Xenharmonic Wiki

13edo: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,004 edits
No edit summary
Tags: Mobile edit Mobile web edit
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Text replacement - "Ups and Downs Notation" to "Ups and downs notation"
Tags: Mobile edit Mobile web edit
 
(243 intermediate revisions by 30 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| de =  
| de = 13-EDO
| en = 13edo
| en = 13edo
| es =  
| es =  
| ja = 13平均律
| ja = 13平均律
}}
}}
__FORCETOC__
{{Infobox ET}}
{{ED intro}}


=13edo: 13 equal divisions of the octave=
== Theory ==
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth [[prime_numbers|prime]] edo, following [[11edo|11edo]] and coming before [[17edo|17edo]]. 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.
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.


As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, 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.
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.  


{| class="wikitable"
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.
 
=== Odd harmonics ===
 
{{Harmonics in equal|13}}
 
=== Subsets and supersets ===
13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
 
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 ==
[[: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"
|-
|-
! | Degree
! #
! | Cents
! Cents
! | Approximated 21-limit Ratios*
! 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
![[Erv Wilson's Linear Notations|Erv Wilson]]
! | Archaeotonic
! Archaeotonic
! | Oneirotonic
(Heptatonic 2nd-generated)
! | [[26edo|26edo]] names
! Oneirotonic
! | Kentaku
(Octatonic 5th-generated)
![[26edo]] names
(subset
 
notation)
! Fox-Raven<br>(J = 360Hz)
! Pseudo-Diatonic<br>Category
! Audio
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.00
| 0.00
| style="text-align:center;" | 1/1
|[[1/1]]
| style="text-align:center;" | H
| H
| style="text-align:center;" | C
| C
| style="text-align:center;" | C
| C
| style="text-align:center;" | C
| C
| style="text-align:center;" | J
| J
| Unison
|[[File:piano_0_1edo.mp3]]
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 92.31
| 92.31
| style="text-align:center;" | 17/16, 18/17, 19/18, 20/19, 21/20, 22/21
|[[17/16]], [[18/17]], [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| style="text-align:center;" | β
| β
| style="text-align:center;" | C#/Db
| C#/Db
| style="text-align:center;" | C#/Db
| C#/Db
| style="text-align:center;" | Cx/Dbb
| Cx/Dbb
| style="text-align:center;" | J#/Kb
| J#/Kb
| Minor second
|[[File:piano_1_13edo.mp3]]
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 184.615
| 184.62
| style="text-align:center;" | 9/8, 10/9, 11/10, 19/17, 21/19
|[[9/8]], [[10/9]], [[11/10]], [[19/17]], [[21/19]]
| style="text-align:center;" | A
| A
| style="text-align:center;" | D
| D
| style="text-align:center;" | D
| D
| style="text-align:center;" | D
| D
| style="text-align:center;" | K
| K
| Major second
|[[File:piano_2_13edo.mp3]]
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 276.92
| 276.92
| style="text-align:center;" | 7/6, 13/11, 20/17, 19/16, 22/19
|[[7/6]], [[13/11]], [[20/17]], [[19/16]], [[22/19]]
| style="text-align:center;" | δ
| δ
| style="text-align:center;" | D#/Eb
| D#/Eb
| style="text-align:center;" | D#/Eb
| D#/Eb
| style="text-align:center;" | Dx/Ebb
| Dx/Ebb
| style="text-align:center;" | K#/Lb
| L
| Minor third
|[[File:piano_3_13edo.mp3]]
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 369.23
| 369.23
| style="text-align:center;" | 5/4, 11/9, 16/13, 26/21
|[[5/4]], [[11/9]], [[16/13]], [[26/21]]
| style="text-align:center;" | C
| C
| style="text-align:center;" | E
| E
| style="text-align:center;" | E
| E
| style="text-align:center;" | E
| E
| style="text-align:center;" | L
| L#/Mb
| Major third
|[[File:piano_4_13edo.mp3]]
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 461.54
| 461.54
| style="text-align:center;" | 13/10, 17/13, 21/16, 22/17
|[[13/10]], [[17/13]], [[21/16]], [[22/17]]
| style="text-align:center;" | B
| B
| style="text-align:center;" | E#/Fb
| E#/Fb
| style="text-align:center;" | F
| F
| style="text-align:center;" | Ex/Fb
| Ex/Fb
| style="text-align:center;" | M
| M
| Minor fourth
|[[File:piano_5_13edo.mp3]]
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 553.85
| 553.85
| style="text-align:center;" | 11/8, 18/13, 26/19
|[[11/8]], [[18/13]], [[26/19]]
| style="text-align:center;" | ε
| ε
| style="text-align:center;" | F
| F
| style="text-align:center;" | F#/Gb
| F#/Gb
| style="text-align:center;" | F#
| F#
| style="text-align:center;" | M#/Nb
| M#/Nb
| Major fourth/Minor tritone
|[[File:piano_6_13edo.mp3]]
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 646.15
| 646.15
| style="text-align:center;" | 16/11, 13/9, 19/13
|[[16/11]], [[13/9]], [[19/13]]
| style="text-align:center;" | D
| D
| style="text-align:center;" | F#/Gb
| F#/Gb
| style="text-align:center;" | G
| G
| style="text-align:center;" | Gb
| Gb
| style="text-align:center;" | N
| N
| Minor fifth/Major tritone
|[[File:piano_7_13edo.mp3]]
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 738.46
| 738.46
| style="text-align:center;" | 17/11, 20/13, 26/17, 32/21
|[[17/11]], [[20/13]], [[26/17]], [[32/21]]
| style="text-align:center;" | γ
| γ
| style="text-align:center;" | G
| G
| style="text-align:center;" | G#/Hb
| G#/Hb
| style="text-align:center;" | G#
| G#
| style="text-align:center;" | N#/Ob
| O
| Major fifth
|[[File:piano_8_13edo.mp3]]
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 830.77
| 830.77
| style="text-align:center;" | 8/5, 13/8, 18/11, 21/13
|[[8/5]], [[13/8]], [[18/11]], [[21/13]]
| style="text-align:center;" | F
| F
| style="text-align:center;" | G#/Ab
| G#/Ab
| style="text-align:center;" | H
| H
| style="text-align:center;" | Ab
| Ab
| style="text-align:center;" | O
| O#/Pb
| Minor sixth
|[[File:piano_9_13edo.mp3]]
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 923.08
| 923.08
| style="text-align:center;" | [[17/10|17/10]], [[12/7|12/7]], [[22/13|22/13]], [[19/11|19/11]]
|[[17/10]], [[12/7]], [[22/13]], [[19/11]]
| style="text-align:center;" | E
| E
| style="text-align:center;" | A
| A
| style="text-align:center;" | A
| A
| style="text-align:center;" | A#
| A#
| style="text-align:center;" | P
| P
| Major sixth
|[[File:piano_10_13edo.mp3]]
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 1015.385
| 1015.38
| style="text-align:center;" | 9/5, 16/9, 20/11, 34/19, 38/21
|[[9/5]], [[16/9]], [[20/11]], [[34/19]], [[38/21]]
| style="text-align:center;" | α
| α
| style="text-align:center;" | A#/Bb
| A#/Bb
| style="text-align:center;" | A#/Bb
| A#/Bb
| style="text-align:center;" | Bb
| Bb
| style="text-align:center;" | P#/Qb
| Q
| Minor seventh
|[[File:piano_11_13edo.mp3]]
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 1107.69
| 1107.69
| style="text-align:center;" | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
|[[17/9]], [[19/10]], [[21/11]], [[32/17]], [[36/19]], [[40/21]]
| style="text-align:center;" | G
| G
| style="text-align:center;" | B/Cb
| B/Cb
| style="text-align:center;" | B
| B
| style="text-align:center;" | B#/Cbb
| B#/Cbb
| style="text-align:center;" | Q
| Q#/Jb
| Major seventh
|[[File:piano_12_13edo.mp3]]
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 1200
| 1200.00
| style="text-align:center;" | 2/1
|[[2/1]]
| style="text-align:center;" | H
| H
| style="text-align:center;" | C/B#
| C/B#
| style="text-align:center;" | C
| C
| style="text-align:center;" | C
| C
| style="text-align:center;" | J
| J
| Octave
|[[File:piano_1_1edo.mp3]]
|}
|}
*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.
<references />


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.
== 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:


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".
...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"
{| 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
|-
|-
! | Degree
| 13
! | Cents
| 1200
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]] using the narrow 5th of 7\13,
| 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.
 
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".


with major wider than minor
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.
! colspan="3" | Up/down notation using the narrow 5th of 7\13,


with major narrower than minor
{| class="wikitable center-all right-2"
|-
! #
! 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" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0
| 0
| style="text-align:center;" | perfect unison
| perfect unison
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | D
| D
| style="text-align:center;" | perfect unison
| perfect unison
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | D
| D
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 92
| 92
| style="text-align:center;" | up unison, minor 2nd
| up unison, minor 2nd
| style="text-align:center;" | ^1, m2
| ^1, m2
| style="text-align:center;" | ^D, E
| ^D, E
| style="text-align:center;" | up unison, major 2nd
| up unison, major 2nd
| style="text-align:center;" | ^1, M2
| ^1, M2
| style="text-align:center;" | ^D, E
| ^D, E
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 185
| 185
| style="text-align:center;" | upminor 2nd, minor 3rd
| upminor 2nd, minor 3rd
| style="text-align:center;" | ^m2, m3
| ^m2, m3
| style="text-align:center;" | ^E, Fb
| ^E, Fb
| style="text-align:center;" | upmajor 2nd, major 3rd
| upmajor 2nd, major 3rd
| style="text-align:center;" | ^M2, M3
| ^M2, M3
| style="text-align:center;" | ^E, F#
| ^E, F#
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 277
| 277
| style="text-align:center;" | downmajor 2nd, upminor 3rd
| downmajor 2nd, upminor 3rd
| style="text-align:center;" | vM2, ^m3
| vM2, ^m3
| style="text-align:center;" | vE#, ^Fb
| vE#, ^Fb
| style="text-align:center;" | downminor 2nd, upmajor 3rd
| downminor 2nd, upmajor 3rd
| style="text-align:center;" | vm2, ^M3
| vm2, ^M3
| style="text-align:center;" | vEb, ^F#
| vEb, ^F#
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 369
| 369
| style="text-align:center;" | major 2nd, downmajor 3rd
| major 2nd, downmajor 3rd
| style="text-align:center;" | M2, vM3
| M2, vM3
| style="text-align:center;" | E#, vF
| E#, vF
| style="text-align:center;" | minor 2nd, downminor 3rd
| minor 2nd, downminor 3rd
| style="text-align:center;" | m2, vm3
| m2, vm3
| style="text-align:center;" | Eb, vF
| Eb, vF
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 462
| 462
| style="text-align:center;" | major 3rd, down 4th
| major 3rd, down 4th
| style="text-align:center;" | M3, v4
| M3, v4
| style="text-align:center;" | F, vG
| F, vG
| style="text-align:center;" | minor 3rd, down 4th
| minor 3rd, down 4th
| style="text-align:center;" | m3, v4
| m3, v4
| style="text-align:center;" | F, vG
| F, vG
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 554
| 554
| style="text-align:center;" | perfect 4th, down 5th
| perfect 4th, down 5th
| style="text-align:center;" | P4, v5
| P4, v5
| style="text-align:center;" | G, vA
| G, vA
| style="text-align:center;" | perfect 4th, down 5th
| perfect 4th, down 5th
| style="text-align:center;" | P4, v5
| P4, v5
| style="text-align:center;" | G, vA
| G, vA
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 646
| 646
| style="text-align:center;" | up 4th, perfect 5th
| up 4th, perfect 5th
| style="text-align:center;" | ^4, P5
| ^4, P5
| style="text-align:center;" | ^G, A
| ^G, A
| style="text-align:center;" | up 4th, perfect 5th
| up 4th, perfect 5th
| style="text-align:center;" | ^4, P5
| ^4, P5
| style="text-align:center;" | ^G, A
| ^G, A
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 738
| 738
| style="text-align:center;" | up 5th, minor 6th
| up 5th, minor 6th
| style="text-align:center;" | ^5, m6
| ^5, m6
| style="text-align:center;" | ^A, B
| ^A, B
| style="text-align:center;" | up 5th, major 6th
| up 5th, major 6th
| style="text-align:center;" | ^5, M6
| ^5, M6
| style="text-align:center;" | ^A, B
| ^A, B
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 831
| 831
| style="text-align:center;" | upminor 6th, minor 7th
| upminor 6th, minor 7th
| style="text-align:center;" | ^m6, m7
| ^m6, m7
| style="text-align:center;" | ^B, Cb
| ^B, Cb
| style="text-align:center;" | upmajor 6th, major 7th
| upmajor 6th, major 7th
| style="text-align:center;" | ^M6, M7
| ^M6, M7
| style="text-align:center;" | ^B, C#
| ^B, C#
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 923
| 923
| style="text-align:center;" | downmajor 6th, upminor 7th
| downmajor 6th, upminor 7th
| style="text-align:center;" | vM6, ^m7
| vM6, ^m7
| style="text-align:center;" | vB#, ^Cb
| vB#, ^Cb
| style="text-align:center;" | downminor 6th, upmajor 7th
| downminor 6th, upmajor 7th
| style="text-align:center;" | vm6, ^M7
| vm6, ^M7
| style="text-align:center;" | vBb, ^C#
| vBb, ^C#
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 1015
| 1015
| style="text-align:center;" | major 6th, downmajor 7th
| major 6th, downmajor 7th
| style="text-align:center;" | M6, vM7
| M6, vM7
| style="text-align:center;" | B#, vC
| B#, vC
| style="text-align:center;" | minor 6th, downminor 7th
| minor 6th, downminor 7th
| style="text-align:center;" | m6, vm7
| m6, vm7
| style="text-align:center;" | Bb, vC
| Bb, vC
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 1108
| 1108
| style="text-align:center;" | major 7th, down 8ve
| major 7th, down 8ve
| style="text-align:center;" | M7, v8
| M7, v8
| style="text-align:center;" | C, vD
| C, vD
| style="text-align:center;" | minor 7th, down 8ve
| minor 7th, down 8ve
| style="text-align:center;" | m7, v8
| m7, v8
| style="text-align:center;" | C, vD
| C, vD
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 1200
| 1200
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | D
| D
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | D
| D
|}
|}


This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 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|13 edo 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>


== 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.]]


[[:File:13ed2-001.svg|13ed2-001.svg]]
=== 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 ==
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.
 
== 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]].
 
Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.
 
See also: [[9edϕ]]
 
{| class="wikitable center-all"
|+Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| 2<sup>ϕ</sup> / ϕ
| 0.858
|-
| ϕ
| 2.321
|-
| 2<sup>ϕ</sup>
| 3.179
|}
 
== Scales ==
{{Main | 13edo scales }}


=Scales in 13edo=
Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):
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 <u>[[Degree|degree]]s of</u> 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
* archaeotonic [[6L 1s]] 2222221 (2\13, 1\1)
* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\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.


[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]]
[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]]
Line 342: Line 1,172:
~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson
~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson


Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO 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.


=Harmony in 13edo=
== Harmony in 13edo ==
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (<u>[[13edo#top|degree]]s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N_subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.
Contrary to popular belief, consonant harmony is possible in 13edo, but it requires a radically different approach than that used in 12edo (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12edo within 13edo is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13edo. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12edo, since the strongest dissonances in 13edo are near the middle of the octave (<u>[[13edo#top|degree]]s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13edo is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N_subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26edo.


By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos.This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.
By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.


The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13 edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.


Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.
Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.
Line 369: Line 1,199:
[[File:13_edo_45921_chord.mp3]]
[[File:13_edo_45921_chord.mp3]]


=Notational and Compositional Approaches to 13edo=
== Notational and compositional approaches ==
13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.
13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.


==The Cryptic Ruse Methods==
=== The Cryptic Ruse Methods ===
13edo offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are [[Rothenberg_propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.
13edo offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L 1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.


To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. The 2\13-based heptatonic has been named "archeotonic" after the "Old Ones" that rule the Dreamlands, and the 5\13-based octatonic has been named "oneirotonic" after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands.
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 Archaeotonic Scale===
==== Modes and harmony in the archaeotonic scale ====
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 13edo.
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.


[[File:Archeotonic.png|alt=Archeotonic.png|Archeotonic.png]]
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: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.


There may be other concordant harmonies possible in this scale that do not represent segments of the overtone 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 ====
The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.


===Modes and Harmony in the Oneirotonic Scale===
Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.
Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.


Line 393: 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.


==The Kentaku (aka William Lynch) Method for Octatonic Notation==
== Mapping to standard keyboards ==
Normally, 13edo can be notated by adding an accidental between E and F. For some reading the same staff with the same letters but in different places can be mind boggling and lead to confusion. That's why some have recommended different options.
 
13edo may be better suited to be notated by using the 8 notes of the MOS father[8] as the notes except that similarly to meantone, the 4th mode of Father[8] is the default "Basic" mode of 13edo and is notated with new letters. A new set of letters may benefit by giving 13edo a fresh perspective. [[William_Lynch|William Lynch]] proposes using the letters JKLMNOPQ as the notes of the 4th mode of Father[8] with J being an 11/8 from a normal C, in order to give it a more personal identify rather than hearing it as C. The remaining letters of 13 edo are notated with sharps and flats like 12edo. The whole alphabet is written J-J#-K-K#-L-M-M#-N-N#-O-P-P#-Q-J . There are no accidentals between LM, OP, and QJ. The idea behind using the 4th mode rather than the 1st is that this mode contains a warmer sounding shape because it has a flat sixth instead of a natural one so William has chosen this to be the default mode rather than Dylathian.
 
This new approach simply modifies the way 13edo is systematized and notated but is based on the exact same 8 tone scale that Cryptic Ruse uses.
 
[[File:13_edo_alphabet.PNG|alt=13 edo alphabet.PNG|13 edo alphabet.PNG]]
 
Notation written on the staff looks nearly the same as 12edo notation, except that because there are 8 letters instead of 7, the notes offset themselves to where space-to-space and line-to-line are now octaves rather than line-to-space. This makes learning the system much easier because a letter that is on a line is always on a line.
 
13EDO Chromatic Scale in Octatonic Notation:
 
[[File:8_tone_scale_13_edo.PNG|alt=8 tone scale 13 edo.PNG|8 tone scale 13 edo.PNG]]
 
[[File:13_edo_chromatic_scale.PNG|alt=13 edo chromatic scale.PNG|13 edo chromatic scale.PNG]]
 
Also, this approach allows for a circle of fourths to be produced:
 
[[File:13_ET_circle_of_fourths.PNG|alt=13 ET circle of fourths.PNG|13 ET circle of fourths.PNG]]
 
Listen to Ryonian on J (F#):
 
[[File:13_edo_Ryo_mode.mp3]]
 
Listen to Dylathian on J: (F#)
 
[[File:13_edo_8_tone_mode_1.mp3]]
 
[[Kentaku's_Approach_to_13EDO|More on William Lynch's 13 EDO octaton approach]]
==Inthar's thoughts on 13edo==
Note on terminology: I often use "cofourth", "cothird", and "cosecond" for octave complements of a fourth, third or second in place of sixth, seventh and eighth. The terms describe their melodic behavior being similar to octave complements of fourths, thirds, and seconds, and work in both archaeotonic and oneirotonic contexts.
 
Both archaeotonic and oneirotonic modes are partly analogous to diatonic modes, though some of them sound more like combinations of different modes.
 
===Archaeotonic===
The archaeotonic scale is overall brighter, more "majory" and more concordant than the oneirotonic scale: there are more 4:5:9 chords and chords involving the 11th and 13th harmonics, and extremely dissonant intervals such as 16/11 and 32/21 are less common.
 
===Oneirotonic===
Seconds and thirds are similar in consonance to 12edo seconds and thirds, and similarly cothirds and coseconds are similar to diatonic sixths and sevenths.
 
Perfect fourths (21/16) are dissonant, but they work a lot like diatonic perfect fourths do e.g. in "sus24" chords that resolve down to thirds, and can also be spread out to make convincing 4:9:21 chords. Minor fifths (approximating 11/8) work like tritones and they like to resolve inward to a third. Major fifths (16/11) are the opposite: they like to resolve outward to a cothird. Unlike in 12edo, there is a major difference in quality between fourths and fifths, and their octave inversions. Perfect fourths and minor fifths are more consonant than their inversions major fifths and perfect cofourths; they can also both be spread out to make them more consonant, whereas their inversions cannot.
 
The diminished fourth can work either like the diatonic diminished fourth, or (uniquely in 13edo) serve as an extra 5/4 in the scale and can be part of extra consonant chords (such as O-J-K-M, representing both 8:10:11:13 and 13:16:18:21, but it only represents 13:16:18:21 in other oneirotonic-supporting tunings such as [[31edo]]).
 
Basic chord progressions can move by perfect fourths or major seconds: J major-M minor-P minor-Ob major-J major (in Ilarnekian) or J major-K major-O major-M major-J major (in Dylathian)
 
=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.


{| class="wikitable"
{| class="wikitable"
|-
|-
| | 1
| 1
| | 6
| 6
| | 11
| 11
| | 3
| 3
| | 8
| 8
| | (13)
| (13)
| | 5
| 5
| | 10
| 10
| | 2
| 2
| | 7
| 7
| | 12
| 12
| | 4
| 4
| | 9
| 9
| | 1
| 1
| | Place in Chain of 738.5 cent intervals
| Place in Chain of 738.5 cent intervals
|-
|-
| | X
| X
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | X
| X
| | Marked are the octatonic scales (X=Sarnathian)
| Marked are the octatonic scales (X=Sarnathian)
|-
|-
| |  
|  
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | X
| X
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| |  
|  
|-
|-
| |  
|  
| | *
| *
| |  
|  
| | X
| X
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| |  
|  
|-
|-
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | X
| X
| | *
| *
| |  
|  
| |  
|  
|-
|-
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| | X
| X
| | *
| *
| |  
|  
| | *
| *
| | *
| *
| |  
|  
| | *
| *
| |  
|  
| |  
|  
|-
|-
| | '''D'''
| '''D'''
| | Eb
| Eb
| | E
| E
| | '''F'''
| '''F'''
| | Gb
| Gb
| |  
|  
| | '''G'''
| '''G'''
| | Ab
| Ab
| | '''A'''
| '''A'''
| | Bb
| Bb
| | B
| B
| | '''C'''
| '''C'''
| | Db
| Db
| | '''D'''
| '''D'''
| | Keeps the pentatonic scale on the white keys
| Keeps the pentatonic scale on the white keys
|-
|-
| | '''A'''
| '''A'''
| | Bb
| Bb
| | B
| B
| | '''C'''
| '''C'''
| | Db
| Db
| |  
|  
| | '''D'''
| '''D'''
| | Eb
| Eb
| | '''E'''
| '''E'''
| | F
| F
| | Gb
| Gb
| | '''G'''
| '''G'''
| | Ab
| Ab
| | '''A'''
| '''A'''
| |  
|  
|-
|-
| | '''E'''
| '''E'''
| | F
| F
| | Gb
| Gb
| | '''G'''
| '''G'''
| | Ab
| Ab
| |  
|  
| | '''A'''
| '''A'''
| | Bb
| Bb
| | '''B'''
| '''B'''
| | C
| C
| | Db
| Db
| | '''D'''
| '''D'''
| | Eb
| Eb
| | '''E'''
| '''E'''
| |  
|  
|-
|-
| | '''B'''
| '''B'''
| | C
| C
| | Db
| Db
| | '''D'''
| '''D'''
| | Eb
| Eb
| |  
|  
| | '''E'''
| '''E'''
| | F
| F
| | '''Gb'''
| '''Gb'''
| | G
| G
| | Ab
| Ab
| | '''A'''
| '''A'''
| | Bb
| Bb
| | '''B'''
| '''B'''
| |  
|  
|-
|-
| | C
| C
| | Db
| Db
| | D
| D
| | Eb
| Eb
| | E
| E
| |  
|  
| | F
| F
| | Gb
| Gb
| | G
| G
| | Ab
| Ab
| | A
| A
| | Bb
| Bb
| | B
| B
| | C
| C
| | Puts the missing key between a semitone
| Puts the missing key between a semitone
|-
|-
| | G
| G
| | Ab
| Ab
| | A
| A
| | Bb
| Bb
| | B
| B
| |  
|  
| | C
| C
| | Db
| Db
| | D
| D
| | Eb
| Eb
| | E
| E
| | F
| F
| | Gb
| Gb
| | G
| G
| | (if that were to be valuable in any way)
| (if that were to be valuable in any way)
|}
|}


The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.
The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.


=Commas=
== Regular temperament properties ==
13 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)
=== Uniform maps ===
{{Uniform map|edo=13}}
 
=== Commas ===
13et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)


{| class="wikitable"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]], Fokker 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, Avicenna's enharmonic diesis
|-
|-
! | Comma
| 7
! | Monzo
| [[64/63]]
! | Cents
| {{monzo| 6 -2 0 -1 }}
![[Color notation/Temperament Names|Color Name]]
| 27.26
! | Name 1
| Ru
! | Name 2
| Septimal comma, Archytas' comma, Leipziger Komma
! | Name 3
|-
|-
| style="text-align:center;" | 2109375/2097152
| 7
| |<nowiki> | -21 3 7 </nowiki>&gt;
| [[64827/64000]]
| style="text-align:right;" | 10.06
| {{monzo| -9 3 -3 4 }}
| style="text-align:center;" |Lasepyo
| 22.23
| style="text-align:center;" | Semicomma
| Laquadzo-atrigu
| style="text-align:center;" | Fokker Comma
| Squalentine comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 1029/1000
| 7
| |<nowiki> | -3 1 -3 3 </nowiki>&gt;
| [[3125/3087]]
| style="text-align:right;" | 49.49
| {{monzo| 0 -2 5 -3 }}
| style="text-align:center;" |Trizogu
| 21.18
| style="text-align:center;" | Keega
| Triru-aquinyo
| style="text-align:center;" |
| Gariboh comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 525/512
| 7
| |<nowiki> | -9 1 2 1 </nowiki>&gt;
| [[3136/3125]]
| style="text-align:right;" | 43.41
| {{monzo| 6 0 -5 2 }}
| style="text-align:center;" |Lazoyoyo
| 6.08
| style="text-align:center;" | Avicennma
| Zozoquingu
| style="text-align:center;" | Avicenna's Enharmonic Diesis
| Hemimean comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 64/63
| 11
| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
| [[56/55]]
| style="text-align:right;" | 27.26
| {{monzo| 3 0 -1 1 -1 }}
| style="text-align:center;" |Ru
| 31.19
| style="text-align:center;" | Septimal Comma
| Luzogu
| style="text-align:center;" | Archytas' Comma
| Undecimal diesis
| style="text-align:center;" | Leipziger Komma
|-
|-
| style="text-align:center;" | 64827/64000
| 11
| |<nowiki> | -9 3 -3 4 </nowiki>&gt;
| [[121/120]]
| style="text-align:right;" | 22.23
| {{monzo| -3 -1 -1 0 2 }}
| style="text-align:center;" |Laquadzo-atrigu
| 14.37
| style="text-align:center;" | Squalentine
| Lologu
| style="text-align:center;" |
| Biyatisma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 3125/3087
| 11
| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| [[441/440]]
| style="text-align:right;" | 21.18
| {{monzo| -3 2 -1 2 -1 }}
| style="text-align:center;" |Triru-aquinyo
| 3.93
| style="text-align:center;" | Gariboh
| Luzozogu
| style="text-align:center;" |
| Werckisma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 3136/3125
| 13
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| [[40/39]]
| style="text-align:right;" | 6.08
| {{monzo| 3 -1 1 0 0 -1 }}
| style="text-align:center;" |Zozoquingu
| 43.83
| style="text-align:center;" | Hemimean
| Thuyo
| style="text-align:center;" |
| Tridecimal minor diesis
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 121/120
| 13
| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| [[105/104]]
| style="text-align:right;" | 14.37
| {{monzo| -3 1 1 1 0 -1 }}
| style="text-align:center;" |Lologu
| 16.57
| style="text-align:center;" | Biyatisma
| Thuzoyo
| style="text-align:center;" |
| Animist comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 441/440
| 13
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| [[169/168]]
| style="text-align:right;" | 3.93
| {{monzo| -3 -1 0 -1 0 2 }}
| style="text-align:center;" |Luzozogu
| 10.27
| style="text-align:center;" | Werckisma
| Thothoru
| style="text-align:center;" |
| Buzurgisma
| style="text-align:center;" |
|}
|}
<references/>


==Animism==
=== Animism ===
The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=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 735: 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


=Guitar=
== Introductory materials ==
<ul><li>[[13EDO_Scales_and_Chords_for_Guitar|13EDO Scales and Chords for Guitar]]</li></ul>
=== By Inthar ===
 
* [[File:13edo_1MC.mp3|270px]] 13edo example composition ([[File:13edo_1MC_score.pdf|score]])
=Compositions=
<span style="">[http://www.microtonalmusic.net/audio/slowdance13edo.mp3 Slow Dance]</span> by [http://danielthompson.blogspot.com/ Daniel Thompson]
 
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3 Prelude in 13ET]</span> by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] - [http://soonlabel.com/xenharmonic/archives/2449 Organ version]
 
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3 Two-Part Invention in 13ET]</span> by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]
 
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3 Triskaidekaphobia]</span> by [http://soonlabel.com/xenharmonic/archives/2642 Herman Miller]<span style=""> - </span>[http://soonlabel.com/xenharmonic/archives/2642 Baroque ensemble version]
 
[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265 Spikey Hair in 13tET] <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3 play]</span> by [[Andrew_Heathwaite|Andrew Heathwaite]]
 
''[[:File:Fuzz_Reef.mp3|Fuzz Reef]]'' in Glacial[7] by Chuckles McGee
 
''[[:File:Lunar_Approach.mp3|Lunar Approach]]'' in Father[8] by Chuckles McGee
 
[http://www.elvenminstrel.com/music/tuning/equal/13equal/13tet.htm Upsidedown and Backwards: Explorations in 13-tone Equal Temperament] by [http://www.elvenminstrel.com/ David J. Finnamore]
 
<span style="">[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3 Comets Over Flatland 11]</span> by [[Randy_Winchester|Randy Winchester]]
 
[http://archive.org/details/ineedSynthetiklove (iNeed) SyNthetikLove] [http://archive.org/download/ineedSynthetiklove/iNeed_SyNthetikLove.mp3 play] by Jon Lyle Smith
 
[https://soundcloud.com/jonlylesmith/strange-strange-day Strange, Strange Day] [http://micro.soonlabel.com/gene_ward_smith/Others/Jlsimith/__Strange__Strange_Day_by_jonlylesmith.mp3 play] by Jon Lyle Smith
 
[https://soundcloud.com/jonlylesmith/stare-into-the-sun Stare (into The Sun)] [http://micro.soonlabel.com/gene_ward_smith/Others/Jlsimith/__Stare__into_The_Sun__by_jonlylesmith.mp3 play] by Jon Lyle Smith
 
<span style="">[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3 Prelude for Synthesizer in 13 Equal]</span> by [[Chris_Vaisvil|Chris Vaisvil]]
 
<span style="">[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3 Muon Catalyzed Fusion]</span> by [[Chris_Vaisvil|Chris Vaisvil]]
 
<span style="">[http://www.seraph.it/dep/det/13Miles.mp3 13 Miles]</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html blog entry])
 
<span style="">[http://www.seraph.it/dep/det/FastAndFurious13.mp3 Fast And Furious 13]</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html blog entry])
 
<span style="">[http://www.seraph.it/dep/det/SazDul13.mp3 SazDul 13]</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html blog entry])
 
<span style="">[http://www.seraph.it/dep/det/Thirteenstan.mp3 Thirteenstan]</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html blog entry])
 
<span style="">[http://www.seraph.it/dep/det/Concertina13.mp3 Concertina 13]</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html blog entry])
 
[http://www.seraph.it/dep/det/HocusBogus13.mp3 Hocus-Bogus 13] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/3c72765dabcf0381cfba6e53b9adccab-179.html blog entry])
 
[http://www.seraph.it/dep/det/MP13.mp3 MP13] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/0039a08cabcf860c813c37912c8898a0-183.html blog entry])
 
[http://www.seraph.it/dep/det/Berimbau13.mp3 Berimbau 13] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/6b9292c1c0c8b46af3fd0d906d887183-184.html blog entry])
 
[http://youtu.be/-6Q5XXvZhKE Tenori Blofeld (video)] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/8f73ac37040c73a7d86101cd50af3c6d-188.html blog entry])
 
[http://www.seraph.it/dep/det/FromTheDiaryOfThe13thFly.mp3 From The Diary Of The 13th Fly] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/7d977261410acf3eb3f6bce49b04f296-206.html blog entry])


[http://www.seraph.it/dep/det/Lotus13.mp3 Lotus 13] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/e63d2a478d5d422027d27eb6f9dc59f5-255.html blog entry])
==== Oneirotonic Modal Studies ====
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Illarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian


[http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/ Liber Stellarum] by [http://votta.wordpress.com Alfredo Votta]
== Music ==
{{Main| 13edo/Music}}
{{Catrel|13edo tracks}}


[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3 For Claudi] by [http://soonlabel.com/xenharmonic/archives/1788 Margo Schulter]
== See also ==
* [[13EDO Scales and Chords for Guitar]]
* [[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


[https://youtu.be/7ohvxMjsuXs Autogynephilia by Diamond Doll]
== 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.


[[Category:13-tone]]
[[Category:13-tone scales]]
[[Category:13edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Prime EDO]]
[[Category:Teentuning]]
[[Category:Teentuning]]
[[Category:Theory]]
[[Category:Pages with internal sound examples]]
[[Category:Oneirotonic]]


[[Category:todo:unify_precision]]
{{Todo|cleanup|add rank 2 temperaments table}}
Retrieved from "https://en.xen.wiki/w/13edo"