Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

which would imply a modification of the major key representation into

TMaj(k) = w1 .CMa'j(k) ' + w2 .(a'.CMaj(k +1) +(1 -a )'.Cmin(k + 1)) + w3 .(b .CMaj(k- 1)+ (1- b ).Cmin(k - 1)).

4.4 Center-Of-Effect-Generator

Recall from subsection 4 that we are dealing with the tone space $TON ES = F$ of note names. Furthermore we choose the space of tone profiles $CHORDS = P ro(F)$ in order to define a Riemann-Logic. Recall also that in the spiral array model three levels of description are merged together. Especially chords are represented as ’centers of effect’ of weighted tone sets. Hence, instead of interpreting harmonic loci within the space of tone profiles we can study the representations of tone profiles within the spiral array model. Let K denote the carrier of a tone profile $sum X = k (- K ai .k$ . The representation $sum CE(X) := k (- K ai .P (k)$ is a point within the convex closure of the point set $P (K)$ in the euclidean space $3 R$ . Chew sucessfully applied this Center of Effect generation map $CE$ to problems of key finding as well as to chord root finding(c.f. Chew (2000), pp. 99 - 138 as well as 158 respectively). It is natural to also try it in the determination of harmonic loci within the space $3 Cfunc < R$ .

We propose the following Riemann Logics both of which translate high distances into small values near $0$ and small distances into values near $1$ :

1 RL(X |\ H)inv := 1+-||CE(X)---H||- RL(X |\ H)exp := exp(-||CE(X) - H||).

Remark 6 In her applications Chew (2000) obtains the coefficients $0 < a \< 1 i$ within tone profiles $X = sum ak˙ k (- K i$ from score data. Especially, she encodes tone durations and proposes also to consider metric weights. This is a general method which has to be used in connection with the other approaches too.

5 Fred Lerdahl’s model of Chordal/Regional Space

The present section reflects upon selected aspects from Fred Lerdahl’s study of harmonic pathways in a harmonic configuration space which Lerdahl calls the chordal/regional space. Our investigations do not address Lerdahl’s theoretical framework (c.f. Lehrdahl, 2001) as a whole but concentrate on his attempt to combine a principle of hierarchy with a principle of shortest path and we focus on the theoretical and practical problems which arise from this attempt for the definition of distances between harmonic loci. The following subsections present elementary investigations into Lerdahl’s approach.