Mathematics of composing
The "Mathematics of Composing" explores the intricate relationship between mathematics and music throughout history. Beginning with ancient thinkers such as Pythagoras and Plato, who viewed music as closely linked to numerical relationships, the connection has evolved significantly. During the Renaissance, composers like Johannes Ockeghem developed canons that employed mathematical transformations to derive new musical lines from a single melody. Compositional techniques became more structured, with figures like Jean-Philippe Rameau asserting that music must adhere to mathematical principles.
In later eras, particularly the twentieth century, composers embraced mathematical formalisms to navigate and create in a post-tonal landscape. Concepts like serialism, introduced by Arnold Schoenberg, utilized permutations of the 12 chromatic pitches for thematic development, while set theory provided analytical tools for modern compositions. Additionally, genres like "math rock" emerged, reflecting complex rhythms and structures rooted in mathematical concepts. Overall, the mathematics of composing reveals how numerical ideas have shaped musical creation and theory across centuries, inviting diverse interpretations and innovations in the art form.
Mathematics of composing
Summary: Mathematics and music developed in tandem and composition is firmly grounded in mathematics.
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 modern standardized musical notation thought to have begun in the Renaissance, compositional craft became more highly developed. Compositions intertwined with mathematical patterns were particularly highly regarded.
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, writing, “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.” Fugal composition techniques in the high Baroque period were highly mathematical. 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 tonal structures of the romantic era. There are even subgenres of rock music (started in the 1980s) called “math rock” and “mathcore” (after metalcore, a fusion of heavy metal and hardcore punk), which uses complex and atypical rhythmic structures, angular melodies, unusual time signatures, and changing meters. Metalcore, in particular, also uses harmonic dissonance. In another example, Robert Schneider composed a mathematical score for a play in 2009. He said:
I wrote a composition called ‘Reverie in Prime Time Signatures,’ that is obviously written in prime time signatures, that is, only prime numbers of beats per measure. Also the piece has kind of a sophisticated middle section that encodes some ancient Greek mathematics related to prime numbers in musical form, that I am proud of.
The Renaissance Canon
During the Renaissance, 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, which means a simple duplication with later onset time, as in the nursery rhyme round “Row, Row, Row Your Boat.”
Mathematical Transformations in Composition
In addition to standard musical notation, music can be represented mathematically as a sequence of points in an algebraic structure. A musical composition can be represented as a sequence of points from the module M over the cyclic groups of integers Zp
with the coordinates representing (respectively) onset time, pitch, duration, and loudness. For example, the 12 notes of the chromatic scale would be represented in the second coordinate by Z12. In this schematic, if a point (x1, x2, x3, x4) in a musical motif were repeated later at a different volume level, the repetition would differ in the first and last coordinate and would be represented as (x+α,x1, x2,x3,x4+β), where α is the time shift and β is the amount of the volume difference.
Inversion takes the form (x1, 2α-x2, x3, x4). Mensuration, as in the canons of Ockegham, is written (x1,x2,α• x3, x4). Transformations of this form were used extensively in the Renaissance and Baroque eras and played a fundamental role in post-tonal era of the twentieth century.
Mathematical Structure in Atonal Music
At the turn of the twentieth century, music theorists and composers looked for new organizing principles on which atonal music could be structured. Groundbreaking composer Arnold Schoenberg turned to the idea of “serialism,” in which a given permutation of the 12 chromatic pitches constitutes the basis for a composition. The new organizing principle called for the 12 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 Anton Webern, Pierre Boulez, and Karlheinz 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.
Bibliography
Beran, Jan. Statistics in Musicology. Boca Raton, FL: Chapman & Hall/CRC Press, 2003.
Forte, Allen. The Structure of Atonal Music. New Haven, CT: Yale University Press, 1973.
Grout, Donald Jay. A History of Western Music. New York: Norton, 1980.
Temperley, David. Music and Probability. Cambridge, MA: MIT Press, 2010.
Xenakis, Iannis. Formalized Music: Thought and Mathematics in Composition. Hillsdale, NY: Pendragon Press, 1992.