Old and New Isometries between Pc sets in the Planet-4D Model

2020

Two differing explanations of the cosmos existed in the ancient world, and they co-existed for hundreds of years. Planetary harmonies can be traced from Plato's Republic (c. 380 BCE) to Macrobius (c. 420 CE), a period of around eight centuries. The Planets also appear in Plato's Timaeus, where they follow the circle of the Different, in orbits along the ecliptic that delineates the constellations of the Zodiac. Intersecting the circle of the Different is the circle of the Same, and these two circles meet in the shape of an X, at two symmetrically opposite locations in the sky. Differing cosmic constructs -one musical, the other three-dimensionaltraveled together from Plato to Cicero to Macrobius, whose Commentary on the Dream of Scipio transmitted the intricacies of Plato's cosmos into the Renaissance and beyond.

2012

Parsimonious voice leading is a term, first used by Richard Cohn, to describe non-diatonic motion among triads that will preserve as many common tones as possible, while limiting the distance traveled by the voice that does move to a tone or, better yet, a semitone. Some scholars have applied these principles to seventh chords, laying the groundwork for this study, which strives toward a reasonably comprehensive, usable model for musical analysis. Rather than emphasizing mathematical proofs, as a number of approaches have done, this study relies on two- and three-dimensional geometric visualizations and spatial analogies to describe pitch-class and harmonic relationships. These geometric realizations are based on the organization of the neo-Riemannian Tonnetz, but they expand and apply the organizational principles of the Tonnetz to seventh sonorities. It allows for the descriptive “mapping” or prescriptive “navigation” of harmonic paths through a defined space. The viability of the theoretical model is examined in analyses of passages from the repertoire of Frédéric Chopin. These passages exhibit a harmonic syntax that is often difficult to analyze as anything other than “tonally unstable” or “transitional.” This study seeks to analyze these passages in terms of what they are, rather than what they are not.

Voronoi Diagrams in …, 2007

We introduce the use of weighted medial graph structures to represent sounds and musical events. Such structures permit the building of hierarchical neighborhood relationships (symmetries) while mapping time to a weight (of a note, or event). This is useful to relate perceptual organisations (groupings in the sense of Gestalt theory) both visually and aurally. We treat time in a cinematographic way, effectively using a spacetime representation. The use of colour, modulation (of waveforms), orientation, speed, etc., allows us to integrate multiple musical features in our dynamic diagrams. We provide illustrations, including animations, of our ideas.

This paper documents a piece of practice-based research into the manipulation of data axes within audiovisual music composition. A theoretical and historical context is outlined , and a working methodology is put forward. Musical parameters are mapped over time using a combination of Processing and SuperCollider, to a pyramid structure which is manipulated to reorient the geometric musical forms contained within. This work results in Precession, a piece of audiovisual concert music a copy of which is available at youtube.com/user/edwright440. The piece is briefly evaluated and suggestions as to how this and similar works could be developed are put forward.

Would intelligent beings in another galaxy use musical scales with the same mathematical basis as ours? If the philosopher Plato had known of galaxies, he would surely have answered yes. In his Timaeus he made creation start with the division of the World Soul in the ratios of a Greek scale, which then became the model for the movements of the stars and planets and everything elsethe music of the spheres.

Acta Musicologica, 2015

Abstract: The aim of this essay is to create a geometrical link between the music theory and the mathematics of the early 17th century by studying and comparing diagrams which directly or indirectly refer to mathematical logarithms.The focus is on the relationships between ratios of numbers referring to sounds and related concepts of perception. The relationship between frequency and pitch is a paradigmatic case of the Weber-Fechner law of psychophysics, stating that equal frequency ratios are perceived as equally sized musical intervals. The Weber-Fechner law maintains that many perceptual phenomena are logarithmic by their very nature.The circular diagrams studied here are by Descartes (1618), Robert Fludd (1618) and Jost Bürgi (1620). Descartes’s diagrams have recently attracted the attention of authors from different fields. A second type of geometric diagrams related to musical arithmetic is looked at in the final section of this article.

"Let no one ignorant of geometry enter" this quote according to legend, was inscribed on the entrance to Plato’s Academy in Athens. A theory of musical Platonism is outlined, firstly a universal theory in which Xenakis’ outside time structures and temporal structures are considered to be built from pitch and duration quanta. Secondly, a transcendental musical Platonism is described that regards Xenakis’ in-time structures as in a direct relationship to a Platonic Form that may be modelled as a sequence of pitch and duration ratios. Ideas derived from Gödel’s use of integers built from successive prime numbers raised to a power, a Gödel number, are used to symbolically encode in-time structures of pre-existing music. This encoding of music in a musical integer is regarded as a model of the music’s in-time structure that is in a direct relationship to the Platonic Form of the music. This leads to a number based, geometric interpretation of musical Platonism that is of interest to composers, music theorists and philosophers. Keywords Music, Philosophy, Logic, Plato, Xenakis, Gödel

2009

Musical theory about the structure and morphology of Western tonality is quite difficult to teach to young children, due to the relatively complex mathematical concepts behind tonality. Children usually grasp the concepts of musical harmony intuitively through listening to music examples. Placing the 12 notes of the well-tempered scale into a spatial arrangement, in which the proximity of these notes represents their mutual harmonic relationship, would allow to link physical motion through a spatial area with the exploration of music tonality. Music theorists have postulated the Circle of Fifth, the “Spiral Array”, and the “Tonnetz” as paradigms for spatial arrangements of music notes which allow mapping the distance between notes onto their “mutual consonance”. These approaches mostly have been of qualitative nature, leaving the actual numeric parameters of the spatial description undetermined. In this paper, these parameters have been determined, leading to a concrete numerical de...

This work retrieves part of the first principles of the ancient Egyptian architectonic mathematics of material and ethereal cosmic systems of numbers in plane and spherical geometry, which are presented in thirty-nine mathematical equations. It is part of the Architecture of God on depicting the initial structure and the architectonic workability of a solar system similar to the eternal ethereal system of the throne of one and its twenty-four musical seats, its four guards that each has six wings, and its seven working heavens. It is related to the theory of pyramid design and the works of the great king Suphis Nagos (Sphinx) Armageddon, i.e., the falcon of the Mighty God, and the friend of Gephry-La.

Aplimat 2019- 18th Conference on Applied Mathematics (5-7 February 2019), Bratislava, Slovak University of Technology, 2019

The present contribution examines the characteristics and reasons for the recurrence in graphic language of plane tessellation that can be assimilated to isometric orthogonal axonometry: a peculiar case where the two-dimensional configuration acquires a three-dimensional meaning. Known since ancient times, its suggestive effects were used in a series of artistic expressions in the twentieth century as well as in the graphic language of today’s music videos.