22edo: Difference between revisions

Line 36:

! Approximate Ratios<ref group="note">{{sg|2.3.5.7.11.17 subgroup}}</ref>

! colspan="3" | [[Ups and Downs Notation]]

! colspan="3" | [[SKULO interval names|SKULO notation]] (K = 1)

! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}

! Audio

|-

Line 166:

| vA4, ^d5

| vG#, ^Ab

| comma-narrow augmented 4th comma-wide diminished 5th

| comma-narrow augmented 4th comma-wide diminished 5th

| kA4 Kd5

| kA4 Kd5

| kG#, KAb

| [[File:0-600.000c_2edo.mp3]]

Line 304:

! colspan="2" | [[Ups and downs notation|Ups and Downs Notation]]

|-

![[5L 2s|Diatonic Interval Names]]

! [[5L 2s|Diatonic Interval Names]]

!Note Names on D

! Note Names on D

|-

| 0

Line 452:

Porcupine 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. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.

The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.

The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.

=== Pentatonic Notation ===

In Pentatonic Notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.

The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.

The keyboard runs {{nowrap|D * * * * F * * * G * * * A * * * * C * * * D}}.

=== Decatonic Notation ===

The Decatonic Notation is 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. 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

Chain 1: {{nowrap|C G D A E}}

Chain 2: γ δ α ε β

Chain 2: {{nowrap|γ δ α ε β}}

The alphabet is, in ascending order: C δ D ε E γ G α A β C

The alphabet is, in ascending order: {{nowrap|C δ D ε E γ G α A β C}}

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.

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.

=== Sagittal Notation ===

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:

When 22edo is treated as generated by a cycle of its fifths, the natural notes {{nowrap|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:

[[File:22edo.png|alt=22edo.png|22edo.png]]

Line 542:

| d2

| Eb

| double-aug unison, double-dim sub3rd

| double-aug unison, double-dim sub3rd

| AA1, dds3

| AA1, dds3

| Dx, Fb3

| Dx, Fb3

| natural 2nd

| N2

Line 642:

| M3

| F#

| double-aug sub3rd, double-dim 4thoid

| double-aug sub3rd, double-dim 4thoid

| AAs3, dd4d

| AAs3, dd4d

| Fx, Gbb

| Fx, Gbb

| natural 4th

| N4

Line 717:

| 11

| 600

| p-Augmented Fourth, p-diminished Fifth, Half-Octave

| p-Augmented Fourth, p-diminished Fifth, Half-Octave

| A4, HO

| aug 4th, dim 5th

| aug 4th, dim 5th

| A4, d5

| Gx, Abb

| Gx, Abb

| double-aug 4thoid, double-dim 5thoid

| double-aug 4thoid, double-dim 5thoid

| AA4d, dd5d

| AA4d, dd5d

| Gx, Abb

| Gx, Abb

| natural 6th

| N6

Line 731:

| vA4, ^d5

| vG#, ^Ab

| comma-narrow augmented 4th

| comma-narrow augmented 4th comma-wide diminished 5th

comma-wide diminished 5th

| kA4 Kd5

| kA4

Kd5

| kG#, KAb

|-

Line 804:

Line 802:

| m6

| Bb

| double-aug 5thoid, double-dim sub7th

| double-aug 5thoid, double-dim sub7th

| AA5d, dds7

| AA5d, dds7

| Ax, Cb3

| Ax, Cb3

| flat 8th

| f8

Line 904:

Line 902:

| A7

| C#

| double-aug sub7th, double-dim octave

| double-aug sub7th, double-dim octave

| AAs7, dd8

| AAs7, dd8

| Cx, Dbb

| Cx, Dbb

| natural 10th

| N10

Line 990:

Line 988:

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

Line 1,002:

Line 1,001:

| {{monzo| 35 -22 }}

| {{mapping| 22 35 }}

| −2.25

| 2.25

| 4.12

Line 1,009:

Line 1,008:

| 250/243, 2048/2025

| {{mapping| 22 35 51 }}

| −0.86

| 2.70

| 4.94

Line 1,016:

Line 1,015:

| 50/49, 64/63, 245/243

| {{mapping| 22 35 51 62 }}

| −1.80

| 2.85

| 5.23

Line 1,023:

Line 1,022:

| 50/49, 55/54, 64/63, 99/98

| {{mapping| 22 35 51 62 76 }}

| −1.11

| 2.90

| 5.33

Line 1,030:

Line 1,029:

| 50/49, 55/54, 64/63, 85/84, 99/98

| {{mapping| 22 35 51 62 76 90 }}

| −1.09

| 2.65

| 4.87

Line 1,045:

Line 1,044:

{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

Line 1,298:

Line 1,297:

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

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Temperaments

Line 1,304:

Line 1,303:

| 1

| 1\22

| [[Escapade]] / [[escaped]] [[Chromo]] [[Ceratitid]]

| [[Escapade]] / [[escaped]] [[Chromo]] [[Ceratitid]]

|-

| 1

Line 1,324:

Line 1,323:

| 2

| 1\22

| [[Shrutar]] / hemipaj [[Comic]]

| [[Shrutar]] / hemipaj [[Comic]]

|-

| 2

Line 1,336:

Line 1,335:

| 2

| 4\22

| [[Astrology]] [[Antikythera]] [[Wizard]]

| [[Astrology]] [[Antikythera]] [[Wizard]]

|-

| 2

Line 1,344:

Line 1,343:

| 11

| 1\22

| [[Undeka]] [[Hendecatonic]]

| [[Undeka]] [[Hendecatonic]]

|}

==Scales==

== Scales ==

''See [[22edo modes]]''.

==Tetrachords ==

== Tetrachords ==

''See [[22edo tetrachords]].''

==Chord names==

== Chord names ==

Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:

Line 1,448:

Line 1,447:

* [[Chords of orwell]]

==Instruments==

== Instruments ==

===Keyboards===

=== Keyboards ===

[[File:22-tone halberstadt layout.png|link=https://en.xen.wiki/w/File:22-tone%20halberstadt%20layout.png|alt=|frameless]]

@@ Line 36: / Line 36: @@
 ! Approximate Ratios<ref group="note">{{sg|2.3.5.7.11.17 subgroup}}</ref>
 ! colspan="3" | [[Ups and Downs Notation]]
-! colspan="3" | [[SKULO interval names|SKULO notation]] (K = 1)
+! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}
 ! Audio
 |-
@@ Line 166: / Line 166: @@
 | vA4, ^d5
 | vG#, ^Ab
-| comma-narrow augmented 4th<br>comma-wide diminished 5th
+| comma-narrow augmented 4th<br />comma-wide diminished 5th
-| kA4<br>Kd5
+| kA4<br />Kd5
 | kG#, KAb
 | [[File:0-600.000c_2edo.mp3]]
@@ Line 304: / Line 304: @@
 ! colspan="2" | [[Ups and downs notation|Ups and Downs Notation]]
 |-
-![[5L 2s|Diatonic Interval Names]]
+! [[5L 2s|Diatonic Interval Names]]
-!Note Names<br />on D
+! Note Names<br />on D
 |-
 | 0
@@ Line 452: / Line 452: @@
 Porcupine 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. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.
-The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.
+The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.
 === Pentatonic Notation ===
 In Pentatonic Notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.
-The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.
+The keyboard runs {{nowrap|D * * * * F * * * G * * * A * * * * C * * * D}}.
 === Decatonic Notation ===
 The Decatonic Notation is 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. 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
+Chain 1: {{nowrap|C G D A E}}
-Chain 2: γ δ α ε β
+Chain 2: {{nowrap|γ δ α ε β}}
-The alphabet is, in ascending order: C δ D ε E γ G α A β C
+The alphabet is, in ascending order: {{nowrap|C δ D ε E γ G α A β C}}
-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.
+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&ndash;D is a fifth, and so is γ&ndash;δ.
 === Sagittal Notation ===
-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:
+When 22edo is treated as generated by a cycle of its fifths, the natural notes {{nowrap|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:
 [[File:22edo.png|alt=22edo.png|22edo.png]]
@@ Line 542: / Line 542: @@
 | d2
 | Eb
-| double-aug unison, <br>double-dim sub3rd
+| double-aug unison,<br />double-dim sub3rd
-| AA1, <br>dds3
+| AA1,<br />dds3
-| Dx, <br>Fb<span style="vertical-align: super;">3 </span>
+| Dx,<br />Fb<span style="vertical-align: super;">3</span>
 | natural 2nd
 | N2
@@ Line 642: / Line 642: @@
 | M3
 | F#
-| double-aug sub3rd, <br>double-dim 4thoid
+| double-aug sub3rd,<br />double-dim 4thoid
-| AAs3, <br>dd4d
+| AAs3,<br />dd4d
-| Fx, <br>Gbb
+| Fx,<br />Gbb
 | natural 4th
 | N4
@@ Line 717: / Line 717: @@
 | 11
 | 600
-| p-Augmented Fourth, <br>p-diminished Fifth, <br>Half-Octave
+| p-Augmented Fourth,<br />p-diminished Fifth,<br />Half-Octave
 | A4, HO
-| aug 4th, <br>dim 5th
+| aug 4th, <br />dim 5th
 | A4, d5
-| Gx, <br>Abb
+| Gx, <br />Abb
-| double-aug 4thoid, <br>double-dim 5thoid
+| double-aug 4thoid,<br />double-dim 5thoid
-| AA4d, <br>dd5d
+| AA4d, <br />dd5d
-| Gx, <br>Abb
+| Gx, <br />Abb
 | natural 6th
 | N6
@@ Line 731: / Line 731: @@
 | vA4, ^d5
 | vG#, ^Ab
-| comma-narrow augmented 4th
+| comma-narrow augmented 4th<br />comma-wide diminished 5th
-comma-wide diminished 5th
+| kA4<br />Kd5
-| kA4
-Kd5
 | kG#, KAb
 |-
@@ Line 804: / Line 802: @@
 | m6
 | Bb
-| double-aug 5thoid, <br>double-dim sub7th
+| double-aug 5thoid,<br />double-dim sub7th
-| AA5d, <br>dds7
+| AA5d,<br />dds7
-| Ax, <br>Cb<span style="vertical-align: super;">3</span>
+| Ax,<br />Cb<span style="vertical-align: super;">3</span>
 | flat 8th
 | f8
@@ Line 904: / Line 902: @@
 | A7
 | C#
-| double-aug sub7th, <br>double-dim octave
+| double-aug sub7th,<br />double-dim octave
-| AAs7, <br>dd8
+| AAs7,<br />dd8
-| Cx, <br>Dbb
+| Cx,<br />Dbb
 | natural 10th
 | N10
@@ Line 990: / Line 988: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 1,002: / Line 1,001: @@
 | {{monzo| 35 -22 }}
 | {{mapping| 22 35 }}
-| −2.25
+| &minus;2.25
 | 2.25
 | 4.12
@@ Line 1,009: / Line 1,008: @@
 | 250/243, 2048/2025
 | {{mapping| 22 35 51 }}
-| −0.86
+| &minus;0.86
 | 2.70
 | 4.94
@@ Line 1,016: / Line 1,015: @@
 | 50/49, 64/63, 245/243
 | {{mapping| 22 35 51 62 }}
-| −1.80
+| &minus;1.80
 | 2.85
 | 5.23
@@ Line 1,023: / Line 1,022: @@
 | 50/49, 55/54, 64/63, 99/98
 | {{mapping| 22 35 51 62 76 }}
-| −1.11
+| &minus;1.11
 | 2.90
 | 5.33
@@ Line 1,030: / Line 1,029: @@
 | 50/49, 55/54, 64/63, 85/84, 99/98
 | {{mapping| 22 35 51 62 76 90 }}
-| −1.09
+| &minus;1.09
 | 2.65
 | 4.87
@@ Line 1,045: / Line 1,044: @@
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic limit|Prime<br>limit]]
+! [[Harmonic limit|Prime<br />limit]]
 ! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
@@ Line 1,298: / Line 1,297: @@
 {| class="wikitable center-1 center-2"
 |-
-! Periods<br>per 8ve
+! Periods<br />per 8ve
 ! Generator
 ! Temperaments
@@ Line 1,304: / Line 1,303: @@
 | 1
 | 1\22
-| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
+| [[Escapade]] / [[escaped]]<br />[[Chromo]]<br />[[Ceratitid]]
 |-
 | 1
@@ Line 1,324: / Line 1,323: @@
 | 2
 | 1\22
-| [[Shrutar]] / hemipaj<br>[[Comic]]
+| [[Shrutar]] / hemipaj<br />[[Comic]]
 |-
 | 2
@@ Line 1,336: / Line 1,335: @@
 | 2
 | 4\22
-| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
+| [[Astrology]]<br />[[Antikythera]]<br />[[Wizard]]
 |-
 | 2
@@ Line 1,344: / Line 1,343: @@
 | 11
 | 1\22
-| [[Undeka]]<br>[[Hendecatonic]]
+| [[Undeka]]<br />[[Hendecatonic]]
 |}
-==Scales==
+== Scales ==
 ''See [[22edo modes]]''.
-==Tetrachords ==
+== Tetrachords ==
 ''See [[22edo tetrachords]].''
-==Chord names==
+== Chord names ==
 Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
@@ Line 1,448: / Line 1,447: @@
 * [[Chords of orwell]]
-==Instruments==
+== Instruments ==
-===Keyboards===
+=== Keyboards ===
 [[File:22-tone halberstadt layout.png|link=https://en.xen.wiki/w/File:22-tone%20halberstadt%20layout.png|alt=|frameless]]