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

Proof: The chord $ext(M )$ belongs to all three chordsets $Ext(M0),Ext(M )$ and ${ext(M )}$ . Hence, the corresponding intensions contain no other constant tone perspectives than those in $M0$ , i.e., the three monoids indeed belong to the same fiber $ext-1(ext(M ))$ . Further these chord sets satisfy the inclusions ${ext(M )}( _ Ext(M ) (_ Ext(M0)$ , since $M0 (_ M$ . Hence the corresponding intensions have inclusions in the reverse order. Finally, the inclusions hold independent of the particular choice of $M (- ext- 1(ext(M ))$ , defining the same $M0$ and $ext(M )$ . $[]$

Definition 10 A chord $X$ is called poor if the identity $0 1$ and the constant tone perspectives $x 0,x (- X$ , are its only selfperspectives, i.e., if $X 0 int(X) = 0 U { 1}$ . A chord $X$ is said to be primitive if its selfperspectives are shared by all of its superchords, i.e., if $Ext(int(X)) = Super(X) := {Y |X (_ Y }$ .

The primitive morphemes, generated by primitive chords $X$ , have the structure $(int(X), Super(X))$ . Poor chords are always primitive, but there are also non-poor primitive chords. To be more precise, among the 157 chord classes there are 31 classes, whose chords are primitive, but there are only 5 classes of poor chords. The latter are represented by

{0,1,3},{0,1,2,4},{0,1,2,5},{0,1,2,3,5},{0,1,2,4,5}.

In order to get the full picture of all harmonic morphemes one has to investigate all the saturated monoids within the partially ordered sets $ext-1(X)$ , where $X$ runs through representatives of the 157 chord classes. For this it is sufficient to calculate representatives for the conjugation classes $MON s(X)/conjX$ of saturated monoids within $MON s(X) := MON s /~\ ext-1(X)$ under the conjugation action of the selfsymmetries $* ºA (X)$ of $X$ .

These calculations have been carried out for all morphemes

(Int(Ext(M )),Ext(M ), with M0 /= Ø

by the help of a Mathematica notebook. On the whole, there are 25364 such morphemes. The detailed results cannot be published within this article.³

³: Contact the author in order to get a textfile with the results or the program.

What remains, is the calculation of those morphemes with no constant tone perspectives in their intensions. The saturated groups have first been described as ”musical” groups in Mazzola (1985). The general case can be studied in a refined way in terms of global morphemes (see following subsection).

3.4 Harmonic Topology

The following topological constructions are studied by Guerino Mazzola (cf. Mazzola, 2002, chapter 24) in the much more general situation of functorial local compositions and their endomorphisms. However, we recall some aspects within the narrow context of chords and chord perspectives.

For a given set $M (_ ºA$ of tone perspectives let $M < := {M ' (- MON |M ' )_ M }$ denote the set of all supermonoids of $M$ . These sets $M <$ are called basic monoid neighbourhoods. Similarly, we call chord extensions $Ext(M )$ basic chord neighbourhoods.