Guerino%20Mazzola

1
Grothendiecks Coconut Parabolaand the Grand
Unification ofMusic Theories

• Guerino Mazzola
• U ETH Zürich
• i2musics
• guerino_at_mazzola.ch
• www.encyclospace.org

2
hammer and chisel
The shell softens
3

• Two instances of Grothendiecks soft coconut
  method
• Unscrewing of identityCounterpoint Theory
• Heating up conceptsUnification ofCounterpoint
  and Harmony

4
Birkhäuser 20021368 pages, hardcover incl.
CD-ROM 128. / CHF 188. ISBN
3-7643-5731-2 English
www.encyclospace.org
5
CD
M. W.
6
scheme
7
Unscrewing of identity Counterpoint Theory
8
unscrewing of identity
9
Ernst Tittel Der neue Gradus Lehrbuch des
strengen Satzes nach Johann Joseph Fux
unscrewing of identity
10
Ÿ12 ? Ÿ3 x Ÿ4 z gt (z mod 3, -z
mod4) 4.u3.v lt (u,v)
unscrewing of identity
11
unscrewing of identity
12
Consonance-dissonance dichotomy
unscrewing of identity
Ÿ12 K? D disjoint, K D 6 K 0, 3,
4, 7, 8, 9, D 1, 2, 5, 6, 10, 11
Ke Ÿ12 e.0, 3, 4, 7, 8, 9 consonant
intervals
De Ÿ12 e.1, 2, 5, 6, 10, 11 dissonant
intervals
13
?
unscrewing of identity
punctus contra punctum vertical and
horizontal!
14
unscrewing of identity
Rigid identity Ke ?De ØDe Ke Soften
here! Idea Soften negation!
15
unscrewing of identity
d(x,y) min. major/minor thirds from x to y
16
unscrewing of identity
180 inversion Refl. fourth circle 90minor
third chain 120major third chain
17
(K/D) is a strong dichotomy, i.e. there is
exactly one (invertible) symmetry ya.xb of the
torus which exchanges K and D, i.e.y5.x2 This
is the autocomplementarity function AC AC(0)
2AC(3) 5AC(4) 8AC(7) 1AC(8) 6AC(9)
11AC2 Id
While improvising, I had this idea...
unscrewing of identity
18
Proposition Among the 34 classes of
dichotomies, there are 6 strong classes. The
distances among the members of one half (or the
other) of such a dichotomy are class invariants
and characterize these classes
unscrewing of identity
19
span
unscrewing of identity
diameter
20
unscrewing of identity
Ke Ÿ12 e.0, 3, 4, 7, 8, 9 consonances
De Ÿ12 e.1, 2, 5, 6, 10, 11 dissonances
21
?
unscrewing of identity
22
unscrewing of identity
g(Ke)
g(De)
g Ÿ12e Ÿ12econtrapuntal symmetry g ea
e.b.(u e.v)u 1,5,7,11
23
unscrewing of identity
24
Contrapuntal symmetries are local
unscrewing of identity
25
Allowed transition for the major scale
unscrewing of identity
The Topos of Music Table O.2 pp.1217/18
26
unscrewing of identity
Paralles of fifths are always forbidden
27
unscrewing of identity
28
unscrewing of identity
29
unscrewing of identity
V(Event) (S/Sq,S/Sa,S/Sb) vigilance vector
30
unscrewing of identity
111
31
unscrewing of identity
32
Jonathan Winson Hippocampal Gate Hypothesis
unscrewing of identity
Elton John and Diana
Music is a key to unconscious emotional contents
33
unscrewing of identity
K 0, 3, 4, 7, 8, 9, 11 (add leading note
11 to consonances)
34
K 0, 3, 4, 7, 8, 9, 11 class 60 ragas
-gt melakarta 72 scales mela scale Nr. 15
0, 3, 4, 7, 8 , 9, 1class 61
unscrewing of identity
Do counterpoint with the major dichotomy on
exotic scales! Write a counterpoint deformation
program (K/D)2(I/J)!
35
unscrewing of identity
36
Lifting of concepts Unification ofCounterpoint
and Harmony
37
Arnold Schönberg Harmonielehre (1911)
Old Tonality Neutral Degrees (IC, VIC)
Modulation Degrees (IIF, IVF, VIIF)
New Tonality Cadence Degrees (IIF VF)
heating up concepts

• What is the considered set of tonalities?
• What is a degree?
• What is a cadence?
• What is the modulation mechanism?
• How do these structures determine the modulation
  degrees?

38
Ludwig van Beethoven op.106/Allegro/124-127 Inve
rsiondb G(3) E b(3)
heating up concepts
450
39
heating up concepts
40
heating up concepts
modulation S(3) T(3) cadence symmetry
41
heating up concepts
42
Thesis The modulation structure of op. 106 is
governed by the inner symmetries of the
diminished seventh chord C -7 c, e, g,
bb in the role of the admitted modulation
forces.
heating up concepts
43
Modeling Riemann Harmony (Thomas Noll)
heating up concepts
Trans(Dt,Tc) lt f? Ÿ12 _at_ Ÿ12 fDt Tc gt
_at_ means affine maps
44
Ÿ12
Ÿ12
? Dt, Tc
? 0 _at_ Ÿ12
constant tones
prime intervals
heating up concepts
differential
constant intervals
?
45
Ke, De
heating up concepts
Trans(Dt,Tc)
Trans(Ke,Ke)
46
Birkhäuser 20021368 pages, hardcover incl.
CD-ROM 128. / CHF 188. ISBN
3-7643-5731-2 English
www.encyclospace.org