Guerino Mazzola

U & ETH ZürichInternet Institute for Music Science

guerino@mazzola.chwww.encyclospace.org

The Cognitive Relevance of the Mathematical Counterpoint Modelin Human Depth EEG

Birkhäuser 2002Birkhäuser 20021368 pages, hardcover 1368 pages, hardcover incl. CD-ROMincl. CD-ROM€ € 128.– / CHF 188.–128.– / CHF 188.–ISBN 3-7643-5731-2ISBN 3-7643-5731-2EnglishEnglish

www.encyclospace.orgwww.encyclospace.org

K/D Symmetry inK/D Symmetry inHuman Depth EEGHuman Depth EEG

Extension to ExoticExtension to ExoticInterval DichotomiesInterval Dichotomies

Rules of CounterpointRules of CounterpointFollowing J.J. FuxFollowing J.J. Fux

Ernst Tittel: Der neue GradusLehrbuch des strengen Satzes nachJohann Joseph Fux

ŸŸ12 12 ŸŸ3 3 xx ŸŸ44

z ~> (z mod 3, -z mod4)z ~> (z mod 3, -z mod4)4.u+3.v <~ (u,v)4.u+3.v <~ (u,v)

11

10

8

1

2

3 4

567

9

0

ŸŸ12 12 ŸŸ1212[[]= ]= ŸŸ1212[X]/(X[X]/(X22))

c+c+. . ŸŸ1212

ccc+c+.d.d

25

minor thirdminor third

10

major thirdmajor third

d(x,y) = d(x,y) = min. # major/minor thirdsmin. # major/minor thirdsfrom x to yfrom x to y

900

1800

900 1200

180 =inversionRefl. =fourth circle90=minor third chain120=major third chain

ŸŸ12 12 = K= K D disjoint, #K = #D = 6 D disjoint, #K = #D = 6 K = {0,3,4,7,8,9}, D ={1,2,5,6,10,11} K = {0,3,4,7,8,9}, D ={1,2,5,6,10,11}

(Marked) dichotomy = (K/D)(Marked) dichotomy = (K/D)

Consonance-dissonance dichotomyConsonance-dissonance dichotomy

(K/D) is a (K/D) is a strong dichotomystrong dichotomy, i.e.,, i.e.,there is exactly one (invertible) symmetrythere is exactly one (invertible) symmetryy=a.x+b y=a.x+b of the torus which exchanges K and D, i.e.,of the torus which exchanges K and D, i.e.,y=5.x+2y=5.x+2

This is the This is the autocomplementarity functionautocomplementarity function AC: AC:

AC(0) = 2AC(0) = 2AC(3) = 5AC(3) = 5AC(4) = 8AC(4) = 8AC(7) = 1AC(7) = 1AC(8) = 6AC(8) = 6AC(9) = 11AC(9) = 11

ACAC22 = Id = Id

Proposition:Proposition:Among the 34 classes of marked dichotomies,Among the 34 classes of marked dichotomies,there are 6 strong classes. there are 6 strong classes. The distances among the members of The distances among the members of one half (or the other) of such a dichotomyone half (or the other) of such a dichotomyare class invariants and characterize these classes:are class invariants and characterize these classes:

(K/D)(K/D)

(I/J)(I/J)

(K/D)(K/D) AC(x) = 5x+2AC(x) = 5x+2

(I/J)(I/J)AC(x) = 11x+5AC(x) = 11x+5

spanspan

diameterdiameter

= = ŸŸ1212 + + = consonances = consonances

DD = = ŸŸ1212 + +{1,2,5,6,10,11} = dissonances{1,2,5,6,10,11} = dissonances

e e .2.2.5.5

KKDD

„„punctus punctus contra contra punctum“ =punctum“ =vertical and horizontal!vertical and horizontal!

??

g(g(KK))g(g(DD))

g: ŸŸ1212[[] ≈ ] ≈ ŸŸ1212[[]]contrapuntal symmetrycontrapuntal symmetryg = eg = ea+ a+ .b.b.(u+ .(u+ .v).v)u = 1,5,7,11u = 1,5,7,11

„„punctus punctus contra contra Punctum“ =Punctum“ =vertical and horizontal!vertical and horizontal!

Contrapuntal symmetries are localContrapuntal symmetries are local

The Topos of MusicThe Topos of MusicTable O.2Table O.2pp.1217/18pp.1217/18

Allowed transition for the major scaleAllowed transition for the major scale

Paralles of fifths are always forbiddenParalles of fifths are always forbidden

V(Event) = (S/SV(Event) = (S/S,S/S,S/S,S/S,S/S))= = vigilance vectorvigilance vector

EventEvent

44 88 1414 404050 Hz50 Hz

PowerPower

FrequencyFrequency

V(Event)V(Event)

0.68 sec

1:11

Jonathan Winson: Hippocampal Gate HypothesisJonathan Winson: Hippocampal Gate Hypothesis

Music is a key to unconscious emotional contentsMusic is a key to unconscious emotional contentsElton Johnand Diana

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

K* = {0,3,4,7,8,9,11} K* = {0,3,4,7,8,9,11} class # 60^class # 60^

ragas -> melakarta: 72 scales ragas -> melakarta: 72 scales

mela scale Nr. 15 = {0,3,4,7,8,9,1}mela scale Nr. 15 = {0,3,4,7,8,9,1}class #61^class #61^

K*K* mela 15mela 15

Do counterpoint with the major dichotomy on exotic scales!Do counterpoint with the major dichotomy on exotic scales!Write a counterpoint deformation program (K/D)2(I/J)!Write a counterpoint deformation program (K/D)2(I/J)!