Music and Mathematics: Algorithmic Composition

The process of music composition can be, at times, a severely trying exercise. There seem to be endless techniques and approaches that a composer can use to further their artistic pursuits. One particular method may seem peculiar or fascinating depending on your view of music, this method being algorithmic composition (music being created through algorithms, or precise step-by-step mathematical instruction).

Algorithmic composition dates all the way back to the time of Mozart (eighteenth century) through usage of “musical dice games” which utilized matrices to create music. How this would occur is that vertical columns in the matrix represented certain measures in a traditional eight measure phrase (which tends to be uniform for most classical era works), and the horizontal columns would show all outcomes that could be caused by two throws of dice. Then, to create a second theme, another matrix would be used.

Moving into the era that algorithmic composition found its greatest progress, the age of computers, it is revealed how this composition technique became the revolutionary style that it is. The initial algorithmic compositions were a result of breakthroughs by the mathematician Joseph Schillinger in 1948. What Schillinger created were designs that would allow for machines to create music. Leon Theremin put Schillinger’s ideas into practice by building the Rhythmicon (a machine that played various rhythmic ideas) and the Musamaton (an instrument with the capability to reorganize existing music pieces).  Around 1955, the first programs in algorithmic composition were created at the University of Illinois by Lejaren Hiller and Leonard Isaacson. Essentially the studies were the computerization of the “musical dice games” of the eighteenth century composers like Mozart, with numbers being selected from “musical note tables.”  Eventually what was created out of this experimentation was in fact a piece for string quartet entitled Illiac Suite for String Quartet (although the piece was created via algorithms, it was transcribed for  western notation for performance purposes). This particular piece wound up being published in the music journal New Music Quarterly, in turn solidifying the road to progress algorithmic composition was taking.

Arguably the most famous composer of algorithmic music was the Greek composer Iannis Xenakis. Xenakis was educated in engineering, became a member of the Resistance against Soviet military action in the region and eventually worked as an architect before turning to music composition. Much of Xenakis’ pieces are composed in two parts, the first being initially created as a mathematical model, and the second being brought to life in a computer. Some of the mathematical ideas Xenakis used in his music included probability laws, stochastics, Markovian chains, game theory, group theory, set theory, Boolean algebra, and Gaussian distributions as formalizations. Although some music scholars deny it and propose an alternate theory, most scholars agree that indeterminacy is a driving force behind Xenakis’ work. He (Xenakis) even states in his book Formulized Music (p. 9) that “As a result of the impasse in serial music, as well as other causes, I originated in 1954 a music constructed from the principle of indeterminism.” Songs of Xenakis that show this idea are pieces like ST/10-1, 080262 (which was actually premiered at the French headquarters of IBM in 1962), ST/48-1, 240162, Atrées, and Morsima-Amorsima.

It shall be interesting to see how, in the current age of computers and technology, the process of algorithmic composition progresses. The revolutionary ideas highlighted in this blog post were created many years ago, far before the incredible advancements in technology we enjoy in our day-to-day lives. Only time will tell what advancements can be made in algorithmic composition, as musical progression is often a drawn-out process, but I believe it will be worth the wait.

(Note: It was truly impossible to feature every ounce of history and technical achievements by algorithmic composers, as this is a mere blog post. For further reading on this subject, I encourage you to look at books of David Cope, whose work helped me immensely in writing this post. I would specifically recommend the works The Algorithmic Composer and Virtual Music: Computer Synthesis of Musical Style).


18 thoughts on “Music and Mathematics: Algorithmic Composition

  1. Sometimes the most tedious part of composing is the endless choices…I like the idea of leaving it to chance to choose parameters to finish a piece…that or a good deadline!

  2. One of my most memorable experiences was watching Bernard Haitink conducting the KCO (Koninglijk Concertgebouw Orkest) in the beautiful Concertgebouw in Amsterdam. One could watch the music floating above the orchestra as he teased out the notes and created a visual mosaic above the heads of the musicians. So many ways to appreciate music. Thank you for this posting and for visiting my site which is also like visiting my sight.

  3. Thank you for this post Derek. I studied with M. Lloyd Tew in Hawaii and he was across all that you refer to. He too used computers to compose, even programming the rules of counterpoint and percentages of specific ‘uses’ (ie, 3rd in the bass) to produce Bach like results. His book for ‘vectoral’ composition is yet another way of looking at composition and/or analysis.

  4. I studied with Joseph Schwantner who had a high regard for Xenakis and the math behind his work. It’s fascinating stuff. But we all play from some built-in algorithms, right? The harmonic series is a pattern, and even composers like Hindemith revered and followed the inner patterns and laws of the physics of music. It’s all a bit over my head though!

  5. Glenn Gould, within the confines of the Classical Canon, was continually rejudging and reinterpreting the set formatting of the major composers, often to the delight of some and the anger of many.

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