glareanÕs dodecachordon revisited mcgill 2013.pdf · to every well-formed n-scale there are n...
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To every well-formed N-scale there are N modes
Guidonian Modes
Glarean-Zarlino Modes
Donnerstag, 13. Juni 2013
To every well-formed N-scale there are N modes
bad conjugate
Guidonian Modes
Glarean-Zarlino Modes
Donnerstag, 13. Juni 2013
plain adjoint twisted adjoint
Lattice Paths in Regenerʼs Generic Note Space
aaba|aab yx|yxyxy aaba|aab xy|xyxyy
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
Donnerstag, 13. Juni 2013
aaba|aab aaba|aab
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Donnerstag, 13. Juni 2013
aaba|aab aaba|aab
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Automorphism f of F2:
Donnerstag, 13. Juni 2013
aaba|aab aaba|aab
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
x = f(a) = aaba
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Automorphism f of F2:
Donnerstag, 13. Juni 2013
aaba|aab aaba|aab
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
x = f(a) = aaba y = f(b) = aab
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Automorphism f of F2:
Donnerstag, 13. Juni 2013
aaba|aab y-1x|y-1xy-1xy-1 aaba|aab
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
x = f(a) = aaba y = f(b) = aab
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Automorphism f of F2:
Donnerstag, 13. Juni 2013
aaba|aab y-1x|y-1xy-1xy-1 aaba|aab x-1y|x-1yx-1yy
C
D
E
F#
F
G
A
H
C’
C
D
E
B
F
G
A
H
C’
x = f(a) = aaba y = f(b) = aab
plain adjoint twisted adjoint
The Species of the Fifth and the Fourth
Automorphism f of F2:
Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
Donnerstag, 13. Juni 2013
plain adjoint
aab)a aab)a aab)a(b-1a-1a-1 (b-1a-1a-1 (b-1a-1a-1 b-1a-1a-1aaa
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
aab
aab
Donnerstag, 13. Juni 2013
Question:The morphisms nicely generate the interval patterns. But we don’t have access to the notes yet. Is there a transformational approach to ?
Answer:
Donnerstag, 13. Juni 2013
Common Finalis Modes (“Tropes”): The lattice-path transformations are
applied to the same initial lattice path.
Donnerstag, 13. Juni 2013
Common Origin (“White Note”) Modes: The lattice-path transformations are
applied to the different initial lattice paths.
Donnerstag, 13. Juni 2013
Lattice Path Transformations have Linear Adjoints
Geometric Interpretation (here in 3 Dimenions: Example from Arnoux and Ito)
Donnerstag, 13. Juni 2013
The initial lattice path can be mapped to the dual space
and we can apply the adjoints E(fi)* of the 6 lattice path transformations E(fi) ...
Donnerstag, 13. Juni 2013
... and obtain the associated foldings (with notes)
C
F#B E
A D
G
C C
C C C
Donnerstag, 13. Juni 2013