Cris Forster established The Chrysalis Foundation, a public 501(c)3 nonprofit foundation. The Chrysalis Foundation supports instrument builder and composer Cris Forster, who wrote Musical Mathematics.


Musical Mathematics

on the art and science of acoustic instruments


Table of Contents



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Also available from the publisher at Chronicle Books, San Francisco.


© 2000–2020 Cristiano M.L. Forster
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David Canrights Foreword


          I met Cris Forster more than thirty years ago. Shortly thereafter, I saw him perform Song of Myself, his setting of Walt Whitman poems from Leaves of Grass. His delivery was moving and effective. Several of the poems were accompanied by his playing on unique instruments — one an elegant box with many steel strings and moveable bridges, a bit like a koto in concept; the other had a big wheel with strings like spokes from offset hubs, and he rotated the wheel as he played and intoned the poetry. I was fascinated.

          Since that time, Cris has built several more instruments of his own design. Each shows exquisite care in conception and impeccable craftsmanship in execution. And of course, they are a delight to hear. Part of what makes them sound so good is his deep understanding of how acoustic musical instruments work, and part is due to his skill in working the materials to his exacting standards.

          But another important aspect of their sound, and indeed one of the main reasons Cris could not settle for standard instruments, is that his music uses scales and harmonies that are not found in the standard Western system of intonation (with each octave divided into twelve equal semitones, called equal temperament). Rather, his music employs older notions of consonance, which reach back as far as ancient Greek music and to other cultures across the globe, based on what is called just intonation. Here, the musical intervals that make up the scales and chords are those that occur naturally in the harmonic series of overtones, in stretched flexible strings, and in organ pipes, for example.

          In just intonation, the octave is necessarily divided into unequal parts. In comparison to equal temperament, the harmonies of just intonation have been described as smoother, sweeter, and/or more powerful. Many theorists consider just intonation to be the standard of comparison for consonant intervals. There has been a resurgence of interest in just intonation since the latter part of the twentieth century, spurred by such pioneers as Harry Partch and Lou Harrison. Even so, the community of just intonation composers remains comparatively quite small, and the subset of those who employ only acoustic instruments is much smaller still. I know of no other living composer who has created such a large and varied ensemble of high quality just intoned acoustical instruments, and a body of music for them, as Cris Forster.

          Doing what he has done is not easy, far from it. The long process of developing his instruments has required endless experimentation and careful measurement, as well as intense study of the literature on acoustics of musical instruments. In this way Cris has developed deep and rich knowledge of how to design and build instruments that really work. Also, in the service of his composing, Cris has studied the history of intonation practices, not only in the Western tradition, but around the world.

          This book is his generous offering of all that hard-earned knowledge, presented as clearly as he can make it, for all of you who have an interest in acoustic musical instrument design and/or musical scales over time and space. The unifying theme is how mathematics applies to music, in both the acoustics of resonant instruments and the analysis of musical scales. The emphasis throughout is to show how to use these mathematical tools, without requiring any background in higher mathematics; all that is required is the ability to do arithmetic on a pocket calculator, and to follow Cris’ clear step-by-step instructions and examples. Any more advanced mathematical tools required, such as logarithms, are carefully explained with many illustrative examples.

          The first part of the book contains practical information on how to design and build musical instruments, starting from first principles of vibrating sound sources of various kinds. The ideas are explained clearly and thoroughly. Many beautiful figures have been carefully conceived to illuminate the concepts. And when Cris gives, say, formulas for designing flutes, it’s not just something he read in a book somewhere (though he has carefully studied many books); rather, you can be sure it is something he has tried out: he knows it works from direct experience. While some of this information can be found (albeit in a less accessible form) in other books on musical acoustics, other information appears nowhere else. For example, Cris developed a method for tuning the overtones of marimba bars that results in a powerful, unique tone not found in commercial instruments. Step-by-step instructions are given for applying this technique (see Chapter 6). Another innovation is Cris’ introduction of a new unit of mass, the “mica,” that greatly simplifies calculations using lengths measured in inches. And throughout Cris gives careful explanations, in terms of physical principles, that make sense based on one’s physical intuition and experience.

          The latter part of the book surveys the development of musical notions of consonance and scale construction. Chapter 10 traces Western ideas about intonation, from Pythagoras finding number in harmony, through “meantone” and then “well-temperament” in the time of J. S. Bach, up to modern equal temperament. The changing notions of which intervals were considered consonant when, and by whom, makes a fascinating story. Chapter 11 looks at the largely independent (though sometimes parallel) development of musical scales and tunings in various Eastern cultures, including China, India, Indonesia, as well as Persian, Arabian, and Turkish musical traditions. As far as possible, Cris relies on original sources, to which he brings his own analysis and explication. To find all of these varied scales compared and contrasted in a single work is unique in my experience.

          The book concludes with two short chapters on specific original instruments. One introduces the innovative instruments Cris has designed and built for his music. Included are many details of construction and materials, and also scores of his work that demonstrate his notation for the instruments. The last chapter encourages the reader (with explicit plans) to build a simple stringed instrument (a “canon”) with completely adjustable tuning, to directly explore the tunings discussed in the book. In this way, the reader can follow in the tradition of Ptolemy, of learning about music through direct experimentation, as has Cris Forster.

David R. Canright, Ph.D.
Del Rey Oaks, California
January 2010