are >in phase<: The pitch classes can be identified with the fundamentals of the minor keys. The explanation lies in the relatively high overall symbolic duration of the fundamental pitch class in the symbolic duration profile in minor compared to Major. Also the overall symbolic duration of the Fugues in minor is longer than the one in Major. We conclude that the minor keys induce the circle of fifths in the pitch classes.

In Figure 5 the singular values to the factors are shown. They indicate how much of the variance in the data is captured if correspondence analysis projects the data onto the factors with highest singular values. It is interesting that the explanatory values, e.g., the singular values, of the two most prominent factors in WTC I (Fugues) are almost equal, in contrast to the other singular values, whose explanatory value is much smaller.

Toroidal Model of Inter-Key Relations. Figure 3 displays the projection of keys and pitch classes onto the plane spanned by the two most prominent factors. How can we visualize the projection onto the first four factors? We represent points on each of the planes spanned by the first & second and third & fourth factor, respectively, in polar co-ordinates, i.e., by their polar angle and by their distance to the origin. We then plot their angle in the 1-2-plane against their angle in the 3-4-plane (upper Figure 6). Topologically, we can think of the resulting body as the surface of a torus, which can be parameterized by two angles. Upper Figure 6 can be viewed as a torus if we consider vertical and horizontal periodicity, i.e., we glue together the upper and lower side as well as the right and left side. The three circles of fifths then meander around the torus three times as indicated by the solid, dashed, and dotted lines. In addition to the relationship regarding fifths in upper Figure 6 we see that the projection on the 3. and 4. factor contains information about the inter relation between Major keys and their parallel and relative minor keys.