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

Figure 3:

Symbolic durations of keys and pitch classes are derived from the score of the Fugues of Bach’s WTC I and then projected onto the factors of correspondence analysis. Upper: The emerged circle of fifths is lined out for Major (solid) and minor (dashed) keys (”m” denoting minor). Middle: As above pitch classes appear in the order of the circle of fifths. Lower: The biplot of keys and pitch classes derived from putting both transparent upper graphs on top of each other. We observe that the pitch classes are close to the corresponding minor keys.

3.1 Circle of Fifths from Score

Humdrum (CCARH, 2003) is a digital format that is aimed at fully and precisely representing the essential content of a composition as it is obvious from notation in a score. All fugues of Bach’s WTC are encoded in Humdrum **kern format. Instead of analyzing the co-occurrence table of frequencies of keys and pitch classes we look at the overall symbolic duration of pieces in a particular key and the overall symbolic duration of pitch classes across all 24 Fugues for WTC I and for WTC II. Symbolic duration means that it is measured in multiples and fractions of quarter notes, rather than in seconds. Measuring duration in seconds would imply that the results would vary a lot depending on the choice of tempo. But the issue of tempo is highly discussed in Bach.

The correspondence analysis of WTC (Figure 3) reveals a two-dimensional structure which allows for an adequate representation in a plot based on the first two factors corresponding to the two largest singular values (Figure 5). The 24 pitch class frequency profiles are optimally projected onto a 2-dimensional plane, such that the $x2-$ distance between profiles is minimally distorted and the $x2-$ distance of the 2-dimensional projections matches the original profile distance as well as possible. In the Fugues of WTC, the circle of fifths emerges clearly and homogeneously (upper Figure 3). Even though in the upper Figure 3 some inter-key distances are smaller (D $b$ -A $b$ ) than others (D-A), due to different $x2$ -distances between pitch class prominence profiles in these pieces. In addition the minor keys form a circle of fifths inside the circle of fifths of the major keys. This shows that the pitch prominence patterns for major keys are more distinct than for minor keys according to the metric employed.