Abstract
Music is a testing challenge for formal information systems. Here we apply the full power of category theory to the challenge, involving the topos for data structuring and the monad for process. The topos handles many aspects of the data for a performance including the score and variants, the orchestral players, the conductor and the supporting infrastructure such as funding bodies. The monad as process controls the adjointness between the functors representing articulation and intonation, based on perceived activity in the brain in professional musicians. We present a musical performance as a categorical composition over time signatures that proceed in successive adjoint steps with the monad looking back and its associated comonad looking forward. The physical complexity of each musical sound operates in its respective time-frame, represented by a limit, as a colimit. The formalism can be implemented in a functional programming language such as Haskell.
Highlights
Much work has been done on computer representations of music at the physical level
Earlier the emphasis had been on the pitch classes being treated as sets of elements, each element being a note within the item
The transformational approach extended this technique by adding a transition from one pitch class to another to capture the dynamic possibilities within a musical piece
Summary
Much work has been done on computer representations of music at the physical level. Developments by Klumpenhouwer such as K-nets [10] provide a way for representing transformations from one pitch-class to another. Problems occurred with the sets representing the graphs, resulting in their replacement by the category of relations REL [28] This facilitates handling relationships but is inferior to the pullback, which can be locally Cartesian closed and adaptable to a topos view. A verbal presentation of the work was given at the 6th World Congress on Universal Logic, held at the University of Vichy, France, in June 2018 [31]
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