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

The following set of parameters has been selected as an optimal solution to all these contraints (c.f. Chew, 2000, pp. 94 - 97):

V~ -2 radius and height (1, 15) internal chord weights w = (0.6025,0.2930,0,1045) u = (0.6011,0.2121,0.1868) internal key weights w = n = w = (0.6025,0.2930,0,1045) mode preferences (34, 34)

Chew’s empirical applications of this tuned model to key finding can be re-interpreted as a concrete case of the general framework presented here. The harmonic configuration space is then the space of the key-representations

Ckey = {TMaj(k) |k (- Z} U {Tmin(k)|k (- Z}

and the harmonic tensor is given by the Euclidean distance. We see in subsection 4.4 below, how the ’Center of Effect Generation’ can be viewed as a special kind of Riemann Logic.

4.3 Proposal for a Riemann Function Space

We complete this review of Chew’s spiral array model by proposing a harmonic configuration space $3 Cfunc < R$ of tonal functions, which are represented by ’centers of effect’ on the line between the key representation and the chord representation of the prototypical triad associated with that tonal function. A user defined function parameter $0 < t < 1$ regulates the relative influence of the key center with respect to the chord center:

T (k) := t.TMaj(k) +(1- t).CMaj(k) D(k) := t.TMaj(k) +(1- t).CMaj(k + 1) S(k) := t.TMaj(k) +(1- t).CMaj(k - 1) t(k) := t.Tmin(k) +(1- t).Cmin(k) d#(k) := t.Tmin(k) +(1+ t).CMaj(k + 1) d(k) := t.Tmin(k) +(1- t).Cmin(k+ 1) s(k) := t.Tmin(k) +(1- t).Cmin(k- 1) s#(k) := t.Tmin(k) +(1- t).CMaj(k - 1)

The configuration space

U Cfunc = {T(k),D(k),S(k),t(k),d#(k),d(k),s(k),s#(k)} k (- Z

equipped with the Euclidean distance in $3 R$ is a straight forward generalization of $Ckey$ . For applications in romantic music one may also include the functions

D (k) := t.T (k) +(1- t).C (k + 1) Sb(k) := t.TMaj(k) +(1- t).Cmin(k - 1) b Maj min