Throughout the history of western music, composers have utilized mathematical techniques in creating musical works. From Pythagoras, Plato, and Ptolemy in ancient Greece to the sixth century music theorist Boethius, music was thought to be a corollary of arithmetic. With the widespread development of written musical notation beginning in the Renaissance Age, compositional craft became more and more highly developed. Especially prized were compositions intertwined with mathematical patterns. The eighteenth century composer and theorist Jean-Philippe Rameau was unequivocal in his views on the connection between mathematics and music in his 1722 Treatise on Harmony: “Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics.” With fugal composition techniques in the high baroque period, mathematical techniques in musical composition reached a high point. The classical and romantic eras, characterized by a movement away from polyphonic music, produced less obvious mathematically oriented composition technique. In the twentieth century however, mathematical formalisms were fundamental as replacements for the outdated tonal structures of the romantic era.
The Renaissance Canon
During the Renaissance Age, as musical craft was becoming highly prized, mathematical devices were developed to a considerable degree by Northern European composers. In the canons of Johannes Ockeghem a single melodic voice provides the basis by which one or more additional voices are composed according to various mathematical transformations of the original: mirror reflection of musical intervals (inversion), time translation, mirror reflection in time (retrograde), or a non-unit time scaling (mensuration canon). Composers of this period understood the word canon to mean a rule by which secondary voices could be derived from a given melody, in contrast to our modern usage of the word meaning a simple duplication with later onset time, as in the nursery rhyme round “Row, Row, Row Your Boat”.
Bach: The Canon Master
J.S. Bach was undoubtedly a master of canonic composition. Bach’s canons challenged performers to solve puzzles he set before them. Examples abound in A Musical Offering (BWV 1079), a collection of pieces written for King Frederick the Great of Prussia in 1747. At that time, Frederick was the employer of Bach’s son C.P.E. Bach. The first of two Canon a 2 (canon for two voices) from Musical Offering appears to be written for a single voice, but with two different clef symbols: one at the beginning of the first measure, and one at the end of the last. The canon, then, is to be performed with one voice reading from beginning to end, the other voice reading from end to end. In this small piece, Bach provides an example of retrograde or cancrizan (crab) canon. The puzzle in the second Canon a 2 is even more cleverly concealed: a single line with two clef signs in the first measure, one upside down. The cryptic instruction Quaerendo invenietis "Seek and ye shall find” is inscribed at the top of the manuscript. After deducing that the second, inverted clef sign indicates that the second voice of the canon is to proceed in inversion, the performer is left to “seek” the appropriate time translation at which the second voice should begin.
Other examples of Bach’s masterful canonic treatment are BWV 1074: Kanon zu vier Stimmen, which with its numerous key signatures, clefs, and repeat signs can be played from any viewing angle, and BWV 1072: Trias Harmonica, a single line of manuscript that when fully realized yields an eight-voice canon.
A late example of the technical canonic craft is the Der Spiegel-Duett (mirror duet) for two violins, attributed to Mozart, which maintains the retrograde inversion transformation for its entire 63 measures. An example of table canon, the sheet of music for this piece is placed on a table between two violinists. The players read the music from opposite sides: one right side up from beginning to end, and the other upside down from end to beginning. The musical artistry in this mechanistic device lies in the harmonic consistency that is maintained throughout the work.
A General Representation of Mathematical Transformations in Composition
A musical composition can be represented as a sequence of points from the module M over the cyclic groups of integers Zp:
M = Zp1 × Zp2 × Zp3 × Zp4
with the coordinates representing (respectively) onset time, pitch, duration, and loudness. For example, the twelve notes of the chromatic scale would be represented in the second coordinate by Z12. In this schematic, a point (x1, x2, x3, x4) in a musical motif would be represented as (x1+a, x2, x3, x4+β) in a later repetition of that motif at a different volume level. Similarly, a point in the transposed motif would be (x1, x2+a, x3, x4). Retrograde is represented as (μ-x1, x2, x3, x4). Inversion takes the form (x1, 2a-x2, x3, x4). Mensuration, as in the canons of Ockegham, is written (x1, x2, a×x3, x4). Transformations of this form were used extensively in the Renaissance and Baroque eras and, as we will see, played a fundamental role in post-tonal era of the twentieth-century.
Mathematical Structure in Atonal Music
At the turn of the twentieth century, as musical composition moved further from traditional concepts of tonality, music theorists and composers looked for new organizing principles on which atonal music could be structured. Older formalisms emphasized tonal center, repetition and development of themes. In a conscious movement away from such notions, the groundbreaking composer Arnold Schoenberg turned to the idea of serialism, in which a given permutation of the twelve chromatic pitches constitutes the basis for a composition. The new organizing principle called for the twelve pitches of this tone row to be used, singly or as chords at the discretion of the composer, always in the order specified by the row. When the notes of the row have been used, the process repeats from the beginning of the row. Composers like Webern, Boulez, and Stockhausen consciously used geometric transformations of onset time, pitch, duration, and loudness as mechanisms for applying the tone row in compositions. In the latter half of the twentieth century, set theoretic methods on pitch class sets dominated the theoretical discussion. Predicated on the notions of octave equivalence and the equally tempered scale, Howard Hanson and Allen Forte developed mathematical analysis tools that brought a sense of theoretical cohesion to seemingly intractable modern compositions.
Another mathematical approach to composition without tonality is known as Aleatoric Music, or chance music. This technique encompasses a wide range of spontaneous influences in both composition and performance. One notable exploration of aleatoric music can be seen in the stochastic compositions of Iannis Xenakis from the 1950s. Xenakis’s stochastic composition technique, in which musical scores are produced by following various probability models, was realized in the orchestral works “Metastasis” and “Pithoprakta” which were subsequently performed as ballet music in a work by George Ballanchine.
FURTHER READINGS:
Beran, Jan. Statistics in Musicology. Boca Raton: Chapman & Hall/CRC, 2003.
Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973
Grout, Donald Jay. A History of Western Music. New York: Norton, 1980.
Xenakis, Iannis. Formalized Music: Thought and Mathematics in Composition. Hillsdale, NY: Pendragon Press, 1992
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