- 440 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory 
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In the co-occurrence table above pitch classes are represented as columns of accumulated symbolic durations in the different keys, that means accumulated symbolic durations in each fugue of WTC I, since in WTC I a one-to-one correspondence of Fugues and the 24 keys is given. The same factors with maximal singular values as in the upper Figure 3 are used to optimally project the 24-dimensional pitch class vectors upon a plane. We observe the pitch classes forming a circle of fifths as well (middle Figure 3). We can now consider the biplot (lower Figure 3) by putting the transparent plot of pitch classes (middle Figure 3) on top of the plot of keys (upper Figure 3). We have three circles of fifths, one each for the Major and minor keys and one for the pitch classes. We change the co-ordinates of the factor plane to polar co-ordinates in terms of a polar angle (on the circle of fifths) and the distance to the origin. Consider the angles of both the Major and minor keys relative to the angles of the fundamental pitch class (Figure 4). The plot shows two almost straight parallel lines, reflecting that pitch classes and keys proceed through the circle of fifths with almost constant offset. The graph for the minor keys is almost the identity, indicating that pitch classes and keys


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