GlareanÕs Dodecachordon Revisited...GlareanÕs Dodecachordon Revisited Donnerstag, 13. Juni 2013 Points of Departure Donnerstag, 13. Juni 2013 To every well-formed N-scale there are

Glarean’s Dodecachordon Revisited

Donnerstag, 13. Juni 2013

Points of Departure


To every well-formed N-scale there are N modes



Guidonian Modes



Guidonian Modes

Glarean-Zarlino Modes



bad conjugate

Guidonian Modes

Glarean-Zarlino Modes


4

Modes and their Plain Adjoints


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’


aaba|aab aaba|aab

C

D

E

F#

F

G

A

H

C’

C

D

E

B

F

G

A

H

C’


The Species of the Fifth and the Fourth


aaba|aab aaba|aab

C

D

E

F#

F

G

A

H

C’

C

D

E

B

F

G

A

H

C’



Automorphism f of F2:


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





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





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’






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’






plain adjoint

a b-1


plain adjointDonnerstag, 13. Juni 2013

plain adjoint

aaba


plain adjoint

aaba

aaba


plain adjoint

aaba

aaba

aaba


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


plain adjoint

aaba

aaba

aaba

b -1a -1a -1

b -1a -1a -1

b -1a -1a -1


plain adjoint

aaba

aaba

aaba

b -1a -1a -1

b -1a -1a -1

b -1a -1a -1

b -1a -1a -1


plain adjoint

aaba aaba aabab-1a-1a-1 b-1a-1a-1 b-1a-1a-1 b-1a-1a-1


plain adjoint

aab)a aab)a aab)a(b-1a-1a-1 (b-1a-1a-1 (b-1a-1a-1 b-1a-1a-1aaa


plain adjoint

b-1a-1a-1aaa


plain adjoint

b-1a-1a-1aaa


plain adjoint

b-1(aa)-1aa)(a


plain adjoint

b-1(aa)-1aa)(a


twisted adjointDonnerstag, 13. Juni 2013

twisted adjoint

a-1b

-1a-1a

-1


twisted adjointaa

b

a-1b

-1a-1a

-1


twisted adjointaa

b

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1


twisted adjointaa

b

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1

aab


twisted adjointaa

b

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1aa

b


twisted adjointaa

b

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1aa

b

aab


twisted adjointaa

b

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1

a-1b

-1a-1a

-1aa

b

aab

aab


twisted adjoint

aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab


twisted adjoint

aaba-1b-1a-1a-1 a-1b-1a-1a-1 a-1b-1a-1a-1aab aab aab


twisted adjoint

aaba-1 a-1 a-1


twisted adjoint

aaba-1 a-1 a-1


twisted adjoint

aaba-1 a-1 a-1 b


twisted adjoint

aaba-1 a-1 a-1 b


twisted adjoint

a-1 b


twisted adjoint

a-1b


twisted adjoint

a-1b


twisted adjoint

a-1b



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:


Sturmian Morphisms generate Lattice-path Transformations


Common Finalis Modes (“Tropes”): The lattice-path transformations are

applied to the same initial lattice path.


Common Origin (“White Note”) Modes: The lattice-path transformations are

applied to the different initial lattice paths.


Lattice Path Transformations have Linear Adjoints

Geometric Interpretation (here in 3 Dimenions: Example from Arnoux and Ito)


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) ...


... and obtain the associated foldings (with notes)

C

F#B E

A D

G

C C

C C C



Documents

GlareanÕs Dodecachordon Revisited...GlareanÕs Dodecachordon Revisited Donnerstag, 13. Juni 2013 Points of Departure Donnerstag, 13. Juni 2013 To every well-formed N-scale there are