Art of the Marbler
Film made in 1970 by Bedfordshire Record Office of Cockerell marbling.
The musical realm is a promising area in which to expect to find nontrivial topological structures. This paper describes several kinds of metrics on musical data, and explores the implications of these metrics in two ways: via techniques of classical topology where the metric space of all-possible musical data can be described explicitly, and via modern data-driven ideas of persistent homology which calculates the Betti-number bar-codes of individual musical works. Both analyses are able to recover three well known topological structures in music: the circle of notes (octave-reduced scalar structures), the circle of fifths, and the rhythmic repetition of timelines. Applications to a variety of musical works (for example, folk music in the form of standard MIDI files) are presented, and the bar codes show many interesting features. Examples show that individual pieces may span the complete space (in which case the classical and the data-driven analyses agree), or they may span only part of the space.
People naturally identify the rhythm of music as they tap their feet and sway in time with the beat. Underlying such motions is an act of cognition that is not easily reproduced in a computer program or automated by machine. ‘Rhythm and Transforms’ asks (and answers) the question: How can we build a device that can ‘tap its foot’ along with the music? The result is a tool for detecting and measuring the temporal aspects of a musical performance: the periodicities, the regularities (and irregularities), the beat, the rhythm. The impact of such a ‘rhythm meter’ on music theory and on the design of sound processing electronics is described. It allows discussion of the relationship between cognitive processing of temporal information and mathematical techniques used to describe and understand regularities in data.This book will interest engineers and others interested in the design of audio devices such as musical synthesizers, special effects devices, drum machines, and electronic keyboards. It will be useful to musicians and composers who exploit computer-based tools; arrangers, musicologists, and others interested in musical analysis; and those interested in the way the ear works, and how this influences the types of sound patterns we like to listen to.
Moon Watching I Big Bay Lagoon, Madeline Island, WI. 2014