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 guerino@mazzola.ch www.encyclospace.org

Birkhäuser 2002 1368 pages, hardcover incl. CD-ROM € 128.– / CHF 188.– ISBN 3-7643-5731-2 English www.encyclospace.org

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

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

Ÿ12  Ÿ3 x Ÿ4 z ~> (z mod 3, -z mod4) 4.u+3.v <~ (u,v) 8 11 4 3 10 8 1 2 3 4 5 6 7 9

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

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

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

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)

(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

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)

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

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

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

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

Contrapuntal symmetries are local

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

Paralles of fifths are always forbidden

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

1:11 0.68 sec

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

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

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)!