glareanÕs dodecachordon revisitedmarianamontiel.gsucreate.org/conferences/mcm mcgill 2013.pdf ·...
TRANSCRIPT
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Glarean’s Dodecachordon Revisited
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Points of Departure
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To every well-formed N-scale there are N modes
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To every well-formed N-scale there are N modes
Guidonian Modes
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To every well-formed N-scale there are N modes
Guidonian Modes
Glarean-Zarlino Modes
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To every well-formed N-scale there are N modes
bad conjugate
Guidonian Modes
Glarean-Zarlino Modes
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4
Modes and their Plain Adjoints
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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’
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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
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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:
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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:
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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:
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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:
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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:
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plain adjoint
a b-1
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plain adjointDonnerstag, 13. Juni 2013
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plain adjoint
aaba
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plain adjoint
aaba
aaba
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plain adjoint
aaba
aaba
aaba
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plain adjoint
aaba
aaba
aaba
b -1a -1a -1Donnerstag, 13. Juni 2013
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plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
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plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
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plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
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plain adjoint
aaba aaba aabab-1a-1a-1 b-1a-1a-1 b-1a-1a-1 b-1a-1a-1
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plain adjoint
aab)a aab)a aab)a(b-1a-1a-1 (b-1a-1a-1 (b-1a-1a-1 b-1a-1a-1aaa
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plain adjoint
b-1a-1a-1aaa
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plain adjoint
b-1a-1a-1aaa
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plain adjoint
b-1(aa)-1aa)(a
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plain adjoint
b-1(aa)-1aa)(a
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twisted adjointDonnerstag, 13. Juni 2013
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twisted adjoint
a-1b
-1a-1a
-1
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twisted adjointaa
b
a-1b
-1a-1a
-1
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twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
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twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
aab
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twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
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twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
aab
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twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
aab
aab
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twisted adjoint
aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab
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twisted adjoint
aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab
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twisted adjoint
aaba-1 a-1 a-1
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twisted adjoint
aaba-1 a-1 a-1
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twisted adjoint
aaba-1 a-1 a-1 b
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twisted adjoint
aaba-1 a-1 a-1 b
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twisted adjoint
a-1 b
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twisted adjoint
a-1b
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twisted adjoint
a-1b
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twisted adjoint
a-1b
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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:
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Sturmian Morphisms generate Lattice-path Transformations
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Common Finalis Modes (“Tropes”): The lattice-path transformations are
applied to the same initial lattice path.
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Common Origin (“White Note”) Modes: The lattice-path transformations are
applied to the different initial lattice paths.
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Lattice Path Transformations have Linear Adjoints
Geometric Interpretation (here in 3 Dimenions: Example from Arnoux and Ito)
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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) ...
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... and obtain the associated foldings (with notes)
C
F#B E
A D
G
C C
C C C
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Donnerstag, 13. Juni 2013