The Topos of Music I: Theory

This chapter describes the overall extension of music-related activities in the space-time of human existence. The basic scope of the systematization as described in this book is declared. Our selection of corresponding fundamental scientific domains is not meant as a qualitative judgment over other scientific or artistic domains; it merely names the scientific pillars of the musical realm.

This chapter deals with an ontological orientation in the subject of music. It was already contended in chapter 1 that music is communication, has meaning and mediates on the physical level between its mental and psychological levels. Such an orientation is topographic in nature since it offers a number of ontological \dimensions" and \coordinates" to profile musicological discourse and helps avoiding misplaced or blurred arguments. The topography involves three mutually independent dimensions: communication, reality, and semiosis. The local nature of this orientation scheme is discussed.

This chapter introduces the difficult subject of musical ontology. Such a discussion is substantial for a reconciliation of traditional musicology and innovative perspectives in cognitive and computational musicology.

This chapter introduces the paradigm of experimental humanities. Such a perspective is basic to all computational, resp. effective, methods in musicology. We discuss the parallel to the epochal Galilean change from speculative to experimental natural sciences.

Well-conceived information produces knowledge. But conceiving means building concepts, and this amounts to having well-structured access modes. This enforces the development of concept-oriented access modalities to given information, including associated concept spaces. Music is an excellent field to exemplify such knowledge spaces. It has become virtually impossible to navigate through music information without developing powerful concept formats.

This chapter introduces the universal data format of denotators to describe musical objects. Denotators generalize the structures of local compositions in mathematical music theory as well as the data model used in the RUBATO® software. They do, however, not include deeper semantic layers which are related to physical, psychic, social or religious meaning. Their semiotic structure represents an elaborate denotative baseline and resides in a purely mental level usually attributed to mathematical objects.

Local compositions are introduced as elementary objects of music. They derive from powerset denotators. It is shown that all denotators may be transformed into local compositions. A special type of local compositions can be defined from a fixed set of denotators of a given “ambient space”. With these so-called objective local compositions the problem of universal construction of new concepts from given ones cannot be solved. This imposes a deeper theory of non-objective functorial local compositions. The basic vocabulary as well as an introductory list of common local compositions—such as scales, ordinary and fractal chords, meters, rhythms, and motives—are presented. The chapter concludes with a discussion of tangent objects in local theory, a concept framework which leads to alterations and related results by Mason and Mazzola.

After having introduced the objects of local theory, their relations are discussed and formalized. Thereby, symmetry is a key concept—however in its modern version which by far exceeds traditional axial or rotational symmetries. The core process of symmetry leads to structure-conserving transformations which we rebuild in the context of local compositions and the underlying forms.

This chapter reviews and completes the inherent paradigm change from “object-based” to functorial mathematics which was initiated by category theory, and completed in the celebrated Yoneda Lemma. Beyond an apparant technical innocence, this lemma introduced a revolution in understanding structures of general types, and transcending pure mathematics. Its implications touch general hermeneutics as well as esthetics of art, and—quite paradoxically!—principles of “object-oriented” programming.

Paradigmatic classification deals with formation of classes of objects which belong to specific paradigms. Here, paradigms are fields of equivalence or association. In a more general setting than in mathematics, equivalence is not necessarily understood as being a transitive relation. We give motivation of the paradigm concept from musicology, semiotics and poetology, and mathematics. Our taxonomy yields two types of paradigms: by transformations and by similarity—however, in practice, they often appear in mixed form. In a mathematical perspective, the first type is covered by group theory, the second by topology. This means that fuzzy concepts in the humanities are not a priori useless, they can be incorporated into exact reasoning by means of a refined paradigmatic reconstruction.

This chapter deals with groups of symmetries, their action and orbits as musicological and mathematical concepts. Elementary local compositions—chords, self-addressed chords, and motives are classified under group actions. Enumeration theory of orbits of local compositions in finite $$\mathbb{Z}$$ -modules—including traditional pitch class sets and motives—is presented and discussed for its implications towards a “Big Science” in music. Follows a discussion of group-theoretical methods in composition and theory, including a review of the American tradition and recent developments.

In general, transformations will conserve interior relations of objects, but not their ‘absolute’ position, or site, in the ambient space. This aspect is covered by the topological perspective. In its infancy, mathematical topology was in fact called “analysis situs”. It deals with the general question of what it means to be in the vicinity of an object. In music, topological considerations are of central importance since slight deformations of objects to neighboring objects are standard identification concepts|though never handled with the necessary care. The point in making these structures precise lies in the sharpening of a fundamental descriptive tool, and in the semantic potential which topology induces. We make explicit the latter topic in a discussion of the problem of topological classification of sounds.

The categories ObLoc and Loc are representations of local objects, i.e., of objects that are not composed of proper “parts”. However, music mainly deals with compound objects. The adequate concept of a global composition is defined. The corresponding vocabulary of elementary global music objects is described, including ecclesiastical modes, tridadic degrees, meters, rhythms, motives and themes.

Global perspectives deal with relations among global music objects. We introduce this subject together with its musical motivation and append the formal definition of morphisms among global compositions. This leads to the categories ObGlob of objective and Glob of functorial global compositions. We describe the combinatorial aspects of globality and associated functors, as well as corresponding geometric classification tools: nerves and simplicial weights.

Global classification relies on two concepts: affine functions and resolutions of global compositions. These constructs are discussed and exemplified. We derive classifying spaces and compare them to the situation in the Dreiding{Dress{Haegi theory of molecules: The latter are deduced from global compositions by additional structures concerning orientation, distances and angles (bilinear and exterior forms). It is therefore possible to view “molecules” as being global compositions with additional constraints; their musical meaning is discussed.

This chapter exposes criteria for characterizing interpretable compositions in terms of classifying spaces. This is a central issue since interpretability yields access to instrumental parameters for the physical ‘rendering’ of a compositional structure. In contrast to general classification, interpretable molecular structures are difficult to classify. We also review global enumeration theory as well as global American Set Theory.

Contrary to the seemingly bookkeping character of the concept of classification, the subject is deeply tied to esthetics. This is explicated and illustrated with a detailed example.

Denotators are purely mathematical structures which do not specify “what is the case” and what is not. This chapter deals with the existence problem of music-related objects in contrast to mathematical ‘fiction’. This amounts to loading mathematical constructs with an additional semiotic signification process in order to express “which denotators are the case”. These existence specifications instantiate an interface between mental potentiality and historical actuality. It reveals two fundamentally different existentialities, termed “textual” and “paratextual” signification, respectively. The former involves predicates defined by classical extension over specific denotators, whereas the latter transcends pure extensionality and thus points into domains of open semiosis. Both, textual and paratextual predicates are essential enrichments of mathematical constructs: The platonic ontology is thereby supplemented by a differentiation which cannot be reduced to pure “mentality”. The variety of textual predicates follows certain construction rules of logical and geometric nature and is founded in a triply typed set of “atomic” predicates of (a) mathematical, (b) musical, and (c) deictic types.

This chapter is a conceptual synthesis of the previous achievements. We show that the overall structure of the category Glob of global compositions carries a Grothendieck pretopology via finite covering families. It is well known that such a pretopology generates a Grothendieck topology J and therefore a Lawvere{Tierny topology j on the presheaf topos Glob@. We discuss the associated instances, such as the subobject classifier sheaf Ω, and the subtopos Sh(J; Glob) of sheaves.

As a compensation to the abstract nature of general topos theory, some principles for visualizing such abstract objects are mandatory, in particular in view of implementations on computerized knowledge bases. We give an account of such principles as they are being applied in graphical interface design.

Metrics and rhythmics are an excellent elementary test-case for global structures in music. We shall critically review two commonly known approaches: the Riemann and the Jackendoff–Lerdahl theories. We then develop the concepts of global time structures and their topologies, including associated weight functions.

This chapter is not only a good test-case for the mathematization of elementary music concepts, it is above all a refined study of turning fuzzy concept sketches of the humanities into precise and consistent frameworks—without the expected side-effect of “terrible mathematical simplification”. In the present case of motives, our topic is—within the general task of grasping motivic phenomena—the construction of Rudolph Reti's immanent motif analysis.

The present part on harmony is a traditionally dominant and extended portion of music theory. Therefore, it is adequate to review some of the important approaches to harmony. This chapter is however far from a complete synthesis of harmony and its history. We have selected three representative approaches which are systematically elaborate and theoretically founded: H. Riemann, P. Hindemith, and H. Schenker. The following overview concentrates on the divergence between claim and realization, and it does, once again, lay bare the enormous difficulty to set up a precise discourse about music without—héelas—the power of mathematical language. Also this critique is not thought to be a preliminary to something which in the subsequent chapters of this part will be perfectly solved by mathematical music theory. The discourse simply tries to persuade music theorists that a) the commonly cultivated status quo of the subject is scientifically unacceptable, and b) that mathematically sharpened concepts, constructs, and models can show ways to more in-depth and precise understanding of harmony—without banning it to history and “atonal” negation. Generic harmony is a universal perspective of music, and it is unscientific as well as near-sighted if not antimusical to abandon harmonic paradigms instead of embedding them into a diachronically and synchronically open, unified, and universal concept framework. To be clear, the main question is not to defend or instantiate any ideology of harmony—this is the unhappy business of Pythagorean fundamentalists—but to investigate its possible semiotic functions in musical works and their communicative explication, to develop an adequate language, and to propose consistent and sound models of harmonic processes.

This chapter introduces a systematic correspondence between chords and symmetries. It lays the morphological fundament for a semantic theory of harmonic functions which will be exposed in the following chapter. Essentially, the idea of such a correspondence is to carry over the structural discourse on harmony to richer objects by an intermediate switch to ‘richer’ addresses.

This chapter is about “understanding” aggregates of pitches in their combination on the syntagmatic/paradigmatic axes. This requires constructions of targets of such an understanding, i.e., harmonic semantics. The present theory succeeds in a (re)construction of function-theoretic semantics which is based on the paradigms associated with classical tonal functions. We discuss different approaches, among others the morphological theory of Noll and the approach of Mazzola based on chains of triads.

Cadences are shorthand representations of tonalities. (We do not discuss the other meanings—e.g., the solo cadence in the sense of a concert climax|of this typically homonymic term.) There is a variety of approaches to realize such a representation. We give an explicit definition of the concept of a cadence with respect to varied addresses and ambient spaces. In particular, we present the very classical cadences, those related to self-addressed function theory, and more exotic self-addressed cadences which relate to symmetries rather than to tones or sets of such objects.

This chapter deals with the central issue of modulation between two given tonalities. It involves explicit models of tonalities, of cadences and—even more crucial—of the transition process from one tonality to its successor. The present model involves the analogy to elementary particle physics: Modulation is viewed as a ‘force interactionș between two ‘tonality particles’ which is mediated by a ‘modulation quantum’. The model allows for a complete calculation of fundamental degrees of modulation in congruence with Arnold Schönbergs harmony [948]. The model is realized for diatonic tonalities in 12-tempered and just tuning. It has been extended to all 7-element scales in 12-tempered tuning and to a number of scales in just tuning. The 12-tempered extension reveals a privileged position of the diatonic scale with regard to this modulation theory. We conclude the chapter with a discussion of the basic role of modulation models and their application to optimize harmonic paths in the sense of Section 27.2.

This chapter deals with illustrations of the modulation model in chapter 80. It treats short and longer examples from Bach to modern jazz. This exercise should make clear the methodological background of these musicological experiments and illustrate the theory exposed in chapter 4.

The ideas of tangent objects described in Section 7.5 are applied to define contrapuntal intervals as tangential “arrows” from the cantus firmus to the discantus tone. This formalism fits with the idea that the discantus is a kind of melodic variation around the cantus firmus line. The ring structure of the set of such arrows is discussed and motivated from the musical perspective.

For contrapuntal composition and theory, consonant and dissonant intervals are a dichotomic concept. We present the mathematical restatement and fundamental properties of the basic concept of an interval dichotomy. For the classification of dichotomies, a strong condition on unique symmetries of polarity between the two halves of dichotomies is added. It reveals a distinguished role of the consonance/dissonance dichotomy of classical counterpoint and of the major dichotomy (associated with the major scale).We discuss evidence of the consonance/dissonance dichotomy from theoretical and empirical points of view and open the discourse to an intercultural perspective guided by the classification of dichotomies, as investigated by Jens Hichert [466].

This chapter presents the counterpoint model in form of a counterpoint theorem which guarantees the existence and exhibits an arsenal of admitted contrapuntal steps that come in extremely close to the rules of classical counterpoint. The theorem is based on the concept of a contrapuntal symmetry and follows the paradigm of local symmetries as a rationale for forces in physics. Because of its generic concept framework, the theorem, which in this general form1 was proved by Jens Hichert [466], is also valid for non-European scales. We discuss these extensions.