Mathematics and musicology have a long-standing relationship, but it is less productive than it might be. Reasons for this are explored in case studies of the ‘gap-fill’ melodic principle and of motivic analysis. In the first case empirical results do not unequivocally support the principle, but it continues to be used by musicologists. In the second, mathematical and computational approaches are found to differ significantly from those of music analysis in their purpose and effect. Other differences between the disciplines are examined in the use of metaphor in musical discourse, and misunderstandings over the role of abstraction and estimation. Throughout, human factors are found to confound proper communication. I propose that better interdisciplinary research could arise from honesty about limitations, effort in understanding each other’s disciplines, and humility about achievements.
The response to Guerino Mazzola defends the use of imprecise concepts in some circumstances, particularly in the light of the impossibility of a precise definition of the domain of music. A possible contribution of mathematics to music through the demonstration of relationships between formulations of different theories is envisaged, specifically those of species counterpoint and functional harmony.
In response to the contribution to this volume by Geraint Wiggins, I ask what music theory is for, and argue that, through education and the activity of composers, it has an influence on the very music it aims to describe. I defend Schoenberg from the charge of ignoring musical perception, and claim his music is successful even if not in the way he had envisaged. The music theoretic enterprise, including its mathematical branch, has an effect on musical culture, but that effect might be difficult to predict.
Apparently contrasting ontologies of music as a psychological-cognitive entity or as having independent existence are shown not to be mutually exclusive. They are related to orthogonal dimensions of empirical and rational approaches in research. The best research employs both approaches. Mathematical and computational researchers are urged to explore research which moves away from the abstract, rational approach and makes greater use of empirical data. Suggestions are given for future research.