Xenharmonic Wiki - User contributions [en]

19edo chords

2025-12-18T21:18:17Z

YoVariable: /* Augmented chords */ More spacing

In contrast to [[12edo]] chords, [[19edo]] has four instead of the usual two main tertian chord qualities, which opens up completely new territory for eager musicians/microtonalists to explore.

19edo approximates intervals with factors of 2 ([[2/1]]), 3 ([[3/2]]), 5 ([[5/4]], [[5/3]], [[6/5]]), and some intervals involving 7 ([[9/7]], [[27/14]]) quite well. This essentially means that normal chords, like in 12edo, can be represented nicely in 19edo.

Despite that [[enharmonic equivalence]] works differently in 19edo, pitches can be written down with [[Chain-of-fifths notation|standard notation]].

== Triads ==
Note that the cent values of the intervals are approximated. For detailed numbers, see [[19edo]].
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major
| C
| C–E–G
| 0–6–11
| 0–379–695
| [[File:C_(19-EDO).mp3|frameless]]
|-
| Minor
| Cm, Cmin
| C–E♭–G
| 0–5–11
| 0–316–695
| [[File:Cm (19-EDO).mp3|frameless]]
|-
| Supermajor, Major sharp 3
| Csmaj, Csaj, CS, C(♯3), Cmaj(♯3)
| C–E♯–G
| 0–7–11
| 0–442–695
| [[File:Csmaj (19-EDO).mp3|frameless]]
|-
| Subminor, Minor flat 3
| Csmin, Csin, Cs, Cmin(♭3)
| C–E𝄫–G
| 0–4–11
| 0–253–695
| [[File:Csmin (19-EDO).mp3|frameless]]
|-
| Sus4
| Csus4
| C–F–G
| 0–8–11
| 0–505–695
| [[File:Csus4 (19-EDO).mp3|frameless]]
|-
| Sus2
| Csus2
| C–D–G
| 0–3–11
| 0–189–695
| [[File:Csus2 (19-EDO).mp3|frameless]]
|-
| Diminished
| Cdim, C°
| C–E♭–G♭
| 0–5–10
| 0–316–632
| [[File:Cdim (19-EDO).mp3|frameless]]
|-
| Harmonic diminished,
Otonal diminished
| Cm(𝄫5), Cmin(𝄫5), Cd+
| C–E♭–G𝄫
| 0–5–9
| 0–316–568
| {{Todo|inline=1| add audio }}
|-
| Augmented, Magic
| Caug, C+, CJ
| C–E–G♯
| 0–6–12
| 0–379–758
| [[File:Caug (19-EDO).mp3|frameless]]
|}

== Tetrads (sixth/seventh chords) ==
Because of interesting new features – the supermajor seventh and "harmonic" seventh/augmented sixth – new tetrads are possible while existing ones can be preserved.

=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major seventh
| Cmaj7, CM7
| C–E–G–B
| 0–6–11–17
| 0–379–695–1074
| [[File:Cmaj7 (19-EDO).mp3|frameless]]
|-
| Dominant seventh
| C7
| C–E–G–B♭
| 0–6–11–16
| 0–379–695–1011
| [[File:C7 (19-EDO).mp3|frameless]]
|-
| Harmonic seventh,
Major subminor seventh
| Ch7, C(s7), C(𝄫7)
| C–E–G–B𝄫
| 0–6–11–15
| 0–379–695–947
| [[File:Ch7 (19-EDO).mp3|frameless]]
|-
| Major sixth
| C6
| C–E–G–A
| 0–6–11–14
| 0–379–695–884
| [[File:C6 (19-EDO).mp3|frameless]]
|}

=== Minor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Minor seventh
| Cm7
| 0-5-11-16
| 0-316-695-1011
| [[File:Cm7 (19-EDO).mp3|frameless]]
|-
| Minor major seventh
| Cmmaj7, CmM7
| 0-5-11-17
| 0-316-695-1074
| [[File:Cmmaj7 (19-EDO).mp3|frameless]]
|-
| Minor augmented sixth
| Cm+6, Cm(S6), Cm(♯6)
| 0-5-11-15
| 0-316-695-947
| [[File:Cm+6 (19-EDO).mp3|frameless]]
|-
| Minor sixth
| Cm6
| 0-5-11-14
| 0-316-695-884
| [[File:Cm6 (19-EDO).mp3|frameless]]
|-
| Minor seven flat six (NT aeolian seven)
| Cm7(♭6) [Faeol7]
| 0-5-13-16
| 0-316-821-1011
| [[File:Cm7b6 (19-EDO).mp3|frameless]]
|}

=== Supermajor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Supermajor seventh
| Csmaj7, Csaj7, CS(S7), Cmaj7(♯3, ♯7)
| 0-7-11-18
| 0-442-695-1137
| [[File:Csmaj7 (19-EDO).mp3|frameless]]
|}

=== Subminor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Subminor seventh
| Csmin7, Csin7, Cs(s7), Cs(𝄫7), Cmin7(♭3, ♭7)
| 0-4-11-15
| 0-253-695-947
| [[File:Csmin7 (19-EDO).mp3|frameless]]
|}

=== Diminished chords ===
{| class="wikitable"
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Diminished seventh (fully diminished)
| Cdim7, C°7
| 0-5-10-15
| 0-316-632-947
| [[File:Cdim7 (19-EDO).mp3|frameless]]
|-
| Minor seven flat five (half-diminished)
| Cm7(♭5), Cø7
| 0-5-10-16
| 0-316-632-1011
| [[File:Cm7b5 (19-EDO).mp3|frameless]]
|}

=== Augmented chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Augmented seven, Magic seven
| Caug(7), CJ(7), C+(7), C7(♯5)
| 0-6-12-16
| 0-379-758-1011
| [[File:Caug7 (19-EDO).mp3|frameless]]
|-
| Supermajor seven sharp five, Magic supermajor seventh, Magic sharp seven, Augmented sharp seven, Augmented supermajor seventh, Major sharp five sharp seven
| Csmaj7(♯5), Csaj7(♯5), CJ(S7), Caug(♯7), Cmaj(♯5, ♯7)
| 0-6-12-18
| 0-379-758-1137
| [[File:Cmaj7-5 (19-EDO).mp3|frameless]]
|}

== Pentads (ninth chords) ==
=== Major chords ===
=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Major ninth
| Cmaj9, CM9
| 0-6-11-17-22
| 0-379-695-1074-1389
| [[File:Cmaj9 (19-EDO).mp3| frameless]]
|-
| Dominant ninth
| C9
| 0-6-11-16-22
| 0-379-695-1011-1389
| [[File:C9 (19-EDO).mp3| frameless]]
|-
| Dominant seven flat nine
| C7(♭9)
| 0-6-11-16-21
| 0-379-695-1011-1326
| [[File:C7b9 (19-EDO).mp3| frameless]]
|-
| Harmonic ninth,
Major subminor seven nine
| Ch9, C(s7, 9), C(𝄫7, 9)
| 0-6-11-15-22
| 0-379-695-947-1389
| [[File:Ch9 (19-EDO).mp3| frameless]]
|-
| Harmonic seven flat nine
Major subminor seven flat nine
| Ch7(♭9), C(s7, ♭9), C(𝄫7, ♭9)
| 0-6-11-15-21
| 0-379-695-947-1326
| [[File:Ch7b9 (19-EDO).mp3| frameless]]
|}

== Ups and downs notation ==
Various [[19edo]] triads, 6th and 7th chords, named via [[Ups and downs notation|ups and downs]]. Not meant to be exhaustive, but this list does demonstrate the basic rules for naming. The aug 6th and the dim 7th are the same interval, and chords that use that interval can be named as either a 6th chord or a 7th chord.

Highly implausible chords are written as a more plausible [[Chord homonym|homonym]], e.g. 0-8-12 = C4(a5) becomes 8-12-19 = F(d3). "a" stands for augmented and "d" stands for diminished.

{| class="wikitable"
|-
! colspan="2" |third ---->
! d3
! m3
! M3
! a3/d4
! P4
|-
! colspan="2" |Triads with P5
| C(d3)
| Cm
| C
| C(a3)
| C4
|-
! rowspan="2" |Other
triads
!d5
|Cd(d3)
|Cd
|C(b5)
|C(a3b5)
|C4(b5)
|-
!a5
|''G#(a3)''
|''G#a(a3)''
|Ca
|Ca(a3)
|''F(d3)''
|-
! rowspan="5" |Tetrads
with a

P5
! M6
| C6(d3)
| Cm6
| C6
| C6(a3)
| C4,6
|-
! A6
d7
| C(d3)#6
C(d3)d7
| Cm#6
Cmd7
| C,#6
C,d7
| C(a3)#6
C(a3)d7
| C4#6
C4d7
|-
! m7
| C7(d3)
| Cm7
| C7
| C7(a3)
| C4,7
|-
! M7
| CM7(d3)
| CmM7
| CM7
| CM7(a3)
| C4M7
|-
! A7
| Ca7(d3)
| Cm#7
| C,#7
| C(a3)#7
| C4#7
|}

A comma (the actual punctuation mark ",") is spoken as "add", thus C,v7 is "C add-down-seven". The only exception is when a comma separates two numbers, as in C4,7 which is "C four-seven". A comma is written, and "add" is spoken, whenever not doing so would cause confusion with another chord.

4:5:6:7 = C E G vBb is named C add-dim7. To get a shorter name for this important chord, one could call it a harmonic7 chord, or one could borrow from [[color notation]] to call it a har7 chord, written Ch7. Names for subharmonic chords can be similarly shortened.

{| class="wikitable"
|-
! Chord
! Notes
! colspan="2" | Ups and downs name
! colspan="2" | Color name
|-
| 4:5:6:7
| C E G Bbb
| C add-dim-7
| C,d7
| C har7
| Ch7
|-
|4:5:6:7:9
|C E G Bbb D
|C nine dim-7
|C9(d7)
|C har9
|Ch9
|-
| 7:6:5:4
| C Ebb Gbb Bbb
| C dim-3 dim-7 double-dim5
| C(d3)d7(dd5)
| C sub7
| Cs7
|-
| 12:10:8:7
| C Eb G A#
| C minor sharp-6
| Cm#6
| C sub6
| Cs6
|-
|9:7:6:5:4
|C E# G Bb D
|C nine aug-3
|C9(a3)
|Csub9
|Cs9
|}

== See also ==
* [[15edo chord names]]
* [[22edo chord names]]
* [[24edo chord names]]
* [[31edo chord names]]
* [[41edo chord names]]
* [[Kite Guitar chord shapes (downmajor tuning)]]

[[Category:19edo]]
[[Category:Chords]]

19edo chords

2025-12-18T21:17:24Z

YoVariable: /* Augmented chords */ Spacing

In contrast to [[12edo]] chords, [[19edo]] has four instead of the usual two main tertian chord qualities, which opens up completely new territory for eager musicians/microtonalists to explore.

19edo approximates intervals with factors of 2 ([[2/1]]), 3 ([[3/2]]), 5 ([[5/4]], [[5/3]], [[6/5]]), and some intervals involving 7 ([[9/7]], [[27/14]]) quite well. This essentially means that normal chords, like in 12edo, can be represented nicely in 19edo.

Despite that [[enharmonic equivalence]] works differently in 19edo, pitches can be written down with [[Chain-of-fifths notation|standard notation]].

== Triads ==
Note that the cent values of the intervals are approximated. For detailed numbers, see [[19edo]].
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major
| C
| C–E–G
| 0–6–11
| 0–379–695
| [[File:C_(19-EDO).mp3|frameless]]
|-
| Minor
| Cm, Cmin
| C–E♭–G
| 0–5–11
| 0–316–695
| [[File:Cm (19-EDO).mp3|frameless]]
|-
| Supermajor, Major sharp 3
| Csmaj, Csaj, CS, C(♯3), Cmaj(♯3)
| C–E♯–G
| 0–7–11
| 0–442–695
| [[File:Csmaj (19-EDO).mp3|frameless]]
|-
| Subminor, Minor flat 3
| Csmin, Csin, Cs, Cmin(♭3)
| C–E𝄫–G
| 0–4–11
| 0–253–695
| [[File:Csmin (19-EDO).mp3|frameless]]
|-
| Sus4
| Csus4
| C–F–G
| 0–8–11
| 0–505–695
| [[File:Csus4 (19-EDO).mp3|frameless]]
|-
| Sus2
| Csus2
| C–D–G
| 0–3–11
| 0–189–695
| [[File:Csus2 (19-EDO).mp3|frameless]]
|-
| Diminished
| Cdim, C°
| C–E♭–G♭
| 0–5–10
| 0–316–632
| [[File:Cdim (19-EDO).mp3|frameless]]
|-
| Harmonic diminished,
Otonal diminished
| Cm(𝄫5), Cmin(𝄫5), Cd+
| C–E♭–G𝄫
| 0–5–9
| 0–316–568
| {{Todo|inline=1| add audio }}
|-
| Augmented, Magic
| Caug, C+, CJ
| C–E–G♯
| 0–6–12
| 0–379–758
| [[File:Caug (19-EDO).mp3|frameless]]
|}

== Tetrads (sixth/seventh chords) ==
Because of interesting new features – the supermajor seventh and "harmonic" seventh/augmented sixth – new tetrads are possible while existing ones can be preserved.

=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major seventh
| Cmaj7, CM7
| C–E–G–B
| 0–6–11–17
| 0–379–695–1074
| [[File:Cmaj7 (19-EDO).mp3|frameless]]
|-
| Dominant seventh
| C7
| C–E–G–B♭
| 0–6–11–16
| 0–379–695–1011
| [[File:C7 (19-EDO).mp3|frameless]]
|-
| Harmonic seventh,
Major subminor seventh
| Ch7, C(s7), C(𝄫7)
| C–E–G–B𝄫
| 0–6–11–15
| 0–379–695–947
| [[File:Ch7 (19-EDO).mp3|frameless]]
|-
| Major sixth
| C6
| C–E–G–A
| 0–6–11–14
| 0–379–695–884
| [[File:C6 (19-EDO).mp3|frameless]]
|}

=== Minor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Minor seventh
| Cm7
| 0-5-11-16
| 0-316-695-1011
| [[File:Cm7 (19-EDO).mp3|frameless]]
|-
| Minor major seventh
| Cmmaj7, CmM7
| 0-5-11-17
| 0-316-695-1074
| [[File:Cmmaj7 (19-EDO).mp3|frameless]]
|-
| Minor augmented sixth
| Cm+6, Cm(S6), Cm(♯6)
| 0-5-11-15
| 0-316-695-947
| [[File:Cm+6 (19-EDO).mp3|frameless]]
|-
| Minor sixth
| Cm6
| 0-5-11-14
| 0-316-695-884
| [[File:Cm6 (19-EDO).mp3|frameless]]
|-
| Minor seven flat six (NT aeolian seven)
| Cm7(♭6) [Faeol7]
| 0-5-13-16
| 0-316-821-1011
| [[File:Cm7b6 (19-EDO).mp3|frameless]]
|}

=== Supermajor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Supermajor seventh
| Csmaj7, Csaj7, CS(S7), Cmaj7(♯3, ♯7)
| 0-7-11-18
| 0-442-695-1137
| [[File:Csmaj7 (19-EDO).mp3|frameless]]
|}

=== Subminor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Subminor seventh
| Csmin7, Csin7, Cs(s7), Cs(𝄫7), Cmin7(♭3, ♭7)
| 0-4-11-15
| 0-253-695-947
| [[File:Csmin7 (19-EDO).mp3|frameless]]
|}

=== Diminished chords ===
{| class="wikitable"
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Diminished seventh (fully diminished)
| Cdim7, C°7
| 0-5-10-15
| 0-316-632-947
| [[File:Cdim7 (19-EDO).mp3|frameless]]
|-
| Minor seven flat five (half-diminished)
| Cm7(♭5), Cø7
| 0-5-10-16
| 0-316-632-1011
| [[File:Cm7b5 (19-EDO).mp3|frameless]]
|}

=== Augmented chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Augmented seven,
Magic seven
| Caug(7), CJ(7), C+(7), C7(♯5)
| 0-6-12-16
| 0-379-758-1011
| [[File:Caug7 (19-EDO).mp3|frameless]]
|-
| Supermajor seven sharp five,
Magic supermajor seventh,
Magic sharp seven,
Augmented sharp seven,
Augmented supermajor seventh,
Major sharp five sharp seven
| Csmaj7(♯5), Csaj7(♯5), CJ(S7), Caug(♯7), Cmaj(♯5, ♯7)
| 0-6-12-18
| 0-379-758-1137
| [[File:Cmaj7-5 (19-EDO).mp3|frameless]]
|}

== Pentads (ninth chords) ==
=== Major chords ===
=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Major ninth
| Cmaj9, CM9
| 0-6-11-17-22
| 0-379-695-1074-1389
| [[File:Cmaj9 (19-EDO).mp3| frameless]]
|-
| Dominant ninth
| C9
| 0-6-11-16-22
| 0-379-695-1011-1389
| [[File:C9 (19-EDO).mp3| frameless]]
|-
| Dominant seven flat nine
| C7(♭9)
| 0-6-11-16-21
| 0-379-695-1011-1326
| [[File:C7b9 (19-EDO).mp3| frameless]]
|-
| Harmonic ninth,
Major subminor seven nine
| Ch9, C(s7, 9), C(𝄫7, 9)
| 0-6-11-15-22
| 0-379-695-947-1389
| [[File:Ch9 (19-EDO).mp3| frameless]]
|-
| Harmonic seven flat nine
Major subminor seven flat nine
| Ch7(♭9), C(s7, ♭9), C(𝄫7, ♭9)
| 0-6-11-15-21
| 0-379-695-947-1326
| [[File:Ch7b9 (19-EDO).mp3| frameless]]
|}

== Ups and downs notation ==
Various [[19edo]] triads, 6th and 7th chords, named via [[Ups and downs notation|ups and downs]]. Not meant to be exhaustive, but this list does demonstrate the basic rules for naming. The aug 6th and the dim 7th are the same interval, and chords that use that interval can be named as either a 6th chord or a 7th chord.

Highly implausible chords are written as a more plausible [[Chord homonym|homonym]], e.g. 0-8-12 = C4(a5) becomes 8-12-19 = F(d3). "a" stands for augmented and "d" stands for diminished.

{| class="wikitable"
|-
! colspan="2" |third ---->
! d3
! m3
! M3
! a3/d4
! P4
|-
! colspan="2" |Triads with P5
| C(d3)
| Cm
| C
| C(a3)
| C4
|-
! rowspan="2" |Other
triads
!d5
|Cd(d3)
|Cd
|C(b5)
|C(a3b5)
|C4(b5)
|-
!a5
|''G#(a3)''
|''G#a(a3)''
|Ca
|Ca(a3)
|''F(d3)''
|-
! rowspan="5" |Tetrads
with a

P5
! M6
| C6(d3)
| Cm6
| C6
| C6(a3)
| C4,6
|-
! A6
d7
| C(d3)#6
C(d3)d7
| Cm#6
Cmd7
| C,#6
C,d7
| C(a3)#6
C(a3)d7
| C4#6
C4d7
|-
! m7
| C7(d3)
| Cm7
| C7
| C7(a3)
| C4,7
|-
! M7
| CM7(d3)
| CmM7
| CM7
| CM7(a3)
| C4M7
|-
! A7
| Ca7(d3)
| Cm#7
| C,#7
| C(a3)#7
| C4#7
|}

A comma (the actual punctuation mark ",") is spoken as "add", thus C,v7 is "C add-down-seven". The only exception is when a comma separates two numbers, as in C4,7 which is "C four-seven". A comma is written, and "add" is spoken, whenever not doing so would cause confusion with another chord.

4:5:6:7 = C E G vBb is named C add-dim7. To get a shorter name for this important chord, one could call it a harmonic7 chord, or one could borrow from [[color notation]] to call it a har7 chord, written Ch7. Names for subharmonic chords can be similarly shortened.

{| class="wikitable"
|-
! Chord
! Notes
! colspan="2" | Ups and downs name
! colspan="2" | Color name
|-
| 4:5:6:7
| C E G Bbb
| C add-dim-7
| C,d7
| C har7
| Ch7
|-
|4:5:6:7:9
|C E G Bbb D
|C nine dim-7
|C9(d7)
|C har9
|Ch9
|-
| 7:6:5:4
| C Ebb Gbb Bbb
| C dim-3 dim-7 double-dim5
| C(d3)d7(dd5)
| C sub7
| Cs7
|-
| 12:10:8:7
| C Eb G A#
| C minor sharp-6
| Cm#6
| C sub6
| Cs6
|-
|9:7:6:5:4
|C E# G Bb D
|C nine aug-3
|C9(a3)
|Csub9
|Cs9
|}

== See also ==
* [[15edo chord names]]
* [[22edo chord names]]
* [[24edo chord names]]
* [[31edo chord names]]
* [[41edo chord names]]
* [[Kite Guitar chord shapes (downmajor tuning)]]

[[Category:19edo]]
[[Category:Chords]]

19edo chords

2025-12-12T01:16:40Z

YoVariable: Added new chord names and changed the "maj7♯5" to "saj7♯5" or "J(S7)".

In contrast to [[12edo]] chords, [[19edo]] has four instead of the usual two main tertian chord qualities, which opens up completely new territory for eager musicians/microtonalists to explore.

19edo approximates intervals with factors of 2 ([[2/1]]), 3 ([[3/2]]), 5 ([[5/4]], [[5/3]], [[6/5]]), and some intervals involving 7 ([[9/7]], [[27/14]]) quite well. This essentially means that normal chords, like in 12edo, can be represented nicely in 19edo.

Despite that [[enharmonic equivalence]] works differently in 19edo, pitches can be written down with [[Chain-of-fifths notation|standard notation]].

== Triads ==
Note that the cent values of the intervals are approximated. For detailed numbers, see [[19edo]].
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major
| C
| C–E–G
| 0–6–11
| 0–379–695
| [[File:C_(19-EDO).mp3|frameless]]
|-
| Minor
| Cm, Cmin
| C–E♭–G
| 0–5–11
| 0–316–695
| [[File:Cm (19-EDO).mp3|frameless]]
|-
| Supermajor, Major sharp 3
| Csmaj, Csaj, CS, C(♯3), Cmaj(♯3)
| C–E♯–G
| 0–7–11
| 0–442–695
| [[File:Csmaj (19-EDO).mp3|frameless]]
|-
| Subminor, Minor flat 3
| Csmin, Csin, Cs, Cmin(♭3)
| C–E𝄫–G
| 0–4–11
| 0–253–695
| [[File:Csmin (19-EDO).mp3|frameless]]
|-
| Sus4
| Csus4
| C–F–G
| 0–8–11
| 0–505–695
| [[File:Csus4 (19-EDO).mp3|frameless]]
|-
| Sus2
| Csus2
| C–D–G
| 0–3–11
| 0–189–695
| [[File:Csus2 (19-EDO).mp3|frameless]]
|-
| Diminished
| Cdim, C°
| C–E♭–G♭
| 0–5–10
| 0–316–632
| [[File:Cdim (19-EDO).mp3|frameless]]
|-
| Harmonic diminished,
Otonal diminished
| Cm(𝄫5), Cmin(𝄫5), Cd+
| C–E♭–G𝄫
| 0–5–9
| 0–316–568
| {{Todo|inline=1| add audio }}
|-
| Augmented, Magic
| Caug, C+, CJ
| C–E–G♯
| 0–6–12
| 0–379–758
| [[File:Caug (19-EDO).mp3|frameless]]
|}

== Tetrads (sixth/seventh chords) ==
Because of interesting new features – the supermajor seventh and "harmonic" seventh/augmented sixth – new tetrads are possible while existing ones can be preserved.

=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Notes
! Steps
! Cents
! Audio
|-
| Major seventh
| Cmaj7, CM7
| C–E–G–B
| 0–6–11–17
| 0–379–695–1074
| [[File:Cmaj7 (19-EDO).mp3|frameless]]
|-
| Dominant seventh
| C7
| C–E–G–B♭
| 0–6–11–16
| 0–379–695–1011
| [[File:C7 (19-EDO).mp3|frameless]]
|-
| Harmonic seventh,
Major subminor seventh
| Ch7, C(s7), C(𝄫7)
| C–E–G–B𝄫
| 0–6–11–15
| 0–379–695–947
| [[File:Ch7 (19-EDO).mp3|frameless]]
|-
| Major sixth
| C6
| C–E–G–A
| 0–6–11–14
| 0–379–695–884
| [[File:C6 (19-EDO).mp3|frameless]]
|}

=== Minor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Minor seventh
| Cm7
| 0-5-11-16
| 0-316-695-1011
| [[File:Cm7 (19-EDO).mp3|frameless]]
|-
| Minor major seventh
| Cmmaj7, CmM7
| 0-5-11-17
| 0-316-695-1074
| [[File:Cmmaj7 (19-EDO).mp3|frameless]]
|-
| Minor augmented sixth
| Cm+6, Cm(S6), Cm(♯6)
| 0-5-11-15
| 0-316-695-947
| [[File:Cm+6 (19-EDO).mp3|frameless]]
|-
| Minor sixth
| Cm6
| 0-5-11-14
| 0-316-695-884
| [[File:Cm6 (19-EDO).mp3|frameless]]
|-
| Minor seven flat six (NT aeolian seven)
| Cm7(♭6) [Faeol7]
| 0-5-13-16
| 0-316-821-1011
| [[File:Cm7b6 (19-EDO).mp3|frameless]]
|}

=== Supermajor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Supermajor seventh
| Csmaj7, Csaj7, CS(S7), Cmaj7(♯3, ♯7)
| 0-7-11-18
| 0-442-695-1137
| [[File:Csmaj7 (19-EDO).mp3|frameless]]
|}

=== Subminor chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Subminor seventh
| Csmin7, Csin7, Cs(s7), Cs(𝄫7), Cmin7(♭3, ♭7)
| 0-4-11-15
| 0-253-695-947
| [[File:Csmin7 (19-EDO).mp3|frameless]]
|}

=== Diminished chords ===
{| class="wikitable"
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Diminished seventh (fully diminished)
| Cdim7, C°7
| 0-5-10-15
| 0-316-632-947
| [[File:Cdim7 (19-EDO).mp3|frameless]]
|-
| Minor seven flat five (half-diminished)
| Cm7(♭5), Cø7
| 0-5-10-16
| 0-316-632-1011
| [[File:Cm7b5 (19-EDO).mp3|frameless]]
|}

=== Augmented chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Augmented seven,
Magic seven
| Caug(7), CJ(7), C+(7), C7(♯5)
| 0-6-12-16
| 0-379-758-1011
| [[File:Caug7 (19-EDO).mp3|frameless]]
|-
| Supermajor seven sharp five,
Magic supermajor seventh,
Magic sharp seven,
Augmented sharp seven,
Augmented supermajor seventh,
Major sharp five sharp seven
| Csmaj7(♯5), Csaj7(♯5), CJ(S7), Caug(♯7), Cmaj(♯5, ♯7)
| 0-6-12-18
| 0-379-758-1137
| [[File:Cmaj7-5 (19-EDO).mp3|frameless]]
|}

== Pentads (ninth chords) ==
=== Major chords ===
=== Major chords ===
{| class="wikitable"
|-
! Chord name
! Symbol
! Steps
! Cents
! Audio
|-
| Major ninth
| Cmaj9, CM9
| 0-6-11-17-22
| 0-379-695-1074-1389
| [[File:Cmaj9 (19-EDO).mp3| frameless]]
|-
| Dominant ninth
| C9
| 0-6-11-16-22
| 0-379-695-1011-1389
| [[File:C9 (19-EDO).mp3| frameless]]
|-
| Dominant seven flat nine
| C7(♭9)
| 0-6-11-16-21
| 0-379-695-1011-1326
| [[File:C7b9 (19-EDO).mp3| frameless]]
|-
| Harmonic ninth,
Major subminor seven nine
| Ch9, C(s7, 9), C(𝄫7, 9)
| 0-6-11-15-22
| 0-379-695-947-1389
| [[File:Ch9 (19-EDO).mp3| frameless]]
|-
| Harmonic seven flat nine
Major subminor seven flat nine
| Ch7(♭9), C(s7, ♭9), C(𝄫7, ♭9)
| 0-6-11-15-21
| 0-379-695-947-1326
| [[File:Ch7b9 (19-EDO).mp3| frameless]]
|}

== Ups and downs notation ==
Various [[19edo]] triads, 6th and 7th chords, named via [[Ups and downs notation|ups and downs]]. Not meant to be exhaustive, but this list does demonstrate the basic rules for naming. The aug 6th and the dim 7th are the same interval, and chords that use that interval can be named as either a 6th chord or a 7th chord.

Highly implausible chords are written as a more plausible [[Chord homonym|homonym]], e.g. 0-8-12 = C4(a5) becomes 8-12-19 = F(d3). "a" stands for augmented and "d" stands for diminished.

{| class="wikitable"
|-
! colspan="2" |third ---->
! d3
! m3
! M3
! a3/d4
! P4
|-
! colspan="2" |Triads with P5
| C(d3)
| Cm
| C
| C(a3)
| C4
|-
! rowspan="2" |Other
triads
!d5
|Cd(d3)
|Cd
|C(b5)
|C(a3b5)
|C4(b5)
|-
!a5
|''G#(a3)''
|''G#a(a3)''
|Ca
|Ca(a3)
|''F(d3)''
|-
! rowspan="5" |Tetrads
with a

P5
! M6
| C6(d3)
| Cm6
| C6
| C6(a3)
| C4,6
|-
! A6
d7
| C(d3)#6
C(d3)d7
| Cm#6
Cmd7
| C,#6
C,d7
| C(a3)#6
C(a3)d7
| C4#6
C4d7
|-
! m7
| C7(d3)
| Cm7
| C7
| C7(a3)
| C4,7
|-
! M7
| CM7(d3)
| CmM7
| CM7
| CM7(a3)
| C4M7
|-
! A7
| Ca7(d3)
| Cm#7
| C,#7
| C(a3)#7
| C4#7
|}

A comma (the actual punctuation mark ",") is spoken as "add", thus C,v7 is "C add-down-seven". The only exception is when a comma separates two numbers, as in C4,7 which is "C four-seven". A comma is written, and "add" is spoken, whenever not doing so would cause confusion with another chord.

4:5:6:7 = C E G vBb is named C add-dim7. To get a shorter name for this important chord, one could call it a harmonic7 chord, or one could borrow from [[color notation]] to call it a har7 chord, written Ch7. Names for subharmonic chords can be similarly shortened.

{| class="wikitable"
|-
! Chord
! Notes
! colspan="2" | Ups and downs name
! colspan="2" | Color name
|-
| 4:5:6:7
| C E G Bbb
| C add-dim-7
| C,d7
| C har7
| Ch7
|-
|4:5:6:7:9
|C E G Bbb D
|C nine dim-7
|C9(d7)
|C har9
|Ch9
|-
| 7:6:5:4
| C Ebb Gbb Bbb
| C dim-3 dim-7 double-dim5
| C(d3)d7(dd5)
| C sub7
| Cs7
|-
| 12:10:8:7
| C Eb G A#
| C minor sharp-6
| Cm#6
| C sub6
| Cs6
|-
|9:7:6:5:4
|C E# G Bb D
|C nine aug-3
|C9(a3)
|Csub9
|Cs9
|}

== See also ==
* [[15edo chord names]]
* [[22edo chord names]]
* [[24edo chord names]]
* [[31edo chord names]]
* [[41edo chord names]]
* [[Kite Guitar chord shapes (downmajor tuning)]]

[[Category:19edo]]
[[Category:Chords]]

93/67

2025-11-29T09:13:23Z

YoVariable: Added approximation chart

{{Infobox Interval
| Name = banana phone fourth, sexacontaheptimal augmented fourth
| Color name = sisuthiwo 3rd, 67u31o3
}}
'''93/67''', the '''banana phone fourth''' or '''sexacontaheptimal augmented fourth''', is a [[67-limit]] '''narrow [[tritone]]''' measuring about 567.7¢. It represents the interval between the 67th and 93rd harmonics in the [[overtone scales|harmonic series]].

The narrow tritone in [[19edo]] (9\19) approximates it quite well, only being about 0.738¢ sharper than 93/67.

===Etymology===
This interval was named by [[User:YoVariable|YoVariable]] in 2025. The origin of the name comes from Gen Alpha slang, namely associated with the numbers 67, 93, and the song "What Is This Diddy Blud Doing On The Calculator". The lyrics "banana phone ding, ding, dong" and "six seven" naturally lead 93/67 to be called the "banana phone fourth", despite 93 not being mentioned in the song at all. The origin of the name has nothing to do with the fruit or an actual phone.

== Approximation ==
{{Interval edo approximation|93/67}}
== See also ==
* [[Gallery of just intervals]]

[[Category:67-limit intervals]]
[[Category:Tritone]]
[[Category:Rational intervals]]

93/67

2025-11-29T09:10:32Z

YoVariable: NINETY THREE, SIX SEVEN

{{Infobox Interval
| Name = banana phone fourth, sexacontaheptimal augmented fourth
| Color name = sisuthiwo 3rd, 67u31o3
}}
'''93/67''', the '''banana phone fourth''' or '''sexacontaheptimal augmented fourth''', is a [[67-limit]] '''narrow [[tritone]]''' measuring about 567.7¢. It represents the interval between the 67th and 93rd harmonics in the [[overtone scales|harmonic series]].

The narrow tritone in [[19edo]] (9\19) approximates it quite well, only being about 0.738¢ sharper than 93/67.

===Etymology===
This interval was named by [[User:YoVariable|YoVariable]] in 2025. The origin of the name comes from Gen Alpha slang, namely associated with the numbers 67, 93, and the song "What Is This Diddy Blud Doing On The Calculator". The lyrics "banana phone ding, ding, dong" and "six seven" naturally lead 93/67 to be called the "banana phone fourth", despite 93 not being mentioned in the song at all. The origin of the name has nothing to do with the fruit or an actual phone.

[[Category:67-limit intervals]]
[[Category:Tritone]]
[[Category:Gallery of just intervals]]

343edo

2025-10-30T02:20:34Z

YoVariable: Added hyperlink to “YoVariable” mention

{{Infobox ET}}
{{ED intro}}

== Theory ==
343edo is only [[consistent]] to the [[3-odd-limit]] since its errors of [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large. To start with, consider the 2.9.15.7 [[subgroup]], where it [[tempering out|tempers out]] 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.

For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]] (gen. 97\343, per. 343\343). The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072.

=== Odd harmonics ===
{{Harmonics in equal|343}}

=== Subsets and supersets ===
Since 343 factors into 73, 343edo has [[7edo]] and [[49edo]] as its subsets. [[686edo]], which doubles it, gives a good correction to the harmonics 3 and 5.

== Use as a NEJI ==
Of all n-[[afdo]]s where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, [[476afdo]] (''7x17x2x2'') approximates 343edo with the least [[relative error]]. (''See [[User:BudjarnLambeth/Approximating 343edo in afdos|Approximating 343edo in afdos]].)''

343edo could be approximated into 476afdo as a [[neji]] scale. Doing so would make it an over-17-by-7 scale (when viewed through a [[primodal]] lens). (''[[User:BudjarnLambeth/Approximating 343edo in afdos#Scala file|Scala file]].)''

It would make sense to use smaller over-17, over-7, or over-17-by-7 JI scales as subsets of this neji.

== Regular temperament properties ==
343edo is on the [[optimal ET sequence]] of [[gammy]] temperament (343be, 10\343 generator, 2/1 period), [[protolangwidge]] temperament (343, 200\343 g, 2/1 p) and [[anthoine]] temperament (343dd, 110\343 g, 2/1 p).

343edo might potentially be useful for [[49th-octave temperaments]] ''(see [[Fractional-octave temperaments]])'', this is something which hasn't been explored yet.

{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| -1087 343 }}
| {{mapping| 343 1087 }}
| +0.1569
| 0.1569
| 4.48
|-
| 2.9.5
| {{monzo| -27 -1 13 }}, {{monzo| 40 -28 21 }}
| {{mapping| 343 1087 796 }}
| +0.3162
| 0.2592
| 7.41
|-
| 2.9.5.7
| 118098/117649, 7381125/7340032, 9765625/9680832
| {{mapping| 343 1087 796 963 }}
| +0.2130
| 0.2869
| 8.20
|}

== Octave stretch or compression ==
If one is using 343edo, it's probably either for a specific temperament, or because of its good [[prime]]s 2, 7, 17 and 19, which will inform how one might want to [[octave stretch]] or compress it.

; Using for a temperament
[[TE]] octave stretch:
* For 13-limit gammy
** Octave size: 1200.437{{c}}

* For 7-limit anthoine
** Octave size: 1199.630{{c}}

; Using for primes 2, 7, 17, 19
If one is using 343 for its accurate 2.7.17.19 intervals, one will probably not want to use 343edo with [[wart]]s a, d, g or h.

That leaves the following [[TE]] tunings for the [[19-limit]]:

* 343cf
** Octave size: 1199.643{{c}}
** TE error: 0.363{{c}}/octave

* 343c
** Octave size: 1199.761{{c}}
** TE error: 0.382{{c}}/octave

* 343 (patent val)
** Octave size: 1199.950{{c}}
** TE error: 0.395{{c}}/octave

* 343e
** Octave size: 1200.076{{c}}
** TE error: 0.418{{c}}/octave

* 343f
** Octave size: 1199.831{{c}}
** TE error: 0.431{{c}}/octave

* 343ce
** Octave size: 1199.888{{c}}
** TE error: 0.461{{c}}/octave
{{Harmonics in cet|3.498|intervals=odd|title=Odd harmonics in TE-tuned 343cf}}

== Scales ==
343edo includes every 49edo scale (see [[49edo#Scales]]).

==== Lucite[23] ====
'''Lucite[23]''' is a 23-tone [[MOS scale]] discovered by [[Gordon Wery]] in October 2025:
* 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13

; Properties
It is very similar to [[23edo]] and can be used as a [[well temperament]] of 23edo.

In his post on Discord describing it, Wery said of the scale:

"''Basically a more complicated version of 23edo, centered around a more minor (less neutral) anti-diatonic scale.''

''This scale has a sort of glassy quality, ample neutral seconds, and two sets of dual fifths--a true dual dual fifth scale.''"

It is [[generator|generated]] by 30\343 (104.956{{c}}).

Lucite[23] can be generalised into a 17-limit [[regular temperament]] called '''[[User:BudjarnLambeth/Regular temperament interpretation of lucite23|lucite temperament]]'''.

; Naming
Lucite is another name for acrylic glass.

Wery named the temperament "lucite" because musically, it sounds like frosted glass (perhaps to do with the timbre/partials of struck glass).

Some coincidences that make the name "lucite" particularly fitting
* Lucite is often installed in double layers in building, and lucite temperament has two sizes of perfect fifth-like interval.
* Lucite[23] is close to ripple[23], but turned inside out; and lucite is reflective and clear like water, but solid instead of liquid
* Lucite is an especially lightweight material, and lucite temperament is lightweight in the way it only needs 18 generators to reach every 17-limit prime.

; Subsets
* [[Modmos]] of lucite[6]: 60 60 30 40 93 60

=== Other MOS scales ===
* Amity[7]: 52 52 45 52 45 52 45
* Amity[11]: 45 7 45 45 7 45 7 45 45 7 42
* Amity[18]: 7 38 7 38 7 7 38 7 7 38 7 38 7 7 38 7 38 7
* Amity[25]: 7 31 7 7 7 31 7 7 31 7 7 7 31 7 7 7 31 7 7 31 7 7 7 31 7
* Amity[32]: 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7
* Amity[39]: 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7
* Amity[53]: 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7
** ''Try approximating scales fron 53edo ([[53edo#Scales]]) within the amity[53] scale''
* Amity[99]: 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3
** ''Try approximating scales fron 99edo ([[99edo#Scales]]) within the amity[99] scale''
* Lucite[23]: 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13
** ''Try approximating scales fron 23edo ([[23edo#Scales]]) within the lucite[23] scale''
* Lucite[34]: 13 4 13 13 4 13 13 4 13 13 4 13 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13
** ''Try approximating scales fron 34edo ([[34edo#Scales]]) within the lucite[34] scale''

=== 343ed16 ===
'''343ed16''' is contained within 343edo (it is every 4th step of 343edo). It is like [[86edo]] with the [[octave stretching|octave stretched]] by 3.5 [[cents]].

It is quite similar to [[136edt]].

Compared to 86edo it improves harmonics 3, 5, 7 and 11. Its mappings of multiple-of-2 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.

Many temperaments and scales from 86edo can be used here in 343ed16 too.
{{Harmonics in equal|343|16|1|intervals=integer|columns=12}}
{{Harmonics in equal|86|2|1|intervals=integer|columns=12|collapsed=1|title=86edo for comparison}}

=== 34.3edo ===
'''34.3edo''' is contained within 343edo (it is every 10th step of 343edo). It is like [[34edo]] with the [[octave shrinking|octave compressed]] by 11.51 [[cents]].

It has a step size of 34.985{{c}}.

It was discovered by chaseofspades513 and [[YoVariable]] on the [[Xenharmonic Alliance]] Discord server and further described by [[Gordon Wery]].

Compared to 34edo it improves harmonics 7, 11 and 13, at the expense of 2, 3 and 5. Its mappings of multiple-of-2 and multiple-of-3 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.

Many temperaments and [[34edo#Scales|scales from 34edo]] can be used here in 34.3edo too.
{{Harmonics in cet|34.985|intervals=integer|columns=12|title=Approximation of harmonics in 34.3edo}}
{{Harmonics in equal|34|2|1|intervals=integer|columns=12|collapsed=1|title=34edo for comparison}}

=== Scales approximated from JI ===
* [[4 of 7-17-19-21-51 pentany]]: 96 50 55 96 46 (sounds like minor pentatonic)
* [[4 of 7-17-19-21-51 by 3/2 tetrapentany]]: 3 9 46 4 34 21 29 8 47 3 9 46 38 46
* [[7-17-19-21 hexany]]: 50 46 50 55 87 55 (sounds like minor hexatonic)
* [[7-17-19-21 by 3/2 trihexany]]: 3 47 8 38 12 38 8 47 3 46 9 29 9 46
* [[9afdo]]: 40 37 34 32 30 28 52 47 43
* [[18afdo]]: 21 19 19 18 17 17 16 16 15 15 14 14 27 25 24 23 22 21
* [[36afdo]]: 11 10 9 10 9 10 9 9 8 9 8 9 8 8 8 8 7 8 7 8 7 7 7 7 14 13 13 12 12 12 12 11 11 11 11 10
* [[72afdo]]: 5 6 5 5 4 5 5 5 5 4 5 5 4 5 4 5 4 4 5 4 4 4 5 4 4 4 4 4 4 4 4 4 4 3 4 4 4 3 4 4 3 4 4 3 4 3 4 3 7 7 6 7 6 7 6 6 6 6 6 6 6 6 6 5 6 5 6 5 6 5 5 5

=== Other scales ===
* [[Equiheptatonic]] (as from [[7edo]]): 49 49 49 49 49 49 49
* [[49edo]]: 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

== Music ==
; [[Budjarn Lambeth]]
* [https://youtu.be/aWqdWHSk5J4 ''Odd Findings in the Caves''] (2025) - ''uses two copies of lucite[23]''

User:YoVariable

2025-10-12T03:34:54Z

YoVariable:

Hello! My name is YoVariable (she/her), but you can just call me Variable. I am a xenharmonic musician who focuses on [[Meantone|meantone]] edos like [[19edo|19]], [[31edo|31]], and [[43edo]], and [[Superpyth|superpyth]] edos like [[22edo|22]], [[27edo|27]], and [[49edo]]. You may find me editing superpyth or meantone edo pages occasionally.

I also created the "Exploring 22edo" series (TBD) on YouTube. I have taken an interest in schismatic edos, like [[41edo|41]] and [[53edo]], but tend to avoid them due to their size. I focus on the practicality of each edo, as I am a pianist and guitarist (I also play the Lumatone), so I tend to stay within the [[13edo]] to [[46edo]] range. If I want to play in a larger edo, I would use subsets.

I am semi-active in [[Gordon Wery]]'s livestreams and stream occasionally on YouTube and Twitch. My Discord is YoVariable.

== External links ==
* YouTube: https://www.youtube.com/@YoVariable
* Twitch: https://twitch.tv/yovariable

49edo

2025-09-02T02:07:33Z

YoVariable: /* Music */ Added new 49edo cover

{{Infobox ET}}
{{ED intro}}

== Theory ==
49edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]), and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit.

=== Harmonics ===
{{Harmonics in equal|49}}

=== Subsets and supersets ===
Since 49 factors into {{factorization|49}}, 49edo contains [[7edo]] as its only nontrivial subset. 49edo is the first square edo with a [[enfactoring|non-enfactored]] diatonic fifth.

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.000
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 24.490
| [[50/49]]
| {{UDnote|step=1}}
|-
| 2
| 48.980
| ''[[28/27]]'', [[36/35]], ''[[49/48]]'', ''[[81/80]]''
| {{UDnote|step=2}}
|-
| 3
| 73.469
| [[22/21]], [[25/24]], ''[[33/32]]''
| {{UDnote|step=3}}
|-
| 4
| 97.959
| ''[[16/15]]'', [[21/20]]
| {{UDnote|step=4}}
|-
| 5
| 122.449
| [[15/14]]
| {{UDnote|step=5}}
|-
| 6
| 146.939
| [[12/11]]
| {{UDnote|step=6}}
|-
| 7
| 171.429
| [[10/9]], [[11/10]]
| {{UDnote|step=7}}
|-
| 8
| 195.918
| [[28/25]]
| {{UDnote|step=8}}
|-
| 9
| 220.408
| [[8/7]], ''[[9/8]]''
| {{UDnote|step=9}}
|-
| 10
| 244.898
| 125/108, 144/125
| {{UDnote|step=10}}
|-
| 11
| 269.388
| [[7/6]]
| {{UDnote|step=11}}
|-
| 12
| 293.878
| [[25/21]], [[33/28]]
| {{UDnote|step=12}}
|-
| 13
| 318.367
| [[6/5]]
| {{UDnote|step=13}}
|-
| 14
| 342.857
| [[11/9]]
| {{UDnote|step=14}}
|-
| 15
| 367.347
| [[27/22]]
| {{UDnote|step=15}}
|-
| 16
| 391.837
| [[5/4]]
| {{UDnote|step=16}}
|-
| 17
| 416.327
| [[14/11]]
| {{UDnote|step=17}}
|-
| 18
| 440.816
| [[9/7]]
| {{UDnote|step=18}}
|-
| 19
| 465.306
| 125/96, ''162/125''
| {{UDnote|step=19}}
|-
| 20
| 489.796
| [[4/3]], ''[[21/16]]''
| {{UDnote|step=20}}
|-
| 21
| 514.286
| [[75/56]]
| {{UDnote|step=21}}
|-
| 22
| 538.776
| [[15/11]], ''[[27/20]]''
| {{UDnote|step=22}}
|-
| 23
| 563.265
| [[11/8]]
| {{UDnote|step=23}}
|-
| 24
| 587.755
| [[7/5]]
| {{UDnote|step=24}}
|-
| 25
| 612.245
| [[10/7]]
| {{UDnote|step=25}}
|-
| 26
| 636.735
| [[16/11]]
| {{UDnote|step=26}}
|-
| 27
| 661.244
| [[22/15]], ''[[40/27]]''
| {{UDnote|step=27}}
|-
| 28
| 685.714
| [[112/75]]
| {{UDnote|step=28}}
|-
| 29
| 710.204
| [[3/2]], ''[[32/21]]''
| {{UDnote|step=29}}
|-
| 30
| 734.694
| ''125/81'', 192/125
| {{UDnote|step=30}}
|-
| 31
| 759.184
| [[14/9]]
| {{UDnote|step=31}}
|-
| 32
| 783.673
| [[11/7]]
| {{UDnote|step=32}}
|-
| 33
| 808.163
| [[8/5]]
| {{UDnote|step=33}}
|-
| 34
| 832.653
| [[44/27]]
| {{UDnote|step=34}}
|-
| 35
| 857.143
| [[18/11]]
| {{UDnote|step=35}}
|-
| 36
| 881.633
| [[5/3]]
| {{UDnote|step=36}}
|-
| 37
| 906.122
| [[42/25]], [[56/33]]
| {{UDnote|step=37}}
|-
| 38
| 930.612
| [[12/7]]
| {{UDnote|step=38}}
|-
| 39
| 955.102
| 125/72, 216/125
| {{UDnote|step=39}}
|-
| 40
| 979.592
| [[7/4]], ''[[16/9]]''
| {{UDnote|step=40}}
|-
| 41
| 1004.082
| [[25/14]]
| {{UDnote|step=41}}
|-
| 42
| 1028.571
| [[9/5]], [[20/11]]
| {{UDnote|step=42}}
|-
| 43
| 1053.061
| [[11/6]]
| {{UDnote|step=43}}
|-
| 44
| 1077.551
| [[28/15]]
| {{UDnote|step=44}}
|-
| 45
| 1102.041
| ''[[15/8]]'', [[40/21]]
| {{UDnote|step=45}}
|-
| 46
| 1126.531
| [[21/11]], [[48/25]], ''[[64/33]]''
| {{UDnote|step=46}}
|-
| 47
| 1151.020
| ''[[27/14]]'', [[35/18]], ''[[96/49]]'', ''[[160/81]]''
| {{UDnote|step=47}}
|-
| 48
| 1175.510
| [[49/25]]
| {{UDnote|step=48}}
|-
| 49
| 1200.000
| [[2/1]]
| {{UDnote|step=49}}
|}
<nowiki />* Based on 49edo's 11-limit patent val {{val| 49 78 114 138 170 }} mapping

== Notation ==
=== Ups and downs notation ===
49edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}

Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}

=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:49-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 589 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
default [[File:49-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:49-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 534 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
default [[File:49-EDO_Revo_Sagittal.svg]]
</imagemap>

== Approximation to JI ==
[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|49}}

=== Zeta peaks ===
The strongest [[The Riemann zeta function and tuning|local zeta peak]] around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for [[Triple BP]].

== Approximation to irrational intervals ==
=== Acoustic ϕ and ϕϕ−1 ===
49edo has a very close approximation of both [[acoustic phi]] and ϕϕ-1, a kind of logarithmic phi that divides [[acoustic phi]] logarithmically by phi ([[Logarithmic phi|instead of dividing 2/1]]).

ϕϕ-1 has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales within [[acoustic phi]].

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
! #\49
|-
| {{nowrap|ϕ / ϕϕ−1 {{=}} ϕ(2 − ϕ)}}
| 0.155
| 13
|-
| ϕ
| −0.437
| 34
|-
| ϕϕ−1
| −0.592
| 21
|}

Not until [[592edo|592]] do we find a better edo in terms of relative error on these two intervals (but whose edo-steps upon which these intervals are mapped are not based on the Fibonacci sequence, unlike 49edo).

=== Music ===
* [https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕϕ−1. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5

== 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 (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 78 -49 }}
| {{mapping| 49 78 }}
| −2.60
| 2.60
| 10.62
|-
| 2.3.5
| 15625/15552, 20480/19683
| {{mapping| 49 78 114 }}
| −2.53
| 2.12
| 8.69
|-
| 2.3.5.7
| 64/63, 245/243, 3125/3087
| {{mapping| 49 78 114 138 }}
| −2.85
| 1.92
| 7.87
|-
| 2.3.5.7.11
| 64/63, 100/99, 245/243, 1331/1323
| {{mapping| 49 78 114 138 170 }}
| −2.97
| 1.74
| 7.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 1\49
| 24.5
| 99/98
| [[Sengagen]]
|-
| 1
| 4\49
| 98.0
| 16/15
| [[Passion]]
|-
| 1
| 6\49
| 146.9
| 12/11
| [[Bohpier]]
|-
| 1
| 8\49
| 195.9
| 28/25
| [[Didacus]]
|-
| 1
| 11\49
| 269.4
| 7/6
| [[Infraorwell]]
|-
| 1
| 12\49
| 293.9
| 25/21
| [[Kleiboh]]
|-
| 1
| 13\49
| 318.4
| 6/5
| [[Catalan]]
|-
| 1
| 16\49
| 391.8
| 5/4
| [[Magus]]
|-
| 1
| 17\49
| 416.3
| 14/11
| [[Sqrtphi]]
|-
| 1
| 18\49
| 440.8
| 9/7
| [[Clyde]]
|-
| 1
| 19\49
| 465.3
| 55/36
| [[Semisept]]
|-
| 1
| 20\49
| 489.8
| 4/3
| [[Superpyth]]
|-
| rowspan="2" | 7
| rowspan="2" | 20\49 (1\49)
| rowspan="2" | 489.8 (24.5)
| 4/3 (250/243)
| [[Sevond]] (49)
|-
| 4/3 (25/24)
| style="text-align: left;" | [[Seville]] (49c)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
=== MOS scales ===
{{main|List of MOS scales in 49edo}}

== Instruments ==
=== Lumatone ===
* [[Lumatone mapping for 49edo]]

=== Skip fretting ===
'''Skip fretting system 49 3 7''' is a [[skip fretting]] system for [[49edo]]. All examples are for 5-string bass.

; Harmonics
1/1: string 2 open

2/1: not easily accessible

3/2: string 4 fret 5 and string 1 fret 12

5/4: string 3 fret 3

7/4: string 3 fret 11

11/8: string 3 fret 5

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation (Demo, January 2022)''] (2023)

; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023)

; [[YoVariable]]
* [https://www.youtube.com/watch?v=GHslu-ZWspk The Cure - ''Boys Don't Cry'' (cover)] (2025)

[[Category:Archytas]]
[[Category:Ares]]
[[Category:Listen]]
[[Category:Superpyth]]

26edo

2025-06-25T09:59:28Z

YoVariable: Fixed typo “edoteps” —> “edosteps”

{{interwiki
| de = 26-EDO
| en = 26edo
| es =
| ja =
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
26edo has a [[3/2|perfect fifth]] of about 692 cents and [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s|diatonic scale]].

In the [[7-limit]], it tempers out [[50/49]], [[525/512]], and [[875/864]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log27.

26edo's minor sixth (1.6158) is very close to {{nowrap|''φ'' ≈ 1.6180}} (i.e. the golden ratio).

With a fifth of 15 steps, it can be equally divided into 3 or 5, supporting [[slendric]] temperament and [[bleu]] temperament respectively.

The structure of 26edo is an interesting beast, with various approaches relating it to various rank-2 temperaments.

# In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which provides an interesting structure but unsatisfying intonation due mainly to the poorly tuned thirds. Extending meantone harmony to the 7-limit is quite intuitive; for example, augmented becomes supermajor, and diminished becomes subminor. Simple mappings for harmonics up to 13 are also achieved.
# As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].
# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and {{monzo| -3 0 0 6 -4 }}. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The {{monzo| -3 0 0 6 -4 }} comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.
# We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
# It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.

Its step of 46.2{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible. In other words, there is a common perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]].

=== Odd harmonics ===
{{Harmonics in equal|26}}

=== Subsets and supersets ===
26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing harmonics 5 and 9 through 23 (including direct approximations) with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics.

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! Degrees
! [[Cent]]s
! Approximate ratios<ref group="note">{{sg|limit=13-limit}}</ref>
! Interval name
! Example in D
! [[SKULO interval names|SKULO]] [[SKULO interval names|Interval name]]
! Example in D
! colspan="2" | [[Solfege|Solfeges]]
|-
| 0
| 0.00
| 1/1
| P1
| D
| P1
| D
| da
| do
|-
| 1
| 46.15
| [[33/32]], [[49/48]], [[36/35]], [[25/24]]
| A1
| D#
| A1, S1
| D#, SD
| du
| di
|-
| 2
| 92.31
| [[21/20]], [[22/21]], [[26/25]]
| d2
| Ebb
| sm2
| sEb
| fro
| rih
|-
| 3
| 138.46
| [[12/11]], [[13/12]], [[14/13]], [[16/15]]
| m2
| Eb
| m2
| Eb
| fra
| ru
|-
| 4
| 184.62
| [[9/8]], [[10/9]], [[11/10]]
| M2
| E
| M2
| E
| ra
| re
|-
| 5
| 230.77
| [[8/7]], [[15/13]]
| A2
| E#
| SM2
| SE
| ru
| ri
|-
| 6
| 276.92
| [[7/6]], [[13/11]], [[33/28]]
| d3
| Fb
| sm3
| sF
| no
| ma
|-
| 7
| 323.08
| [[135/112]], [[6/5]]
| m3
| F
| m3
| F
| na
| me
|-
| 8
| 369.23
| [[5/4]], [[11/9]], [[16/13]]
| M3
| F#
| M3
| F#
| ma
| muh/mi
|-
| 9
| 415.38
| [[9/7]], [[14/11]], [[33/26]]
| A3
| Fx
| SM3
| SF#
| mu
| maa
|-
| 10
| 461.54
| [[21/16]], [[13/10]]
| d4
| Gb
| s4
| sG
| fo
| fe
|-
| 11
| 507.69
| [[75/56]], [[4/3]]
| P4
| G
| P4
| G
| fa
| fa
|-
| 12
| 553.85
| [[11/8]], [[18/13]]
| A4
| G#
| A4
| G#
| fu/pa
| fu
|-
| 13
| 600.00
| [[7/5]], [[10/7]]
| AA4, dd5
| Gx, Abb
| SA4, sd5
| SG#, sAb
| pu/sho
| fi/se
|-
| 14
| 646.15
| [[16/11]], [[13/9]]
| d5
| Ab
| d5
| Ab
| sha/so
| su
|-
| 15
| 692.31
| [[112/75]], [[3/2]]
| P5
| A
| P5
| A
| sa
| sol
|-
| 16
| 738.46
| [[32/21]], [[20/13]]
| A5
| A#
| S5
| SA
| su
| si
|-
| 17
| 784.62
| [[11/7]], [[14/9]]
| d6
| Bbb
| sm6
| sBb
| flo
| leh
|-
| 18
| 830.77
| [[13/8]], [[8/5]]
| m6
| Bb
| m6
| Bb
| fla
| le/lu
|-
| 19
| 876.92
| [[5/3]], [[224/135]]
| M6
| B
| M6
| B
| la
| la
|-
| 20
| 923.08
| [[12/7]], [[22/13]]
| A6
| B#
| SM6
| SB
| lu
| li
|-
| 21
| 969.23
| [[7/4]], [[26/15]]
| d7
| Cb
| sm7
| sC
| tho
| ta
|-
| 22
| 1015.38
| [[9/5]], [[16/9]], [[20/11]]
| m7
| C
| m7
| C
| tha
| te
|-
| 23
| 1061.54
| [[11/6]], [[13/7]], [[15/8]], [[24/13]]
| M7
| C#
| M7
| C#
| ta
| tu/ti
|-
| 24
| 1107.69
| [[21/11]], [[25/13]], [[40/21]]
| A7
| Cx
| SM7
| SC#
| tu
| to
|-
| 25
| 1153.85
| [[64/33]], [[96/49]], [[35/18]], [[48/25]]
| d8
| Db
| d8, s8
| Db, sD
| do
| da
|-
| 26
| 1200.00
| 2/1
| P8
| D
| P8
| D
| da
| do
|}

=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align:center;"
|-
! Quality
! Color
! Monzo Format
! Examples
|-
| diminished
| zo
| {a, b, 0, 1}
| 7/6, 7/4
|-
| minor
| fourthward wa
| {a, b}, b < -1
| 32/27, 16/9
|-
| "
| gu
| {a, b, -1}
| 6/5, 9/5
|-
| major
| yo
| {a, b, 1}
| 5/4, 5/3
|-
| "
| fifthward wa
| {a, b}, b > 1
| 9/8, 27/16
|-
| augmented
| ru
| {a, b, 0, -1}
| 9/7, 12/7
|}

All 26edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Spelling certain chords properly may require triple sharps and flats, especially if the tonic is anything other than the 11 keys in the Eb-C# range. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
! [[Kite's color notation|Color of the 3rd]]
! JI chord
! Notes as Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo
| 6:7:9
| 0-6-15
| C Ebb G
| C(b3) or C(d3)
| C flat-three or C dim-three
|-
| gu
| 10:12:15
| 0-7-15
| C Eb G
| Cm
| C minor
|-
| yo
| 4:5:6
| 0-8-15
| C E G
| C
| C major or C
|-
| ru
| 14:18:21
| 0-9-15
| C E# G
| C(#3) or C(A3)
| C sharp-three or C aug-three
|}

For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].

== Notation ==
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[12edo#Sagittal notation|12]], and [[19edo#Sagittal notation|19]], is a subset of the notation for [[52edo#Sagittal notation|52-EDO]], and is a superset of the notation for [[13edo#Sagittal notation|13-EDO]].

==== Evo flavor ====
<imagemap>
File:26-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:26-EDO_Evo_Sagittal.svg]]
</imagemap>

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
<imagemap>
File:26-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 511 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:26-EDO_Revo_Sagittal.svg]]
</imagemap>

== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|26}}

=== Zeta peak index ===
{{ZPI
| zpi = 100
| steps = 25.9356996537225
| step size = 46.2682717652372
| tempered height = 5.545073
| pure height = 4.164318
| integral = 1.031155
| gap = 14.793013
| octave = 1202.97506589617
| consistent = 14
| distinct = 9
}}

== Approximation to irrational intervals ==
26edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| 2ϕ / ϕ
| 0.858
|-
| ϕ
| 2.321
|-
| π
| 2.820
|-
| 2ϕ
| 3.179
|-
| π/ϕ
| 5.141
|-
| 2ϕ / π
| 5.999
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma basis]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -41 26 }}
| {{mapping| 26 41 }}
| +3.043
| 3.05
| 6.61
|-
| 2.3.5
| 81/80, 78125/73728
| {{mapping| 26 41 60 }}
| +4.489
| 3.22
| 6.98
|-
| 2.3.5.7
| 50/49, 81/80, 405/392
| {{mapping| 26 41 60 73 }}
| +3.324
| 3.44
| 7.45
|-
| 2.3.5.7.11
| 45/44, 50/49, 81/80, 99/98
| {{mapping| 26 41 60 73 90 }}
| +2.509
| 3.48
| 7.53
|-
| 2.3.5.7.11.13
| 45/44, 50/49, 65/64, 78/77, 81/80
| {{mapping| 26 41 60 73 90 96 }}
| +2.531
| 3.17
| 6.87
|-
| 2.3.5.7.11.13.17
| 45/44, 50/49, 65/64 78/77, 81/80, 85/84
| {{mapping| 26 41 60 73 90 96 106 }}
| +2.613
| 2.94
| 6.38
|-
| 2.3.5.7.11.13.17.19
| 45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84
| {{mapping| 26 41 60 73 90 96 106 110 }}
| +2.894
| 2.85
| 6.18
|}
* 26et is lower in relative error than any previous equal temperaments in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]] (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are [[27edo|27eg]], 27eg, [[29edo|29g]], and [[46edo|46]], respectively.

=== Rank-2 Temperaments ===
* [[List of 26et rank two temperaments by badness]]
* [[List of edo-distinct 26et rank two temperaments]]

Important mos scales include (in addition to ones found in [[13edo]]):
* [[Flattone]][7] (diatonic) 4443443 (15\26, 1\1)
* [[Flattone]][12] (chromatic) 313131331313 (15\26, 1\1)
* [[Flattone]][19] (enharmonic) 2112112112121121121 (15\26, 1\1)
* [[Orgone]][7] 5525252 (7\26, 1\1)
* [[Orgone]][11] 32322322322 (7\26, 1\1)
* [[Orgone]][15] 212212221222122 (7\26, 1\1)
* [[Lemba]][6] 553553 (5\26, 1\2)
* [[Lemba]][10] 3232332323 (5\26, 1\2)
* [[Lemba]][16] 2122122121221221 (5\26, 1\2)

{| class="wikitable center-all left-3"
|-
! Periods per 8ve
! Generator
! Temperaments
|-
| 1
| 1\26
| [[Quartonic]] / [[quarto]]
|-
| 1
| 3\26
| [[Glacier]] / [[bleu]] / [[jerome]] / [[secund]]
|-
| 1
| 5\26
| [[Cynder]] / [[mothra]]
|-
| 1
| 7\26
| [[Orgone]] / [[superkleismic]]
|-
| 1
| 9\26
| [[Wesley]] / [[roman]]
|-
| 1
| 11\26
| [[Flattone]] / [[flattertone]]
|-
| 2
| 1\26
| [[Elvis]]
|-
| 2
| 2\26
| [[Injera]]
|-
| 2
| 3\26
| [[Fifive]] / [[crepuscular]]
|-
| 2
| 4\26
| [[Dubbla]] [[Unidec]] / [[hendec]]
|-
| 2
| 5\26
| [[Lemba]]
|-
| 2
| 6\26
| [[Doublewide]] / [[cavalier]]
|-
| 13
| 1\26
| [[Triskaidekic]]
|}

=== Hendec in 26et ===
[[Hendec]], the 13-limit {{nowrap|26 & 46}} temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy.

=== Commas ===
26et [[tempering out|tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 26 41 60 73 90 96 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name(s)
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymos comma, meantone comma
|-
| 5
| <abbr title="381520424476945831628649898809/381469726562500000000000000000">(60 digits)</abbr>
| {{monzo| -17 62 -35 }}
| 0.23
| Quadla-sepquingu
| [[Senior comma]]
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Sarurutriyo
| Octagar comma
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| <abbr title="201768035/201326592">(18 digits)</abbr>
| {{monzo| -26 -1 1 9 }}
| 3.79
| Latritrizo-ayo
| [[Wadisma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Thuzoyo
| Animist comma
|}

== Scales ==
=== Orgone temperament ===
[[Andrew Heathwaite]] first proposed [[Orgonia|orgone]] temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:

The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L_3s|4L 3s (mish)]].

The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.

The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. [[MOSScales|MOS]] of type [[4L_7s|4L 7s]].

The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108 1200.

The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.

[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]]

=== Additional scalar bases available ===
Since the perfect 5th in 26edo spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).

-Igs

=== MOS scales ===
''See [[List of MOS scales in 26edo]]''

== Instruments ==
James Fenn's midi keyboard.
[[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]]

== Literature ==
[http://www.ronsword.com Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.]

== Music ==
{{Catrel|26edo tracks}}

=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=LUNOFjiyZ0Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=dlXFoIoc_uk "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=K7oTEXgmdKY ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

=== 21st century ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
* [https://www.youtube.com/watch?v=f5eYIH3TO4o ''Moondust''] (2024)

; [[Jim Aikin]]
* [http://midiguru.wordpress.com/2013/01/06/public-rituals/ Public Rituals « Jim Aikin's Oblong Blob ''The Triumphal Procession of Nebuchadnezzar''] (2013)

; [[Beheld]]
* [https://www.youtube.com/watch?v=0WbLTtDZUms ''Damp vibe''] (2022)

; [[benyamind]]
* [https://www.youtube.com/watch?v=H1hYI2hBcEU ''Cinematic music in 26-tone equal temperament''] (2024)

; [[Cameron Bobro]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/LittleFugueIn26_CBobro.mp3 Little Fugue in 26]{{dead link}}

; [[User:CellularAutomaton|CellularAutomaton]]
* [https://cellularautomaton.bandcamp.com/track/innerstate ''Innerstate''] (2024)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/two-pairs-of-socks-26edo Two Pairs of Socks (26edo)]{{dead link}}
* [https://cityoftheasleep.bandcamp.com/track/between-the-branes-26edo Between the Branes (26edo)]{{dead link}}

; [[Zach Curley]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3 Guitar Serenade in Q Major]{{dead link}}

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/FxTxQ0ayDpg ''Microtonal Improvisation in 26edo''] (2023)

; [[User:Eboone|Ebooone]]
* [https://www.youtube.com/watch?v=KvaEyzCuBwA ''26-EDO Nocturne No. 1 in F♯ Minor''] (2024)

; [[Francium]]
* [https://www.youtube.com/watch?v=4i6no5-zwKQ ''Eskalation''] (2022)
* [https://www.youtube.com/watch?v=tIZjchfF2Iw ''Dark Forest''] (2023)
* [https://www.youtube.com/watch?v=FRd_sLuTpQQ ''Lembone''] (2024)
* [https://www.youtube.com/watch?v=XzQ09i6RBsg ''Happy Birthday in 26edo''] (2024)

; [[IgliashonJones|Igliashon Jones]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3 A Time-Yellowed Photo of the Cliffs Hangs on the Wall]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks]{{dead link}}

; [[Melopœia]]
* [https://melopoeia.bandcamp.com/album/ainulindal Ainulindalë] (2016) – A text to music translation of Tolkien's Silmarillion using 26edo.
* [https://melopoeia.bandcamp.com/album/valaquenta Valaquenta] (2023)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=siNzE3d_WsQ Canon 3-in-1 on a ground "The Tempest", in 26edo] (2023)
* [https://youtube.com/shorts/uh7auNGakk4 Greensleeves for three soprano saxes and baroque bassoon, in 26edo] (2023)
* [https://www.youtube.com/watch?v=r0jCdHEZpzM Claudi Meneghin - Suite (Prelude, Variations, Fugue) in 26edo, for Synth & Baroque Bassoon] (2023)
* [https://www.youtube.com/watch?v=rjo3X1-D57Y Canon 3-in-1 on a Ground for Baroque Ensemble] (2023)

; [[Herman Miller]]
* [https://sites.google.com/site/teamouse/26-et-Pianoteq-Bechstein.mp3 Etude in 26-tone equal temperament]{{dead link}}

; [[Shaahin Mohajeri]]
* [http://www.96edo.com/music/micro900607.mp3 Microtonal music in 26-EDO]{{dead link}}

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=43WoQd_UO7w ''Getting to 100'' (with David Sinclair)] (2021)
* [https://www.youtube.com/watch?v=TRloI6ONcmw ''Primitive Mountain''] (2022)
* "Into Thin Air" from ''The Vanishing Bus'' (2024) – [https://open.spotify.com/track/4R2cBwidLwfy5QsdjcFRvi Spotify] | [https://mundoworld.bandcamp.com/track/into-thin-air Bandcamp] | [https://www.youtube.com/watch?v=cekBZ3Ql69U YouTube]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=37KM1roKOMQ ''Edolian - Riemann''] (2020)
* [https://www.youtube.com/watch?v=Y0-Bf2ePfzY ''Redvault''] (2021)

; [[Ray Perlner]]
* [https://youtu.be/rivfU8Rw4IM Scherzo in 26 EDO for Oboe, Horn, and Organ] (2020)
* [https://www.youtube.com/watch?v=hsd00wrSJnE Octatonic Groove] (2021)
* [https://www.youtube.com/watch?v=xhn5lz2cB-4 A Little Prog Rock in 26EDO] (2023)

; [[Sevish]]
* [https://www.youtube.com/watch?v=Kh_C5Va5jDE ''Yeah Groove''] (2022)

; [[Jon Lyle Smith]]
* [http://archive.org/details/UnderTheHeatdome under the heatDome]{{dead link}} [http://archive.org/download/UnderTheHeatdome/under_the_heatDome.mp3 play]{{dead link}}

; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=pJOlZ9sHCjk ''Languor Study''] (2022)

; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=OjW8dgooG9Q ''Spring''] (2024)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016)

== See also ==
* [[Lumatone mapping for 26edo]]

== Notes ==
<references group="note" />

[[Category:Listen]]
[[Category:Twentuning]]
[[Category:Meantone]]
[[Category:Flattone]]
[[Category:Mothra]]
[[Category:Unidec]]

User:YoVariable

2024-12-21T04:02:47Z

YoVariable: Re-added Discord

Hello! My name is YoVariable (she/her), but you can just call me Variable. I am a xenharmonic musician who focuses on [[Meantone|meantone]] edos like [[19edo|19]], [[31edo|31]], and [[43edo]], and [[Superpyth|superpyth]] edos like [[22edo|22]], [[27edo|27]], and [[49edo]]. You may find me editing superpyth or meantone edo pages occasionally.

I also created the "Exploring 22edo" series (TBD) on YouTube. I have taken an interest in schismatic edos, like [[41edo|41]] and [[53edo]], but tend to avoid them due to their size. I focus on the practicality of each edo, as I am a pianist and guitarist (I also play the Lumatone), so I tend to stay within the [[13edo]] to [[46edo]] range. If I want to play in a larger edo, I would use subsets.

I am frequently active in [[Gordon Wery]]'s livestreams and stream occasionally on YouTube and Twitch. My Discord is YoVariable.

== External links ==
* YouTube: https://www.youtube.com/@YoVariable
* Twitch: https://twitch.tv/yovariable

User:YoVariable

2024-12-21T03:59:12Z

YoVariable: Updated info

Hello! My name is YoVariable (she/her), but you can just call me Variable. I am a xenharmonic musician who focuses on [[Meantone|meantone]] edos like [[19edo|19]], [[31edo|31]], and [[43edo]], and [[Superpyth|superpyth]] edos like [[22edo|22]], [[27edo|27]], and [[49edo]]. You may find me editing superpyth or meantone edo pages occasionally.

I also created the "Exploring 22edo" series (TBD) on YouTube. I have taken an interest in schismatic edos, like [[41edo|41]] and [[53edo]], but tend to avoid them due to their size. I focus on the practicality of each edo, as I am a pianist and guitarist (I also play the Lumatone), so I tend to stay within the [[13edo]] to [[46edo]] range. If I want to play in a larger edo, I would use subsets.

I am frequently active in [[Gordon Wery]]'s livestreams and stream occasionally on YouTube and Twitch.

== External links ==
* YouTube: https://www.youtube.com/@YoVariable
* Twitch: https://twitch.tv/yovariable

Gordon Wery

2024-12-21T03:49:54Z

YoVariable: Created Gordon Wery page

'''Gordon Wery''', also known as '''stalefleas''' or '''xenpilled''', is a composer and pianist who has made music in xenharmonic tunings, such as [[14edo]], [[31edo]], and [[13-limit]] just intonation. He is currently focusing on learning [[22edo]] on his Linnstrument and produces music on his microtonal livestreams.

== External links ==
* [https://gordonwery.bandcamp.com Bandcamp]
* [https://twitch.tv/xenpilled Twitch]
* [https://youtube.com/@gordonwery YouTube]

{{stub}}
[[Category:People]][[Category:Composers]]

31edo/Music

2024-12-20T15:07:24Z

YoVariable: Added 31edo song "Open Source Your Mind or Switch to Gentoo"

{{breadcrumb}}
This is a collection of pieces in [[31edo]].
__FORCETOC__
== Modern renderings ==
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=lWqp6a3MGPY "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=1R0GE3p-XMk ''Komm, süßer Tod'', BWV 478] (1736) – arranged for organ, rendered by Francium (2023)
* [https://www.youtube.com/watch?v=u-PEhbSOh74 "Fugue" from ''Great Fantasia and Fugue in G minor'', BWV 542] (''c''. 1714–1720) – arranged for jazz band, rendered by Claudi Meneghin (2020)
* [https://www.youtube.com/watch?v=3bvlz6RgKng ''Canzona'', BWV 588] (1703–1707) – rendered by Claudi Meneghin (2018)
* [https://www.youtube.com/watch?v=BH2fhy9EB7Y ''Das alte Jahr vergangen ist'', BWV 614] (''c''. 1711–1713) – 31edo, arranged for cornet, recorder, bassoon, cello, rendered by Claudi Meneghin (2020)
* [https://www.youtube.com/watch?v=oHRg3ALv0yc ''Fantasy and Fugue'', BWV 906] (''c''. 1738) – harpsichord, completed and rendered by Claudi Meneghin (2019)
* [https://www.youtube.com/watch?v=y2lWbo-nHMw ''Fugue on a theme by Albinoni'', BWV 951] (1714–1717) – transcribed for organ, rendered by Claudi Meneghin (2019)
* [https://www.youtube.com/watch?v=dUr5_vl48uw 1st movement from ''Gamba Sonata'', BWV 1029] (late 1730s – early 1740s) – MIDI organ interpretation, rendered by Claudi Meneghin (2011)
* [https://www.youtube.com/watch?v=4AlEqLBEkpA 1st movement from ''Gamba Sonata'', BWV 1029] (late 1730s – early 1740s) – transcribed for gamba and organ, rendered by Claudi Meneghin (2019)
* [https://www.youtube.com/watch?v=AzdLJMB2naI "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – tuned into 31edo implementation of Vicentino's 1/4-comma adaptive just intonation, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=WMZAXxuKMKw "The Endelessly Rising Canon" from ''The Musical Offering'', BWV 1079] (1747) – with the shepard-tone effect, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=juxsijd_4vI "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=4U1I5MjNDhg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – arranged for baroque wind ensemble, rendered by Claudi Meneghin (2018)
* [https://www.youtube.com/watch?v=Zpz_tFcUP4M "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – transcribed into two versions, one for string and one for organ, rendered by Claudi Meneghin (2019)
* [https://www.youtube.com/watch?v=KqVFA5In008 "Contrapunctus 14" from ''The Art of Fugue'', BWV 1080] (1742–1749) – completed and rendered by Claudi Meneghin (2020)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=Dq8t_skMUMM ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=p59tAr_4IgM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|Frédéric Chopin}}
* [https://www.youtube.com/watch?v=jcJclT5C1_k ''Nocturne No. 1 in B flat minor'', op. 9 Nr. 1] (1830–1831) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=uMJ9zLOkBTg ''Nocturne No. 20 in C sharp minor''] (1830) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=FO9ihziyL5c "Waterfall" Étude from ''12 Études'', op. 10] (1829–1832) – rendered by YoVariable (2023)

; {{W|Antonín Dvořák}}
* [https://www.youtube.com/watch?v=X6-2H88WrVI ''Slavonic Dance'', Op. 72 No. 2] – rendered by Claudi Meneghin (2018)

; {{W|John Eccles (composer)|John Eccles}}
* [https://www.youtube.com/watch?v=cP2JVH5-Y-A ''The Mad Lover – Aire on a Ground''] – arranged for oboe and continuo, rendered by Claudi Meneghin (2024)

; {{W|George Frideric Handel}}
* [https://www.youtube.com/watch?v=OMhzs2Zmw2U ''Suite in D minor HWV 428 for Harpsichord - Allemande''] (1720) – rendered by Claudi Meneghin (2024)

; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=jDVXLrJh8C4 ''Maple Leaf Rag''] (1899) – arranged for harpsichord, rendered by Claudi Meneghin (2024)

; {{W|Jean-Marie Leclair}}
* [https://www.youtube.com/watch?v=f28Nbionyss "Chaconne" from ''Deuxième Récréation de Musique'', Op. 8] – rendered by Claudi Meneghin (2021)

; {{W|Felix Mendelssohn}}
* [https://www.youtube.com/watch?v=7q0T_rcc4Vk #1 and #3 from ''7 Characteristic Pieces'', Op. 7] – arranged for organ, rendered by Claudi Meneghin (2018)

; {{W|Johann Pachelbel}}
* [https://www.youtube.com/watch?v=Y-Jhdnx2xDE ''Canon in D''] – arranged for 3 Oboes and bassoon, rendered by Claudi Meneghin (2023)

; {{W|Pier Domenico Paradies}}
* [https://www.youtube.com/watch?v=GIUHtuHaMp0 "Allegro" from ''Harspichord sonata in A'', a.k.a. "Toccata"] – played at midi organ, rendered by Claudi Meneghin (2011)

; {{W|Domenico Scarlatti}}
* [https://www.youtube.com/watch?v=6JvxnRPlDa4 ''Keybord Sonata in E Major'', K. 380] – rendered by Francium (2023)

; {{W|Pyotr Ilyich Tchaikovsky}}
* [https://www.youtube.com/watch?v=h_Us8f4lUSg "Danse des Mirlitons" from ''The Nutcracker Suite''] – arranged for brass ensemble, rendered by Claudi Meneghin (2018)
* [https://www.youtube.com/watch?v=DAzKunp1gzI Excerpts from ''The Nutcracker Suite''] – arranged for harpsichord, rendered by Claudi Meneghin (2018)

; {{W|Lodovico Grossi da Viadana}}
* [https://www.youtube.com/watch?v=3FRKoB2JfJ4 ''Sinfonia La Padovana''] – arranged for brass quartet and organ, rendered by Claudi Meneghin (2018)

== 20th century ==
; [[Henk Badings]]
* Preludium en fuga voor 31-toonsorgel (1952)
* Contrasten (H. De Vries) voor gemengd koor a cappella (1952)
* Preludium en fuga IV voor 31-toonsorgel (1954)
* Suite van kleine stukken voor 31-toonsorgel (1954)
* Reeks van kleine klankstukken in selectieve toonsystemen, voor 31-toonsorgel (1957)
* Sonata no 2 for two violins (1967) [https://www.youtube.com/watch?v=zZv-jUCynRU YouTube]
* Sonata no 3 for two violins (1967)
* String Quartet (1967)
* Concerto for two violins and orchestra in 31-tone temperament (1969)

; [[Anton de Beer]]
* Method for the 31-note keyboard (1961)
* Instructive Sonatina (1964)
* Small pieces (1967)
* "Speelmusiek" for two violins and archiphone (1971)
* Music for the archiphone (1973)

; [[Ivor Darreg]]
* Three preludes for 31-tone guitar

; [[Jan van Dijk]]
* Vier harmonisch-melodische intonatie-oefeningen voor strijkkwartet (1946)
* Acht stukken in elementaire geslachten van Euler (orgel) (1948)
* Vierstemmige intonatie-oefeningen voor gemengd koor a cappella, bij een studieboek van prof. Fokker (1949)
* Musica per organo trentunisono I (1950/1951)
** I. Quattro pezzi per organo trentunisono solo
** II. Quattro pezzi per organo e archi
** III. Dieci pezzi per organo e strumenti diversi
** IV. Concerto per organo e orchestra
** V. Canzone in genere enharmonico vocale per organo e canto ad lib.
* Deuntje (J. Van den Vondel) voor dubbel gemengd koor a cappella
* Hymne (P. Minderaa) voor gemengd koor a cappella
* Musica per organo trentunisono II: Sette pezzi per organo solo
* Concerto per trombone, violino e violoncello

; [[Adriaan Fokker]]
* Zes bagatelen op tien toetsen
* Preludium chromaticum
* Kalenderblaadjes in de kringloop van acht
* Passacaglia
* In generibus Leonhardi Euleri
* Kringspiegelingen

; [[Eugen Frischknecht]]
* Drei Stücke für 31-Ton-Orgel

; [[Jaap Geraedts]]
* Zes studies in Euler's toongeslachten voor fluit, en voor twee fluiten

; [[John Strong Glasier]]
* Trio for viola (1979)

; [[Alois Hába]]
* String Quartet No. 16 in the fifth-tone system, opus 88 (1967)

; [[Anthon van der Horst]]
* Suite voor 31-toonsorgel, op. 60

; [[Hans Kox]]
* Drie stukken voor viool solo, in (3^3•7^2)
* Vues des anges (R. Rilke), voor viool en bariton
* Passacaglia en koraal, voor 31-toonsorgel
* Vier stukken voor strijkkwartet, 31-toons
* Vier didaktische stukken, voor twee trompetten en een trombone
* Serenade for two violins

; [[Ton de Leeuw]]
* Elektronische studie (1957)

; [[Joel Mandelbaum]]
* Ten studies in 31-tone temperament for organ (1963)
* The Dybbuk, Opera (Act III) (1963)
* Three Songs for soprano, two violins and archiphone (1971)

; [[Mats Öljare]]
* Åppelblomman and Valley of the Lepers (2000)

; [[Richard Orton]]
* Mosaics, for 31-tone organ (1965)

; [[Oedoen Partos]]
* Three fantasies for two violins (1972)

; [[Alan Ridout]]
* Partita for cello solo (1959)
* Music for 31-tone organ (1960)
* Trio for strings, I. Chants, II. Dances, III. Variations (1961)
* Sonata for two violins (1965)
* Animula vagula blandula (Hadrian), for violin and baritone (1965)

; [[Peter Schat]]
* Collages, voor 31-toonsorgel, I. Contrastes, II. Canons, III. Clusters (1962)

; [[Jon Lyle Smith]]
* [http://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 Aire #2 in 31-equal temperament]

; [[Alphonse Stallaert]]
* Vier liederen voor twee violen (1965)

; [[Paul chr. van Westering]]
* Zes inventies voor 31-toonsorgel (1951)
* Variates over "Merck toch, hoe sterck", voor 31-toonsorgel (1951)

; [[Ivan Wyschnegradsky]]
* Etude ultrachromatique pour l'orgue trente-et-unisonique

== 21st century ==
; [[Alefian]]
* [https://www.youtube.com/watch?v=dTHBDDjH4Bo ''I Remember''] (2024)

; [[avoset]]
* from ''2023 Collection: DARKMODE''
** "paranoia" – [https://avoset.bandcamp.com/track/paranoia Bandcamp] | [https://www.youtube.com/watch?v=e-q654bM2s8 YouTube]
** "distill" – [https://avoset.bandcamp.com/track/distill Bandcamp] | [https://www.youtube.com/watch?v=OiUD6sNTyUw YouTube]
** "effervesce" – [https://avoset.bandcamp.com/track/effervesce Bandcamp] | [https://www.youtube.com/watch?v=Q47ipJ3NWdg YouTube]

; [[Beheld]]
* [https://youtu.be/mndlbJXItHk ''Vacant vibe''] (2022)

; [[benyamind]]
* ''We Are Earth'' (2018) – [https://youtu.be/1a-yMGCkjes YouTube] | [https://benyamind.bandcamp.com/track/we-are-earth-31-edo Bandcamp] | [https://soundcloud.com/benyamind/we-are-earth SoundCloud]
* [https://www.youtube.com/watch?v=3OMqtmlAkp0 ''Diesis''] (2024)

; [[Stevie Boyes]]
* [https://www.youtube.com/watch?v=YnRMlI3r7Os ''Horizon''] (2021)

; [[cadmium]]
* [https://cadecomposer.bandcamp.com/track/opening-31-edo "opening" from ''THIRD EYE''] (2023)

; [[Circular17]]
* ''Mystère et tolérance'' (2020) – [https://d.tube/v/circular17/QmbyDumQJNH3MYZvphivVPSELW3XSkQPDt2CtWqdD6giTm DTube] | [https://soundcloud.com/johann-elsass/mystere-et-tolerance-31-et SoundCloud]
* [https://d.tube/v/circular17/QmfGU62ozp4GcuGVbSx9QnmkJLtnHvZjtyJ91B3un447Hf ''Wave from the past''] (2020)
* [https://d.tube/v/circular17/QmTgajhiSEC5mB6C1YZLxzeiTUatRTaeZfxrV7ZusJMjU9 ''Curieuse planète''] (2020)
* [https://d.tube/v/circular17/QmPc5BymhqPhJdbfZFNUJn1wasGsHYsvNmvrQaM8FRwScc ''Deep but not too much''] (2020)
* [https://d.tube/v/circular17/QmWEmGM3WLBActuddEKK4VRay4LMNy4e8LA7FPKtjHz58N ''Heal''] (2020)
* ''Désir mimétique'' (2020) – [https://d.tube/v/circular17/QmbfqgGKFVkEYaaiacqY7VJdjez713sQaYQgMPHJiiRbEM DTube] | [https://soundcloud.com/johann-elsass/desir-mimetique SoundCloud] – in meantone, virtually 31edo

; [[City of the Asleep]]
* ''I Stand Hopeless Before the Gray Sea'' (2005) – [https://web.archive.org/web/20201127015341/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2031-tET-I%20Stand%20Hopeless%20Before%20the%20Gray%20Sea.mp3 play] | [https://cityoftheasleep.bandcamp.com/track/31edo-i-stand-hopeless-before-the-gray-sea Bandcamp]

; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=GjepsRQ-a6g ''Self Contained Universe (Reprise) - OneShot''] (2022)
* [https://www.youtube.com/watch?v=ggp0WjgNvaM ''bo en - my time''] (2023)
* [https://www.youtube.com/watch?v=VD4hXcllkzI ''microtonal waltz in 31edo''] (2023)
* [https://www.youtube.com/watch?v=UxllyiD9tqk ''Liana Flores - rises the moon''] (2023)
* [https://www.youtube.com/watch?v=IDKbmF5qbck ''The Stanley Parable's Elevator Music''] (2023)
* [https://www.youtube.com/watch?v=Yi62leuDoJw ''Mad Father - Old Doll''] (2023)
* [https://www.youtube.com/watch?v=nA_GJ_2OCBE ''A World I Built For You - KinitoPET''] (2024)

; [[Diamond Doll]]
* ''Q: IS ART REVOLUTIONARY? A: LOL NO'' (2019) – [https://open.spotify.com/album/25PTpkMSKBueji3m80L21v?si=ga82okggSdqvoVMQVSOG4g Spotify] | [https://youtu.be/C_W1obqEryU YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/q-is-art-revolutionary-a-lol-no Bandcamp]
* ''Cool Guitar Girl'' (2019) – [https://open.spotify.com/track/6KBZAaPQu2A1NTbbaO4FH2?si=d4ff435363064786 Spotify] | [https://youtu.be/TL0xCZG-FWI Youtube] | [https://helloitsdiamonddoll.bandcamp.com/track/cool-guitar-girl Bandcamp]
* ''Hello, My Name's Diamond Doll'' (2021) – [https://open.spotify.com/track/02hYX9uAKNk7uBr5lLnXIG?si=908b94e033b44cdd Spotify] | [https://www.youtube.com/watch?v=zQdGuBjuv7g Youtube] | [https://helloitsdiamonddoll.bandcamp.com/track/hello-my-names-diamond-doll Bandcamp]

; [[E8 Heterotic]]
* [https://youtu.be/Q2bmXl3zgdE?si=GxrYA7S8TTRcwRg2 "Elements - Air"] from ''Elements'' (2020)

; [[Paul Erlich]], [[Kite Giedraitis]], and [[Dylan Fox]]
* [https://www.youtube.com/watch?v=7rm4RkFo1vI "Lively Up Yourself" cover] on 31edo guitar

; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=wvLLV5qle48 ''Glitterdance''] (2019)
* [https://www.youtube.com/watch?v=8Ol2gzkcprE ''Divinate''] (2019)
* [https://www.youtube.com/watch?v=CqxQPmC1tZA ''Apophenia''] (2024)

; [[Francium]]
* [https://www.youtube.com/watch?v=HJ1XigMsjzI ''Serenity''] (2022)
* [https://www.youtube.com/watch?v=PxwrLncsyR4 ''Calm Optimism''] (2022)
* [https://www.youtube.com/watch?v=Eia5W_Y5SLQ ''Super Mario Kart Rainbow Road''] (2022)
* [https://www.youtube.com/watch?v=H700ffaXU8k ''Fantasia No. 10 in A major''] (2023)
* [https://www.youtube.com/watch?v=SOmdm3DLmjE ''Chiwawa''] (2024) – würschmidt[16] in 31edo tuning
* [https://www.youtube.com/watch?v=SeIVEg3H8iA ''Christmas Moon''] (2024)

; [[Adam Freese]]
* [https://www.youtube.com/watch?v=RmLBdvfr6Q0 ''She Was Like a Daylily''] (2021)
* [https://www.youtube.com/watch?v=FLLhPPqSgXs ''Midwestern Breeze''] (2022)

; [[Hear Between The Lines]]
* from ''Radical Tenderness'' (2023)
** "Match" [https://www.youtube.com/watch?v=ZMBXo33uuKQ YouTube] | [https://open.spotify.com/track/47bhCXtWnMWZGMsZ1N9wZE Spotify]
** "Time To Call My Friend" (feat. Clara Lucas) [https://www.youtube.com/watch?v=t_MumneEC80 Youtube] | [https://open.spotify.com/track/2qfD7nwUOaXO3sF50mOWwZ Spotify]
** "Colourblind" [https://www.youtube.com/watch?v=tNLlIo4TCOs YouTube] | [https://open.spotify.com/track/2aNhfUR6PwKghQP6Su31iM Spotify]
** "Frippe's Nocturne" (feat. Samantha Wright) [https://www.youtube.com/watch?v=1d_E-RSQ-MQ YouTube] | [https://open.spotify.com/track/6MymC4mkW9Qwb5gf30dlr8 Spotify]
** "Between Dreams" (feat. Clara Lucas) [https://open.spotify.com/track/6xC0UTYqe6wD4GqaAw2Prd Spotify]
** "Radical Tenderness" (feat. Philipp Gerschlauer) [https://www.youtube.com/watch?v=BklV5d75jHo YouTube] | [https://open.spotify.com/track/0bXUiC3AOsQ2JhpAoX40Ut Spotify]
** "Observing The Yard" (feat. Steven James Taylor) [https://open.spotify.com/track/5MF4YJbboHDzp0DEM7I0wM Spotify]

; [[Andrew Heathwaite]]
* [[Earwig]] (2012)

; [[User:TromboneBoi9|Andrew Kahler (TromboneBoi9, Sir Semiflat)]]
* [https://www.youtube.com/watch?v=KfLfdHin5_U ''Torn Gamelan'' – ''study of the supermajor second in 31-TET''] (2023)

; [[Aryaman Manish Joshi]]
* [https://youtube.com/watch?v=JBK5SfeWuQs ''Free Among The Stars''] (2023)

; [[Lucilla]]
* [https://www.youtube.com/watch?v=clHJY7exVec ''Ultrajoy''] (2023)

; [[Mandrake]]
* [https://www.youtube.com/watch?v=DxSFtKM10zU ''Cascade''] (2023)

; [[Claudi Meneghin]]
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2012/03/Claudi_Meneghin_Chaconne_G_001.mp3 ''Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’)'']{{dead link}}
* [https://www.youtube.com/watch?v=j4MqILRm2xs ''The Padanian Fugue''] (2014) – Fugue on The Mother's Curse Theme in 31edo, for organ
* [https://www.youtube.com/watch?v=ooSWPmEtdQY ''Fugue on Greensleeves Theme''] (2018) – for organ in 31edo
* [https://www.youtube.com/watch?v=WLPEIc17eHk ''Fuga sobre el Cant dels Ocells'' ("Fugue on the Song of the Birds")] (2018) – fugue on the Catalan traditional theme
* [https://www.youtube.com/watch?v=UAu0M_N-knc ''Shave and a Haircut''] (2019) – a fugue in 31edo for two recorders, bass oboe, bassoon
* [https://www.youtube.com/watch?v=1Dl2TAqMmno ''El Noi de La Mare''] (2019) – jazz 31edo harmonisation of a Catalan traditional xmas song
* [https://www.youtube.com/watch?v=PGXTygDLvnI ''Old Mac Donald Wrote a Fugue''] (2019) – for recorder, flute, English horn, bassoon, in 31edo
* [https://www.youtube.com/watch?v=3jZbRtgpiHk ''Little Twilight Music''] (2020) – for brass and timpani, in 31edo
* [https://www.youtube.com/watch?v=B599TRn_foo ''El Cant dels Ocells''] (2020) – 31edo organ arrangement
* [https://www.youtube.com/watch?v=fKKnDJJJyXU ''The Miracle Canon, 3-in-1 on a Ground''] (2021) – for recorder, oboe, cornet, and continuo
* [https://www.youtube.com/watch?v=z_GkwGpELYw ''Fantasy "Almost a Fugue"''] (2021) – on a theme by Giuliani, for string quartet
* [https://www.youtube.com/watch?v=PkeJ6a-Ls9I ''Fugue on St Louis Blues''] (2023) – for baroque ensemble, in 31edo
* [https://www.youtube.com/watch?v=PKqua00QSA0 ''El Testament de n'Amèlia'' ("Amelia's Will")] (2023) – Catalan folk song arranged for baroque ensemble

; [[miaxia]]
* [https://www.youtube.com/watch?v=i7cF48p7AGQ ''plarna''] (2023)
* [https://www.youtube.com/watch?v=FjTatLLr2lk ''xenharmonic generative music''] (2023)
* [https://www.youtube.com/watch?v=YNaZpoPr-to ''Elunara''] (2024)

; [[James Mulvale]]
* [https://www.youtube.com/watch?v=-SCWPxDiq4s ''Squallish''] (2023)
* [https://www.youtube.com/watch?v=HKSl-afxWNA ''くつろいだ (Kutsuroida)''] (2023)
* [https://www.youtube.com/watch?v=73nKIueZpak ''Soleado''] (2023)

; [[Mundoworld]]
* "Royal Radio" from ''Truth in Progress'' (2023) – [https://www.youtube.com/watch?v=boKNUWK6i9o YouTube] | [https://mundoworld.bandcamp.com/track/royal-radio Bandcamp]
* "Where Is It Now?" from ''The Vanishing Bus'' (2024) – [https://open.spotify.com/track/4lfRfGs9mV0xbHChLlzFzo Spotify] | [https://mundoworld.bandcamp.com/track/where-is-it-now Bandcamp] | [https://www.youtube.com/watch?v=WpZE5UPRsDo YouTube]

; [[Nick, The NRG]]
* [https://www.youtube.com/watch?v=McS7b4llk_0 ''The Werewolf Who Wears A Tuxedo''] (2018)
* ''That Beautiful Beat'' (2022) – [https://www.youtube.com/watch?v=OcMz8xQzPus YouTube] | [https://nick-the-nrg.bandcamp.com/track/that-beautiful-beat-brazilian-lofi Bandcamp]
* ''Oranges, Grapes, and Raspberries'' (2022) – [https://www.youtube.com/watch?v=9ItYMl2D8Lo YouTube] | [https://nick-the-nrg.bandcamp.com/track/oranges-grapes-and-raspberries-groovy-31-edo-microtonal-pop-with-edited-vocal-beatboxing-samples Bandcamp]
* ''SUPACHOOOOORD!!!'' (2023) – [https://www.youtube.com/watch?v=8GniFqVBp4Q YouTube] | [https://nick-the-nrg.bandcamp.com/track/supachooooord-31-edo-lofi-hip-hop Bandcamp]

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=OqGuFvUaH68 ''instability''] (2024)

; [[NullPointerException Music]]
* [https://youtu.be/Ogt0ankXLts "Coda"] from the 1edo to 31edo album ''Edolian'' (2020)

; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=r1mat9f1DZ0 ''Fanfare and Toccata''] (2014)
* [https://www.youtube.com/watch?v=MkElrZwRIR4 Philip Glass's ''Orphée's Bedroom'' on the Lumatone] (2021)
* [https://www.youtube.com/watch?v=maRDg2AXV8w ''Urban Nocturne''] (2022)

; [[Matt Pollock]]
* [https://mattpollock.bandcamp.com/album/six-harmonies ''Six Harmonies''] (2024) Six-track album in 31edo.

; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca/ site])
* ''Meantone Suite III in C Minor'' (2016) – [https://youtu.be/tJHtortXVoM YouTube] | [http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-III-C-Minor.pdf score]
* ''Meantone Suite IV in F Major'' (2016) – [https://youtu.be/O4C46mhji7I YouTube] | [http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-IV-F-Major.pdf score]

; [[Benton Roark]]
* [https://www.youtube.com/watch?v=v4SxhKNopxA ''Asterion''] (2016)
* [https://www.youtube.com/watch?v=9ZozXzKOf8o ''Sundog''] (2020)

; [[Seasons Of]]
* [https://www.youtube.com/watch?v=L_cx4MylmXk ''Cycles''] (2020)

; [[William Sethares]]
* [http://launch.groups.yahoo.com/group/MakeMicroMusic/message/16277 Conversation about the Moon]{{dead link}} (2007)

; [[Jon Lyle Smith]]
* [https://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 ''Aire #2 in 31-equal temperament'']{{dead link}}

; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=AUaiIYHAde0 ''Some hopeful chords with N64 strings''] (2022)

; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/enharmonic-melody-for-guitar ''Enharmonic melody for guitar''] (2013)
* [https://soundcloud.com/camtaylor-1/what-use-is-a-boy ''What use is a boy''] (2013)
* [https://soundcloud.com/camtaylor-1/back-to-31-hyperchromatic ''Back to 31: Hyperchromatic progression on C^''] (2014)

; [[User:Tristanbay|Tristan Bay]]
* [https://youtu.be/d3V3mM4XzFE?t=8m52s ''Wallaby''] from ''Bonus Room'' (2024)

; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=CBmYRoej2yQ ''Lamentation of the Geological Methane Buildup''] (2024)

; [[Georg Vogel]]
* [https://www.youtube.com/watch?v=wZ7rMsE1ia8 ''Vonamoe''] (2023) – performed on a replica Clavemusicum Omnitonum
* [https://www.youtube.com/watch?v=oqjkwanfItk ''Felfed''] (2024) – performed on a replica Clavemusicum Omnitonum
* [https://www.youtube.com/watch?v=Xcz7DWETiew ''Gnardnotteil''] (2024) – performed on a replica Clavemusicum Omnitonum

; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=PGagADh7HdI ''Time to Call My Friend''] (2023), cover, originally by Hear Between The Lines

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=8-55rnqoZVg ''Huygens in Love''] (2021)
* [https://www.youtube.com/watch?v=U_V951pXI6U ''Delta Phase Canvas - Trial 1''] (2023)
* [https://www.youtube.com/watch?v=31-GywJ9Av8 ''Maybe Don't Eat the Plants on This World''] (2023)

; [[Xotla]]
* [https://www.youtube.com/watch?v=xCgpOhm-HZU ''Fern Frond's Pond''] (2016)
* [https://www.youtube.com/watch?v=PLOpkUQd4RI ''Lycanthrope''] (2016)
* from ''Nano Particular'' (2019)
** "Glacial Lightning" – [https://open.spotify.com/track/27iSxCgpCMSxLBBSwXbSsb Spotify] | [https://xotla.bandcamp.com/track/glacial-lightning-31edo Bandcamp] | [https://www.youtube.com/watch?v=gF4LxKaN2bI YouTube]
** "Interference Patterns" – [https://open.spotify.com/track/6moHxa5z5EehBult4ei1YB Spotify] | [https://xotla.bandcamp.com/track/interference-patterns-31edo Bandcamp] | [https://www.youtube.com/watch?v=FAsZ7p1VNEw YouTube]
* "Astro Alum" from ''Science Fraction'' (2022) – [https://open.spotify.com/track/0JfFjW4fEnKkSYHKkjEJrT Spotify] | [https://xotla.bandcamp.com/track/astro-alum-31edo Bandcamp] | [https://www.youtube.com/watch?v=W35S-PCvA7c YouTube]
* ''Planting Seeds of Microtones'' (2022) – [https://open.spotify.com/album/6eiYDwdAbxCzYtrXxX7YKT Spotify] | [https://xotla.bandcamp.com/album/planting-seeds-of-microtones Bandcamp] | [https://youtube.com/playlist?list=PL4HmfPDldHXvtjyCJpWg6K41vJJcfBmZ5 YouTube] – 10-piece album in 31edo
** "Got Better Things To Do" · "Call Me So Late" · "Keep Me Waiting For You" · "Wanna Be Left Alone" · "Haunted By The Ghosts" · "Glass Half-Full (Or Empty?) · "The Seasons Are Changing" · "Never Wanna Get Used To The Sound" · "Pieces Of The Puzzle" · "No Matter What (You're Thinking)"
* from ''Microtonal Allsorts'' (2023)
** "Nothing, Always" – [https://open.spotify.com/track/1dFZkgZ8AEzusm4c0F9tOa Spotify] | [https://xotla.bandcamp.com/track/nothing-always-31edo Bandcamp] | [https://www.youtube.com/watch?v=rdnDpXdfohM YouTube]
** "Ephemeral Arrival" – [https://open.spotify.com/track/3q5n2lgEvivfnMaKwoVEgY Spotify] | [https://xotla.bandcamp.com/track/ephemeral-arrival-31edo Bandcamp] | [https://www.youtube.com/watch?v=01jDsj4Al3Y YouTube]
** "Anemosis Null" – [https://open.spotify.com/track/1qYvaTMG0YSX6kDyKj72NO Spotify] | [https://xotla.bandcamp.com/track/anemosis-null-31edo Bandcamp] | [https://www.youtube.com/watch?v=jLbaGTsxfl8 YouTube]
** "Tortuous And Long Forgotten" – [https://open.spotify.com/track/7A4516MMPMXbIC5AQrYCOv Spotify] | [https://xotla.bandcamp.com/track/tortuous-and-long-forgotten-31edo Bandcamp] | [https://www.youtube.com/watch?v=O5JKpqo9Ab8 YouTube]
* "Big Band Bug" from ''Jazzbeetle'' (2023) – [https://xotla.bandcamp.com/track/big-band-bug-31edo Bandcamp] | [https://www.youtube.com/watch?v=-eU2Ix-HiAA YouTube]

;; [[YoVariable]] and [[rhapsody_mdr]]
* [https://www.youtube.com/watch?v=MJUP3u7UE7A ''Open Source Your Mind or Switch to Gentoo''] (2024)

[[Category:31edo]]
[[Category:Listen]]

YoVariable

2024-12-16T00:24:21Z

YoVariable: Added Category: Wiki editors

#REDIRECT [[User:YoVariable]]

[[Category:People]]
[[Category:Composers]]
[[Category: Wiki editors]]

YoVariable

2024-12-16T00:21:28Z

YoVariable: Redirected page to User:YoVariable

#REDIRECT [[User:YoVariable]]

[[Category:People]]
[[Category:Composers]]

22edo/Music

2024-12-15T23:54:45Z

YoVariable: /* 21st century */ Added 22edo cover of Rain by Rob Scallon

{{breadcrumb}}
This is a collection of pieces in [[22edo]].
== Modern renderings ==
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=o36TKOyoWh0 "Prelude No. 1 in C major" from ''The Well-Tempered Clavier I''] (1722) – rendered by Francium (2024)

; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=-6UKt_zboBA ''Maple Leaf Rag''] (1899) – with Syntonic Comma Adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)

; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=eyQk4ZGCwyA ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2024)
* [https://www.youtube.com/watch?v=Mill7SDoFKI ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by YoVariable (2024)

== 20th century ==
; [[Ivor Darreg]]
* On the Enharmonic Tetrachord (1975) ([https://www.youtube.com/watch?v=OdohUkbuaGA played by Juhani Nuorvala on lumatone])

== 21st century ==
; [[Abnormality]]
* [https://www.youtube.com/watch?v=_vAhYicHHf8 ''Bubbles''] (2024)
* [https://www.youtube.com/watch?v=H1y2CXbiNjY ''Stardust''] (2024)

; [[Alefian]]
* [https://www.youtube.com/watch?v=fYN5g63Tsdc ''raindance''] (2024)

; [[Jacob Barton]] ([[Metaclown]])
* [https://soundcloud.com/metaclown/couples-therapy ''Couples' Therapy''] (2016)

; [[Mike Battaglia]]
* [https://www.youtube.com/watch?v=WMtp9Wk0tO0 Improvisation in 22-equal temperament] (2011)

; [[Bevkcan]]
* [https://youtu.be/-WNMyuCEGCM?feature=shared ''Nightsoarer''] (2020)

; [[Blendy Wave]]
* [https://www.youtube.com/watch?v=Jjn5S5MS2FM ''Languid Lavender''] (2023)
* [https://www.youtube.com/watch?v=w4fU3O7SwaQ ''Marigold''] (2023)
* [https://www.youtube.com/watch?v=PwQ55yIjw_g ''Amethyst Sky''] (2023)
* [https://www.youtube.com/watch?v=GKH0RL5zFR4 ''Ceramic''] (2023)

; [[Brendan Byrnes]]
* [https://soundcloud.com/ilevens/tracks Tracks of ILEVENS] - all their tracks on SoundCloud are tagged with 22edo
* [https://www.youtube.com/watch?v=qHHv3mwJTlg Short piece and demonstration] (video) (electric guitar) (2014)
* [http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3 ''22 EDO Guitar Etude'']{{dead link}} ([https://brendanbyrnes.bandcamp.com/ Bandcamp user page])
* [https://brendanbyrnes.bandcamp.com/track/llurion-4 ''Llurion''] ([https://youtu.be/IgDdTNAa7Ls on YouTube]) from his 2017 album ''Neutral Paradise''
* [https://brendanbyrnes.bandcamp.com/track/hysteria-3 ''Hysteria''] ([https://youtu.be/U5BZ2KncKs8 on YouTube]) from his 2017 album ''Neutral Paradise''
* [https://youtu.be/XS6wxEtttU8 ''Unreachable Island''] from his 2020 album ''Realism''

; [[cadmium]]
* [https://cadecomposer.bandcamp.com/track/pulse-22-edo "pulse" from ''THIRD EYE''] (2023)

; [[Circular17]]
* [https://d.tube/v/circular17/QmWDXi7hgSwZF9kRUUXUkCjEz8BMepoxehM9mRhUecTubQ ''Good devil''] (2020)
* [https://d.tube/v/circular17/QmazZ9NBed2LoJb1bauNuEsAztah6Jir2VVrX2wiG6rwVm ''Wave from the past''] (2020)

; [[Diamond Doll]]
* ''Little Brother'' (2020) – [https://open.spotify.com/track/4UqXCEKsKR5VdggZrr1cWo?si=353860112a574071 Spotify]| [https://youtu.be/0NtKxk8Aaz0 YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/little-brother Bandcamp]
* ''Together For Never'' (2021) – [https://open.spotify.com/track/5A0gpKuvfJmxKoAlUmD5cY?si=652a2647932b4449 Spotify] | [https://www.youtube.com/watch?v=Z_-6ywGVnEk YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/together-for-never Bandcamp]
* ''Music For Men'' (2022) – [https://open.spotify.com/track/5CxPDgwJTv6Nak61uIIzdJ?si=1d6cf16bd04a4b3a Spotify] | [https://www.youtube.com/watch?v=v19qgo4oVvY YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/music-for-men Bandcamp]
* ''My Cheerleader'' (2022) – [https://open.spotify.com/track/2XKjpZjRgZYCmhOUuVvQ8V?si=8da0894501fe4ea5 Spotify] | [https://www.youtube.com/watch?v=hpsIClESmBM YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/my-cheerleader Bandcamp]
* ''Alexandria's Genesis'' (2021) – [https://open.spotify.com/track/6cnJpBOzdj1GygkfW4zsVo?si=c0d4dc233c9d456f Spotify] | [https://www.youtube.com/watch?v=-XdPetzxUwA YouTube]
* ''Prized Performer'' (2021) – [https://open.spotify.com/track/2kyqzYe5lbdLpty43TZosN?si=220061b9a46c4299 Spotify] | [https://www.youtube.com/watch?v=H-yYcH0qwC0 YouTube]
* ''Hot Like Goths In The Summer'' (2021) – [https://open.spotify.com/track/3Tvu4ZZr1TK3DqNginyWeD?si=841b1308023446bc Spotify] | [https://www.youtube.com/watch?v=1oWL6spG_r4 YouTube]

; [[dotuXil]]
* [https://dotuxil.bandcamp.com/track/porky-spaceflight Porky Spaceflight] (2024)
* [https://dotuxil.bandcamp.com/track/enter-nil Enter Nil] (2024)

; [[E8 Heterotic]]
* [https://youtu.be/f4IhlN8T7oE?si=oSj3Z4DQFiEfLhjX "Elements - Earth"] from ''Elements'' (2020) – porcupine[7] in 22edo tuning

; [[Paul Erlich]]
* [http://www.tallkite.com/words/Tibia.mp3 ''Tibia'']
**[https://www.youtube.com/watch?v=1Hz7J-1rK7E As rendered by Francium]
** 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]]
* [https://www.youtube.com/watch?v=lO5xSjIHyMg Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity] - YouTube (2011)

; [[Paul Erlich]] and [[Ara Sarkissian]]
* [https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 ''Glassic''] (2007)
* [http://lumma.org/tuning/erlich/decatonic-swing.mp3 ''Decatonic Swing''] (jazz)

; [[Fallen Eyelash]]
* [https://falleneyelash.bandcamp.com/album/from-there ''From There...''] (2023) Full album in 22edo

; [[Francium]]
* [https://www.youtube.com/watch?v=RVE0EBNMqBg ''Asana''] (2022)
* [https://www.youtube.com/watch?v=zOVAafn5YnE ''Twinkle''] (2023)
* [https://www.youtube.com/watch?v=mVnkS7ENOXc ''Fugue in C Porcupine''] (2023)
* "Dancing Halluzinated Crazyness" from ''Mysteries'' (2023) – [https://open.spotify.com/track/1tJ59w4gAYPDytm4f616Hp Spotify] | [https://francium223.bandcamp.com/track/dancing-halluzinated-crazyness Bandcamp] | [https://www.youtube.com/watch?v=k3HDYZ0aGh8 YouTube]
* "The Dark Side In Me" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/6dQ2nuv2K8kzCbxIROoyNy Spotify] | [https://francium223.bandcamp.com/track/the-dark-side-in-me Bandcamp] | [https://www.youtube.com/watch?v=snSydL7XMQU YouTube]
* "Lenticular" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/1Ymbjkjg5wUTG0v33PvQlQ Spotify] | [https://francium223.bandcamp.com/track/lenticular Bandcamp] | [https://www.youtube.com/watch?v=BWYiWIy0qVU YouTube]

; [[Jake Freivald]]
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs ''Chord sequence in Paul Erlich's 22 EDO decatonic major''] (2014)
* [https://soundcloud.com/jdfreivald/porcupine-comma-pump ''Porcupine Comma Pump''] (2012)

; [[Frédéric Gagné]]
* [https://youtu.be/BQluy3k3cpQ ''Jetlag''] (2022)

; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus ''Mass in 22edo - Sanctus''] (2015)
* [https://soundcloud.com/lillianhearne/mass-in-22edo-agnus-dei ''Mass in 22edo - Agnus Dei''] (2015)

; [[Andrew Heathwaite]]
* [https://soundclick.com/share?songid=8839058 ''where words are said to mean''], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. (2010) This & the following pieces by Andrew are for 22-tone guitar & voice.
* [https://soundclick.com/share?songid=9101704 ''I've come with a bucket of roses''] (orwell[9]: 3 2 3 2 3 2 3 2 2). (2010)
* [https://soundclick.com/share?songid=9101705 ''one drop of rain''] (orwell[9]). (2010)
* [https://soundclick.com/share?songid=8839060 ''being a''] (porcupine[8]: 3 1 3 3 3 3 3). (2010)
* [https://soundclick.com/share?songid=8839071 ''my own house''] (a pelog-flavored subset of orwell[9]: 3 2 7 3 7). (2010)

; [[Jake Huryn]]
* [https://www.youtube.com/watch?v=jagxI__W-Mg ''Palinkalin Viharo (Flowers in the Mist)'']{{dead link}} ([https://drive.google.com/file/d/0BwJHTddN0-rdUFdwMEtfYnFJZ0E/view Score]{{dead link}}); uses 11edo machine[6], 22edo orwell[9]

; [[Igliashon Jones]]
* [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'']{{dead link}} (synth with electric guitar)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3 ''Numerology'']{{dead link}} (progressive metal)
* [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'']{{dead link}} (progressive metal)

; [[JJ - Composer]]
* [https://www.youtube.com/watch?v=MA3HQSt_9Ag ''Nocturne #2 in B down minor (22 EDO)''] (2024)

; [[JUMBLE]]
* [https://www.youtube.com/watch?v=nE222orRSXE ''Brass''] (2023)
* [https://www.youtube.com/watch?v=uABD4OgwXIM ''Stained Glass''] (2023)
* [https://www.youtube.com/watch?v=YGL1AHCju5M ''Forbidden Bay''] (2023)
* [https://www.youtube.com/watch?v=AYenXbvg1qI ''Stingray''] (2023)
* [https://www.youtube.com/watch?v=dlD1a9DffPg ''Rose''] (2024)
* [https://www.youtube.com/watch?v=jiGF1qqKT6s ''BORROMEA!''] (2024)
* [https://www.youtube.com/watch?v=cz96vCjShZo ''RAY'S SECRET SOUND TEST!''] (2024)
* [https://www.youtube.com/watch?v=GqkaJNEk75g ''UFO CRUISIN'''] (2024)
* [https://www.youtube.com/watch?v=zZqfBUV_Utk ''BUBBLEGUM POP''] (2024)
* [https://www.youtube.com/watch?v=XAZp2iTFAzg ''A Z U R E''] (2024)
* [https://www.youtube.com/watch?v=hlNBhSdN2Jw ''revenge bedtime procrastination''] (2024)

; [[Claudi Meneghin]]
* [http://soonlabel.com/xenharmonic/archives/1145 ''Canon 2 in 1 upon a ground (22edo)'']{{dead link}}
* [https://www.youtube.com/watch?v=WJnM1IOhe58 ''Chaconne l'Escalier''] (2019)
* [https://www.youtube.com/watch?v=W6Y9pwNqwo0 ''Microtonal Canon 2 in 1''] (2019)
* [https://www.youtube.com/watch?v=w1uLtNAgZcs ''Fugue on The Lick, for Organ and Jazz Band, in Orwell (22edo)] (2022)

; [[Joseph Monzo]]
* [https://www.youtube.com/watch?v=OfMwN3yduqo ''Beach Scene, from Darks and Whites OST''] (2006)
* [https://www.youtube.com/watch?v=BF28U5D25nw ''Pajara10 Jazz. 22edo pajara10''] (2021)
* [https://www.youtube.com/watch?v=bYBo9fw6hRU ''Pajara10 Jazz, tuned in 22edo''] (2021)
* [https://www.youtube.com/watch?v=jJc6hI29Sfc ''Hedgehog8 March''] (2021)
* [https://www.youtube.com/watch?v=yVs5KLf5Knw ''Hedgehog8 Chorale''] (2021)
* [https://www.youtube.com/watch?v=9FUeARqqv-w ''The Right Cable''] (2021)
* [https://www.youtube.com/watch?v=u4RL-GQni_U ''Hedgehog14 Study Chorale''] (2021)
* [https://www.youtube.com/watch?v=OgWCjhI3WiY ''Hedgehog March''] (2021)
* [https://www.youtube.com/watch?v=JdhDYs2l29Y ''Hedgehog8 Chorale''] (2021)
* [https://www.youtube.com/watch?v=1ki_g6Xf2Ps ''22edo Hedgehog14 Study''] (2021)
* [https://www.youtube.com/watch?v=jrVzWznaD4U ''Hedgehog8 Etude 1:01''] (2021)
* [https://www.youtube.com/watch?v=azVOFSuOJiI ''Hedgehog8 Etude 1:08''] (2021)
* [https://www.youtube.com/watch?v=9x6345FEG1k ''Doublewide-10 Study''] (2021)
* [https://www.youtube.com/watch?v=GFStvOFcLaM ''Threesome''] (2022)
* [https://www.youtube.com/watch?v=beDT-RMQ_Jw ''Orwell9 Sonatina, for flute and piano''] (2022)
* [https://www.youtube.com/watch?v=qLk6abIv420 ''Orwell9 Sonatina, for flute and piano (up/down notation)''] (2022)
* [https://www.youtube.com/watch?v=U7F9U89FdtE ''Doublewide10 BossaNova 0108''] (2022)
* [https://www.youtube.com/watch?v=z1mMwviDtMo ''Doublewide10 BossaNova 0117''] (2022)

; [[James Mulvale]]
* [https://www.youtube.com/watch?v=IEb9llRrjEY ''Breathe''] (2023)
* [https://www.youtube.com/watch?v=hv3VGJRVPwM ''Hygge''] (2023)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=XNQCLj9O8O8 ''Contact''] (2021)

; [[MÜÜR]] ([https://muur-proj.web.app/ site])
* [https://www.youtube.com/watch?v=Qgb59snzMII ''Nenio reala''] (2020)
* [https://www.youtube.com/watch?v=MuoZQqvR-gc ''Imzadi''] (2018)
* [https://www.youtube.com/watch?v=sK8lVDyvakE ''Imzadi (alie)''] (2018)

; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=bFgviybbpxw ''yo wtf is that''] (2022)

; [[Alex Ness]]
* [https://drive.google.com/drive/folders/0BwsXD8q2VCYUNGZJOGRzSVdhRjg Rose, liz, printemps, verdure] (in 22edo with stretched octaves) (2017)

; [[norokusi]]
* [https://www.youtube.com/watch?v=4EXxue7iIw4 ''Piano Sonata n. 853''] (2020)
* [https://www.youtube.com/watch?v=WsTadXC8Jc0 ''22 Preludes''] (2020)
* [https://www.youtube.com/watch?v=9dOSiZhtoIo ''Piano Sonata n. 873''] (2021)
* [https://www.youtube.com/watch?v=kV0vR52b7Kg ''Suite No. 2 for fortepiano''] (2021)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=Aph-53clkE8 ''Transcendancy''] (2020)
* [https://www.youtube.com/watch?v=dlwy84sy_WM ''Edolian - The Descent''] (2020)

; [[User:Phanomium|Phanomium]]
;* [https://www.youtube.com/watch?v=SQ_FQBjx2LA Sharpened Blade] (2024)
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=TmOP4sNo7e0 ''Philip Glass, 'Modern Love Waltz' on Lumatone (22edo)''] (2022)
* [https://www.youtube.com/watch?v=1y4Ry1sbJ-E ''Prelude in Pajara''] (2021)
* [https://www.youtube.com/watch?v=1tTetMKn1Eg ''Selections from the Pärnu Codex''] (2021)
* [https://www.youtube.com/watch?v=qMDfQZg-tRQ ''Sonata for Violin and Keyboard''] (2019)
* [https://www.youtube.com/watch?v=7Gw38p9euPc ''esa&jusa: Coendou''] (2024)

; [[Mats Öljare]]
* Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - [http://www.angelfire.com/mo/oljare/midicomp.html MIDI files] {{dead link}}
** [[:File:sunday3.pdf|Sagittal score of Sunday Pipes]]

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=wYgGP50D4bA ''Octatonic Groove''] (2020) – jubilismic[8] in 22edo tuning
* [https://www.youtube.com/watch?v=alLfAvQFXro ''Fugue in 22EDO Porcupine<nowiki>[</nowiki>7<nowiki>]</nowiki> sLsssss "Badgerian"''] (2022)
* [https://www.youtube.com/watch?v=hZ6Mh-Sachs ''Organ fugue in 22EDO Bohlen-Pierce-Stearns<nowiki>[</nowiki>9<nowiki>]</nowiki> (Hedgehog extension) LsLsLssLs "Walker II"''] (2024)

; [[Sevish]]
* "[[Dirty Drummer]]", from ''[[Golden Hour]]'' (2010)
* "[[Ganymede]]", from ''Golden Hour'' (2010)
* "[[Ambrosia]]", from ''[[Human Astronomy]]'' (2010)
* "[[Movement]]", from ''Human Astronomy'' (2010) ([[polysystemic]])
* "[[Earthling]]", from ''[[Rhythm and Xen]]'' (2015) (polysystemic)
* "[[Guano Sequence]]", from ''Rhythm and Xen'' (2015)
* "[[Spellbound]]", from ''Rhythm and Xen'' (2015)
* "[[MK-SUPERDUPER (track)|MK-SUPERDUPER]]", from ''[[MK-SUPERDUPER]]'' (2016)
* "[[The Sky Are Sick]]", from ''MK-SUPERDUPER'' (2016)
* "[[Gleam]]", from ''[[Harmony Hacker]]'' (2017)
* "[[We Can't Be Blamed]]", from ''Harmony Hacker'' (2017)
* "[[Horizons]]", from ''[[Horixens]]'' (2019)
* "[[Dream Up]]", from ''[[Bubble]]'' (2021) (polysystemic)
* "[[Some Things Must]]", from ''Bubble'' (2021)
* "[[Some Things Must Reprise]]", from ''Bubble'' (2021)
* "[[Starfish]]", from ''Bubble'' (2021) (0:00-3:36)
* "[[Big Numb]]", from ''[[Formless Shadows]]'' (2021)

; [[Gene Ward Smith]]
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] – possibly a redux on {{w|Modest Mussorgsky}}'s ''{{w|Night on Bald Mountain}}''

; [[The Stern Brocot Band]]
* [https://www.youtube.com/watch?v=oNJr1YOOqF8 ''Yak Butter''], 1L 6s MOS, compressed period/generator (2012)

; [[Redrick Sultan]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3 ''Recurring Mimosa'']{{dead link}} ([https://soundcloud.com/redrick-sultan/recurring-mimosa SoundCloud user page]) (2013)

; {{W|Rob Scallon}}
* [https://www.youtube.com/watch?v=IOXEwo4kzok ''Rain''] (2008) – cover by YoVariable (2024)

; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=awkDvtj9-gQ ''Melody for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=xmwz7387v9c ''Gymnopedie for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=yD3vw0x3RpA ''Prelude for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=QBXhVEJl-KI ''Helter-Skelter Fugue''] (2022)
* [https://www.youtube.com/watch?v=hipRpvksdyE ''Toccata in D(?)''] (2022)

; [[Joel Grant Taylor]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''12-22hexachordal Dirge'']{{dead link}}

; [[Togenom]]
* "Will and Representation" from ''Xenharmonics, Vol. 5'' (2024) – [https://open.spotify.com/track/04Byr52VRogiIlJL8AzMe7 Spotify] | [https://togenom.bandcamp.com/track/will-and-representation Bandcamp] | [https://www.youtube.com/watch?v=EqOURDIzGSc YouTube]

; [[Gabriel Torre]]
* [https://www.youtube.com/watch?v=dhBoQEokwqw ''Stork''] (2019)
* [https://www.youtube.com/watch?v=hZq53oKDKZo ''Octopus Rhythm''] (2020)
* [https://www.youtube.com/watch?v=2uuR3kTgSG4 ''Pastel''] (2020)

; [[Triaam]]
* [https://www.youtube.com/watch?v=c9gGZ9sf780 ''Void Joyride''] (2023)

; [[User:Tristanbay|Tristan Bay]]
* [https://youtu.be/H47nsivr-TU ''That''] (2024)

; [[Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=267 ''My Crazy Aunt Sophie''] ([http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3 play]). Blatantly xenharmonic piano. (2010)
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] in [[hedgehog14|hedgehog[14]]] tuned to 22edo. (2012)
* [http://chrisvaisvil.com/?p=2494 ''The Capture and Release of the Fairy''] ([http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3 play]) (2012)
* [http://chrisvaisvil.com/?p=2523 ''From the Sky Islands They Came''] ([http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3 play]) (2012)
* [http://chrisvaisvil.com/smoke-filled-bar/ ''Smoke Filled Bar''] ([http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play]) (2012)
* [http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/ ''The Saharan Pump''] (2013)
* [http://micro.soonlabel.com/22-ET/20150910_22edo.mp3 ''22 edo electric guitar duet''] (2015)
* [http://chrisvaisvil.com/for-the-sunset/ ''For the Sunset''] - 22edo rock ensemble (2016)

; [[Nick Vuci]]
* [https://en.xen.wiki/images/0/0b/NickVuci-20230426-22edo-PorcupinePrelude1.mp3 Porcupine Prelude 1 (22edo)] (2023)
* [https://en.xen.wiki/images/3/39/NickVuci-20230518-22edo-PorcupinePrelude2.mp3 Porcupine Prelude 2 (22edo)] (2023)
* [https://en.xen.wiki/images/b/bd/NickVuci-20230521-22edo-PorcupinePrelude3.mp3 Porcupine Prelude 3 (22edo)] (2023)
* [https://en.xen.wiki/images/0/0b/NickVuci-20230523-22edo-Praeambulum.mp3 Porcupine Praeambulum (22edo)] (2023)
* [https://en.xen.wiki/images/2/26/NickVuci-20230531-22edo-PorcupineChoraleWithPrelude.mp3 Porcupine Chorale with Prelude "Nature's Lament" (22edo)] (2023)

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=kpx7mSXfO78 ''Sincerely I Have Loved You''] (2021)

; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3 ''Comets Over Flatland 17'']{{dead link}}

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version ''Emancipate Pitch!''] (2016)

; [[x2st]]
* [https://www.youtube.com/watch?v=lIKmwjtBBdo ''Voyage''] (2022)
* [https://www.youtube.com/watch?v=88YNWzs1GDE ''Voyage II''] (2023)

; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=BlKepZSOL5w ''Hypertension''] (2022)
* [https://www.youtube.com/watch?v=eHuqp6x62-M ''Le Cauchemar de la Guerre''] (2022)

; [[Xotla]]
* [https://www.youtube.com/watch?v=k5QDWFng-4A ''Amanita''] (2016)
* "Perfect Dystopia" from ''Microtones & Garden Gnomes'' (2017) – [https://xotla.bandcamp.com/track/perfect-dystopia-mixed-intonation Bandcamp] | [https://youtu.be/x_5YqboXfAA?si=3WT9lQmnoyp0QNxZ YouTube] – in part, the other part being in [[22edo]]
* from ''UnFound'' (2018)
** "Inflorescence" – [https://xotla.bandcamp.com/track/inflorescence-22edo Bandcamp]
** "Ephemeral Glade" – [https://xotla.bandcamp.com/track/ephemeral-glade-22edo Bandcamp] | [https://youtu.be/QiFoQ-KJ8zc?si=CP3t3elUsuEyccZJ YouTube]
* from ''Micro Biological'' (2018)
** "Catalysis" – [https://open.spotify.com/track/46QmV2tlPqOhucz3rKPfRK Spotify] | [https://xotla.bandcamp.com/track/catalysis-22edo Bandcamp] | [https://youtu.be/vHzHPzGMD1k?si=8qHLzUhR7-ltddQa YouTube]
** "Nodal Plane" – [https://open.spotify.com/track/1fuvjoPm5IP9MH8SrSMa7C Spotify] | [https://xotla.bandcamp.com/track/nodal-plane-22edo-29edo Bandcamp] | [https://youtu.be/3CPbU-W-sgg?si=cZGe_HhACJ24CzE2 YouTube] – in part, the other part being in [[29edo]]
** "Viscosity" – [https://open.spotify.com/track/6ZZgNKleOfnFPZEIGdEFkN Spotify] | [https://xotla.bandcamp.com/track/viscosity-22edo Bandcamp]
** "Refractive" – [https://open.spotify.com/track/7F38dDvcWYKeuRyZlMh8YP Spotify] | [https://xotla.bandcamp.com/track/refractive-22edo Bandcamp]
* from ''Nano Particular'' (2019)
** "Dark Pillars" – [https://open.spotify.com/track/2nRGb75AFAHY2T0mmcdzCk Spotify] | [https://xotla.bandcamp.com/track/dark-pillars-22edo Bandcamp] | [https://youtu.be/kYJ9wTR58pc?si=8vOfVqYMUzVqHCwg YouTube]
** "Floating Apart" – [https://open.spotify.com/track/4gbWPhzav3lIIdtJodDm8b Spotify] | [https://xotla.bandcamp.com/track/floating-apart-22edo Bandcamp] | [https://youtu.be/DmnWziyF17o?si=oU6a0kBeB3m9EYoq YouTube]
* "Light Pillars" from ''Pico Metric'' (2019) – [https://open.spotify.com/track/5Vqw6LWP85iGSAy9ngO3LM Spotify] | [https://xotla.bandcamp.com/track/light-pillars-22edo Bandcamp] | [https://youtu.be/dW1kQtMFlHI?si=x979_S--Kr5XDZf6 YouTube]
* "Time Mayfly" from ''Jazzbeetle'' (2023) – [https://xotla.bandcamp.com/track/time-mayfly-22edo Bandcamp] | [https://youtu.be/BsUjxujkBc4?si=XD2hDoHDGAmvgl2N YouTube]

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=EAFbQHL8b9w ''Twenty two''] (2019)

[[Category:22edo]]
[[Category:Listen]]

22edo/Music

2024-12-15T22:37:15Z

YoVariable: New piece added ---> "Nocturne #2 in B down minor"

{{breadcrumb}}
This is a collection of pieces in [[22edo]].
== Modern renderings ==
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=o36TKOyoWh0 "Prelude No. 1 in C major" from ''The Well-Tempered Clavier I''] (1722) – rendered by Francium (2024)

; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=-6UKt_zboBA ''Maple Leaf Rag''] (1899) – with Syntonic Comma Adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)

; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=eyQk4ZGCwyA ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2024)
* [https://www.youtube.com/watch?v=Mill7SDoFKI ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by YoVariable (2024)

== 20th century ==
; [[Ivor Darreg]]
* On the Enharmonic Tetrachord (1975) ([https://www.youtube.com/watch?v=OdohUkbuaGA played by Juhani Nuorvala on lumatone])

== 21st century ==
; [[Abnormality]]
* [https://www.youtube.com/watch?v=_vAhYicHHf8 ''Bubbles''] (2024)
* [https://www.youtube.com/watch?v=H1y2CXbiNjY ''Stardust''] (2024)

; [[Alefian]]
* [https://www.youtube.com/watch?v=fYN5g63Tsdc ''raindance''] (2024)

; [[Jacob Barton]] ([[Metaclown]])
* [https://soundcloud.com/metaclown/couples-therapy ''Couples' Therapy''] (2016)

; [[Mike Battaglia]]
* [https://www.youtube.com/watch?v=WMtp9Wk0tO0 Improvisation in 22-equal temperament] (2011)

; [[Bevkcan]]
* [https://youtu.be/-WNMyuCEGCM?feature=shared ''Nightsoarer''] (2020)

; [[Blendy Wave]]
* [https://www.youtube.com/watch?v=Jjn5S5MS2FM ''Languid Lavender''] (2023)
* [https://www.youtube.com/watch?v=w4fU3O7SwaQ ''Marigold''] (2023)
* [https://www.youtube.com/watch?v=PwQ55yIjw_g ''Amethyst Sky''] (2023)
* [https://www.youtube.com/watch?v=GKH0RL5zFR4 ''Ceramic''] (2023)

; [[Brendan Byrnes]]
* [https://soundcloud.com/ilevens/tracks Tracks of ILEVENS] - all their tracks on SoundCloud are tagged with 22edo
* [https://www.youtube.com/watch?v=qHHv3mwJTlg Short piece and demonstration] (video) (electric guitar) (2014)
* [http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3 ''22 EDO Guitar Etude'']{{dead link}} ([https://brendanbyrnes.bandcamp.com/ Bandcamp user page])
* [https://brendanbyrnes.bandcamp.com/track/llurion-4 ''Llurion''] ([https://youtu.be/IgDdTNAa7Ls on YouTube]) from his 2017 album ''Neutral Paradise''
* [https://brendanbyrnes.bandcamp.com/track/hysteria-3 ''Hysteria''] ([https://youtu.be/U5BZ2KncKs8 on YouTube]) from his 2017 album ''Neutral Paradise''
* [https://youtu.be/XS6wxEtttU8 ''Unreachable Island''] from his 2020 album ''Realism''

; [[cadmium]]
* [https://cadecomposer.bandcamp.com/track/pulse-22-edo "pulse" from ''THIRD EYE''] (2023)

; [[Circular17]]
* [https://d.tube/v/circular17/QmWDXi7hgSwZF9kRUUXUkCjEz8BMepoxehM9mRhUecTubQ ''Good devil''] (2020)
* [https://d.tube/v/circular17/QmazZ9NBed2LoJb1bauNuEsAztah6Jir2VVrX2wiG6rwVm ''Wave from the past''] (2020)

; [[Diamond Doll]]
* ''Little Brother'' (2020) – [https://open.spotify.com/track/4UqXCEKsKR5VdggZrr1cWo?si=353860112a574071 Spotify]| [https://youtu.be/0NtKxk8Aaz0 YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/little-brother Bandcamp]
* ''Together For Never'' (2021) – [https://open.spotify.com/track/5A0gpKuvfJmxKoAlUmD5cY?si=652a2647932b4449 Spotify] | [https://www.youtube.com/watch?v=Z_-6ywGVnEk YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/together-for-never Bandcamp]
* ''Music For Men'' (2022) – [https://open.spotify.com/track/5CxPDgwJTv6Nak61uIIzdJ?si=1d6cf16bd04a4b3a Spotify] | [https://www.youtube.com/watch?v=v19qgo4oVvY YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/music-for-men Bandcamp]
* ''My Cheerleader'' (2022) – [https://open.spotify.com/track/2XKjpZjRgZYCmhOUuVvQ8V?si=8da0894501fe4ea5 Spotify] | [https://www.youtube.com/watch?v=hpsIClESmBM YouTube] | [https://helloitsdiamonddoll.bandcamp.com/track/my-cheerleader Bandcamp]
* ''Alexandria's Genesis'' (2021) – [https://open.spotify.com/track/6cnJpBOzdj1GygkfW4zsVo?si=c0d4dc233c9d456f Spotify] | [https://www.youtube.com/watch?v=-XdPetzxUwA YouTube]
* ''Prized Performer'' (2021) – [https://open.spotify.com/track/2kyqzYe5lbdLpty43TZosN?si=220061b9a46c4299 Spotify] | [https://www.youtube.com/watch?v=H-yYcH0qwC0 YouTube]
* ''Hot Like Goths In The Summer'' (2021) – [https://open.spotify.com/track/3Tvu4ZZr1TK3DqNginyWeD?si=841b1308023446bc Spotify] | [https://www.youtube.com/watch?v=1oWL6spG_r4 YouTube]

; [[dotuXil]]
* [https://dotuxil.bandcamp.com/track/porky-spaceflight Porky Spaceflight] (2024)
* [https://dotuxil.bandcamp.com/track/enter-nil Enter Nil] (2024)

; [[E8 Heterotic]]
* [https://youtu.be/f4IhlN8T7oE?si=oSj3Z4DQFiEfLhjX "Elements - Earth"] from ''Elements'' (2020) – porcupine[7] in 22edo tuning

; [[Paul Erlich]]
* [http://www.tallkite.com/words/Tibia.mp3 ''Tibia'']
**[https://www.youtube.com/watch?v=1Hz7J-1rK7E As rendered by Francium]
** 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]]
* [https://www.youtube.com/watch?v=lO5xSjIHyMg Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity] - YouTube (2011)

; [[Paul Erlich]] and [[Ara Sarkissian]]
* [https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 ''Glassic''] (2007)
* [http://lumma.org/tuning/erlich/decatonic-swing.mp3 ''Decatonic Swing''] (jazz)

; [[Fallen Eyelash]]
* [https://falleneyelash.bandcamp.com/album/from-there ''From There...''] (2023) Full album in 22edo

; [[Francium]]
* [https://www.youtube.com/watch?v=RVE0EBNMqBg ''Asana''] (2022)
* [https://www.youtube.com/watch?v=zOVAafn5YnE ''Twinkle''] (2023)
* [https://www.youtube.com/watch?v=mVnkS7ENOXc ''Fugue in C Porcupine''] (2023)
* "Dancing Halluzinated Crazyness" from ''Mysteries'' (2023) – [https://open.spotify.com/track/1tJ59w4gAYPDytm4f616Hp Spotify] | [https://francium223.bandcamp.com/track/dancing-halluzinated-crazyness Bandcamp] | [https://www.youtube.com/watch?v=k3HDYZ0aGh8 YouTube]
* "The Dark Side In Me" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/6dQ2nuv2K8kzCbxIROoyNy Spotify] | [https://francium223.bandcamp.com/track/the-dark-side-in-me Bandcamp] | [https://www.youtube.com/watch?v=snSydL7XMQU YouTube]
* "Lenticular" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/1Ymbjkjg5wUTG0v33PvQlQ Spotify] | [https://francium223.bandcamp.com/track/lenticular Bandcamp] | [https://www.youtube.com/watch?v=BWYiWIy0qVU YouTube]

; [[Jake Freivald]]
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs ''Chord sequence in Paul Erlich's 22 EDO decatonic major''] (2014)
* [https://soundcloud.com/jdfreivald/porcupine-comma-pump ''Porcupine Comma Pump''] (2012)

; [[Frédéric Gagné]]
* [https://youtu.be/BQluy3k3cpQ ''Jetlag''] (2022)

; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus ''Mass in 22edo - Sanctus''] (2015)
* [https://soundcloud.com/lillianhearne/mass-in-22edo-agnus-dei ''Mass in 22edo - Agnus Dei''] (2015)

; [[Andrew Heathwaite]]
* [https://soundclick.com/share?songid=8839058 ''where words are said to mean''], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. (2010) This & the following pieces by Andrew are for 22-tone guitar & voice.
* [https://soundclick.com/share?songid=9101704 ''I've come with a bucket of roses''] (orwell[9]: 3 2 3 2 3 2 3 2 2). (2010)
* [https://soundclick.com/share?songid=9101705 ''one drop of rain''] (orwell[9]). (2010)
* [https://soundclick.com/share?songid=8839060 ''being a''] (porcupine[8]: 3 1 3 3 3 3 3). (2010)
* [https://soundclick.com/share?songid=8839071 ''my own house''] (a pelog-flavored subset of orwell[9]: 3 2 7 3 7). (2010)

; [[Jake Huryn]]
* [https://www.youtube.com/watch?v=jagxI__W-Mg ''Palinkalin Viharo (Flowers in the Mist)'']{{dead link}} ([https://drive.google.com/file/d/0BwJHTddN0-rdUFdwMEtfYnFJZ0E/view Score]{{dead link}}); uses 11edo machine[6], 22edo orwell[9]

; [[Igliashon Jones]]
* [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'']{{dead link}} (synth with electric guitar)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3 ''Numerology'']{{dead link}} (progressive metal)
* [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'']{{dead link}} (progressive metal)

; [[JJ - Composer]]
* [https://www.youtube.com/watch?v=MA3HQSt_9Ag ''Nocturne #2 in B down minor (22 EDO)''] (2024)

; [[JUMBLE]]
* [https://www.youtube.com/watch?v=nE222orRSXE ''Brass''] (2023)
* [https://www.youtube.com/watch?v=uABD4OgwXIM ''Stained Glass''] (2023)
* [https://www.youtube.com/watch?v=YGL1AHCju5M ''Forbidden Bay''] (2023)
* [https://www.youtube.com/watch?v=AYenXbvg1qI ''Stingray''] (2023)
* [https://www.youtube.com/watch?v=dlD1a9DffPg ''Rose''] (2024)
* [https://www.youtube.com/watch?v=jiGF1qqKT6s ''BORROMEA!''] (2024)
* [https://www.youtube.com/watch?v=cz96vCjShZo ''RAY'S SECRET SOUND TEST!''] (2024)
* [https://www.youtube.com/watch?v=GqkaJNEk75g ''UFO CRUISIN'''] (2024)
* [https://www.youtube.com/watch?v=zZqfBUV_Utk ''BUBBLEGUM POP''] (2024)
* [https://www.youtube.com/watch?v=XAZp2iTFAzg ''A Z U R E''] (2024)
* [https://www.youtube.com/watch?v=hlNBhSdN2Jw ''revenge bedtime procrastination''] (2024)

; [[Claudi Meneghin]]
* [http://soonlabel.com/xenharmonic/archives/1145 ''Canon 2 in 1 upon a ground (22edo)'']{{dead link}}
* [https://www.youtube.com/watch?v=WJnM1IOhe58 ''Chaconne l'Escalier''] (2019)
* [https://www.youtube.com/watch?v=W6Y9pwNqwo0 ''Microtonal Canon 2 in 1''] (2019)
* [https://www.youtube.com/watch?v=w1uLtNAgZcs ''Fugue on The Lick, for Organ and Jazz Band, in Orwell (22edo)] (2022)

; [[Joseph Monzo]]
* [https://www.youtube.com/watch?v=OfMwN3yduqo ''Beach Scene, from Darks and Whites OST''] (2006)
* [https://www.youtube.com/watch?v=BF28U5D25nw ''Pajara10 Jazz. 22edo pajara10''] (2021)
* [https://www.youtube.com/watch?v=bYBo9fw6hRU ''Pajara10 Jazz, tuned in 22edo''] (2021)
* [https://www.youtube.com/watch?v=jJc6hI29Sfc ''Hedgehog8 March''] (2021)
* [https://www.youtube.com/watch?v=yVs5KLf5Knw ''Hedgehog8 Chorale''] (2021)
* [https://www.youtube.com/watch?v=9FUeARqqv-w ''The Right Cable''] (2021)
* [https://www.youtube.com/watch?v=u4RL-GQni_U ''Hedgehog14 Study Chorale''] (2021)
* [https://www.youtube.com/watch?v=OgWCjhI3WiY ''Hedgehog March''] (2021)
* [https://www.youtube.com/watch?v=JdhDYs2l29Y ''Hedgehog8 Chorale''] (2021)
* [https://www.youtube.com/watch?v=1ki_g6Xf2Ps ''22edo Hedgehog14 Study''] (2021)
* [https://www.youtube.com/watch?v=jrVzWznaD4U ''Hedgehog8 Etude 1:01''] (2021)
* [https://www.youtube.com/watch?v=azVOFSuOJiI ''Hedgehog8 Etude 1:08''] (2021)
* [https://www.youtube.com/watch?v=9x6345FEG1k ''Doublewide-10 Study''] (2021)
* [https://www.youtube.com/watch?v=GFStvOFcLaM ''Threesome''] (2022)
* [https://www.youtube.com/watch?v=beDT-RMQ_Jw ''Orwell9 Sonatina, for flute and piano''] (2022)
* [https://www.youtube.com/watch?v=qLk6abIv420 ''Orwell9 Sonatina, for flute and piano (up/down notation)''] (2022)
* [https://www.youtube.com/watch?v=U7F9U89FdtE ''Doublewide10 BossaNova 0108''] (2022)
* [https://www.youtube.com/watch?v=z1mMwviDtMo ''Doublewide10 BossaNova 0117''] (2022)

; [[James Mulvale]]
* [https://www.youtube.com/watch?v=IEb9llRrjEY ''Breathe''] (2023)
* [https://www.youtube.com/watch?v=hv3VGJRVPwM ''Hygge''] (2023)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=XNQCLj9O8O8 ''Contact''] (2021)

; [[MÜÜR]] ([https://muur-proj.web.app/ site])
* [https://www.youtube.com/watch?v=Qgb59snzMII ''Nenio reala''] (2020)
* [https://www.youtube.com/watch?v=MuoZQqvR-gc ''Imzadi''] (2018)
* [https://www.youtube.com/watch?v=sK8lVDyvakE ''Imzadi (alie)''] (2018)

; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=bFgviybbpxw ''yo wtf is that''] (2022)

; [[Alex Ness]]
* [https://drive.google.com/drive/folders/0BwsXD8q2VCYUNGZJOGRzSVdhRjg Rose, liz, printemps, verdure] (in 22edo with stretched octaves) (2017)

; [[norokusi]]
* [https://www.youtube.com/watch?v=4EXxue7iIw4 ''Piano Sonata n. 853''] (2020)
* [https://www.youtube.com/watch?v=WsTadXC8Jc0 ''22 Preludes''] (2020)
* [https://www.youtube.com/watch?v=9dOSiZhtoIo ''Piano Sonata n. 873''] (2021)
* [https://www.youtube.com/watch?v=kV0vR52b7Kg ''Suite No. 2 for fortepiano''] (2021)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=Aph-53clkE8 ''Transcendancy''] (2020)
* [https://www.youtube.com/watch?v=dlwy84sy_WM ''Edolian - The Descent''] (2020)

; [[User:Phanomium|Phanomium]]
;* [https://www.youtube.com/watch?v=SQ_FQBjx2LA Sharpened Blade] (2024)
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=TmOP4sNo7e0 ''Philip Glass, 'Modern Love Waltz' on Lumatone (22edo)''] (2022)
* [https://www.youtube.com/watch?v=1y4Ry1sbJ-E ''Prelude in Pajara''] (2021)
* [https://www.youtube.com/watch?v=1tTetMKn1Eg ''Selections from the Pärnu Codex''] (2021)
* [https://www.youtube.com/watch?v=qMDfQZg-tRQ ''Sonata for Violin and Keyboard''] (2019)
* [https://www.youtube.com/watch?v=7Gw38p9euPc ''esa&jusa: Coendou''] (2024)

; [[Mats Öljare]]
* Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - [http://www.angelfire.com/mo/oljare/midicomp.html MIDI files] {{dead link}}
** [[:File:sunday3.pdf|Sagittal score of Sunday Pipes]]

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=wYgGP50D4bA ''Octatonic Groove''] (2020) – jubilismic[8] in 22edo tuning
* [https://www.youtube.com/watch?v=alLfAvQFXro ''Fugue in 22EDO Porcupine<nowiki>[</nowiki>7<nowiki>]</nowiki> sLsssss "Badgerian"''] (2022)
* [https://www.youtube.com/watch?v=hZ6Mh-Sachs ''Organ fugue in 22EDO Bohlen-Pierce-Stearns<nowiki>[</nowiki>9<nowiki>]</nowiki> (Hedgehog extension) LsLsLssLs "Walker II"''] (2024)

; [[Sevish]]
* "[[Dirty Drummer]]", from ''[[Golden Hour]]'' (2010)
* "[[Ganymede]]", from ''Golden Hour'' (2010)
* "[[Ambrosia]]", from ''[[Human Astronomy]]'' (2010)
* "[[Movement]]", from ''Human Astronomy'' (2010) ([[polysystemic]])
* "[[Earthling]]", from ''[[Rhythm and Xen]]'' (2015) (polysystemic)
* "[[Guano Sequence]]", from ''Rhythm and Xen'' (2015)
* "[[Spellbound]]", from ''Rhythm and Xen'' (2015)
* "[[MK-SUPERDUPER (track)|MK-SUPERDUPER]]", from ''[[MK-SUPERDUPER]]'' (2016)
* "[[The Sky Are Sick]]", from ''MK-SUPERDUPER'' (2016)
* "[[Gleam]]", from ''[[Harmony Hacker]]'' (2017)
* "[[We Can't Be Blamed]]", from ''Harmony Hacker'' (2017)
* "[[Horizons]]", from ''[[Horixens]]'' (2019)
* "[[Dream Up]]", from ''[[Bubble]]'' (2021) (polysystemic)
* "[[Some Things Must]]", from ''Bubble'' (2021)
* "[[Some Things Must Reprise]]", from ''Bubble'' (2021)
* "[[Starfish]]", from ''Bubble'' (2021) (0:00-3:36)
* "[[Big Numb]]", from ''[[Formless Shadows]]'' (2021)

; [[Gene Ward Smith]]
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] – possibly a redux on {{w|Modest Mussorgsky}}'s ''{{w|Night on Bald Mountain}}''

; [[The Stern Brocot Band]]
* [https://www.youtube.com/watch?v=oNJr1YOOqF8 ''Yak Butter''], 1L 6s MOS, compressed period/generator (2012)

; [[Redrick Sultan]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3 ''Recurring Mimosa'']{{dead link}} ([https://soundcloud.com/redrick-sultan/recurring-mimosa SoundCloud user page]) (2013)

; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=awkDvtj9-gQ ''Melody for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=xmwz7387v9c ''Gymnopedie for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=yD3vw0x3RpA ''Prelude for microtonal keyboard''] (2022)
* [https://www.youtube.com/watch?v=QBXhVEJl-KI ''Helter-Skelter Fugue''] (2022)
* [https://www.youtube.com/watch?v=hipRpvksdyE ''Toccata in D(?)''] (2022)

; [[Joel Grant Taylor]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''12-22hexachordal Dirge'']{{dead link}}

; [[Togenom]]
* "Will and Representation" from ''Xenharmonics, Vol. 5'' (2024) – [https://open.spotify.com/track/04Byr52VRogiIlJL8AzMe7 Spotify] | [https://togenom.bandcamp.com/track/will-and-representation Bandcamp] | [https://www.youtube.com/watch?v=EqOURDIzGSc YouTube]

; [[Gabriel Torre]]
* [https://www.youtube.com/watch?v=dhBoQEokwqw ''Stork''] (2019)
* [https://www.youtube.com/watch?v=hZq53oKDKZo ''Octopus Rhythm''] (2020)
* [https://www.youtube.com/watch?v=2uuR3kTgSG4 ''Pastel''] (2020)

; [[Triaam]]
* [https://www.youtube.com/watch?v=c9gGZ9sf780 ''Void Joyride''] (2023)

; [[User:Tristanbay|Tristan Bay]]
* [https://youtu.be/H47nsivr-TU ''That''] (2024)

; [[Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=267 ''My Crazy Aunt Sophie''] ([http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3 play]). Blatantly xenharmonic piano. (2010)
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] in [[hedgehog14|hedgehog[14]]] tuned to 22edo. (2012)
* [http://chrisvaisvil.com/?p=2494 ''The Capture and Release of the Fairy''] ([http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3 play]) (2012)
* [http://chrisvaisvil.com/?p=2523 ''From the Sky Islands They Came''] ([http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3 play]) (2012)
* [http://chrisvaisvil.com/smoke-filled-bar/ ''Smoke Filled Bar''] ([http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play]) (2012)
* [http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/ ''The Saharan Pump''] (2013)
* [http://micro.soonlabel.com/22-ET/20150910_22edo.mp3 ''22 edo electric guitar duet''] (2015)
* [http://chrisvaisvil.com/for-the-sunset/ ''For the Sunset''] - 22edo rock ensemble (2016)

; [[Nick Vuci]]
* [https://en.xen.wiki/images/0/0b/NickVuci-20230426-22edo-PorcupinePrelude1.mp3 Porcupine Prelude 1 (22edo)] (2023)
* [https://en.xen.wiki/images/3/39/NickVuci-20230518-22edo-PorcupinePrelude2.mp3 Porcupine Prelude 2 (22edo)] (2023)
* [https://en.xen.wiki/images/b/bd/NickVuci-20230521-22edo-PorcupinePrelude3.mp3 Porcupine Prelude 3 (22edo)] (2023)
* [https://en.xen.wiki/images/0/0b/NickVuci-20230523-22edo-Praeambulum.mp3 Porcupine Praeambulum (22edo)] (2023)
* [https://en.xen.wiki/images/2/26/NickVuci-20230531-22edo-PorcupineChoraleWithPrelude.mp3 Porcupine Chorale with Prelude "Nature's Lament" (22edo)] (2023)

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=kpx7mSXfO78 ''Sincerely I Have Loved You''] (2021)

; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3 ''Comets Over Flatland 17'']{{dead link}}

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version ''Emancipate Pitch!''] (2016)

; [[x2st]]
* [https://www.youtube.com/watch?v=lIKmwjtBBdo ''Voyage''] (2022)
* [https://www.youtube.com/watch?v=88YNWzs1GDE ''Voyage II''] (2023)

; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=BlKepZSOL5w ''Hypertension''] (2022)
* [https://www.youtube.com/watch?v=eHuqp6x62-M ''Le Cauchemar de la Guerre''] (2022)

; [[Xotla]]
* [https://www.youtube.com/watch?v=k5QDWFng-4A ''Amanita''] (2016)
* "Perfect Dystopia" from ''Microtones & Garden Gnomes'' (2017) – [https://xotla.bandcamp.com/track/perfect-dystopia-mixed-intonation Bandcamp] | [https://youtu.be/x_5YqboXfAA?si=3WT9lQmnoyp0QNxZ YouTube] – in part, the other part being in [[22edo]]
* from ''UnFound'' (2018)
** "Inflorescence" – [https://xotla.bandcamp.com/track/inflorescence-22edo Bandcamp]
** "Ephemeral Glade" – [https://xotla.bandcamp.com/track/ephemeral-glade-22edo Bandcamp] | [https://youtu.be/QiFoQ-KJ8zc?si=CP3t3elUsuEyccZJ YouTube]
* from ''Micro Biological'' (2018)
** "Catalysis" – [https://open.spotify.com/track/46QmV2tlPqOhucz3rKPfRK Spotify] | [https://xotla.bandcamp.com/track/catalysis-22edo Bandcamp] | [https://youtu.be/vHzHPzGMD1k?si=8qHLzUhR7-ltddQa YouTube]
** "Nodal Plane" – [https://open.spotify.com/track/1fuvjoPm5IP9MH8SrSMa7C Spotify] | [https://xotla.bandcamp.com/track/nodal-plane-22edo-29edo Bandcamp] | [https://youtu.be/3CPbU-W-sgg?si=cZGe_HhACJ24CzE2 YouTube] – in part, the other part being in [[29edo]]
** "Viscosity" – [https://open.spotify.com/track/6ZZgNKleOfnFPZEIGdEFkN Spotify] | [https://xotla.bandcamp.com/track/viscosity-22edo Bandcamp]
** "Refractive" – [https://open.spotify.com/track/7F38dDvcWYKeuRyZlMh8YP Spotify] | [https://xotla.bandcamp.com/track/refractive-22edo Bandcamp]
* from ''Nano Particular'' (2019)
** "Dark Pillars" – [https://open.spotify.com/track/2nRGb75AFAHY2T0mmcdzCk Spotify] | [https://xotla.bandcamp.com/track/dark-pillars-22edo Bandcamp] | [https://youtu.be/kYJ9wTR58pc?si=8vOfVqYMUzVqHCwg YouTube]
** "Floating Apart" – [https://open.spotify.com/track/4gbWPhzav3lIIdtJodDm8b Spotify] | [https://xotla.bandcamp.com/track/floating-apart-22edo Bandcamp] | [https://youtu.be/DmnWziyF17o?si=oU6a0kBeB3m9EYoq YouTube]
* "Light Pillars" from ''Pico Metric'' (2019) – [https://open.spotify.com/track/5Vqw6LWP85iGSAy9ngO3LM Spotify] | [https://xotla.bandcamp.com/track/light-pillars-22edo Bandcamp] | [https://youtu.be/dW1kQtMFlHI?si=x979_S--Kr5XDZf6 YouTube]
* "Time Mayfly" from ''Jazzbeetle'' (2023) – [https://xotla.bandcamp.com/track/time-mayfly-22edo Bandcamp] | [https://youtu.be/BsUjxujkBc4?si=XD2hDoHDGAmvgl2N YouTube]

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=EAFbQHL8b9w ''Twenty two''] (2019)

[[Category:22edo]]
[[Category:Listen]]

30edo

2024-11-21T06:06:27Z

YoVariable: Corrected "Pythagorean minor sedond" to "Pythagorean minor second"

{{Infobox ET}}
{{EDO intro|30}}
== Theory ==
{{Harmonics in equal|30}}

Its [[patent val]] is a doubled version of the patent val for [[15edo]] through the 11-limit, so 30 can be viewed as a [[contorted]] version of 15. In the 13-limit it supplies the optimal patent val for [[Trienstonic_clan#Quindecic|quindecic temperament]].
[[File:Plot30.png|alt=plot30.png|thumb|A plot of the Z function around 30.]]
However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30edo can be viewed inconsistently, as having a 9/1 at 95\30 as well as 96\30.

Instead of the 18\30 fifth of 720 cents, 30edo also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30edo is used for pelogic, 5\30 can again be used inconsistently as a 9/8.

=== Subsets and supersets ===
30edo has subset edos {{EDOs|1, 2, 3, 5, 6, 10, 15}} and it is a [[largely composite]] edo.

30edo is the 3rd [[wikipedia:primorial|primorial]] edo, being the product of first three primes and thus the smallest number with three distinct prime factors. As a corollary, 30edo is the smallest EDO that supports [[perfectly balanced]] scales that are minimal and not equally spaced. See the article on perfect balance.

== Intervals ==
{| class="wikitable center-all"
|-
! Step
! [[Cent]]s
! colspan="3" | [[Ups and downs notation]]
! [[Armodue_theory|Armodue Notation]]
|-
| 0
| 0
| P1
| unison, minor 2nd
| D, Eb
| 1
|-
| 1
| 40
| ^1, ^m2
| up unison, upminor 2nd
| ^D, ^Eb
| 2b
|-
| 2
| 80
| ^^1, v~2
| dup unison, downmid 2nd
| ^^D, ^^Eb
| 9#
|-
| 3
| 120
| ~2
| mid 2nd
| v3E
| 1#
|-
| 4
| 160
| ^~2
| upmid 2nd
| vvE
| 2
|-
| 5
| 200
| vM2
| downmajor 2nd
| vE
| 3b
|-
| 6
| 240
| M2, m3
| major 2nd, minor 3rd
| E, F
| 1x, 4bb
|-
| 7
| 280
| ^m3
| upminor 3rd
| ^F
| 2#
|-
| 8
| 320
| v~3
| downmid 3rd
| ^^F
| 3
|-
| 9
| 360
| ~3
| mid 3rd
| ^3F, v3F#
| 4b
|-
| 10
| 400
| ^~3
| upmid 3rd
| vvF#
| 5b
|-
| 11
| 440
| vM3, v4
| downmajor 3rd, down 4th
| vF#, vG
| 3#
|-
| 12
| 480
| M3, P4
| major 3rd, perfect 4th
| F#, G
| 4
|-
| 13
| 520
| ^4
| up 4th
| ^G
| 5
|-
| 14
| 560
| v~4, v~d5
| downmid 4th, downmid 5th
| ^^G, ^^Ab
| 6b
|-
| 15
| 600
| ~4, ~5
| mid 4th, mid 5th
| ^3G, v3A
| 4#
|-
| 16
| 640
| ^~4, ^~5
| upmid 4th, upmid 5th
| vvG#, vvA
| 5#
|-
| 17
| 680
| v5
| down 5th
| vA
| 6
|-
| 18
| 720
| P5, m6
| perfect 5th, minor 6th
| A, Bb
| 7b
|-
| 19
| 760
| ^5, ^m6
| up 5th, upminor 6th
| ^A, ^Bb
| 5x, 8bb
|-
| 20
| 800
| v~6
| downmid 6th
| ^^Bb
| 6#
|-
| 21
| 840
| ~6
| mid 6th
| v3B
| 7
|-
| 22
| 880
| ^~6
| upmid 6th
| vvB
| 8b
|-
| 23
| 920
| vM6
| downmajor 6th
| vB
| 6x, 9bb
|-
| 24
| 960
| M6. m7
| major 6th, minor 7th
| B, C
| 7#
|-
| 25
| 1000
| ^m7
| upminor 7th
| ^C
| 8
|-
| 26
| 1040
| v~7
| downmid 7th
| ^^C
| 9b
|-
| 27
| 1080
| ~7
| mid 7th
| ^3C
| 1b
|-
| 28
| 1120
| ^~7, vv8
| upmid 7th, dud 8ve
| vvC#, vvD
| 8#
|-
| 29
| 1160
| vM7, v8
| downmajor 7th, down 8ve
| vC#, vD
| 9
|-
| 30
| 1200
| P8
| major 7th, 8ve
| C#, D
| 1
|}

== Commas ==
30 EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 30 48 70 84 104 111 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic Limit|Prime Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean minor second
|-
| 5
| [[250/243]]
| {{monzo| 1 -5 3 }}
| 49.17
| Triyo
| Maximal diesis, Porcupine comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma, semicomma majeur
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[64827/64000]]
| {{monzo| -9 3 -3 4 }}
| 22.23
| Laquadzo-atrigu
| Squalentine
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|}
<references/>

== Rank-2 temperaments ==
As 30edo is largely composite, only 7, 11 and 13 steps create [[MOS scale]]s that cover every interval using one period per octave.

7/30 produces [[Chromatic_pairs#Lovecraft|Lovecraft]], in which 2 generators is a moderately sharp [[11/8]], 3 a near perfect [[13/8]] and 5 the familiar mildly flat [[9/8]] from [[12edo]], creating the possibility of ignoring the 3rd & 5th entirely to use those harmonics as the primary building blocks of harmony in a similar way to [[orgone]].

11 produces a flat [[sensi]] scale. 13 is an excellent higher order [[Pelogic_family#Mavila|Mavila]] tuning that functions the closest to the familiar diatonic scale you can get in this edo.

== Scales ==

=== MOS scales ===

* [[Lovecraft5|Lovecraft[5]]] - 77772
* [[Lovecraft9|Lovecraft[9]]] - 525252522
* [[Lovecraft13|Lovecraft[13]]] - 3223223223222
* Lovecraft[17] - 22221222122212221
* [[Sensi5|Sensi[5]]] - 83838
* [[Sensi8|Sensi[8]]] - 53353353
* [[Sensi11|Sensi[11]]] - 33323332332
* [[Sensi19|Sensi[19]]] - 2121212212121221212
* Mavila[5] - 94944
* Mavila[7] - 5445444
* Mavila[9] - 444414441
* Mavila[16] - 3131313113131311
* Mavila[23] - 21121121121112112112111

=== Subsets of [[Mavila]][16] ===

* Arcade (approximated from [[32afdo]]): 9 3 5 8 5
* [[Blackened skies]] (approximated from [[Compton]] in [[72edo]]): 8 5 2 3 2 8 2
* Carousel (this is the original/default tuning): 9 4 4 9 4
* Dewdrops (this is the original/default tuning): 4 4 4 5 4 4 5
* Geode (approximated from [[6afdo]]): 7 6 4 9 4
* [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]): 7 5 2 3 5 3 5
* Lost phantom (this is the original/default tuning): 8 5 2 2 6 2 5
* Mechanical (approximated from [[16afdo]]): 7 2 8 8 5
* Mushroom (approximated from [[30afdo]]): 7 5 5 3 10
* Nightdrive (this is the original/default tuning): 8 5 4 9 4
* Pelagic (this is the original/default tuning): 8 4 2 4 7 5
* Bathypelagic (this is the original/default tuning): 8 4 2 3 8 5
* Underpass (approximated from [[10afdo]]): 8 9 5 3 5
* Volcanic (approximated from [[16afdo]]): 3 6 8 8 5

=== Subsets of [[15edo]] ===

* Augmented[6] MOS: 8 2 8 2 8 2
* Equipentatonic (exact from [[5edo]]): 6 6 6 6 6
* Rockpool (approximated from [[47zpi]]): 2 8 2 6 6 6

=== Other notable scales ===

* Approximation of [[Pelog]] lima: 3 4 10 3 10
* [[Amiot]] scale

== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.
=== Fully delta-rational triads ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
!Steps
!Delta signature
!Least-squares error
|-
|0,1,2
| +1 +1
|0.00024
|-
|0,1,3
| +1 +2
|0.00052
|-
|0,1,4
| +1 +3
|0.00081
|-
|0,2,3
| +2 +1
|0.00046
|-
|0,2,4
| +1 +1
|0.00100
|-
|0,3,4
| +3 +1
|0.00068
|-
|0,3,5
| +3 +2
|0.00146
|-
|0,3,7
| +2 +3
|0.00144
|-
|0,3,11
| +1 +3
|0.00055
|-
|0,4,11
| +1 +2
|0.00035
|-
|0,4,14
| +1 +3
|0.00131
|-
|0,5,8
| +3 +2
|0.00055
|-
|0,6,16
| +1 +2
|0.00037
|-
|0,7,13
| +1 +1
|0.00033
|-
|0,7,23
| +1 +3
|0.00021
|-
|0,10,25
| +1 +2
|0.00064
|-
|0,11,17
| +3 +2
|0.00061
|-
|0,11,27
| +1 +2
|0.00065
|-
|0,12,17
| +2 +1
|0.00102
|-
|0,12,29
| +1 +2
|0.00133
|-
|0,13,23
| +1 +1
|0.00028
|-
|0,14,29
| +2 +3
|0.00017
|-
|0,15,19
| +3 +1
|0.00068
|-
|0,20,25
| +3 +1
|0.00084
|}

=== Partially delta-rational tetrads ===
{|class="mw-collapsible mw-collapsed class="wikitable sortable"
!Steps
!Delta signature
!Least-squares error
|-
|0,1,2,3
| +1 +? +1
|0.00064
|-
|0,1,2,4
| +1 +? +2
|0.00114
|-
|0,1,3,4
| +1 +? +1
|0.00097
|-
|0,1,4,5
| +1 +? +1
|0.00131
|-
|0,1,14,15
| +2 +? +3
|0.00134
|-
|0,1,14,16
| +1 +? +3
|0.00144
|-
|0,1,15,16
| +2 +? +3
|0.00097
|-
|0,1,15,17
| +1 +? +3
|0.00098
|-
|0,1,16,17
| +2 +? +3
|0.00060
|-
|0,1,16,18
| +1 +? +3
|0.00050
|-
|0,1,17,18
| +2 +? +3
|0.00021
|-
|0,1,17,19
| +1 +? +3
|0.00002
|-
|0,1,18,19
| +2 +? +3
|0.00018
|-
|0,1,18,20
| +1 +? +3
|0.00047
|-
|0,1,19,20
| +2 +? +3
|0.00058
|-
|0,1,19,21
| +1 +? +3
|0.00098
|-
|0,1,20,21
| +2 +? +3
|0.00099
|-
|0,1,20,22
| +1 +? +3
|0.00149
|-
|0,1,21,22
| +2 +? +3
|0.00141
|-
|0,1,27,28
| +1 +? +2
|0.00128
|-
|0,1,28,29
| +1 +? +2
|0.00086
|-
|0,2,3,4
| +2 +? +1
|0.00094
|-
|0,2,4,5
| +2 +? +1
|0.00133
|-
|0,2,5,10
| +1 +? +3
|0.00133
|-
|0,2,6,11
| +1 +? +3
|0.00036
|-
|0,2,7,12
| +1 +? +3
|0.00063
|-
|0,2,10,11
| +3 +? +2
|0.00133
|-
|0,2,11,12
| +3 +? +2
|0.00089
|-
|0,2,11,14
| +1 +? +2
|0.00084
|-
|0,2,12,13
| +3 +? +2
|0.00044
|-
|0,2,12,15
| +1 +? +2
|0.00005
|-
|0,2,13,14
| +3 +? +2
|0.00002
|-
|0,2,13,16
| +1 +? +2
|0.00095
|-
|0,2,14,15
| +3 +? +2
|0.00049
|-
|0,2,15,16
| +3 +? +2
|0.00098
|-
|0,2,15,19
| +1 +? +3
|0.00149
|-
|0,2,16,17
| +3 +? +2
|0.00147
|-
|0,2,16,18
| +2 +? +3
|0.00121
|-
|0,2,16,20
| +1 +? +3
|0.00053
|-
|0,2,17,19
| +2 +? +3
|0.00043
|-
|0,2,17,21
| +1 +? +3
|0.00046
|-
|0,2,18,20
| +2 +? +3
|0.00036
|-
|0,2,18,22
| +1 +? +3
|0.00147
|-
|0,2,19,21
| +2 +? +3
|0.00117
|-
|0,3,4,5
| +3 +? +1
|0.00118
|-
|0,3,4,8
| +2 +? +3
|0.00071
|-
|0,3,5,9
| +2 +? +3
|0.00050
|-
|0,3,6,11
| +1 +? +2
|0.00117
|-
|0,3,7,12
| +1 +? +2
|0.00017
|-
|0,3,8,15
| +1 +? +3
|0.00125
|-
|0,3,9,16
| +1 +? +3
|0.00024
|-
|0,3,16,17
| +2 +? +1
|0.00137
|-
|0,3,16,20
| +1 +? +2
|0.00139
|-
|0,3,16,22
| +1 +? +3
|0.00003
|-
|0,3,17,18
| +2 +? +1
|0.00085
|-
|0,3,17,19
| +1 +? +1
|0.00100
|-
|0,3,17,20
| +2 +? +3
|0.00066
|-
|0,3,17,21
| +1 +? +2
|0.00006
|-
|0,3,17,23
| +1 +? +3
|0.00148
|-
|0,3,18,19
| +2 +? +1
|0.00031
|-
|0,3,18,20
| +1 +? +1
|0.00005
|-
|0,3,18,21
| +2 +? +3
|0.00055
|-
|0,3,18,22
| +1 +? +2
|0.00131
|-
|0,3,19,20
| +2 +? +1
|0.00025
|-
|0,3,19,21
| +1 +? +1
|0.00092
|-
|0,3,20,21
| +2 +? +1
|0.00081
|-
|0,3,21,22
| +2 +? +1
|0.00139
|-
|0,3,24,29
| +1 +? +3
|0.00063
|-
|0,4,5,12
| +1 +? +2
|0.00139
|-
|0,4,5,15
| +1 +? +3
|0.00038
|-
|0,4,7,12
| +2 +? +3
|0.00062
|-
|0,4,8,13
| +2 +? +3
|0.00101
|-
|0,4,10,19
| +1 +? +3
|0.00023
|-
|0,4,11,12
| +3 +? +1
|0.00147
|-
|0,4,11,17
| +1 +? +2
|0.00078
|-
|0,4,12,13
| +3 +? +1
|0.00099
|-
|0,4,12,14
| +3 +? +2
|0.00137
|-
|0,4,12,15
| +1 +? +1
|0.00122
|-
|0,4,12,18
| +1 +? +2
|0.00104
|-
|0,4,13,14
| +3 +? +1
|0.00049
|-
|0,4,13,15
| +3 +? +2
|0.00044
|-
|0,4,13,16
| +1 +? +1
|0.00005
|-
|0,4,14,15
| +3 +? +1
|0.00002
|-
|0,4,14,16
| +3 +? +2
|0.00052
|-
|0,4,14,17
| +1 +? +1
|0.00136
|-
|0,4,15,16
| +3 +? +1
|0.00054
|-
|0,4,15,17
| +3 +? +2
|0.00149
|-
|0,4,15,23
| +1 +? +3
|0.00101
|-
|0,4,16,17
| +3 +? +1
|0.00107
|-
|0,4,16,24
| +1 +? +3
|0.00101
|-
|0,4,17,21
| +2 +? +3
|0.00089
|-
|0,4,18,22
| +2 +? +3
|0.00074
|-
|0,4,20,25
| +1 +? +2
|0.00030
|-
|0,4,22,29
| +1 +? +3
|0.00041
|-
|0,5,6,9
| +3 +? +2
|0.00051
|-
|0,5,6,18
| +1 +? +3
|0.00011
|-
|0,5,8,16
| +1 +? +2
|0.00028
|-
|0,5,9,15
| +2 +? +3
|0.00030
|-
|0,5,10,12
| +2 +? +1
|0.00110
|-
|0,5,10,14
| +1 +? +1
|0.00027
|-
|0,5,10,21
| +1 +? +3
|0.00084
|-
|0,5,11,13
| +2 +? +1
|0.00017
|-
|0,5,11,15
| +1 +? +1
|0.00137
|-
|0,5,12,14
| +2 +? +1
|0.00078
|-
|0,5,14,21
| +1 +? +2
|0.00095
|-
|0,5,15,22
| +1 +? +2
|0.00136
|-
|0,5,15,25
| +1 +? +3
|0.00006
|-
|0,5,17,22
| +2 +? +3
|0.00112
|-
|0,5,18,23
| +2 +? +3
|0.00093
|-
|0,5,20,29
| +1 +? +3
|0.00014
|-
|0,5,21,27
| +1 +? +2
|0.00137
|-
|0,5,22,23
| +3 +? +1
|0.00135
|-
|0,5,22,28
| +1 +? +2
|0.00093
|-
|0,5,23,24
| +3 +? +1
|0.00073
|-
|0,5,23,25
| +3 +? +2
|0.00075
|-
|0,5,23,26
| +1 +? +1
|0.00020
|-
|0,5,24,25
| +3 +? +1
|0.00009
|-
|0,5,24,26
| +3 +? +2
|0.00045
|-
|0,5,24,27
| +1 +? +1
|0.00144
|-
|0,5,25,26
| +3 +? +1
|0.00057
|-
|0,5,26,27
| +3 +? +1
|0.00124
|-
|0,6,7,21
| +1 +? +3
|0.00086
|-
|0,6,8,13
| +1 +? +1
|0.00079
|-
|0,6,9,14
| +1 +? +1
|0.00120
|-
|0,6,10,17
| +2 +? +3
|0.00091
|-
|0,6,10,23
| +1 +? +3
|0.00141
|-
|0,6,11,20
| +1 +? +2
|0.00026
|-
|0,6,13,16
| +3 +? +2
|0.00142
|-
|0,6,14,17
| +3 +? +2
|0.00003
|-
|0,6,14,26
| +1 +? +3
|0.00138
|-
|0,6,16,24
| +1 +? +2
|0.00146
|-
|0,6,17,23
| +2 +? +3
|0.00136
|-
|0,6,17,25
| +1 +? +2
|0.00135
|-
|0,6,18,22
| +1 +? +1
|0.00112
|-
|0,6,18,24
| +2 +? +3
|0.00113
|-
|0,6,19,21
| +2 +? +1
|0.00066
|-
|0,6,19,23
| +1 +? +1
|0.00086
|-
|0,6,20,22
| +2 +? +1
|0.00048
|-
|0,7,8,10
| +3 +? +1
|0.00111
|-
|0,7,8,11
| +2 +? +1
|0.00095
|-
|0,7,8,12
| +3 +? +2
|0.00035
|-
|0,7,8,19
| +1 +? +2
|0.00121
|-
|0,7,9,11
| +3 +? +1
|0.00020
|-
|0,7,9,12
| +2 +? +1
|0.00039
|-
|0,7,9,13
| +3 +? +2
|0.00139
|-
|0,7,10,12
| +3 +? +1
|0.00074
|-
|0,7,10,25
| +1 +? +3
|0.00144
|-
|0,7,11,19
| +2 +? +3
|0.00075
|-
|0,7,13,23
| +1 +? +2
|0.00005
|-
|0,7,14,28
| +1 +? +3
|0.00034
|-
|0,7,15,20
| +1 +? +1
|0.00136
|-
|0,7,16,21
| +1 +? +1
|0.00097
|-
|0,7,18,25
| +2 +? +3
|0.00134
|-
|0,7,18,27
| +1 +? +2
|0.00030
|-
|0,7,21,24
| +3 +? +2
|0.00028
|-
|0,7,22,25
| +3 +? +2
|0.00146
|-
|0,7,26,28
| +2 +? +1
|0.00102
|-
|0,7,27,29
| +2 +? +1
|0.00032
|-
|0,8,10,22
| +1 +? +2
|0.00143
|-
|0,8,10,27
| +1 +? +3
|0.00088
|-
|0,8,12,21
| +2 +? +3
|0.00022
|-
|0,8,13,19
| +1 +? +1
|0.00129
|-
|0,8,14,18
| +3 +? +2
|0.00099
|-
|0,8,14,20
| +1 +? +1
|0.00142
|-
|0,8,15,17
| +3 +? +1
|0.00054
|-
|0,8,15,18
| +2 +? +1
|0.00001
|-
|0,8,15,19
| +3 +? +2
|0.00101
|-
|0,8,15,26
| +1 +? +2
|0.00130
|-
|0,8,16,18
| +3 +? +1
|0.00053
|-
|0,8,19,29
| +1 +? +2
|0.00117
|-
|0,8,22,27
| +1 +? +1
|0.00033
|-
|0,9,10,15
| +3 +? +2
|0.00013
|-
|0,9,10,29
| +1 +? +3
|0.00029
|-
|0,9,12,19
| +1 +? +1
|0.00028
|-
|0,9,12,25
| +1 +? +2
|0.00000
|-
|0,9,16,28
| +1 +? +2
|0.00005
|-
|0,9,19,25
| +1 +? +1
|0.00028
|-
|0,9,20,22
| +3 +? +1
|0.00136
|-
|0,9,20,23
| +2 +? +1
|0.00109
|-
|0,9,20,24
| +3 +? +2
|0.00025
|-
|0,9,21,23
| +3 +? +1
|0.00015
|-
|0,9,21,24
| +2 +? +1
|0.00068
|-
|0,9,22,24
| +3 +? +1
|0.00109
|-
|0,10,11,19
| +1 +? +1
|0.00107
|-
|0,10,12,16
| +2 +? +1
|0.00147
|-
|0,10,13,17
| +2 +? +1
|0.00052
|-
|0,10,13,24
| +2 +? +3
|0.00042
|-
|0,10,15,20
| +3 +? +2
|0.00006
|-
|0,10,17,24
| +1 +? +1
|0.00005
|-
|0,10,25,28
| +2 +? +1
|0.00138
|-
|0,10,25,29
| +3 +? +2
|0.00048
|-
|0,10,26,28
| +3 +? +1
|0.00028
|-
|0,10,26,29
| +2 +? +1
|0.00061
|-
|0,10,27,29
| +3 +? +1
|0.00111
|-
|0,11,12,15
| +3 +? +1
|0.00121
|-
|0,11,13,16
| +3 +? +1
|0.00032
|-
|0,11,13,25
| +2 +? +3
|0.00127
|-
|0,11,15,23
| +1 +? +1
|0.00130
|-
|0,11,17,21
| +2 +? +1
|0.00085
|-
|0,11,18,22
| +2 +? +1
|0.00138
|-
|0,11,20,25
| +3 +? +2
|0.00095
|-
|0,11,22,29
| +1 +? +1
|0.00143
|-
|0,12,14,23
| +1 +? +1
|0.00005
|-
|0,12,16,22
| +3 +? +2
|0.00141
|-
|0,12,17,20
| +3 +? +1
|0.00014
|-
|0,12,22,26
| +2 +? +1
|0.00081
|-
|0,12,24,29
| +3 +? +2
|0.00014
|-
|0,13,14,28
| +2 +? +3
|0.00109
|-
|0,13,15,20
| +2 +? +1
|0.00132
|-
|0,13,16,21
| +2 +? +1
|0.00137
|-
|0,13,18,27
| +1 +? +1
|0.00000
|-
|0,13,21,24
| +3 +? +1
|0.00013
|-
|0,14,16,23
| +3 +? +2
|0.00035
|-
|0,14,19,24
| +2 +? +1
|0.00067
|-
|0,14,23,29
| +3 +? +2
|0.00104
|-
|0,14,25,28
| +3 +? +1
|0.00040
|-
|0,15,16,20
| +3 +? +1
|0.00142
|-
|0,15,23,28
| +2 +? +1
|0.00083
|-
|0,16,18,24
| +2 +? +1
|0.00117
|-
|0,16,19,23
| +3 +? +1
|0.00076
|-
|0,17,20,28
| +3 +? +2
|0.00099
|-
|0,17,21,27
| +2 +? +1
|0.00067
|-
|0,17,22,26
| +3 +? +1
|0.00042
|-
|0,18,25,29
| +3 +? +1
|0.00042
|-
|0,19,20,29
| +3 +? +2
|0.00033
|-
|0,20,22,29
| +2 +? +1
|0.00124
|-
|0,21,23,28
| +3 +? +1
|0.00012
|}

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=uSpDz2Dmksw ''microtonal improvisation in 30edo''] (2023)

; [[Todd Harrop]]
* [https://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces ''Fifteen Short Pieces'']

; [[Micronaive]]
* [https://youtu.be/tAxEetp1TaE ''No.27.62'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=6gydbVD7Xdc ''Edolian - Shift''] (2020)

== Related pages ==
* [[Lumatone mapping for 30edo]]
* [[Mavila]]

[[Category:Pelogic]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Listen]]

224/135

2024-11-09T22:52:26Z

YoVariable: Changed "minor" to "major" in description

{{Infobox Interval
| Name = septimal narrow major sixth, marvelous major sixth
| Color name = zg7, zogu 7th
}}

'''224/135''', the '''septimal narrow major sixth''', or '''marvelous major sixth''', is a [[7-limit]] interval that forms the difference between [[7/4]] and [[135/128]], and between [[16/9]] and [[15/14]]. Given that 135/128 is a type of chromatic semitone, this means 224/135 functionally doubles as a type of diminished seventh.

== See also ==

* [[135/112]] -- its octave complement
* [[Gallery of just intervals]]

43edo

2024-11-02T05:04:45Z

YoVariable: /* Intervals */ Added interval names

{{Infobox ET}}
{{EDO intro|43}}

== History ==
The French Baroque acoustician {{w|Joseph Sauveur}}, who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling one step a '''méride'''. Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it.<ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>

The composer [[Juhan Puhm]] uses 43edo in some of his fortepiano suites and prefers it to [[31edo]].

== Theory ==
43edo tempers out [[81/80]] in the 5-limit, and as such it is strongly associated with [[meantone]]. Specifically, it is (for all practical purposes) equivalent to [[1/5-comma meantone]], as it tunes the major third sharp of [[5/4]] and perfect fifth flat of [[3/2]] by slightly more than four cents on both of them. It also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that six chromatic semitones make a perfect fourth and eight minor seconds make a major sixth.

Except for 9/7, 11/9, 14/9, and 18/11, all [[15-odd-limit]] intervals have [[consistent]] approximations in 43edo, making it an excellent tuning in the 7-, 11-, and 13-limit. In the 7-limit, it supports septimal meantone, as it tempers out [[126/125]], [[225/224]], and [[3136/3125]]. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[Huygens temperament|Huygens]]. In the 13-limit it supports [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.

43edo's patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not meantone. [[Thuja]] is also a possibility, whose 11-limit extension makes five 11/8s stack to a major third (i.e. {{nowrap|(11/8)5 → 5/1}}), with [[mos]]es of 15 and 28.

=== Prime harmonics ===
{{Harmonics in equal|43}}
Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results. Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarily, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly.

=== Divisors ===
43edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].

== Intervals ==
The distance from C to C♯ is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp.
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Approximate 17-limit Ratios
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0.000
| 1/1
| P1
| perfect unison
| D
|-
| 1
| 27.907
| ''36/35'', 50/49, 64/63, 65/64, 66/65
| ^1, d2
| up unison, dim 2nd
| ^D, Ebb
|-
| 2
| 55.814
| ''49/48'', 33/32
| vA1, ^d2
| downaug unison, updim 2nd
| vD#, ^Ebb
|-
| 3
| 83.721
| 25/24, 21/20, ''28/27'', 22/21, ''18/17''
| A1, vm2
| aug 1sn, downminor 2nd
| D#, vEb
|-
| 4
| 111.628
| 16/15, 15/14, 17/16
| m2
| minor 2nd
| Eb
|-
| 5
| 139.535
| 12/11, 13/12, 14/13
| ^m2
| upminor 2nd
| ^Eb
|-
| 6
| 167.442
| 11/10
| vM2
| downmajor 2nd
| vE
|-
| 7
| 195.349
| 9/8, 10/9
| M2
| major 2nd
| E
|-
| 8
| 223.256
| 8/7
| ^M2, d3
| upmajor 2nd, dim 3rd
| ^E, Fb
|-
| 9
| 251.163
| 15/13
| vA2, ^d3
| downaug 2nd, updim 3rd
| vE#, ^Fb
|-
| 10
| 279.070
| 7/6, 13/11
| A2, vm3
| aug 2nd, downminor 3rd
| E#, vF
|-
| 11
| 306.977
| 6/5
| m3
| minor 3rd
| F
|-
| 12
| 334.884
| 39/32, 17/14
| ^m3
| upminor 3rd
| ^F
|-
| 13
| 362.791
| 16/13, 21/17, ''11/9''
| vM3
| downmajor 3rd
| vF#
|-
| 14
| 390.698
| 5/4
| M3
| major 3rd
| F#
|-
| 15
| 418.605
| ''9/7'', 14/11
| ^M3, d4
| upmajor 3rd, dim 4th
| ^F#, Gb
|-
| 16
| 446.512
| 13/10
| vA3, ^d4
| downaug 3rd, updim 4th
| vFx, ^Gb
|-
| 17
| 474.419
| 21/16
| v4
| down 4th
| vG
|-
| 18
| 502.326
| 4/3
| P4
| perfect 4th
| G
|-
| 19
| 530.233
| 15/11
| ^4
| up 4th
| ^G
|-
| 20
| 558.140
| 11/8, 18/13
| vA4
| downaug 4th
| vG#
|-
| 21
| 586.047
| 45/32, 7/5, 24/17
| A4, vd5
| aug 4th, downdim 5th
| G#, ^Ab
|-
| 22
| 613.953
| 64/45, 10/7, 17/12
| ^A4, d5
| upaug 4th, dim 5th
| ^G#, Ab
|-
| 23
| 641.860
| 16/11, 13/9
| ^d5
| updim 5th
| ^Ab
|-
| 24
| 669.767
| 22/15
| v5
| down 5th
| vA
|-
| 25
| 697.674
| 3/2
| P5
| perfect 5th
| A
|-
| 26
| 725.581
| 32/21
| ^5
| up 5th
| ^A
|-
| 27
| 753.488
| 20/13
| vA5, ^d6
| downaug 5th, updim 6th
| vA#, ^Bbb
|-
| 28
| 781.395
| ''14/9'', 11/7
| A5, vm6
| aug 5th, downminor 6th
| A#, vBb
|-
| 29
| 809.302
| 8/5
| m6
| minor 6th
| Bb
|-
| 30
| 837.209
| 13/8, 34/21, ''18/11''
| ^m6
| upminor 6th
| ^Bb
|-
| 31
| 865.116
| 64/39, 28/17
| vM6
| downmajor 6th
| vB
|-
| 32
| 893.023
| 5/3
| M6
| major 6th
| B
|-
| 33
| 920.930
| 12/7, 22/13
| ^M6, d7
| upmajor 6th, dim 7th
| ^B, Cb
|-
| 34
| 948.837
| 26/15
| vA6, ^d7
| downaug 6th, updim 7th
| vB#, ^Cb
|-
| 35
| 976.744
| 7/4
| A6, vm7
| aug 6th, downminor 7th
| B#, vC
|-
| 36
| 1004.651
| 16/9, 9/5
| m7
| minor 7th
| C
|-
| 37
| 1032.558
| 20/11
| ^m7
| upminor 7th
| ^C
|-
| 38
| 1060.465
| 11/6, 24/13, 13/7
| vM7
| downmajor 7th
| vC#
|-
| 39
| 1088.372
| 15/8, 28/15, 32/17
| M7
| major 7th
| C#
|-
| 40
| 1116.279
| 48/25, 40/21, ''27/14'', 21/11, ''17/9''
| ^M7, d8
| upmajor 7th, dim 8ve
| ^C#, Db
|-
| 41
| 1144.186
| ''96/49'', 64/33
| vA7, ^d8
| downaug 7th, updim 8ve
| vCx, ^Db
|-
| 42
| 1172.093
| ''35/18'', 49/25, 63/32, 65/33, 128/65
| A7, v8
| aug 7th, down 8ve
| Cx, vD
|-
| 43
| 1200.000
| 2/1
| P8
| perfect 8ve
| D
|}

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation #Chords and Chord Progressions]].

== JI approximation ==
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]
=== Interval mappings ===
{{Q-odd-limit intervals|43}}

== Notation ==
=== Red-Blue Notation ===
Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A♯ and B♭ are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.

For people who aren't colorblind, a red-note/blue-note system (similar to that proposed for [[36edo]]) can be used. (Note that this is different than Kite's [[color notation]].) Now we have the following sequence of notes, each separated by one meride: {{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|A♯}}, {{colored note|A♯}}, {{colored note|B♭}}, {{colored note|red|B♭}}, {{colored note|blue|B}}, {{colored note|B}}. (Note that red sharps or blue flats are enharmonically equivalent to simpler notes: {{colored note|red|A♯}} is enharmonic to B♭, and {{colored note|blue|B♭}} is actually just A♯).

The diatonic semitone is four steps, so for the region between B and C, we can use: {{colored note|B}}, {{colored note|C♭}}, {{colored note|blue|B♯}} / {{colored note|red|C♭}} (they are enharmonic equivalents), {{colored note|B♯}}, and {{colored note|C}}. All of the notes in 43edo therefore have unambiguous names except for {{colored note|blue|B♯}} / {{colored note|red|C♭}}, and {{colored note|blue|E♯}} / {{colored note|red|F♭}}. It might also be possible to design special symbols for those two notes (resembling a cross between the letters B and C in the former case, and E and F in the latter).

If {{colored note|red|C♭}} and {{colored note|blue|B♯}} (and {{colored note|red|F♭}} / {{colored note|blue|E♯}}) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a ''completely'' unambiguous red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.

=== Ups and downs notation ===
The third-sharps and third-flats can also be notated using [[ups and downs notation]] and extended [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals:
{{Sharpness-sharp3}}
The notes between A and B can then be notated as A, A{{naturalup}}, A{{sharpdown}}, A♯, B♭, B{{flatup}}, B{{naturaldown}}, B. Note that A♯ is enharmonic to B{{flatdown}}, and B♭ is enharmonic to A{{sharpup}}.

The notes from B to C are B, C♭, B{{sharpdown}} / C{{flatup}}, B♯, and C. Similarily, the notes from E to F are E, F♭, E{{sharpdown}} / F{{flatup}}, E♯, and F. As with the red/blue note system described above, all notes in 43edo therefore have unambiguous names except for B{{sharpdown}} / C{{flatup}} and E{{sharpdown}} / F{{flatup}}.

Double or even triple arrows may arise if the arrows are taken to have their own layer of enharmonic spellings.

=== Sagittal ===
The following table shows [[sagittal notation]] accidentals in one apotome for 43do.

{| class="wikitable center-all"
! Steps
| 0
| 1
| 2
| 3
|-
! Symbol
| [[File:Sagittal natural.png]]
| [[File:Sagittal tai.png]]
| [[File:Sagittal sharp tao.png]]
| [[File:Sagittal sharp.png]]
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -68 43 }}
| [{{val| 43 68 }}]
| +1.35
| 1.35
| 4.84
|-
| 2.3.5
| 81/80, 50331648/48828125
| [{{val| 43 68 100 }}]
| +0.27
| 1.88
| 6.75
|-
| 2.3.5.7
| 81/80, 126/125, 17280/16807
| [{{val| 43 68 100 121 }}]
| −0.51
| 2.11
| 7.56
|-
| 2.3.5.7.11
| 81/80, 99/98, 126/125, 864/847
| [{{val| 43 68 100 121 149 }}]
| −0.80
| 1.98
| 7.08
|-
| 2.3.5.7.11.13
| 78/77, 81/80, 99/98, 126/125, 144/143
| [{{val| 43 68 100 121 149 159 }}]
| −0.52
| 1.91
| 6.85
|-
| 2.3.5.7.11.13.17
| 78/77, 81/80, 99/98, 120/119, 126/125, 144/143
| [{{val| 43 68 100 121 149 159 176 }}]
| −0.52
| 1.81
| 6.49
|-
| 2.3.5.7.11.13.17.19
| 78/77, 81/80, 99/98, 120/119, 126/125, 135/133, 144/143
| [{{val| 43 68 100 121 149 159 176 183 }}]
| −0.87
| 1.77
| 6.34
|}

=== Commas ===
This is a partial list of the 19-limit [[commas]] that 43edo [[tempers out]] with its patent [[val]], {{val| 43 68 100 121 149 159 176 183 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime Limit]]
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| <abbr title="328256967394537077627/295147905179352825856">(42 digits)</abbr>
| {{monzo| -68 43 }}
| 184.07
| Tribilawa
| 43-comma
|-
| 5
| <abbr title="254803968/244140625">(18 digits)</abbr>
| {{monzo| 20 5 -12 }}
| 74.01
| Saquadtrigu
| [[Hypovishnuzma]]
|-
| 5
| <abbr title="50331648/48828125">(16 digits)</abbr>
| {{monzo| 24 1 -11 }}
| 52.50
| Salegu
| [[Magus comma]]
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 5
| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>
| {{monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom family|Vavoom comma]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| <abbr title="2500000/2470629">(14 digits)</abbr>
| {{monzo| 5 -1 7 -7 }}
| 20.46
| Sepruyo
| [[Merman|Mermisma]]
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.80
| Zotrigu
| Starling comma
|-
| 7
| <abbr title="2097152/2083725">(14 digits)</abbr>
| {{monzo| 21 -5 -2 -3 }}
| 11.12
| Satriru-agugu
| [[Bronzisma]]
|-
| 7
| <abbr title="257298363/256000000">(18 digits)</abbr>
| {{monzo| -14 7 -6 6 }}
| 8.76
| Latribizogu
| [[Historisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 11
| [[1350/1331]]
| {{monzo| 1 3 2 0 -3}}
| 24.54
| Trilu-ayoyo
| Large Tetracot diesis
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3}}
| 3.03
| Triluyo
| Wizardharry comma, pine comma
|-
| 11
| <abbr title="131072/130977">(12 digits)</abbr>
| {{monzo| 17 -5 0 -2 -1 }}
| 1.26
| Salururu
| [[Olympic comma]]
|-
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|-
| 13
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[169/168]]
| {{monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
| 13
| <abbr title="373248/371293">(12 digits)</abbr>
| {{monzo| 9 6 0 0 0 -5 }}
| 9.09
| Quinthu
| [[Glacier comma]]
|-
| 13
| [[364/363]]
| {{monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Tholuluzo
| Minor minthma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[2080/2079]]
| {{monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Tholuruyo
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[120/119]]
| {{monzo| 3 1 1 -1 0 0 -1 }}
| 14.49
| Suruyo
| Lynchisma
|-
| 17
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Sotholugu
| Minor naiadma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[273/272]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 6.35
| Suthozo
| Tannisma
|-
| 17
| [[715/714]]
| {{monzo| -1 -1 1 -1 1 1 -1 }}
| 2.42
| Sutholoruyo
| September comma
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[209/208]]
| {{monzo| -4 0 0 0 1 -1 0 1 }}
| 8.30
| Nothulo
| Yama comma
|-
| 19
| [[210/209]]
| {{monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.26
| Nuluzoyo
| Spleen comma
|}
<references group="note"/>

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator (Reduced)
! Cents (Reduced)
! Associated Ratio (Reduced)
! Temperament
|-
| 1
| 1\43
| 27.91
| 64/63
| [[Arch]]
|-
| 1
| 2\43
| 55.81
| 33/32
| [[Escapade]]
|-
| 1
| 4\43
| 111.63
| 16/15
| [[Vavoom]]
|-
| 1
| 5\43
| 139.53
| 13/12
| [[Jerome]]
|-
| 1
| 6\43
| 167.44
| 11/10
| [[Superpine]]
|-
| 1
| 7\43
| 195.35
| 28/25
| [[Didacus]]
|-
| 1
| 8\43
| 223.26
| 8/7
| [[Kumonga]]
|-
| 1
| 9\43
| 251.16
| 15/13
| [[Hemimeantone]]
|-
| 1
| 10\43
| 279.07
| 75/64
| [[Decipentic]]
|-
| 1
| 11\43
| 334.88
| 17/14
| [[Cohemimabila]]
|-
| 1
| 13\43
| 362.79
| 16/13
| [[Submajor]] (43e) / interpental (43)
|-
| 1
| 14\43
| 390.70
| 5/4
| [[Amigo]]
|-
| 1
| 16\43
| 446.51
| 13/10
| [[Supersensi]]
|-
| 1
| 18\43
| 502.33
| 4/3
| [[Meantone]]
|-
| 1
| 19\43
| 530.23
| 15/11
| [[Amavil]]
|-
| 1
| 20\43
| 558.14
| 11/8
| [[Thuja]]
|-
| 1
| 21\43
| 586.05
| 7/5
| [[Merman]]
|}

== Detemperaments ==
=== Ringer 43 ===
The metaphorical color palette that the intervals of 43edo present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 109-limit except 71 and 89) is Ringer 43, a [[Ringer scale]] with 43 notes per octave period:
<pre>
55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110
</pre>
Or equivalently in the form of reduced, [[rooted interval]]s:

65/64, 33/32, 67/64, 17/16, 69/64, 35/32, 9/8, 73/64, 37/32, 75/64, 19/16, 39/32, 79/64, 5/4, 41/32, 83/64, 21/16, 43/32, 87/64, 11/8, 45/32, 91/64, 23/16, 47/32, 3/2, 97/64, 49/32, 25/16, 51/64, 13/8, 53/32, 27/16, 109/64, 55/32, 7/4, 57/32, 29/16, 59/32, 15/8, 61/32, 31/16, 63/32, 2/1

== Scales ==

=== MOS scales ===
{{main|List of MOS scales in 43edo}}

=== Harmonic scales ===
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
* 7\43 (195.349¢) stands in for frequency ratio [[9/8]] (203.910¢) and [[10/9]] (182.404¢).
* 6\43 (156.522¢) stands in for [[11/10]] (165.004¢)
* 5\46 (130.435¢) stands in for [[12/11]] (150.637¢), [[13/12]] (138.573¢), and [[14/13]] (128.298¢).
* 4\43 (111.628¢) stands in for [[15/14]] (119.443¢) and [[16/15]] (111.731¢).
{| class="wikitable center-all"
|-
! Harmonic
! Note (starting from C)
|-
! 1
| style="font-size: 16px;" | C
|-
! 3
| style="font-size: 16px;" | G
|-
! 5
| style="font-size: 16px;" | E
|-
! 7
| style="font-size: 16px;" | A♯, B{{flatdown|36}}
|-
! 9
| style="font-size: 16px;" | D
|-
! 11
| style="font-size: 16px;" | E𝄪, F{{sharpdown|36}}, F{{naturalup2|36}}
|-
! 13
| style="font-size: 16px;" | B♭♭♭, A{{flatup|36}}
|-
! 15
| style="font-size: 16px;" | B
|}

=== Subsets of Meantone ===

'''Major scales'''
* Ionian Pentatonic: 14 4 7 14 4
* Major: 7 7 4 7 7 7 4
* Major Pentatonic: 7 7 11 7 11

'''Minor scales'''
* Minor: 7 4 7 7 4 7 7
* Minor Harmonic: 7 4 7 7 4 10 4
* Minor Harmonic Pentatonic: 7 4 14 14 4
* Minor Hexatonic: 7 4 7 7 11 7
* Minor Melodic: 7 4 7 7 7 7 4
* Minor Pentatonic: 11 7 7 11 7

'''Modal scales'''
* Dorian: 7 4 7 7 7 4 7
* Locrian: 4 7 7 4 7 7 7
* Lydian: 7 7 7 4 7 7 4
* Mixolydian: 7 7 4 7 7 4 7
* Mixolydian Harmonic: 14 4 7 4 7 7
* Mixolydian Pentatonic: 14 4 7 11 7
* Phrygian: 4 7 7 7 4 7 7
* Phrygian Dominant: 4 10 4 7 4 7 7
* Phrygian Dominant Hexatonic: 4 10 4 7 11 7
* Phrygian Dominant Pentatonic: 14 4 7 4 14
* Phrygian Pentatonic: 4 7 14 4 14

'''Blues scales'''
* Blues Aeolian Hexatonic: 11 7 4 3 4 14
* Blues Aeolian Pentatonic I: 11 7 7 4 14
* Blues Aeolian Pentatonic II: 11 14 4 7 7
* Blues Bright Double Harmonic: 4 10 4 7 4 7 3 4
* Blues Dark Double Harmonic: 7 4 7 4 3 4 10 4
* Blues Dorian Hexatonic: 11 7 7 7 4 7
* Blues Dorian Pentatonic: 11 14 7 4 7
* Blues Dorian Septatonic: 11 7 4 3 7 4 7
* Blues Harmonic Hexatonic: 7 4 7 7 14 4
* Blues Harmonic Septatonic: 11 7 4 3 4 10 4
* Blues Leading: 11 7 4 3 11 3 4
* Blues Minor: 11 7 4 3 11 7
* Blues Minor Maj7: 11 7 4 3 14 4
* Blues Pentachordal: 7 4 7 4 3 18
* Hyperblue Dorian: 11 7 2 5 9 2 7
* Hyperblue Harmonic: 11 7 2 5 3 12 3

'''Others'''
* Akebono I: 7 4 14 7 11
* Dominant Pentatonic: 7 7 11 11 7
* Double Harmonic: 4 10 4 7 4 10 4
* Hirajoshi: 7 4 14 4 14
* Javanese Pentachordal: 4 7 11 3 18
* Picardy Hexatonic: 7 7 4 7 4 14
* Picardy Pentatonic: 7 7 11 4 14

=== Other notable scales ===
* [[Magnetosphere scale]] (approximated from [[Hexany 1728]]): 4 10 11 11 7

== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=v3mBkctQ4SI ''Prelude in C minor'', BWV 999] (1717–1723) – transposed into E minor, arranged for Organ and rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=E7W-t2KDeSs "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=qyWfguU0iQM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=GkuUVQYpjo4 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=UYaZZXUrGeA ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|George Frideric Handel}}
* [https://www.youtube.com/watch?v=l5g9XvUNaVg ''Suite in D minor HWV 428 for Harpsichord - Allemande''] (1720) – rendered by Claudi Meneghin (2024)

; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=nV_jmn31Kiw ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)

; {{W|Shirō Sagisu}}
* [https://www.youtube.com/watch?v=WyU38ESUBqo ''Pensées Intimes''] – rendered by [[MortisTheneRd]] (2024)

=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=pALxebjbhZo ''microtonal improvisation in 43edo''] (2023)

; [[Peter Kosmorsky]]
* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]] [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3]{{dead link}} (late 2011) – in meantone

; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
* ''Meantone Suite V in D Minor'' (2017) – [https://www.youtube.com/watch?v=I68hwh45CyQ YouTube] | [http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-V-D-Minor.pdf score]

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=xN-QzI6OgGs ''Time Travel''] (2021)

; [[Xotla]]
* "Beebounce" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/4PzANNtxXsNEsdApnYKgHK Spotify] | [https://xotla.bandcamp.com/track/beebounce-43edo Bandcamp] | [https://youtu.be/EZIg5fojFfE YouTube] – jazzy big band electronic hybrid

==Instruments==
*[[Lumatone mapping for 43edo]]
*[[Skip fretting system 43 2 9]]

===Keyboards===
A possible isomorphic keyboard layout for 43edo:
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]]

==References==
<references />

== External links ==
=== Articles ===
* [http://tonalsoft.com/enc/m/meride.aspx méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-43Et-EMT-43EBMT.pdf ''Harmonic Resources of 43Et EMT and 43EBMT''] by Juhan Puhm (2018)

=== Diagrams ===
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-43Et.pdf ''Keys and Modes of 43Et''] by Juhan Puhm (2016)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-43Et.pdf ''Keyboard Mapping for 43Et''] by Juhan Puhm (2017)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-43Et.pdf ''Mapping Range for 43Et''] by Juhan Puhm (2017)

[[Category:Meantone]]
[[Category:Quartismic]]
[[Category:Historical]]
[[Category:Listen]]

19edo

2024-09-18T11:54:17Z

YoVariable: /* Standard notation */ Added sharpness step offsets

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{EDO intro|19}}
== Theory ==
=== History ===
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone|{{frac|1|3}}-comma meantone]], in which the fifth is 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

=== As an approximation of other temperaments ===
19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth, though [[46edo]] is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy, and is the fifth (after 12) [[zeta integral edo]]. It is less successful in the [[7-limit]] (but still better than 12et), as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles, and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5 and 7 are not only much farther from just than they are in 19, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to employ [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29 cents, and a step size of between 63.2 and 63.4 cents would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57 cents, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. A more extreme option would be [[11edf]], which has octaves stretched by 12.47 cents.

=== As a means of extending harmony ===
Because 19edo allows for more blended, consonant harmonies than 12edo does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 cents off [[23/16]].

In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Prime harmonics ===
{{Harmonics in equal|19}}

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
! [[Degree]]
! [[Cent]]s
! [[Interval region|Interval Region]]
! Approximated [[Just intonation|JI]] Intervals<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
| 0
| 0.00
| Unison (prime)
| [[1/1]]
| Do
| unison
| P1
|-
| 1
| 63.16
| Augmented unison
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| super unison, subminor second
| S1, sm2
|-
| 2
| 126.32
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| minor second
| m2
|-
| 3
| 189.47
| Major second
| [[9/8]], [[10/9]]
| Re
| major second
| M2
|-
| 4
| 252.63
| Augmented second Diminished third
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.79
| Minor third
| [[6/5]]
| Me
| minor third
| m3
|-
| 6
| 378.95
| Major third
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| major third
| M3
|-
| 7
| 442.11
| Augmented third
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.26
| Perfect fourth
| [[4/3]], [[75/56]]
| Fa
| perfect fourth
| P4
|-
| 9
| 568.42
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| augmented fourth
| A4
|-
| 10
| 631.58
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
| Se
| diminished fifth
| d5
|-
| 11
| 694.74
| Perfect fifth
| [[3/2]], [[112/75]]
| So
| perfect fifth
| P5
|-
| 12
| 757.89
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| super fifth, subminor sixth
| S5, sm6
|-
| 13
| 821.05
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
| Le
| minor sixth
| m6
|-
| 14
| 884.21
| Major sixth
| [[5/3]]
| La
| major sixth
| M6
|-
| 15
| 947.37
| Augmented sixth Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.53
| Minor seventh
| [[9/5]], [[16/9]]
| Te
| minor seventh
| m7
|-
| 17
| 1073.68
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| major seventh
| M7
|-
| 18
| 1136.84
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.00
| Octave
| [[2/1]]
| Do
| octave
| P8
|}

=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center"
! Quality
! [[Color name|Color Name]]
! Monzo Format
! Examples
|-
| diminished
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| rowspan="2" | minor
| fourthward wa
| (a, b), b < -1
| 32/27, 16/9
|-
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| rowspan="2" | major
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| fifthward wa
| (a, b), b > 1
| 9/8, 27/16
|-
| augmented
| ru
| (a, b, 0, -1)
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3), Cmin(♭3), C(d3)
| C subminor, C minor flat-three, C diminished-three
|-
| gu
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm, Cmin
| C minor
|-
| yo
| 4:5:6
| 0–6–11
| C–E–G
| C, Cmaj
| C, C major
|-
| ru
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3), Cmaj(♯3), C(A3)
| C supermajor, C major sharp-three, C augmented-three
|-
|
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| C(h7), Cadd(d7), Cmaj(add(d7))
| C harmonic 7, C (major) add dim-seven
|-
|
| 1/(4:5:6:7) = 1:6/5:3/2:12/7
| 0–5–11–15
| C–E♭–G–A♯
| Cm(♯6), Cm(A6), Cm(add(♯6)), Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios.

For a more complete list, see [[19edo Chord Names]] and [[Ups and downs notation #Chords and Chord Progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfege, or sargam. Note that D# and Eb are two different notes.

{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard Notation]]
|-
! [[5L 2s|Diatonic Interval Names]]
! Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.16
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.32
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.47
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.63
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.79
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.95
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.11
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.26
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.42
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.58
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.74
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.89
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.05
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.21
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.37
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.53
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.68
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.84
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.00
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

{{Sharpness-sharp1}}

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic Notation of 19edo
! [[Degree]]
! [[Cent]]s
! Interval Names
|-
| 0
| 0.00
| P1
|-
| 1
| 63.16
| A1, m2
|-
| 2
| 126.32
| M2, m3
|-
| 3
| 189.47
| M3
|-
| 4
| 252.63
| m4, A3
|-
| 5
| 315.79
| M4, m5
|-
| 6
| 378.95
| M5
|-
| 7
| 442.11
| A5, d6
|-
| 8
| 505.26
| P6
|-
| 9
| 568.42
| A6, m7
|-
| 10
| 631.58
| M7, d8
|-
| 11
| 694.74
| P8
|-
| 12
| 757.89
| A8, m9
|-
| 13
| 821.05
| M9, m10
|-
| 14
| 884.21
| M10
|-
| 15
| 947.37
| m11, A10
|-
| 16
| 1010.53
| M11, m12
|-
| 17
| 1073.68
| M12
|-
| 18
| 1136.84
| A12, d13
|-
| 19
| 1200.00
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| [{{val| 19 30 }}]
| +2.28
| 2.28
| 3.61
|-
| 2.3.5
| 81/80, 3125/3072
| [{{val| 19 30 44 }}]
| +2.58
| 1.91
| 3.02
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| [{{val| 19 30 44 53 }}]
| +3.85
| 2.76
| 4.35
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| [{{val| 19 30 44 53 70 }}]
| +4.14
| 2.53
| 3.99
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| [{{val| 19 30 44 53 70 86 }}]
| +3.32
| 2.93
| 4.64
|}

19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – ''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.

19et is prominent in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|13|18.5|19.5}}

=== Commas ===
19et [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime Limit]]
! [[Ratio]]<ref group="note">{{rd|10}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references/>

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic-kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! MOSes
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[rhinocerus]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] / [[pycnic]] [[Triton]]
|}

== Scales ==
=== MOS scales ===
==== Octave-equivalent mosses ====
* [[meantone]] pentatonic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2
* Meantone harmonic major: 3 3 2 3 2 4 2
* chromatic octave species - Meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
*[[Pinetone#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
*[[Pinetone#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
*[[Pinetone#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Arto and Tendo Theory]]
* [[Lumatone mapping for 19edo]]

=== Notes ===
<references group="note" />

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

[[Category:19-tone scales]]
[[Category:Godzilla]]
[[Category:Golden meantone]]
[[Category:Kleismic]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Negri]]
[[Category:Semaphore]]
[[Category:Sensi]]
[[Category:Teentuning]]
[[Category:Historical]]

36/25

2024-09-05T16:25:35Z

YoVariable: Fixed typo “diminished”

{{Infobox Interval
| Name = classic(al) diminished fifth, diptolemaic diminished fifth
| Color name = gg5, gugu 5th
| Sound = jid_36_25_pluck_adu_dr220.mp3
}}

'''36/25''', the '''classic(al) diminished fifth''' or '''diptolemaic diminished fifth''', is the [[5-limit]] tritone produced by stacking two [[6/5]] minor thirds. This means it is justly tuned in [[1/3-comma meantone]], and almost perfectly in [[19edo]]. It is [[325/324]] (5.3 cents) flat of [[13/9]], and the interval between [[10/9]] and [[8/5]] or [[5/4]] and [[9/5]] respectively.

== See also ==
* [[25/18]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Meantone]]
[[Category:todo:expand]]

201edo

2024-08-28T08:09:04Z

YoVariable: Changed any "210" to "201"

{{Infobox ET}}
{{EDO intro|201}}

201edo is only [[consistent]] to the [[5-odd-limit]], and [[harmonic]] [[3/1|3]] is about halfway between its steps.

Using the [[patent val]], it [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 25 -26 7 }} in the 5-limit; [[245/243]], [[50421/50000]], and [[2100875/2097152]] in the 7-limit; [[385/384]], [[896/891]], 1331/1323, and 47432/46875 in the 11-limit; [[196/195]], [[325/324]], [[2080/2079]], [[2200/2197]], and 3146/3125 in the 13-limit.

Using the 201e val, it tempers out [[441/440]], [[2200/2187]], [[3388/3375]], and [[65536/65219]] in the 11-limit; 196/195, 325/324, [[352/351]], [[1001/1000]], and 106496/105875 in the 13-limit.

Using the 201de val, it tempers out [[4000/3969]], [[10976/10935]], and 4194304/4134375 in the 7-limit; [[540/539]], 896/891, 1375/1372, and 234375/234256 in the 11-limit; 325/324, 352/351, [[364/363]], [[640/637]], and [[4394/4375]] in the 13-limit (supporting the [[pluto]] temperament).

Using the 201b val, it tempers out 1990656/1953125 (valentine comma) and {{monzo| -31 24 -3 }} in the 5-limit; [[126/125]], [[1029/1024]], and {{monzo| -2 19 0 -10 }} in the 7-limit; 540/539, 1944/1925, 2835/2816, and 483153/480200 in the 11-limit; [[1287/1280]], [[1575/1573]], [[1716/1715]], 2200/2197, and 3146/3125 in the 13-limit.

Using the 201bcf val, it tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| -56 31 3 }} in the 5-limit; 1029/1024, [[250047/250000]], and 273375/268912 in the 7-limit; 385/384, 441/440, [[4000/3993]], and 295245/290521 in the 11-limit; [[351/350]], 975/968, 1287/1280, [[1573/1568]], and 10935/10816 in the 13-limit.

=== Odd harmonics ===
{{Harmonics in equal|201}}

=== Subsets and supersets ===
Since 201 factors into {{factorization|201}}, 201edo contains [[3edo]] and [[67edo]] as its subsets. [[402edo]], which doubles it, provides a good correction to the approximation of harmonic 3.

19edo modes

2024-08-26T00:53:39Z

YoVariable: /* Miscellaneous scales */ Spelling correction

__FORCETOC__
Some scales and modes available in [[19edo|19edo]]. Please add more, discovered or newly-composed!

=Note names in 19edo=
The charts below primarily use the following note names (based on standard meantone/Pythagorean notation):

{| class="wikitable"
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 1
| style="text-align:center;" | 2
| style="text-align:center;" | 3
| style="text-align:center;" | 4
| style="text-align:center;" | 5
| style="text-align:center;" | 6
| style="text-align:center;" | 7
| style="text-align:center;" | 8
| style="text-align:center;" | 9
| style="text-align:center;" | 10
| style="text-align:center;" | 11
| style="text-align:center;" | 12
| style="text-align:center;" | 13
| style="text-align:center;" | 14
| style="text-align:center;" | 15
| style="text-align:center;" | 16
| style="text-align:center;" | 17
| style="text-align:center;" | 18
|-
| style="text-align:center;" | C
| style="text-align:center;" | Ç

C# (Δb)
| style="text-align:center;" | ΔÇ# (Db)
| style="text-align:center;" | D
| style="text-align:center;" | Þ

D# (εb, Sb)
| style="text-align:center;" | ε, S

Þ# (Eb)
| style="text-align:center;" | E
| style="text-align:center;" | E# (Fb)
| style="text-align:center;" | F
| style="text-align:center;" | ϕ

F# (Γb)
| style="text-align:center;" | Γϕ# (Gb)
| style="text-align:center;" | G
| style="text-align:center;" | Z

G# (αb)
| style="text-align:center;" | αZ# (Ab)
| style="text-align:center;" | A
| style="text-align:center;" | Æ

A# (βb, Bb)
| style="text-align:center;" | Æ#β ,B(b)
| style="text-align:center;" | B, H
| style="text-align:center;" | B#, H# (Cb)
|}
Note that E#=Fb and B#=Cb, but other notes are not enharmonic (for example, C# and Db are different; and E# is not the same as F). Double flats and sharps may be used as in "standard" notation; for example; Fx=F##=Gb and Bbb=A#.

Other systems of notation are of course possible, and may be preferred depending on the scales you are working with!

==[[Meantone|Meantone]]==

'''meantone[5]''' ([[2L_3s|2L+3s]] / [[Modal_UDP_Notation|chroma-positive generator]] = 8\19 ~= 4/3)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | comments
!audio
|-
| | 4|0
| | 5 3 5 3 3
| | 0 5 8 13 16
| | C Eb F Ab Bb
| | minor pentatonic (2)
|[[File:Minor 2 Pentatonic (19-EDO).mp3|frameless]]
|-
| | 3|1
| | 5 3 3 5 3
| | 0 5 8 11 16
| | C Eb F G Bb
| | minor pentatonic
|[[File:Minor Pentatonic (19-EDO).mp3|frameless]]
|-
| | 2|2
| | 3 5 3 5 3
| | 0 3 8 11 16
| | C D F G Bb
| | "mixolydian" pentatonic
|[[File:Mixolydian Pentatonic (19-EDO).mp3|frameless]]
|-
| | 1|3
| | 3 5 3 3 5
| | 0 3 8 11 14
| | C D F G A
| | "suspended" pentatonic
|[[File:Suspended Pentatonic (19-EDO).mp3|frameless]]
|-
| | 0|4
| | 3 3 5 3 5
| | 0 3 6 11 14
| | C D E G A
| | major pentatonic
|[[File:Major Pentatonic (19-EDO).mp3|frameless]]
|}

'''meantone[7]''' ([[5L_2s|5L+2s]] / CPG = 11\19 ~= 3/2)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | mode name
!audio
|-
| | 6|0
| | 3 3 3 2 3 3 2
| | 0 3 6 9 11 14 17
| | C D E F# G A B
| | Lydian
|[[File:Lydian (19-EDO).mp3|frameless]]
|-
| | 5|1
| | 3 3 2 3 3 3 2
| | 0 3 6 8 11 14 17
| | C D E F G A B
| | Ionian
|[[File:Ionian (19-EDO).mp3|frameless]]
|-
| | 4|2
| | 3 3 2 3 3 2 3
| | 0 3 6 8 11 14 16
| | C D E F G A Bb
| | Mixolydian
|[[File:Mixolydian (19-EDO).mp3|frameless]]
|-
| | 3|3
| | 3 2 3 3 3 2 3
| | 0 3 5 8 11 14 16
| | C D Eb F G A Bb
| | Dorian
|[[File:Dorian (19-EDO).mp3|frameless]]
|-
| | 2|4
| | 3 2 3 3 2 3 3
| | 0 3 5 8 11 13 16
| | C D Eb F G Ab Bb
| | Aeolian
|[[File:Aeolian (19-EDO).mp3|frameless]]
|-
| | 1|5
| | 2 3 3 3 2 3 3
| | 0 2 5 8 11 13 16
| | C Db Eb F G Ab Bb
| | Phrygian
|[[File:Phrygian (19-EDO).mp3|frameless]]
|-
| | 0|6
| | 2 3 3 2 3 3 3
| | 0 2 5 8 10 13 16
| | C Db Eb F Gb Ab Bb
| | Locrian
|[[File:Locrian (19-EDO).mp3|frameless]]
|}

'''meantone[12]''' ([[7L_5s|7L+5s]] / CPG = 8\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | comments
|-
| | 11|0
| | 2 2 1 2 1 2 2 1 2 1 2 1
| | 0 2 4 5 7 8 10 12 13 15 16 18
| | C Db D# Eb E# F Gb G# Ab A# Bb B#

C Δ Þ S E# F Γ Z α Æ β B#
| |
|-
| | 10|1
| | 2 2 1 2 1 2 1 2 2 1 2 1
| | 0 2 4 5 7 8 10 11 13 15 16 18
| | C Db D# Eb E# F Gb G Ab A# Bb B#

C Δ Þ S E# F Γ G α Æ β B#
| |
|-
| | 9|2
| | 2 1 2 2 1 2 1 2 2 1 2 1
| | 0 2 3 5 7 8 10 11 13 15 16 18
| | C Db D Eb E# F Gb G Ab A# Bb B#

C Δ D S E# F Γ G α Æ β B#
| |
|-
| | 8|3
| | 2 1 2 2 1 2 1 2 1 2 2 1
| | 0 2 3 5 7 8 10 11 13 14 16 18
| | C Db D Eb E# F Gb G Ab A Bb B#

C Δ D S E# F Γ G α A β B#
| |
|-
| | 7|4
| | 2 1 2 1 2 2 1 2 1 2 2 1
| | 0 2 3 5 6 8 10 11 13 14 16 18
| | C Db D Eb E F Gb G Ab A Bb B#

C Δ D S E F Γ G α A β B#
| |
|-
| | 6|5
| | 2 1 2 1 2 2 1 2 1 2 1 2
| | 0 2 3 5 6 8 10 11 13 14 16 17
| | C Db D Eb E F Gb G Ab A Bb B

C Δ D S E F Γ G α A β B
| | Yasser's Supradiatonic

(twelfths of 12-5 chords form C Ionian mode, elevenths of 11-4 chords form C Aeolian mode)
|-
| | 5|6
| | 2 1 2 1 2 1 2 2 1 2 1 2
| | 0 2 3 5 6 8 9 11 13 14 16 17
| | C Db D Eb E F F# G Ab A Bb B

C Δ D S E F ϕ G α A β B
| | fifths of 12-5 chords form C Ionian mode
|-
| | 4|7
| | 1 2 2 1 2 1 2 2 1 2 1 2
| | 0 1 3 5 6 8 9 11 13 14 16 17
| | C C# D Eb E F F# G Ab A Bb B

C Ç D S E F ϕ G α A β B
| | octaves of 11-5 chords form C Mixolydian mode
|-
| | 3|8
| | 1 2 2 1 2 1 2 1 2 2 1 2
| | 0 1 3 5 6 8 9 11 12 14 16 17
| | C C# D Eb E F F# G G# A Bb B

C Ç D S E F ϕ G Z A β B
| |
|-
| | 2|9
| | 1 2 1 2 2 1 2 1 2 2 1 2
| | 0 1 3 4 6 8 9 11 12 14 16 17
| | C C# D D# E F F# G G# A Bb B

C Ç D Þ E F ϕ G Z A β B
| | octaves of 12-5 chords form C Ionian mode
|-
| | 1|10
| | 1 2 1 2 2 1 2 1 2 1 2 2
| | 0 1 3 4 6 8 9 11 12 14 15 17
| | C C# D D# E F F# G G# A A# B

C Ç D Þ E F ϕ G Z A Æ B
| | roots of 12-5 chords form C Ionian mode
|-
| | 0|11
| | 1 2 1 2 1 2 2 1 2 1 2 2
| | 0 1 3 4 6 7 9 11 12 14 15 17
| | C C# D D# E E# F# G G# A A# B

C Ç D Þ E Fb ϕ G Z A Æ B
| |
|}

=Miscellaneous scales=

2 3 1 4 1 4 4 - E# F# G# Ab B Cb D - Cartunes

1 1 6 3 1 1 6 - E# F Gbb A# B# C Dbb E# - enharmonic approximation

5 3 1 2 4 1 3 - C Eb F F# G A# B - Hyperblue Dorian (approximated from [[31edo]])

=MOS scales and modes of rank-2 temperaments=

==[[Negri|Negri]]==

'''negri[9]''' ([[1L_8s|1L+8s]] / [[Modal_UDP_Notation|chroma-positive generator]] = 17\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | comments
|-
| | 8|0
| | 3 2 2 2 2 2 2 2 2
| | 0 3 5 7 9 11 13 15 17
| | C D Eb E# F# G Ab A# B
| |
|-
| | 7|1
| | 2 3 2 2 2 2 2 2 2
| | 0 2 5 7 9 11 13 15 17
| | C Db Eb E# F# G Ab A# B
| |
|-
| | 6|2
| | 2 2 3 2 2 2 2 2 2
| | 0 2 4 7 9 11 13 15 17
| | C Db D# E# F# G Ab A# B
| |
|-
| | 5|3
| | 2 2 2 3 2 2 2 2 2
| | 0 2 4 6 9 11 13 15 17
| | C Db D# E F# G Ab A# B
| |
|-
| | 4|4
| | 2 2 2 2 3 2 2 2 2
| | 0 2 4 6 8 11 13 15 17
| | C Db D# E F G Ab A# B
| | symmetrical, has both 4/3 and 3/2
|-
| | 3|5
| | 2 2 2 2 2 3 2 2 2
| | 0 2 4 6 8 10 13 15 17
| | C Db D# E F Gb Ab A# B
| |
|-
| | 2|6
| | 2 2 2 2 2 2 3 2 2
| | 0 2 4 6 8 10 12 15 17
| | C Db D# E F Gb G# A# B
| |
|-
| | 1|7
| | 2 2 2 2 2 2 2 3 2
| | 0 2 4 6 8 10 12 14 17
| | C Db D# E F Gb G# A B
| |
|-
| | 0|8
| | 2 2 2 2 2 2 2 2 3
| | 0 2 4 6 8 10 12 14 16
| | C Db D# E F Gb G# A Bb
| |
|}

'''negri[10]''' ([[9L_1s|9L+1s]] / CPG = 2\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 9|0
| | 2 2 2 2 2 2 2 2 2 1
| | 0 2 4 6 8 10 12 14 16 18
| | C Db D# E F Gb G# A Bb B#
|-
| | 8|1
| | 2 2 2 2 2 2 2 2 1 2
| | 0 2 4 6 8 10 12 14 16 17
| | C Db D# E F Gb G# A Bb B
|-
| | 7|2
| | 2 2 2 2 2 2 2 1 2 2
| | 0 2 4 6 8 10 12 14 15 17
| | C Db D# E F Gb G# A A# B
|-
| | 6|3
| | 2 2 2 2 2 2 1 2 2 2
| | 0 2 4 6 8 10 12 13 15 17
| | C Db D# E F Gb G# Ab A# B
|-
| | 5|4
| | 2 2 2 2 2 1 2 2 2 2
| | 0 2 4 6 8 10 11 13 15 17
| | C Db D# E F Gb G Ab A# B
|-
| | 4|5
| | 2 2 2 2 1 2 2 2 2 2
| | 0 2 4 6 8 9 11 13 15 17
| | C Db D# E F F# G Ab A# B
|-
| | 3|6
| | 2 2 2 1 2 2 2 2 2 2
| | 0 2 4 6 7 9 11 13 15 17
| | C Db D# E E# F# G Ab A# B
|-
| | 2|7
| | 2 2 1 2 2 2 2 2 2 2
| | 0 2 4 5 7 9 11 13 15 17
| | C Db D# Eb E# F# G Ab A# B
|-
| | 1|8
| | 2 1 2 2 2 2 2 2 2 2
| | 0 2 3 5 7 9 11 13 15 17
| | C Db D Eb E# F# G Ab A# B
|-
| | 0|9
| | 1 2 2 2 2 2 2 2 2 2
| | 0 1 3 5 7 9 11 13 15 17
| | C C# D Eb E# F# G Ab A# B
|}

==[[deutone|Deutone]]==

'''deutone[6]''' ([[1L_5s|1L+5s ]]/ [[Modal_UDP_Notation|chroma-positive generator]] = 16\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 5|0
| | 4 3 3 3 3 3
| | 0 4 7 10 13 16
| | C D# E# Gb Ab Bb
|-
| | 4|1
| | 3 4 3 3 3 3
| | 0 3 7 10 13 16
| | C D E# Gb Ab Bb
|-
| | 3|2
| | 3 3 4 3 3 3
| | 0 3 6 10 13 16
| | C D E Gb Ab Bb
|-
| | 2|3
| | 3 3 3 4 3 3
| | 0 3 6 9 13 16
| | C D E F# Ab Bb
|-
| | 1|4
| | 3 3 3 3 4 3
| | 0 3 6 9 12 16
| | C D E F# G# Bb
|-
| | 0|5
| | 3 3 3 3 3 4
| | 0 3 6 9 12 15
| | C D E F# G# A#
|}

'''deutone[7]''' ([[6L_1s|6L+1s ]]/ CPG = 3\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 6|0
| | 3 3 3 3 3 3 1
| | 0 3 6 9 12 15 18
| | C D E F# G# A# B#
|-
| | 5|1
| | 3 3 3 3 3 1 3
| | 0 3 6 9 12 15 16
| | C D E F# G# A# Bb
|-
| | 4|2
| | 3 3 3 3 1 3 3
| | 0 3 6 9 12 13 16
| | C D E F# G# Ab Bb
|-
| | 3|3
| | 3 3 3 1 3 3 3
| | 0 3 6 9 10 13 16
| | C D E F# Gb Ab Bb
|-
| | 2|4
| | 3 3 1 3 3 3 3
| | 0 3 6 7 10 13 16
| | C D E E# Gb Ab Bb
|-
| | 1|5
| | 3 1 3 3 3 3 3
| | 0 3 4 7 10 13 16
| | C D D# E# Gb Ab Bb
|-
| | 0|6
| | 1 3 3 3 3 3 3
| | 0 1 4 7 10 13 16
| | C C# D# E# Gb Ab Bb
|}

'''deutone[13]''' ([[6L_7s|6L+7s ]]/ CPG = 3\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 12|0
| | 2 1 2 1 2 1 2 1 2 1 2 1 1
| | 0 2 3 5 6 8 9 11 12 14 15 17 18
| | C Db D Eb E F F# G G# A A# B B#
|-
| | 11|1
| | 2 1 2 1 2 1 2 1 2 1 1 2 1
| | 0 2 3 5 6 8 9 11 12 14 15 16 18
| | C Db D Eb E F F# G G# A A# Bb B#
|-
| | 10|2
| | 2 1 2 1 2 1 2 1 1 2 1 2 1
| | 0 2 3 5 6 8 9 11 12 13 15 16 18
| | C Db D Eb E F F# G G# Ab A# Bb B#
|-
| | 9|3
| | 2 1 2 1 2 1 1 2 1 2 1 2 1
| | 0 2 3 5 6 8 9 10 12 13 15 16 18
| | C Db D Eb E F F# Gb G# Ab A# Bb B#
|-
| | 8|4
| | 2 1 2 1 1 2 1 2 1 2 1 2 1
| | 0 2 3 5 6 7 9 10 12 13 15 16 18
| | C Db D Eb E E# F# Gb G# Ab A# Bb B#
|-
| | 7|5
| | 2 1 1 2 1 2 1 2 1 2 1 2 1
| | 0 2 3 4 6 7 9 10 12 13 15 16 18
| | C Db D D# E E# F# Gb G# Ab A# Bb B#
|-
| | 6|6
| | 1 2 1 2 1 2 1 2 1 2 1 2 1
| | 0 1 3 4 6 7 9 10 12 13 15 16 18
| | C C# D D# E E# F# Gb G# Ab A# Bb B#
|-
| | 5|7
| | 1 2 1 2 1 2 1 2 1 2 1 1 2
| | 0 1 3 4 6 7 9 10 12 13 15 16 17
| | C C# D D# E E# F# Gb G# Ab A# Bb B
|-
| | 4|8
| | 1 2 1 2 1 2 1 2 1 1 2 1 2
| | 0 1 3 4 6 7 9 10 12 13 14 16 17
| | C C# D D# E E# F# Gb G# Ab A Bb B
|-
| | 3|9
| | 1 2 1 2 1 2 1 1 2 1 2 1 2
| | 0 1 3 4 6 7 9 10 11 13 14 16 17
| | C C# D D# E E# F# Gb G Ab A Bb B
|-
| | 2|10
| | 1 2 1 2 1 1 2 1 2 1 2 1 2
| | 0 1 3 4 6 7 8 10 11 13 14 16 17
| | C C# D D# E E# F Gb G Ab A Bb B
|-
| | 1|11
| | 1 2 1 1 2 1 2 1 2 1 2 1 2
| | 0 1 3 4 5 7 8 10 11 13 14 16 17
| | C C# D D# Eb E# F Gb G Ab A Bb B
|-
| | 0|12
| | 1 1 2 1 2 1 2 1 2 1 2 1 2
| | 0 1 2 4 5 7 8 10 11 13 14 16 17
| | C C# Db D# Eb E# F Gb G Ab A Bb B
|}

==[[Semaphore_and_Godzilla|Godzilla / Semaphore]]==

'''godzilla[5]''' ([[4L_1s|4L+1s ]]/ [[Modal_UDP_Notation|chroma-positive generator]] = 4\19 ~= 8/7 ~= 7/6)

{| class="wikitable"
|-
! | UDP
! | mode steps
! | mode degrees
! | note names
|-
| | 4|0
| | 4 4 4 4 3
| | 0 4 8 12 16
| | C D# F G# Bb
|-
| | 3|1
| | 4 4 4 3 4
| | 0 4 8 12 15
| | C D# F G# A#
|-
| | 2|2
| | 4 4 3 4 4
| | 0 4 8 11 15
| | C D# F G A#
|-
| | 1|3
| | 4 3 4 4 4
| | 0 4 7 11 15
| | C D# E# G A#
|-
| | 0|4
| | 3 4 4 4 4
| | 0 3 7 11 15
| | C D E# G A#
|}

'''godzilla[9]''' ([[5L_4s|5L+4s ]]/ CPG = 15\19 ~= 7/4 ~= 12/7)

{| class="wikitable"
|-
! | UDP
! | mode steps
! | mode degrees
! | note names
|-
| | 8|0
| | 3 3 1 3 1 3 1 3 1
| | 0 3 6 7 10 11 14 15 18
| | C D E E# Gb G A A# B#
|-
| | 7|1
| | 3 1 3 3 1 3 1 3 1
| | 0 3 4 7 10 11 14 15 18
| | C D D# E# Gb G A A# B#
|-
| | 6|2
| | 3 1 3 1 3 3 1 3 1
| | 0 3 4 7 8 11 14 15 18
| | C D D# E# F G A A# B#
|-
| | 5|3
| | 3 1 3 1 3 1 3 3 1
| | 0 3 4 7 8 11 12 15 18
| | C D D# E# F G G# A# B#
|-
| | 4|4
| | 3 1 3 1 3 1 3 1 3
| | 0 3 4 7 8 11 12 15 16
| | C D D# E# F G G# A# Bb
|-
| | 3|5
| | 1 3 3 1 3 1 3 1 3
| | 0 1 4 7 8 11 12 15 16
| | C C# D# E# F G G# A# Bb
|-
| | 2|6
| | 1 3 1 3 3 1 3 1 3
| | 0 1 4 5 8 11 12 15 16
| | C C# D# Eb F G G# A# Bb
|-
| | 1|7
| | 1 3 1 3 1 3 3 1 3
| | 0 1 4 5 8 9 12 15 16
| | C C# D# Eb F F# G# A# Bb
|-
| | 0|8
| | 1 3 1 3 1 3 1 3 3
| | 0 1 4 5 8 9 12 13 16
| | C C# D# Eb F F# G# Ab Bb
|}

'''godzilla[14]''' ([[5L_9s|5L+9s ]]/ CPG = 4\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 13|0
| | 2 1 2 1 1 2 1 1 2 1 1 2 1 1
| | 0 2 3 5 6 7 9 10 11 13 14 15 17 18
| | C Db D Eb E E# F# Gb G Ab A A# B B#
|-
| | 12|1
| | 2 1 1 2 1 2 1 1 2 1 1 2 1 1
| | 0 2 3 4 6 7 9 10 11 13 14 15 17 18
| | C Db D D# E E# F# Gb G Ab A A# B B#
|-
| | 11|2
| | 2 1 1 2 1 1 2 1 2 1 1 2 1 1
| | 0 2 3 4 6 7 8 10 11 13 14 15 17 18
| | C Db D D# E E# F Gb G Ab A A# B B#
|-
| | 10|3
| | 2 1 1 2 1 1 2 1 1 2 1 2 1 1
| | 0 2 3 4 6 7 8 10 11 12 14 15 17 18
| | C Db D D# E E# F Gb G G# A A# B B#
|-
| | 9|4
| | 2 1 1 2 1 1 2 1 1 2 1 1 2 1
| | 0 2 3 4 6 7 8 10 11 12 14 15 16 18
| | C Db D D# E E# F Gb G G# A A# Bb B#
|-
| | 8|5
| | 1 2 1 2 1 1 2 1 1 2 1 1 2 1
| | 0 1 3 4 6 7 8 10 11 12 14 15 16 18
| | C C# D D# E E# F Gb G G# A A# Bb B#
|-
| | 7|6
| | 1 2 1 1 2 1 2 1 1 2 1 1 2 1
| | 0 1 3 4 5 7 8 10 11 12 14 15 16 18
| | C C# D D# Eb E# F Gb G G# A A# Bb B#
|-
| | 6|7
| | 1 2 1 1 2 1 1 2 1 2 1 1 2 1
| | 0 1 3 4 5 7 8 9 11 12 14 15 16 18
| | C C# D D# Eb E# F F# G G# A A# Bb B#
|-
| | 5|8
| | 1 2 1 1 2 1 1 2 1 1 2 1 2 1
| | 0 1 3 4 5 7 8 9 11 12 13 15 16 18
| | C C# D D# Eb E# F F# G G# Ab A# Bb B#
|-
| | 4|9
| | 1 2 1 1 2 1 1 2 1 1 2 1 1 2
| | 0 1 3 4 5 7 8 9 11 12 13 15 16 17
| | C C# D D# Eb E# F F# G G# Ab A# Bb B
|-
| | 3|10
| | 1 1 2 1 2 1 1 2 1 1 2 1 1 2
| | 0 1 2 4 5 7 8 9 11 12 13 15 16 17
| | C C# Db D# Eb E# F F# G G# Ab A# Bb B
|-
| | 2|11
| | 1 1 2 1 1 2 1 2 1 1 2 1 1 2
| | 0 1 2 4 5 6 8 9 11 12 13 15 16 17
| | C C# Db D# Eb E F F# G G# Ab A# Bb B
|-
| | 1|12
| | 1 1 2 1 1 2 1 1 2 1 2 1 1 2
| | 0 1 2 4 5 6 8 9 10 12 13 15 16 17
| | C C# Db D# Eb E F F# Gb G# Ab A# Bb B
|-
| | 0|13
| | 1 1 2 1 1 2 1 1 2 1 1 2 1 2
| | 0 1 2 4 5 6 8 9 10 12 13 14 16 17
| | C C# Db D# Eb E F F# Gb G# Ab A Bb B
|}

==[[Kleismic_family|Kleismic / Hanson]]==

'''kleismic[7]''' ([[4L_3s|4L+3s ]]/ [[Modal_UDP_Notation|chroma-positive generator]] = 14\19 ~= 5/3)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | comments
|-
| | 6|0
| | 4 4 1 4 1 4 1
| | 0 4 8 9 13 14 18
| | C D# F F# Ab A B#
| | has 4/3
|-
| | 5|1
| | 4 1 4 4 1 4 1
| | 0 4 5 9 13 14 18
| | C D# Eb F# Ab A B#
| |
|-
| | 4|2
| | 4 1 4 1 4 4 1
| | 0 4 5 9 10 14 18
| | C D# Eb F# Gb A B#
| |
|-
| | 3|3
| | 4 1 4 1 4 1 4
| | 0 4 5 9 10 14 15
| | C D# Eb F# Gb A A#
| | symmetrical
|-
| | 2|4
| | 1 4 4 1 4 1 4
| | 0 1 5 9 10 14 15
| | C C# Eb F# Gb A A#
| |
|-
| | 1|5
| | 1 4 1 4 4 1 4
| | 0 1 5 6 10 14 15
| | C C# Eb E Gb A A#
| |
|-
| | 0|6
| | 1 4 1 4 1 4 4
| | 0 1 5 6 10 11 15
| | C C# Eb E Gb G A#
| | has 3/2
|}

'''kleismic[11]''' ([[4L_7s|4L+7s ]]/ CPG = 14\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
! | comments
|-
| | 10|0
| | 3 1 3 1 1 3 1 1 3 1 1
| | 0 3 4 7 8 9 12 13 14 17 18
| | C D D# E# F F# G# Ab A B B#
| |
|-
| | 9|1
| | 3 1 1 3 1 3 1 1 3 1 1
| | 0 3 4 5 8 9 12 13 14 17 18
| | C D D# Eb F F# G# Ab A B B#
| |
|-
| | 8|2
| | 3 1 1 3 1 1 3 1 3 1 1
| | 0 3 4 5 8 9 10 13 14 17 18
| | C D D# Eb F F# Gb Ab A B B#
| |
|-
| | 7|3
| | 3 1 1 3 1 1 3 1 1 3 1
| | 0 3 4 5 8 9 10 13 14 15 18
| | C D D# Eb F F# Gb Ab A A# B#
| |
|-
| | 6|4
| | 1 3 1 3 1 1 3 1 1 3 1
| | 0 1 4 5 8 9 10 13 14 15 18
| | C C# D# Eb F F# Gb Ab A A# B#
| |
|-
| | 5|5
| | 1 3 1 1 3 1 3 1 1 3 1
| | 0 1 4 5 6 9 10 13 14 15 18
| | C C# D# Eb E F# Gb Ab A A# B#
| | symmetrical, has neither 4/3 nor 3/2
|-
| | 4|6
| | 1 3 1 1 3 1 1 3 1 3 1
| | 0 1 4 5 6 9 10 11 14 15 18
| | C C# D# Eb E F# Gb G A A# B#
| |
|-
| | 3|7
| | 1 3 1 1 3 1 1 3 1 1 3
| | 0 1 4 5 6 9 10 11 14 15 16
| | C C# D# Eb E F# Gb G A A# Bb
| |
|-
| | 2|8
| | 1 1 3 1 3 1 1 3 1 1 3
| | 0 1 2 5 6 9 10 11 14 15 16
| | C C# Db Eb E F# Gb G A A# Bb
| |
|-
| | 1|9
| | 1 1 3 1 1 3 1 3 1 1 3
| | 0 1 2 5 6 7 10 11 14 15 16
| | C C# Db Eb E E# Gb G A A# Bb
| |
|-
| | 0|10
| | 1 1 3 1 1 3 1 1 3 1 3
| | 0 1 2 5 6 7 10 11 12 15 16
| | C C# Db Eb E E# Gb G G# A# Bb
| |
|}

'''kleismic[15]''' ([[4L_3s|4L+11s ]]/ CPG = 14\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 14|0
| | 2 1 1 2 1 1 1 2 1 1 1 2 1 1 1
| | 0 2 3 4 6 7 8 9 11 12 13 14 16 17 18
| | C Db D D# E E# F F# G G# Ab A Bb B B#
|-
| | 13|1
| | 2 1 1 1 2 1 1 2 1 1 1 2 1 1 1
| | 0 2 3 4 5 7 8 9 11 12 13 14 16 17 18
| | C Db D D# Eb E# F F# G G# Ab A Bb B B#
|-
| | 12|2
| | 2 1 1 1 2 1 1 1 2 1 1 2 1 1 1
| | 0 2 3 4 5 7 8 9 10 12 13 14 16 17 18
| | C Db D D# Eb E# F F# Gb G# Ab A Bb B B#
|-
| | 11|3
| | 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1
| | 0 2 3 4 5 7 8 9 10 12 13 14 15 17 18
| | C Db D D# Eb E# F F# Gb G# Ab A A# B B#
|-
| | 10|4
| | 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1
| | 0 1 3 4 5 7 8 9 10 12 13 14 15 17 18
| | C C# D D# Eb E# F F# Gb G# Ab A A# B B#
|-
| | 9|5
| | 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1
| | 0 1 3 4 5 6 8 9 10 12 13 14 15 17 18
| | C C# D D# Eb E F F# Gb G# Ab A A# B B#
|-
| | 8|6
| | 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
| | 0 1 3 4 5 6 8 9 10 11 13 14 15 17 18
| | C C# D D# Eb E F F# Gb G Ab A A# B B#
|-
| | 7|7
| | 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1
| | 0 1 3 4 5 6 8 9 10 11 13 14 15 16 18
| | C C# D D# Eb E F F# Gb G Ab A A# Bb B#
|-
| | 6|8
| | 1 1 2 1 1 2 1 1 1 2 1 1 1 2 1
| | 0 1 2 4 5 6 8 9 10 11 13 14 15 16 18
| | C C# Db D# Eb E F F# Gb G Ab A A# Bb B#
|-
| | 5|9
| | 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1
| | 0 1 2 4 5 6 7 9 10 11 13 14 15 16 18
| | C C# Db D# Eb E E# F# Gb G Ab A A# Bb B#
|-
| | 4|10
| | 1 1 2 1 1 1 2 1 1 1 2 1 1 2 1
| | 0 1 2 4 5 6 7 9 10 11 12 14 15 16 18
| | C C# Db D# Eb E E# F# Gb G G# A A# Bb B#
|-
| | 3|11
| | 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2
| | 0 1 2 4 5 6 7 9 10 11 12 14 15 16 17
| | C C# Db D# Eb E E# F# Gb G G# A A# Bb B
|-
| | 2|12
| | 1 1 1 2 1 1 2 1 1 1 2 1 1 1 2
| | 0 1 2 3 5 6 7 9 10 11 12 14 15 16 17
| | C C# Db D Eb E E# F# Gb G G# A A# Bb B
|-
| | 1|13
| | 1 1 1 2 1 1 1 2 1 1 2 1 1 1 2
| | 0 1 2 3 5 6 7 8 10 11 12 14 15 16 17
| | C C# Db D Eb E E# F Gb G G# A A# Bb B
|-
| | 0|14
| | 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2
| | 0 1 2 3 5 6 7 8 10 11 12 13 15 16 17
| | C C# Db D Eb E E# F Gb G G# Ab A# Bb B
|}

==[[Magic|Magic]]==

'''magic[7]''' ([[3L_4s|3L+4s]] / [[Modal_UDP_Notation|chroma-positive generator]] = 6\19 ~=5/4)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 6|0
| | 5 1 5 1 5 1 1
| | 0 5 6 11 12 17 18
| | C Eb E G G# B B#
|-
| | 5|1
| | 5 1 5 1 1 5 1
| | 0 5 6 11 12 13 18
| | C Eb E G G# Ab B#
|-
| | 4|2
| | 5 1 1 5 1 5 1
| | 0 5 6 7 12 13 18
| | C Eb E E# G# Ab B#
|-
| | 3|3
| | 1 5 1 5 1 5 1
| | 0 1 6 7 12 13 18
| | C C# E E# G# Ab B#
|-
| | 2|4
| | 1 5 1 5 1 1 5
| | 0 1 6 7 12 13 14
| | C C# E E# G# Ab A
|-
| | 1|5
| | 1 5 1 1 5 1 5
| | 0 1 6 7 8 13 14
| | C C# E E# F Ab A
|-
| | 0|6
| | 1 1 5 1 5 1 5
| | 0 1 2 7 8 13 14
| | C C# Db E# F Ab A
|}

'''magic[10]''' ([[3L_7s|3L+7s]] / CPG = 6\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 9|0
| | 4 1 1 4 1 1 4 1 1 1
| | 0 4 5 6 10 11 12 16 17 18
| | C D# Eb E Gb G G# Bb B B#
|-
| | 8|1
| | 4 1 1 4 1 1 1 4 1 1
| | 0 4 5 6 10 11 12 13 17 18
| | C D# Eb E Gb G G# Ab B B#
|-
| | 7|2
| | 4 1 1 1 4 1 1 4 1 1
| | 0 4 5 6 7 11 12 13 17 18
| | C D# Eb E E# G G# Ab B B#
|-
| | 6|3
| | 1 4 1 1 4 1 1 4 1 1
| | 0 1 5 6 7 11 12 13 17 18
| | C C# Eb E E# G G# Ab B B#
|-
| | 5|4
| | 1 4 1 1 4 1 1 1 4 1
| | 0 1 5 6 7 11 12 13 14 18
| | C C# Eb E E# G G# Ab A B#
|-
| | 4|5
| | 1 4 1 1 1 4 1 1 4 1
| | 0 1 5 6 7 8 12 13 14 18
| | C C# Eb E E# F G# Ab A B#
|-
| | 3|6
| | 1 1 4 1 1 4 1 1 4 1
| | 0 1 2 6 7 8 12 13 14 18
| | C C# Db E E# F G# Ab A B#
|-
| | 2|7
| | 1 1 4 1 1 4 1 1 1 4
| | 0 1 2 6 7 8 12 13 14 15
| | C C# Db E E# F G# Ab A A#
|-
| | 1|8
| | 1 1 4 1 1 1 4 1 1 4
| | 0 1 2 6 7 8 9 13 14 15
| | C C# Db E E# F F# Ab A A#
|-
| | 0|9
| | 1 1 1 4 1 1 4 1 1 4
| | 0 1 2 3 7 8 9 13 14 15
| | C C# Db D E# F F# Ab A A#
|}

'''magic[13]''' ([[3L_10s|3L+10s]] / CPG = 6\19 ~=5/4)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 12|0
| | 3 1 1 1 3 1 1 1 3 1 1 1 1
| | 0 3 4 5 6 9 10 11 12 15 16 17 18
| | C D D# Eb E F# Gb G G# A# Bb B B#
|-
| | 11|1
| | 3 1 1 1 3 1 1 1 1 3 1 1 1
| | 0 3 4 5 6 9 10 11 12 13 16 17 18
| | C D D# Eb E F# Gb G G# Ab Bb B B#
|-
| | 10|2
| | 3 1 1 1 1 3 1 1 1 3 1 1 1
| | 0 3 4 5 6 7 10 11 12 13 16 17 18
| | C D D# Eb E E# Gb G G# Ab Bb B B#
|-
| | 9|3
| | 1 3 1 1 1 3 1 1 1 3 1 1 1
| | 0 1 4 5 6 7 10 11 12 13 16 17 18
| | C C# D# Eb E E# Gb G G# Ab Bb B B#
|-
| | 8|4
| | 1 3 1 1 1 3 1 1 1 1 3 1 1
| | 0 1 4 5 6 7 10 11 12 13 14 17 18
| | C C# D# Eb E E# Gb G G# Ab A B B#
|-
| | 7|5
| | 1 3 1 1 1 1 3 1 1 1 3 1 1
| | 0 1 4 5 6 7 8 11 12 13 14 17 18
| | C C# D# Eb E E# F G G# Ab A B B#
|-
| | 6|6
| | 1 1 3 1 1 1 3 1 1 1 3 1 1
| | 0 1 2 5 6 7 8 11 12 13 14 17 18
| | C C# Db Eb E E# F G G# Ab A B B#
|-
| | 5|7
| | 1 1 3 1 1 1 3 1 1 1 1 3 1
| | 0 1 2 5 6 7 8 11 12 13 14 15 18
| | C C# Db Eb E E# F G G# Ab A A# B#
|-
| | 4|8
| | 1 1 3 1 1 1 1 3 1 1 1 3 1
| | 0 1 2 5 6 7 8 9 12 13 14 15 18
| | C C# Db Eb E E# F F# G# Ab A A# B#
|-
| | 3|9
| | 1 1 1 3 1 1 1 3 1 1 1 3 1
| | 0 1 2 3 6 7 8 9 12 13 14 15 18
| | C C# Db D E E# F F# G# Ab A A# B#
|-
| | 2|10
| | 1 1 1 3 1 1 1 3 1 1 1 1 3
| | 0 1 2 3 6 7 8 9 12 13 14 15 16
| | C C# Db D E E# F F# G# Ab A A# Bb
|-
| | 1|11
| | 1 1 1 3 1 1 1 1 3 1 1 1 3
| | 0 1 2 3 6 7 8 9 10 13 14 15 16
| | C C# Db D E E# F F# Gb Ab A A# Bb
|-
| | 0|12
| | 1 1 1 1 3 1 1 1 3 1 1 1 3
| | 0 1 2 3 4 7 8 9 10 13 14 15 16
| | C C# Db D D# E# F F# Gb Ab A A# Bb
|}

'''magic[16]''' ([[3L_13s|3L+13s]] / CPG = 6\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 15|0
| | 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
| | 0 2 3 4 5 6 8 9 10 11 12 14 15 16 17 18
| | C Db D D# Eb E F F# Gb G G# A A# Bb B B#
|-
| | 14|1
| | 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1
| | 0 2 3 4 5 6 8 9 10 11 12 13 15 16 17 18
| | C Db D D# Eb E F F# Gb G G# Ab A# Bb B B#
|-
| | 13|2
| | 2 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1
| | 0 2 3 4 5 6 7 9 10 11 12 13 15 16 17 18
| | C Db D D# Eb E E# F# Gb G G# Ab A# Bb B B#
|-
| | 12|3
| | 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1
| | 0 1 3 4 5 6 7 9 10 11 12 13 15 16 17 18
| | C C# D D# Eb E E# F# Gb G G# Ab A# Bb B B#
|-
| | 11|4
| | 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1
| | 0 1 3 4 5 6 7 9 10 11 12 13 14 16 17 18
| | C C# D D# Eb E E# F# Gb G G# Ab A Bb B B#
|-
| | 10|5
| | 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1 1
| | 0 1 3 4 5 6 7 8 10 11 12 13 14 16 17 18
| | C C# D D# Eb E E# F Gb G G# Ab A Bb B B#
|-
| | 9|6
| | 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
| | 0 1 2 4 5 6 7 8 10 11 12 13 14 16 17 18
| | C C# Db D# Eb E E# F Gb G G# Ab A Bb B B#
|-
| | 8|7
| | 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1
| | 0 1 2 4 5 6 7 8 10 11 12 13 14 15 17 18
| | C C# Db D# Eb E E# F Gb G G# Ab A A# B B#
|-
| | 7|8
| | 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1
| | 0 1 2 4 5 6 7 8 9 11 12 13 14 15 17 18
| | C C# Db D# Eb E E# F F# G G# Ab A A# B B#
|-
| | 6|9
| | 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1
| | 0 1 2 3 5 6 7 8 9 11 12 13 14 15 17 18
| | C C# Db D Eb E E# F F# G G# Ab A A# B B#
|-
| | 5|10
| | 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1
| | 0 1 2 3 5 6 7 8 9 11 12 13 14 15 16 18
| | C C# Db D Eb E E# F F# G G# Ab A A# Bb B#
|-
| | 4|11
| | 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1
| | 0 1 2 3 5 6 7 8 9 10 12 13 14 15 16 18
| | C C# Db D Eb E E# F F# Gb G# Ab A A# Bb B#
|-
| | 3|12
| | 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1
| | 0 1 2 3 4 6 7 8 9 10 12 13 14 15 16 18
| | C C# Db D D# E E# F F# Gb G# Ab A A# Bb B#
|-
| | 2|13
| | 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2
| | 0 1 2 3 4 6 7 8 9 10 12 13 14 15 16 17
| | C C# Db D D# E E# F F# Gb G# Ab A A# Bb B
|-
| | 1|14
| | 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2
| | 0 1 2 3 4 6 7 8 9 10 11 13 14 15 16 17
| | C C# Db D D# E E# F F# Gb G Ab A A# Bb B
|-
| | 0|15
| | 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2
| | 0 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17
| | C C# Db D D# Eb E# F F# Gb G Ab A A# Bb B
|}

==[[Sensi|Sensi]]==

'''sensi[5]''' ([[3L_2s|3L+2s]] / [[Modal_UDP_Notation|chroma-positive generator]] = 12\19)

{| class="wikitable"
|-
! | UDP
! | mode steps
! | mode degrees
! | note names
|-
| | 4|0
| | 5 5 2 5 2
| | 0 5 10 12 17
| | C Eb Gb G# B
|-
| | 3|1
| | 5 2 5 5 2
| | 0 5 7 12 17
| | C Eb E# G# B
|-
| | 2|2
| | 5 2 5 2 5
| | 0 5 7 12 14
| | C Eb E# G# A
|-
| | 1|3
| | 2 5 5 2 5
| | 0 2 7 12 14
| | C Db E# G# A
|-
| | 0|4
| | 2 5 2 5 5
| | 0 2 7 9 14
| | C Db E# F# A
|}

'''sensi[8]''' ([[3L_5s|3L+5s]] / CPG = 12\19)

{| class="wikitable"
|-
! | UDP
! | mode steps
! | mode degrees
! | note names
|-
| | 7|0
| | 3 2 3 2 2 3 2 2
| | 0 3 5 8 10 12 15 17
| | C D Eb F Gb G# A# B
|-
| | 6|1
| | 3 2 2 3 2 3 2 2
| | 0 3 5 7 10 12 15 17
| | C D Eb E# Gb G# A# B
|-
| | 5|2
| | 3 2 2 3 2 2 3 2
| | 0 3 5 7 10 12 14 17
| | C D Eb E# Gb G# A B
|-
| | 4|3
| | 2 3 2 3 2 2 3 2
| | 0 2 5 7 10 12 14 17
| | C Db Eb E# Gb G# A B
|-
| | 3|4
| | 2 3 2 2 3 2 3 2
| | 0 2 5 7 9 12 14 17
| | C Db Eb E# F# G# A B
|-
| | 2|5
| | 2 3 2 2 3 2 2 3
| | 0 2 5 7 9 12 14 16
| | C Db Eb E# F# G# A Bb
|-
| | 1|6
| | 2 2 3 2 3 2 2 3
| | 0 2 4 7 9 12 14 16
| | C Db D# E# F# G# A Bb
|-
| | 0|7
| | 2 2 3 2 2 3 2 3
| | 0 2 4 7 9 11 14 16
| | C Db D# E# F# G A Bb
|}

'''sensi[11]''' ([[8L_3s|8L+3s]] / CPG = 7\19)

{| class="wikitable"
|-
! | UDP
! | mode steps
! | mode degrees
! | note names
|-
| | 10|0
| | 2 2 2 1 2 2 2 1 2 2 1
| | 0 2 4 6 7 9 11 13 14 16 18
| | C Db D# E E# F# G Ab A Bb B#
|-
| | 9|1
| | 2 2 2 1 2 2 1 2 2 2 1
| | 0 2 4 6 7 9 11 12 14 16 18
| | C Db D# E E# F# G G# A Bb B#
|-
| | 8|2
| | 2 2 1 2 2 2 1 2 2 2 1
| | 0 2 4 5 7 9 11 12 14 16 18
| | C Db D# Eb E# F# G G# A Bb B#
|-
| | 7|3
| | 2 2 1 2 2 2 1 2 2 1 2
| | 0 2 4 5 7 9 11 12 14 16 17
| | C Db D# Eb E# F# G G# A Bb B
|-
| | 6|4
| | 2 2 1 2 2 1 2 2 2 1 2
| | 0 2 4 5 7 9 10 12 14 16 17
| | C Db D# Eb E# F# Gb G# A Bb B
|-
| | 5|5
| | 2 1 2 2 2 1 2 2 2 1 2
| | 0 2 3 5 7 9 10 12 14 16 17
| | C Db D Eb E# F# Gb G# A Bb B
|-
| | 4|6
| | 2 1 2 2 2 1 2 2 1 2 2
| | 0 2 3 5 7 9 10 12 14 15 17
| | C Db D Eb E# F# Gb G# A A# B
|-
| | 3|7
| | 2 1 2 2 1 2 2 2 1 2 2
| | 0 2 3 5 7 8 10 12 14 15 17
| | C Db D Eb E# F Gb G# A A# B
|-
| | 2|8
| | 1 2 2 2 1 2 2 2 1 2 2
| | 0 1 3 5 7 8 10 12 14 15 17
| | C C# D Eb E# F Gb G# A A# B
|-
| | 1|9
| | 1 2 2 2 1 2 2 1 2 2 2
| | 0 1 3 5 7 8 10 12 13 15 17
| | C C# D Eb E# F Gb G# Ab A# B
|-
| | 0|10
| | 1 2 2 1 2 2 2 1 2 2 2
| | 0 1 3 5 6 8 10 12 13 15 17
| | C C# D Eb E F Gb G# Ab A# B
|}

==[[Meantone_family#Liese|Liese]] / [[Marvel_temperaments#Triton|Triton]]==

'''liese[5]''' ([[2L_3s|2L+3s]] / [[Modal_UDP_Notation|chroma-positive generator]] = 9\19 ~= 7/5)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 4|0
| | 8 1 8 1 1
| | 0 8 9 17 18
| | C F F# B B#
|-
| | 3|1
| | 8 1 1 8 1
| | 0 8 9 10 18
| | C F F# Gb B#
|-
| | 2|2
| | 1 8 1 8 1
| | 0 1 9 10 18
| | C C# F# Gb B#
|-
| | 1|3
| | 1 8 1 1 8
| | 0 1 9 10 11
| | C C# F# Gb G
|-
| | 0|4
| | 1 1 8 1 8
| | 0 1 2 10 11
| | C C# Db Gb G
|}

'''liese[7]''' ([[2L_5s|2L+5s]] / CPG = 9\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 6|0
| | 7 1 1 7 1 1 1
| | 0 7 8 9 16 17 18
| | C E# F F# Bb B B#
|-
| | 5|1
| | 7 1 1 1 7 1 1
| | 0 7 8 9 10 17 18
| | C E# F F# Gb B B#
|-
| | 4|2
| | 1 7 1 1 7 1 1
| | 0 1 8 9 10 17 18
| | C C# F F# Gb B B#
|-
| | 3|3
| | 1 7 1 1 1 7 1
| | 0 1 8 9 10 11 18
| | C C# F F# Gb G B#
|-
| | 2|4
| | 1 1 7 1 1 7 1
| | 0 1 2 9 10 11 18
| | C C# Db F# Gb G B#
|-
| | 1|5
| | 1 1 7 1 1 1 7
| | 0 1 2 9 10 11 12
| | C C# Db F# Gb G G#
|-
| | 0|6
| | 1 1 1 7 1 1 7
| | 0 1 2 3 10 11 12
| | C C# Db D Gb G G#
|}

'''liese[9]''' ([[2L_7s|2L+7s]] / CPG = 9\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 8|0
| | 6 1 1 1 6 1 1 1 1
| | 0 6 7 8 9 15 16 17 18
| | C E E# F F# A# Bb B B#
|-
| | 7|1
| | 6 1 1 1 1 6 1 1 1
| | 0 6 7 8 9 10 16 17 18
| | C E E# F F# Gb Bb B B#
|-
| | 6|2
| | 1 6 1 1 1 6 1 1 1
| | 0 1 7 8 9 10 16 17 18
| | C C# E# F F# Gb Bb B B#
|-
| | 5|3
| | 1 6 1 1 1 1 6 1 1
| | 0 1 7 8 9 10 11 17 18
| | C C# E# F F# Gb G B B#
|-
| | 4|4
| | 1 1 6 1 1 1 6 1 1
| | 0 1 2 8 9 10 11 17 18
| | C C# Db F F# Gb G B B#
|-
| | 3|5
| | 1 1 6 1 1 1 1 6 1
| | 0 1 2 8 9 10 11 12 18
| | C C# Db F F# Gb G G# B#
|-
| | 2|6
| | 1 1 1 6 1 1 1 6 1
| | 0 1 2 3 9 10 11 12 18
| | C C# Db D F# Gb G G# B#
|-
| | 1|7
| | 1 1 1 6 1 1 1 1 6
| | 0 1 2 3 9 10 11 12 13
| | C C# Db D F# Gb G G# Ab
|-
| | 0|8
| | 1 1 1 1 6 1 1 1 6
| | 0 1 2 3 4 10 11 12 13
| | C C# Db D D# Gb G G# Ab
|}

'''liese[11]''' ([[2L_9s|2L+9s]] / CPG = 9\19)

{| class="wikitable"
|-
! | UDP
! | steps
! | degrees
! | note names
|-
| | 10|0
| | 5 1 1 1 1 5 1 1 1 1 1
| | 0 5 6 7 8 9 14 15 16 17 18
| | C Eb E E# F F# A A# Bb B B#
|-
| | 9|1
| | 5 1 1 1 1 1 5 1 1 1 1
| | 0 5 6 7 8 9 10 15 16 17 18
| | C Eb E E# F F# Gb A# Bb B B#
|-
| | 8|2
| | 1 5 1 1 1 1 5 1 1 1 1
| | 0 1 6 7 8 9 10 15 16 17 18
| | C C# E E# F F# Gb A# Bb B B#
|-
| | 7|3
| | 1 5 1 1 1 1 1 5 1 1 1
| | 0 1 6 7 8 9 10 11 16 17 18
| | C C# E E# F F# Gb G Bb B B#
|-
| | 6|4
| | 1 1 5 1 1 1 1 5 1 1 1
| | 0 1 2 7 8 9 10 11 16 17 18
| | C C# Db E# F F# Gb G Bb B B#
|-
| | 5|5
| | 1 1 5 1 1 1 1 1 5 1 1
| | 0 1 2 7 8 9 10 11 12 17 18
| | C C# Db E# F F# Gb G G# B B#
|-
| | 4|6
| | 1 1 1 5 1 1 1 1 5 1 1
| | 0 1 2 3 8 9 10 11 12 17 18
| | C C# Db D F F# Gb G G# B B#
|-
| | 3|7
| | 1 1 1 5 1 1 1 1 1 5 1
| | 0 1 2 3 8 9 10 11 12 13 18
| | C C# Db D F F# Gb G G# Ab B#
|-
| | 2|8
| | 1 1 1 1 5 1 1 1 1 5 1
| | 0 1 2 3 4 9 10 11 12 13 18
| | C C# Db D D# F# Gb G G# Ab B#
|-
| | 1|9
| | 1 1 1 1 5 1 1 1 1 1 5
| | 0 1 2 3 4 9 10 11 12 13 14
| | C C# Db D D# F# Gb G G# Ab A
|-
| | 0|10
| | 1 1 1 1 1 5 1 1 1 1 5
| | 0 1 2 3 4 5 10 11 12 13 14
| | C C# Db D D# Eb Gb G G# Ab A
|}

MOS scales of sizes 13, 15, and 17 also exist, and follow a similar pattern:

'''liese[13]''' 4111114111111 etc.

'''liese[15]''' 311111131111111 etc.

'''liese[17]''' 21111111211111111 etc.

[[Category:19edo]]
[[Category:modes]]

19edo

2024-08-26T00:46:02Z

YoVariable: Adjusted formatting

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{EDO intro|19}}
== Theory ==
=== History ===
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone|{{frac|1|3}}-comma meantone]], in which the fifth is 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

=== As an approximation of other temperaments ===
19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth, though [[46edo]] is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy, and is the fifth (after 12) [[zeta integral edo]]. It is less successful in the [[7-limit]] (but still better than 12et), as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles, and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5 and 7 are not only much farther from just than they are in 19, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to employ [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29 cents, and a step size of between 63.2 and 63.4 cents would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57 cents, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. A more extreme option would be [[11edf]], which has octaves stretched by 12.47 cents.

=== As a means of extending harmony ===
Because 19edo allows for more blended, consonant harmonies than 12edo does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 cents off [[23/16]].

In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Prime harmonics ===
{{Harmonics in equal|19}}

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
! [[Degree]]
! [[Cent]]s
! [[Interval region|Interval Region]]
! Approximated [[Just intonation|JI]] Intervals<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
| 0
| 0.00
| Unison (prime)
| [[1/1]]
| Do
| unison
| P1
|-
| 1
| 63.16
| Augmented unison
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| super unison, subminor second
| S1, sm2
|-
| 2
| 126.32
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| minor second
| m2
|-
| 3
| 189.47
| Major second
| [[9/8]], [[10/9]]
| Re
| major second
| M2
|-
| 4
| 252.63
| Augmented second Diminished third
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.79
| Minor third
| [[6/5]]
| Me
| minor third
| m3
|-
| 6
| 378.95
| Major third
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| major third
| M3
|-
| 7
| 442.11
| Augmented third
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.26
| Perfect fourth
| [[4/3]], [[75/56]]
| Fa
| perfect fourth
| P4
|-
| 9
| 568.42
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| augmented fourth
| A4
|-
| 10
| 631.58
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
| Se
| diminished fifth
| d5
|-
| 11
| 694.74
| Perfect fifth
| [[3/2]], [[112/75]]
| So
| perfect fifth
| P5
|-
| 12
| 757.89
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| super fifth, subminor sixth
| S5, sm6
|-
| 13
| 821.05
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
| Le
| minor sixth
| m6
|-
| 14
| 884.21
| Major sixth
| [[5/3]]
| La
| major sixth
| M6
|-
| 15
| 947.37
| Augmented sixth Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.53
| Minor seventh
| [[9/5]], [[16/9]]
| Te
| minor seventh
| m7
|-
| 17
| 1073.68
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| major seventh
| M7
|-
| 18
| 1136.84
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.00
| Octave
| [[2/1]]
| Do
| octave
| P8
|}

=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center"
! Quality
! [[Color name|Color Name]]
! Monzo Format
! Examples
|-
| diminished
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| rowspan="2" | minor
| fourthward wa
| (a, b), b < -1
| 32/27, 16/9
|-
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| rowspan="2" | major
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| fifthward wa
| (a, b), b > 1
| 9/8, 27/16
|-
| augmented
| ru
| (a, b, 0, -1)
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3), Cmin(♭3), C(d3)
| C subminor, C minor flat-three, C diminished-three
|-
| gu
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm, Cmin
| C minor
|-
| yo
| 4:5:6
| 0–6–11
| C–E–G
| C, Cmaj
| C, C major
|-
| ru
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3), Cmaj(♯3), C(A3)
| C supermajor, C major sharp-three, C augmented-three
|-
|
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| C(h7), Cadd(d7), Cmaj(add(d7))
| C harmonic 7, C (major) add dim-seven
|-
|
| 1/(4:5:6:7) = 1:6/5:3/2:12/7
| 0–5–11–15
| C–E♭–G–A♯
| Cm(♯6), Cm(A6), Cm(add(♯6)), Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios.

For a more complete list, see [[19edo Chord Names]] and [[Ups and downs notation #Chords and Chord Progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfege, or sargam. Note that D# and Eb are two different notes.

{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard Notation]]
|-
! [[5L 2s|Diatonic Interval Names]]
! Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.16
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.32
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.47
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.63
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.79
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.95
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.11
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.26
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.42
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.58
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.74
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.89
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.05
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.21
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.37
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.53
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.68
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.84
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.00
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic Notation of 19edo
! [[Degree]]
! [[Cent]]s
! Interval Names
|-
| 0
| 0.00
| P1
|-
| 1
| 63.16
| A1, m2
|-
| 2
| 126.32
| M2, m3
|-
| 3
| 189.47
| M3
|-
| 4
| 252.63
| m4, A3
|-
| 5
| 315.79
| M4, m5
|-
| 6
| 378.95
| M5
|-
| 7
| 442.11
| A5, d6
|-
| 8
| 505.26
| P6
|-
| 9
| 568.42
| A6, m7
|-
| 10
| 631.58
| M7, d8
|-
| 11
| 694.74
| P8
|-
| 12
| 757.89
| A8, m9
|-
| 13
| 821.05
| M9, m10
|-
| 14
| 884.21
| M10
|-
| 15
| 947.37
| m11, A10
|-
| 16
| 1010.53
| M11, m12
|-
| 17
| 1073.68
| M12
|-
| 18
| 1136.84
| A12, d13
|-
| 19
| 1200.00
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| [{{val| 19 30 }}]
| +2.28
| 2.28
| 3.61
|-
| 2.3.5
| 81/80, 3125/3072
| [{{val| 19 30 44 }}]
| +2.58
| 1.91
| 3.02
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| [{{val| 19 30 44 53 }}]
| +3.85
| 2.76
| 4.35
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| [{{val| 19 30 44 53 70 }}]
| +4.14
| 2.53
| 3.99
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| [{{val| 19 30 44 53 70 86 }}]
| +3.32
| 2.93
| 4.64
|}

19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – ''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.

19et is prominent in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|13|18.5|19.5}}

=== Commas ===
19et [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime Limit]]
! [[Ratio]]<ref group="note">{{rd|10}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references/>

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic-kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! MOSes
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[rhinocerus]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] / [[pycnic]] [[Triton]]
|}

== Scales ==
=== MOS scales ===
==== Octave-equivalent mosses ====
* [[meantone]] pentatonic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2
* Meantone harmonic major: 3 3 2 3 2 4 2
* chromatic octave species - Meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
*[[Pinetone#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
*[[Pinetone#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
*[[Pinetone#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Arto and Tendo Theory]]
* [[Lumatone mapping for 19edo]]

=== Notes ===
<references group="note" />

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

[[Category:19-tone scales]]
[[Category:Golden meantone]]
[[Category:Kleismic]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Negri]]
[[Category:Sensi]]
[[Category:Teentuning]]
[[Category:Historical]]

19edo

2024-08-26T00:44:33Z

YoVariable: /* Intervals */ Added secondary interval names

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{EDO intro|19}}
== Theory ==
=== History ===
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone|{{frac|1|3}}-comma meantone]], in which the fifth is 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

=== As an approximation of other temperaments ===
19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth, though [[46edo]] is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy, and is the fifth (after 12) [[zeta integral edo]]. It is less successful in the [[7-limit]] (but still better than 12et), as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles, and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5 and 7 are not only much farther from just than they are in 19, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to employ [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29 cents, and a step size of between 63.2 and 63.4 cents would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57 cents, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. A more extreme option would be [[11edf]], which has octaves stretched by 12.47 cents.

=== As a means of extending harmony ===
Because 19edo allows for more blended, consonant harmonies than 12edo does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 cents off [[23/16]].

In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Prime harmonics ===
{{Harmonics in equal|19}}

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
! [[Degree]]
! [[Cent]]s
! [[Interval region|Interval Region]]
! Approximated [[Just intonation|JI]] Intervals<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
| 0
| 0.00
| Unison (prime)
| [[1/1]]
| Do
| unison
| P1
|-
| 1
| 63.16
| Augmented unison
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| super unison, subminor second
| S1, sm2
|-
| 2
| 126.32
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| minor second
| m2
|-
| 3
| 189.47
| Major second
| [[9/8]], [[10/9]]
| Re
| major second
| M2
|-
| 4
| 252.63
| Augmented second
Diminished third
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.79
| Minor third
| [[6/5]]
| Me
| minor third
| m3
|-
| 6
| 378.95
| Major third
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| major third
| M3
|-
| 7
| 442.11
| Augmented third
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.26
| Perfect fourth
| [[4/3]], [[75/56]]
| Fa
| perfect fourth
| P4
|-
| 9
| 568.42
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| augmented fourth
| A4
|-
| 10
| 631.58
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
| Se
| diminished fifth
| d5
|-
| 11
| 694.74
| Perfect fifth
| [[3/2]], [[112/75]]
| So
| perfect fifth
| P5
|-
| 12
| 757.89
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| super fifth, subminor sixth
| S5, sm6
|-
| 13
| 821.05
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
| Le
| minor sixth
| m6
|-
| 14
| 884.21
| Major sixth
| [[5/3]]
| La
| major sixth
| M6
|-
| 15
| 947.37
| Augmented sixth
Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.53
| Minor seventh
| [[9/5]], [[16/9]]
| Te
| minor seventh
| m7
|-
| 17
| 1073.68
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| major seventh
| M7
|-
| 18
| 1136.84
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.00
| Octave
| [[2/1]]
| Do
| octave
| P8
|}

=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center"
! Quality
! [[Color name|Color Name]]
! Monzo Format
! Examples
|-
| diminished
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| rowspan="2" | minor
| fourthward wa
| (a, b), b < -1
| 32/27, 16/9
|-
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| rowspan="2" | major
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| fifthward wa
| (a, b), b > 1
| 9/8, 27/16
|-
| augmented
| ru
| (a, b, 0, -1)
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3), Cmin(♭3), C(d3)
| C subminor, C minor flat-three, C diminished-three
|-
| gu
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm, Cmin
| C minor
|-
| yo
| 4:5:6
| 0–6–11
| C–E–G
| C, Cmaj
| C, C major
|-
| ru
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3), Cmaj(♯3), C(A3)
| C supermajor, C major sharp-three, C augmented-three
|-
|
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| C(h7), Cadd(d7), Cmaj(add(d7))
| C harmonic 7, C (major) add dim-seven
|-
|
| 1/(4:5:6:7) = 1:6/5:3/2:12/7
| 0–5–11–15
| C–E♭–G–A♯
| Cm(♯6), Cm(A6), Cm(add(♯6)), Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios.

For a more complete list, see [[19edo Chord Names]] and [[Ups and downs notation #Chords and Chord Progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfege, or sargam. Note that D# and Eb are two different notes.

{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard Notation]]
|-
! [[5L 2s|Diatonic Interval Names]]
! Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.16
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.32
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.47
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.63
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.79
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.95
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.11
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.26
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.42
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.58
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.74
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.89
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.05
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.21
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.37
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.53
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.68
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.84
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.00
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic Notation of 19edo
! [[Degree]]
! [[Cent]]s
! Interval Names
|-
| 0
| 0.00
| P1
|-
| 1
| 63.16
| A1, m2
|-
| 2
| 126.32
| M2, m3
|-
| 3
| 189.47
| M3
|-
| 4
| 252.63
| m4, A3
|-
| 5
| 315.79
| M4, m5
|-
| 6
| 378.95
| M5
|-
| 7
| 442.11
| A5, d6
|-
| 8
| 505.26
| P6
|-
| 9
| 568.42
| A6, m7
|-
| 10
| 631.58
| M7, d8
|-
| 11
| 694.74
| P8
|-
| 12
| 757.89
| A8, m9
|-
| 13
| 821.05
| M9, m10
|-
| 14
| 884.21
| M10
|-
| 15
| 947.37
| m11, A10
|-
| 16
| 1010.53
| M11, m12
|-
| 17
| 1073.68
| M12
|-
| 18
| 1136.84
| A12, d13
|-
| 19
| 1200.00
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| [{{val| 19 30 }}]
| +2.28
| 2.28
| 3.61
|-
| 2.3.5
| 81/80, 3125/3072
| [{{val| 19 30 44 }}]
| +2.58
| 1.91
| 3.02
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| [{{val| 19 30 44 53 }}]
| +3.85
| 2.76
| 4.35
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| [{{val| 19 30 44 53 70 }}]
| +4.14
| 2.53
| 3.99
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| [{{val| 19 30 44 53 70 86 }}]
| +3.32
| 2.93
| 4.64
|}

19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – ''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.

19et is prominent in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|13|18.5|19.5}}

=== Commas ===
19et [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime Limit]]
! [[Ratio]]<ref group="note">{{rd|10}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references/>

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic-kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! MOSes
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[rhinocerus]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] / [[pycnic]] [[Triton]]
|}

== Scales ==
=== MOS scales ===
==== Octave-equivalent mosses ====
* [[meantone]] pentatonic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2
* Meantone harmonic major: 3 3 2 3 2 4 2
* chromatic octave species - Meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
*[[Pinetone#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
*[[Pinetone#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
*[[Pinetone#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Arto and Tendo Theory]]
* [[Lumatone mapping for 19edo]]

=== Notes ===
<references group="note" />

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

[[Category:19-tone scales]]
[[Category:Golden meantone]]
[[Category:Kleismic]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Negri]]
[[Category:Sensi]]
[[Category:Teentuning]]
[[Category:Historical]]

Xendergarten

2024-08-23T14:24:26Z

YoVariable: Updated Xendergarten Discord Server Link

'''Xendergarten''' is a comedy podcast about xenharmonic and microtonal music and discussion created by [[User:ks26|groundfault]] and [[Userminusone]] in 2023.

== Episodes ==
{| class="wikitable sortable center-1"
|-
! No. !! Title !! style="width:8em;" | Host(s) !! Guest(s) !! Music artists !! Release date
|-
| 0 || [https://youtu.be/0tTK67Om6NA Gasenlightened: Episode 0 - the lost episode] || rowspan="11" | [[Userminusone]], [[User:ks26|groundfault]] || Domin, [[Inthar]], kgrodunds, Nicolai, Nosigntoday, Riffy, [[Stephen Weigel]], xenoindex || (None) || April 1, 2023
|-
| 1 || [https://youtu.be/99yNcPi32EY Episode 1 - Thick Thives] ([https://youtu.be/j02K5TGZ3fM highlight reel]) || kgrodunds, Inthar, Nosignstoday, Riffy, [[User:Tristanbay|Tristan Bay]], dotu_mtr||(None)||March 14, 2023
|-
| 2 || [https://youtu.be/05CdorjwL4E Why do they call it JI when J-you of in the comma temper of out EDO temper the schisma] || Dead Shaman, Domin, kam, Carmen Parker, Nosignstoday, xenoindex || Userminusone, groundfault, Carmen Parker, Wrenharmonic, Stephen Weigel, Animusic, [[Sevish]], [[benyamind]], Dead Shaman || May 11, 2023
|-
| 3a || [https://youtu.be/NV2876Svlos Rhythms are not the goal of my plugin] || [[User:AraMax|AraMax]], Inthar, Dead Shaman, kgrodunds, kam, Riffy, Stephen Weigel, Tristan Bay, xenoindex || Stephen Weigel, Userminusone, Nick Vuci, Benjamin D, HEHEHE I AM A SUPAHSTAR SAGA, xenoindex, Sevish || June 10, 2023
|-
| 3b || [https://youtu.be/r61sowvODyA The Purgatory Pile] || AraMax, Inthar, Dead Shaman, kgrodunds, kam, Riffy, Stephen Weigel, xenoindex || Wrenharmonic, Stephen Weigel, Userminusone, groundfault || June 10, 2023
|-
| 4 || [https://youtu.be/dXzkMG9dE2s Green and a Half Pilled] || inthar, dotu_mtr, xenoindex, Riffy, Dead Shaman || (None) || August 27, 2023
|-
| 5 || [https://youtu.be/vDb0fETpyYM Live From Lake Doramos] || Dead Shaman, xenoindex, dotu_mtr, Riffy, Wrenharmonic, Tristan Bay, Adam Freese || (None) || August 30, 2023
|-
| 6 || [https://youtu.be/2-rJe0FNTag Planck IQ] || Dead Shaman, xenoindex, dotu_mtr, Tristan Bay || (None) || October 9, 2023
|-
| 7 || [https://youtu.be/fMz1vRSFZ1g Desert Island Pain] || kam, Tristan Bay, Dead Shaman, dotu_mtr, xenoindex, inthar || (todo) || December 16, 2023
|-
|8
|[https://www.youtube.com/watch?v=Dv9VgKTE4tU The most violent thing I've ever seen]
|inthar, Tristan Bay, Aramax, Dead Shaman, kam, dotu_mtr, Abnormality, Wrenharmonic
|(todo)
|January 28, 2024
|-
|9a
|[https://www.youtube.com/watch?v=dZLXpOlXTl8 I don't think, therefore I am not]
|Tristan Bay, Aramax, Dead Shaman, dotu_mtr, Abnormality, Wrenharmonic
|(None)
|February 7, 2024
|}

== External links ==
* [https://www.youtube.com/@XendergartenPodcast Xendergarten on YouTube]
* [https://discord.gg/NYJKwqz38w Official Xendergarten Discord Server]

[[Category:Podcasts]]

Kite's ups and downs notation

2024-07-24T20:16:46Z

YoVariable: /* Examples: EDOs 12-24 */ Added v to A1 that was supposed to be vA1 in the 22edo section

== Definition ==
Ups and Downs (or ^v) is a notation system invented by [[Kite Giedraitis]] that can notate almost every [[EDO|EDO]]. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.

Ups and downs can also notate any [[Tour of Regular Temperaments|rank-2 temperament]], although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a [[comma]] that has been tempered, but not tempered out. For example, in [[Porcupine|Triyo aka Porcupine]], an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyo is supported by both 15-edo and 22-edo, and both EDOs map 81/80 to one EDOstep. Thus if Triyo is tuned to 15-edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22-edo, the up means both 81/80 and 1\22. If not tuned to an EDO at all, then the up only means 81/80. Thus a piece written in Triyo can be converted to a piece written in 22-edo by simply writing "22-edo" on the top of the page.

Ups and downs can also be used to notate rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11. See [[Ups and Downs Notation for Rank-3 JI]].

'''This page only discusses notation of EDOs.''' However the notation of chords and chord progressions applies to all situations. For notation of rank-2 and rank-3 temperaments, see the [[pergen|pergens]] article.

For more on EDO notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf '''Notation guide for edos 5-72'''], which also covers chord names, slash chords, staff notation, key signatures, and scale trees.

==Explanation -- a 22-edo example==

To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDOstep. C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-edo as long as you remember that C# and Db are different notes.

In contrast, 22-edo is hard to notate because 7 fifths reduces to three EDOsteps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the 5/4, usually a major 3rd, becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really confusing. A notation system should work in every key!

The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up-arrow symbol to mean "sharpened by one EDOstep". 22-edo can be written C - Db - ^Db - vD - D - Eb - ^Eb - vE - E - F etc. The notes are pronounced up-D-flat, down-D, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. The up or down comes before the note name to make naming chords easy.

The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - ^C - vC# - C# - D - ^D - vD# - D# - E etc.

The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

=== Relative notation and interval arithmetic ===
Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22-edo intervals: P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^4/d5 - vA4/^d5 - A4/v5 - P5 etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22-edo key. The '''plain''' notes (those without ups or downs) always form a chain of fifths.

A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:
{| class="wikitable"
|+
!
!interval between

two notes
!note plus

an interval
!sum of two

intervals
|-
!conventional
|C to E = M3
|C + M3 = E
|M2 + M2 = M3
|-
! rowspan="2" |with ups

and downs
|^C to E = vM3
|^C + M3 = ^E
|^M2 + M2 = ^M3
|-
|C to ^E = ^M3
|C + ^M3 = ^E
|M2 + vM2 = vM3
|-
!(cancelling)
|^C to ^E = M3
|^C + vM3 = E
|^M2 + vM2 = M3
|-
!(combining)
|^C to vE = vvM3
|^C + ^M3 = ^^E
|vM2 + vM2 = vvM3
|}
The same logic holds for a note minus an interval (C - vm3 = ^A) or one interval minus another interval (M3 - vM2 = ^M2).

=== "Arrow" as a term for EDOstep ===
Up and down are short for up-arrow and down-arrow, and arrow refers to both. Sometimes the name of a notation symbol comes to mean that which the symbol indicates. Just as "bar" (the vertical line that separates measures) has come to mean "measure", "[[arrow]]" has also come to mean "EDOstep".

=== Enharmonic equivalents ===
Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-edo, Db can be spelled ^C or vB# or even ^^B (double-up B, or '''dup''' B for short, rhymes with "cup").

From the [[Pergen|pergens]] article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic intervals are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank."

Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two enharmonic intervals are needed to define the notation: v3A1 and vm2. Either interval can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v3A1 = vE. Any combination of these two enharmonic intervals is also an enharmonic interval, for example their sum v4M2. Thus ^^F = vvG (double-down G, or '''dud''' G for short, rhymes with "cud").

In larger edos, triple-arrows, quadruple-arrows, etc. can occur, and are names thusly:
{| class="wikitable"
|+words for multiple arrows
!1 arrow
!2 arrows
!3 arrows
!4 arrows
!5 arrows
|-
|up
|dup (rhymes
with "up")
|trup (rhymes
with "up")
|quup
("kwup")
|quip
|-
|down
|dud
|trud (rhymes
with "dud")
|quud
("kwud")
|quid
|}

=== Staff Notation ===
For staff notation, put an arrow to the left of the note and any sharp or flat it might have. Like sharps and flats, an arrow applies to any similar note that follows in the measure. If C is upped, any other C in the same octave inherits the up. If an up-C is followed by a down-C, the down-arrow cancels the up-arrow.

But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?

There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.
{| class="wikitable"
|+
! rowspan="2" |start with
this
! colspan="6" |turn it into this
|-
!C
!^C
!^^C
!C#
!^C#
!^^C#
|-
!C
|
|^
|^^
|#
|^#
|^^#
|-
!^C
|<big>♮</big>
|
|^^
|#
|^#
|^^#
|-
!^^C
|<big>♮</big>
|^
|
|#
|^#
|^^#
|-
!C#
|<big>♮</big>
|^
|^^
|
|^#
|^^#
|-
!^C#
|<big>♮</big>
|^
|^^
|#
|
|^^#
|-
!^^C#
|<big>♮</big>
|^
|^^
|#
|^#
|
|}
See [[Kite Guitar originals#Cancelling rules]] for another way.

For more on staff notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide for EDOs 5-72].

=== Key signatures ===
Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo has the key of Bbb with a key signature of Bbb Ebb Ab Db Gb Cb Fb. Some EDOs have upped/downed tonics, e.g. 24-edo has the key of vD with a key signature of F# C# (v). The (v) is a "global down" that downs all 7 notes of the vD scale. See also [[Kite Guitar originals#Scales and key signatures]] for the use of '''arrow stacks'''.

=== Placement of the arrow ===
It might seem more natural to place the arrow after the note, for example B^ or Bb^. But the arrow must come first, to make chord names unambiguous. Otherwise B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)

The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.

=== Further notes ===
EDO intervals are often written as 7\22. This can also be written as vM3\22. This is useful when comparing EDOs, e.g. vM3\22 vs. vM3\15.

==Examples: EDOs 12-24==

Sharp-1, flat-2, etc. refer to the [[sharpness]], the number of arrows made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.

A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs.

13-edo and 18-edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, as 13b and 18b.

There are four flat-N edos on this list. 16-edo and 23-edo are flat-1, 18b is flat-2 and 13b is flat-3. There are two ways to notate such edos: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. Both notations are shown. In the 2nd notation, note that a fifth above B is Fb, not F#.

12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |[[12-edo|'''12-edo''']]
sharp-1
|'''D'''
|D#/Eb
|'''E'''
|'''F'''
|F#/Gb
|'''G'''
|G#/Ab
|'''A'''
|A#/Bb
|'''B'''
|'''C'''
|C#/Db
|'''D'''
|-
|P1
|A1/m2
|M2
|m3
|M3
|P4
|A4/d5
|P5
|m6
|M6
|m7
|M7
|P8
|}
There are two ways to notate 13b-edo. The enharmonic intervals for the 1st notation are ^3A1 and vM2. For the 2nd they are v3A1 and vm2.
{| class="wikitable" style="text-align:center;"
| rowspan="4" |'''[[13-edo|13b-edo]]'''
flat-3
| rowspan="2" |sharp lowers the pitch,
major narrower than minor
|'''D'''
|'''E'''
|^E/F#
|vEb/^F#
|Eb/vF
|'''F'''
|'''G'''
|'''A'''
|'''B'''
|^B/C#
|vBb/^C#
|Bb/vC
|'''C'''
|'''D'''
|-
|P1
|M2
|^M2/M3
|vm2/^M3
|m2/vm3
|m3
|P4
|P5
|M6
|^M6/M7
|vm6/^M7
|m6/vm7
|m7
|P8
|-
| rowspan="2" |sharp raises the pitch,
major wider than minor
|'''D'''
|'''E'''
|^E/Fb
|vE#/^Fb
|E#/vF
|'''F'''
|'''G'''
|'''A'''
|'''B'''
|^B/Cb
|vB#/^Cb
|B#/vC
|'''C'''
|'''D'''
|-
|P1
|m2
|^m2/m3
|vM2/^m3
|M2/vM3
|M3
|P4
|P5
|m6
|^m6/m7
|vM6/^m7
|M6/vM7
|M7
|P8
|}

Because every 14-edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14-edo contains 2 rings of 7-edo: an up/down-ring and a plain-ring. Enharmonic intervals: A1 and vvm2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[14-edo]]'''
sharp-0
|'''D'''
|^D/vE
|'''E'''
|^E/vF
|'''F'''
|^F/vG
| '''G'''
|^G/vA
|'''A'''
|^A/vB
|'''B'''
|^B/vC
|'''C'''
|^C/vD
| '''D'''
|-
|1
|^1/v2
|2
|^2/v3
|3
|^3/v4
|4
|^4/v5
|5
|^5/v6
|6
|^6/v7
|7
|^7/v8
|8
|}
15-edo contains 3 rings of 5-edo: an up-ring, a down-ring, and a plain-ring. Enharmonic intervals: v3A1 and m2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[15-edo]]'''
sharp-3
|'''D'''
|^D
| vE
|'''E/F'''
| ^F
| vG
| '''G'''
|^G
|vA
|'''A'''
|^A
|vB
| '''B/C'''
| ^C
| vD
| '''D'''
|-
|P1
| ^m2
| vM2
| M2/m3
| ^m3
|vM3
| M3/P4
| ^4
| v5
| P5
| ^m6
| vM6
| M6/m7
| ^m7
| vM7
| P8
|}
16-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either AA2 or dd2.
{| class="wikitable" style="text-align:center;"
| rowspan="4" |'''[[16-edo]]'''
flat-1
| rowspan="2" |sharp lowers the pitch,
major narrower than minor
|'''D'''
| Db/E#
| '''E'''
| Eb
| F#
| '''F'''
| Fb/G#
| '''G'''
| Gb/A#
| '''A'''
|Ab/B#
| '''B'''
|Bb
| C#
| '''C'''
| Cb/D#
| '''D'''
|-
|P1
| A2
| M2
| m2/A3
| M3
| m3
| d3/A4
| P4
| d4/A5
| P5
| d5/A6
| M6
| m6/A7
| M7
| m7
| d7
| P8
|-
| rowspan="2" | sharp raises the pitch,
major wider than minor
|'''D'''
| D#/Eb
| '''E'''
| E#
| Fb
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| B#
| Cb
| '''C'''
| C#/Db
| '''D'''
|-
|P1
| d2
| m2
| M2
| m3
| M3
| A3
| P4
| A4/d5
| P5
| d6
| m6
| M6/d7
| m7
| M7
| A7
| P8
|}
17-edo is sharp-2 and thus has mid intervals. Enharmonic intervals: vvA1 and vm2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[17edo|17-edo]]'''
sharp-2
|'''D'''
| ^D/Eb
| D#/vE
| '''E'''
| '''F'''
|^F/Gb
| F#/vG
| '''G'''
| ^G/Ab
| G#/vA
| '''A'''
| ^A/Bb
| A#/vB
| '''B'''
| '''C'''
| ^C/Db
| C#/vD
| '''D'''
|-
|P1
| ^1/m2
| A1/~2
| M2
| m3
| ~3
| M3
| P4
| ^4/~4/d5
| A4/v5/~5
| P5
| m6
| ~6
| M6
| m7
| ~7
| M7
| P8
|}
18b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic intervals: either ^^A1 and vvM2, or vvA1 and vvm2.
{| class="wikitable" style="text-align:center;"
| rowspan="4" |'''[[18-edo|18b-edo]]'''
flat-2
| rowspan="2" |sharp lowers,
major is narrower
|'''D'''
| ^D/vE
| '''E'''
| ^E
| Eb/F#
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| Bb/C#
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
|P1
| ^1/vM2
| M2
| ~2
| m2/M3
| ~3
| m3
| ^m3/v4
| P4
| ^4/v5
| P5
| ^5/vM6
| M6
| ~6
| m6/M7
| ~7
| m7
| ^m2/d8
| P8
|-
| rowspan="2" | sharp raises,
major is wider
|'''D'''
| ^D/vE
| '''E'''
|^E
| E#/Fb
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| B#/Cb
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
|P1
| ^1/vm2
| m2
| ~2
| M2/m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/v5
| P5
| ^5/vm6
| m6
| ~6
| M6/m7
| ~7
| M7
| ^M7/d8
| P8
|}
19-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: dd2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[19-edo]]'''
sharp-1
|'''D'''
| D#
| Eb
| '''E'''
| E#/Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#/Cb
| '''C'''
| C#
| Db
| '''D'''
|-
|P1
| d2
| m2
| M2
| d3
| m3
| M3
| A3
| P4
| A4
| d5
| P5
| A5
| m6
| M6
| d7
| m7
| M7
| A7
| P8
|}
20-edo contains 4 rings of 5-edo: an up-ring, a down-ring, a dup/dud-ring, and a plain-ring. Enharmonic intervals: v4A1 and m2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[20-edo]]'''
sharp-4
|'''D'''
| ^D
| ^^D/vvE
| vE
| '''E/F'''
| ^F
| ^^F/vvG
| vG
| '''G'''
| ^G
| ^^G/vvA
| vA
| '''A'''
| ^A
| ^^A/vvB
| vB
| '''B/C'''
| ^C
| ^^C/vvD
| vD
| '''D'''
|-
|P1/m2
| ^m2
| ~2
| vM2
| M2/m3
| ^m3
| ~3
| vM3
| M3/P4
| ^4
|~4/~5
| v5
| P5/m6
| ^m6
| ~6
| vM6
| M6/m7
| ^m7
| ~7
| vM7
| P8
|}
Because every 21-edo interval is perfect, the quality can be omitted. 21-edo contains 3 rings of 7-edo: an up-ring, a down-ring and a plain-ring. Enharmonic intervals: A1 and v3m2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[21-edo]]'''
sharp-0
|'''D'''
| ^D
| vE
| '''E'''
| ^E
| vF
| '''F'''
| ^F
| vG
| '''G'''
| ^G
| vA
| '''A'''
| ^A
| vB
| '''B'''
| ^B
| vC
| '''C'''
| ^C
| vD
| '''D'''
|-
|1
| ^1
| v2
| 2
| ^2
| v3
| 3
| ^3
| v4
| 4
| ^4
| v5
| 5
| ^5
| v6
| 6
| ^6
| v7
| 7
| ^7
| v8
| 8
|}
22-edo is sharp-3. Enharmonic intervals: v3A1 and vm2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" |'''[[22-edo]]'''
sharp-3
|'''D'''
| ^D/Eb
| vD#/^Eb
| D#/vE
| '''E'''
| '''F'''
| ^F/Gb
| vF#/^Gb
| F#/vG
| '''G'''
| ^G/Ab
| vG#/^Ab
| G#/vA
| '''A'''
| etc.
|-
|P1
| ^1/m2
| vA1/^m2
| vM2
| M2
| m3
| ^m3
| vM3
| M3
| P4
| ^4/d5
| vA4/^d5
| A4/v5
| P5
| etc.
|}
23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A32 or d32.
{| class="wikitable" style="text-align:center;"
| rowspan="4" |'''[[23-edo]]'''
flat-1
| rowspan="2" |sharp lowers,
major is narrower
| '''D'''
| Db
| E#
| '''E'''
| Eb
| Ebb/Fx
| F#
| '''F'''
| Fb
| G#
| '''G'''
| Gb
| A#
| '''A'''
| Ab
| B#
| '''B'''
| Bb
| Bbb/Cx
| C#
| '''C'''
| Cb
| D#
| '''D'''
|-
| P1
| d1
| A2
| M2
| m2
| d2/A3
| M3
| m3
| d3
| A4
| P4
| d4
| A5
| P5
| d5
| A6
| M6
| m6
| d6/A7
| M7
| m7
| d7
| A8
| P8
|-
| rowspan="2" | sharp raises,
major is wider
| '''D'''
| D#
| Eb
| '''E'''
| E#
| Ex/Fbb
| Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#
| Bx/Cbb
| Cb
| '''C'''
| C#
| Db
| '''D'''
|-
| P1
| A1
| d2
| m2
| M2
| A2/d3
| m3
| M3
| A3
| d4
| P4
| A4
| d5
| P5
| A5
| d6
| m6
| M6
| A6/d7
| m7
| M7
| A7
| d8
| P8
|}
24-edo contains 2 rings of 12-edo: an up/down-ring and a plain-ring. Enharmonic intervals: vvA1 and d2.
{| class="wikitable" style="text-align:center;"
| rowspan="2" | '''[[24-edo]]'''
sharp-2
| '''D'''
| ^D/vEb
| D#/Eb
| ^D#/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F
| F#/Gb
| vG
| '''G'''
| ^G/vAb
| G#/Ab
| ^G#/vA
| '''A'''
| etc.
|-
|P1
| ^1/vm2
| A1/m2
| ~2
| M2
| ^M2/vm3
| m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/~4
| A4/d5
| v5/~5
| P5
| etc.
|}

== Extremely Large EDOs ==
In theory, every edo can be notated with ups and downs only. For example, in 159-edo, 11/8 above C would be ^7F. But large exponents can be avoided if the edo is multi-ring (if the circle of 5ths doesn't include every note). Ups and downs are used within a ring, and lifts and drops (/ and \, see the [[pergen]] article) are used to label each ring. 159-edo has 3 rings of 53-edo. Many people who work with 159-edo are familiar with 53-edo, and can read the lifts and drops as small inflections of the familiar 53-edo notation. 11/8 becomes ^^/F. See the [[159edo notation#Ups-and-Downs-based notation|159edo notation]] page for the complete notation. In this situation, "arrow" refers to 1\53 and "slant" refers to 1\159.

Even if the edo isn't multi-ring, lifts and drops can still be used for single EDOsteps and ups and downs can be used for a group of EDOsteps. See [[311edo#Ups and downs notation]].

==Chords and Chord Progressions==

Chord names are based on jazz chord names. See Jim Aiken's book ''A Player's Guide to Chords & Harmony''. Alterations are enclosed in parentheses, additions never are. Examples:

* [[19edo Chord Names]]
* [[22edo Chord Names]]
* [[24edo Chord Names]]
* [[31edo Chord Names]]
* [[41edo Chord Names]]
* [[Kite Guitar Chord Shapes (downmajor tuning)]]

In perfect EDOs (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following list of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".

An arrow between the chord root and the chord type (e.g. C^m7) raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A mid-something chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord is affected: '''6th''' - root - '''3rd''' - 5th - '''7th''' - 9th - '''11th''' - 13th. Note that the 6th is affected, but the 13th is not.

The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in larger EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.

Every conventional chord can accept such a "global" arrow, with one exception: it's pointless for a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down-5", it means C(v5) = C E vG.

Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, '''never use lower case roman numerals''' for minor chords, because both vIIm and VIIm would be written vii.

The major chord and various alterations of it:
* C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")
* C ^E G = C^ = "C up" or "C upmajor"
* C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")
* C vvE G = Cvv = "C dud" or "C dudmajor" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 30, 37, 44, 51, etc. C^~ = "C upmid")
This table shows how altering the 3rd or the 5th affects the name of the triad. The conventional abbreviations for aug and dim are + and o. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.
{| class="wikitable" style="text-align:center;"
|+
!
!major
!minor
!sus4
!sus2
! colspan="2" |augmented
! colspan="2" |diminished
|-
!what's downed
!C E G
!C Eb G
!C F G
!C D G
! colspan="2" |C E G#
! colspan="2" |C Eb Gb
|-
!nothing
|C
|Cm
|C4
|C2
|Ca
|C+
|Cd
|Co
|-
!3rd
|Cv
|Cvm
|Cv4
|Cv2
|Cva
|Cv+
|Cvd
|Cvo
|-
!5th
|C(v5)
|Cm(v5)
|C4(v5)
|C2(v5)
|Ca(v5)
|C+(v5)
|Cd(v5)
|Co(v5)
|-
!3rd, 5th
|Cv(v5)
|Cvm(v5)
|Cv4(v5)
|Cv2(v5)
|Cva(v5)
|Cv+(v5)
|Cvd(v5)
|Cvo(v5)
|}
Many EDOs have notes between the major 3rd and the perfect 4th, creating triads impossible in 12-edo, such as:
* C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
* C E# G = C(A3) or C(#3) = "C aug-three" or "C sus-sharp-three"
* C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
* C D# G = C(A2) or C(#2) = "C aug-two" or "C sus-sharp-two"
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.

'''Sixth and seventh chords:'''

If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th.
* C E G Bb = C7 = "C seven"
* C vE G Bb = Cv,7 = "C down add-seven"
* C E G vBb = C,v7 = "C add down-seven"
* C vE G vBb = Cv7 = "C down seven"
All 7th chords follow this same pattern. Likewise, if a 6th is not a P4 or A4 above the 3rd, it's an "add-6" chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can both be replaced with "dim add-7", written Cd,7.

In the table below, if a chord is '''bolded''', the comma (the actual punctuation mark, not the interval) must be spoken as "add".
{| class="wikitable" style="text-align:center;"
|+
!
!maj7
!dom7
!min7
! colspan="3" |dim-add-7 or min7(b5) or half-dim
! colspan="2" |dim7
!maj6
!min6
|-
!what's downed
!C E G B
!C E G Bb
!C Eb G Bb
! colspan="3" |C Eb Gb Bb
! colspan="2" |C Eb Gb Bbb
!C E G A
!C Eb G A
|-
!nothing
|CM7
|C7
|Cm7
|'''Cd,7'''
|Cm7(b5)
|Cø
|Cd7
|Co7
|C6
|Cm6
|-
!3rd
|'''Cv,M7'''
|'''Cv,7'''
|'''Cvm,7'''
|'''Cvd,7'''
|'''Cvm,7(b5)'''
|Cø(v3)
|Cvd,d7
|Cvo,d7
|'''Cv,6'''
|'''Cvm,6'''
|-
!5th
|CM7(v5)
|C7(v5)
|Cm7(v5)
|Cd(v5)7
|Cm7(vb5)
|Cø(v5)
|Cd7(v5)
|Co7(v5)
|C6(v5)
|Cm6(v5)
|-
!6th/7th
|'''C,vM7'''
|'''C,v7'''
|Cmv7
|Cdv7
|Cmv7(b5)
|Cø(v7)
|Cdvd7
|Covd7
|'''C,v6'''
|Cmv6
|-
!3rd, 5th
|'''Cv,M7(v5)'''
|'''Cv,7(v5)'''
|'''Cvm,7(v5)'''
|Cvd(v5)7
|'''Cvm,7(vb5)'''
|Cø(v3v5)
|Cvd(v5)d7
|Cvo(v5)d7
|Cv(v5)6
|Cvm(v5)6
|-
!3rd, 6th/7th
|CvM7
|Cv7
|Cvm7
|Cvdv7
|Cvm7(b5)
|Cvø
|Cvd7
|Cvo7
|Cv6
|Cvm6
|-
!5th, 6th/7th
|C,vM7(v5)
|'''C,v7(v5)'''
|Cmv7(v5)
|Cd(v5)v7
|Cmv7(vb5)
|Cø(v5v7)
|Cd(v5)vd7
|Co(v5)vd7
|C(v5)v6
|Cm(v5)v6
|-
!3rd, 5th, 6th/7th
|CvM7(v5)
|Cv7(v5)
|Cvm7(v5)
|Cvd(v5)v7
|Cvm7(vb5)
|Cvø(v5)
|Cvd7(v5)
|Cvo7(v5)
|Cv6(v5)
|Cvm6(v5)
|}
Various unusual tetrads:
* C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")
* C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
* C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
* C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)
* C E G B# = C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"
* C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"
'''Ninth chords:'''

In '''bolded''' chords, the comma punctuation is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk '''*''', only the first comma is spoken as "add".
{| class="wikitable" style="text-align:center;"
|+
!
!add9
!maj9
!dom9
!min9
!dom7b9
!maj6/9
!min6/9
|-
!what's downed
!C E G D
!C E G B D
!C E G Bb D
!C Eb G Bb D
!C E G Bb Db
!C E G A D
!C Eb G A D
|-
!nothing
|'''C,9'''
|CM9
|C9
|Cm9
|C7b9
|C6,9
|Cm6,9
|-
!3rd
|'''Cv,9'''
|CM9(v3)
|C9(v3)
|Cm9(v3)
|'''Cv,7b9'''
|'''Cv,6,9 *'''
|'''Cvm,6,9 *'''
|-
!5th
|'''C,9(v5)'''
|CM9(v5)
|C9(v5)
|Cm9(v5)
|C7(v5)b9
|C6(v5)9
|Cm6,9(v5)
|-
!6th/7th
| ------
|CM9(v7)
|C9(v7)
|Cm9(v7)
|'''C,v7b9'''
|'''C,v6,9 *'''
|Cmv6,9
|-
!9th
|'''C,v9'''
|CM7v9
|C7v9
|Cm7v9
|C7vb9
|C6v9
|Cm6v9
|-
!3rd, 5th
|'''Cv,9(v5)'''
|CM9(v3v5)
|C9(v3v5)
|Cm9(v3v5)
|'''Cv,7b9(v5)'''
|Cv(v5)6,9
|Cvm(v5)6,9
|-
!3rd, 6th/7th
| ------
|CvM9
|Cv9
|Cvm9
|Cv7b9
|Cv6,9
|Cvm6,9
|-
!3rd, 9th
|Cv,v9
|'''Cv,M7v9''' or
CM7(v3)v9
|'''Cv,7v9''' or
C7(v3)v9
|'''Cvm,7v9''' or
Cm7(v3)v9
|'''Cv,7vb9''' or
C7(v3)vb9
|'''Cv,6v9''' or
C6(v3)v9
|'''Cvm,6v9''' or

Cm6(v3)v9
|-
!5th, 6th/7th
| ------
|CM9(v5v7)
|C9(v5v7)
|Cm9(v5v7)
|C(v5)v7b9
|C(v5)v6,9
|Cm(v5)v6,9
|-
!5th, 9th
|C(v5)v9
|CM7(v5)v9
|C7(v5)v9
|Cm7(v5)v9
|C7(v5)vb9
|C6(v5)v9
|Cm6(v5)v9
|-
!6th/7th, 9th
| ------
|'''C,vM7v9'''
|'''C,v7v9'''
|Cmv7v9
|'''C,v7vb9'''
|'''C,v6v9'''
|Cmv6v9
|-
!3rd, 5th, 6th/7th
| ------
|CvM9(v5)
|Cv9(v5)
|Cvm9(v5)
|Cv7(v5)b9
|Cv6(v5)9
|Cvm6(v5)9
|-
!3rd, 5th, 9th
|Cv(v5)v9
|Cv(v5)M7v9 or
CM7(v3v5)v9
|Cv(v5)7v9 or
C7(v3v5)v9
|Cvm(v5)7v9 or
Cm7(v3v5)v9
|Cv(v5)7vb9 or
C7(v3v5)b9
|Cv(v5)6v9 or
C6(v3v5)v9
|Cvm(v5)6v9 or
Cm6(v3v5)v9
|-
!3rd, 6th/7th, 9th
| ------
|CvM7v9
|Cv7v9
|Cvm7v9
|Cv7vb9
|Cv6v9
|Cvm6v9
|-
!5th, 6th/7th, 9th
| ------
|C(v5)vM7v9
|C(v5)v7v9
|Cm(v5)v7v9
|C(v5)v7vb9
|C(v5)v6v9
|Cm(v5)v6v9
|-
!3rd, 5th, 6th/7th, 9th
| ------
|CvM7(v5)v9
|Cv7(v5)v9
|Cvm7(v5)v9
|Cv7(v5)vb9
|Cv6(v5)v9
|Cvm6(v5)v9
|}
==Cross-EDO considerations==

In 22-edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19-edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22-edo major chords sound ru (7-under) and 19-edo major chords sound yo (5-over).

A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any EDO, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22-edo, the chord that you need sounds like a ru chord.

In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is downmajor, it doesn't work in that progression.

Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over, thus 7/4). In 19-edo, it sounds gu (5-under, thus 9/5).

==Ups and downs solfege==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down. See [[Uniform solfege|Uniform Solfege]].
== See also ==
* [[Alternative symbols for ups and downs notation]]

[[Category:Ups and Downs Notation| ]] 
[[Category:Notation]]

{{todo|intro}}

19edo

2024-07-18T17:59:40Z

YoVariable: Changed 5/3 from being a "minor 6th" to a "major 6th"

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{EDO intro|19}}
== Theory ==
=== History ===
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed [[1/3-comma meantone|{{frac|1|3}}-comma meantone]], in which the fifth is 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

=== As an approximation of other temperaments ===
19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth, though [[46edo]] is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy, and is the fifth (after 12) [[zeta integral edo]]. It is less successful in the [[7-limit]] (but still better than 12et), as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles, and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5 and 7 are not only much farther from just than they are in 19, but fairly sharp already. 19edo's [[negri]], [[sensi]] and [[semaphore]] scales have many 13-limit chords. (You can think of the sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to employ [[octave stretching]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29 cents, and a step size of between 63.2 and 63.4 cents would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57 cents, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. A more extreme option would be [[11edf]], which has octaves stretched by 12.47 cents.

=== As a means of extending harmony ===
Because 19edo allows for more blended, consonant harmonies than 12edo does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 cents off [[23/16]].

In addition, [[Joseph Yasser]] talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]] (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Prime harmonics ===
{{Harmonics in equal|19}}

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
! [[Degree]]
! [[Cent]]s
! [[Interval region|Interval Region]]
! Approximated [[Just intonation|JI]] Intervals*
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
| 0
| 0.00
| Unison (prime)
| [[1/1]]
| Do
| unison
| P1
|-
| 1
| 63.16
| Augmented unison
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| super unison, subminor second
| S1, sm2
|-
| 2
| 126.32
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| minor second
| m2
|-
| 3
| 189.47
| Major second
| [[9/8]], [[10/9]]
| Re
| major second
| M2
|-
| 4
| 252.63
| Diminished third
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.79
| Minor third
| [[6/5]]
| Me
| minor third
| m3
|-
| 6
| 378.95
| Major third
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| major third
| M3
|-
| 7
| 442.11
| Augmented third
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.26
| Perfect fourth
| [[4/3]], [[75/56]]
| Fa
| perfect fourth
| P4
|-
| 9
| 568.42
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| augmented fourth
| A4
|-
| 10
| 631.58
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
| Se
| diminished fifth
| d5
|-
| 11
| 694.74
| Perfect fifth
| [[3/2]], [[112/75]]
| So
| perfect fifth
| P5
|-
| 12
| 757.89
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| super fifth, subminor 6th
| S5, sm6
|-
| 13
| 821.05
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
| Le
| minor sixth
| m6
|-
| 14
| 884.21
| Major sixth
| [[5/3]]
| La
| major sixth
| M6
|-
| 15
| 947.37
| Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.53
| Minor seventh
| [[9/5]], [[16/9]]
| Te
| minor seventh
| m7
|-
| 17
| 1073.68
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| major seventh
| M7
|-
| 18
| 1136.84
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.00
| Octave
| [[2/1]]
| Do
| octave
| P8
|}
<nowiki>*</nowiki> based on treating 19edo as a 2.3.5.7.13 subgroup temperament; other approaches are possible.

=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center"
! Quality
! [[Color name|Color Name]]
! Monzo Format
! Examples
|-
| diminished
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| rowspan="2" | minor
| fourthward wa
| (a, b), b < -1
| 32/27, 16/9
|-
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| rowspan="2" | major
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| fifthward wa
| (a, b), b > 1
| 9/8, 27/16
|-
| augmented
| ru
| (a, b, 0, -1)
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3), Cmin(♭3), C(d3)
| C subminor, C minor flat-three, C diminished-three
|-
| gu
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm, Cmin
| C minor
|-
| yo
| 4:5:6
| 0–6–11
| C–E–G
| C, Cmaj
| C, C major
|-
| ru
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3), Cmaj(♯3), C(A3)
| C supermajor, C major sharp-three, C augmented-three
|-
|
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| C(h7), Cadd(d7), Cmaj(add(d7))
| C harmonic 7, C (major) add dim-seven
|-
|
| 1/(4:5:6:7) = 1:6/5:3/2:12/7
| 0–5–11–15
| C–E♭–G–A♯
| Cm(♯6), Cm(A6), Cm(add(♯6)), Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios.

For a more complete list, see [[19edo Chord Names]] and [[Ups and downs notation #Chords and Chord Progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfege, or sargam. Note that D# and Eb are two different notes.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 19edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard Notation]]
|-
! [[5L 2s|Diatonic Interval Names]]
! Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.16
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.32
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.47
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.63
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.79
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.95
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.11
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.26
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.42
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.58
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.74
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.89
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.05
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.21
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.37
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.53
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.68
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.84
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.00
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+style="white-space:nowrap" | Dodecatonic Notation of 19edo
! [[Degree]]
! [[Cent]]s
! Interval Names
|-
| 0
| 0.00
| P1
|-
| 1
| 63.16
| A1, m2
|-
| 2
| 126.32
| M2, m3
|-
| 3
| 189.47
| M3
|-
| 4
| 252.63
| m4, A3
|-
| 5
| 315.79
| M4, m5
|-
| 6
| 378.95
| M5
|-
| 7
| 442.11
| A5, d6
|-
| 8
| 505.26
| P6
|-
| 9
| 568.42
| A6, m7
|-
| 10
| 631.58
| M7, d8
|-
| 11
| 694.74
| P8
|-
| 12
| 757.89
| A8, m9
|-
| 13
| 821.05
| M9, m10
|-
| 14
| 884.21
| M10
|-
| 15
| 947.37
| m11, A10
|-
| 16
| 1010.53
| M11, m12
|-
| 17
| 1073.68
| M12
|-
| 18
| 1136.84
| A12, d13
|-
| 19
| 1200.00
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| [{{val| 19 30 }}]
| +2.28
| 2.28
| 3.61
|-
| 2.3.5
| 81/80, 3125/3072
| [{{val| 19 30 44 }}]
| +2.58
| 1.91
| 3.02
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| [{{val| 19 30 44 53 }}]
| +3.85
| 2.76
| 4.35
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| [{{val| 19 30 44 53 70 }}]
| +4.14
| 2.53
| 3.99
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| [{{val| 19 30 44 53 70 86 }}]
| +3.32
| 2.93
| 4.64
|}

19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – ''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.

19et is prominent in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|13|18.5|19.5}}

=== Commas ===
19et [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| 19-comma
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references/>

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic-kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! MOSes
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[rhinocerus]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] / [[pycnic]] [[Triton]]
|}

== Scales ==
=== MOS scales ===
==== Octave-equivalent mosses ====
* [[meantone]] pentatonic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2
* Meantone harmonic major: 3 3 2 3 2 4 2
* chromatic octave species - Meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
*[[Pinetone#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
*[[Pinetone#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
*[[Pinetone#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Arto and Tendo Theory]]
* [[Lumatone mapping for 19edo]]

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

[[Category:19-tone scales]]
[[Category:Golden meantone]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Kleismic]]
[[Category:Teentuning]]
[[Category:Historical]]

22edo

2024-07-12T20:01:23Z

YoVariable: Moved 22edo "Notation" category to below the "Intervals" category

{{interwiki
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

==Theory==
=== Prime harmonics===
{{Harmonics in equal|22|columns=11}}

===History===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

===Overview to JI approximation quality===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

===Subsets and supersets===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

==Intervals==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
!Degree
!Cents
!Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
!Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
| Km2
| KEb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
|aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
| kM2
|kE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
| major 2nd
|M2
|E
| major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
| minor 3rd
|m3
|F
| minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| ^m3
|^F
|classic minor 3rd
| Km3
|KF
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
| vM3
| vF#
|classic major 3rd
| kM3
| kF#
|[[File:0-381.818c_22edo.mp3]]
|-
|8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
| major 3rd
|M3
|F#
| major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|[[File:0-545.455c_11edo.mp3]]
|-
|11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th comma-wide diminished 5th
|kA4 Kd5
|kG#, KAb
|[[File:0-600.000c_2edo.mp3]]
|-
|12
|654.545
|[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|comma-narrow 5th
|k5
|kA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
| minor 6th
|m6
|Bb
| minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
| ^m6
| ^Bb
|classic minor 6th
| Km6
| KBb
|[[File:0-818.182c_22edo.mp3]]
|-
|16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| vM6
|vB
|classic major 6th
| kM6
|kB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
| major 6th
|M6
|B
| major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
| minor 7th
|m7
|C
| minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
| Km7
|kC
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
| kM7
| kC#
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
|M7, v8
|C#, vD
| major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==Notation==
===Ups and Downs Notation===
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that Eb and D# are different notes and that Eb is lower in pitch than D#.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 54.545
| Minor second (m2) Up-unison (^1)
| Eb ^D
|-
| 2
| 109.091
| Upminor 2nd (^m2) Down-augmented unison (vA1) Diminished third (d3)
| ^Eb vD# Fb
|-
| 3
| 163.636
| Downmajor second (vM2) Augmented unison (A1)
| vE D#
|-
| 4
| 218.182
| '''Major second (M2)''' Up-augmented unison (^A1) Downminor third (vm3)
| '''E''' ^D# vF
|-
| 5
| 272.727
| Upmajor second (^M2) '''Minor third (m3)'''
| ^E '''F'''
|-
| 6
| 327.273
| '''Upminor third (^m3)''' Diminished fourth (d4)
| '''^F''' Gb
|-
| 7
| 381.818
| '''Downmajor third (vM3)''' Augmented second (A2) Up-diminished fourth (^d4)
| '''vF#''' E# ^Gb
|-
| 8
| 436.364
| '''Major third (M3)''' Up-augmented second (^A2) Down-fourth (v4)
| '''F#''' ^E# vG
|-
| 9
| 490.909
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 10
| 545.455
| Up-fourth (^4) Diminished fifth (d5)
| ^G Ab
|-
| 11
| 600.000
| Down-augmented fourth (vA4) Up-diminished fifth (^d5)
| vG# ^Ab
|-
| 12
| 654.545
| Augmented fourth (A4) Down-fifth (v5)
| G# vA
|-
| 13
| 709.091
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 14
| 763.636
| Up-fifth (^5) Minor sixth (m6)
| ^A Bb
|-
| 15
| 818.182
| Down-augmented fifth (vA5) Upminor sixth (^m6)
| vA# ^Bb
|-
| 16
| 872.727
| Augmented fifth (A5) '''Downmajor sixth (vM6)'''
| A# '''vB'''
|-
| 17
| 927.273
| '''Major sixth (M6)''' Up-augmented fifth (^A5) Downminor seventh (vm7)
| '''B''' ^A# vC
|-
| 18
| 981.818
| '''Minor seventh (m7)''' Upmajor sixth (^M6) Down-diminished octave (vd8)
| '''C''' ^B vDb
|-
| 19
| 1036.364
| '''Upminor seventh (^m7)''' Diminished octave (d8)
| '''^C''' Db
|-
| 20
| 1090.909
| Downmajor seventh (vM7) Up-diminished octave (^d8) Augmented sixth (A6)
| vC# ^Db B#
|-
| 21
| 1145.455
| Major seventh (M7) Down-octave (v8)
| C# vD
|-
| 22
| 1200.000
| '''Perfect octave (P8)'''
| '''D'''
|}

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:

{{Sharpness-sharp3}}

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

=== 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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
|0
|Natural Unison
|1
|perfect unison
|P1
| D
|perfect unison
|P1
|D
|natural 1st
|N1
| C
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|-
|1
|55
|s-minor second
|sm2
|aug unison
|A1
|D#
|aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|up-unison, minor 2nd
| ^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
|3
| 164
| p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|major 2nd
|M2
|E
|major 2nd
|M2
|E
|-
|5
|273
|s-minor third
|sm3
|dim 3rd
|d3
|Fb
|major sub3rd
| Ms3
|F
|sharp 3rd
| s3
|D#
|minor 3rd
|m3
|F
|minor 3rd
|m3
| F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
|f4
|εb
|upminor 3rd
| ^m3
| ^F
| classic minor 3rd
|Km3
|KF
|-
| 7
|382
|p-Major third
| pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
|natural 4th
|N4
| ε
|downmajor 3rd
|vM3
| vF#
| classic major 3rd
|kM3
|kF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
| dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
|s4, f5
|ε#, Eb
|major 3rd
|M3
|F#
|major 3rd
|M3
|F#
|-
| 9
| 491
|Natural Fourth
|4, N4
|minor 4th
|m4
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|10
|545
| p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
|M4
|G#
| aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
|A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
|Gx, Abb
|natural 6th
| N6
|γ
| downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
|m5
|Ab
|dim 5thoid
|d5d
|Ab
| sharp 6th, flat 7th
|s6, f7
|γ#, Gb
|aug 4th, down-5th
|A4, v5
|G#, vA
| comma-narrow 5th
|k5
|kA
|-
|13
| 709
|Natural Fifth
|5, N5
|major 5th
|M5
|A
|perfect 5thoid
|P5d
|A
|natural 7th
|N7
|G
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|-
|14
|764
| s-minor sixth
|sm6
|aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
|G#
| minor 6th
|m6
|Bb
|minor 6th
| m6
| Bb
|-
| 15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
| double-aug 5thoid, double-dim sub7th
|AA5d, dds7
| Ax, Cb3
|flat 8th
|f8
|αb
|upminor 6th
|^m6
|^Bb
| classic minor 6th
| Km6
|KBb
|-
|16
|873
|p-Major sixth
|pM6
|major 6th
|M6
|B
| dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
| downmajor 6th
|vM6
|vB
|classic major 6th
|kM6
|kB
|-
| 17
|927
| s-Major sixth
|sM6
|aug 6th
|A6
|B#
|minor sub7th
|ms7
|Cb
| sharp 8th, flat 9th
|s8, f9
|α#, Ab
|major 6th
|M6
|B
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
|m7
| dim 7th
|d7
|Cb
|major sub7th
| Ms7
|C
|natural 9th
| N9
|A
|minor 7th
|m7
| C
| minor 7th
| m7
|C
|-
|19
|1036
| p-Major seventh
| pM7
|perfect 7th
| P7
|C
| aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
|1091
|p-Augmented seventh
|pA7
|aug 7th
|A7
|C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
|N10
| β
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
|21
|1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
| sharp 10th
|s10
|β#, Cb
|major 7th, down 8ve
|M7, v8
|C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
|22
|1200
| Octave
|8
|perfect octave
| P8
|D
|perfect octave
|P8
|D
|natural 11th
|N11
|C
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|}

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|22}}

==Defining features ==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
|−2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| −0.86
|2.70
|4.94
|-
|2.3.5.7
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
|−1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps ===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
| 7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
|Trizo-agu
| Hemimage comma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell comma
|-
|7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry comma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
| Kalisma
|-
|13
|[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
|26.84
|Thoyo
|Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
|Tholuru
|Negustma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|13
|[[31213/31104]]
|{{monzo| -7 -5 0 4 0 1 }}
|6.06
|Thoquadzo
|Praveensma
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
|Twizzler comma
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
|1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
|[[Porcupine]]
|-
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

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

{| class="wikitable center-all"
|-
!Quality
![[Color name]]
![[Monzo]] Format
!Examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
|7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
|upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|Color of the 3rd]]
!JI Chord
!Notes as edosteps
!Notes of C chord
!Written name
!Spoken name
|-
|zo
|6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
|gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
|C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
|14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

==Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

==References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-07-11T12:39:34Z

YoVariable: /* Ups and Downs Notation */Fixed typo with Augmented fourth (A4)

{{interwiki
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

==Theory==
=== Prime harmonics===
{{Harmonics in equal|22|columns=11}}

===History===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

===Overview to JI approximation quality===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

===Subsets and supersets===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

==Intervals==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
!Degree
!Cents
!Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
!Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
| Km2
| KEb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
|aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
| kM2
|kE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
| major 2nd
|M2
|E
| major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
| minor 3rd
|m3
|F
| minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| ^m3
|^F
|classic minor 3rd
| Km3
|KF
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
| vM3
| vF#
|classic major 3rd
| kM3
| kF#
|[[File:0-381.818c_22edo.mp3]]
|-
|8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
| major 3rd
|M3
|F#
| major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|[[File:0-545.455c_11edo.mp3]]
|-
|11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th comma-wide diminished 5th
|kA4 Kd5
|kG#, KAb
|[[File:0-600.000c_2edo.mp3]]
|-
|12
|654.545
|[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|comma-narrow 5th
|k5
|kA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
| minor 6th
|m6
|Bb
| minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
| ^m6
| ^Bb
|classic minor 6th
| Km6
| KBb
|[[File:0-818.182c_22edo.mp3]]
|-
|16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| vM6
|vB
|classic major 6th
| kM6
|kB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
| major 6th
|M6
|B
| major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
| minor 7th
|m7
|C
| minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
| Km7
|kC
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
| kM7
| kC#
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
|M7, v8
|C#, vD
| major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==Notation==
===Ups and Downs Notation===
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that Eb and D# are different notes and that Eb is lower in pitch than D#.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 54.545
| Minor second (m2) Up-unison (^1)
| Eb ^D
|-
| 2
| 109.091
| Upminor 2nd (^m2) Down-augmented unison (vA1) Diminished third (d3)
| ^Eb vD# Fb
|-
| 3
| 163.636
| Downmajor second (vM2) Augmented unison (A1)
| vE D#
|-
| 4
| 218.182
| '''Major second (M2)''' Up-augmented unison (^A1) Downminor third (vm3)
| '''E''' ^D# vF
|-
| 5
| 272.727
| Upmajor second (^M2) '''Minor third (m3)'''
| ^E '''F'''
|-
| 6
| 327.273
| '''Upminor third (^m3)''' Diminished fourth (d4)
| '''^F''' Gb
|-
| 7
| 381.818
| '''Downmajor third (vM3)''' Augmented second (A2) Up-diminished fourth (^d4)
| '''vF#''' E# ^Gb
|-
| 8
| 436.364
| '''Major third (M3)''' Up-augmented second (^A2) Down-fourth (v4)
| '''F#''' ^E# vG
|-
| 9
| 490.909
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 10
| 545.455
| Up-fourth (^4) Diminished fifth (d5)
| ^G Ab
|-
| 11
| 600.000
| Down-augmented fourth (vA4) Up-diminished fifth (^d5)
| vG# ^Ab
|-
| 12
| 654.545
| Augmented fourth (A4) Down-fifth (v5)
| G# vA
|-
| 13
| 709.091
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 14
| 763.636
| Up-fifth (^5) Minor sixth (m6)
| ^A Bb
|-
| 15
| 818.182
| Down-augmented fifth (vA5) Upminor sixth (^m6)
| vA# ^Bb
|-
| 16
| 872.727
| Augmented fifth (A5) '''Downmajor sixth (vM6)'''
| A# '''vB'''
|-
| 17
| 927.273
| '''Major sixth (M6)''' Up-augmented fifth (^A5) Downminor seventh (vm7)
| '''B''' ^A# vC
|-
| 18
| 981.818
| '''Minor seventh (m7)''' Upmajor sixth (^M6) Down-diminished octave (vd8)
| '''C''' ^B vDb
|-
| 19
| 1036.364
| '''Upminor seventh (^m7)''' Diminished octave (d8)
| '''^C''' Db
|-
| 20
| 1090.909
| Downmajor seventh (vM7) Up-diminished octave (^d8) Augmented sixth (A6)
| vC# ^Db B#
|-
| 21
| 1145.455
| Major seventh (M7) Down-octave (v8)
| C# vD
|-
| 22
| 1200.000
| '''Perfect octave (P8)'''
| '''D'''
|}

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|22}}

==Defining features ==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
|−2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| −0.86
|2.70
|4.94
|-
|2.3.5.7
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
|−1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps ===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
| 7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
|Trizo-agu
| Hemimage comma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell comma
|-
|7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry comma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
| Kalisma
|-
|13
|[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
|26.84
|Thoyo
|Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
|Tholuru
|Negustma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|13
|[[31213/31104]]
|{{monzo| -7 -5 0 4 0 1 }}
|6.06
|Thoquadzo
|Praveensma
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
|Twizzler comma
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
|1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
|[[Porcupine]]
|-
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

==Notation==
===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

=== 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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Ups and Downs Notation===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:

{{Sharpness-sharp3}}

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
|0
|Natural Unison
|1
|perfect unison
|P1
| D
|perfect unison
|P1
|D
|natural 1st
|N1
| C
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|-
|1
|55
|s-minor second
|sm2
|aug unison
|A1
|D#
|aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|up-unison, minor 2nd
| ^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
|3
| 164
| p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|major 2nd
|M2
|E
|major 2nd
|M2
|E
|-
|5
|273
|s-minor third
|sm3
|dim 3rd
|d3
|Fb
|major sub3rd
| Ms3
|F
|sharp 3rd
| s3
|D#
|minor 3rd
|m3
|F
|minor 3rd
|m3
| F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
|f4
|εb
|upminor 3rd
| ^m3
| ^F
| classic minor 3rd
|Km3
|KF
|-
| 7
|382
|p-Major third
| pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
|natural 4th
|N4
| ε
|downmajor 3rd
|vM3
| vF#
| classic major 3rd
|kM3
|kF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
| dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
|s4, f5
|ε#, Eb
|major 3rd
|M3
|F#
|major 3rd
|M3
|F#
|-
| 9
| 491
|Natural Fourth
|4, N4
|minor 4th
|m4
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|10
|545
| p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
|M4
|G#
| aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
|A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
|Gx, Abb
|natural 6th
| N6
|γ
| downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
|m5
|Ab
|dim 5thoid
|d5d
|Ab
| sharp 6th, flat 7th
|s6, f7
|γ#, Gb
|aug 4th, down-5th
|A4, v5
|G#, vA
| comma-narrow 5th
|k5
|kA
|-
|13
| 709
|Natural Fifth
|5, N5
|major 5th
|M5
|A
|perfect 5thoid
|P5d
|A
|natural 7th
|N7
|G
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|-
|14
|764
| s-minor sixth
|sm6
|aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
|G#
| minor 6th
|m6
|Bb
|minor 6th
| m6
| Bb
|-
| 15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
| double-aug 5thoid, double-dim sub7th
|AA5d, dds7
| Ax, Cb3
|flat 8th
|f8
|αb
|upminor 6th
|^m6
|^Bb
| classic minor 6th
| Km6
|KBb
|-
|16
|873
|p-Major sixth
|pM6
|major 6th
|M6
|B
| dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
| downmajor 6th
|vM6
|vB
|classic major 6th
|kM6
|kB
|-
| 17
|927
| s-Major sixth
|sM6
|aug 6th
|A6
|B#
|minor sub7th
|ms7
|Cb
| sharp 8th, flat 9th
|s8, f9
|α#, Ab
|major 6th
|M6
|B
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
|m7
| dim 7th
|d7
|Cb
|major sub7th
| Ms7
|C
|natural 9th
| N9
|A
|minor 7th
|m7
| C
| minor 7th
| m7
|C
|-
|19
|1036
| p-Major seventh
| pM7
|perfect 7th
| P7
|C
| aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
|1091
|p-Augmented seventh
|pA7
|aug 7th
|A7
|C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
|N10
| β
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
|21
|1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
| sharp 10th
|s10
|β#, Cb
|major 7th, down 8ve
|M7, v8
|C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
|22
|1200
| Octave
|8
|perfect octave
| P8
|D
|perfect octave
|P8
|D
|natural 11th
|N11
|C
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|}

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

{| class="wikitable center-all"
|-
!Quality
![[Color name]]
![[Monzo]] Format
!Examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
|7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
|upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|Color of the 3rd]]
!JI Chord
!Notes as edosteps
!Notes of C chord
!Written name
!Spoken name
|-
|zo
|6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
|gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
|C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
|14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

==Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

==References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-07-11T02:28:00Z

YoVariable: /* Ups and Downs Notation */ Adjusted text spacing of new 22edo notation section

{{interwiki
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

==Theory==
=== Prime harmonics===
{{Harmonics in equal|22|columns=11}}

===History===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

===Overview to JI approximation quality===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

===Subsets and supersets===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

==Intervals==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
!Degree
!Cents
!Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
!Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
| Km2
| KEb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
|aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
| kM2
|kE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
| major 2nd
|M2
|E
| major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
| minor 3rd
|m3
|F
| minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| ^m3
|^F
|classic minor 3rd
| Km3
|KF
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
| vM3
| vF#
|classic major 3rd
| kM3
| kF#
|[[File:0-381.818c_22edo.mp3]]
|-
|8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
| major 3rd
|M3
|F#
| major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|[[File:0-545.455c_11edo.mp3]]
|-
|11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th comma-wide diminished 5th
|kA4 Kd5
|kG#, KAb
|[[File:0-600.000c_2edo.mp3]]
|-
|12
|654.545
|[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|comma-narrow 5th
|k5
|kA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
| minor 6th
|m6
|Bb
| minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
| ^m6
| ^Bb
|classic minor 6th
| Km6
| KBb
|[[File:0-818.182c_22edo.mp3]]
|-
|16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| vM6
|vB
|classic major 6th
| kM6
|kB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
| major 6th
|M6
|B
| major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
| minor 7th
|m7
|C
| minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
| Km7
|kC
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
| kM7
| kC#
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
|M7, v8
|C#, vD
| major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==Notation==
===Ups and Downs Notation===
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that Eb and D# are different notes and that Eb is lower in pitch than D#.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 54.545
| Minor second (m2) Up-unison (^1)
| Eb ^D
|-
| 2
| 109.091
| Upminor 2nd (^m2) Down-augmented unison (vA1) Diminished third (d3)
| ^Eb vD# Fb
|-
| 3
| 163.636
| Downmajor second (vM2) Augmented unison (A1)
| vE D#
|-
| 4
| 218.182
| '''Major second (M2)''' Up-augmented unison (^A1) Downminor third (vm3)
| '''E''' ^D# vF
|-
| 5
| 272.727
| Upmajor second (^M2) '''Minor third (m3)'''
| ^E '''F'''
|-
| 6
| 327.273
| '''Upminor third (^m3)''' Diminished fourth (d4)
| '''^F''' Gb
|-
| 7
| 381.818
| '''Downmajor third (vM3)''' Augmented second (A2) Up-diminished fourth (^d4)
| '''vF#''' E# ^Gb
|-
| 8
| 436.364
| '''Major third (M3)''' Up-augmented second (^A2) Down-fourth (v4)
| '''F#''' ^E# vG
|-
| 9
| 490.909
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 10
| 545.455
| Up-fourth (^4) Diminished fifth (d5)
| ^G Ab
|-
| 11
| 600.000
| Down-augmented fourth (vA4) Up-diminished fifth (^d5)
| vG# ^Ab
|-
| 12
| 654.545
| Augmented fourth (A5) Down-fifth (v5)
| G# vA
|-
| 13
| 709.091
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 14
| 763.636
| Up-fifth (^5) Minor sixth (m6)
| ^A Bb
|-
| 15
| 818.182
| Down-augmented fifth (vA5) Upminor sixth (^m6)
| vA# ^Bb
|-
| 16
| 872.727
| Augmented fifth (A5) '''Downmajor sixth (vM6)'''
| A# '''vB'''
|-
| 17
| 927.273
| '''Major sixth (M6)''' Up-augmented fifth (^A5) Downminor seventh (vm7)
| '''B''' ^A# vC
|-
| 18
| 981.818
| '''Minor seventh (m7)''' Upmajor sixth (^M6) Down-diminished octave (vd8)
| '''C''' ^B vDb
|-
| 19
| 1036.364
| '''Upminor seventh (^m7)''' Diminished octave (d8)
| '''^C''' Db
|-
| 20
| 1090.909
| Downmajor seventh (vM7) Up-diminished octave (^d8) Augmented sixth (A6)
| vC# ^Db B#
|-
| 21
| 1145.455
| Major seventh (M7) Down-octave (v8)
| C# vD
|-
| 22
| 1200.000
| '''Perfect octave (P8)'''
| '''D'''
|}

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|22}}

==Defining features ==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
|−2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| −0.86
|2.70
|4.94
|-
|2.3.5.7
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
|−1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps ===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
| 7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
|Trizo-agu
| Hemimage comma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell comma
|-
|7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry comma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
| Kalisma
|-
|13
|[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
|26.84
|Thoyo
|Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
|Tholuru
|Negustma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|13
|[[31213/31104]]
|{{monzo| -7 -5 0 4 0 1 }}
|6.06
|Thoquadzo
|Praveensma
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
|Twizzler comma
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
|1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
|[[Porcupine]]
|-
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

==Notation==
===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

=== 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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Ups and Downs Notation===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:

{{Sharpness-sharp3}}

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
|0
|Natural Unison
|1
|perfect unison
|P1
| D
|perfect unison
|P1
|D
|natural 1st
|N1
| C
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|-
|1
|55
|s-minor second
|sm2
|aug unison
|A1
|D#
|aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|up-unison, minor 2nd
| ^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
|3
| 164
| p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|major 2nd
|M2
|E
|major 2nd
|M2
|E
|-
|5
|273
|s-minor third
|sm3
|dim 3rd
|d3
|Fb
|major sub3rd
| Ms3
|F
|sharp 3rd
| s3
|D#
|minor 3rd
|m3
|F
|minor 3rd
|m3
| F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
|f4
|εb
|upminor 3rd
| ^m3
| ^F
| classic minor 3rd
|Km3
|KF
|-
| 7
|382
|p-Major third
| pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
|natural 4th
|N4
| ε
|downmajor 3rd
|vM3
| vF#
| classic major 3rd
|kM3
|kF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
| dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
|s4, f5
|ε#, Eb
|major 3rd
|M3
|F#
|major 3rd
|M3
|F#
|-
| 9
| 491
|Natural Fourth
|4, N4
|minor 4th
|m4
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|10
|545
| p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
|M4
|G#
| aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
|A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
|Gx, Abb
|natural 6th
| N6
|γ
| downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
|m5
|Ab
|dim 5thoid
|d5d
|Ab
| sharp 6th, flat 7th
|s6, f7
|γ#, Gb
|aug 4th, down-5th
|A4, v5
|G#, vA
| comma-narrow 5th
|k5
|kA
|-
|13
| 709
|Natural Fifth
|5, N5
|major 5th
|M5
|A
|perfect 5thoid
|P5d
|A
|natural 7th
|N7
|G
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|-
|14
|764
| s-minor sixth
|sm6
|aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
|G#
| minor 6th
|m6
|Bb
|minor 6th
| m6
| Bb
|-
| 15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
| double-aug 5thoid, double-dim sub7th
|AA5d, dds7
| Ax, Cb3
|flat 8th
|f8
|αb
|upminor 6th
|^m6
|^Bb
| classic minor 6th
| Km6
|KBb
|-
|16
|873
|p-Major sixth
|pM6
|major 6th
|M6
|B
| dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
| downmajor 6th
|vM6
|vB
|classic major 6th
|kM6
|kB
|-
| 17
|927
| s-Major sixth
|sM6
|aug 6th
|A6
|B#
|minor sub7th
|ms7
|Cb
| sharp 8th, flat 9th
|s8, f9
|α#, Ab
|major 6th
|M6
|B
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
|m7
| dim 7th
|d7
|Cb
|major sub7th
| Ms7
|C
|natural 9th
| N9
|A
|minor 7th
|m7
| C
| minor 7th
| m7
|C
|-
|19
|1036
| p-Major seventh
| pM7
|perfect 7th
| P7
|C
| aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
|1091
|p-Augmented seventh
|pA7
|aug 7th
|A7
|C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
|N10
| β
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
|21
|1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
| sharp 10th
|s10
|β#, Cb
|major 7th, down 8ve
|M7, v8
|C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
|22
|1200
| Octave
|8
|perfect octave
| P8
|D
|perfect octave
|P8
|D
|natural 11th
|N11
|C
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|}

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

{| class="wikitable center-all"
|-
!Quality
![[Color name]]
![[Monzo]] Format
!Examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
|7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
|upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|Color of the 3rd]]
!JI Chord
!Notes as edosteps
!Notes of C chord
!Written name
!Spoken name
|-
|zo
|6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
|gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
|C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
|14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

==Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

==References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-07-10T20:37:56Z

YoVariable: /* Ups and Downs Notation */ (This section was created to clarify note spellings using Ups and Downs even though there is a separate notation section towards the bottom of the page).

{{interwiki
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

==Theory==
=== Prime harmonics===
{{Harmonics in equal|22|columns=11}}

===History===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

===Overview to JI approximation quality===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

===Subsets and supersets===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

==Intervals==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
!Degree
!Cents
!Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
!Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
| Km2
| KEb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
|aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
| kM2
|kE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
| major 2nd
|M2
|E
| major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
| minor 3rd
|m3
|F
| minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| ^m3
|^F
|classic minor 3rd
| Km3
|KF
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
| vM3
| vF#
|classic major 3rd
| kM3
| kF#
|[[File:0-381.818c_22edo.mp3]]
|-
|8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
| major 3rd
|M3
|F#
| major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|[[File:0-545.455c_11edo.mp3]]
|-
|11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th comma-wide diminished 5th
|kA4 Kd5
|kG#, KAb
|[[File:0-600.000c_2edo.mp3]]
|-
|12
|654.545
|[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|comma-narrow 5th
|k5
|kA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
| minor 6th
|m6
|Bb
| minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
| ^m6
| ^Bb
|classic minor 6th
| Km6
| KBb
|[[File:0-818.182c_22edo.mp3]]
|-
|16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| vM6
|vB
|classic major 6th
| kM6
|kB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
| major 6th
|M6
|B
| major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
| minor 7th
|m7
|C
| minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
| Km7
|kC
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
| kM7
| kC#
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
|M7, v8
|C#, vD
| major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==Notation==
===Ups and Downs Notation===
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that Eb and D# are different notes and that Eb is lower in pitch than D#.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names on D
|-
|0
|0.00
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 54.545
|Minor second (m2) Up-unison (^1)
|Eb ^D
|-
|2
|109.091
|Upminor 2nd (^m2) Down-augmented unison (vA1)
Diminished third (d3)
|^Eb vD#
Fb
|-
|3
|163.636
|Downmajor second (vM2) Augmented unison (A1)
|vE D#
|-
|4
|218.182
| '''Major second (M2)'''
Up-augmented unison (^A1) Downminor third (vm3)
|'''E'''
^D# vF
|-
|5
|272.727
|Upmajor second (^M2) '''Minor third (m3)'''
|^E '''F'''
|-
| 6
|327.273
|'''Upminor third (^m3)''' Diminished fourth (d4)
|'''^F''' Gb
|-
|7
|381.818
|'''Downmajor third (vM3)''' Augmented second (A2)
Up-diminished fourth (^d4)
|'''vF#''' E#
^Gb
|-
|8
| 436.364
|'''Major third (M3)'''
Up-augmented second (^A2)

Down-fourth (v4)
|'''F#'''
^E#

vG
|-
|9
|490.909
| '''Perfect fourth (P4)'''
|'''G'''
|-
|10
|545.455
|Up-fourth (^4)
Diminished fifth (d5)
|^G Ab
|-
|11
| 600.000
|Down-augmented fourth (vA4)
Up-diminished fifth (^d5)
| vG#
^Ab
|-
|12
|654.545
|Augmented fourth (A5) Down-fifth (v5)
|G# vA
|-
|13
|709.091
|'''Perfect fifth (P5)'''
|'''A'''
|-
|14
| 763.636
|Up-fifth (^5) Minor sixth (m6)
|^A
Bb
|-
|15
| 818.182
|Down-augmented fifth (vA5)
Upminor sixth (^m6)
|vA#
^Bb
|-
|16
|872.727
|Augmented fifth (A5)
'''Downmajor sixth (vM6)'''
|A# '''vB'''
|-
| 17
|927.273
|'''Major sixth (M6)'''
Up-augmented fifth (^A5)

Downminor seventh (vm7)
|'''B'''
^A#

vC
|-
|18
|981.818
|'''Minor seventh (m7)'''
Upmajor sixth (^M6)

Down-diminished octave (vd8)
|'''C'''
^B

vDb
|-
|19
|1036.364
|'''Upminor seventh (^m7)'''
Diminished octave (d8)
|'''^C'''
Db
|-
|20
|1090.909
|Downmajor seventh (vM7)
Up-diminished octave (^d8)

Augmented sixth (A6)
|vC#
^Db

B#
|-
|21
|1145.455
|Major seventh (M7)
Down-octave (v8)
|C#
vD
|-
|22
|1200.000
|'''Perfect octave (P8)'''
|'''D'''
|}

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|22}}

==Defining features ==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
|−2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| −0.86
|2.70
|4.94
|-
|2.3.5.7
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
|−1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps ===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
| 7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
|Trizo-agu
| Hemimage comma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell comma
|-
|7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry comma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
| Kalisma
|-
|13
|[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
|26.84
|Thoyo
|Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
|Tholuru
|Negustma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|13
|[[31213/31104]]
|{{monzo| -7 -5 0 4 0 1 }}
|6.06
|Thoquadzo
|Praveensma
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
|Twizzler comma
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
|1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
|[[Porcupine]]
|-
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

==Notation==
===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

=== 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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Ups and Downs Notation===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:

{{Sharpness-sharp3}}

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
|0
|Natural Unison
|1
|perfect unison
|P1
| D
|perfect unison
|P1
|D
|natural 1st
|N1
| C
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|-
|1
|55
|s-minor second
|sm2
|aug unison
|A1
|D#
|aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|up-unison, minor 2nd
| ^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
|3
| 164
| p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|major 2nd
|M2
|E
|major 2nd
|M2
|E
|-
|5
|273
|s-minor third
|sm3
|dim 3rd
|d3
|Fb
|major sub3rd
| Ms3
|F
|sharp 3rd
| s3
|D#
|minor 3rd
|m3
|F
|minor 3rd
|m3
| F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
|f4
|εb
|upminor 3rd
| ^m3
| ^F
| classic minor 3rd
|Km3
|KF
|-
| 7
|382
|p-Major third
| pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
|natural 4th
|N4
| ε
|downmajor 3rd
|vM3
| vF#
| classic major 3rd
|kM3
|kF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
| dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
|s4, f5
|ε#, Eb
|major 3rd
|M3
|F#
|major 3rd
|M3
|F#
|-
| 9
| 491
|Natural Fourth
|4, N4
|minor 4th
|m4
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|10
|545
| p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
|M4
|G#
| aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
|A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
|Gx, Abb
|natural 6th
| N6
|γ
| downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
|m5
|Ab
|dim 5thoid
|d5d
|Ab
| sharp 6th, flat 7th
|s6, f7
|γ#, Gb
|aug 4th, down-5th
|A4, v5
|G#, vA
| comma-narrow 5th
|k5
|kA
|-
|13
| 709
|Natural Fifth
|5, N5
|major 5th
|M5
|A
|perfect 5thoid
|P5d
|A
|natural 7th
|N7
|G
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|-
|14
|764
| s-minor sixth
|sm6
|aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
|G#
| minor 6th
|m6
|Bb
|minor 6th
| m6
| Bb
|-
| 15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
| double-aug 5thoid, double-dim sub7th
|AA5d, dds7
| Ax, Cb3
|flat 8th
|f8
|αb
|upminor 6th
|^m6
|^Bb
| classic minor 6th
| Km6
|KBb
|-
|16
|873
|p-Major sixth
|pM6
|major 6th
|M6
|B
| dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
| downmajor 6th
|vM6
|vB
|classic major 6th
|kM6
|kB
|-
| 17
|927
| s-Major sixth
|sM6
|aug 6th
|A6
|B#
|minor sub7th
|ms7
|Cb
| sharp 8th, flat 9th
|s8, f9
|α#, Ab
|major 6th
|M6
|B
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
|m7
| dim 7th
|d7
|Cb
|major sub7th
| Ms7
|C
|natural 9th
| N9
|A
|minor 7th
|m7
| C
| minor 7th
| m7
|C
|-
|19
|1036
| p-Major seventh
| pM7
|perfect 7th
| P7
|C
| aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
|1091
|p-Augmented seventh
|pA7
|aug 7th
|A7
|C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
|N10
| β
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
|21
|1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
| sharp 10th
|s10
|β#, Cb
|major 7th, down 8ve
|M7, v8
|C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
|22
|1200
| Octave
|8
|perfect octave
| P8
|D
|perfect octave
|P8
|D
|natural 11th
|N11
|C
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|}

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

{| class="wikitable center-all"
|-
!Quality
![[Color name]]
![[Monzo]] Format
!Examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
|7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
|upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|Color of the 3rd]]
!JI Chord
!Notes as edosteps
!Notes of C chord
!Written name
!Spoken name
|-
|zo
|6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
|gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
|C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
|14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

==Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

==References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-07-10T20:30:26Z

YoVariable: Added a separate 22edo notation section

{{interwiki
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

==Theory==
=== Prime harmonics===
{{Harmonics in equal|22|columns=11}}

===History===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

===Overview to JI approximation quality===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

===Subsets and supersets===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

==Intervals==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
!Degree
!Cents
!Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
!Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
| Km2
| KEb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
|aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
| kM2
|kE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
| major 2nd
|M2
|E
| major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
| minor 3rd
|m3
|F
| minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| ^m3
|^F
|classic minor 3rd
| Km3
|KF
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
| vM3
| vF#
|classic major 3rd
| kM3
| kF#
|[[File:0-381.818c_22edo.mp3]]
|-
|8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
| major 3rd
|M3
|F#
| major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|[[File:0-545.455c_11edo.mp3]]
|-
|11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th comma-wide diminished 5th
|kA4 Kd5
|kG#, KAb
|[[File:0-600.000c_2edo.mp3]]
|-
|12
|654.545
|[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|comma-narrow 5th
|k5
|kA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
| minor 6th
|m6
|Bb
| minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
| ^m6
| ^Bb
|classic minor 6th
| Km6
| KBb
|[[File:0-818.182c_22edo.mp3]]
|-
|16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| vM6
|vB
|classic major 6th
| kM6
|kB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
| major 6th
|M6
|B
| major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
| minor 7th
|m7
|C
| minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
| Km7
|kC
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
| kM7
| kC#
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
|M7, v8
|C#, vD
| major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==Notation==
===Ups and Downs Notation===
Standard Pythagorean ur browser has no SVG suppor can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that Eb and D# are different notes and that Eb is lower in pitch than D#.

{| class="wikitable right-1 right-2 center-3 center-4"
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names on D
|-
|0
|0.00
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 54.545
|Minor second (m2) Up-unison (^1)
|Eb ^D
|-
|2
|109.091
|Upminor 2nd (^m2) Down-augmented unison (vA1)
Diminished third (d3)
|^Eb vD#
Fb
|-
|3
|163.636
|Downmajor second (vM2) Augmented unison (A1)
|vE D#
|-
|4
|218.182
| '''Major second (M2)'''
Up-augmented unison (^A1) Downminor third (vm3)
|'''E'''
^D# vF
|-
|5
|272.727
|Upmajor second (^M2) '''Minor third (m3)'''
|^E '''F'''
|-
| 6
|327.273
|'''Upminor third (^m3)''' Diminished fourth (d4)
|'''^F''' Gb
|-
|7
|381.818
|'''Downmajor third (vM3)''' Augmented second (A2)
Up-diminished fourth (^d4)
|'''vF#''' E#
^Gb
|-
|8
| 436.364
|'''Major third (M3)'''
Up-augmented second (^A2)

Down-fourth (v4)
|'''F#'''
^E#

vG
|-
|9
|490.909
| '''Perfect fourth (P4)'''
|'''G'''
|-
|10
|545.455
|Up-fourth (^4)
Diminished fifth (d5)
|^G Ab
|-
|11
| 600.000
|Down-augmented fourth (vA4)
Up-diminished fifth (^d5)
| vG#
^Ab
|-
|12
|654.545
|Augmented fourth (A5) Down-fifth (v5)
|G# vA
|-
|13
|709.091
|'''Perfect fifth (P5)'''
|'''A'''
|-
|14
| 763.636
|Up-fifth (^5) Minor sixth (m6)
|^A
Bb
|-
|15
| 818.182
|Down-augmented fifth (vA5)
Upminor sixth (^m6)
|vA#
^Bb
|-
|16
|872.727
|Augmented fifth (A5)
'''Downmajor sixth (vM6)'''
|A# '''vB'''
|-
| 17
|927.273
|'''Major sixth (M6)'''
Up-augmented fifth (^A5)

Downminor seventh (vm7)
|'''B'''
^A#

vC
|-
|18
|981.818
|'''Minor seventh (m7)'''
Upmajor sixth (^M6)

Down-diminished octave (vd8)
|'''C'''
^B

vDb
|-
|19
|1036.364
|'''Upminor seventh (^m7)'''
Diminished octave (d8)
|'''^C'''
Db
|-
|20
|1090.909
|Downmajor seventh (vM7)
Up-diminished octave (^d8)

Augmented sixth (A6)
|vC#
^Db

B#
|-
|21
|1145.455
|Major seventh (M7)
Down-octave (v8)
|C#
vD
|-
|22
|1200.000
|'''Perfect octave (P8)'''
|'''D'''
|}

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|22}}

==Defining features ==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
|−2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| −0.86
|2.70
|4.94
|-
|2.3.5.7
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
|−1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps ===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
| 7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
|Trizo-agu
| Hemimage comma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell comma
|-
|7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry comma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
| Kalisma
|-
|13
|[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
|26.84
|Thoyo
|Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
|Tholuru
|Negustma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|13
|[[31213/31104]]
|{{monzo| -7 -5 0 4 0 1 }}
|6.06
|Thoquadzo
|Praveensma
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
|Twizzler comma
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
|1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
|[[Porcupine]]
|-
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

==Notation==
===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

=== 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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Ups and Downs Notation===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:

{{Sharpness-sharp3}}

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
|0
|Natural Unison
|1
|perfect unison
|P1
| D
|perfect unison
|P1
|D
|natural 1st
|N1
| C
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|-
|1
|55
|s-minor second
|sm2
|aug unison
|A1
|D#
|aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|up-unison, minor 2nd
| ^1, m2
|^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
|3
| 164
| p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|major 2nd
|M2
|E
|major 2nd
|M2
|E
|-
|5
|273
|s-minor third
|sm3
|dim 3rd
|d3
|Fb
|major sub3rd
| Ms3
|F
|sharp 3rd
| s3
|D#
|minor 3rd
|m3
|F
|minor 3rd
|m3
| F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
|f4
|εb
|upminor 3rd
| ^m3
| ^F
| classic minor 3rd
|Km3
|KF
|-
| 7
|382
|p-Major third
| pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
|natural 4th
|N4
| ε
|downmajor 3rd
|vM3
| vF#
| classic major 3rd
|kM3
|kF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
| dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
|s4, f5
|ε#, Eb
|major 3rd
|M3
|F#
|major 3rd
|M3
|F#
|-
| 9
| 491
|Natural Fourth
|4, N4
|minor 4th
|m4
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|10
|545
| p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
|M4
|G#
| aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|up-4th, dim 5th
|^4, d5
|^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
|A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
|Gx, Abb
|natural 6th
| N6
|γ
| downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
|m5
|Ab
|dim 5thoid
|d5d
|Ab
| sharp 6th, flat 7th
|s6, f7
|γ#, Gb
|aug 4th, down-5th
|A4, v5
|G#, vA
| comma-narrow 5th
|k5
|kA
|-
|13
| 709
|Natural Fifth
|5, N5
|major 5th
|M5
|A
|perfect 5thoid
|P5d
|A
|natural 7th
|N7
|G
|perfect 5th
|P5
|A
|perfect 5th
|P5
|A
|-
|14
|764
| s-minor sixth
|sm6
|aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
|G#
| minor 6th
|m6
|Bb
|minor 6th
| m6
| Bb
|-
| 15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
| double-aug 5thoid, double-dim sub7th
|AA5d, dds7
| Ax, Cb3
|flat 8th
|f8
|αb
|upminor 6th
|^m6
|^Bb
| classic minor 6th
| Km6
|KBb
|-
|16
|873
|p-Major sixth
|pM6
|major 6th
|M6
|B
| dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
| downmajor 6th
|vM6
|vB
|classic major 6th
|kM6
|kB
|-
| 17
|927
| s-Major sixth
|sM6
|aug 6th
|A6
|B#
|minor sub7th
|ms7
|Cb
| sharp 8th, flat 9th
|s8, f9
|α#, Ab
|major 6th
|M6
|B
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
|m7
| dim 7th
|d7
|Cb
|major sub7th
| Ms7
|C
|natural 9th
| N9
|A
|minor 7th
|m7
| C
| minor 7th
| m7
|C
|-
|19
|1036
| p-Major seventh
| pM7
|perfect 7th
| P7
|C
| aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
|1091
|p-Augmented seventh
|pA7
|aug 7th
|A7
|C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
|N10
| β
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
|21
|1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
| sharp 10th
|s10
|β#, Cb
|major 7th, down 8ve
|M7, v8
|C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
|22
|1200
| Octave
|8
|perfect octave
| P8
|D
|perfect octave
|P8
|D
|natural 11th
|N11
|C
|perfect octave
|P8
|D
|perfect 8ve
|P8
|D
|}

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

{| class="wikitable center-all"
|-
!Quality
![[Color name]]
![[Monzo]] Format
!Examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
|7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
|upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|Color of the 3rd]]
!JI Chord
!Notes as edosteps
!Notes of C chord
!Written name
!Spoken name
|-
|zo
|6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
|gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
|C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
|14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

==Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

==References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

User:YoVariable

2024-07-10T14:51:16Z

YoVariable:

Hello! My name is Variable (she/her) and I love to explore xenharmonic music. My Discord is YoVariable. I love meantone-based EDOs, like [[19edo|19]], [[31edo|31]], and [[43edo]] and superpyth EDOs like [[17edo|17]], [[22edo|22]], and [[27edo]].

You may find me editing superpyth or meantone EDO pages occasionally.

== External links ==
YouTube: https://www.youtube.com/@YoVariable

User:YoVariable

2024-05-12T05:10:45Z

YoVariable:

Hello! My name is Variable (she/her/hers) and I love to explore xenharmonic music. My Discord is YoVariable. I love meantone-based EDOs, like [[19edo|19]], [[31edo|31]], and [[43edo]] and superpyth EDOs like [[17edo|17]], [[22edo|22]], and [[27edo]].

You may find me editing superpyth or meantone EDO pages occasionally.

== External links ==
YouTube: https://www.youtube.com/@YoVariable

22edo

2024-04-28T04:25:58Z

YoVariable: Cleaned up 22edo history, overview to JI approximation quality, and subsets and supersets.

{{interwiki
| de = 22-EDO
| en = 22edo
| es =
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

== Theory ==
=== Prime harmonics ===
{{Harmonics in equal|22|columns=11}}

=== History ===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.

=== Overview to JI approximation quality ===
The 22edo 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 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]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22edo, unlike 12 and 19, is not a [[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, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

=== Subsets and supersets ===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

== Intervals ==
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}

{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
! Audio
|-
| 0
| 0.000
| [[1/1]]
| perfect unison
| P1
| D
| perfect unison
| P1
| D
| [[File:0-0.000c_P1.mp3]]
|-
| 1
| 54.545
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
| comma-wide unison, minor 2nd
| K1, m2
| KD, Eb
| [[File:0-54.545c_22edo.mp3]]
|-
| 2
| 109.091
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
| classic minor 2nd
| Km2
| KEb
| [[File:0-109.091c_11edo.mp3]]
|-
| 3
| 163.636
| [[12/11]], [[11/10]], [[10/9]]
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
| classic/comma-narrow major 2nd
| kM2
| kE
| [[File:0-163.636c_22edo.mp3]]
|-
| 4
| 218.182
| [[9/8]], [[17/15]], [[8/7]]
| major 2nd
| M2
| E
| major 2nd
| M2
| E
| [[File:0-218.182c_11edo.mp3]]
|-
| 5
| 272.727
| [[20/17]], [[7/6]]
| minor 3rd
| m3
| F
| minor 3rd
| m3
| F
| [[File:0-272.727c_22edo.mp3]]
|-
| 6
| 327.273
| [[6/5]], [[17/14]], [[11/9]]
| upminor 3rd
| ^m3
| ^F
| classic minor 3rd
| Km3
| KF
| [[File:0-327.273c_11edo.mp3]]
|-
| 7
| 381.818
| [[5/4]], [[96/77]]
| downmajor 3rd
| vM3
| vF#
| classic major 3rd
| kM3
| kF#
| [[File:0-381.818c_22edo.mp3]]
|-
| 8
| 436.364
| [[14/11]], [[9/7]], [[22/17]]
| major 3rd
| M3
| F#
| major 3rd
| M3
| F#
| [[File:0-436.364c_11edo.mp3]]
|-
| 9
| 490.909
| [[4/3]]
| perfect 4th
| P4
| G
| perfect 4th
| P4
| G
| [[File:0-490.909c_22edo.mp3]]
|-
| 10
| 545.455
| [[15/11]], [[11/8]]
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
| comma-wide 4th
| K4
| KG
| [[File:0-545.455c_11edo.mp3]]
|-
| 11
| 600.000
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
| comma-narrow augmented 4th comma-wide diminished 5th
| kA4 Kd5
| kG#, KAb
| [[File:0-600.000c_2edo.mp3]]
|-
| 12
| 654.545
| [[16/11]], [[22/15]]
| aug 4th, down-5th
| A4, v5
| G#, vA
| comma-narrow 5th
| k5
| kA
| [[File:0-654.545c_11edo.mp3]]
|-
| 13
| 709.091
| [[3/2]]
| perfect 5th
| P5
| A
| perfect 5th
| P5
| A
| [[File:0-709.091c_22edo.mp3]]
|-
| 14
| 763.636
| [[17/11]], [[14/9]], [[11/7]]
| minor 6th
| m6
| Bb
| minor 6th
| m6
| Bb
| [[File:0-763.636c_11edo.mp3]]
|-
| 15
| 818.182
| [[8/5]], [[77/48]]
| upminor 6th
| ^m6
| ^Bb
| classic minor 6th
| Km6
| KBb
| [[File:0-818.182c_22edo.mp3]]
|-
| 16
| 872.727
| [[18/11]], [[28/17]], [[5/3]]
| downmajor 6th
| vM6
| vB
| classic major 6th
| kM6
| kB
| [[File:0-872.727c_11edo.mp3]]
|-
| 17
| 927.273
| [[17/10]], [[12/7]]
| major 6th
| M6
| B
| major 6th
| M6
| B
| [[File:0-927.273c_22edo.mp3]]
|-
| 18
| 981.818
| [[7/4]], [[30/17]], [[16/9]]
| minor 7th
| m7
| C
| minor 7th
| m7
| C
| [[File:0-981.818c_11edo.mp3]]
|-
| 19
| 1036.364
| [[9/5]], [[11/6]], [[20/11]]
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
| classic minor 7th
| Km7
| kC
| [[File:0-1036.364c_22edo.mp3]]
|-
| 20
| 1090.909
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
| classic major 7th
| kM7
| kC#
| [[File:0-1090.909c_11edo.mp3]]
|-
| 21
| 1145.455
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
| M7, v8
| C#, vD
| major 7th / comma-narrow 8ve
| M7 / k8
| C#, kD
| [[File:0-1145.455c_22edo.mp3]]
|-
| 22
| 1200.000
| [[2/1]]
| perfect octave
| P8
| D
| perfect 8ve
| P8
| D
| [[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
=== 15-odd-limit interval mappings ===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" | 15-odd-limit intervals by direct approximation (even if inconsistent)
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[9/7]], [[14/9]]
| 1.280
| 2.3
|-
| [[11/10]], [[20/11]]
| 1.368
| 2.5
|-
| [[15/8]], [[16/15]]
| 2.640
| 4.8
|-
| '''[[5/4]], [[8/5]]'''
| '''4.496'''
| '''8.2'''
|-
| [[7/6]], [[12/7]]
| 5.856
| 10.7
|-
| '''[[11/8]], [[16/11]]'''
| '''5.863'''
| '''10.7'''
|-
| '''[[3/2]], [[4/3]]'''
| '''7.136'''
| '''13.1'''
|-
| [[15/11]], [[22/15]]
| 8.504
| 15.6
|-
| [[15/14]], [[28/15]]
| 10.352
| 19.0
|-
| [[5/3]], [[6/5]]
| 11.631
| 21.3
|-
| '''[[7/4]], [[8/7]]'''
| '''12.992'''
| '''23.8'''
|-
| [[11/6]], [[12/11]]
| 12.999
| 23.8
|-
| [[9/8]], [[16/9]]
| 14.272
| 26.2
|-
| [[13/11]], [[22/13]]
| 16.482
| 30.2
|-
| [[7/5]], [[10/7]]
| 17.488
| 32.1
|-
| [[13/10]], [[20/13]]
| 17.850
| 32.7
|-
| ''[[13/9]], [[18/13]]''
| ''17.928''
| ''32.9''
|-
| [[9/5]], [[10/9]]
| 18.767
| 34.4
|-
| [[11/7]], [[14/11]]
| 18.856
| 34.6
|-
| ''[[13/7]], [[14/13]]''
| ''19.207''
| ''35.2''
|-
| [[11/9]], [[18/11]]
| 20.135
| 36.9
|-
| '''[[13/8]], [[16/13]]'''
| '''22.346'''
| '''41.0'''
|-
| [[15/13]], [[26/15]]
| 24.986
| 45.8
|-
| ''[[13/12]], [[24/13]]''
| ''25.064''
| ''46.0''
|}
{{15-odd-limit|22}}

== Defining features ==

=== Septimal vs syntonic comma ===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

=== 5-limit commas ===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas ===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

=== Other features ===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| 2.85
| 5.23
|-
| 2.3.5.7.11
| 50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
| -1.11
| 2.90
| 5.33
|-
| 2.3.5.7.11.17
| 50/49, 55/54, 64/63, 85/84, 99/98
| [{{val| 22 35 51 62 76 90 }}]
| -1.09
| 2.65
| 4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

=== Uniform maps ===
{{Uniform map|13|21.5|22.5}}

=== Commas ===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name
|-
| 3
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| {{monzo| 35 -22 }}
| 156.98
| Trisawa
| 22-comma
|-
| 5
| [[250/243]]
| {{monzo| 1 -5 3 }}
| 49.17
| Triyo
| Porcupine comma, maximal diesis
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]]
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 5
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| {{monzo| -53 10 16 }}
| 0.57
| Quadla-quadquadyo
| [[Kwazy comma]]
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[2430/2401]]
| {{monzo| 1 5 1 -4 }}
| 20.79
| Quadru-ayo
| Nuwell comma
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Trizo-agu
| Hemimage comma
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell comma
|-
| 7
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell comma
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo| -6 -8 2 5 }}
| 1.12
| Quinzo-ayoyo
| [[Wizma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry comma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[31213/31104]]
| {{monzo| -7 -5 0 4 0 1 }}
| 6.06
| Thoquadzo
| Praveensma
|-
| 31
| [[125/124]]
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler comma
|}
<references />

=== Rank-2 temperaments ===
* [[List of 22et rank two temperaments by badness]]
* [[List of 22et rank two temperaments by complexity]]
* [[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods per octave
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
| [[Porcupine]]
|-
| 1
| 5\22
| [[Orwell]] (22) / blair (22) / winston (22f)
|-
| 1
| 7\22
| [[Magic]] / telepathy
|-
| 1
| 9\22
| [[Superpyth]] / [[suprapyth]]
|-
| 2
| 1\22
| [[Shrutar]] / hemipaj [[Comic]]
|-
| 2
| 2\22
| [[Srutal]] / [[pajara]] / pajarous
|-
| 2
| 3\22
| [[Hedgehog]] / [[echidna]]
|-
| 2
| 4\22
| [[Astrology]] [[Antikythera]] [[Wizard]]
|-
| 2
| 5\22
| [[Doublewide]] / fleetwood
|-
| 11
| 1\22
| [[Undeka]] [[Hendecatonic]]
|}

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

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

== Notation ==
=== Superpyth/Porcupine Notation ===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

=== Porcupine Notation ===
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.

=== 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.

=== 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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

=== 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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

=== Ups and Downs Notation ===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

=== Comparison of 22edo notation systems ===

{| class="wikitable center-all right-2"
|-
! [[Degree]]
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" | Pentatonic
! colspan="3" | Decatonic
! colspan="3" | [[Ups and downs notation|Ups and Downs]]
! colspan="3" |[[SKULO interval names]]
|-
| 0
| 0
| Natural Unison
| 1
| perfect unison
| P1
| D
| perfect unison
| P1
| D
| natural 1st
| N1
| C
| perfect unison
| P1
| D
|perfect unison
|P1
|D
|-
| 1
| 55
| s-minor second
| sm2
| aug unison
| A1
| D#
| aug unison
| A1
| D#
| flat 2nd
| f2
| C#, δb
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
|comma-wide unison, minor 2nd
|K1, m2
|KD, Eb
|-
| 2
| 109
| p-diminished second
| pd2
| dim 2nd
| d2
| Eb
| double-aug unison, double-dim sub3rd
| AA1, dds3
| Dx, Fb3 
| natural 2nd
| N2
| δ
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
|classic minor 2nd
|Km2
|KEb
|-
| 3
| 164
| p-minor second
| pm2
| perfect 2nd
| P2
| E
| dim sub3rd
| ds3
| Fbb
| sharp 2nd, flat 3rd
| s2, f3
| δ#, Db
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
|classic/comma-narrow major 2nd
|kM2
|kE
|-
| 4
| 218
| (s/p) Major second
| M2
| aug 2nd
| A2
| E#
| minor sub3rd
| ms3
| Fb
| natural 3rd
| N3
| D
| major 2nd
| M2
| E
|major 2nd
|M2
|E
|-
| 5
| 273
| s-minor third
| sm3
| dim 3rd
| d3
| Fb
| major sub3rd
| Ms3
| F
| sharp 3rd
| s3
| D#
| minor 3rd
| m3
| F
|minor 3rd
|m3
|F
|-
| 6
| 327
| p-minor third
| pm3
| minor 3rd
| m3
| F
| aug sub3rd
| As3
| F#
| flat 4th
| f4
| εb
| upminor 3rd
| ^m3
| ^F
|classic minor 3rd
|Km3
|KF
|-
| 7
| 382
| p-Major third
| pM3
| major 3rd
| M3
| F#
| double-aug sub3rd, double-dim 4thoid
| AAs3, dd4d
| Fx, Gbb
| natural 4th
| N4
| ε
| downmajor 3rd
| vM3
| vF#
|classic major 3rd
|kM3
|kF#
|-
| 8
| 436
| s-Major third
| sM3
| aug 3rd, dim 4th
| A3, d4
| Fx, Gb
| dim 4thoid
| d4d
| Gb
| sharp 4th, flat 5th
| s4, f5
| ε#, Eb
| major 3rd
| M3
| F#
|major 3rd
|M3
|F#
|-
| 9
| 491
| Natural Fourth
| 4, N4
| minor 4th
| m4
| G
| perfect 4thoid
| P4d
| G
| natural 5th
| N5
| E
| perfect 4th
| P4
| G
|perfect 4th
|P4
|G
|-
| 10
| 545
| p-Major fourth, s-dim fifth
| pM4, sd5
| major 4th
| M4
| G#
| aug 4thoid
| A4d
| G#
| sharp 5th, flat 6th
| s5, f6
| E#, γb
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
|comma-wide 4th
|K4
|KG
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
| A4, HO
| aug 4th, dim 5th
| A4, d5
| Gx, Abb
| double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
| Gx, Abb
| natural 6th
| N6
| γ
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
|comma-narrow augmented 4th
comma-wide diminished 5th
|kA4
Kd5
|kG#, KAb
|-
| 12
| 655
| p-minor Fifth, s-aug Fourth
| pm5, sA4
| minor 5th
| m5
| Ab
| dim 5thoid
| d5d
| Ab
| sharp 6th, flat 7th
| s6, f7
| γ#, Gb
| aug 4th, down-5th
| A4, v5
| G#, vA
|comma-narrow 5th
|k5
|kA
|-
| 13
| 709
| Natural Fifth
| 5, N5
| major 5th
| M5
| A
| perfect 5thoid
| P5d
| A
| natural 7th
| N7
| G
| perfect 5th
| P5
| A
|perfect 5th
|P5
|A
|-
| 14
| 764
| s-minor sixth
| sm6
| aug 5th, dim 6th
| A5, d6
| A#, Bbb
| aug 5thoid
| A5d
| A#
| sharp 7th
| s7
| G#
| minor 6th
| m6
| Bb
|minor 6th
|m6
|Bb
|-
| 15
| 818
| p-minor sixth
| pm6
| minor 6th
| m6
| Bb
| double-aug 5thoid, double-dim sub7th
| AA5d, dds7
| Ax, Cb3
| flat 8th
| f8
| αb
| upminor 6th
| ^m6
| ^Bb
|classic minor 6th
|Km6
|KBb
|-
| 16
| 873
| p-Major sixth
| pM6
| major 6th
| M6
| B
| dim sub7th
| ds7
| Cbb
| natural 8th
| N8
| α
| downmajor 6th
| vM6
| vB
|classic major 6th
|kM6
|kB
|-
| 17
| 927
| s-Major sixth
| sM6
| aug 6th
| A6
| B#
| minor sub7th
| ms7
| Cb
| sharp 8th, flat 9th
| s8, f9
| α#, Ab
| major 6th
| M6
| B
|major 6th
|M6
|B
|-
| 18
| 982
| (s/p) minor seventh
| m7
| dim 7th
| d7
| Cb
| major sub7th
| Ms7
| C
| natural 9th
| N9
| A
| minor 7th
| m7
| C
|minor 7th
|m7
|C
|-
| 19
| 1036
| p-Major seventh
| pM7
| perfect 7th
| P7
| C
| aug sub7th
| As7
| C#
| sharp 9th, flat 10th
| s9, f10
| A#, βb
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
|classic minor 7th
|Km7
|kC
|-
| 20
| 1091
| p-Augmented seventh
| pA7
| aug 7th
| A7
| C#
| double-aug sub7th, double-dim octave
| AAs7, dd8
| Cx, Dbb
| natural 10th
| N10
| β
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
|classic major 7th
|kM7
|kC#
|-
| 21
| 1145
| s-Major seventh
| sM7
| dim 8ve
| d8
| Db
| dim octave
| d8
| Db
| sharp 10th
| s10
| β#, Cb
| major 7th, down 8ve
| M7, v8
| C#, vD
|major 7th / comma-narrow 8ve
|M7 / k8
|C#, kD
|-
| 22
| 1200
| Octave
| 8
| perfect octave
| P8
| D
| perfect octave
| P8
| D
| natural 11th
| N11
| C
| perfect octave
| P8
| D
|perfect 8ve
|P8
|D
|}

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

{| class="wikitable center-all"
|-
! Quality
! [[Color name]]
! [[Monzo]] Format
! Examples
|-
| rowspan="2" |minor
| zo
| [a b 0 1>
| 7/6, 7/4
|-
| fourthward wa
| [a b> where b < -1
| 32/27, 16/9
|-
| upminor
| gu
| [a b -1>
| 6/5, 9/5
|-
| downmajor
| yo
| [a b 1>
| 5/4, 5/3
|-
| rowspan="2" |major
| fifthward wa
| [a b> where b > 1
| 9/8, 27/16
|-
| ru
| [a b 0 -1>
| 9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Notes as edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo
| 6:7:9
| 0-5-13
| C Eb G
| Cm
| C minor
|-
| gu
| 10:12:15
| 0-6-13
| C ^Eb G
| C^m
| C upminor
|-
| yo
| 4:5:6
| 0-7-13
| C vE G
| Cv
| C downmajor or C down
|-
| ru
| 14:18:21
| 0-8-13
| C E G
| C
| C major or C
|}

Examples:

* 0-4-13 = C D G = C2
* 0-9-13 = C F G = C4
* 0-10-13 = C ^F G = C^4 or C(^4)
* 0-5-10 = C Eb Gb = Cd = Cdim
* 0-5-11 = C Eb ^Gb = Cd(^5)
* 0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

* [[22edo Chord Names]]
* [[22 EDO Chords]]
* [[Ups and Downs Notation #Chords and Chord Progressions]]
* [[Chords of orwell]]

== Music ==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

== Related pages ==
* [[Lumatone mapping for 22edo]]
* [[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
* [[22edo/Eliora's approach|22edo/Eliora's Approach]]

== Further reading ==
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
* [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]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

== References ==
# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
# Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-03-11T16:04:38Z

YoVariable: Adjusted headings

{{interwiki
| de = 22-EDO
| en = 22edo
| es =
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

== Theory ==
=== History ===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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''.

=== Overview to JI approximation quality ===
The 22et 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 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]] to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22et, unlike 12 and [[19edo|19]], is not a [[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.

22et can also be treated as adding harmonics 3 and 5 to 11edo's 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

22et is very close to an extended "quarter-comma archy", 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]]).

=== Prime harmonics ===
{{Harmonics in equal|22|columns=11}}

=== Subsets and supersets ===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

== Intervals ==
{{See also| 22edo solfege }}

{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! Audio
|-
| 0
| 0.000
| [[1/1]]
| perfect unison
| P1
| D
| [[File:0-0.000c_P1.mp3]]
|-
| 1
| 54.545
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
| [[File:0-54.545c_22edo.mp3]]
|-
| 2
| 109.091
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
| [[File:0-109.091c_11edo.mp3]]
|-
| 3
| 163.636
| [[12/11]], [[11/10]], [[10/9]]
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
| [[File:0-163.636c_22edo.mp3]]
|-
| 4
| 218.182
| [[9/8]], [[17/15]], [[8/7]]
| major 2nd
| M2
| E
| [[File:0-218.182c_11edo.mp3]]
|-
| 5
| 272.727
| [[20/17]], [[7/6]]
| minor 3rd
| m3
| F
| [[File:0-272.727c_22edo.mp3]]
|-
| 6
| 327.273
| [[6/5]], [[17/14]], [[11/9]]
| upminor 3rd
| ^m3
| ^F
| [[File:0-327.273c_11edo.mp3]]
|-
| 7
| 381.818
| [[5/4]], [[96/77]]
| downmajor 3rd
| vM3
| vF#
| [[File:0-381.818c_22edo.mp3]]
|-
| 8
| 436.364
| [[14/11]], [[9/7]], [[22/17]]
| major 3rd
| M3
| F#
| [[File:0-436.364c_11edo.mp3]]
|-
| 9
| 490.909
| [[4/3]]
| perfect 4th
| P4
| G
| [[File:0-490.909c_22edo.mp3]]
|-
| 10
| 545.455
| [[15/11]], [[11/8]]
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
| [[File:0-545.455c_11edo.mp3]]
|-
| 11
| 600.000
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
| [[File:0-600.000c_2edo.mp3]]
|-
| 12
| 654.545
| [[16/11]], [[22/15]]
| aug 4th, down-5th
| A4, v5
| G#, vA
| [[File:0-654.545c_11edo.mp3]]
|-
| 13
| 709.091
| [[3/2]]
| perfect 5th
| P5
| A
| [[File:0-709.091c_22edo.mp3]]
|-
| 14
| 763.636
| [[17/11]], [[14/9]], [[11/7]]
| minor 6th
| m6
| Bb
| [[File:0-763.636c_11edo.mp3]]
|-
| 15
| 818.182
| [[8/5]], [[77/48]]
| upminor 6th
| ^m6
| ^Bb
| [[File:0-818.182c_22edo.mp3]]
|-
| 16
| 872.727
| [[18/11]], [[28/17]], [[5/3]]
| downmajor 6th
| vM6
| vB
| [[File:0-872.727c_11edo.mp3]]
|-
| 17
| 927.273
| [[17/10]], [[12/7]]
| major 6th
| M6
| B
| [[File:0-927.273c_22edo.mp3]]
|-
| 18
| 981.818
| [[7/4]], [[30/17]], [[16/9]]
| minor 7th
| m7
| C
| [[File:0-981.818c_11edo.mp3]]
|-
| 19
| 1036.364
| [[9/5]], [[11/6]], [[20/11]]
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
| [[File:0-1036.364c_22edo.mp3]]
|-
| 20
| 1090.909
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
| [[File:0-1090.909c_11edo.mp3]]
|-
| 21
| 1145.455
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
| M7, v8
| C#, vD
| [[File:0-1145.455c_22edo.mp3]]
|-
| 22
| 1200.000
| [[2/1]]
| perfect octave
| P8
| D
| [[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

== Approximation to JI ==
=== 15-odd-limit interval mappings ===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" | 15-odd-limit intervals by direct approximation (even if inconsistent)
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[9/7]], [[14/9]]
| 1.280
| 2.3
|-
| [[11/10]], [[20/11]]
| 1.368
| 2.5
|-
| [[15/8]], [[16/15]]
| 2.640
| 4.8
|-
| '''[[5/4]], [[8/5]]'''
| '''4.496'''
| '''8.2'''
|-
| [[7/6]], [[12/7]]
| 5.856
| 10.7
|-
| '''[[11/8]], [[16/11]]'''
| '''5.863'''
| '''10.7'''
|-
| '''[[3/2]], [[4/3]]'''
| '''7.136'''
| '''13.1'''
|-
| [[15/11]], [[22/15]]
| 8.504
| 15.6
|-
| [[15/14]], [[28/15]]
| 10.352
| 19.0
|-
| [[5/3]], [[6/5]]
| 11.631
| 21.3
|-
| '''[[7/4]], [[8/7]]'''
| '''12.992'''
| '''23.8'''
|-
| [[11/6]], [[12/11]]
| 12.999
| 23.8
|-
| [[9/8]], [[16/9]]
| 14.272
| 26.2
|-
| [[13/11]], [[22/13]]
| 16.482
| 30.2
|-
| [[7/5]], [[10/7]]
| 17.488
| 32.1
|-
| [[13/10]], [[20/13]]
| 17.850
| 32.7
|-
| ''[[13/9]], [[18/13]]''
| ''17.928''
| ''32.9''
|-
| [[9/5]], [[10/9]]
| 18.767
| 34.4
|-
| [[11/7]], [[14/11]]
| 18.856
| 34.6
|-
| ''[[13/7]], [[14/13]]''
| ''19.207''
| ''35.2''
|-
| [[11/9]], [[18/11]]
| 20.135
| 36.9
|-
| '''[[13/8]], [[16/13]]'''
| '''22.346'''
| '''41.0'''
|-
| [[15/13]], [[26/15]]
| 24.986
| 45.8
|-
| ''[[13/12]], [[24/13]]''
| ''25.064''
| ''46.0''
|}
{{15-odd-limit|22}}

=== Selected 17-limit intervals ===
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]

== Defining features ==

=== Septimal vs syntonic comma ===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

=== 5-limit commas ===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas ===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

=== Other features ===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| 2.85
| 5.23
|-
| 2.3.5.7.11
| 50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
| -1.11
| 2.90
| 5.33
|-
| 2.3.5.7.11.17
| 50/49, 55/54, 64/63, 85/84, 99/98
| [{{val| 22 35 51 62 76 90 }}]
| -1.09
| 2.65
| 4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

=== Uniform maps ===
{{Uniform map|13|21.5|22.5}}

=== Commas ===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name
|-
| 3
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| {{monzo| 35 -22 }}
| 156.98
|
|
|-
| 5
| [[250/243]]
| {{monzo| 1 -5 3 }}
| 49.17
| Triyo
| Porcupine comma
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]]
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 5
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| {{monzo| -53 10 16 }}
| 0.57
| Quadla-quadquadyo
| [[Kwazy]]
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[2430/2401]]
| {{monzo| 1 5 1 -4 }}
| 20.79
| Quadru-ayo
| Nuwell
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Trizo-agu
| Hemimage
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell
|-
| 7
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo| -6 -8 2 5 }}
| 1.12
| Quinzo-ayoyo
| [[Wizma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 31
| [[125/124]]
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler
|}
<references />

=== Rank-2 temperaments ===
* [[List of 22et rank two temperaments by badness]]
* [[List of 22et rank two temperaments by complexity]]
* [[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods per octave
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
| [[Porcupine]]
|-
| 1
| 5\22
| [[Orwell]] (22) / blair (22) / winston (22f)
|-
| 1
| 7\22
| [[Magic]] / telepathy
|-
| 1
| 9\22
| [[Superpyth]] / [[suprapyth]]
|-
| 2
| 1\22
| [[Shrutar]] / hemipaj [[Comic]]
|-
| 2
| 2\22
| [[Srutal]] / [[pajara]] / pajarous
|-
| 2
| 3\22
| [[Hedgehog]] / [[echidna]]
|-
| 2
| 4\22
| [[Astrology]] [[Antikythera]] [[Wizard]]
|-
| 2
| 5\22
| [[Doublewide]] / fleetwood
|-
| 11
| 1\22
| [[Undeka]] [[Hendecatonic]]
|}

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

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

== Notation ==
=== Superpyth/Porcupine Notation ===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

=== Porcupine Notation ===
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.

=== 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.

=== 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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

=== 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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

=== Ups and Downs Notation ===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

=== Comparison of 22edo notation systems ===

{| class="wikitable center-all right-2"
|-
! [[Degree]]
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" | Pentatonic
! colspan="3" | Decatonic
! colspan="3" | Sagittal
! colspan="3" | Ups and Downs
|-
| 0
| 0
| Natural Unison
| 1
| perfect unison
| P1
| D
| perfect unison
| P1
| D
| natural 1st
| N1
| C
|
|
|
| perfect unison
| P1
| D
|-
| 1
| 55
| s-minor second
| sm2
| aug unison
| A1
| D#
| aug unison
| A1
| D#
| flat 2nd
| f2
| C#, δb
|
|
|
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
|-
| 2
| 109
| p-diminished second
| pd2
| dim 2nd
| d2
| Eb
| double-aug unison, double-dim sub3rd
| AA1, dds3
| Dx, Fb3 
| natural 2nd
| N2
| δ
|
|
|
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
|-
| 3
| 164
| p-minor second
| pm2
| perfect 2nd
| P2
| E
| dim sub3rd
| ds3
| Fbb
| sharp 2nd, flat 3rd
| s2, f3
| δ#, Db
|
|
|
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
|-
| 4
| 218
| (s/p) Major second
| M2
| aug 2nd
| A2
| E#
| minor sub3rd
| ms3
| Fb
| natural 3rd
| N3
| D
|
|
|
| major 2nd
| M2
| E
|-
| 5
| 273
| s-minor third
| sm3
| dim 3rd
| d3
| Fb
| major sub3rd
| Ms3
| F
| sharp 3rd
| s3
| D#
|
|
|
| minor 3rd
| m3
| F
|-
| 6
| 327
| p-minor third
| pm3
| minor 3rd
| m3
| F
| aug sub3rd
| As3
| F#
| flat 4th
| f4
| εb
|
|
|
| upminor 3rd
| ^m3
| ^F
|-
| 7
| 382
| p-Major third
| pM3
| major 3rd
| M3
| F#
| double-aug sub3rd, double-dim 4thoid
| AAs3, dd4d
| Fx, Gbb
| natural 4th
| N4
| ε
|
|
|
| downmajor 3rd
| vM3
| vF#
|-
| 8
| 436
| s-Major third
| sM3
| aug 3rd, dim 4th
| A3, d4
| Fx, Gb
| dim 4thoid
| d4d
| Gb
| sharp 4th, flat 5th
| s4, f5
| ε#, Eb
|
|
|
| major 3rd
| M3
| F#
|-
| 9
| 491
| Natural Fourth
| 4, N4
| minor 4th
| m4
| G
| perfect 4thoid
| P4d
| G
| natural 5th
| N5
| E
|
|
|
| perfect 4th
| P4
| G
|-
| 10
| 545
| p-Major fourth, s-dim fifth
| pM4, sd5
| major 4th
| M4
| G#
| aug 4thoid
| A4d
| G#
| sharp 5th, flat 6th
| s5, f6
| E#, γb
|
|
|
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
| A4, HO
| aug 4th, dim 5th
| A4, d5
| Gx, Abb
| double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
| Gx, Abb
| natural 6th
| N6
| γ
|
|
|
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
|-
| 12
| 655
| p-minor Fifth, s-aug Fourth
| pm5, sA4
| minor 5th
| m5
| Ab
| dim 5thoid
| d5d
| Ab
| sharp 6th, flat 7th
| s6, f7
| γ#, Gb
|
|
|
| aug 4th, down-5th
| A4, v5
| G#, vA
|-
| 13
| 709
| Natural Fifth
| 5, N5
| major 5th
| M5
| A
| perfect 5thoid
| P5d
| A
| natural 7th
| N7
| G
|
|
|
| perfect 5th
| P5
| A
|-
| 14
| 764
| s-minor sixth
| sm6
| aug 5th, dim 6th
| A5, d6
| A#, Bbb
| aug 5thoid
| A5d
| A#
| sharp 7th
| s7
| G#
|
|
|
| minor 6th
| m6
| Bb
|-
| 15
| 818
| p-minor sixth
| pm6
| minor 6th
| m6
| Bb
| double-aug 5thoid, double-dim sub7th
| AA5d, dds7
| Ax, Cb3
| flat 8th
| f8
| αb
|
|
|
| upminor 6th
| ^m6
| ^Bb
|-
| 16
| 873
| p-Major sixth
| pM6
| major 6th
| M6
| B
| dim sub7th
| ds7
| Cbb
| natural 8th
| N8
| α
|
|
|
| downmajor 6th
| vM6
| vB
|-
| 17
| 927
| s-Major sixth
| sM6
| aug 6th
| A6
| B#
| minor sub7th
| ms7
| Cb
| sharp 8th, flat 9th
| s8, f9
| α#, Ab
|
|
|
| major 6th
| M6
| B
|-
| 18
| 982
| (s/p) minor seventh
| m7
| dim 7th
| d7
| Cb
| major sub7th
| Ms7
| C
| natural 9th
| N9
| A
|
|
|
| minor 7th
| m7
| C
|-
| 19
| 1036
| p-Major seventh
| pM7
| perfect 7th
| P7
| C
| aug sub7th
| As7
| C#
| sharp 9th, flat 10th
| s9, f10
| A#, βb
|
|
|
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
|-
| 20
| 1091
| p-Augmented seventh
| pA7
| aug 7th
| A7
| C#
| double-aug sub7th, double-dim octave
| AAs7, dd8
| Cx, Dbb
| natural 10th
| N10
| β
|
|
|
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
|-
| 21
| 1145
| s-Major seventh
| sM7
| dim 8ve
| d8
| Db
| dim octave
| d8
| Db
| sharp 10th
| s10
| β#, Cb
|
|
|
| major 7th, down 8ve
| M7, v8
| C#, vD
|-
| 22
| 1200
| Octave
| 8
| perfect octave
| P8
| D
| perfect octave
| P8
| D
| natural 11th
| N11
| C
|
|
|
| perfect octave
| P8
| D
|}

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

{| class="wikitable center-all"
|-
! Quality
! [[Color name]]
! [[Monzo]] Format
! Examples
|-
| rowspan="2" |minor
| zo
| [a b 0 1>
| 7/6, 7/4
|-
| fourthward wa
| [a b> where b < -1
| 32/27, 16/9
|-
| upminor
| gu
| [a b -1>
| 6/5, 9/5
|-
| downmajor
| yo
| [a b 1>
| 5/4, 5/3
|-
| rowspan="2" |major
| fifthward wa
| [a b> where b > 1
| 9/8, 27/16
|-
| ru
| [a b 0 -1>
| 9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Notes as edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo
| 6:7:9
| 0-5-13
| C Eb G
| Cm
| C minor
|-
| gu
| 10:12:15
| 0-6-13
| C ^Eb G
| C^m
| C upminor
|-
| yo
| 4:5:6
| 0-7-13
| C vE G
| Cv
| C downmajor or C down
|-
| ru
| 14:18:21
| 0-8-13
| C E G
| C
| C major or C
|}

Examples:

* 0-4-13 = C D G = C2
* 0-9-13 = C F G = C4
* 0-10-13 = C ^F G = C^4 or C(^4)
* 0-5-10 = C Eb Gb = Cd = Cdim
* 0-5-11 = C Eb ^Gb = Cd(^5)
* 0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

* [[22edo Chord Names]]
* [[22 EDO Chords]]
* [[Ups and Downs Notation #Chords and Chord Progressions]]
* [[Chords of orwell]]

== Music ==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

== Related pages ==
* [[Lumatone mapping for 22edo]]
* [[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
* [[22edo/Eliora's approach|22edo/Eliora's Approach]]

== Further reading ==
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
* [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]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

== References ==
# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
# Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-03-11T07:04:20Z

YoVariable: Updated the Ups and Downs Notation in /* Notation */

{{interwiki
| de = 22-EDO
| en = 22edo
| es =
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

== Theory ==
=== History ===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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''.

=== Overview to JI approximation quality ===
The 22et 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 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]] to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22et, unlike 12 and [[19edo|19]], is not a [[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.

22et can also be treated as adding harmonics 3 and 5 to 11edo's 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its 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.

22et is very close to an extended "quarter-comma archy", 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]]).

=== Prime harmonics ===
{{Harmonics in equal|22|columns=11}}

=== Subsets and supersets ===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

== Intervals ==
{{See also| 22edo solfege }}

{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! Audio
|-
| 0
| 0.000
| [[1/1]]
| perfect unison
| P1
| D
| [[File:0-0.000c_P1.mp3]]
|-
| 1
| 54.545
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
| [[File:0-54.545c_22edo.mp3]]
|-
| 2
| 109.091
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
| [[File:0-109.091c_11edo.mp3]]
|-
| 3
| 163.636
| [[12/11]], [[11/10]], [[10/9]]
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
| [[File:0-163.636c_22edo.mp3]]
|-
| 4
| 218.182
| [[9/8]], [[17/15]], [[8/7]]
| major 2nd
| M2
| E
| [[File:0-218.182c_11edo.mp3]]
|-
| 5
| 272.727
| [[20/17]], [[7/6]]
| minor 3rd
| m3
| F
| [[File:0-272.727c_22edo.mp3]]
|-
| 6
| 327.273
| [[6/5]], [[17/14]], [[11/9]]
| upminor 3rd
| ^m3
| ^F
| [[File:0-327.273c_11edo.mp3]]
|-
| 7
| 381.818
| [[5/4]], [[96/77]]
| downmajor 3rd
| vM3
| vF#
| [[File:0-381.818c_22edo.mp3]]
|-
| 8
| 436.364
| [[14/11]], [[9/7]], [[22/17]]
| major 3rd
| M3
| F#
| [[File:0-436.364c_11edo.mp3]]
|-
| 9
| 490.909
| [[4/3]]
| perfect 4th
| P4
| G
| [[File:0-490.909c_22edo.mp3]]
|-
| 10
| 545.455
| [[15/11]], [[11/8]]
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
| [[File:0-545.455c_11edo.mp3]]
|-
| 11
| 600.000
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
| [[File:0-600.000c_2edo.mp3]]
|-
| 12
| 654.545
| [[16/11]], [[22/15]]
| aug 4th, down-5th
| A4, v5
| G#, vA
| [[File:0-654.545c_11edo.mp3]]
|-
| 13
| 709.091
| [[3/2]]
| perfect 5th
| P5
| A
| [[File:0-709.091c_22edo.mp3]]
|-
| 14
| 763.636
| [[17/11]], [[14/9]], [[11/7]]
| minor 6th
| m6
| Bb
| [[File:0-763.636c_11edo.mp3]]
|-
| 15
| 818.182
| [[8/5]], [[77/48]]
| upminor 6th
| ^m6
| ^Bb
| [[File:0-818.182c_22edo.mp3]]
|-
| 16
| 872.727
| [[18/11]], [[28/17]], [[5/3]]
| downmajor 6th
| vM6
| vB
| [[File:0-872.727c_11edo.mp3]]
|-
| 17
| 927.273
| [[17/10]], [[12/7]]
| major 6th
| M6
| B
| [[File:0-927.273c_22edo.mp3]]
|-
| 18
| 981.818
| [[7/4]], [[30/17]], [[16/9]]
| minor 7th
| m7
| C
| [[File:0-981.818c_11edo.mp3]]
|-
| 19
| 1036.364
| [[9/5]], [[11/6]], [[20/11]]
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
| [[File:0-1036.364c_22edo.mp3]]
|-
| 20
| 1090.909
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
| [[File:0-1090.909c_11edo.mp3]]
|-
| 21
| 1145.455
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| major 7th, down 8ve
| M7, v8
| C#, vD
| [[File:0-1145.455c_22edo.mp3]]
|-
| 22
| 1200.000
| [[2/1]]
| perfect octave
| P8
| D
| [[File:0-1200.000c_P8.mp3]]
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

== Approximation to JI ==
=== 15-odd-limit interval mappings ===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" | 15-odd-limit intervals by direct approximation (even if inconsistent)
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[9/7]], [[14/9]]
| 1.280
| 2.3
|-
| [[11/10]], [[20/11]]
| 1.368
| 2.5
|-
| [[15/8]], [[16/15]]
| 2.640
| 4.8
|-
| '''[[5/4]], [[8/5]]'''
| '''4.496'''
| '''8.2'''
|-
| [[7/6]], [[12/7]]
| 5.856
| 10.7
|-
| '''[[11/8]], [[16/11]]'''
| '''5.863'''
| '''10.7'''
|-
| '''[[3/2]], [[4/3]]'''
| '''7.136'''
| '''13.1'''
|-
| [[15/11]], [[22/15]]
| 8.504
| 15.6
|-
| [[15/14]], [[28/15]]
| 10.352
| 19.0
|-
| [[5/3]], [[6/5]]
| 11.631
| 21.3
|-
| '''[[7/4]], [[8/7]]'''
| '''12.992'''
| '''23.8'''
|-
| [[11/6]], [[12/11]]
| 12.999
| 23.8
|-
| [[9/8]], [[16/9]]
| 14.272
| 26.2
|-
| [[13/11]], [[22/13]]
| 16.482
| 30.2
|-
| [[7/5]], [[10/7]]
| 17.488
| 32.1
|-
| [[13/10]], [[20/13]]
| 17.850
| 32.7
|-
| ''[[13/9]], [[18/13]]''
| ''17.928''
| ''32.9''
|-
| [[9/5]], [[10/9]]
| 18.767
| 34.4
|-
| [[11/7]], [[14/11]]
| 18.856
| 34.6
|-
| ''[[13/7]], [[14/13]]''
| ''19.207''
| ''35.2''
|-
| [[11/9]], [[18/11]]
| 20.135
| 36.9
|-
| '''[[13/8]], [[16/13]]'''
| '''22.346'''
| '''41.0'''
|-
| [[15/13]], [[26/15]]
| 24.986
| 45.8
|-
| ''[[13/12]], [[24/13]]''
| ''25.064''
| ''46.0''
|}
{{15-odd-limit|22}}

=== Selected 17-limit intervals ===
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]

== Defining features ==

=== Septimal vs syntonic comma ===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma ===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

=== 5-limit commas ===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas ===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

=== Other features ===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| 2.85
| 5.23
|-
| 2.3.5.7.11
| 50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
| -1.11
| 2.90
| 5.33
|-
| 2.3.5.7.11.17
| 50/49, 55/54, 64/63, 85/84, 99/98
| [{{val| 22 35 51 62 76 90 }}]
| -1.09
| 2.65
| 4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

=== Uniform maps ===
{{Uniform map|13|21.5|22.5}}

=== Commas ===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name
|-
| 3
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| {{monzo| 35 -22 }}
| 156.98
|
|
|-
| 5
| [[250/243]]
| {{monzo| 1 -5 3 }}
| 49.17
| Triyo
| Porcupine comma
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]]
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 5
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| {{monzo| -53 10 16 }}
| 0.57
| Quadla-quadquadyo
| [[Kwazy]]
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[2430/2401]]
| {{monzo| 1 5 1 -4 }}
| 20.79
| Quadru-ayo
| Nuwell
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Trizo-agu
| Hemimage
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell
|-
| 7
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo| -6 -8 2 5 }}
| 1.12
| Quinzo-ayoyo
| [[Wizma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 31
| [[125/124]]
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler
|}
<references />

=== Rank-2 temperaments ===
* [[List of 22et rank two temperaments by badness]]
* [[List of 22et rank two temperaments by complexity]]
* [[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods per octave
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
| 1
| 3\22
| [[Porcupine]]
|-
| 1
| 5\22
| [[Orwell]] (22) / blair (22) / winston (22f)
|-
| 1
| 7\22
| [[Magic]] / telepathy
|-
| 1
| 9\22
| [[Superpyth]] / [[suprapyth]]
|-
| 2
| 1\22
| [[Shrutar]] / hemipaj [[Comic]]
|-
| 2
| 2\22
| [[Srutal]] / [[pajara]] / pajarous
|-
| 2
| 3\22
| [[Hedgehog]] / [[echidna]]
|-
| 2
| 4\22
| [[Astrology]] [[Antikythera]] [[Wizard]]
|-
| 2
| 5\22
| [[Doublewide]] / fleetwood
|-
| 11
| 1\22
| [[Undeka]] [[Hendecatonic]]
|}

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

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

== Notation ==
=== Superpyth/Porcupine Notation ===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

=== Porcupine Notation ===
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.

=== 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.

=== 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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

=== 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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

=== Ups and Downs Notation ===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

=== Comparison of 22edo notation systems ===

{| class="wikitable center-all right-2"
|-
! [[Degree]]
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" | Pentatonic
! colspan="3" | Decatonic
! colspan="3" | Sagittal
! colspan="3" | Ups and Downs
|-
| 0
| 0
| Natural Unison
| 1
| perfect unison
| P1
| D
| perfect unison
| P1
| D
| natural 1st
| N1
| C
|
|
|
| perfect unison
| P1
| D
|-
| 1
| 55
| s-minor second
| sm2
| aug unison
| A1
| D#
| aug unison
| A1
| D#
| flat 2nd
| f2
| C#, δb
|
|
|
| up-unison, minor 2nd
| ^1, m2
| ^D, Eb
|-
| 2
| 109
| p-diminished second
| pd2
| dim 2nd
| d2
| Eb
| double-aug unison, double-dim sub3rd
| AA1, dds3
| Dx, Fb3 
| natural 2nd
| N2
| δ
|
|
|
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vD#, ^Eb
|-
| 3
| 164
| p-minor second
| pm2
| perfect 2nd
| P2
| E
| dim sub3rd
| ds3
| Fbb
| sharp 2nd, flat 3rd
| s2, f3
| δ#, Db
|
|
|
| aug 1sn, downmajor 2nd
| A1, vM2
| D#, vE
|-
| 4
| 218
| (s/p) Major second
| M2
| aug 2nd
| A2
| E#
| minor sub3rd
| ms3
| Fb
| natural 3rd
| N3
| D
|
|
|
| major 2nd
| M2
| E
|-
| 5
| 273
| s-minor third
| sm3
| dim 3rd
| d3
| Fb
| major sub3rd
| Ms3
| F
| sharp 3rd
| s3
| D#
|
|
|
| minor 3rd
| m3
| F
|-
| 6
| 327
| p-minor third
| pm3
| minor 3rd
| m3
| F
| aug sub3rd
| As3
| F#
| flat 4th
| f4
| εb
|
|
|
| upminor 3rd
| ^m3
| ^F
|-
| 7
| 382
| p-Major third
| pM3
| major 3rd
| M3
| F#
| double-aug sub3rd, double-dim 4thoid
| AAs3, dd4d
| Fx, Gbb
| natural 4th
| N4
| ε
|
|
|
| downmajor 3rd
| vM3
| vF#
|-
| 8
| 436
| s-Major third
| sM3
| aug 3rd, dim 4th
| A3, d4
| Fx, Gb
| dim 4thoid
| d4d
| Gb
| sharp 4th, flat 5th
| s4, f5
| ε#, Eb
|
|
|
| major 3rd
| M3
| F#
|-
| 9
| 491
| Natural Fourth
| 4, N4
| minor 4th
| m4
| G
| perfect 4thoid
| P4d
| G
| natural 5th
| N5
| E
|
|
|
| perfect 4th
| P4
| G
|-
| 10
| 545
| p-Major fourth, s-dim fifth
| pM4, sd5
| major 4th
| M4
| G#
| aug 4thoid
| A4d
| G#
| sharp 5th, flat 6th
| s5, f6
| E#, γb
|
|
|
| up-4th, dim 5th
| ^4, d5
| ^G, Ab
|-
| 11
| 600
| p-Augmented Fourth, p-diminished Fifth, Half-Octave
| A4, HO
| aug 4th, dim 5th
| A4, d5
| Gx, Abb
| double-aug 4thoid, double-dim 5thoid
| AA4d, dd5d
| Gx, Abb
| natural 6th
| N6
| γ
|
|
|
| downaug 4th, updim 5th
| vA4, ^d5
| vG#, ^Ab
|-
| 12
| 655
| p-minor Fifth, s-aug Fourth
| pm5, sA4
| minor 5th
| m5
| Ab
| dim 5thoid
| d5d
| Ab
| sharp 6th, flat 7th
| s6, f7
| γ#, Gb
|
|
|
| aug 4th, down-5th
| A4, v5
| G#, vA
|-
| 13
| 709
| Natural Fifth
| 5, N5
| major 5th
| M5
| A
| perfect 5thoid
| P5d
| A
| natural 7th
| N7
| G
|
|
|
| perfect 5th
| P5
| A
|-
| 14
| 764
| s-minor sixth
| sm6
| aug 5th, dim 6th
| A5, d6
| A#, Bbb
| aug 5thoid
| A5d
| A#
| sharp 7th
| s7
| G#
|
|
|
| minor 6th
| m6
| Bb
|-
| 15
| 818
| p-minor sixth
| pm6
| minor 6th
| m6
| Bb
| double-aug 5thoid, double-dim sub7th
| AA5d, dds7
| Ax, Cb3
| flat 8th
| f8
| αb
|
|
|
| upminor 6th
| ^m6
| ^Bb
|-
| 16
| 873
| p-Major sixth
| pM6
| major 6th
| M6
| B
| dim sub7th
| ds7
| Cbb
| natural 8th
| N8
| α
|
|
|
| downmajor 6th
| vM6
| vB
|-
| 17
| 927
| s-Major sixth
| sM6
| aug 6th
| A6
| B#
| minor sub7th
| ms7
| Cb
| sharp 8th, flat 9th
| s8, f9
| α#, Ab
|
|
|
| major 6th
| M6
| B
|-
| 18
| 982
| (s/p) minor seventh
| m7
| dim 7th
| d7
| Cb
| major sub7th
| Ms7
| C
| natural 9th
| N9
| A
|
|
|
| minor 7th
| m7
| C
|-
| 19
| 1036
| p-Major seventh
| pM7
| perfect 7th
| P7
| C
| aug sub7th
| As7
| C#
| sharp 9th, flat 10th
| s9, f10
| A#, βb
|
|
|
| upminor 7th, dim 8ve
| ^m7, d8
| ^C, Db
|-
| 20
| 1091
| p-Augmented seventh
| pA7
| aug 7th
| A7
| C#
| double-aug sub7th, double-dim octave
| AAs7, dd8
| Cx, Dbb
| natural 10th
| N10
| β
|
|
|
| downmajor 7th, updim 8ve
| vM7, ^d8
| vC#, ^Db
|-
| 21
| 1145
| s-Major seventh
| sM7
| dim 8ve
| d8
| Db
| dim octave
| d8
| Db
| sharp 10th
| s10
| β#, Cb
|
|
|
| major 7th, down 8ve
| M7, v8
| C#, vD
|-
| 22
| 1200
| Octave
| 8
| perfect octave
| P8
| D
| perfect octave
| P8
| D
| natural 11th
| N11
| C
|
|
|
| perfect octave
| P8
| D
|}

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

{| class="wikitable center-all"
|-
! Quality
! [[Color name]]
! [[Monzo]] Format
! Examples
|-
| rowspan="2" |minor
| zo
| [a b 0 1>
| 7/6, 7/4
|-
| fourthward wa
| [a b> where b < -1
| 32/27, 16/9
|-
| upminor
| gu
| [a b -1>
| 6/5, 9/5
|-
| downmajor
| yo
| [a b 1>
| 5/4, 5/3
|-
| rowspan="2" |major
| fifthward wa
| [a b> where b > 1
| 9/8, 27/16
|-
| ru
| [a b 0 -1>
| 9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Notes as edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo
| 6:7:9
| 0-5-13
| C Eb G
| Cm
| C minor
|-
| gu
| 10:12:15
| 0-6-13
| C ^Eb G
| C^m
| C upminor
|-
| yo
| 4:5:6
| 0-7-13
| C vE G
| Cv
| C downmajor or C down
|-
| ru
| 14:18:21
| 0-8-13
| C E G
| C
| C major or C
|}

Examples:

* 0-4-13 = C D G = C2
* 0-9-13 = C F G = C4
* 0-10-13 = C ^F G = C^4 or C(^4)
* 0-5-10 = C Eb Gb = Cd = Cdim
* 0-5-11 = C Eb ^Gb = Cd(^5)
* 0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

* [[22edo Chord Names]]
* [[22 EDO Chords]]
* [[Ups and Downs Notation #Chords and Chord Progressions]]
* [[Chords of orwell]]

== Music ==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

== Related pages ==
* [[Lumatone mapping for 22edo]]
* [[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
* [[22edo/Eliora's approach|22edo/Eliora's Approach]]

== Further reading ==
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
* [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]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

== References ==
# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
# Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

22edo

2024-03-10T08:40:39Z

YoVariable: /* Intervals */ Added audio to each 22edo interval and made the Ups and Downs Notation easier to read.

{{interwiki
| de = 22-EDO
| en = 22edo
| es =
| ja = 22平均律
}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

== Theory ==
=== History ===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist [https://en.wikipedia.org/wiki/Robert_Holford_Macdowall_Bosanquet| R. H. M. Bosanquet]. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosanquet 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''.

=== Overview to JI approximation quality ===
The 22et 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 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]] to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22et, unlike 12 and [[19edo|19]], is not a [[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.

22et can also be treated as adding harmonics 3 and 5 to 11edo'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 its 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.

22et is very close to an extended "quarter-comma archy", 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]]).

=== Prime harmonics ===
{{Harmonics in equal|22|columns=11}}

=== Subsets and supersets ===
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). 11edo is interesting for sounding melodically very similar to 12edo (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 notation]], 11 can be notated as every other note of 22.

== Intervals ==
{{See also| 22edo solfege }}

{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate Ratios*
! colspan="3" |[[Ups and Downs Notation]]
! Audio
|-
|0
|0.000
|[[1/1]]
|perfect unison
|P1
|D
|[[File:0-0.000c_P1.mp3]]
|-
|1
|54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
|^1, m2
|^D, Eb
|[[File:0-54.545c_22edo.mp3]]
|-
|2
|109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
|vA1, ^m2
|vD#, ^Eb
|[[File:0-109.091c_11edo.mp3]]
|-
|3
|163.636
|[[12/11]], [[11/10]], [[10/9]]
| aug 1sn, downmajor 2nd
|A1, vM2
|D#, vE
|[[File:0-163.636c_22edo.mp3]]
|-
|4
|218.182
|[[9/8]], [[17/15]], [[8/7]]
|major 2nd
|M2
|E
|[[File:0-218.182c_11edo.mp3]]
|-
|5
|272.727
|[[20/17]], [[7/6]]
|minor 3rd
|m3
|F
|[[File:0-272.727c_22edo.mp3]]
|-
|6
|327.273
|[[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
|^m3
|^F
|[[File:0-327.273c_11edo.mp3]]
|-
|7
|381.818
|[[5/4]], [[96/77]]
|downmajor 3rd
|vM3
|vF#
|[[File:0-381.818c_22edo.mp3]]
|-
| 8
|436.364
|[[14/11]], [[9/7]], [[22/17]]
|major 3rd
|M3
|F#
|[[File:0-436.364c_11edo.mp3]]
|-
|9
|490.909
|[[4/3]]
|perfect 4th
| P4
|G
|[[File:0-490.909c_22edo.mp3]]
|-
|10
|545.455
|[[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|[[File:0-545.455c_11edo.mp3]]
|-
| 11
|600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
| vG#, ^Ab
|[[File:0-600.000c_2edo.mp3]]
|-
| 12
|654.545
|
[[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|[[File:0-654.545c_11edo.mp3]]
|-
|13
|709.091
|[[3/2]]
|perfect 5th
|P5
|A
|[[File:0-709.091c_22edo.mp3]]
|-
|14
|763.636
|[[17/11]], [[14/9]], [[11/7]]
|minor 6th
|m6
|Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|15
|818.182
|[[8/5]], [[77/48]]
|upminor 6th
|^m6
| ^Bb
|[[File:0-818.182c_22edo.mp3]]
|-
| 16
|872.727
|[[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
|vM6
|vB
|[[File:0-872.727c_11edo.mp3]]
|-
|17
|927.273
|[[17/10]], [[12/7]]
|major 6th
|M6
|B
|[[File:0-927.273c_22edo.mp3]]
|-
|18
|981.818
|[[7/4]], [[30/17]], [[16/9]]
|minor 7th
|m7
|C
|[[File:0-981.818c_11edo.mp3]]
|-
|19
|1036.364
|[[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
|^m7, d8
|^C, Db
|[[File:0-1036.364c_22edo.mp3]]
|-
|20
|1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
|vM7, ^d8
|vC#, ^Db
|[[File:0-1090.909c_11edo.mp3]]
|-
|21
|1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|major 7th, down 8ve
|M7, v8
|C#, vD
|[[File:0-1145.455c_22edo.mp3]]
|-
|22
|1200.000
|[[2/1]]
|perfect octave
|P8
|D
|[[File:0-1200.000c_P8.mp3]]
|}

<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

==JI approximation==
===15-odd-limit interval mappings===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |15-odd-limit intervals by direct approximation (even if inconsistent)
!Interval, complement
! Error (abs, [[Cent|¢]])
!Error (rel, [[Relative cent|%]])
|-
|[[9/7]], [[14/9]]
|1.280
| 2.3
|-
|[[11/10]], [[20/11]]
|1.368
| 2.5
|-
|[[15/8]], [[16/15]]
|2.640
|4.8
|-
|'''[[5/4]], [[8/5]]'''
|'''4.496'''
|'''8.2'''
|-
|[[7/6]], [[12/7]]
|5.856
|10.7
|-
|'''[[11/8]], [[16/11]]'''
|'''5.863'''
|'''10.7'''
|-
|
'''[[3/2]], [[4/3]]'''
|'''7.136'''
|'''13.1'''
|-
|[[15/11]], [[22/15]]
|8.504
|15.6
|-
|[[15/14]], [[28/15]]
|10.352
|19.0
|-
|[[5/3]], [[6/5]]
|11.631
|21.3
|-
|'''[[7/4]], [[8/7]]'''
|'''12.992'''
|
'''23.8'''
|-
|[[11/6]], [[12/11]]
|12.999
|23.8
|-
|[[9/8]], [[16/9]]
|14.272
|26.2
|-
|[[13/11]], [[22/13]]
|16.482
|30.2
|-
|[[7/5]], [[10/7]]
|17.488
|32.1
|-
|[[13/10]], [[20/13]]
|17.850
|32.7
|-
|''[[13/9]], [[18/13]]''
|''17.928''
|''32.9''
|-
|[[9/5]], [[10/9]]
|18.767
|34.4
|-
|[[11/7]], [[14/11]]
| 18.856
|34.6
|-
|''[[13/7]], [[14/13]]''
|''19.207''
|''35.2''
|-
|[[11/9]], [[18/11]]
|20.135
|36.9
|-
|'''[[13/8]], [[16/13]]'''
|'''22.346'''
|'''41.0'''
|-
|[[15/13]], [[26/15]]
|24.986
|45.8
|-
|''[[13/12]], [[24/13]]''
|''25.064''
|''46.0''
|}
{{15-odd-limit|22}}

===Selected 17-limit intervals===
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]

==Defining features==

===Septimal vs syntonic comma===
Possibly the most striking characteristic of 22edo 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 [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo 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]].

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 22et. 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 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

=== Porcupine comma===
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

===5-limit commas===
Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

===7-limit commas ===
In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (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 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas===
In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property – although it should be noted that the related [[159edo]] ''does''.

===Other features ===
The 164¢ "flat minor whole tone" is a key interval in 22edo, 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 22edo 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.

22edo also supports the [[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 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

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 notation]], 11 can be notated as every other note of 22.

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| 35 -22 }}
|[{{val| 22 35 }}]
| -2.25
|2.25
|4.12
|-
|2.3.5
|250/243, 2048/2025
|[{{val| 22 35 51 }}]
| -0.86
| 2.70
|4.94
|-
|2.3.5.7
|50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
| -1.80
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
| -1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
| -1.09
|2.65
|4.87
|}

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].

===Uniform maps===
{{Uniform map|13|21.5|22.5}}

===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|
{{monzo| 35 -22 }}
|156.98
|
|
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
|Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
| 5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
| Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|
[[Kwazy]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
| {{monzo| 6 -2 0 -1 }}
|27.26
|Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
|21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
| Quadru-ayo
|Nuwell
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
| Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
|7.71
|Ruyoyo
|Marvel comma
|-
|7
|[[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
|6.48
| Trizo-agu
|Hemimage
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.36
|Saruru-atrigu
|Porwell
|-
| 7
|[[65625/65536]]
|{{monzo| -16 1 5 1 }}
|2.35
|Lazoquinyo
|Horwell
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
|Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
| Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.39
|Satrilu-aruru
|Orgonisma
|-
|11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
|4.50
|Lozoyo
|Keenanisma
|-
|11
|[[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
|3.21
|Lururuyo
|Swetisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.03
|Triluyo
|Wizardharry
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
|Kalisma
|-
|13
|
[[65/64]]
|{{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
|13
|[[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
|22.34
| Tholuru
|Negustma
|-
| 13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
|19.13
|Thozogu
|Superleap
|-
|31
|[[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
|Thiwutriyo
|Twizzler
|}
<references />

===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
!Periods per octave
!Generator
!Temperaments
|-
| 1
|1\22
|[[Sensamagic clan #Sensa|Sensa]] [[Chromo]] [[Ceratitid]]
|-
|1
|3\22
|[[Porcupine]]
|-
| 1
|5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|
[[Magic]] / telepathy
|-
|1
| 9\22
|
[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj [[Comic]]
|-
|2
|2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]] [[Antikythera]] [[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]] [[Hendecatonic]]
|}

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

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

==Notation==
===Superpyth/Porcupine Notation===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine 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.

===Porcupine Notation===
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.

===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.

===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 2: γ δ α ε β

The alphabet is, in ascending order: 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.

===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:

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

We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

[[File:22edo Sagittal.png|800px]]

===Ups and Downs Notation===

Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[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]]

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

[[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]]

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.

[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]

Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.

<gallery mode="slideshow">
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>

===Comparison of 22edo notation systems===

{| class="wikitable center-all right-2"
|-
![[Degree]]
![[cent|Cents]]
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" |Porcupine
! colspan="3" |Pentatonic
! colspan="3" |Decatonic
! colspan="3" |Sagittal
! colspan="3" |Ups and Downs
|-
|0
| 0
|Natural Unison
|1
|perfect unison
|P1
|D
|perfect unison
|P1
|D
|natural 1st
|N1
|C
|
|
|
|perfect unison
|P1
|D
|-
|1
|55
| s-minor second
|sm2
|aug unison
|A1
|D#
| aug unison
|A1
|D#
|flat 2nd
|f2
|C#, δb
|
|
|
|minor 2nd
|m2
|Eb
|-
|2
|109
|p-diminished second
|pd2
|dim 2nd
|d2
|Eb
|double-aug unison, double-dim sub3rd
|AA1, dds3
|Dx, Fb3 
|natural 2nd
|N2
|δ
|
|
|
|upminor 2nd
| ^m2
| ^Eb
|-
|3
|164
|p-minor second
|pm2
|perfect 2nd
|P2
|E
|dim sub3rd
|ds3
|Fbb
|sharp 2nd, flat 3rd
|s2, f3
|δ#, Db
|
|
|
|downmajor 2nd
| vM2
|vE
|-
|4
|218
|(s/p) Major second
|M2
|aug 2nd
|A2
|E#
|minor sub3rd
|ms3
|Fb
|natural 3rd
|N3
|D
|
|
|
| major 2nd
| M2
|E
|-
|5
|273
|s-minor third
|sm3
| dim 3rd
| d3
|Fb
|major sub3rd
|Ms3
|F
|sharp 3rd
|s3
|D#
|
|
|
|minor 3rd
|m3
|F
|-
|6
|327
|p-minor third
|pm3
|minor 3rd
|m3
|F
|aug sub3rd
|As3
|F#
|flat 4th
| f4
|εb
|
|
|
|upminor 3rd
| ^m3
|^F
|-
| 7
|382
|p-Major third
|pM3
|major 3rd
|M3
|F#
|double-aug sub3rd, double-dim 4thoid
|AAs3, dd4d
|Fx, Gbb
| natural 4th
|N4
|ε
|
|
|
|downmajor 3rd
|vM3
|vF#
|-
|8
|436
|s-Major third
|sM3
|aug 3rd, dim 4th
|A3, d4
|Fx, Gb
|dim 4thoid
| d4d
|Gb
|sharp 4th, flat 5th
| s4, f5
|ε#, Eb
|
|
|
|major 3rd
|M3
|F#
|-
|9
|491
|Natural Fourth
|4, N4
|minor 4th
| m4
|G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
| E
|
|
|
|perfect fourth
|P4
|G
|-
|10
|545
|p-Major fourth, s-dim fifth
|pM4, sd5
|major 4th
| M4
|G#
|aug 4thoid
|A4d
|G#
|sharp 5th, flat 6th
|s5, f6
|E#, γb
|
|
|
|up-4th, dim 5th
|^4, d5
|^G, Ab
|-
|11
|600
|p-Augmented Fourth,
p-diminished Fifth
Half-Octave
| A4, HO
|aug 4th, dim 5th
|A4, d5
|Gx, Abb
|double-aug 4thoid, double-dim 5thoid
|AA4d, dd5d
|Gx, Abb
| natural 6th
| N6
|γ
|
|
|
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|-
|12
|655
| p-minor Fifth, s-aug Fourth
|pm5, sA4
|minor 5th
| m5
|Ab
|dim 5thoid
|d5d
|Ab
|sharp 6th, flat 7th
|s6, f7
| γ#, Gb
|
|
|
|aug 4th, down-5th
|A4, v5
|G#, vA
|-
|13
| 709
|Natural Fifth
|5, N5
| major 5th
|M5
|A
|perfect 5thoid
| P5d
|A
|natural 7th
|N7
|G
|
|
|
|perfect 5th
|P5
|A
|-
|14
|764
|s-minor sixth
|sm6
| aug 5th, dim 6th
|A5, d6
|A#, Bbb
|aug 5thoid
|A5d
|A#
|sharp 7th
|s7
| G#
|
|
|
|minor 6th
|m6
|Bb
|-
|15
|818
|p-minor sixth
|pm6
|minor 6th
|m6
|Bb
|double-aug 5thoid, double-dim sub7th
| AA5d, dds7
|Ax, Cb3
|flat 8th
|f8
|αb
|
|
|
|upminor 6th
|^m6
|^Bb
|-
| 16
|873
|p-Major sixth
|pM6
|major 6th
| M6
|B
|dim sub7th
|ds7
|Cbb
|natural 8th
|N8
|α
|
|
|
| downmajor 6th
| vM6
|vB
|-
|17
|927
|s-Major sixth
|sM6
|aug 6th
|A6
|B#
| minor sub7th
| ms7
|Cb
|sharp 8th, flat 9th
|s8, f9
|α#, Ab
|
|
|
|major 6th
|M6
|B
|-
|18
|982
|(s/p) minor seventh
| m7
|dim 7th
|d7
|Cb
| major sub7th
| Ms7
|C
| natural 9th
|N9
|A
|
|
|
|minor 7th
|m7
|C
|-
|19
|1036
|p-Major seventh
|pM7
|perfect 7th
|P7
|C
|aug sub7th
|As7
|C#
|sharp 9th, flat 10th
|s9, f10
|A#, βb
|
|
|
|upminor 7th
|^m7
|^C
|-
|20
| 1091
|p-Augmented seventh
| pA7
|aug 7th
|A7
| C#
|double-aug sub7th, double-dim octave
|AAs7, dd8
|Cx, Dbb
|natural 10th
| N10
| β
|
|
|
|downmajor 7th
|vM7
|vC#
|-
|21
| 1145
|s-Major seventh
|sM7
|dim 8ve
|d8
|Db
|dim octave
|d8
|Db
|sharp 10th
|s10
|β#, Cb
|
|
|
| major 7th
|M7
|C#
|-
|22
|1200
| Octave
|8
|perfect octave
|P8
|D
|perfect octave
| P8
|D
|natural 11th
|N11
|C
|
|
|
|perfect octave
|P8
|D
|}

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

{| class="wikitable center-all"
|-
!quality
![[color name]]
![[monzo]] format
!examples
|-
| rowspan="2" |minor
|zo
|[a b 0 1>
| 7/6, 7/4
|-
|fourthward wa
|[a b> where b < -1
|32/27, 16/9
|-
| upminor
|gu
|[a b -1>
|6/5, 9/5
|-
|downmajor
|yo
|[a b 1>
|5/4, 5/3
|-
| rowspan="2" |major
|fifthward wa
|[a b> where b > 1
|9/8, 27/16
|-
|ru
|[a b 0 -1>
|9/7, 12/7
|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

{| class="wikitable center-all"
|-
![[Kite's color notation|color of the 3rd]]
! JI chord
!notes as edosteps
! notes of C chord
! written name
!spoken name
|-
|zo
| 6:7:9
|0-5-13
|C Eb G
|Cm
|C minor
|-
| gu
|10:12:15
|0-6-13
|C ^Eb G
|C^m
| C upminor
|-
|yo
|4:5:6
|0-7-13
|C vE G
|Cv
|C downmajor or C down
|-
|ru
| 14:18:21
|0-8-13
|C E G
|C
|C major or C
|}

Examples:

*0-4-13 = C D G = C2
*0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
* 0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

*[[22edo Chord Names]]
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]

==Music ==
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}

==Related pages==
*[[Lumatone mapping for 22edo]]
*[[William_Lynch's_Thoughts_on_Septimal_Harmony_and_22_EDO|William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]

== Further reading==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
*[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]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]

== References==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://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

[[Category:Twentuning]]
[[Category:Alpharabian]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]

File:0-1200.000c P8.mp3

2024-03-10T08:35:41Z

YoVariable:

File:0-1145.455c 22edo.mp3

2024-03-10T08:34:43Z

YoVariable:

File:0-1090.909c 11edo.mp3

2024-03-10T08:34:08Z

YoVariable:

File:0-1036.364c 22edo.mp3

2024-03-10T08:33:09Z

YoVariable:

File:0-981.818c 11edo.mp3

2024-03-10T08:32:28Z

YoVariable:

File:0-927.273c 22edo.mp3

2024-03-10T08:31:21Z

YoVariable:

File:0-872.727c 11edo.mp3

2024-03-10T08:30:34Z

YoVariable:

File:0-818.182c 22edo.mp3

2024-03-10T08:27:12Z

YoVariable:

File:0-763.636c 11edo.mp3

2024-03-10T08:26:44Z

YoVariable:

File:0-709.091c 22edo.mp3

2024-03-10T08:24:48Z

YoVariable: