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

Step 2:

Monoids $M$ like $* Z2$ are categories with one object and monoid actions $m : M × T --> T$ correspond to functors $[m] : M --> Sets$ into the category of sets. Functor categories of this kind are the most prominent examples of topoi. They are cartesian closed and have a subobject classifier (or truthvalue-object) $_O_$ together with an arrow $T : 1-- > _O_$ . The subobject-classifier of our concrete example consists of those three subsets $F alse = Ø,F alseBut = {0},True = {0,1}$ of $* Z2$ which are closed under multiplication with arbitrary $* Z2$ -elements. In contrast, the category $Sets$ of sets can be identified with the category of functors $o. F : --> Sets$ with one arrow $.$ only. The subobject classifier of this category consists of the two subsets $o. F alse = Ø,T rue = = {.}$ .

3.3 Analytical Example: Scriabin’s Study Op. 65 No. 3

In this final subsection we present a simple analysis of the harmonic vocabulary in Scriabin’s last study Op. 65 No. 3 on the basis of a suitable transformational logics. Callender (1998) discusses this vocabulary with respect to parsimonious voice leading by looking at entire chords (pcs). Here we take a closer look at the distribution of tones in the left and the right hand, respectively. Remarkably, throughout the piece the harmonic material of the left hand is constituted by the transpositions of a single chord, namely ${0,4,10}$ (a dominant seventh chord with omitted fifth.) In the spirit of Hugo Riemann’s view on Harmony we consider this chord (and its transpositions) as »consonant prime chords« and study the added right hand tones as »dissonant«. However, we do not simply label them as $False$ in the context of classical logics. Instead, we investigate their refined Truth values in the sense of $F alseBut$ , i.e. > $F alse$ up to suitable transformations which make them $True$ <. ⁸

⁸: Noll and Brand (2004) (this volume) contains a rather precise survey of the morphological theory behind this analysis as well as a discussion of its relation to Riemannian functional harmony

We consider the circle of fifths $Z12$ identifying $0$ with $C$ , $1$ with $G$ , ..., and $11$ with $F$ . The transformations which constitute our logics are inner perspectives of the chord {0, 4, 10}. Generally, we define a small universe of 144 tone perspectives in terms of (affine) maps $b a : Z12-- > Z12$ given by $b a(z) := a .z + b$ . Here, $a$ and $b$ themselves are elements of $Z12$ . Concatenation of two such tone perspectives $b a$ and $d c$ is given by the formula $d b d+cb c o a = ac$ . Inner perspectives map the chord {0, 4, 10} into itself. The identity $0 1$ obviously does so, and so do the three projections $0 0$ , $4 0$ and $10 0$ . However, there are other inner perspectives which are less obvious, like $10 3$ and $4 8$ . Remarkably, we can generate the three projections $0 0$ , $4 0$ and $10 0$ starting from these two tone perspectives according to the concatenation formula above:

0 4+8.10 4 10 100 = 10+3.48.3 = 180 o 43 4 0 = 4+8.0 3.8 = 4 3o0 8 4 4 10 0 = 8 .0 = 8 o 0 = 8o 8o 3.

Using the two generating tone perspectives $48$ , $103$ and the identity $01$ we can introduce five different truth values $T *,P *,R*,L*,C*$ in order to assign a >measure