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== Intervals ==
== Intervals ==
[[:File:13edo-chromatic-scale.mid|13edo chromatic ascending and descending scale on C (MIDI)]]
[[File:13_Edo_chromatic_scale_on_J.mp3]]
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
Line 27: Line 30:
! Cents
! Cents
! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>
! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>
! [[Erv Wilson's Linear Notations|Erv Wilson]]
![[Erv Wilson's Linear Notations|Erv Wilson]]
! Archaeotonic
! Archaeotonic
(Heptatonic 2nd-generated)
(Heptatonic 2nd-generated)
! Oneirotonic
! Oneirotonic
(Octatonic 5th-generated)
(Octatonic 5th-generated)
! [[26edo]] names
![[26edo]] names
(subset
(subset


Line 42: Line 45:
| 0
| 0
| 0.00
| 0.00
| [[1/1]]
|[[1/1]]
| H
| H
| C
| C
Line 49: Line 52:
| J
| J
| Unison
| Unison
| [[File:piano_0_1edo.mp3]]
|[[File:piano_0_1edo.mp3]]
|-
|-
| 1
| 1
| 92.31
| 92.31
| [[17/16]], [[18/17]], [[19/18]], [[20/19]], [[21/20]], [[22/21]]
|[[17/16]], [[18/17]], [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| β
| β
| C#/Db
| C#/Db
Line 60: Line 63:
| J#/Kb
| J#/Kb
| Minor second
| Minor second
| [[File:piano_1_13edo.mp3]]
|[[File:piano_1_13edo.mp3]]
|-
|-
| 2
| 2
| 184.62
| 184.62
| [[9/8]], [[10/9]], [[11/10]], [[19/17]], [[21/19]]
|[[9/8]], [[10/9]], [[11/10]], [[19/17]], [[21/19]]
| A
| A
| D
| D
Line 71: Line 74:
| K
| K
| Major second
| Major second
| [[File:piano_2_13edo.mp3]]
|[[File:piano_2_13edo.mp3]]
|-
|-
| 3
| 3
| 276.92
| 276.92
| [[7/6]], [[13/11]], [[20/17]], [[19/16]], [[22/19]]
|[[7/6]], [[13/11]], [[20/17]], [[19/16]], [[22/19]]
| δ
| δ
| D#/Eb
| D#/Eb
Line 82: Line 85:
| L
| L
| Minor third
| Minor third
| [[File:piano_3_13edo.mp3]]
|[[File:piano_3_13edo.mp3]]
|-
|-
| 4
| 4
| 369.23
| 369.23
| [[5/4]], [[11/9]], [[16/13]], [[26/21]]
|[[5/4]], [[11/9]], [[16/13]], [[26/21]]
| C
| C
| E
| E
Line 93: Line 96:
| L#/Mb
| L#/Mb
| Major third
| Major third
| [[File:piano_4_13edo.mp3]]
|[[File:piano_4_13edo.mp3]]
|-
|-
| 5
| 5
| 461.54
| 461.54
| [[13/10]], [[17/13]], [[21/16]], [[22/17]]
|[[13/10]], [[17/13]], [[21/16]], [[22/17]]
| B
| B
| E#/Fb
| E#/Fb
Line 104: Line 107:
| M
| M
| Minor fourth
| Minor fourth
| [[File:piano_5_13edo.mp3]]
|[[File:piano_5_13edo.mp3]]
|-
|-
| 6
| 6
| 553.85
| 553.85
| [[11/8]], [[18/13]], [[26/19]]
|[[11/8]], [[18/13]], [[26/19]]
| ε
| ε
| F
| F
Line 115: Line 118:
| M#/Nb
| M#/Nb
| Major fourth/Minor tritone
| Major fourth/Minor tritone
| [[File:piano_6_13edo.mp3]]
|[[File:piano_6_13edo.mp3]]
|-
|-
| 7
| 7
| 646.15
| 646.15
| [[16/11]], [[13/9]], [[19/13]]
|[[16/11]], [[13/9]], [[19/13]]
| D
| D
| F#/Gb
| F#/Gb
Line 126: Line 129:
| N
| N
| Minor fifth/Major tritone
| Minor fifth/Major tritone
| [[File:piano_7_13edo.mp3]]
|[[File:piano_7_13edo.mp3]]
|-
|-
| 8
| 8
| 738.46
| 738.46
| [[17/11]], [[20/13]], [[26/17]], [[32/21]]
|[[17/11]], [[20/13]], [[26/17]], [[32/21]]
| γ
| γ
| G
| G
Line 137: Line 140:
| O
| O
| Major fifth
| Major fifth
| [[File:piano_8_13edo.mp3]]
|[[File:piano_8_13edo.mp3]]
|-
|-
| 9
| 9
| 830.77
| 830.77
| [[8/5]], [[13/8]], [[18/11]], [[21/13]]
|[[8/5]], [[13/8]], [[18/11]], [[21/13]]
| F
| F
| G#/Ab
| G#/Ab
Line 148: Line 151:
| O#/Pb
| O#/Pb
| Minor sixth
| Minor sixth
| [[File:piano_9_13edo.mp3]]
|[[File:piano_9_13edo.mp3]]
|-
|-
| 10
| 10
| 923.08
| 923.08
| [[17/10]], [[12/7]], [[22/13]], [[19/11]]
|[[17/10]], [[12/7]], [[22/13]], [[19/11]]
| E
| E
| A
| A
Line 159: Line 162:
| P
| P
| Major sixth
| Major sixth
| [[File:piano_10_13edo.mp3]]
|[[File:piano_10_13edo.mp3]]
|-
|-
| 11
| 11
| 1015.38
| 1015.38
| [[9/5]], [[16/9]], [[20/11]], [[34/19]], [[38/21]]
|[[9/5]], [[16/9]], [[20/11]], [[34/19]], [[38/21]]
| α
| α
| A#/Bb
| A#/Bb
Line 170: Line 173:
| Q
| Q
| Minor seventh
| Minor seventh
| [[File:piano_11_13edo.mp3]]
|[[File:piano_11_13edo.mp3]]
|-
|-
| 12
| 12
| 1107.69
| 1107.69
| [[17/9]], [[19/10]], [[21/11]], [[32/17]], [[36/19]], [[40/21]]
|[[17/9]], [[19/10]], [[21/11]], [[32/17]], [[36/19]], [[40/21]]
| G
| G
| B/Cb
| B/Cb
Line 181: Line 184:
| Q#/Jb
| Q#/Jb
| Major seventh
| Major seventh
| [[File:piano_12_13edo.mp3]]
|[[File:piano_12_13edo.mp3]]
|-
|-
| 13
| 13
| 1200.00
| 1200.00
| [[2/1]]
|[[2/1]]
| H
| H
| C/B#
| C/B#
Line 192: Line 195:
| J
| J
| Octave
| Octave
| [[File:piano_1_1edo.mp3]]
|[[File:piano_1_1edo.mp3]]
|}
|}
<references/>
<references />


== Notations ==
== Notations ==
Only the first two notations are backwards-compatible. They both allow conventional notation to be used, including the staff, note names, relative notation, etc. And they allow a piece in conventional notation to be translated to 13edo. They both use the conventional genchain of fifths:
There are seven categories of notation. Only the first two categories are backwards-compatible. They both allow conventional notation to be used, including the staff, note names, relative notation, chord names, etc. And they both allow a piece in conventional notation to be translated to 13edo. They both use the conventional genchain of fifths:


...Db - Ab - Eb - Bb - F - C - G - D -A - E - B - F#- C# - G# - D#...
...Db - Ab - Eb - Bb - F - C - G - D -A - E - B - F#- C# - G# - D#...


...d8 - d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 - A1...
...d8 - d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 - A1...
Except the first version of the second notation swaps sharp and flat, major and minor, and augmented and diminished.


=== Heptatonic 5th-generated (wide 5th) ===
=== Heptatonic 5th-generated (wide 5th) ===
13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.
13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.
Enharmonic intervals: ^m2, vvvvA1


{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
Line 303: Line 310:


The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".
The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".
The first approach has enharmonic intervals of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.


{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
Line 308: Line 317:
! #
! #
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
|-
|-
Line 439: Line 448:


=== Pentatonic 5th-generated ===
=== Pentatonic 5th-generated ===
The degrees are named unison, subthird, fourthoid, fifthoid, subseventh and octoid.
The degrees are named unison, subthird, fourthoid, fifthoid, subseventh and octoid.  


'''D * * E * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)
Keyboard: '''D * * E * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)


D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
Enharmonic interval: dds3
 
{| class="wikitable"
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d (s = sub-, d = -oid)
|+notes/intervals in melodic order (s = sub-, d = -oid)
 
!0
pentatonic genchain of fifths:
!1
 
!2
...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
!3
 
!4
...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|D
|D#
|Eb
|E
|E#
Gb
|G
|G#
|Ab
|A
|A#
Cb
|C
|C#
|Db
|D
|-
|P1
|A1
ds3
|ms3
|Ms3
|As3
d4d
|P4d
|A4d
|d5d
|P5d
|A5d
ds7
|ms7
|Ms7
|As7
d8d
|P8d
|}
{| class="wikitable"
|+notes/intervals in genchain order (s = sub-, d = -oid)
!...
!-8
!-7
!-6
!-5
!-4
!-3
!-2
!-1
!0
!1
!2
!3
!4
!5
!6
!7
!8
!...
|-
|...
|Ebb
|Cb
|Gb
|Db
|Ab
|Eb
|C
|G
|D
|A
|E
|C#
|G#
|D#
|A#
|E#
|Cx
|...
|-
|...
|ds3
|ds7
|d4d
|d8d
|d5d
|ms3
|ms7
|P4d
|P1
|P5d
|Ms3
|Ms7
|A4d
|A1
|A5d
|As3
|As7
|...
|}


=== Octatonic 5th-generated (oneirotonic) ===
=== Octatonic 5th-generated (oneirotonic) ===
'''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)  
Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)  


A - A#/Bb - B - C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - H - H#/Ab - A
Enharmonic interval: d2
 
{| class="wikitable"
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
|+notes/intervals in melodic order
 
!0
octotonic genchain of sixths:
!1
 
!2
...D# - A# - F# - C# - H# - E - B - G - D - A - F - C - H - Eb - Bb - Gb - Db - Ab...
!3
 
!4
...M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7...
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|A
|A#
Bb
|B
|C
|C#
Db
|D
|D#
Eb
|E
|F
|F#
Gb
|G
|H
|H#
Ab
|A
|-
|P1
|A1
m2
|M2
|m3
|M3
|P4
|m5
|M5
|P6
|m7
|M7
|m8
|M8
d9
|P9
|}
{| class="wikitable"
|+notes/intervals in genchain order
!...
!-8
!-7
!-6
!-5
!-4
!-3
!-2
!-1
!0
!1
!2
!3
!4
!5
!6
!7
!8
!...
|-
|...
|D#
|A#
|F#
|C#
|H#
|E
|B
|G
|D
|A
|F
|C
|H
|Eb
|Bb
|Gb
|Db
|...
|-
|...
|A1
|A6
|M3
|M8
|M5
|M2
|M7
|P4
|P1
|P6
|m3
|m8
|m5
|m2
|m7
|d4
|d8
|...
|}


=== Heptatonic 2nd-generated (archaeotonic) ===
=== Heptatonic 2nd-generated (archaeotonic) ===
'''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)
Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)


D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
Enharmonic interval: dd2
 
{| class="wikitable"
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
|+notes/intervals in melodic order
 
!0
genchain of seconds:
!1
 
!2
...Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#...
!3
 
!4
...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3...
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|D
|D#
Eb
|E
|E#
Fb
|F
|F#
Gb
|G
|A
|A#
Bb
|B
|B#
Cb
|C
|C#
Db
|D
|-
|P1
|A1
d2
|P2
|m3
|M3
|m4
|M4
|m5
|M5
|m6
|M6
|P7
|A7
d8
|P8
|}
{| class="wikitable"
|+notes/intervals in genchain order
!...
!-7
!-6
!-5
!-4
!-3
!-2
!-1
!0
!1
!2
!3
!4
!5
!6
!7
!...
|-
|...
|Db
|Eb
|Fb
|Gb
|A
|B
|C
|D
|E
|F
|G
|A#
|B#
|C#
|D#
|...
|-
|...
|d8
|d2
|m3
|m4
|m5
|m6
|P7
|P1
|P2
|M3
|M4
|M5
|M6
|A7
|A1
|...
|}


=== Heptatonic 3rd-generated (mosh) ===
=== Heptatonic 3rd-generated (mosh) ===
This notation requires ups and downs because 7 perfect thirds octave-reduces to 2 edosteps, not 1.
This notation requires ups and downs because 7 perfect thirds octave-reduces to 2 edosteps, not 1.


'''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)
Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)


D - E - ^E/Fb - E#/vF - F - G - ^G/Ab - G#/vA - A - B - ^B/Cb - B#/vC - C - D
Enharmonic intervals: vvA1, vm2, dd3
 
{| class="wikitable"
P1 - m2 - ~2/d3 - M2/v3 - P3 - m4 - ~4/m5 - M4/~5 - M5 - P6 - ^6/m7 - A6/~7 - M7 - P8
|+notes/intervals in melodic order
 
!0
genchain of thirds:
!1
 
!2
...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#...
!3
 
!4
...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|D
|E
|^E
Fb
|E#
vF
|F
|G
|^G
Ab
|G#
vA
|A
|B
|^B
Cb
|B#
vC
|C
|D
|-
|P1
|m2
|~2
d3
|M2
v3
|P3
|m4
|~4
m5
|M4
~5
|M5
|P6
|^6
m7
|A6
~7
|M7
|P8
|}
{| class="wikitable"
|+notes/intervals in genchain order
!...
!-7
!-6
!-5
!-4
!-3
!-2
!-1
!0
!1
!2
!3
!4
!5
!6
!7
!...
|-
|...
|Db
|Fb
|Ab
|Cb
|E
|G
|B
|D
|F
|A
|C
|E#
|G#
|B#
|D#
|...
|-
|...
|d8
|d3
|m5
|m7
|m2
|m4
|P6
|P1
|P3
|M5
|M7
|M2
|M4
|A6
|A1
|...
|}


=== 26edo subset ===
=== 26edo subset ===
This notation uses every other name of [[26edo]]. There are no perfect 4ths or 5ths, only augmented and diminished ones. There are two versions of this notation. One has only three natural notes (C, D and E) and the other one has only four (F, G, A and B).
This notation uses every other note name of [[26edo]]. There are no perfect 4ths or 5ths, only augmented and diminished ones. There are no minor 2nds or augmented 1sns.


'''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 15\26 = 7.5\13 = perfect 5th)
There are two versions of the absolute notation. One has only three natural notes (C, D and E) and the other one has only four (F, G, A and B).


[[:File:13edo-chromatic-scale.mid|13edo chromatic ascending and descending scale on C (MIDI)]]
Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)


[[File:13_Edo_chromatic_scale_on_J.mp3]]
Enharmonic interval: ddd2
{| class="wikitable"
|+notes/intervals in melodic order
!
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
!version #1
|D
|D##
Ebb
|E
|Fb
|F#
|Gb
|G#
|Ab
|A#
|Bb
|B#
|C
|C##
Dbb
|D
|-
!version #2
|D#
|Eb
|E#
|F
|F##
Gbb
|G
|G##
Abb
|A
|A##
Bbb
|B
|Cb
|C#
|Db
|D#
|-
!
|P1
|d2
|M2
|d3
|M3
|d4
|A4
|d5
|A5
|m6
|A6
|m7
|A7
|P8
|}
{| class="wikitable"
|+notes/intervals in genchain order
!
!...
!-7
!-6
!-5
!-4
!-3
!-2
!-1
!0
!1
!2
!3
!4
!5
!6
!7
!...
|-
!version #1
|...
|Dbb
|Ebb
|Fb
|Gb
|Ab
|Bb
|C
|D
|E
|F#
|G#
|A#
|B#
|C##
|D##
|...
|-
!version #2
|...
|Gbb
|Abb
|Bbb
|Cb
|Db
|Eb
|F
|G
|A
|B
|C#
|D#
|E#
|F##
|G##
|...
|-
!
|...
|dd8
|d2
|d3
|d4
|d5
|m6
|m7
|P1
|M2
|M3
|A4
|A5
|A6
|A7
|AA1
|...
|}


== JI approximation ==
== JI approximation ==
Retrieved from "https://en.xen.wiki/w/13edo"