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

minor key representation

Tmin(k) = n1 .Cmin(k) + n2 .(a.CMaj(k + 1)+ (1- a).Cmin(k + 1)) + n3 .(b .Cmin(k - 1)+ (1 - b).CMaj(k - 1)),

where $n = (n1,n2,n3)$ is a weight vector of positive real numbers satisfying $n1 + n2 + n3 = 1$ and $0 \< a,b \< 1$ . The point $Tmin(k) (- R3$ is the ’center of effect’ of the five weighted triad representations of the triad collection $kmin(k)$ with respect to the weight vector $n$ and the mode preferences $a,b$ . The weight $a$ controlles the relative influence of the major and the minor dominants to the center of effect of the minor key. Similarly, $b$ controlles the relative influence of the minor and the major subdominants.

Figure 4:

The figure shows the three spirals corresponding to the tones (outer spiral), major triads (middle spiral) and major keys (inner spiral)

4.2 Tuning of the Free Parameters

So far the spiral array representation is defined up to 16 non-negative real variables:

radius and height (r,h) internal chord weights w = (w1,w2,w3) u = (u1,u2,u3) internal key weights w = (w1, w2,w3) n = (u1,u2,u3) mode preferences (a,b)