Централна ротација правилних (и неправилних) музичких полигона

Abstract

The text describes the application of one of the most important isometric transformations to the projected metro-rhythmic entities of individual harmonics of the spectrum. It is a direct isometry called central rotation. Central rotation conditions the hemiola structuring of the meter. Hemiolas are identified with regular and irregular geometric figures (primarily triangles) by means of a partition and the composition (index) number of a particular spectral harmonics. The partition and composition of numbers, which are dealt with in discrete mathematics, on the one hand, and, the technique of horizontal hemiolas, characteristic of the polyphony of the sub-Saharan region, on the other, served as a means of creating methods by which the isometric transformation of central rotation would be realized in (musical) time.

У тексту је описана примена једне од најважнијих изометријских трансформација на пројектоване метро-ритмичке ентитете појединачних хармоника спектра. Реч је о директној изометрији која се зове централна ротација. Централна ротација условљава хемиолно структурисање метра. Хемиоле су идентификоване с правилним и неправилним геометријским фигурама (првенствено троугловима) посредством партиције и композиције (индексног) броја одређеног спектралног хармоника. Партиција и композиција броја којима се бави дискретна математика, с једне стране, и, техника хоризонталних хемиола, својствена полифонији суб-сахарске регије, с друге стране, послужили су као средство за остварење метода којим би се изометријска трансформација централне ротације реализовала у (музичком) времену.

The application of central rotation to the projected metro-rhythmic entities of the harmonics of the spectrum presents the continuation of the study of possibilities of music-mathematical analysis primarily through the prism of planimetry and trigonometry. The text assumes that polygonal lines from the reference systems of the individual harmonics of the spectrum can be represented by integers. Accordingly, geometric figures (polygons) could be identified and defined according to the extent of the harmonics of the spectrum, while the newly acquired spectral geometric figures would be structured by means of a partition. The principle of polygonal structuring of harmonics expressed in integer is elaborated below. For example, polygonal (triangular) number 3 (which would be the reference for the third harmonic) is also a number that is polygonally constructed by partitioning three of its equal sides (1 + 1 + 1), thus forming an equilateral triangle. However, numbers 5 or 7 are not triangular numbers, although they can become numbers that are polygonal (triangular) constructed by partitioning three of their elements (pages) as follows: (a) for number 5: (2 + 1 + 2); and, (b) for number 7: (1 + 3 + 3) or (2 + 2 + 3). Accordingly, in the reference system of the fifth harmonic of the spectrum it is possible to form (spectral) equilateral (sharp) triangles, and in the reference system of the seventh harmonic of the spectrum it is possible to form (even) two (spectral) equilateral triangles - the first would be sharp-angled while the second would be obtuse. With this observation, a method can be developed for polygonal number construction (harmonics), in which the aim would be, above all, for the elements to serve as homothetically growing shapes, both regular and irregular (musical) polygons. Hemiol groups are generally regarded as regular polygons. The text describes how the choice of a rhythmic unit of reference influences the structuring of a hemiol on any metric platform. With regular polygons, which are a reference for each harmonic individually, the principle of rhythmic rotability can be realized by means of isometric transformation, in particular – central rotation. This principle is valuable because we find its etymological roots in the folklore rhythm of Central African and West African music. It is achieved by “moving” the accents that make up the geometric figure (regular polygon) to the adjacent rhythmic unit in the direction: left or right. The temporal along from one accent to another is the site of a regular polygon. This gives the impression of temporal kinematics of an isolated sound entity. This metro-rhythmic investigation is also aimed at examining the identification of irregular geometric figures in a plane. Examples of irregular hemiol groups are associated with irregular polygons – equilateral or triangular triangles. They can be constructed in the metro-rhythmic plane, and more importantly, all isometric transformations, including central rotation, can be performed on them. All the above examples of the rotational motion of horizontal hemiol groups find their constructive application in compositional work.