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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="display: block; text-align: right;">[[:de:22edo Deutsch]] - [[22平均律|日本語]]</span>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
__FORCETOC__
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-24 15:17:23 UTC</tt>.<br>
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: The original revision id was <tt>625328217</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/22edo|Deutsch]] - [[22平均律|日本語]]
&lt;/span&gt;
[[toc]]
----
=Theory=


In music, //22 equal temperament//, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the [[octave]] into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 [[cent]]s. Because it distinguishes 10/9 and 9/8, it's good for 5-limit.
=Theory=
 
In music, ''22 equal temperament'', called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the [[Octave|octave]] into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 [[cent|cent]]s. Because it distinguishes 10/9 and 9/8, it's good for 5-limit.


The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo|19 equal temperament]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo|19 equal temperament]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.


The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak]]. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the [[7-limit|7-]] and [[11-limit]]s to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit[[consistent| consistent]]ly. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular Temperaments#meantone|meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.
The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the [[5-limit|5-limit]] to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak]]. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the [[7-limit|7-]] and [[11-limit|11-limit]]s to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit[[consistent| consistent]]ly. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular_Temperaments#meantone|meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.


22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.
22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.
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22-et is very close to an extended "quarter-comma superpyth", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64:63 instead of the syntonic comma 81:80. Because of this it has nearly pure septimal major thirds (9:7).
22-et is very close to an extended "quarter-comma superpyth", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64:63 instead of the syntonic comma 81:80. Because of this it has nearly pure septimal major thirds (9:7).


==Intervalic Naming Systems==  
==Intervalic Naming Systems==
The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.
The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.


==Intervals by degree (Superpyth/Porcupine)==  
==Intervals by degree (Superpyth/Porcupine)==
|| Degree || Name and Abbreviation || Cents ||= Approximate
 
Ratios* ||
{| class="wikitable"
|| 0 || Natural Unison, 1 || 0 ||= 1/1 ||
|-
|| 1 || s-minor second, sm2 || 54.55 ||= 33/32, 34/33, 32/31 ||
| | Degree
|| 2 || p-diminished second, pd2 || 109.09 ||= 18/17, 17/16, 16/15, 15/14 ||
| | Name and Abbreviation
|| 3 || p-minor second, pm2 || 163.64 ||= 11/10, 10/9, 32/29 ||
| | Cents
|| 4 || (s/p) Major second, M2 || 218.18 ||= 9/8, 8/7, 17/15 ||
| style="text-align:center;" | Approximate
|| 5 || s-minor third, sm3 || 272.73 ||= [[7_6|7/6]], [[20_17|20/17]] ||
 
|| 6 || p-minor third, pm3 || 327.27 ||= 6/5, 17/14, 11/9, 29/24 ||
Ratios*
|| 7 || p-Major third, pM3 || 381.82 ||= 5/4 ||
|-
|| 8 || s-Major third, sM3 || 436.36 ||= 9/7, 14/11, 22/17 ||
| | 0
|| 9 || Natural Fourth, 4, N4 || 490.91 ||= 4/3 ||
| | Natural Unison, 1
|| 10 || p-Major Fourth, pM4
| | 0
s-dim fifth || 545.45 ||= 11/8, 15/11 ||
| style="text-align:center;" | 1/1
|| 11 || Augmented Fourth, A4,
|-
Half-Octave, HO || 600 ||= 7/5, 10/7, 17/12, 24/17 ||
| | 1
|| 12 || p-minor Fifth, pm5
| | s-minor second, sm2
s-aug fourth || 654.55 ||= 16/11, 22/15 ||
| | 54.55
|| 13 || Natural Fifth, 5, N5 || 709.09 ||= 3/2 ||
| style="text-align:center;" | 33/32, 34/33, 32/31
|| 14 || s-minor sixth, sm6 || 763.64 ||= 11/7, 14/9, 17/11 ||
|-
|| 15 || p-minor sixth, pm6 || 818.18 ||= 8/5 ||
| | 2
|| 16 || p-Major sixth, pM6 || 872.73 ||= 5/3, 18/11, 28/17 ||
| | p-diminished second, pd2
|| 17 || s-Major sixth, sM6 || 927.27 ||= [[12_7|12/7]], [[17_10|17/10]] ||
| | 109.09
|| 18 || (s/p) minor seventh, m7 || 981.82 ||= 7/4, 16/9, 30/17 ||
| style="text-align:center;" | 18/17, 17/16, 16/15, 15/14
|| 19 || p-Major seventh, pM7 || 1036.36 ||= 20/11, 9/5, 29/16 ||
|-
|| 20 || p-Augmented Seventh || 1090.91 ||= 15/8, 32/17, 17/9, 28/15 ||
| | 3
|| 21 || s-Major Seventh, sM7 || 1145.45 ||= 33/17, 64/33, 31/16 ||
| | p-minor second, pm2
|| 22 || Octave, 8 || 1200 ||= 2/1 ||
| | 163.64
| style="text-align:center;" | 11/10, 10/9, 32/29
|-
| | 4
| | (s/p) Major second, M2
| | 218.18
| style="text-align:center;" | 9/8, 8/7, 17/15
|-
| | 5
| | s-minor third, sm3
| | 272.73
| style="text-align:center;" | [[7/6|7/6]], [[20/17|20/17]]
|-
| | 6
| | p-minor third, pm3
| | 327.27
| style="text-align:center;" | 6/5, 17/14, 11/9, 29/24
|-
| | 7
| | p-Major third, pM3
| | 381.82
| style="text-align:center;" | 5/4
|-
| | 8
| | s-Major third, sM3
| | 436.36
| style="text-align:center;" | 9/7, 14/11, 22/17
|-
| | 9
| | Natural Fourth, 4, N4
| | 490.91
| style="text-align:center;" | 4/3
|-
| | 10
| | p-Major Fourth, pM4
 
s-dim fifth
| | 545.45
| style="text-align:center;" | 11/8, 15/11
|-
| | 11
| | Augmented Fourth, A4,


22edo intervals can also be notated using [[Ups and Downs Notation|ups and downs]]. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D.
Half-Octave, HO
| | 600
| style="text-align:center;" | 7/5, 10/7, 17/12, 24/17
|-
| | 12
| | p-minor Fifth, pm5
 
s-aug fourth
| | 654.55
| style="text-align:center;" | 16/11, 22/15
|-
| | 13
| | Natural Fifth, 5, N5
| | 709.09
| style="text-align:center;" | 3/2
|-
| | 14
| | s-minor sixth, sm6
| | 763.64
| style="text-align:center;" | 11/7, 14/9, 17/11
|-
| | 15
| | p-minor sixth, pm6
| | 818.18
| style="text-align:center;" | 8/5
|-
| | 16
| | p-Major sixth, pM6
| | 872.73
| style="text-align:center;" | 5/3, 18/11, 28/17
|-
| | 17
| | s-Major sixth, sM6
| | 927.27
| style="text-align:center;" | [[12/7|12/7]], [[17/10|17/10]]
|-
| | 18
| | (s/p) minor seventh, m7
| | 981.82
| style="text-align:center;" | 7/4, 16/9, 30/17
|-
| | 19
| | p-Major seventh, pM7
| | 1036.36
| style="text-align:center;" | 20/11, 9/5, 29/16
|-
| | 20
| | p-Augmented Seventh
| | 1090.91
| style="text-align:center;" | 15/8, 32/17, 17/9, 28/15
|-
| | 21
| | s-Major Seventh, sM7
| | 1145.45
| style="text-align:center;" | 33/17, 64/33, 31/16
|-
| | 22
| | Octave, 8
| | 1200
| style="text-align:center;" | 2/1
|}
 
22edo intervals can also be notated using [[Ups_and_Downs_Notation|ups and downs]]. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D.


Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.
Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.
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Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.
Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.


==Intervals by degree (Ups and Downs, Porcupine and Pentatonic)==  
==Intervals by degree (Ups and Downs, Porcupine and Pentatonic)==
||~ [[Degree]] ||~ Size ([[cent|Cents]]) ||||||~ Ups and downs ||||||~ Porcupine ||||||~ Pentatonic ||
||= 0 ||= 0 ||= perfect unison ||= P1 ||= D ||= perfect unison ||= P1 ||= D ||= perfect unison ||= P1 ||= D ||
||= 1 ||= 55 ||= minor 2nd ||= m2 ||= Eb ||= aug unison ||= A1 ||= D# ||= aug unison ||= A1 ||= D# ||
||= 2 ||= 109 ||= upminor 2nd ||= ^m2 ||= Eb^ ||= dim 2nd ||= d2 ||= Eb ||= double-aug unison,
double-dim sub3rd ||= AA1,
dds3 ||= Dx,
Fb&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ||
||= 3 ||= 164 ||= downmajor 2nd ||= vM2 ||= Ev ||= perfect 2nd ||= P2 ||= E ||= dim sub3rd ||= ds3 ||= Fbb ||
||= 4 ||= 218 ||= major 2nd ||= M2 ||= E ||= aug 2nd ||= A2 ||= E# ||= minor sub3rd ||= ms3 ||= Fb ||
||= 5 ||= 273 ||= minor 3rd ||= m3 ||= F ||= dim 3rd ||= d3 ||= Fb ||= major sub3rd ||= Ms3 ||= F ||
||= 6 ||= 327 ||= upminor 3rd ||= ^m3 ||= F^ ||= minor 3rd ||= m3 ||= F ||= aug sub3rd ||= As3 ||= F# ||
||= 7 ||= 382 ||= downmajor 3rd ||= vM3 ||= F#v ||= major 3rd ||= M3 ||= F# ||= double-aug sub3rd,
double-dim 4thoid ||= AAs3,
dd4d ||= Fx,
Gbb ||
||= 8 ||= 436 ||= major 3rd ||= M3 ||= F ||= aug 3rd, dim 4th ||= A3, d4 ||= Fx, Gb ||= dim 4thoid ||= d4d ||= Gb ||
||= 9 ||= 491 ||= perfect fourth ||= P4 ||= G ||= minor 4th ||= m4 ||= G ||= perfect 4thoid ||= P4d ||= G ||
||= 10 ||= 545 ||= up-4th, dim 5th ||= ^4, d5 ||= G^, Ab ||= major 4th ||= M4 ||= G# ||= aug 4thoid ||= A4d ||= G# ||
||= 11 ||= 600 ||= downaug 4th,
updim 5th ||= vA4, ^d5 ||= G#v,
Ab^ ||= aug 4th,
dim 5th ||= A4, d5 ||= Gx,
Abb ||= double-aug 4thoid,
double-dim 5thoid ||= AA4d,
dd5d ||= Gx,
Abb ||
||= 12 ||= 655 ||= aug 4th, down-5th ||= A4, v5 ||= G#, Av ||= minor 5th ||= m5 ||= Ab ||= dim 5thoid ||= d5d ||= Ab ||
||= 13 ||= 709 ||= perfect 5th ||= P5 ||= A ||= major 5th ||= M5 ||= A ||= perfect 5thoid ||= P5d ||= A ||
||= 14 ||= 764 ||= minor 6th ||= m6 ||= Bb ||= aug 5th, dim 6th ||= A5, d6 ||= A#, Bbb ||= aug 5thoid ||= A5d ||= A# ||
||= 15 ||= 818 ||= upminor 6th ||= ^m6 ||= Bb^ ||= minor 6th ||= m6 ||= Bb ||= double-aug 5thoid,
double-dim sub7th ||= AA5d,
dds7 ||= Ax,
Cb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||
||= 16 ||= 873 ||= downmajor 6th ||= vM6 ||= Bv ||= major 6th ||= M6 ||= B ||= dim sub7th ||= ds7 ||= Cbb ||
||= 17 ||= 927 ||= major 6th ||= M6 ||= B ||= aug 6th ||= A6 ||= B# ||= minor sub7th ||= ms7 ||= Cb ||
||= 18 ||= 982 ||= minor 7th ||= m7 ||= C ||= dim 7th ||= d7 ||= Cb ||= major sub7th ||= Ms7 ||= C ||
||= 19 ||= 1036 ||= upminor 7th ||= ^m7 ||= C^ ||= perfect 7th ||= P7 ||= C ||= aug sub7th ||= As7 ||= C# ||
||= 20 ||= 1091 ||= downmajor 7th ||= vM7 ||= C#v ||= aug 7th ||= A7 ||= C# ||= double-aug sub7th,
double-dim octave ||= AAs7,
dd8 ||= Cx,
Dbb ||
||= 21 ||= 1145 ||= major 7th ||= M7 ||= C# ||= dim 8ve ||= d8 ||= Db ||= dim octave ||= d8 ||= Db ||
||= 22 ||= 1200 ||= perfect octave ||= P8 ||= D ||= perfect octave ||= P8 ||= D ||= perfect octave ||= P8 ||= D ||


Combining ups and downs notation with [[Kite's color notation|color notation]], qualities can be loosely associated with colors:
{| class="wikitable"
||~ quality ||~ color ||~ monzo format ||~ examples ||
|-
||= minor ||= blue ||= {a, b, 0, 1} ||= 7/6, 7/4 ||
! | [[Degree|Degree]]
||= " ||= fourthward white ||= {a, b}, b &lt; -1 ||= 32/27, 16/9 ||
! | Size ([[cent|Cents]])
||= upminor ||= green ||= {a, b, -1} ||= 6/5, 9/5 ||
! colspan="3" | Ups and downs
||= downmajor ||= yellow ||= {a, b, 1} ||= 5/4, 5/3 ||
! colspan="3" | Porcupine
||= major ||= fifthward white ||= {a, b}, b &gt; 1 ||= 9/8, 27/16 ||
! colspan="3" | Pentatonic
||= " ||= red ||= {a, b, 0, -1} ||= 9/7, 12/7 ||
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | D
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | D
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | D
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 55
| style="text-align:center;" | minor 2nd
| style="text-align:center;" | m2
| style="text-align:center;" | Eb
| style="text-align:center;" | aug unison
| style="text-align:center;" | A1
| style="text-align:center;" | D#
| style="text-align:center;" | aug unison
| style="text-align:center;" | A1
| style="text-align:center;" | D#
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 109
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | ^m2
| style="text-align:center;" | Eb^
| style="text-align:center;" | dim 2nd
| style="text-align:center;" | d2
| style="text-align:center;" | Eb
| style="text-align:center;" | double-aug unison,


=Chord Names=  
double-dim sub3rd
| style="text-align:center;" | AA1,
 
dds3
| style="text-align:center;" | Dx,
 
Fb<span style="vertical-align: super;">3 </span>
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 164
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | vM2
| style="text-align:center;" | Ev
| style="text-align:center;" | perfect 2nd
| style="text-align:center;" | P2
| style="text-align:center;" | E
| style="text-align:center;" | dim sub3rd
| style="text-align:center;" | ds3
| style="text-align:center;" | Fbb
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 218
| style="text-align:center;" | major 2nd
| style="text-align:center;" | M2
| style="text-align:center;" | E
| style="text-align:center;" | aug 2nd
| style="text-align:center;" | A2
| style="text-align:center;" | E#
| style="text-align:center;" | minor sub3rd
| style="text-align:center;" | ms3
| style="text-align:center;" | Fb
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 273
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | m3
| style="text-align:center;" | F
| style="text-align:center;" | dim 3rd
| style="text-align:center;" | d3
| style="text-align:center;" | Fb
| style="text-align:center;" | major sub3rd
| style="text-align:center;" | Ms3
| style="text-align:center;" | F
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 327
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | ^m3
| style="text-align:center;" | F^
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | m3
| style="text-align:center;" | F
| style="text-align:center;" | aug sub3rd
| style="text-align:center;" | As3
| style="text-align:center;" | F#
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 382
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | vM3
| style="text-align:center;" | F#v
| style="text-align:center;" | major 3rd
| style="text-align:center;" | M3
| style="text-align:center;" | F#
| style="text-align:center;" | double-aug sub3rd,
 
double-dim 4thoid
| style="text-align:center;" | AAs3,
 
dd4d
| style="text-align:center;" | Fx,
 
Gbb
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 436
| style="text-align:center;" | major 3rd
| style="text-align:center;" | M3
| style="text-align:center;" | F
| style="text-align:center;" | aug 3rd, dim 4th
| style="text-align:center;" | A3, d4
| style="text-align:center;" | Fx, Gb
| style="text-align:center;" | dim 4thoid
| style="text-align:center;" | d4d
| style="text-align:center;" | Gb
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 491
| style="text-align:center;" | perfect fourth
| style="text-align:center;" | P4
| style="text-align:center;" | G
| style="text-align:center;" | minor 4th
| style="text-align:center;" | m4
| style="text-align:center;" | G
| style="text-align:center;" | perfect 4thoid
| style="text-align:center;" | P4d
| style="text-align:center;" | G
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 545
| style="text-align:center;" | up-4th, dim 5th
| style="text-align:center;" | ^4, d5
| style="text-align:center;" | G^, Ab
| style="text-align:center;" | major 4th
| style="text-align:center;" | M4
| style="text-align:center;" | G#
| style="text-align:center;" | aug 4thoid
| style="text-align:center;" | A4d
| style="text-align:center;" | G#
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 600
| style="text-align:center;" | downaug 4th,
 
updim 5th
| style="text-align:center;" | vA4, ^d5
| style="text-align:center;" | G#v,
 
Ab^
| style="text-align:center;" | aug 4th,
 
dim 5th
| style="text-align:center;" | A4, d5
| style="text-align:center;" | Gx,
 
Abb
| style="text-align:center;" | double-aug 4thoid,
 
double-dim 5thoid
| style="text-align:center;" | AA4d,
 
dd5d
| style="text-align:center;" | Gx,
 
Abb
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 655
| style="text-align:center;" | aug 4th, down-5th
| style="text-align:center;" | A4, v5
| style="text-align:center;" | G#, Av
| style="text-align:center;" | minor 5th
| style="text-align:center;" | m5
| style="text-align:center;" | Ab
| style="text-align:center;" | dim 5thoid
| style="text-align:center;" | d5d
| style="text-align:center;" | Ab
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 709
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | P5
| style="text-align:center;" | A
| style="text-align:center;" | major 5th
| style="text-align:center;" | M5
| style="text-align:center;" | A
| style="text-align:center;" | perfect 5thoid
| style="text-align:center;" | P5d
| style="text-align:center;" | A
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 764
| style="text-align:center;" | minor 6th
| style="text-align:center;" | m6
| style="text-align:center;" | Bb
| style="text-align:center;" | aug 5th, dim 6th
| style="text-align:center;" | A5, d6
| style="text-align:center;" | A#, Bbb
| style="text-align:center;" | aug 5thoid
| style="text-align:center;" | A5d
| style="text-align:center;" | A#
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 818
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | ^m6
| style="text-align:center;" | Bb^
| style="text-align:center;" | minor 6th
| style="text-align:center;" | m6
| style="text-align:center;" | Bb
| style="text-align:center;" | double-aug 5thoid,
 
double-dim sub7th
| style="text-align:center;" | AA5d,
 
dds7
| style="text-align:center;" | Ax,
 
Cb<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 873
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | vM6
| style="text-align:center;" | Bv
| style="text-align:center;" | major 6th
| style="text-align:center;" | M6
| style="text-align:center;" | B
| style="text-align:center;" | dim sub7th
| style="text-align:center;" | ds7
| style="text-align:center;" | Cbb
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 927
| style="text-align:center;" | major 6th
| style="text-align:center;" | M6
| style="text-align:center;" | B
| style="text-align:center;" | aug 6th
| style="text-align:center;" | A6
| style="text-align:center;" | B#
| style="text-align:center;" | minor sub7th
| style="text-align:center;" | ms7
| style="text-align:center;" | Cb
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 982
| style="text-align:center;" | minor 7th
| style="text-align:center;" | m7
| style="text-align:center;" | C
| style="text-align:center;" | dim 7th
| style="text-align:center;" | d7
| style="text-align:center;" | Cb
| style="text-align:center;" | major sub7th
| style="text-align:center;" | Ms7
| style="text-align:center;" | C
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 1036
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | ^m7
| style="text-align:center;" | C^
| style="text-align:center;" | perfect 7th
| style="text-align:center;" | P7
| style="text-align:center;" | C
| style="text-align:center;" | aug sub7th
| style="text-align:center;" | As7
| style="text-align:center;" | C#
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 1091
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | vM7
| style="text-align:center;" | C#v
| style="text-align:center;" | aug 7th
| style="text-align:center;" | A7
| style="text-align:center;" | C#
| style="text-align:center;" | double-aug sub7th,
 
double-dim octave
| style="text-align:center;" | AAs7,
 
dd8
| style="text-align:center;" | Cx,
 
Dbb
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 1145
| style="text-align:center;" | major 7th
| style="text-align:center;" | M7
| style="text-align:center;" | C#
| style="text-align:center;" | dim 8ve
| style="text-align:center;" | d8
| style="text-align:center;" | Db
| style="text-align:center;" | dim octave
| style="text-align:center;" | d8
| style="text-align:center;" | Db
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 1200
| style="text-align:center;" | perfect octave
| style="text-align:center;" | P8
| style="text-align:center;" | D
| style="text-align:center;" | perfect octave
| style="text-align:center;" | P8
| style="text-align:center;" | D
| style="text-align:center;" | perfect octave
| style="text-align:center;" | P8
| style="text-align:center;" | D
|}
 
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
 
{| class="wikitable"
|-
! | quality
! | color
! | monzo format
! | examples
|-
| style="text-align:center;" | minor
| style="text-align:center;" | blue
| style="text-align:center;" | {a, b, 0, 1}
| style="text-align:center;" | 7/6, 7/4
|-
| style="text-align:center;" | "
| style="text-align:center;" | fourthward white
| style="text-align:center;" | {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
|-
| style="text-align:center;" | upminor
| style="text-align:center;" | green
| style="text-align:center;" | {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
|-
| style="text-align:center;" | downmajor
| style="text-align:center;" | yellow
| style="text-align:center;" | {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
|-
| style="text-align:center;" | major
| style="text-align:center;" | fifthward white
| style="text-align:center;" | {a, b}, b &gt; 1
| style="text-align:center;" | 9/8, 27/16
|-
| style="text-align:center;" | "
| style="text-align:center;" | red
| style="text-align:center;" | {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
|}
 
=Chord Names=


All 22edo chords can be named using ups and downs notation. Here are the blue, green, yellow and red triads:
All 22edo chords can be named using ups and downs notation. Here are the blue, green, yellow and red triads:
||~ color of the 3rd ||~ JI chord ||~ notes as edosteps ||~ notes of C chord ||~ written name ||~ spoken name ||
 
||= blue ||= 6:7:9 ||= 0-5-13 ||= C Eb G ||= Cm ||= C minor ||
{| class="wikitable"
||= green ||= 10:12:15 ||= 0-6-13 ||= C Eb^ G ||= C.^m ||= C upminor ||
|-
||= yellow ||= 4:5:6 ||= 0-7-13 ||= C Ev G ||= C.v ||= C downmajor or C dot down ||
! | color of the 3rd
||= red ||= 14:18:27 ||= 0-8-13 ||= C E G ||= C ||= C major or C ||
! | JI chord
! | notes as edosteps
! | notes of C chord
! | written name
! | spoken name
|-
| style="text-align:center;" | blue
| style="text-align:center;" | 6:7:9
| style="text-align:center;" | 0-5-13
| style="text-align:center;" | C Eb G
| style="text-align:center;" | Cm
| style="text-align:center;" | C minor
|-
| style="text-align:center;" | green
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 0-6-13
| style="text-align:center;" | C Eb^ G
| style="text-align:center;" | C.^m
| style="text-align:center;" | C upminor
|-
| style="text-align:center;" | yellow
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 0-7-13
| style="text-align:center;" | C Ev G
| style="text-align:center;" | C.v
| style="text-align:center;" | C downmajor or C dot down
|-
| style="text-align:center;" | red
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 0-8-13
| style="text-align:center;" | C E G
| style="text-align:center;" | C
| style="text-align:center;" | C major or C
|}
For C.v, the period is needed because "Cv", spoken as "C down", is either a note, or a major chord Cv Ev Gv.
For C.v, the period is needed because "Cv", spoken as "C down", is either a note, or a major chord Cv Ev Gv.
The period isn't needed in Cm because there's no ups or downs immediately after the note name.
The period isn't needed in Cm because there's no ups or downs immediately after the note name.


0-8-13-18 = C E G Bb = C7 = "C seven"
0-8-13-18 = C E G Bb = C7 = "C seven"
0-7-13-18 = C Ev G Bb = C7(v3) = "C seven, down third"
0-7-13-18 = C Ev G Bb = C7(v3) = "C seven, down third"
0-8-13-21 = C E G B = CM7 = "C major seven"
0-8-13-21 = C E G B = CM7 = "C major seven"
0-7-13-20 = C Ev G Bv = C.vM7 = "C downmajor seven" (the down symbol applies to both the 3rd and the 7th)
0-7-13-20 = C Ev G Bv = C.vM7 = "C downmajor seven" (the down symbol applies to both the 3rd and the 7th)


0-3-13 = C Dv G = C(v2)
0-3-13 = C Dv G = C(v2)
0-4-13 = C D G = C2
0-4-13 = C D G = C2
0-9-13 = C F G = C4
0-9-13 = C F G = C4
0-10-13 = C F^ G = C(^4)
0-10-13 = C F^ G = C(^4)


0-5-10 = C Eb Gb = Cdim
0-5-10 = C Eb Gb = Cdim
0-5-11 = C Eb Gb^ = Cdim(^5)
0-5-11 = C Eb Gb^ = Cdim(^5)
0-5-12 = C Eb Gv = Cm(v5)
0-5-12 = C Eb Gv = Cm(v5)


0-5-10-15 = C Eb Gb Bbb = Cdim7
0-5-10-15 = C Eb Gb Bbb = Cdim7
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7(^5) (the up symbol applies to both the 3rd and the 7th)
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7(^5) (the up symbol applies to both the 3rd and the 7th)
0-5-13-17 = C Eb G A = Cm6
0-5-13-17 = C Eb G A = Cm6


Sometimes doubled ups/downs are unavoidable:
Sometimes doubled ups/downs are unavoidable:
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)


0-8-13-17 = C E G A = C6
0-8-13-17 = C E G A = C6
0-8-13-16 = C E G Av = C(v6)
0-8-13-16 = C E G Av = C(v6)
0-7-13-17 = C Ev G A = C6(v3)
0-7-13-17 = C Ev G A = C6(v3)
0-7-13-16 = C Ev G Av = C.v6 (the down symbol applies to both the 3rd and the 6th)
0-7-13-16 = C Ev G Av = C.v6 (the down symbol applies to both the 3rd and the 6th)


0-5-13-18 = C Eb G Bb = Cm7
0-5-13-18 = C Eb G Bb = Cm7
0-6-13-19 = C Eb^ G Bb^ = C.^m7
0-6-13-19 = C Eb^ G Bb^ = C.^m7
0-8-13-21 = C E G B = CM7
0-8-13-21 = C E G B = CM7
0-7-13-20 = C Ev G Bv = C.vM7
0-7-13-20 = C Ev G Bv = C.vM7


0-5-13-16 = C Eb G Av = Cm(v6)
0-5-13-16 = C Eb G Av = Cm(v6)
0-8-13-19 = C E G Bb^ = C(^7)
0-8-13-19 = C E G Bb^ = C(^7)
0-7-13-18-26 = C Ev G Bb D = C9(v3)
0-7-13-18-26 = C Ev G Bb D = C9(v3)
0-7-13-18-26-32 = C Ev G Bb D F^ = C9(v3,^11)
0-7-13-18-26-32 = C Ev G Bb D F^ = C9(v3,^11)


For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].


==Selected just intervals by error==  
==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 22edo (ordered by absolute error).
The following table shows how [[Just-24|some prominent just intervals]] are represented in 22edo (ordered by absolute error).
||~ Interval, complement ||~ Error (abs., in [[cent|cents]]) ||
||= [[9_7|9/7]], [[14_9|14/9]] ||= 1.280 ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= 1.368 ||
||= [[16_15|16/15]], [[15_8|15/8]] ||= 2.640 ||
||= [[5_4|5/4]], [[8_5|8/5]] ||= 4.496 ||
||= [[7_6|7/6]], [[12_7|12/7]] ||= 5.856 ||
||= [[11_8|11/8]], [[16_11|16/11]] ||= 5.863 ||
||= [[4_3|4/3]], [[3_2|3/2]] ||= 7.136 ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= 8.504 ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 10.352 ||
||= [[6_5|6/5]], [[5_3|5/3]] ||= 11.631 ||
||= [[8_7|8/7]], [[7_4|7/4]] ||= 12.992 ||
||= [[12_11|12/11]], [[11_6|11/6]] ||= 12.999 ||
||= [[9_8|9/8]], [[16_9|16/9]] ||= 14.272 ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= 16.482 ||
||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= 17.850 ||
||= [[18_13|18/13]], [[13_9|13/9]] ||= 17.928 ||
||= [[10_9|10/9]], [[9_5|9/5]] ||= 18.767 ||
||= [[14_11|14/11]], [[11_7|11/7]] ||= 18.856 ||
||= [[14_13|14/13]], [[13_7|13/7]] ||= 19.207 ||
||= [[11_9|11/9]], [[18_11|18/11]] ||= 20.135 ||
||= [[16_13|16/13]], [[13_8|13/8]] ||= 22.346 ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= 24.986 ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= 25.064 ||


[[media type="custom" key="24838814"]]
{| class="wikitable"
|-
! | Interval, complement
! | Error (abs., in [[cent|cents]])
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 1.280
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 1.368
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 2.640
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 4.496
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 5.856
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 5.863
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 7.136
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 8.504
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 10.352
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 11.631
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 12.992
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 12.999
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 14.272
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 16.482
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 17.850
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 17.928
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 18.767
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 18.856
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 19.207
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 20.135
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 22.346
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 24.986
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 25.064
|}


[[file:22ed2-001e.svg]]
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]


See also: [[22edo Solfege]], [[22edo Tetrachords]], [[22 EDO Chords]], [[22edo Modes]]
[[:File:22ed2-001e.svg|22ed2-001e.svg]]


==Properties of 22 equal temperament==
See also: [[22edo_Solfege|22edo Solfege]], [[22edo_tetrachords|22edo Tetrachords]], [[22_EDO_Chords|22 EDO Chords]], [[22edo_Modes|22edo Modes]]


Possibly the most striking characteristic of 22-et to those not used to it is that it does **not** "temper out" the syntonic comma of 81/80, and therefore is not a system of [[Regular Temperaments#meantone|meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12-EDO, 19-EDO, 31-EDO, ... do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].
==Properties of 22 equal temperament==


The diatonic scale it produces is instead derived from [[superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22-EDO. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12-equal and meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.
Possibly the most striking characteristic of 22-et to those not used to it is that it does '''not''' "temper out" the syntonic comma of 81/80, and therefore is not a system of [[Regular_Temperaments#meantone|meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12-EDO, 19-EDO, 31-EDO, ... do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo|34edo]], [[41edo|41edo]] and [[53edo|53edo]].


It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22-EDO supports [[porcupine]] temperament. The generator for porcupine is is a flat minor whole tone of [[10_9|10/9]], two of which is a slightly sharp [[6_5|6/5]], and three of which is a slightly flat [[4_3|4/3]], implying the existence of an equal-step tetrachord, which is characteristic of Porcupine. Porcupine is notable for being the 5-limit temperament lowest in [[badness]] which is //not// approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22-EDO. It forms [[MOSScales|MOS]]'s of 7 and 8, which in 22-EDO are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).
The diatonic scale it produces is instead derived from [[Superpyth|superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L_2s|5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22-EDO. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12-equal and meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.


The 164¢ "flat minor whole tone" is a key interval in 22-EDO, in part because it functions as no less than three different consonant ratios in the [[11-limit]]: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22-EDO can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22-EDO supports [[Porcupine|porcupine]] temperament. The generator for porcupine is is a flat minor whole tone of [[10/9|10/9]], two of which is a slightly sharp [[6/5|6/5]], and three of which is a slightly flat [[4/3|4/3]], implying the existence of an equal-step tetrachord, which is characteristic of Porcupine. Porcupine is notable for being the 5-limit temperament lowest in [[Badness|badness]] which is ''not'' approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22-EDO. It forms [[MOSScales|MOS]]'s of 7 and 8, which in 22-EDO are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).


22-EDO also supports Orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms MOS scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, Orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]] and [[84edo]]. But 22-equal Orwell has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish in 22.
The 164¢ "flat minor whole tone" is a key interval in 22-EDO, in part because it functions as no less than three different consonant ratios in the [[11-limit|11-limit]]: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22-EDO can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.


Other 5-limit commas 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, the [[diatonic tritone]] [[45_32|45/32]], which is a major third above a [[major whole tone]] representing [[9_8|9/8]], is equated to its inverted form, [[64_45|64/45]]. That the magic comma is tempered out means that 22-et is a [[Regular Temperaments#magic|magic]] system, where five major thirds make up a perfect fifth.
22-EDO also supports Orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms MOS scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, Orwell can be tuned more accurately in other temperaments, such as [[31edo|31edo]], [[53edo|53edo]] and [[84edo|84edo]]. But 22-equal Orwell has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish in 22.


In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50_49|50/49]], (the [[jubilee comma]]), and [[64_63|64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et.
Other 5-limit commas 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, the [[diatonic_tritone|diatonic tritone]] [[45/32|45/32]], which is a major third above a [[major_whole_tone|major whole tone]] representing [[9/8|9/8]], is equated to its inverted form, [[64/45|64/45]]. That the magic comma is tempered out means that 22-et is a [[Regular_Temperaments#magic|magic]] system, where five major thirds make up a perfect fifth.


As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo]], in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal]], 11 can be notated as every other note of 22.
In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49|50/49]], (the [[jubilee_comma|jubilee comma]]), and [[64/63|64/63]], (the [[Septimal_comma|septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal_kleisma|septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell_comma|orwell comma]]; and the [[orwell_tetrad|orwell tetrad]] is also a chord of 22-et.


===Rank Two Temperaments===
As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo|11edo]], in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal|Sagittal]], 11 can be notated as every other note of 22.
[[List of 22et rank two temperaments by badness]]
[[List of 22et rank two temperaments by complexity]]
[[List of edo-distinct 22et rank two temperaments]]
||~ Periods
per octave ||~ Period ||~ Generator ||~ Temperaments ||
|| 1 || 22\22 || 1\22 || [[Sensamagic clan#Sensa|Sensa]]/chromo/ceratitid ||
|| 1 || 22\22 || 3\22 || [[Porcupine]] ||
|| 1 || 22\22 || 5\22 || [[Orson]]/[[orwell]]/blair ||
|| 1 || 22\22 || 7\22 || [[Magic]]/telepathy ||
|| 1 || 22\22 || 9\22 || [[Superpyth]]/[[suprapyth]] ||
|| 2 || 11\22 || 1\22 || [[Shrutar]]/hemipaj/comic ||
|| 2 || 11\22 || 2\22 || [[Srutal]]/[[pajara]]/pajarous ||
|| 2 || 11\22 || 3\22 || [[Porcupine family#Hedgehog|Hedgehog]]/[[echidna]] ||
|| 2 || 11\22 || 4\22 || [[Astrology]]/[[wizard]]/[[antikythera]] ||
|| 2 || 11\22 || 5\22 || [[Doublewide]]/fleetwood ||
|| 11 || 2\22 || 1\22 || [[Hendecatonic]]/undeka ||
===Commas===
22 EDO tempers out the following commas. (Note: This assumes the val &lt; 22 35 51 62 76 81 |.)
||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 250/243 || | 1 -5 3 &gt; ||&gt; 49.17 ||= Maximal Diesis ||= Porcupine Comma ||=  ||
||= 3125/3072 || | -10 -1 5 &gt; ||&gt; 29.61 ||= Small Diesis ||= Magic Comma ||=  ||
||= 2048/2025 || | 11 -4 -2 &gt; ||&gt; 19.55 ||= Diaschisma ||=  ||=  ||
||= 2109375/2097152 || | -21 3 7 &gt; ||&gt; 10.06 ||= Semicomma ||= Fokker Comma ||=  ||
||= 9193891/9143623 || | 32 -7 -9 &gt; ||&gt; 9.49 ||= Escapade Comma ||  ||=  ||
||= 4758837/4757272 || | -53 10 16 &gt; ||&gt; 0.57 ||= Kwazy ||=  ||=  ||
||= 50/49 || | 1 0 2 -2 &gt; ||&gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=  ||
||= 64/63 || | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 875/864 || | -5 -3 3 1 &gt; ||&gt; 21.90 ||= Keema ||=  ||=  ||
||= 2430/2401 || | 1 5 1 -4 &gt; ||&gt; 20.79 ||= Nuwell ||=  ||=  ||
||= 245/243 || | 0 -5 1 2 &gt; ||&gt; 14.19 ||= Sensamagic ||=  ||=  ||
||= 1728/1715 || | 6 3 -1 -3 &gt; ||&gt; 13.07 ||= Orwellisma ||= Orwell Comma ||=  ||
||= 225/224 || | -5 2 2 -1 &gt; ||&gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=  ||
||= 10976/10935 || | 5 -7 -1 3 &gt; ||&gt; 6.48 ||= Hemimage ||=  ||=  ||
||= 6144/6125 || | 11 1 -3 -2 &gt; ||&gt; 5.36 ||= Porwell ||=  ||=  ||
||= 65625/65536 || | -16 1 5 1 &gt; ||&gt; 2.35 ||= Horwell ||=  ||=  ||
||= 420175/419904 || | -6 -8 2 5 &gt; ||&gt; 1.12 ||= Wizma ||=  ||=  ||
||= 99/98 || | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=  ||=  ||
||= 100/99 || | 2 -2 2 0 -1 &gt; ||&gt; 17.40 ||= Ptolemisma ||=  ||=  ||
||= 121/120 || | -3 -1 -1 0 2 &gt; ||&gt; 14.37 ||= Biyatisma ||=  ||=  ||
||= 125/124 || |-4 0 3 0 ... -1&gt; ||&gt; 13.91 ||= Twizzler ||  ||  ||
||= 176/175 || | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=  ||=  ||
||= 896/891 || | 7 -4 0 1 -1 &gt; ||&gt; 9.69 ||= Pentacircle ||=  ||=  ||
||= 65536/65219 || | 16 0 0 -2 -3 &gt; ||&gt; 8.39 ||= Orgonisma ||=  ||=  ||
||= 385/384 || | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= Keenanisma ||=  ||=  ||
||= 540/539 || | 2 3 1 -2 -1 &gt; ||&gt; 3.21 ||= Swetisma ||=  ||=  ||
||= 4000/3993 || &lt;| 5 -1 3 0 -3 &gt; ||&gt; 3.03 ||= Wizardharry ||=  ||=  ||
||= 9801/9800 || | -3 4 -2 -2 2 &gt; ||&gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=  ||
||= 91/90 || | -1 -2 -1 1 0 1 &gt; ||&gt; 19.13 ||= Superleap ||=  ||=  ||


===How to Notate 22edo in Sagittal===  
===Rank Two Temperaments===
[[List_of_22et_rank_two_temperaments_by_badness|List of 22et rank two temperaments by badness]]


When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:
[[List_of_22et_rank_two_temperaments_by_complexity|List of 22et rank two temperaments by complexity]]
[[image:22edo.png]]
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.


The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250_243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome).
[[List_of_edo-distinct_22et_rank_two_temperaments|List of edo-distinct 22et rank two temperaments]]


===How to notate 22edo with ups and downs===
{| class="wikitable"
|-
! | Periods


Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
per octave
[[image:Tibia 22edo ups and downs guide 1.png width="800" height="147"]]
! | Period
! | Generator
! | Temperaments
|-
| | 1
| | 22\22
| | 1\22
| | [[Sensamagic_clan#Sensa|Sensa]]/chromo/ceratitid
|-
| | 1
| | 22\22
| | 3\22
| | [[Porcupine|Porcupine]]
|-
| | 1
| | 22\22
| | 5\22
| | [[Orson|Orson]]/[[Orwell|orwell]]/blair
|-
| | 1
| | 22\22
| | 7\22
| | [[Magic|Magic]]/telepathy
|-
| | 1
| | 22\22
| | 9\22
| | [[Superpyth|Superpyth]]/[[Suprapyth|suprapyth]]
|-
| | 2
| | 11\22
| | 1\22
| | [[Shrutar|Shrutar]]/hemipaj/comic
|-
| | 2
| | 11\22
| | 2\22
| | [[Srutal|Srutal]]/[[pajara|pajara]]/pajarous
|-
| | 2
| | 11\22
| | 3\22
| | [[Porcupine_family#Hedgehog|Hedgehog]]/[[Echidna|echidna]]
|-
| | 2
| | 11\22
| | 4\22
| | [[Astrology|Astrology]]/[[wizard|wizard]]/[[antikythera|antikythera]]
|-
| | 2
| | 11\22
| | 5\22
| | [[Doublewide|Doublewide]]/fleetwood
|-
| | 11
| | 2\22
| | 1\22
| | [[Hendecatonic|Hendecatonic]]/undeka
|}


Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:
===Commas===
[[image:Tibia 22edo ups and downs guide 2.png width="800" height="150"]]
22 EDO tempers out the following commas. (Note: This assumes the val &lt; 22 35 51 62 76 81 |.)


A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.
{| class="wikitable"
[[image:Tibia 22edo guide D major.png width="800" height="68"]]
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 250/243
| | | 1 -5 3 &gt;
| style="text-align:right;" | 49.17
| style="text-align:center;" | Maximal Diesis
| style="text-align:center;" | Porcupine Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 3125/3072
| | | -10 -1 5 &gt;
| style="text-align:right;" | 29.61
| style="text-align:center;" | Small Diesis
| style="text-align:center;" | Magic Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2109375/2097152
| | | -21 3 7 &gt;
| style="text-align:right;" | 10.06
| style="text-align:center;" | Semicomma
| style="text-align:center;" | Fokker Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 9193891/9143623
| | | 32 -7 -9 &gt;
| style="text-align:right;" | 9.49
| style="text-align:center;" | Escapade Comma
| |
| style="text-align:center;" |
|-
| style="text-align:center;" | 4758837/4757272
| | | -53 10 16 &gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Kwazy
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 875/864
| | | -5 -3 3 1 &gt;
| style="text-align:right;" | 21.90
| style="text-align:center;" | Keema
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2430/2401
| | | 1 5 1 -4 &gt;
| style="text-align:right;" | 20.79
| style="text-align:center;" | Nuwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 245/243
| | | 0 -5 1 2 &gt;
| style="text-align:right;" | 14.19
| style="text-align:center;" | Sensamagic
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1728/1715
| | | 6 3 -1 -3 &gt;
| style="text-align:right;" | 13.07
| style="text-align:center;" | Orwellisma
| style="text-align:center;" | Orwell Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 10976/10935
| | | 5 -7 -1 3 &gt;
| style="text-align:right;" | 6.48
| style="text-align:center;" | Hemimage
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 6144/6125
| | | 11 1 -3 -2 &gt;
| style="text-align:right;" | 5.36
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 65625/65536
| | | -16 1 5 1 &gt;
| style="text-align:right;" | 2.35
| style="text-align:center;" | Horwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 420175/419904
| | | -6 -8 2 5 &gt;
| style="text-align:right;" | 1.12
| style="text-align:center;" | Wizma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 125/124
| | |-4 0 3 0 ... -1&gt;
| style="text-align:right;" | 13.91
| style="text-align:center;" | Twizzler
| |
| |
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 4000/3993
| | &lt;| 5 -1 3 0 -3 &gt;
| style="text-align:right;" | 3.03
| style="text-align:center;" | Wizardharry
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| style="text-align:center;" |
|}


Paul Erlich's "Tibia" in G, with independent ups and downs:
===How to Notate 22edo in Sagittal===
[[image:Tibia in G for the book-1.png width="800" height="956"]]
[[image:Tibia in G for the book-2.png width="800" height="889"]]


=The Decatonic System=
When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:
The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern
 
==[[#TOC-Decatonic-Alphabet]]Decatonic Alphabet==
The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.


Chain 1: C G D A E
[[File:22edo.png|alt=22edo.png|22edo.png]]
Chain 2: γ δ α ε β


The alphabet is, in ascending order: C δ D ε E γ G α A β C
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.


In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.
The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250/243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome).


===How to notate 22edo with ups and downs===


==Internal links==
Treating [[Ups_and_Downs_Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
* [[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]


==External links==
[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]
* [[http://lumma.org/tuning/erlich/erlich-decatonic.pdf|Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']]
* [[http://porcupinemusic.weebly.com/|"Porcupine Music" - Website Focused on the Development of 22 EDO music ]]


==References==
Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:
*Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
*Bosanquet, R.H.M. [[http://www.webcitation.org/5kjJcrhEx|''On the Hindoo division of the octave, with additions to the theory of higher orders'']], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965


=Music=
[[File:Tibia_22edo_ups_and_downs_guide_2.png|alt=Tibia 22edo ups and downs guide 2.png|800x150px|Tibia 22edo ups and downs guide 2.png]]
* [[@https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version|Stephen Weigel · Dose Of Familiarity/Ode to Microtonality]]
* [[https://soundcloud.com/metaclown/couples-therapy|Couples' Therapy]] by metaclown
* [[@http://soonlabel.com/xenharmonic/archives/1145|Canon 2 in 1 upon a ground (22edo)]] by Claudi Meneghin
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://www.tallkite.com/words/Tibia.mp3|TIBIA]]&lt;/span&gt;&lt;/span&gt; by [[Paul Erlich]]
** Sagittal score of Tibia, [[file:xenharmonic/TIBIA.pdf|in F||\]] or [[file:xenharmonic/tibia in g.pdf|in G]] (contains errors in measures 9, 19 and 20)
** Ups and Downs score of Tibia in G [[file:Tibia in G CORRECTED-1.png|page 1]] [[file:Tibia in G CORRECTED-2.png|page 2 ]](no errors)
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095|Glassic]]&lt;/span&gt;&lt;/span&gt; by Paul Erlich and [[Ara Sarkissian]]
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://lumma.org/tuning/erlich/decatonic-swing.mp3|Decatonic Swing]]&lt;/span&gt;&lt;/span&gt; by Paul Erlich and Ara Sarkissian (jazz)
* [[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3|12-22hexachordal Dirge]] by [[Joel Grant Taylor]]
* [[@https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs|Chord sequence in Paul Erlich's 22 EDO decatonic major]] by [[Jake Freivald]]
* [[https://soundcloud.com/jdfreivald/porcupine-comma-pump|Porcupine Comma Pump]] by [[Jake Freivald]]
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Dragged%20By%20a%20Storm%20Across%20the%20Desert%20Years.mp3|Dragged by a Storm Across the Desert Years]]&lt;/span&gt;&lt;/span&gt; by * [[IgliashonJones|Igliashon Jones]] (synth with electric guitar)
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3|Numerology]]&lt;/span&gt;&lt;/span&gt; by Iglashion Jones (progressive metal)
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Revenge%20of%20the%20Inorganic%20Compounds.mp3|Revenge of the inorganic compounds]]&lt;/span&gt;&lt;/span&gt; by Iglashion Jones (progressive metal)
* [[http://chrisvaisvil.com/?p=267|My Crazy Aunt Sophie]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3|play]]&lt;/span&gt;&lt;/span&gt; by [[Chris Vaisvil]]. Blatantly xenharmonic piano.
* [[http://soundclick.com/share?songid=8839058|where words are said to mean]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+wherewordsaresaidtomean.mp3|play]]&lt;/span&gt;&lt;/span&gt; by [[Andrew Heathwaite]], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This &amp; the following pieces by Andrew are for 22-tone guitar &amp; voice.
* [[http://soundclick.com/share?songid=9101704|I've come with a bucket of roses]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3|play]]&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2).
* [[http://soundclick.com/share?songid=9101705|one drop of rain]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3|play]]&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (orwell-9).
* [[http://soundclick.com/share?songid=8839060|being a]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+beinga.mp3|play]]&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3).
* [[http://soundclick.com/share?songid=8839071|my own house]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3|play]]&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7).
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3|Comets Over Flatland 17]]&lt;/span&gt;&lt;/span&gt; by [[Randy Winchester]]
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3|Night on Porcupine Mountain]]&lt;/span&gt;&lt;/span&gt; Mussorgsky-Smith
* &lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&gt;[[http://www.youtube.com/watch?v=lO5xSjIHyMg|Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity]]&lt;/span&gt;&lt;/span&gt; - youtube
* &lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&gt;[[http://www.youtube.com/watch?v=WMtp9Wk0tO0|Improvisation in 22-equal temperament]]&lt;/span&gt;&lt;/span&gt;, Mike Battaglia - youtube
* Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - [[http://www.angelfire.com/mo/oljare/midicomp.html|MIDI files]] by Mats Öljare
** [[file:xenharmonic/sunday3.pdf|Sagittal score of Sunday Pipes]]
* &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3|Phobos Light]]&lt;/span&gt; by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.
* //[[http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3|The Capture and Release of the Fairy]]// by [[Chris Vaisvil]] =&gt; [[http://chrisvaisvil.com/?p=2494|blog post presentation]]
* //[[http://www.youtube.com/watch?v=oNJr1YOOqF8|Yak Butter]]// by The Stern Brocot Band, 1L6s MOS, compressed period/generator
* [[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_03_-_Dirty_Drummer_vbr.mp3|Dirty Drummer]], Sevish
* [[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_12_-_Ganymede_vbr.mp3|Ganymede]], Sevish (doesn't sound that xen, but it's in 22-edo)
* [[http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3|Ambrosia]], Sevish
* //[[http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3|From the Sky Islands They Came]]// by [[Chris Vaisvil]] =&gt; [[http://chrisvaisvil.com/?p=2523|blog post presentation]]
* [[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3|Smoke Filled Bar]] by [[Chris Vaisvil]] =&gt; [[@http://chrisvaisvil.com/smoke-filled-bar/|blog presentation]]
* [[http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3|Recurring Mimosa]] by [[https://soundcloud.com/redrick-sultan/recurring-mimosa|Redrick Sultan]]
* The Saharan Pump by Chris Vaisvil [[http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/|blog post]]
* [[@http://www.youtube.com/watch?v=qHHv3mwJTlg|Short piece and demonstration]] (video) by [[@http://brendanbyrnes.com/|Brendan Byrnes]] (electric guitar)
* [[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3|22 EDO Guitar Etude]] by [[http://brendanbyrnes.bandcamp.com/|Brendan Byrnes]]
* [[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%20Llurion.mp3|Llurion]] by [[http://brendanbyrnes.bandcamp.com/track/llurion|Brendan Byrnes]]
* [[@https://youtu.be/0VLJXecjYK4|Imzadi]] by [[@http://omega9.github.io/|Omega9]]
* [[http://micro.soonlabel.com/22-ET/20150910_22edo.mp3|22 edo electric guitar duet]] by [[Chris Vaisvil]]
* [[https://soundcloud.com/gareth-hearne/mass-in-22edo-sanctus|Mass in 22edo - Sanctus]] by [[Gareth Hearne]]
* [[https://soundcloud.com/gareth-hearne/mass-in-22edo-agnus-dei|Mass in 22edo - Agnus Dei]] by Gareth Hearne
* [[@http://chrisvaisvil.com/for-the-sunset/|For the Sunset]] - 22 edo rock ensemble by [[Chris Vaisvil]]
* [[https://soundcloud.com/ilevens/tracks|tracks of ILEVENS]] - all their tracks on SoundCloud are tagged with 22edo
* [[@https://drive.google.com/drive/folders/0BwsXD8q2VCYUNGZJOGRzSVdhRjg|Rose, liz, printemps, verdure]] by Alex Ness (in 22edo with stretched octaves)
* [[@https://www.youtube.com/watch?v=jagxI__W-Mg|Palinkalin Viharo (Flowers in the Mist)]] by Jake Huryn ([[@https://drive.google.com/file/d/0BwJHTddN0-rdUFdwMEtfYnFJZ0E/view|Score]]); uses 11edo machine[6], 22edo orwell[9]


[[media type="custom" key="27813215"]]</pre></div>
A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;22edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/22edo"&gt;Deutsch&lt;/a&gt; - &lt;a class="wiki_link" href="/22%E5%B9%B3%E5%9D%87%E5%BE%8B"&gt;日本語&lt;/a&gt;&lt;br /&gt;
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&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Chord Names-Properties of 22 equal temperament-Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Chord Names-Properties of 22 equal temperament-How to Notate 22edo in Sagittal"&gt;How to Notate 22edo in Sagittal&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#The Decatonic System-Decatonic Alphabet"&gt;Decatonic Alphabet&lt;/a&gt;&lt;/div&gt;
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&lt;br /&gt;
In music, &lt;em&gt;22 equal temperament&lt;/em&gt;, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s. Because it distinguishes 10/9 and 9/8, it's good for 5-limit.&lt;br /&gt;
&lt;br /&gt;
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the &lt;a class="wiki_link" href="/Indian"&gt;music theory of India&lt;/a&gt;, Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after &lt;a class="wiki_link" href="/19edo"&gt;19 equal temperament&lt;/a&gt;, and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.&lt;br /&gt;
&lt;br /&gt;
The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta peak&lt;/a&gt;. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the &lt;a class="wiki_link" href="/7-limit"&gt;7-&lt;/a&gt; and &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;s to within 3 cents/oct of error. While &lt;a class="wiki_link" href="/31edo"&gt;31 equal temperament&lt;/a&gt; does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit&lt;a class="wiki_link" href="/consistent"&gt; consistent&lt;/a&gt;ly. Furthermore, 22-et, unlike 12 and &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, is not a &lt;a class="wiki_link" href="/Regular%20Temperaments#meantone"&gt;meantone&lt;/a&gt; system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.&lt;br /&gt;
&lt;br /&gt;
22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.&lt;br /&gt;
&lt;br /&gt;
22-et is very close to an extended &amp;quot;quarter-comma superpyth&amp;quot;, a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64:63 instead of the syntonic comma 81:80. Because of this it has nearly pure septimal major thirds (9:7).&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Theory-Intervalic Naming Systems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervalic Naming Systems&lt;/h2&gt;
The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or &amp;quot;Natural&amp;quot;. Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Theory-Intervals by degree (Superpyth/Porcupine)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Intervals by degree (Superpyth/Porcupine)&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
    &lt;tr&gt;
        &lt;td&gt;Degree&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Name and Abbreviation&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
Ratios*&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Natural Unison, 1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-minor second, sm2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;54.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;33/32, 34/33, 32/31&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-diminished second, pd2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.09&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;18/17, 17/16, 16/15, 15/14&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-minor second, pm2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;163.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11/10, 10/9, 32/29&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(s/p) Major second, M2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;218.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9/8, 8/7, 17/15&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-minor third, sm3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/20_17"&gt;20/17&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-minor third, pm3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;327.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5, 17/14, 11/9, 29/24&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-Major third, pM3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;381.82&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-Major third, sM3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;436.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9/7, 14/11, 22/17&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Natural Fourth, 4, N4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.91&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4/3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-Major Fourth, pM4&lt;br /&gt;
s-dim fifth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;545.45&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11/8, 15/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Augmented Fourth, A4,&lt;br /&gt;
Half-Octave, HO&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7/5, 10/7, 17/12, 24/17&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-minor Fifth, pm5&lt;br /&gt;
s-aug fourth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;654.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;16/11, 22/15&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Natural Fifth, 5, N5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;709.09&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-minor sixth, sm6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;763.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11/7, 14/9, 17/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-minor sixth, pm6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;818.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-Major sixth, pM6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;872.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5/3, 18/11, 28/17&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-Major sixth, sM6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;927.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;, &lt;a class="wiki_link" href="/17_10"&gt;17/10&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(s/p) minor seventh, m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;981.82&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7/4, 16/9, 30/17&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-Major seventh, pM7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1036.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20/11, 9/5, 29/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;p-Augmented Seventh&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1090.91&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15/8, 32/17, 17/9, 28/15&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;s-Major Seventh, sM7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1145.45&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;33/17, 64/33, 31/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Octave, 8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
Paul Erlich's "Tibia" in G, with independent ups and downs:
22edo intervals can also be notated using &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D.&lt;br /&gt;
&lt;br /&gt;
Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.&lt;br /&gt;
&lt;br /&gt;
Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Theory-Intervals by degree (Ups and Downs, Porcupine and Pentatonic)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Intervals by degree (Ups and Downs, Porcupine and Pentatonic)&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
[[File:Tibia_in_G_for_the_book-1.png|alt=Tibia in G for the book-1.png|800x956px|Tibia in G for the book-1.png]]
    &lt;tr&gt;
        &lt;th&gt;&lt;a class="wiki_link" href="/Degree"&gt;Degree&lt;/a&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Size (&lt;a class="wiki_link" href="/cent"&gt;Cents&lt;/a&gt;)&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="3"&gt;Ups and downs&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="3"&gt;Porcupine&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="3"&gt;Pentatonic&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;109&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Eb^&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-aug unison,&lt;br /&gt;
double-dim sub3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;AA1,&lt;br /&gt;
dds3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Dx,&lt;br /&gt;
Fb&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;164&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ev&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim sub3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;ds3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fbb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;218&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor sub3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;ms3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;273&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major sub3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ms3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;327&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug sub3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;As3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;382&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#v&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-aug sub3rd,&lt;br /&gt;
double-dim 4thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;AAs3,&lt;br /&gt;
dd4d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fx,&lt;br /&gt;
Gbb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;436&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 3rd, dim 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A3, d4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fx, Gb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 4thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d4d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;491&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect fourth&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 4thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P4d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;545&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up-4th, dim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^4, d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^, Ab&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 4thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A4d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downaug 4th,&lt;br /&gt;
updim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vA4, ^d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#v,&lt;br /&gt;
Ab^&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 4th,&lt;br /&gt;
dim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A4, d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gx,&lt;br /&gt;
Abb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-aug 4thoid,&lt;br /&gt;
double-dim 5thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;AA4d,&lt;br /&gt;
dd5d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gx,&lt;br /&gt;
Abb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;655&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 4th, down-5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A4, v5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#, Av&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ab&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 5thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d5d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ab&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;709&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 5thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P5d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;764&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 5th, dim 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A5, d6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A#, Bbb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 5thoid&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A5d&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;818&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bb^&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-aug 5thoid,&lt;br /&gt;
double-dim sub7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;AA5d,&lt;br /&gt;
dds7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ax,&lt;br /&gt;
Cb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;873&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bv&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim sub7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;ds7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cbb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;927&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor sub7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;ms7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;982&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;minor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major sub7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ms7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1036&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug sub7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;As7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1091&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#v&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-aug sub7th,&lt;br /&gt;
double-dim octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;AAs7,&lt;br /&gt;
dd8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cx,&lt;br /&gt;
Dbb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1145&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;d8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[File:Tibia_in_G_for_the_book-2.png|alt=Tibia in G for the book-2.png|800x889px|Tibia in G for the book-2.png]]
Combining ups and downs notation with &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;color notation&lt;/a&gt;, qualities can be loosely associated with colors:&lt;br /&gt;


=The Decatonic System=
The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern


&lt;table class="wiki_table"&gt;
==Decatonic Alphabet==
    &lt;tr&gt;
The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.
        &lt;th&gt;quality&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;color&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;monzo format&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;examples&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;minor&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;blue&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b, 0, 1}&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7/6, 7/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;fourthward white&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b}, b &amp;lt; -1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;32/27, 16/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;upminor&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;green&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b, -1}&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6/5, 9/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;downmajor&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;yellow&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b, 1}&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5/4, 5/3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;major&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;fifthward white&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b}, b &amp;gt; 1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9/8, 27/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;red&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;{a, b, 0, -1}&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9/7, 12/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
Chain 1: C G D A E
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Chord Names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Chord Names&lt;/h1&gt;
&lt;br /&gt;
All 22edo chords can be named using ups and downs notation. Here are the blue, green, yellow and red triads:&lt;br /&gt;


Chain 2: γ δ α ε β


&lt;table class="wiki_table"&gt;
The alphabet is, in ascending order: C δ D ε E γ G α A β C
    &lt;tr&gt;
        &lt;th&gt;color of the 3rd&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;JI chord&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;notes as edosteps&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;notes of C chord&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;written name&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;spoken name&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;blue&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6:7:9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0-5-13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C Eb G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cm&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C minor&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;green&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;10:12:15&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0-6-13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C Eb^ G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C.^m&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C upminor&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;yellow&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4:5:6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0-7-13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C Ev G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C.v&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C downmajor or C dot down&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;red&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14:18:27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0-8-13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C E G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C major or C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


For C.v, the period is needed because &amp;quot;Cv&amp;quot;, spoken as &amp;quot;C down&amp;quot;, is either a note, or a major chord Cv Ev Gv.&lt;br /&gt;
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.
The period isn't needed in Cm because there's no ups or downs immediately after the note name.&lt;br /&gt;
&lt;br /&gt;
0-8-13-18 = C E G Bb = C7 = &amp;quot;C seven&amp;quot;&lt;br /&gt;
0-7-13-18 = C Ev G Bb = C7(v3) = &amp;quot;C seven, down third&amp;quot;&lt;br /&gt;
0-8-13-21 = C E G B = CM7 = &amp;quot;C major seven&amp;quot;&lt;br /&gt;
0-7-13-20 = C Ev G Bv = C.vM7 = &amp;quot;C downmajor seven&amp;quot; (the down symbol applies to both the 3rd and the 7th)&lt;br /&gt;
&lt;br /&gt;
0-3-13 = C Dv G = C(v2)&lt;br /&gt;
0-4-13 = C D G = C2&lt;br /&gt;
0-9-13 = C F G = C4&lt;br /&gt;
0-10-13 = C F^ G = C(^4)&lt;br /&gt;
&lt;br /&gt;
0-5-10 = C Eb Gb = Cdim&lt;br /&gt;
0-5-11 = C Eb Gb^ = Cdim(^5)&lt;br /&gt;
0-5-12 = C Eb Gv = Cm(v5)&lt;br /&gt;
&lt;br /&gt;
0-5-10-15 = C Eb Gb Bbb = Cdim7&lt;br /&gt;
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)&lt;br /&gt;
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)&lt;br /&gt;
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7(^5) (the up symbol applies to both the 3rd and the 7th)&lt;br /&gt;
0-5-13-17 = C Eb G A = Cm6&lt;br /&gt;
&lt;br /&gt;
Sometimes doubled ups/downs are unavoidable:&lt;br /&gt;
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)&lt;br /&gt;
&lt;br /&gt;
0-8-13-17 = C E G A = C6&lt;br /&gt;
0-8-13-16 = C E G Av = C(v6)&lt;br /&gt;
0-7-13-17 = C Ev G A = C6(v3)&lt;br /&gt;
0-7-13-16 = C Ev G Av = C.v6 (the down symbol applies to both the 3rd and the 6th)&lt;br /&gt;
&lt;br /&gt;
0-5-13-18 = C Eb G Bb = Cm7&lt;br /&gt;
0-6-13-19 = C Eb^ G Bb^ = C.^m7&lt;br /&gt;
0-8-13-21 = C E G B = CM7&lt;br /&gt;
0-7-13-20 = C Ev G Bv = C.vM7&lt;br /&gt;
&lt;br /&gt;
0-5-13-16 = C Eb G Av = Cm(v6)&lt;br /&gt;
0-8-13-19 = C E G Bb^ = C(^7)&lt;br /&gt;
0-7-13-18-26 = C Ev G Bb D = C9(v3)&lt;br /&gt;
0-7-13-18-26-32 = C Ev G Bb D F^ = C9(v3,^11)&lt;br /&gt;
&lt;br /&gt;
For a more complete list, see &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs"&gt;Ups and Downs Notation - Chord names in other EDOs&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Chord Names-Selected just intervals by error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Selected just intervals by error&lt;/h2&gt;
The following table shows how &lt;a class="wiki_link" href="/Just-24"&gt;some prominent just intervals&lt;/a&gt; are represented in 22edo (ordered by absolute error).&lt;br /&gt;


==Internal links==
<ul><li>[[William_Lynch's_Thoughts_on_Septimal_Harmony_and_22_EDO|William Lynch's Thoughts on Septimal Harmony and 22 EDO]]</li></ul>


&lt;table class="wiki_table"&gt;
==External links==
    &lt;tr&gt;
<ul><li>[http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']</li><li>[http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music ]</li></ul>
        &lt;th&gt;Interval, complement&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Error (abs., in &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;)&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;, &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.280&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_10"&gt;11/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_11"&gt;20/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.368&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.640&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4.496&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.856&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.863&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7.136&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_11"&gt;15/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_15"&gt;22/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.504&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_14"&gt;15/14&lt;/a&gt;, &lt;a class="wiki_link" href="/28_15"&gt;28/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;10.352&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.631&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;, &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;12.992&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/12_11"&gt;12/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_6"&gt;11/6&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;12.999&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14.272&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_11"&gt;13/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_13"&gt;22/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;16.482&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.488&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_13"&gt;20/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.850&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/18_13"&gt;18/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_9"&gt;13/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.928&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;, &lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;18.767&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_11"&gt;14/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_7"&gt;11/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;18.856&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_7"&gt;13/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;19.207&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_9"&gt;11/9&lt;/a&gt;, &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20.135&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_13"&gt;16/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;22.346&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;, &lt;a class="wiki_link" href="/26_15"&gt;26/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;24.986&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_12"&gt;13/12&lt;/a&gt;, &lt;a class="wiki_link" href="/24_13"&gt;24/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;25.064&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
==References==
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24838814?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;24838814&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/22ed2-001e.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
*Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:1821:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/22ed2-001e.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@22ed2-001e.svg&amp;quot; title=&amp;quot;File: 22ed2-001e.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/22ed2-001e.svg/482002024/22ed2-001e.svg" onclick="ws.common.trackFileLink('/file/view/22ed2-001e.svg/482002024/22ed2-001e.svg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="22ed2-001e.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/22ed2-001e.svg/482002024/22ed2-001e.svg" onclick="ws.common.trackFileLink('/file/view/22ed2-001e.svg/482002024/22ed2-001e.svg');" class="filename" title="22ed2-001e.svg"&gt;22ed2-001e.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/22ed2-001e.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/22ed2-001e.svg/482002024/22ed2-001e.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;25 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:1821 --&gt;&lt;br /&gt;
&lt;br /&gt;
See also: &lt;a class="wiki_link" href="/22edo%20Solfege"&gt;22edo Solfege&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo%20Tetrachords"&gt;22edo Tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/22%20EDO%20Chords"&gt;22 EDO Chords&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo%20Modes"&gt;22edo Modes&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Chord Names-Properties of 22 equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Properties of 22 equal temperament&lt;/h2&gt;
&lt;br /&gt;
Possibly the most striking characteristic of 22-et to those not used to it is that it does &lt;strong&gt;not&lt;/strong&gt; &amp;quot;temper out&amp;quot; the syntonic comma of 81/80, and therefore is not a system of &lt;a class="wiki_link" href="/Regular%20Temperaments#meantone"&gt;meantone&lt;/a&gt; temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12-EDO, 19-EDO, 31-EDO, ... do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; and &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
The diatonic scale it produces is instead derived from &lt;a class="wiki_link" href="/superpyth"&gt;superpyth&lt;/a&gt; temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, &lt;a class="wiki_link" href="/5L%202s"&gt;5L 2s&lt;/a&gt;), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22-EDO. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12-equal and meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.&lt;br /&gt;
&lt;br /&gt;
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22-EDO supports &lt;a class="wiki_link" href="/porcupine"&gt;porcupine&lt;/a&gt; temperament. The generator for porcupine is is a flat minor whole tone of &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;, two of which is a slightly sharp &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, and three of which is a slightly flat &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, implying the existence of an equal-step tetrachord, which is characteristic of Porcupine. Porcupine is notable for being the 5-limit temperament lowest in &lt;a class="wiki_link" href="/badness"&gt;badness&lt;/a&gt; which is &lt;em&gt;not&lt;/em&gt; approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22-EDO. It forms &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt;'s of 7 and 8, which in 22-EDO are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).&lt;br /&gt;
&lt;br /&gt;
The 164¢ &amp;quot;flat minor whole tone&amp;quot; is a key interval in 22-EDO, in part because it functions as no less than three different consonant ratios in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22-EDO can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a &amp;quot;neutral-like&amp;quot; quality since they are tempered closer together rather than farther apart as in 12edo.&lt;br /&gt;
&lt;br /&gt;
22-EDO also supports Orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms MOS scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, Orwell can be tuned more accurately in other temperaments, such as &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; and &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;. But 22-equal Orwell has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish in 22.&lt;br /&gt;
&lt;br /&gt;
Other 5-limit commas 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, the &lt;a class="wiki_link" href="/diatonic%20tritone"&gt;diatonic tritone&lt;/a&gt; &lt;a class="wiki_link" href="/45_32"&gt;45/32&lt;/a&gt;, which is a major third above a &lt;a class="wiki_link" href="/major%20whole%20tone"&gt;major whole tone&lt;/a&gt; representing &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, is equated to its inverted form, &lt;a class="wiki_link" href="/64_45"&gt;64/45&lt;/a&gt;. That the magic comma is tempered out means that 22-et is a &lt;a class="wiki_link" href="/Regular%20Temperaments#magic"&gt;magic&lt;/a&gt; system, where five major thirds make up a perfect fifth.&lt;br /&gt;
&lt;br /&gt;
In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both &lt;a class="wiki_link" href="/50_49"&gt;50/49&lt;/a&gt;, (the &lt;a class="wiki_link" href="/jubilee%20comma"&gt;jubilee comma&lt;/a&gt;), and &lt;a class="wiki_link" href="/64_63"&gt;64/63&lt;/a&gt;, (the &lt;a class="wiki_link" href="/septimal%20comma"&gt;septimal comma&lt;/a&gt;), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the &lt;a class="wiki_link" href="/septimal%20kleisma"&gt;septimal kleisma&lt;/a&gt;, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the &lt;a class="wiki_link" href="/orwell%20comma"&gt;orwell comma&lt;/a&gt;; and the &lt;a class="wiki_link" href="/orwell%20tetrad"&gt;orwell tetrad&lt;/a&gt; is also a chord of 22-et.&lt;br /&gt;
&lt;br /&gt;
As 22 is divisible by 11, a 22edo instrument can play any music in &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt;, in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In &lt;a class="wiki_link" href="/Sagittal"&gt;Sagittal&lt;/a&gt;, 11 can be notated as every other note of 22.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Chord Names-Properties of 22 equal temperament-Rank Two Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Rank Two Temperaments&lt;/h3&gt;
&lt;a class="wiki_link" href="/List%20of%2022et%20rank%20two%20temperaments%20by%20badness"&gt;List of 22et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/List%20of%2022et%20rank%20two%20temperaments%20by%20complexity"&gt;List of 22et rank two temperaments by complexity&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2022et%20rank%20two%20temperaments"&gt;List of edo-distinct 22et rank two temperaments&lt;/a&gt;&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Periods&lt;br /&gt;
per octave&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Period&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Temperaments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Sensamagic%20clan#Sensa"&gt;Sensa&lt;/a&gt;/chromo/ceratitid&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Orson"&gt;Orson&lt;/a&gt;/&lt;a class="wiki_link" href="/orwell"&gt;orwell&lt;/a&gt;/blair&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Magic"&gt;Magic&lt;/a&gt;/telepathy&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Superpyth"&gt;Superpyth&lt;/a&gt;/&lt;a class="wiki_link" href="/suprapyth"&gt;suprapyth&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Shrutar"&gt;Shrutar&lt;/a&gt;/hemipaj/comic&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Srutal"&gt;Srutal&lt;/a&gt;/&lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;/pajarous&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Porcupine%20family#Hedgehog"&gt;Hedgehog&lt;/a&gt;/&lt;a class="wiki_link" href="/echidna"&gt;echidna&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Astrology"&gt;Astrology&lt;/a&gt;/&lt;a class="wiki_link" href="/wizard"&gt;wizard&lt;/a&gt;/&lt;a class="wiki_link" href="/antikythera"&gt;antikythera&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Doublewide"&gt;Doublewide&lt;/a&gt;/fleetwood&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Hendecatonic"&gt;Hendecatonic&lt;/a&gt;/undeka&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Chord Names-Properties of 22 equal temperament-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Commas&lt;/h3&gt;
22 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 22 35 51 62 76 81 |.)&lt;br /&gt;


*Bosanquet, R.H.M. [http://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965


&lt;table class="wiki_table"&gt;
=Music=
    &lt;tr&gt;
<ul><li>[https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version Stephen Weigel · Dose Of Familiarity/Ode to Microtonality]</li><li>[https://soundcloud.com/metaclown/couples-therapy Couples' Therapy] by metaclown</li><li>[http://soonlabel.com/xenharmonic/archives/1145 Canon 2 in 1 upon a ground (22edo)] by Claudi Meneghin</li><li><span style=""><span style="">[http://www.tallkite.com/words/Tibia.mp3 TIBIA]</span></span> by [[Paul_Erlich|Paul Erlich]]<ul><li>Sagittal score of Tibia, [[:File:TIBIA.pdf|in F||\]] or [[:File:tibia_in_g.pdf|in G]] (contains errors in measures 9, 19 and 20)</li><li>Ups and Downs score of Tibia in G [[:File:Tibia_in_G_CORRECTED-1.png|page 1]] [[:File:Tibia_in_G_CORRECTED-2.png|page 2 ]](no errors)</li></ul></li><li><span style=""><span style="">[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095 Glassic]</span></span> by Paul Erlich and [[Ara_Sarkissian|Ara Sarkissian]]</li><li><span style=""><span style="">[http://lumma.org/tuning/erlich/decatonic-swing.mp3 Decatonic Swing]</span></span> by Paul Erlich and Ara Sarkissian (jazz)</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] by [[Joel_Grant_Taylor|Joel Grant Taylor]]</li><li>[https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord sequence in Paul Erlich's 22 EDO decatonic major] by [[Jake_Freivald|Jake Freivald]]</li><li>[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump] by [[Jake_Freivald|Jake Freivald]]</li><li><span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Dragged%20By%20a%20Storm%20Across%20the%20Desert%20Years.mp3 Dragged by a Storm Across the Desert Years]</span></span> by * [[IgliashonJones|Igliashon Jones]] (synth with electric guitar)</li><li><span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3 Numerology]</span></span> by Iglashion Jones (progressive metal)</li><li><span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Revenge%20of%20the%20Inorganic%20Compounds.mp3 Revenge of the inorganic compounds]</span></span> by Iglashion Jones (progressive metal)</li><li>[http://chrisvaisvil.com/?p=267 My Crazy Aunt Sophie] <span style=""><span style="">[http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3 play]</span></span> by [[Chris_Vaisvil|Chris Vaisvil]]. Blatantly xenharmonic piano.</li><li>[http://soundclick.com/share?songid=8839058 where words are said to mean] <span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+wherewordsaresaidtomean.mp3 play]</span></span> by [[Andrew_Heathwaite|Andrew Heathwaite]], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This &amp; the following pieces by Andrew are for 22-tone guitar &amp; voice.</li><li>[http://soundclick.com/share?songid=9101704 I've come with a bucket of roses] <span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3 play]</span></span> by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2).</li><li>[http://soundclick.com/share?songid=9101705 one drop of rain] <span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3 play]</span></span> by Andrew Heathwaite (orwell-9).</li><li>[http://soundclick.com/share?songid=8839060 being a] <span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+beinga.mp3 play]</span></span> by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3).</li><li>[http://soundclick.com/share?songid=8839071 my own house] <span style=""><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3 play]</span></span> by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7).</li><li><span style=""><span style="">[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3 Comets Over Flatland 17]</span></span> by [[Randy_Winchester|Randy Winchester]]</li><li><span style=""><span style="">[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 Night on Porcupine Mountain]</span></span> Mussorgsky-Smith</li><li><span style=""><span style="">[http://www.youtube.com/watch?v=lO5xSjIHyMg Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity]</span></span> - youtube</li><li><span style=""><span style="">[http://www.youtube.com/watch?v=WMtp9Wk0tO0 Improvisation in 22-equal temperament]</span></span>, Mike Battaglia - youtube</li><li>Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - [http://www.angelfire.com/mo/oljare/midicomp.html MIDI files] by Mats Öljare<ul><li>[[:File:sunday3.pdf|Sagittal score of Sunday Pipes]]</li></ul></li><li><span style="">[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light]</span> by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.</li><li>''[http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3 The Capture and Release of the Fairy]'' by [[Chris_Vaisvil|Chris Vaisvil]] =&gt; [http://chrisvaisvil.com/?p=2494 blog post presentation]</li><li>''[http://www.youtube.com/watch?v=oNJr1YOOqF8 Yak Butter]'' by The Stern Brocot Band, 1L6s MOS, compressed period/generator</li><li>[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_03_-_Dirty_Drummer_vbr.mp3 Dirty Drummer], Sevish</li><li>[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_12_-_Ganymede_vbr.mp3 Ganymede], Sevish (doesn't sound that xen, but it's in 22-edo)</li><li>[http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3 Ambrosia], Sevish</li><li>''[http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3 From the Sky Islands They Came]'' by [[Chris_Vaisvil|Chris Vaisvil]] =&gt; [http://chrisvaisvil.com/?p=2523 blog post presentation]</li><li>[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [[Chris_Vaisvil|Chris Vaisvil]] =&gt; [http://chrisvaisvil.com/smoke-filled-bar/ blog presentation]</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3 Recurring Mimosa] by [https://soundcloud.com/redrick-sultan/recurring-mimosa Redrick Sultan]</li><li>The Saharan Pump by Chris Vaisvil [http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/ blog post]</li><li>[http://www.youtube.com/watch?v=qHHv3mwJTlg Short piece and demonstration] (video) by [http://brendanbyrnes.com/ Brendan Byrnes] (electric guitar)</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3 22 EDO Guitar Etude] by [http://brendanbyrnes.bandcamp.com/ Brendan Byrnes]</li><li>[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%20Llurion.mp3 Llurion] by [http://brendanbyrnes.bandcamp.com/track/llurion Brendan Byrnes]</li><li>[https://youtu.be/0VLJXecjYK4 Imzadi] by [http://omega9.github.io/ Omega9]</li><li>[http://micro.soonlabel.com/22-ET/20150910_22edo.mp3 22 edo electric guitar duet] by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[https://soundcloud.com/gareth-hearne/mass-in-22edo-sanctus Mass in 22edo - Sanctus] by [[Gareth_Hearne|Gareth Hearne]]</li><li>[https://soundcloud.com/gareth-hearne/mass-in-22edo-agnus-dei Mass in 22edo - Agnus Dei] by Gareth Hearne</li><li>[http://chrisvaisvil.com/for-the-sunset/ For the Sunset] - 22 edo rock ensemble by [[Chris_Vaisvil|Chris Vaisvil]]</li><li>[https://soundcloud.com/ilevens/tracks tracks of ILEVENS] - all their tracks on SoundCloud are tagged with 22edo</li><li>[https://drive.google.com/drive/folders/0BwsXD8q2VCYUNGZJOGRzSVdhRjg Rose, liz, printemps, verdure] by Alex Ness (in 22edo with stretched octaves)</li><li>[https://www.youtube.com/watch?v=jagxI__W-Mg Palinkalin Viharo (Flowers in the Mist)] by Jake Huryn ([https://drive.google.com/file/d/0BwJHTddN0-rdUFdwMEtfYnFJZ0E/view Score]); uses 11edo machine[6], 22edo orwell[9]</li></ul>
        &lt;th&gt;Rational&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Monzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Size (Cents)&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 3&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;250/243&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 -5 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;49.17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Maximal Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Porcupine Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3125/3072&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -10 -1 5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;29.61&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Small Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Magic Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2048/2025&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 11 -4 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;19.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diaschisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2109375/2097152&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -21 3 7 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;10.06&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Semicomma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fokker Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9193891/9143623&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 32 -7 -9 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.49&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Escapade Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4758837/4757272&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -53 10 16 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Kwazy&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;50/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 0 2 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;34.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Tritonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Jubilisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;27.26&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Archytas' Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Leipziger Komma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;875/864&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -5 -3 3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.90&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keema&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2430/2401&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 5 1 -4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;20.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Nuwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;245/243&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 0 -5 1 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;14.19&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Sensamagic&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1728/1715&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 6 3 -1 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.07&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Orwellisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Orwell Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;225/224&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;7.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Kleisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Marvel Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;10976/10935&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 5 -7 -1 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;6.48&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Hemimage&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6144/6125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Porwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;65625/65536&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -16 1 5 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;2.35&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Horwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;420175/419904&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Wizma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;99/98&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Mothwellsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 -2 2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;121/120&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -3 -1 -1 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;14.37&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Biyatisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;125/124&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|-4 0 3 0 ... -1&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.91&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Twizzler&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;896/891&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 7 -4 0 1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pentacircle&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;65536/65219&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 16 0 0 -2 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.39&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Orgonisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;540/539&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 3 1 -2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Swetisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Wizardharry&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9801/9800&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -3 4 -2 -2 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Kalisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gauss' Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;19.13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Superleap&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:22edo]]
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Chord Names-Properties of 22 equal temperament-How to Notate 22edo in Sagittal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;How to Notate 22edo in Sagittal&lt;/h3&gt;
[[Category:edo]]
&lt;br /&gt;
[[Category:listen]]
When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:&lt;br /&gt;
[[Category:theory]]
&lt;!-- ws:start:WikiTextLocalImageRule:1815:&amp;lt;img src=&amp;quot;/file/view/22edo.png/269078624/22edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/22edo.png/269078624/22edo.png" alt="22edo.png" title="22edo.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1815 --&gt;&lt;br /&gt;
[[Category:todo:unify_precision]]
This notation is consistent with Sagittal's notation of 5-limit JI harmony: &amp;quot;major&amp;quot; 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.&lt;br /&gt;
[[Category:twentuning]]
&lt;br /&gt;
[[Category:zeta]]
The division of the apotome into three syntonic commas also indicates 22's tempering out of the &lt;a class="wiki_link" href="/250_243"&gt;porcupine comma&lt;/a&gt; (which is equivalent to three syntonic commas minus a Pythagorean apotome).&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Chord Names-Properties of 22 equal temperament-How to notate 22edo with ups and downs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;How to notate 22edo with ups and downs&lt;/h3&gt;
&lt;br /&gt;
Treating &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; as &amp;quot;fused&amp;quot; with sharps and flats, and never appearing separately:&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1816:&amp;lt;img src=&amp;quot;/file/view/Tibia%2022edo%20ups%20and%20downs%20guide%201.png/602422384/800x147/Tibia%2022edo%20ups%20and%20downs%20guide%201.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 147px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%2022edo%20ups%20and%20downs%20guide%201.png/602422384/800x147/Tibia%2022edo%20ups%20and%20downs%20guide%201.png" alt="Tibia 22edo ups and downs guide 1.png" title="Tibia 22edo ups and downs guide 1.png" style="height: 147px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1816 --&gt;&lt;br /&gt;
&lt;br /&gt;
Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1817:&amp;lt;img src=&amp;quot;/file/view/Tibia%2022edo%20ups%20and%20downs%20guide%202.png/602422386/800x150/Tibia%2022edo%20ups%20and%20downs%20guide%202.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 150px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%2022edo%20ups%20and%20downs%20guide%202.png/602422386/800x150/Tibia%2022edo%20ups%20and%20downs%20guide%202.png" alt="Tibia 22edo ups and downs guide 2.png" title="Tibia 22edo ups and downs guide 2.png" style="height: 150px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1817 --&gt;&lt;br /&gt;
&lt;br /&gt;
A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1818:&amp;lt;img src=&amp;quot;/file/view/Tibia%2022edo%20guide%20D%20major.png/602422382/800x68/Tibia%2022edo%20guide%20D%20major.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 68px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%2022edo%20guide%20D%20major.png/602422382/800x68/Tibia%2022edo%20guide%20D%20major.png" alt="Tibia 22edo guide D major.png" title="Tibia 22edo guide D major.png" style="height: 68px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1818 --&gt;&lt;br /&gt;
&lt;br /&gt;
Paul Erlich's &amp;quot;Tibia&amp;quot; in G, with independent ups and downs:&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1819:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20for%20the%20book-1.png/623289179/800x956/Tibia%20in%20G%20for%20the%20book-1.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 956px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20for%20the%20book-1.png/623289179/800x956/Tibia%20in%20G%20for%20the%20book-1.png" alt="Tibia in G for the book-1.png" title="Tibia in G for the book-1.png" style="height: 956px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1819 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1820:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20for%20the%20book-2.png/623289195/800x889/Tibia%20in%20G%20for%20the%20book-2.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 889px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20for%20the%20book-2.png/623289195/800x889/Tibia%20in%20G%20for%20the%20book-2.png" alt="Tibia in G for the book-2.png" title="Tibia in G for the book-2.png" style="height: 889px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1820 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="The Decatonic System"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;The Decatonic System&lt;/h1&gt;
The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="The Decatonic System-Decatonic Alphabet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:55:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-Decatonic-Alphabet&amp;quot; title=&amp;quot;Anchor: TOC-Decatonic-Alphabet&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-Decatonic-Alphabet"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:55 --&gt;Decatonic Alphabet&lt;/h2&gt;
The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.&lt;br /&gt;
&lt;br /&gt;
Chain 1: C G D A E&lt;br /&gt;
Chain 2: γ δ α ε β&lt;br /&gt;
&lt;br /&gt;
The alphabet is, in ascending order: C δ D ε E γ G α A β C&lt;br /&gt;
&lt;br /&gt;
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="The Decatonic System-Internal links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Internal links&lt;/h2&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/William%20Lynch%27s%20Thoughts%20on%20Septimal%20Harmony%20and%2022%20EDO"&gt;William Lynch's Thoughts on Septimal Harmony and 22 EDO&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="The Decatonic System-External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;External links&lt;/h2&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow"&gt;Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://porcupinemusic.weebly.com/" rel="nofollow"&gt;&amp;quot;Porcupine Music&amp;quot; - Website Focused on the Development of 22 EDO music &lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="The Decatonic System-References"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;References&lt;/h2&gt;
*Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]&lt;br /&gt;
*Bosanquet, R.H.M. &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5kjJcrhEx" rel="nofollow"&gt;''On the Hindoo division of the octave, with additions to the theory of higher orders''&lt;/a&gt;, Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Music&lt;/h1&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version" rel="nofollow" target="_blank"&gt;Stephen Weigel · Dose Of Familiarity/Ode to Microtonality&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/metaclown/couples-therapy" rel="nofollow"&gt;Couples' Therapy&lt;/a&gt; by metaclown&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1145" rel="nofollow" target="_blank"&gt;Canon 2 in 1 upon a ground (22edo)&lt;/a&gt; by Claudi Meneghin&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/words/Tibia.mp3" rel="nofollow"&gt;TIBIA&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;&lt;ul&gt;&lt;li&gt;Sagittal score of Tibia, &lt;a href="http://xenharmonic.wikispaces.com/file/view/TIBIA.pdf/313029038/TIBIA.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/TIBIA.pdf/313029038/TIBIA.pdf');"&gt;in F||\&lt;/a&gt; or &lt;a href="http://xenharmonic.wikispaces.com/file/view/tibia%20in%20g.pdf/313029040/tibia%20in%20g.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/tibia%20in%20g.pdf/313029040/tibia%20in%20g.pdf');"&gt;in G&lt;/a&gt; (contains errors in measures 9, 19 and 20)&lt;/li&gt;&lt;li&gt;Ups and Downs score of Tibia in G &lt;a href="/file/view/Tibia%20in%20G%20CORRECTED-1.png/623289787/Tibia%20in%20G%20CORRECTED-1.png" onclick="ws.common.trackFileLink('/file/view/Tibia%20in%20G%20CORRECTED-1.png/623289787/Tibia%20in%20G%20CORRECTED-1.png');"&gt;page 1&lt;/a&gt; &lt;a href="/file/view/Tibia%20in%20G%20CORRECTED-2.png/623289793/Tibia%20in%20G%20CORRECTED-2.png" onclick="ws.common.trackFileLink('/file/view/Tibia%20in%20G%20CORRECTED-2.png/623289793/Tibia%20in%20G%20CORRECTED-2.png');"&gt;page 2 &lt;/a&gt;(no errors)&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095" rel="nofollow"&gt;Glassic&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Paul Erlich and &lt;a class="wiki_link" href="/Ara%20Sarkissian"&gt;Ara Sarkissian&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/decatonic-swing.mp3" rel="nofollow"&gt;Decatonic Swing&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Paul Erlich and Ara Sarkissian (jazz)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3" rel="nofollow"&gt;12-22hexachordal Dirge&lt;/a&gt; by &lt;a class="wiki_link" href="/Joel%20Grant%20Taylor"&gt;Joel Grant Taylor&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs" rel="nofollow" target="_blank"&gt;Chord sequence in Paul Erlich's 22 EDO decatonic major&lt;/a&gt; by &lt;a class="wiki_link" href="/Jake%20Freivald"&gt;Jake Freivald&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/porcupine-comma-pump" rel="nofollow"&gt;Porcupine Comma Pump&lt;/a&gt; by &lt;a class="wiki_link" href="/Jake%20Freivald"&gt;Jake Freivald&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Dragged%20By%20a%20Storm%20Across%20the%20Desert%20Years.mp3" rel="nofollow"&gt;Dragged by a Storm Across the Desert Years&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by * &lt;a class="wiki_link" href="/IgliashonJones"&gt;Igliashon Jones&lt;/a&gt; (synth with electric guitar)&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3" rel="nofollow"&gt;Numerology&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Iglashion Jones (progressive metal)&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Revenge%20of%20the%20Inorganic%20Compounds.mp3" rel="nofollow"&gt;Revenge of the inorganic compounds&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Iglashion Jones (progressive metal)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=267" rel="nofollow"&gt;My Crazy Aunt Sophie&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;. Blatantly xenharmonic piano.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839058" rel="nofollow"&gt;where words are said to mean&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+wherewordsaresaidtomean.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;, a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This &amp;amp; the following pieces by Andrew are for 22-tone guitar &amp;amp; voice.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow"&gt;I've come with a bucket of roses&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2).&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow"&gt;one drop of rain&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (orwell-9).&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839060" rel="nofollow"&gt;being a&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+beinga.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3).&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow"&gt;my own house&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7).&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 17&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3" rel="nofollow"&gt;Night on Porcupine Mountain&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; Mussorgsky-Smith&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=lO5xSjIHyMg" rel="nofollow"&gt;Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity&lt;/a&gt;&lt;/span&gt;&lt;/span&gt; - youtube&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&gt;&lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=WMtp9Wk0tO0" rel="nofollow"&gt;Improvisation in 22-equal temperament&lt;/a&gt;&lt;/span&gt;&lt;/span&gt;, Mike Battaglia - youtube&lt;/li&gt;&lt;li&gt;Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - &lt;a class="wiki_link_ext" href="http://www.angelfire.com/mo/oljare/midicomp.html" rel="nofollow"&gt;MIDI files&lt;/a&gt; by Mats Öljare&lt;ul&gt;&lt;li&gt;&lt;a href="http://xenharmonic.wikispaces.com/file/view/sunday3.pdf/269076436/sunday3.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/sunday3.pdf/269076436/sunday3.pdf');"&gt;Sagittal score of Sunday Pipes&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3" rel="nofollow"&gt;Phobos Light&lt;/a&gt;&lt;/span&gt; by Chris Vaisvil in Hedgehog[14] &lt;a class="wiki_link" href="/hedgehog14"&gt;tuned&lt;/a&gt; to 22edo.&lt;/li&gt;&lt;li&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3" rel="nofollow"&gt;The Capture and Release of the Fairy&lt;/a&gt;&lt;/em&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2494" rel="nofollow"&gt;blog post presentation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=oNJr1YOOqF8" rel="nofollow"&gt;Yak Butter&lt;/a&gt;&lt;/em&gt; by The Stern Brocot Band, 1L6s MOS, compressed period/generator&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_03_-_Dirty_Drummer_vbr.mp3" rel="nofollow"&gt;Dirty Drummer&lt;/a&gt;, Sevish&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_12_-_Ganymede_vbr.mp3" rel="nofollow"&gt;Ganymede&lt;/a&gt;, Sevish (doesn't sound that xen, but it's in 22-edo)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3" rel="nofollow"&gt;Ambrosia&lt;/a&gt;, Sevish&lt;/li&gt;&lt;li&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3" rel="nofollow"&gt;From the Sky Islands They Came&lt;/a&gt;&lt;/em&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2523" rel="nofollow"&gt;blog post presentation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3" rel="nofollow"&gt;Smoke Filled Bar&lt;/a&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/smoke-filled-bar/" rel="nofollow" target="_blank"&gt;blog presentation&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3" rel="nofollow"&gt;Recurring Mimosa&lt;/a&gt; by &lt;a class="wiki_link_ext" href="https://soundcloud.com/redrick-sultan/recurring-mimosa" rel="nofollow"&gt;Redrick Sultan&lt;/a&gt;&lt;/li&gt;&lt;li&gt;The Saharan Pump by Chris Vaisvil &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/" rel="nofollow"&gt;blog post&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=qHHv3mwJTlg" rel="nofollow" target="_blank"&gt;Short piece and demonstration&lt;/a&gt; (video) by &lt;a class="wiki_link_ext" href="http://brendanbyrnes.com/" rel="nofollow" target="_blank"&gt;Brendan Byrnes&lt;/a&gt; (electric guitar)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3" rel="nofollow"&gt;22 EDO Guitar Etude&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://brendanbyrnes.bandcamp.com/" rel="nofollow"&gt;Brendan Byrnes&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%20Llurion.mp3" rel="nofollow"&gt;Llurion&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://brendanbyrnes.bandcamp.com/track/llurion" rel="nofollow"&gt;Brendan Byrnes&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://youtu.be/0VLJXecjYK4" rel="nofollow" target="_blank"&gt;Imzadi&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://omega9.github.io/" rel="nofollow" target="_blank"&gt;Omega9&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20150910_22edo.mp3" rel="nofollow"&gt;22 edo electric guitar duet&lt;/a&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/gareth-hearne/mass-in-22edo-sanctus" rel="nofollow"&gt;Mass in 22edo - Sanctus&lt;/a&gt; by &lt;a class="wiki_link" href="/Gareth%20Hearne"&gt;Gareth Hearne&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/gareth-hearne/mass-in-22edo-agnus-dei" rel="nofollow"&gt;Mass in 22edo - Agnus Dei&lt;/a&gt; by Gareth Hearne&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/for-the-sunset/" rel="nofollow" target="_blank"&gt;For the Sunset&lt;/a&gt; - 22 edo rock ensemble by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/ilevens/tracks" rel="nofollow"&gt;tracks of ILEVENS&lt;/a&gt; - all their tracks on SoundCloud are tagged with 22edo&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://drive.google.com/drive/folders/0BwsXD8q2VCYUNGZJOGRzSVdhRjg" rel="nofollow" target="_blank"&gt;Rose, liz, printemps, verdure&lt;/a&gt; by Alex Ness (in 22edo with stretched octaves)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=jagxI__W-Mg" rel="nofollow" target="_blank"&gt;Palinkalin Viharo (Flowers in the Mist)&lt;/a&gt; by Jake Huryn (&lt;a class="wiki_link_ext" href="https://drive.google.com/file/d/0BwJHTddN0-rdUFdwMEtfYnFJZ0E/view" rel="nofollow" target="_blank"&gt;Score&lt;/a&gt;); uses 11edo machine[6], 22edo orwell[9]&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
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