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

Proof: Two monoids $M ,M 1 2$ , are not equal, if and only if $M / (- M < 1 2$ or $M / (- M < 2 1$ , hence the topology $IN T$ is $T 0$ . The same holds for any two chords and their minimal neighbourhoods: $X /= X 1 2$ if and only if $X / (- Ext(int(X )) 1 2$ or $X / (- Ext(int(X )) 2 1$ . Hence the topology $EX T$ is $T 0$ .

The closure $M--$ of a monoid $M$ is the complement of the largest open set not containing $M$ , i.e.,

M--= ( U M '<)c = {M '| M '/ (_ M }c = {M '|M ' (_ M }= M > M'/ (_ M

Analogously, we find for chords:

-- U X = ( Ext(int(X')))c = {X'|X (- Ext(int(X'))} [] X/ (- Ext(int(X'))

A set $M$ of monoids is closed, if and only if $M (- M$ implies $M ' (- M$ for all $M ' (_ M$ . Analogously, a set $V$ of chords is closed if and only if $X (- V$ implies $X'( - V$ for all $X'$ satisfying $X (- Ext(int(X'))$ .

The closures $-- X$ for the 157 representatives for all the chord classes are easily calculated. We mention that there are 14 classes, whose chords generate closed singletons $-- X = {X}$ , namely the 1-chords, the 2-chords except ${0,6}$ , the 3- chords except ${0,4,8}$ and the 4-chord ${0,1,2,6}$ ). On the other hand, the only chords generating all their subchords, i.e., $-- X = Sub(X) := {Y (- |CH ||Y (_ X}$ are the 6 classes of affine subspaces of $T$ , namely $0T$ , $6T$ , $4T$ , $3T$ , $2T$ , $T$ .

The general chord neighbourhoods, i.e., unions of basic chord neighbourhoods $Ext(M )$ , provide a good means to define global morphemes.

Definition 12 Let $IN T s$ denote the topology induced by $IN T$ on the subset $MON s$ of all saturated monoids.
A global harmonic morphem is given as a pair $(M, {Ext(M )|M (- M})$ consisting of a set $M$ of saturated monoids as its local intensions and the corresponding family ${Ext(M )|M (- M}$ of basic chord neighbourhoods as its local extensions. The open set $U M( - M Ext(M )$ is called the extensional carrier of $M$ .

As an example, consider the global morpheme determined by $M = {R,L}$ ,

4 0 0 4 0 1 4 1 4 0 R = { 8,4, 0, 0,1}, L = { 3,9, 0, 0,1}.

The open set $Ext(R) U Ext(L)$ being covered by $Ext(R),Ext(L)$ , provides a suitable model for chords representing the same tonal function. Observe that the generated monoid $<R,L>*= int({0,1,4})$ is the intension of the ”C-Major-triad”. The extension $Ext(int({0,1,4})))$ contains only particular superchords of ${0,1,4}$ , but not the »relative-minor«-triad ${0,1,3}$ or the »Leittonwechsel«-triad ${1,4,5}$ . These and other prototypical representatives of the tonic triad ${0,1,4}$ in functional harmony are suitably modelled by the global morpheme

({R,L},{Ext(R),Ext(L)}).