Musical Scales, Integer Partitions, Necklaces, and Polygons
نویسنده
چکیده
A musical scale can be viewed as a subsequence of notes taken from a chromatic sequence. Given integers (N,K) N > K we use particular integer partitions of N into K parts to construct distinguished scales. We show that a natural geometric realization of these scales results in maximal polygons.
منابع مشابه
Geometry and Harmony
A musical scale can be viewed as a subset of notes or pitches taken from a chromatic universe. Given integers (N,K) N > K we use particular integer partitions of N into K parts to construct distinguished sets, or scales. We show that a natural geometric realization of these sets have maximal area, so we call them maximal area sets. We then discuss properties of maximal area sets for the integer...
متن کاملMaximal Area Sets and Harmony
A musical scale can be viewed as a subset of notes or pitches taken from a chromatic universe. For the purposes of this paper we consider a chromatic universe of twelve equally spaced pitches. Given integers (N,K) with N > K we use particular integer partitions of N into K parts to construct distinguished sets, or scales. We show that a natural geometric realization of these sets have maximal a...
متن کاملStirling number of the fourth kind and lucky partitions of a finite set
The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.
متن کاملN-gon Waves - Audio Applications of the Geometry of Regular Polygons in the Time Domain
This paper presents an approach to the generation of periodic signals through the application of Euclidean geometry in the time domain. In particular, the content of this paper is focused on waveforms derived from regular polygons. It is an attempt to show how the geometrical relations and proportions of regular polygons and star polygons with respect to their Schläfli symbol can be made audibl...
متن کاملValidation andAutomatic Repair of Planar Partitions Using a Constrained Triangulation
Planar partitions are frequently used tomodel, among others, land cover, cadastral parcels and administrative boundaries. In practice, they are often stored as a set of individual polygons to which attributes are attached (e.g. with the Simple Features paradigm), causing different errors and inconsistencies (e.g. gaps, overlaps anddisconnectedpolygons), which are introducedduring their creation...
متن کامل