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

parsimony--which, surprisingly, again characterize the major and minor triads in a prominent position among other chords. The morphological approach is dedicated to a special kind phenomenon which is seldom addressed in Neo-Riemannian studies, namely the substitution of certain harmonically significant chords by other chords without changing the harmonic signification.

1.5 Circle Chords -- a Motivating Observation

In his early studies into Mathematical Music Theory Guerino Mazzola pays much attention to the fact, that many elementary musical objects -- including chords -- are suitably modeled as subsets of mathematical modules (Mazzola, 1985, 1990). Mazzola terms these objects local compositions. As such they inherit the structure of their ambient spaces and therefore can be suitably studied in terms of transformations of their ambient spaces. One special instance of this idea is provided by the concept of a circle chord. Such chords

X = {t,f(t),f(f(t)),...}

are generated from a given tone $t (- Z12$ and an affine transformation $f : Z12 --> Z12$ iteratively applied to it. Mazzola points out, that the major and minor triads may be produced in that way. With respect to usual >halftone-step<-encoding where $C = 0,C# = 1,...,B = 11$ we can generate the set ${C,E,G}= {0,4,7}$ by applying $f(t) = 3t+ 7mod 12$ iteratively starting from 0:

0 '--> 7 '--> 4 '--> 7 '--> 4...

From this observation the idea was born to algebraically simulate traditional arguments, where chords are studied in terms of overtone concordances and discordances. The multiplication of fundamental frequencies with >stretching factors< $2,3,4,...$ leads to their overtones. In our algebraic counterpart of this idea we study >stretchings< $mod 12$ in concatenation with translations. The overall strategy is to study chords in terms of concordances of affine transformations of $Z12$ .

2 Basic Investigations

In this section we present basic definitions and facts about tone perspectives, i.e. affine transformations of $Z12$ and the study of chords in terms of tone perspectives.

2.1 Tone Perspectives

Consider the ring $Z 12$ of residue classes of integers modulo 12. Let $T$ denote the 12-tone module - i.e., $Z 12$ understood as a module over itself. Its points are called tones.¹

¹: In a music theoretical application of this model one has to be more careful in dealing with the meaning of the elements of $T$ . One might formally distinguish the carrier sets $Z12$ and $T$ from one another in order to avoid confusion between scalar factors and tones.

Affine endomorphisms $f : T --> T$ of the tone module are called tone