The Musical-Mathematical Mind by Gabriel Pareyon Silvia Pina-Romero Octavio A. Agustín-Aquino & Emilio Lluis-Puebla

Author:Gabriel Pareyon, Silvia Pina-Romero, Octavio A. Agustín-Aquino & Emilio Lluis-Puebla
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In this section we define vector fields associated with pairs of tonalities and which fulfill the conditions explained above. Although such vector fields can be defined for quite general situations of tonality pairings, we want to restrict our attention to the pairing of two tonalities that are one fourth apart from each other, and we may choose the concrete situation of C-major and F-major. For each such tonality T, which we identify with its scale for this special discussion, we define a vector field that is motivated by the unique inner symmetry of T. For this is the inversion , for , it is . To have a simple representation of symmetries and fields, we choose a labeling of the pitch classes in such that . With this notation, and 0 being on top, and 3 to the right of the circular representation (like normal time visualisation), the symmetry is the reflection at the vertical diameter through the pitch class circle. We now represent this reflection as a movement in horizontal direction from left to right, thinking of a -rotation in . This can be represented by a vector field . Similarly, for tonality F, we define its vector field as being the clockwise rotation of by . More generally, if R is a nonsingular linear transformation of , we construct a vector field from X by . The we have if R is the clockwise rotation by . Figure 2 shows these fields in a graphic generated by Mathematica software.

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