The Poetics of Simple Mathematics in Music (review)

Pozzi Escot: The Poetics of Simple Mathematics in Music. Softcover, 1999, ISBN 0-9634500-5-0, 124 pages, illustrated, index; Publication Contact International, 24 Avon Hill, Cambridge, Massachusetts 02140, USA; telephone/fax (617) 868-0215

Do universal principles underlie the construction of musical form? And if so, what do they look like? By means of numerous illustrations and accompanying descriptive prose, composer and theorist Pozzi Escot ventures provocatively in search of answers to these questions. She draws on H. E. Huntley's The Divine Proportion: A Study in Mathematical Beauty (1970), citing his notion "that there is a definite connection between music and mathematics . . . based on the similarity between the deep-seated structure of musical form and that of mathematical ideas." Although all of her analyses include discussion of this "divine proportion" (or Golden Mean), she also investigates ways in which arithmetic, harmonic, and geometric means create compositional structure. In the process, she identifies multi-layered symmetrical constructions present within the organization of pitch and rhythm, discusses the notion of "gnomonic growth," and describes specific techniques that have been used to create linear and non-linear musical structures.

Each of the ten chapters of The Poetics of Simple Mathematics in Music includes introductory material and detailed descriptions of how these specific mathematical structures may be found within selected compositions. This material is followed by charts and graphic diagrams; scores of individual works are included for reference. Ms. Escot has analyzed music by European and American composers including Hildegard von Bingen, Guillaume de Machaut, Franz Schubert, Frederic Chopin, Anton Webern, Ruth Crawford Seeger, Milton Babbitt, Luigi Dallapiccola, and György Ligeti. She has also worked with several compositions from (in Ms. Escot's terms) "Across Mountains and Oceans to Worlds Beyond Europe," specifically, songs of the West African Eve (or Ewe), the North American Kwakiutl and Zuni, and the South American Piro cultures.

Chapter One discusses the interrelationship between mathematical principles (ratios) used in the construction of Gothic cathedrals and those used in the construction of liturgical chants. The perfect ratios of the Pythagoras/Theano school (1:1, 1:2, 2:3, 3:4, 4:5) are explained in reference to the architecture created by the Cistercian order, a reformed Benedictine order founded in 1098 (the year Hildegard was born). The Cistercians emphasized that the purpose of chant was to radiate truth, and, as in the construction of cathedrals, attempts to accomplish this were based on geometrical ratios. Ms. Escot discusses the contributions of St. Augustine and Boethius to the formulation of the principles of mathematical means, and she gives diagrams and accompanying analytical descriptions of four chants by Hildegard von Bingen. Through these analyses the author illustrates how proportional structures were used in the creation of both individual melodic phrases and the ordering of these phrases in relationship to one another. In a sense, this first chapter forms the basis for all that follows, as it includes explanations of almost all of the analytical procedures as well as the mathematical principles that are used in the remaining discussions.

An additional principle, the notion of "gnomonic growth" (which Ms. Escot attributes to the naturalist D'Arcy Thompson, author of On Growth and Form, from 1961), underlies the analyses of Chopin's Prelude No. 1, Dallapiccola's Colore (from the Notebook for Annalibera), and Mr. Ligeti's Continuum. In the case of the first and last of these, the ordering of significant musical events is found to coincide with measured time divisions based on the Fibonacci series. Dallopiccola apparently derived his own numerical series by which he ordered events.

Many of Ms. Escot's geometrical graphic designs have been created from measurements of the individual phrases within a composition by noting pitch relationships according to the onset note of the phrase, the high point ("apex"), the low point ("nadir"), and the pitch at the end ("decay") of the phrase. This data is combined with durational measurements and are together plotted on an x...