Facts and Counterfacts Mathematical Contributions to Music-theoretical Knowledge

The present article argues in favor of a discipline of Mathematical Music Theory (see [16], [18] for earlier attempts). By reviewing and re-interpretating known results, we draw further conclusions and formulate working hypotheses. Especially, we recapitulate a known fact about diatonic triads and seventh chords in connection with an analogous fact about pentatonic and diatonic scales in the 12tone system. Putting these facts together on the basis of a canonical theory of scale generation we discuss some open questions concerning traditional music-theoretical agreements and disagreements. Triads and Seventh Chords understood as third chains in the 7tone diatonic system have the property that diatonic steps of the upper or lower voice correspond to third transpositions of the entire triads. The pentatonic and the diatonic scales as fifth chains in the 12-tone system have the similar property that certain semitone steps correspond to fifth transpositions of the entire scales. The commonality between both situations can be more generally described as a P1-cycle and occurs systematically in the generation of wellformed scales. On the basis of this theory we attempt to combine several music-theoretical concepts in a unified picture. Besides that we describe pseudo-diatonic systems as instances of a model class of the proper diatonic and discuss the counter-factual construction of a faithful recomposition. Acknowledgments: I’m grateful to Bernd Mahr for many inspiring discussions about the epistemology of mathematical modeling, to David Clampitt and Norman Carey for an inspiring correspondence about the interpretation of their theory as well as to my colleagues in Barcelona for their feedback.