Psychology. Herbart’s influence on modern psychology is less direct and presently underestimated. Through his conceptual influence on Bernhard Riemann he even contributed to Einstein’s general relativity.

In the 20th century the scientific investigation of musical and music-related phenomena became more and more differentiated with respect to physical, physiological, psychological, cultural, technological and other perspectives. Consequently, various disciplines offered their own mathematical approaches, models and methods to be applied to the specific music-related phenomena under consideration. Acoustics and sound-technology are perhaps the most successful fields of applied mathematics within the broad field of music. Computer Science, linguistics and cognitive sciences gave rise to mathematical applications as well. However, the desire for direct applications of mathematics to music-theory decreased since music theorists turned away from a questionable concept of >nature< and directed their interest to the cultural relativity and conditionality of music. At the same time mathematics became a source of inspiration and an experimental playground for composers. Likewize, the first mathematically inspired approach to music theory with a noticable broader effect -- known as »atonal set theory« -- was developed for early atonal music.

Alongside the scientific main stream we find several attempts to introduce mathematical ideas into music theory and musicology. We recall Wolfgang Graeser who -- for tragic reasons -- was not able to accomplish the challenging research program he outlined in a fascinating talk at the Beethoven-Centennial 1927 in Vienna.²

Since the late seventies and early eighties of the 20th century there is a growing number of mathematical contributions to >pure< music-theoretical problems. Inspired by the pioneering papers and books of authors like John Clough, David Lewin and Guerino Mazzola (c.f. Clough, 1979; Lewin, 1982; Mazzola, 1985; Lewin, 1987; Mazzola, 1990) a scientific community of mathematically interested music theorists and music-theoretically interested mathematicians have been devoting their work to a renewed transdisciplinary discourse between both fields.

1.2 Epistemological and Pragmatic Considerations

Simply collecting all kinds of mathematical applications to music and related phenomena into a smorgasboard does not automatically lead to an (eventually) emerging academic discipline. As a preliminary step for an epistemological positioning we propose to pragmatically narrow the radius of Mathematical Music Theory to those mathematical approaches to music-theoretical problems whose self-conception cannot be clarified with the help of other disciplines besides music theory itself. Admittedly, this proposal does not anwer to the opening question of this section until we have not shown some interesting research results behind this pragmatically drawn epistemological borderline. But for a moment it may be useful to meditate upon some general common sense arguments.

On the one hand, the general skepticism towards mathematical approaches among music theorists is partially rooted in the more or less unsuccessful attempts