Logics and Music Theory appear in different classifications of the medieval academic curriculum. Logic is part of the trivium (among grammar and rhetoric, while music theory is listed among the more mathematical disciplines : arithmetic, geometry and astronomy). Logics as the study of reasoning underwent a tremendous transformation through a process of formalization and mathematization. Music Theory opened its scope to many non-mathematical aspects (in particular those, traditionally covered by the disciplines of the trivium). This "contrary motion" of research interest offers several meetings points for Logics and Music Theory. One particularly interesting 19th century meeting point shall be the starting point for my talk which then proceeds into 20th century Logics and Mathematical Music Theory.

Moritz Hauptmann (1953) in his treatise "Die Natur der Harmonik und der Metrik : Zur Theorie der Musik" presented some ideas which mark a radical position in the context of this MaMuPhi session. Hauptmann interprets music first of all as a manifestation of human thought. While assuming general dialectical principles behind the activity of human thought he claims that musical mistakes are logical mistakes. According to Hauptmann the unity of a tonality (Tonart) is the result of a dialectical triad. Inspired by the idea to literally interpret the musical triad as a dialectical triad, he loads the names of the intervals octave, fifth and third with the corresponding dialectical meanings. A tonality is a kind of hypertriad, i.e. constituted by three musical triads. Their contiguity via common tones is the source for the Quintbegriff of the tonality, a diremption as the result of conflicting tone meanings. The mediating and unifying Terzbegriff is based on a change of perspective : the state of the tonic triad of being a dominant (relative to the subdominant triad) is turned into the state of having a dominant (relative to the dominant triad).

Hugo Riemann’s (1872 and 1874) "Musikalische Logik" is inspired by Hauptmann’s ideas. Riemann elaborates upon the explanatory power of this dialectical paradigm for the constitution of typical cadences. I will show some traces of the intellectual squeeze on Riemann when he tries to bring both sides together : the dialectical explanation and music-theoretical facts. [Being in Paris I cannot refrain from re-addressing Riemann’s problem with a side glance to the semiotic square].

Riemann’s "Musikalische Logik" and "Musikalische Syntaxis" inspired the recent Neo-Riemannian approaches by David Lewin, Richard Cohn, Clifton Callender, Jay Hook, Tom Fiore and Ramon Satyendra and several others. But these left the original dialectical motivations behind. Yet the transformational approaches of David Lewin and Guerino Mazzola offer new ways to tie up with H. Riemann’s orphaned project of a "musical logic". My 2004 article "The Topos of Triads" is an attempt in this direction. [In my MaMuX-talk (friday december 4) I will clarify the close mathematical links between these investigations on the one hand and the american Neo-Riemannian tradition on the other]. The locial component which enters music theory here, is the internal logical semantics of a topos, even though in a rudimentary way. I will explain and illustrate this in my talk.