.86
| ^^7 <br> v8
| dup 7th, <br> down 8ve
| ^^B <br> vC
| B#/Cb
| Supermajor 7th
| 27/14, 29/15
| 35/18, 68/35
| 63/32
|-
| 21
| 1200.00
| 8
| 8ve
| C
| C
| Octave
| 2/1
| 2/1
| 2/1
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced


&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
= Main page =
= Main page =
== Welcome to the Xenharmonic Wiki! ==
== Welcome to the Xenharmonic Wiki! ==
The Xenharmonic Wiki is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.
The [[Xenharmonic Wiki]] is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.


For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
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* [[Just intonation]] &ndash; Tuning based on [[interval]]s with {{W|rational number}} [[frequency ratio]]s
* [[Just intonation]] &ndash; Tuning based on [[interval]]s with {{W|rational number}} [[frequency ratio]]s
* [[EDO|Equal divisions of the octave]] and other [[equal-step tuning]]s
* [[EDO|Equal divisions of the octave]] and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]] &ndash; Scales with at most two distinct sizes (e.g. {{w|major and minor}}) for each interval class, among [[MOS scale#Equivalent definitions and generalizations|many other things]]
* [[MOS scale|Moment of Symmetry (MOS) scales]] &ndash; Scales with at most two distinct sizes (e.g. {{w|major and minor}}) for each interval class, [[MOS scale#Equivalent definitions and generalizations|among many other things]]
* [[Regular temperaments]] &ndash; Tuning systems that appear the same everywhere, excellent for free modulation; [[equal temperament]]s are a basic example
* [[Regular temperaments]] &ndash; Tuning systems that appear the same everywhere, excellent for free modulation; [[equal temperament]]s are a basic example
* [[Historical temperaments]], such as [[Pythagorean tuning]], [[meantone]] temperaments, and [[well temperament]]s
* [[Historical temperaments]] &ndash; such as [[Pythagorean tuning]], [[meantone]] temperaments, and [[well temperament]]s


== Practical xenharmonics ==
== Practical xenharmonics ==
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