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'''13 equal divisions of the octave''' ('''13edo''') is a tuning system which divides the [[octave]] into 13 equal parts of approximately 92.3 [[cent]]s each.  
== Theory ==
13edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.


== Theory ==
The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.
In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.
 
Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
 
One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]).
 
In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.  


As a temperament of [[21-odd-limit]] [[just intonation]], 13edo has excellent approximations to the 11th and 21st [[harmonic]]s, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|13}}
{{Harmonics in equal|13}}


Line 19: Line 26:
13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].


One step of 13edo is very close to [[135/128]] by direct approximation, in fact one might stress that it is a [[Wikipedia:Continued_fraction|semiconvergent]]. The 5-limit [[aluminium]] temperament realizes this proximity through a regular temperament perspective, and EDOs supporting it (for example, [[494edo]] or [[1547edo]]), combine the sound of 13edo with relative simplicities and precision of 5-limit JI.
The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.)  A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.


== 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 37:
! 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)
! Oneirotonic
! Oneirotonic
! [[26edo]] names
(Octatonic 5th-generated)
! Fox-Raven<br>Notation (J = 360Hz)
![[26edo]] names
(subset
 
notation)
! Fox-Raven<br>(J = 360Hz)
! Pseudo-Diatonic<br>Category
! Pseudo-Diatonic<br>Category
! Audio
! Audio
Line 37: Line 52:
| 0
| 0
| 0.00
| 0.00
| [[1/1]]
|[[1/1]]
| H
| H
| C
| C
Line 44: Line 59:
| 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 55: Line 70:
| 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 66: Line 81:
| 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 77: Line 92:
| 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 88: Line 103:
| 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 99: Line 114:
| 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 110: Line 125:
| 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 121: Line 136:
| 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 132: Line 147:
| 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 143: Line 158:
| 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
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| 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 165: Line 180:
| 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 176: Line 191:
| 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 187: Line 202:
| J
| J
| Octave
| Octave
| [[File:piano_1_1edo.mp3]]
|[[File:piano_1_1edo.mp3]]
|}
|}
<references/>
<references />
 
== Notations ==
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#...
 
...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) ===
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 unisons: v⁴A1, ^m2
 
{| class="wikitable center-all right-2"
|-
! #
! Cents
! colspan="3" |[[Ups and downs notation|Up/down notation]] using the wide 5th of 8\13
|-
| 0
| 0
| perfect unison
| P1
| D
|-
| 1
| 92
| up unison, mid 2nd
| ^1, ~2
| ^D, ^^Eb, vvE
|-
| 2
| 185
| downmajor 2nd, (minor 3rd)
| vM2, (m3)
| vE, (F)
|-
| 3
| 277
| major 2nd, upminor 3rd
| M2, ^m3
| E, ^F
|-
| 4
| 369
| mid 3rd
| ~3
| ^^F, vvF#
|-
| 5
| 462
| perfect 4th
| P4
| G
|-
| 6
| 554
| up 4th, dud 5th
| ^4, vv5
| ^G, vvA
|-
| 7
| 646
| dup 4th, down 5th
| ^^4, v5
| ^^G, vA
|-
| 8
| 738
| perfect 5th
| P5
| A
|-
| 9
| 831
| mid 6th
| ~6
| ^^Bb, vvB
|-
| 10
| 923
| downmajor 6th, minor 7th
| vM6
| vB, C
|-
| 11
| 1015
| (major 6th), upminor 7th
| (M6), ^M7
| (B), ^C
|-
| 12
| 1108
| mid 7th, down 8ve
| ~7, v8
| ^^C, vvC#, vD
|-
| 13
| 1200
| perfect 8ve
| P8
| D
|}
{{Sharpness-sharp4}}
Half-sharps and half-flats can also be used, making the ascending scale:
D E{{Demiflat2}} vE ^F F{{Demisharp2}} G ^G vA A B{{Demiflat2}} vB ^C C{{Demisharp2}} D
 
=== Heptatonic 5th-generated (narrow fifth) ===
The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th avoids the minor 2nd being descending.


== Notation ==
There are two ways to do this. The first way preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
13edo can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.


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 unisons 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 200: Line 327:
! #
! #
! 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 330: Line 457:
|}
|}


This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, and heptatonic 2nd-generated.
=== Pentatonic 5th-generated (3L2s) ===
The degrees are named unison, subthird, fourthoid, fifthoid, subseventh and octoid.  


'''<u>Pentatonic 5th-generated</u>:''' '''D * * E * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)
Keyboard: '''D * F * * 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 unison: dds3
{| class="wikitable"
|+notes/intervals in melodic order (s = sub-, d = -oid)
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|D
|D#
Fb
|F
|F#
|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
!...
|-
|...
|Fb
|Cb
|Gb
|Db
|Ab
|F
|C
|G
|D
|A
|F#
|C#
|G#
|D#
|A#
|Fx
|Cx
|...
|-
|...
|ds3
|ds7
|d4d
|d8d
|d5d
|ms3
|ms7
|P4d
|P1
|P5d
|Ms3
|Ms7
|A4d
|A1
|A5d
|As3
|As7
|...
|}


P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d (s = sub-, d = -oid)
=== Octatonic 5th-generated (5L3s or oneirotonic) ===
Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)  


pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
Enharmonic unison: d2
{| class="wikitable"
|+notes/intervals in melodic order
!0
!1
!2
!3
!4
!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
|...
|}


pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)
=== Heptatonic 2nd-generated (6L1s or archaeotonic) ===
Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)


'''<u>Octatonic 5th-generated</u>:''' '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)
Enharmonic unison: dd2
{| class="wikitable"
|+notes/intervals in melodic order
!0
!1
!2
!3
!4
!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
|...
|}


A - A#/Bb - B - C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - H - H#/Ab - A
=== Heptatonic 3rd-generated (3L4s or mosh) ===
This notation requires ups and downs because 7 perfect thirds octave-reduces to 2 edosteps, not 1.


P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)


octotonic genchain of sixths: ..D# - A# - F# - C# - H# - E - B - G - D - A - F - C - H - Eb - Bb - Gb - Db - Ab...
Enharmonic unisons: vvA1, vm2
{| class="wikitable"
|+notes/intervals in melodic order
!0
!1
!2
!3
!4
!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
|...
|}


octotonic genchain of sixths: ...M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7...
=== 26edo subset ===
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.


'''<u>Heptatonic 2nd-generated</u>:''' '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)
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).


D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)


P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
Enharmonic unison: 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
|Dx
Ebb
|E
|Fb
|F#
|Gb
|G#
|Ab
|A#
|Bb
|B#
|C
|Cx
Dbb
|D
|-
!version #2
|D#
|Eb
|E#
|F
|Fx
Gbb
|G
|Gx
Abb
|A
|Ax
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#
|Cx
|Dx
|...
|-
!version #2
|...
|Gbb
|Abb
|Bbb
|Cb
|Db
|Eb
|F
|G
|A
|B
|C#
|D#
|E#
|Fx
|Gx
|...
|-
!
|...
|dd8
|d2
|d3
|d4
|d5
|m6
|m7
|P1
|M2
|M3
|A4
|A5
|A6
|A7
|AA1
|...
|}


genchain of seconds: ...Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#...
====Sagittal notation====
This notation is a subset of the notations for EDOs [[26edo#Sagittal notation|26]] and [[52edo#Sagittal notation|52]].
=====Evo flavor=====


genchain of seconds: ...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3...
<imagemap>
File:13-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 447 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]
default [[File:13-EDO_Evo_Sagittal.svg]]
</imagemap>


[[:File:13edo-chromatic-scale.mid|13edo chromatic ascending and descending scale on C (MIDI)]]
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
=====Revo flavor=====


[[File:13_Edo_chromatic_scale_on_J.mp3]]
<imagemap>
File:13-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 495 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 495 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]
default [[File:13-EDO_Revo_Sagittal.svg]]
</imagemap>


== JI approximation ==
== Approximation to JI ==
=== Selected 13-odd-limit intervals ===
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]
=== Local zeta peak ===
{{Main | 13edo and optimal octave stretching }}
At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].


== Tuning by ear ==
== Tuning by ear ==
13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.
13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.


== Phi vibes ==
== Approximation to irrational intervals ==
=== Golden ratio ===
13edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].


=== Acoustic phi ===
Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.
13edo has a very close approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] are even closer, but unlike all closer EDOs, 13-EDO has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13-EDO is the only EDO that tempers the ratios of the Fibonacci sequence into a single interval.


See also: [[9edϕ]]
See also: [[9edϕ]]
=== Logarithmic phi ===
As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error.
Not until [[144edo|144]] do we find a better EDO in terms of relative error on these two intervals.
However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2**(phi) ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant phi ≃ 1.6180.
That being said, logarithmic phi has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales.


{| class="wikitable center-all"
{| class="wikitable center-all"
|+Direct mapping
|+Direct approximation
|-
|-
! Interval
! Interval
! Error (abs, [[Cent|¢]])
! Error (abs, [[Cent|¢]])
|-
|-
| 2**(ϕ) / ϕ
| 2<sup>ϕ</sup> / ϕ
| 0.858
| 0.858
|-
|-
Line 400: Line 1,152:
| 2.321
| 2.321
|-
|-
| 2**(ϕ)
| 2<sup>ϕ</sup>
| 3.179
| 3.179
|}
|}
== Local zeta peak ==
{{Main | 13edo and optimal octave stretching }}
At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].


== Scales ==
== Scales ==
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Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):
Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
* archeotonic [[6L 1s]] 2222221 (2\13, 1\1)
* archaeotonic [[6L 1s]] 2222221 (2\13, 1\1)
* [[Chromatic_pairs#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
* [[Chromatic_pairs#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)
* [[No-threes subgroup temperaments#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)
Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two [[degree]]s of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.
Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two [[degree]]s of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.


Line 425: Line 1,173:


Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.
Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.
=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS
3 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS


== Harmony in 13edo ==
== Harmony in 13edo ==
Line 466: Line 1,207:
To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.
To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.


==== Modes and Harmony in the Archeotonic Scale ====
==== Modes and harmony in the archaeotonic scale ====
The 2\13-based heptatonic has been named '''archeotonic''' after the "Old Ones" that rule the Dreamlands. Modes of the archeotonic are named after the individual Old Ones.
The 2\13-based heptatonic has been named '''archaeotonic''' after the "Old Ones" that rule the Dreamlands. Modes of the archaeotonic are named after the individual Old Ones.


A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.
A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.


[[File:Archeotonic.png|alt=Archeotonic.png|Archeotonic.png]]
[[File:Archaeotonic.png|Archaeotonic.png|link=Special:FilePath/Archaeotonic.png]]


Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
Line 477: Line 1,218:
There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.
There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.


==== Modes and Harmony in the Oneirotonic Scale ====
==== Modes and harmony in the oneirotonic scale ====
The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.
The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.


Line 486: Line 1,227:
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.


=== Fox, Inthar and Jaimbee's approach ===
== Mapping to standard keyboards ==
Our approach is based on [[5L 3s]]. In fact, we have an absolute pitch notation for 5L 3s edos called the [[User:Inthar/Fox-Raven notation|Fox-Raven notation]].
 
== Mapping to Standard Keyboards ==
The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.


Line 691: Line 1,429:
== Regular temperament properties ==
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|12.5|13.5}}
{{Uniform map|edo=13}}


=== Commas ===
=== Commas ===
13edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)
13et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime <br> limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
Line 724: Line 1,462:
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma, Avicenna's Enharmonic Diesis
| Avicennma, Avicenna's enharmonic diesis
|-
|-
| 7
| 7
Line 731: Line 1,469:
| 27.26
| 27.26
| Ru
| Ru
| Septimal Comma, Archytas' Comma, Leipziger Komma
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|-
| 7
| 7
Line 738: Line 1,476:
| 22.23
| 22.23
| Laquadzo-atrigu
| Laquadzo-atrigu
| Squalentine
| Squalentine comma
|-
|-
| 7
| 7
Line 745: Line 1,483:
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
Line 787: Line 1,525:
| 16.57
| 16.57
| Thuzoyo
| Thuzoyo
| Animist
| Animist comma
|-
|-
| 13
| 13
Line 799: Line 1,537:


=== Animism ===
=== Animism ===
The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:
The animist comma, 105/104, appears whenever {{nowrap| ~3 × ~5 × ~7 = ~2<sup>3</sup> × ~13 }}… 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:


0 4 5 8 9 13 pentatonic
0 4 5 8 9 13 pentatonic
Line 807: Line 1,545:
0 1 3 4 5 8 9 10 12 13 nonatonic
0 1 3 4 5 8 9 10 12 13 nonatonic


== Introductory Materials ==
== Introductory materials ==
=== By Inthar ===
=== By Inthar ===
* [[File:13edo_1MC.mp3|270px]] 13edo example composition ([[File:13edo_1MC_score.pdf|score]])
* [[File:13edo_1MC.mp3|270px]] 13edo example composition ([[File:13edo_1MC_score.pdf|score]])
Line 826: Line 1,564:


== See also ==
== See also ==
* [[13edo/Inthar's approach|Inthar's approach to 13edo]]
* [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]]
* [[13EDO Scales and Chords for Guitar]]
* [[13EDO Scales and Chords for Guitar]]
* [[Lumatone mapping for 13edo]]
* [[Lumatone mapping for 13edo]]
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo keyboard pieces in a variety of keys and modes)
* Approaches:
** [[Kentaku's Approach to 13EDO|William Lynch's approach]]
** [[User:Inthar/13edo|Inthar's approach]]
* [[Fendo family]] - temperaments closely related to 13edo


== Further reading ==
== Further reading ==
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Triskaidecaphonic Scales for Guitar: Practical Theory and Scales on the Thirteen Equal Divisions of the Octave]''. 2009.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Triskaidecaphonic Scales for Guitar: Practical Theory and Scales on the Thirteen Equal Divisions of the Octave]''. 2009.


[[Category:13edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:13-tone scales]]
[[Category:13-tone scales]]
[[Category:Listen]]
[[Category:Prime EDO]]
[[Category:Teentuning]]
[[Category:Teentuning]]
[[Category:Pages with internal sound examples]]
[[Category:Pages with internal sound examples]]
[[Category:Oneirotonic]]
[[Category:Oneirotonic]]


{{Todo| Cleanup }}
{{Todo|cleanup|add rank 2 temperaments table}}
Retrieved from "https://en.xen.wiki/w/13edo"