1. Internet Institute for Music Science
The Cognitive Relevance of the
Mathematical Counterpoint Model
in Human Depth EEG
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science

2. Birkhäuser 2002 1368 pages, hardcover
incl. CD-ROM
€ 128.– / CHF 188.–
ISBN
English

3. Rules of Counterpoint Following J.J. Fux
K/D Symmetry in Human Depth EEG
Extension to Exotic Interval Dichotomies

4. Ernst Tittel:
Der neue Gradus
Lehrbuch des strengen Satzes nach
Johann Joseph Fux

5. Ÿ12  Ÿ3 x Ÿ4 z ~> (z mod 3, -z mod4) 4.u+3.v <~ (u,v) 8 11 4 310 8 1 2 3 4 5 6 7 9

6. Ÿ12
Ÿ12[e]= Ÿ12[X]/(X2)
c+e. Ÿ12 c
c+e.d

7. 2 5
minor third 10
major third
d(x,y) = min. # major/minor thirds from x to y

8. 900 1800 1200 900 180 =inversion Refl. =fourth circle
90=minor third chain
120=major third chain

9. Consonance-dissonance dichotomy
Ÿ12 = K D disjoint, #K = #D = 6 K = {0,3,4,7,8,9}, D ={1,2,5,6,10,11}
(Marked) dichotomy = (K/D)

10. (K/D) is a strong dichotomy, i.e.,
there is exactly one (invertible) symmetry
y=a.x+b of the torus which exchanges K and D, i.e., y=5.x+2
This is the autocomplementarity function AC:
AC(0) = 2 AC(3) = 5 AC(4) = 8 AC(7) = 1 AC(8) = 6 AC(9) = 11 AC2 = Id

11. Proposition:
Among the 34 classes of marked 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:
(I/J)
(K/D)

12. (K/D)
AC(x) = 5x+2
span
(I/J)
AC(x) = 11x+5
diameter

13. e e.2.5
K = Ÿ12 +e.{0,3,4,7,8,9} = consonances
D = Ÿ12 +e.{1,2,5,6,10,11} = dissonances

14. K D ?
„punctus
contra
punctum“ =
vertical and horizontal!

15. vertical and horizontal!
g(D)
g(K)
„punctus
contra
Punctum“ =
vertical and horizontal!
g: Ÿ12[e] ≈ Ÿ12[e] contrapuntal symmetry
g = ea+ e.b.(u+ e.v) u = 1,5,7,11

16. Contrapuntal symmetries are local

17. Allowed transition for the major scale
The Topos of Music
Table O.2
pp.1217/18

18. Paralles of fifths are always forbidden

21. Event V(Event) = (S/Sq,S/Sa,S/Sb) = vigilance vector   a b V(Event)4 8 14 40
50 Hz
Power   a b
Frequency
V(Event)  a b
V(Event) = (S/Sq,S/Sa,S/Sb)
= vigilance vector

22. 1:11
0.68 sec

24. Music is a key to unconscious emotional contents
Jonathan Winson: Hippocampal Gate Hypothesis
Elton John
and Diana
Music is a key to unconscious emotional contents

25. K* = {0,3,4,7,8,9,11} (add „leading note“ 11 to consonances)

26. Do counterpoint with the major dichotomy on exotic scales!
K* = {0,3,4,7,8,9,11} class # 60^
ragas -> melakarta: 72 scales
mela scale Nr. 15 = {0,3,4,7,8,9,1} class #61^ K*
mela 15
Do counterpoint with the major dichotomy on exotic scales!
Write a counterpoint deformation program (K/D)2(I/J)!