Geometry of Chords
نویسنده
چکیده
This article provides an introduction to basic geometric investigations of the 12-tone sytem and its subsets, the chords. The various definitions and results are intended to lay a theoretical basis for 12-tone-based explorative and empirical research on occidental harmony. The entire approach is motivated by the assumption that the 12 tones do not constitute an arbitrary 12-element set with arbitrary paradigmatic relations, but rather a discrete geometrical space. Transpositions obviously are instances of paradigmatic relations, that are reflected by affine transformations. From the mathematical point of view, the article presents a straightforward elaboration of this observation. From the music theoretical point of view it is of speculative nature and asks for experimental studies.
منابع مشابه
The geometry of musical chords.
A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly sy...
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The chords’ problem is a variant of an old problem of computational geometry: given a set of points of , one can easily build the multiset of the distances between the points of the set but the converse construction is known for a longtime as difficult. The problem that we are going to investigate is also a converse construction with the difference that it no more the one of the distances multi...
متن کاملComment on "The geometry of musical chords".
Tymoczko (Reports, 7 July 2006, p. 72) proposed that the familiar sonorities of Western tonal music cluster around the center of a multidimensional orbifold. However, this is not true for all tonal progressions. When prototypical three-voice cadential progressions by Bach converge on the tonic, the chords migrate from the center to the edge of the orbifold.
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A chord is an edge between two vertices of a cycle that is not an edge on the cycle. If a cycle has at least one chord, then the cycle is called a chorded cycle, and if a cycle has at least two chords, then the cycle is called a doubly chorded cycle. The minimum degree and the minimum degree-sum conditions are given for a graph to contain vertex-disjoint chorded (doubly chorded) cycles containi...
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