Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch

1. The Cognitive Relevance of the Mathematical Counterpoint Model in Human Depth EEG Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org

2. Birkhäuser 20021368 pages, hardcover incl. CD-ROM € 128.– / CHF 188.– ISBN 3-7643-5731-2 English www.encyclospace.org

3. Rules of CounterpointFollowing J.J. Fux K/D Symmetry inHuman Depth EEG Extension to ExoticInterval Dichotomies

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

5. 8 11 4 3 0 7 5 6 2 9 1 10 Ÿ12 Ÿ3 xŸ4 z ~> (z mod 3, -z mod4) 4.u+3.v <~ (u,v)

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

7. 5 2 10 minor third 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) = 2AC(3) = 5AC(4) = 8AC(7) = 1AC(8) = 6AC(9) = 11AC2 = Id

11. (I/J) (K/D) 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:

12. AC(x) = 5x+2 (K/D) (I/J) AC(x) = 11x+5 span 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. D K ? „punctus contra punctum“ = vertical and horizontal!

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

22. 0.68 sec 1:11

24. 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. mela 15 K* 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^ Do counterpoint with the major dichotomy on exotic scales! Write a counterpoint deformation program (K/D)2(I/J)!