What could now be, after all, the program of classification? Its core value is that it deals with understanding musical works. And we should stress that our concept of a musical work is not the narrow one which restricts to those individual opera which--at least in Europe--started to emanate in the Renaissance. It includes as well general musical corpora such as scales, systems, everything that can be represented by means of global compositions.

From the precise parametric description of a work and of its ambiguities, this work appears as a point configuration in a more or less complex space. However this configuration is already a determinate perspective which shows a multitude of relations among its ingredients. It is the composer’s perspective (now including an abstract >composer< or creator of a general musical structure like a scale). For example, the choice of tonality, instrumentation, tempo, etc., are points of view which may or may not pertain to the composition, this is a question of the epoch of creation. But their character can undoubtedly be subject to variation. Among others, here we do address the question of historical instrumentation for early music.

In order to understand the relations among different parts of a composition, and even to simply recognize them, a change of the given perspective is mandatory. If a never seen object must be inspected, what should we do? You walk around it. This is the most common version of Yoneda’s lemma. The analogy to cartography is straightforward: The natural perspective of the landscape in which we live does not coincide with the perspective which meets best our need for orientation. To reach this goal, we preferrably build maps which show the landscape from an infinitely high point.

The same happens to music. You play a piece in slow motion >from very near<, in a zoomed optics, a complex chord is arpeggiated, i.e., viewed from a skew angle, and so forth. This idea of variation of the perspective has in fact been integrated in the compositional thinking of the 20th century, perhaps most prominently by Edgar Varèse, especially in his comments on the composition Intégrales (Varèse, 1960, p.67). There, he invokes a geometric analog of a machine which is able to project a mobile spatial object from variable space angles onto a luminous surface. This latter idea is astonishingly akin to the resolution projection from points in general position to points in special position.

5 Towards Grand Unification

In this section, we shall shortly illustrate on the basis a concrete musicological situation--harmony and counterpoint--why some of the above general concepts have been introduced, and how they create perspectives of unification.

5.1 An Isomorphism Between Instances of Harmony and Counterpoint

Classically, mathematical music theory worked on the pitch class space introduced above. In what follows, we shall slightly adjust it by the »fifth circle«