A New Geometry Of Musical Chords In Interval Representation Dissonance Enrichment Degeneracy And Complementation

This monograph covers a fresh and original look at musical chords. The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition. This condition itself leads to a convenient geometric representation of the chordal space as a simplicial grid. Chords appear as points in this grid and musical inversions of the chord would generate beautiful polyhedra inscribed in concentric spheres centered at the barycenter. The radii of these spheres would effectively quantify the evenness and thus the consonance of the chord. Internal symmetries would collapse these chordal structures into polar or equatorial displays, creating a platform for a thorough degeneracy study. Appropiate morphisms would allow us to navigate through different chordal cardinalities and ultimately to characterise complementary chords.

This monograph covers a fresh and original look at musical chords. The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition. This condition itself leads to a convenient geometric representation of the chordal space as a simplicial grid. Chords appear as points in this grid and musical inversions of the chord would generate beautiful polyhedra inscribed in concentric spheres centered at the barycenter. The radii of these spheres would effectively quantify the evenness and thus the consonance of the chord. Internal symmetries would collapse these chordal structures into polar or equatorial displays, creating a platform for a thorough degeneracy study. Appropiate morphisms would allow us to navigate through different chordal cardinalities and ultimately to characterise complementary chords.

Pendragon Press is proud to offer this new, revised, and expanded edition of Formalized Music, Iannis Xenakis's landmark book of 1971. In addition to three totally new chapters examining recent breakthroughs in music theory, two original computer programs illustrating the actual realization of newly proposed methods of composition, and an appendix of the very latest developments of stochastic synthesis as an invitation to future exploration, Xenakis offers a very critical self-examination of his theoretical propositions and artistic output of the past thirty-five years. This edition of Formalized Music is an essential tool for understanding the man and the thought processes of one of this century's most important and revolutionary musical figures.

The first edition of Connections was chosen by the National Association of Publishers (USA) as the best book in OC Mathematics, Chemistry, and Astronomy OCo Professional and ReferenceOCO in 1991. It has been a comprehensive reference in design science, bringing together in a single volume material from the areas of proportion in architecture and design, tilings and patterns, polyhedra, and symmetry. The book presents both theory and practice and has more than 750 illustrations. It is suitable for research in a variety of fields and as an aid to teaching a course in the mathematics of design. It has been influential in stimulating the burgeoning interest in the relationship between mathematics and design. In the second edition there are five new sections, supplementary, as well as a new preface describing the advances in design science since the publication of the first edition. Contents: Proportion in Architecture; Similarity; The Golden Mean; Graphs; Tilings with Polygons; Two-Dimensional Networks and Lattices; Polyhedra: Platonic Solids; Transformation of the Platonic Solids I; Transformation of the Platonic Solids II; Polyhedra: Space Filling; Isometries and Mirrors; Symmetry of the Plane. Readership: Polytechnic students, architects, designers, mathematicians and general readers."

This book is an adventure into the unexplored territory of French esoteric philosophies and their relation to music. Occultism and esotericism flourished in nineteenth-century France as they did nowhere else. Many philosophers sought the key to the universe, some claimed to have found it, and, in the unitive vision that resulted, music invariably played an important part. These modern Pythagoreans all believed in the Harmony of the Spheres and in the powerful effects of music on the human soul and body. Faced with the challenge of the rationalist Enlightenment, then with that of modern scientism, they adapted their occultism to the prevailing style. A widely published musicologist and authority on esotericism, Godwin is able to give a clear and concise context for these philosophers' often surprising beliefs, and he demonstrates how this "speculative music" influenced composers such as Satie and Debussy, who were familiar with occultism. His long study of music and the Western esoteric tradition makes him uniquely qualified to unravel the strange story of these forgotten sages.

In recent years neo-Riemannian theory has established itself as the leading approach of our time, and has proven particularly adept at explaining features of chromatic music. The Oxford Handbook of Neo-Riemannian Music Theories assembles an international group of leading music theory scholars in an exploration of the music-analytical, theoretical, and historical aspects of this new field.

How we experience space by listening: the concepts of aural architecture, with examples ranging from Gothic cathedrals to surround sound home theater. We experience spaces not only by seeing but also by listening. We can navigate a room in the dark, and "hear" the emptiness of a house without furniture. Our experience of music in a concert hall depends on whether we sit in the front row or under the balcony. The unique acoustics of religious spaces acquire symbolic meaning. Social relationships are strongly influenced by the way that space changes sound. In Spaces Speak, Are You Listening?, Barry Blesser and Linda-Ruth Salter examine auditory spatial awareness: experiencing space by attentive listening. Every environment has an aural architecture.The audible attributes of physical space have always contributed to the fabric of human culture, as demonstrated by prehistoric multimedia cave paintings, classical Greek open-air theaters, Gothic cathedrals, acoustic geography of French villages, modern music reproduction, and virtual spaces in home theaters. Auditory spatial awareness is a prism that reveals a culture's attitudes toward hearing and space. Some listeners can learn to "see" objects with their ears, but even without training, we can all hear spatial geometry such as an open door or low ceiling. Integrating contributions from a wide range of disciplines—including architecture, music, acoustics, evolution, anthropology, cognitive psychology, audio engineering, and many others—Spaces Speak, Are You Listening? establishes the concepts and language of aural architecture. These concepts provide an interdisciplinary guide for anyone interested in gaining a better understanding of how space enhances our well-being. Aural architecture is not the exclusive domain of specialists. Accidentally or intentionally, we all function as aural architects.

The first edition of Ernst Kurth’s Musikpsychologie appeared in 1931, and was regarded by contemporaneous psychologists as no less than the foundation for a new systematic approach to the perception and cognition of music. Time has hardly diminished Kurth’s standing as an original scholar with a distinctive point of view. Music theorists, both in Europe and North America, regard him as an important figure in the history of music theory. Daphne Tan and Christoph Neidhöfer’s first full translation provides English-speaking theorists the opportunity to delve deeper into his ideas. Indeed, Kurth’s concerns – listening habits and habituation, metaphorical language, the limits of memory, and the role of the body in music experience, to name a few – are shared by many in the field today, especially scholars who work at the intersections of music theory, psychology, linguistics, and related disciplines. And while Kurth’s approach lacks the scientific rigour of modern-day empirical musicology, Musikpsychologie nevertheless presents a source of testable hypotheses for those working in the area of music perception and cognition. This translation of Musikpsychologie also has the potential to inspire a new generation of composers, especially through the topics in the second section (energy, force, space, and matter) and, given the inherently interdisciplinary nature of this book and the number of philosophical and scientific sources Kurth incorporates, it will appeal to those interested in the history of science and particularly in the emergence of psychology as an academic discipline in the early 20th century.