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

Guerino Mazzola and collaborators--more transparent and fully controllable by the user. It is not our music-theoretical intention to naively argue in favor of first order transition models. But we do argue that experiments on the basis of such a simple framework may offer useful insights for researchers with different theoretical perspectives: to >advocates< of first order transition models who may strengthen or relativize their arguments and to their critics or to the advocates of more sophisticated approaches who may try to interpret the results in the context of their theory.

The investigations presented in this article led to an extended experimental framework consisting of re-design of the basic HarmoRubette as well as a strategy allowing the user to integrate his/her own approaches.

1.1 Chords, Harmonic Loci, and Harmonic Analyses

A semiological point of view offers a suitable way to describe aspects of musical harmony in terms of a sign system the syntactic layer of which is constituted by a vocabulary $CHORDS$ of chords and the semantic layer of which is constituted by a harmonic configuration space $HARM$ . In the present article we discuss several concrete specifications of these two layers in detail.

The search for principles according to which chord syntagms are associated with configurations of harmonic loci could be labeled as a »study of harmonic signification«. However, we restrict our concept of harmonic signification to the elementary case of context-free significations $X |\ H$ of harmonic loci $H$ by isolated chords $X$ . We do not insist in a two-valued logic behind these significations, i.e. instead of sharply considering a relation $|\ (_ CHORDS × HARM$ with the associated characteristic function $x|\ : CHORDS × HARM --> {0,1}$ we consider a »fuzzy« Riemann Logic $RL : CHORDS × HARM --> R$ attributing (generalized) truth values $RL(X |\H)$ to the significations $X |\ H$ .²

²: There are several motivations behind the term >Riemann Logic<. Firstly, Hugo Riemann himself compared musical activity with logical reasoning. Secondly, does the harmonic ambiguity and especially in Riemannian functional harmony suggest a treatment in terms of fuzzy logic. Thirdly, does the attempt to apply set theory to music--as in the case of »American Set Theory«--naturally imply the critical question for the role of logics. Recall that sets are the semantic models for classical logics. In subsection 2.3 we review a link to Topos Logic applied to harmonic morphemes.

Besides our restriction to elementary significations, we consider only the simplest type of syntagms, namely chord sequences $S = (X ) k k=1,...,n$ and associate them with sequences $S |\H := (X | \ H ) k k k=1,...,n$ of isolated significations of the chords in $S$ . Within our framework we call such a sequence a harmonic analysis of the chord sequence $S$ . Its pure semantic layer, i.e. sequence $H = (H ) k k=1,...,n$ of harmonic loci is called the harmonic path of the analysis $S |\H$ . The term transition refers only to ordered pairs $(H ,H ) k k+1$ of consecutive loci within the semantic layer. i.e. it does not include the underlying chords. We write $H ~ H k k+1$ . Behind this decision there is another strong restriction of our framework: Chord successions are not studied directly (i.e. syntactically), but only indirectly in terms of transitions between signified harmonic loci. Hence, the data actually taken into considera-