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

Lemma 8 The familiy of basic monoid neighbourhoods is closed with respect to intersection: For any $M1,M2 (_ ºA$ let $<M1, M2 >*$ denote the monoid generated from $M1$ and $M2$ . One has

(M1 U M2)< = M <1 /~\ M <2 = <M1, M2 ><*

Similarly, the family of basic chord neighbourhoods is closed with respect to intersection: For any $M ,M < ºA 1 2$ one has

Ext(M U M ) = Ext(M ) /~\ Ext(M ) 1 2 1 2

Definition 11 We introduce the following two topologies on chords and monoids:

The topology $EX T$ on the set $|CH |$ of all chords generated by the family of all basic chord neighbourhoods ${Ext(M )|M (- MON } s$ is called the harmonic extension topology.
The topology $IN T$ on the set $MON$ of all monoids of tone perspectives generated by the family of all basic monoid neighbourhoods ${M < |M (- MON }$ is called the harmonic intension topology.

Proposition 5 The map $int :|CH|--> MON$ is continous with respect to the topologies $EX T$ and $IN T$ . Moreover, $EXT$ is the inverse-image topology of $IN T$ with respect to the map $int$ .

Proof: For any given monoid $M$ and and any given chord $X$ we have

X (- int- 1(M <) <====> int(X) (_ M <====> X (- Ext(M ).

Hence, $int-1(M <) = Ext(M )[]$ .

Remark 7 The map $ext : MON --> |CH |$ is not continous with respect to the topologies $IN T$ and $EX T$ . A general open set $M$ in the topology $IN T$ is characterized by the property $M (- M ===> M ' (- M ( A M ' )_ M )$ . On the other hand, the minimal chord neighbourhood $Ext(int(X))$ of any non-primitive chord $X$ does not contain all superchords of $X$ . If $Y > X$ is such a superchord with $Y / (- Ext(int(X))$ , we have $X -1 0 (- ext (Ext(int(X)))$ and $Y X 0 > 0$ , but $Y -1 0 / (- ext (Ext(int(X)))$ .
The non-continuity of $ext$ reflects the essential difference between chords as objects of the category $CH$ and chords just as sets.

The topologies $IN T$ and $EX T$ are rather exotic ones. In terms of the axioms of separation from general topology $T ,T ,T 0 1 2$ , we have the following characterizations:

Proposition 6 Both topologies $IN T$ and $EX T$ satisfy the $T 0$ -axiom, but not the $T 1$ -axiom. The closures of singletons are explicitly given as follows:

For any $M (- MON$ one has $--- M = M > := {M ' (- MON |M ' (_ M }$ .
For any $X (- |CH |$ one has $-- ' ' X = {X (- |CH ||Ext(int(X )) (_ Ext(int(X))}$ .