This page contains additional references that we found subsequent to our book's publication. We provide some commentary to show how they relate to our book's content.
* G. Touissaint, The Geometry of Music. Taylor & Francis/CRC Press, 2013. This book provides extensive coverage of mathematical techniques of describing cyclic rhythms. Several sections of Chapter 6 from our book made reference to papers by Touissant. Here you can find, in one place, a very thorough treatment of his techniques. It is an excellent place to go to for continuing the study of rhythm.
* H. Cowell, New Musical Resources. Cambridge University Press, 1996. This book is a reprinting of a classic by Henry Cowell, first published in 1930. Here you can find a clear discussion of the relationship between harmonics of tones and music (similar to, and more extensive than our treatment in Chapter 1). There is also an introduction to unusual time signatures (for which the mathematical treatment we gave in Chapter 2 should be of aid). Finally, and perhaps most importantly, there is a beautiful discussion with many musical examples of the close relation between pitch and rhythm (which we introduced in Chapter 6). Cowell's ideas on connecting pitch and rhythm were very influential in creating new music during the twentieth century.
* H. Cowell. Charles Ives. Pages 51 - 71 of Essential Cowell (Ed. D. Higgins), McPherson & Company, 2001. Also pp. 128 - 145 of American Composers on American Music (Ed. H. Cowell), Stanford University Press, 1933. This article is an excellent discussion of Charles Ives' use of unusual time signatures, as well as polyrhythmic and polyharmonic relationships. There is an excellent set of graphs illustrating a type of harmonic phasing in Ives' The Fourth of July (somewhat like Steve Reich's rhythmic phasing, which we discussed in Chapter 6, but applied in the harmonic realm).
* J. Douthett, M. Hyde, and C. Smith (Eds.). Music Theory and Mathematics: Chords, Collections, and Transformations. University of Rochester Press, 2008. This book contains a collection of articles by mathematical music theorists. Many of these articles would be good topics for Independent Study courses or Seminars, once completing our book. For example, the articles by John Clough ("Flip-Flop Circles and Their Groups"), by Jack Douthett ("Filtered Point-Symmetry and Dynamical Voice-Leading"), and Julian Hook ("Signature Transformations"), are important extensions of the music theory discussed in Chapters 3, 6, and 7 of our book.
* P.N. Vassilakis. SRA: A Web-based Research Tool for Spectral and Roughness Analysis of Sound Signals. Proc. SMC'07, 4th Sound and Music Computing Conf., July 2007, Lefkada, Greece. Download This paper contains a good discussion of tonal dissonance. An online program for computing tonal dissonance ("roughness") from musical recordings can be accessed by clicking here.