guerino mazzola u & eth zürich internet institute for music science [email protected]
DESCRIPTION
The Cognitive Relevance of the Mathematical Counterpoint Model in Human Depth EEG. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science [email protected] www.encyclospace.org. Birkhäuser 2002 1368 pages, hardcover incl. CD-ROM € 128.– / CHF 188.– ISBN 3-7643-5731-2 - PowerPoint PPT PresentationTRANSCRIPT
Guerino Mazzola
U & ETH ZürichInternet Institute for Music Science
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)!