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Glarean’s Dodecachordon Revisited
Donnerstag, 13. Juni 2013
Points of Departure
Donnerstag, 13. Juni 2013
To every well-formed N-scale there are N modes
Donnerstag, 13. Juni 2013
To every well-formed N-scale there are N modes
Guidonian Modes
Donnerstag, 13. Juni 2013
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
4
Modes and their Plain Adjoints
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
a b-1
Donnerstag, 13. Juni 2013
plain adjointDonnerstag, 13. Juni 2013
plain adjoint
aaba
Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
aaba
Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
aaba
b -1a -1a -1Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
Donnerstag, 13. Juni 2013
plain adjoint
aaba
aaba
aaba
b -1a -1a -1
b -1a -1a -1
b -1a -1a -1
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
aaba aaba aabab-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
plain adjoint
b-1a-1a-1aaa
Donnerstag, 13. Juni 2013
plain adjoint
b-1a-1a-1aaa
Donnerstag, 13. Juni 2013
plain adjoint
b-1(aa)-1aa)(a
Donnerstag, 13. Juni 2013
plain adjoint
b-1(aa)-1aa)(a
Donnerstag, 13. Juni 2013
twisted adjointDonnerstag, 13. Juni 2013
twisted adjoint
a-1b
-1a-1a
-1
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
aab
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
Donnerstag, 13. Juni 2013
twisted adjointaa
b
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1
a-1b
-1a-1a
-1aa
b
aab
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
twisted adjoint
aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab
Donnerstag, 13. Juni 2013
twisted adjoint
aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab
Donnerstag, 13. Juni 2013
twisted adjoint
aaba-1 a-1 a-1
Donnerstag, 13. Juni 2013
twisted adjoint
aaba-1 a-1 a-1
Donnerstag, 13. Juni 2013
twisted adjoint
aaba-1 a-1 a-1 b
Donnerstag, 13. Juni 2013
twisted adjoint
aaba-1 a-1 a-1 b
Donnerstag, 13. Juni 2013
twisted adjoint
a-1 b
Donnerstag, 13. Juni 2013
twisted adjoint
a-1b
Donnerstag, 13. Juni 2013
twisted adjoint
a-1b
Donnerstag, 13. Juni 2013
twisted adjoint
a-1b
Donnerstag, 13. Juni 2013
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
Sturmian Morphisms generate Lattice-path Transformations
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
Donnerstag, 13. Juni 2013