Asymmetric polygons with maximum area

Abstract

We say that a polygon inscribed in the circle is asymmetric if it contains no two antipodal points being the endpoints of a diameter. Given n diameters of a circle and a positive integer k < n, this paper addresses the problem of computing a maximum area asymmetric k-gon having as vertices k < n endpoints of the given diameters. The study of this type of polygons is motivated by ethnomusiciological applications.

Maths, Computation and Flamenco: Overview and Challenges

“…That is, no two vertices of the polygon are antipodal vertices. The bulería pattern, which is referred to as an asymmetric rhythm in [2], as well as an antipodal rhythm in [1], falls into this category.…”

Scaling and compressing melodies using geometric similarity measures

“…With the increased availability of large musical data comes the need to design methods for searching and analysing this information. Musicological and computational studies on rhythmic and melodic similarity have given rise to a number of geometric problems [1,2,3]. A melody can be codified as a consecutive sequence of musical notes and each note can be represented by a point (point representation) [4] or a horizontal segment (segment representation) with a time/pitch value [5].…”

Computing Melodic Templates in Oral Music Traditions

“…An important subfield of music technology is music information retrieval (MIR), the scientific discipline of retrieving high-level information from music recordings. In particular, several optimization problems have been explored in the context of music analysis, including the modeling of rhythmic structures [22,1,2], harmony improvisation [24], melodic similarity [23] and the detection of melodic patterns [15].…”