Knots, Music and DNA

Abstract

Musical gestures connect the symbolic layer of the score to the physical layer of sound. I focus here on the mathematical theory of musical gestures, and I propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (they apply to knotted proteins, for example), I end the article with a new musical rendition of DNA.

Quantum approaches to music cognition

“…in analogy to the density matrix formalism of statistical quantum mechanics (cf. Mannone (2018) for a related account in musicology). For an arbitrary number N of equally weighted context tones in a chord, we get…”

Shall We (Math and) Dance?

“…Summarizing, we applied to dance categorical formalism formerly used for music [16,17], comparing gestures in music and dance, and highlighting some of their connections. We started from conceptual considerations in Section 2, we discussed metaphorical yet formal applications of functors and commutative diagrams to music and dance in Section 3, and we finally made some references to monoidal categories in Section 4.…”

“…Here, the conductor metaphorically plays the role of a limit, while the audience plays that of a colimit. 16 Arrows appear thus inverted with respect to [17], but the two representations are both valid. See details in [17] about conductor and listener.…”

“…The category of gestures has gestures as objects and gestures of gestures (hypergestures) as the morphisms between them. The composition of hypergestures (paths) is associative up to a path of paths[16] 10. From[5]: "The physical signal of dance consists of a change in position of the dancer body in space with respect to time."…”

“…In salsa, there often are recognizable patterns with clave rhythms between strong beats. The pulse can depend on the specific music style, and it can be the object of future research in itself 16. Limits and colimits are generalizations of products and coproducts, respectively; they are obtained the ones from the others via reversing arrows[12] 17.…”