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

octave (i.e. in a given cyclic group of order $n$ ) drammatically decreases¹⁵

¹⁵: From the perspective of Mazzola’s Mathematical Music Theory these elementary musical structures can be considered as zero-addressed local compositions in the module $/Z12$ . T. Noll’s concept of self-addressed chords, as it has been introduced in Noll (1995), has been one of the first steps towards a generalised theory of chords classification. For a presentation of the theory of classification of self-addressed local structures in the framework of denotators see section 11.3.7 of Mazzola (2003).

, as it is clear from the following figure:

Figure 3:

Number of orbits under the action of the dihedral group for the twelve-tone and for the quarter-tone temperament

We started to implement in OpenMusic some some special families of translation classes of chords which have been considered, by some theorists and composers, as particularly relevant from a music-theoretical perspective. The families of equivalent classes of chords which are entirely implemented are the following :

Limited transposition chords
Self-complementary chords
Auto-inverse chords
Inverse-complementary chords
All-combinatorial chords