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

3.1 Harmonic Morphemes

The considerations of this paragraph are based on the perspectivic incidence relation $H$ between tone perspectives and chords.

º º H = {(f,X)|X (- |CH|,f (- A(X)} < A× |CH |.

Let us take a closer look at this relation in order to generalize the usual study of common chord tones²

²: In functional harmony common chord tones indicate the possibility of functional synonymy of two chords.

to that of common chord perspectives. Technically, we build formal concepts on the basis of perspectivic incidence:

Definition 9 Consider the following two maps:

ºA |CH| Ext : 2 -- > 2 with Ext(M ) := {X | A f (- M : (f,X) (- H}, Int : 2|CH |--> 2ºA with Int(U ) := {f| A X (- U : (f,X) (- H}.

A (formal) harmonic morpheme is an ordered pair $(M, U) (- 2ºA × 2|CH |$ satisfying $Ext(M ) = U$ and $Int(U) = M$ .

We introduce the following terminology: $Ext$ and $Int$ are called the (formal) harmonic extension and intension maps, respectively. The two ”coordinates” $M$ and $U$ of a harmonic morpheme $(M, U)$ are called the intension and the extension of this morpheme. Harmonic morphemes can be obtained in two ways:

Start with any set $U (_ |CH |$ of chords and construct the morpheme
$(Int(U ),Ext(Int(U ))$ . The concatenation $|CH| |CH| Ext o Int : 2 --> 2$ is called the extensional completion
Start with any set $M (_ ºA$ of tone perspectives and construct the morpheme $(Int(Ext(M )),Int(M ))$ . The concatenation $ºA ºA Into Ext : 2 -- > 2$ is called intensional saturation.

3.2 Internal Logics of a Morpheme

Now, consider a fixed harmonic morpheme $(M, U )$ . Our goal is to introduce a suitable structure on the extension $U$ which is controlled by the intension $M$ .

Lemma 7 The intension of $M$ of a harmonic morpheme $(M, U)$ is a monoid.

Proof: If two tone perspectives map a chord into itself then their concatenations do so as well, furthermore is the identity $0 1$ a selfperspective of any chord $X$ . $[]$
The monoid $M$ can be viewed as a (mathematical) category on its own--consisting of a single object, and having its elements as arrows. Instead of viewing $U$ just as a set of chords we study it as a subcategory of the category $M Sets$ of covariant functors from $M$ to $Sets$ . The objects of this category can be identified with monoid actions $m : M × X --> X$ (satisfying $m(g,m(f,x)) = m(g o f,x)$ and $0 mX( 1,x) = x$ for all $f,g (- M$ and $x (- X$ ). The functor $[m]$ corresponding to such an action $m$ sends