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

mathematical models and procedures. Several papers in this volume exemplify this other side of Mathematical Music Theory (see for example Volk, 2004; Buteau, 2004; Nestke, 2004; Noll and Garbers, 2004).

2 >Classical< MaMuTh-Achievements

As the title suggests, this paper is written as a >vade mecum< to Mathematical Music Theory. Instead of listing various investigations I will try to explain just three selected results in a simple way, such that the reader should become interested to study the sources in more detail. The three examples share the characteristics of a >contrafactual experiment<, and all of them belong to the pitch-related subject domain. Norman Carey’s and David Clampitt’s theory of Wellformed Tone Systems emerged within the >Buffalo school< under the influence of John Clough. The Structure Theory of Consonance and Dissonance was developed by Guerino Mazzola who founded the >Zurich school<.³

³: Many contributions of this volume -- including my own -- are inspired by Mazzola’s work and by the discourse of the Zurich-school.

Richard Cohn’s investigation into the relation between transformation and voice leading parsimony is rooted in the >David-Lewin school<.⁴

⁴: The >Buffalo< and the >Lewin school< were in close contact from the very beginning. John Roeder was the first to compare Mazzola’s and Lewin’s rather independent works (cf. Roeder, 1993). Meanwhile the traditions merge into a common scientific discourse.

With regard to the search for inner-musical motivations of musical conventions I find all three results very interesting and promising.

2.1 Wellformed Tone Systems

Many mathematical approaches to tone relations and systems share the assumption that simply balancing the differences in pitch height along a single axis does not grasp the musically constitutive relations between tones. Instead we often find the investigation of (discrete, possibly infinite) tone- or interval vocabularies, which are generated through additive combination from suitably chosen elementary intervals (like octave and fifth, octave and fifth and major third). These generated vocabularies are supposed to be homogenous and are therefore called tone spaces. In our case we are concerned with the pythagorean tone space which is a two dimensional discrete lattice $P = Z .q× Z .o$ (q and o denoting »fifth«- and »octave«-generators respectively).

The concept of a tone system is more elaborate and involves tone spaces together with maps between them. The $pythagorean tone system$ $h : P --> R$ connects the pythagorean tone space $P$ with a real pitch height space $R$ through the pitch height function $3 h(a .q,b .o) = b+ a.log2(2)$ . ⁵

⁵: We refer to Haluska (2003) for a systematic study of such tone systems.

Carey and Clampitt (1989) search explanations for the prominence of certain tone systems, such as the pentatonic, diatonic or twelve tone system. In their investigation they depart from the observation that these systems can be generated by the fifth interval and the operation of octave identification. Hence one may study them with respect to the pythagorean tone system, where they occupy vertical