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

|_ _| |_ _| |_ _| |_ - 1 0 -1 _| |_ o _| |_ -o - d _| 0 - 1 0 p = - p 0 0 1 d d

and

|_ _| |_ _| |_ _| s - o- d s- o -d |_ t _| + |_ - p _| = |_ t- p _| . 0 d d

This way, an inversion with retrograde takes us from

3 3 R -- > R ,(o,p,d) '- --> (s- o- d,t- p,d).

10 Epilogue

An important aspect of MMT is how an application of mathematics to a particular area of the sciences or the humanities (in this case music theory) can enrich mathematics itself with new results, new objects, structures, and models.

In our case, MMT yields new mathematical objects (forms, denotators, global and local compositions, etc.) and motivated the development of the software tools RUBATO and PRESTO. As far as musicology is concerned, the contribution is so revolutionary that we can really talk about scientific musicology.

References

MAC LANE, SAUNDERS (1971). Categories for the Working Mathematician. Springer-Verlag, New York.

MAC LANE, SAUNDERS and MOERDIJK, IEKE (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, New York.

MAZZOLA, GUERINO (2002). The Topos of Music. Birkhäuser, Basel.

MONTIEL, MARIANA (1999). The Denotator. Its Structure, Its Construction and Its Role in Mathematical Music Theory (Masters Thesis). Ph.D. thesis, Facultad de Ciencias, UNAM, Mexico.