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13edo: Difference between revisions

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! [[Erv Wilson's Linear Notations|Erv Wilson]]
! [[Erv Wilson's Linear Notations|Erv Wilson]]
! Archaeotonic
! Archaeotonic
(Heptatonic 2nd-generated)
! Oneirotonic
! Oneirotonic
(Octatonic 5th-generated)
! [[26edo]] names
! [[26edo]] names
! Fox-Raven<br>Notation (J = 360Hz)
(subset
 
notation)
! Fox-Raven<br>(J = 360Hz)
! Pseudo-Diatonic<br>Category
! Pseudo-Diatonic<br>Category
! Audio
! Audio
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<references/>
<references/>


== Notation ==
== Notations ==
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.
Only the first two notations are backwards-compatible. They both allow conventional notation to be used, including the staff, note names, relative notation, etc. And they allow a piece in conventional notation to be translated to 13edo. They both use the conventional genchain of fifths:
 
...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...
 
=== 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.
 
{| 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)
| ^m2, (m3)
| ^E, (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
|}
 
=== 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".
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".
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|}
|}


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


D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
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P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d (s = sub-, d = -oid)
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d (s = sub-, d = -oid)


pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
pentatonic genchain of fifths:  
 
...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...


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)
...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)


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


A - A#/Bb - B - C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - H - H#/Ab - A
A - A#/Bb - B - C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - H - H#/Ab - A
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P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9


octotonic genchain of sixths: ..D# - A# - F# - C# - H# - E - B - G - D - A - F - C - H - Eb - Bb - Gb - Db - Ab...
octotonic genchain of sixths:  
 
...D# - A# - F# - C# - H# - E - B - G - D - A - F - C - H - Eb - Bb - Gb - Db - Ab...


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


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


D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
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P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8


genchain of seconds: ...Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#...
genchain of seconds:  
 
...Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#...
 
...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3...
 
=== Heptatonic 3rd-generated (mosh) ===
This notation requires ups and downs because 7 perfect thirds octave-reduces to 2 edosteps, not 1.
 
'''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)
 
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
 
genchain of thirds:
 
...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#...
 
...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
 
=== 26edo subset ===
This notation uses every other name of [[26edo]]. There are no perfect 4ths or 5ths, only augmented and diminished ones. There are two versions of this notation. One has only three natural notes (C, D and E) and the other one has only four (F, G, A and B).


genchain of seconds: ...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3...
'''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 15\26 = 7.5\13 = perfect 5th)


[[:File:13edo-chromatic-scale.mid|13edo chromatic ascending and descending scale on C (MIDI)]]
[[:File:13edo-chromatic-scale.mid|13edo chromatic ascending and descending scale on C (MIDI)]]
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