“There is more in man and in music than in mathematics, but music includes all that is in mathematics.”—Peter Hoffman
Infotainment is usually thought of as light entertainment peppered with superficial “facts” and forgettable news. Yet another kind of infotainment exists, a musical kind that is based on mathematical algorithms. It is true entertainment that is filled with true information and though it is mathematically modeled none of it is fake.
In the twentieth century interest in the multidisciplinary fields of Information Theory and Cybernetics led to dizzy bursts of creativity when their ideas were applied to making new music. These disciplines applied rigorous math to the study of communication systems and how a signal transmitted from one person can cut through the noise of other spurious signals to be received by another person. They also made explicit the role of feedback inside of a system, how signals can amplify themselves and trigger new signals. All of this was studied complex equations and formulas.
Yet there is nothing new about the relationship between music and math.
Algorithmic music has been made for centuries. It can be traced all the way back to Pythagoras, who thought of music and math as inseparable. If music can be formalized in terms of numbers, music can also be formalized as information or data. The “data” the ancients used to drive their compositions was the movement of the stars. Ptolemy is known to us most for his geocentric view of the cosmos and the ordered spheres the celestial bodies traveled on. Besides being an astronomer Ptolemy was also a systematic musical theorist. He believed that math was the basis for musical intervals and he saw those same intervals at play in the spacing of heavenly bodies, each planet and body corresponding to a certain modes and notes.
Ptolemy was just one of many who believed in the reality of the music of the spheres. Out of these ancient Greek investigations into the nature of music and the cosmos came the first musical systems. The musician who used them was thus a mediator between the cosmic forces of the heavens above and the life of humanity here below.
Western music went through myriad changes across the intervening centuries after Ptolemy. World powers rose and fell, new religions came into being. Out of the mystical monophonic plainchant uttered by Christian monks in candlelit monasteries polyphony arose, and it called for new rules and laws to govern how the multiple voices were to sing together. This was called “canonic” composition. A composer in this era (15th century) would write a line for a single voice. The canonic rule gave the additional singers and voices the necessary instruction. For instance one rule would be to for a second voice to start singing the melody begun by one voice again after a set amount of time. Other rules would denote inversions, retrograde movement, or other practices as applied to the music.
From this basis the rules, voices, and number of instruments were enlarged through the renaissance until the time of the era of “Common Practice”, roughly between 1650 to 1900. This period encompassed baroque music, and the classical, romantic and impressionist movements. The 20th and 21st century are now giving birth to what Alvin Curran has called the New Common Practice.
In the Common Practice Era tonal harmony and counterpoint reigned supreme, and a suite of rhythmic and durational patterns gave form to the music. These were the “algorithmic” sand boxes composers could play in.
The New Common Practice, according to Curran encompasses, “the direct unmediated embracing of sound, all and any sound, as well as the connecting links between sounds, regardless of their origins, histories or specific meanings; by extension, it is the self guided compositional structuring of any number of sound objects of whatever kind sequentially and/or simultaneously in time and in space with any available means.” I’ve begun to think of this New Common Practice as embracing the entire gamut of 20th and 21st century musical practices: serialism, atonality, musique concrete, electronics, solo and collective improvisation, text pieces, and the rest of it.
One vital facet of the New Common Practice is chance operations, or the use of randomizing procedures to create compositions. Chance operations have a direct relation to information theory, but this approach can already be seen making cultural inroads in the 18th century when games of chance had a brief period of popularity among composers and the musical and mathematically literate. These are a direct precursor to the deeper algorithmic musical investigations that have started to flourish in the 20th century.
Much of this original algorithmic music work was done the old school way, with pencil, sheets of paper, and tables of numbers. This was the way composers plotted voice-leading in Western counterpoint. Chance operations have also been used as one way of making algorithmic music, such as the Musikalisches Würfelspiel or musical dice game, a system that used dice to randomly generate music from tables of pre-composed options. These games were quite popular throughout Western Europe in the 18th century and a number of different versions were devised. Some didn’t use dice but just worked on the basis of choosing random numbers.
In his paper on the subject Stephen Hedges wrote how the middle class in Western Europe were at the time enamored with mathematics, a pursuit as much at home in the parlors of the people as in the classroom of professors. "In this atmosphere of investigation and cataloguing, a systematic device that would seem to make it possible for anyone to write music was practically guaranteed popularity.”
The earliest known example was created by Johann Philipp Kirnberger with his "The Ever-Ready Minuet and Polonaise Composer" in 1757. C. P. E. Bach's came out with his musical dice game "A method for making six bars of double counterpoint at the octave without knowing the rules" five years later in 1758. In 1780 Maximilian Stadler published "A table for composing minuets and trios to infinity, by playing with two dice". Mozart was even thought to have gotten in on the dice game in 1792 when an unattributed version made an appearance from his music publisher a year after the composer’s death. This has not been authenticated to be by the maestro’s hand, but as with all games of possibility, there is a chance.
These games may have been one of the many inspirations behind The Glass Bead Game by Herman Hesse. This novel was one of the primary literary inspirations and touchstones for the young Karlheinz Stockhausen. The Glass Bead Game portrays a far future culture devoted to a mystical understanding of music. It was at the center of the culture of the Castalia, that fictional province or state devoted to the pursuit of pure knowledge.
As Robin Maconie put it the Glass Bead Game itself appears to be “an elusive amalgam of plainchant, rosary, abacus, staff notation, medieval disputation, astronomy, chess, and a vague premonition of computer machine code… In terms suggesting more than a passing acquaintance with Alan Turing’s 1936 paper ‘On Computable Numbers’, the author described a game played in England and Germany, invented at the Musical Academy of Cologne, representing the quintessence of intellectuality and art, and also known as ‘Magic Theater’.”
Hesse wrote his book between 1931 and 1943. The interdisciplinary game at the heart of the book prefigures Claude Shannon’s explosive Information Theory which was established in his 1948 paper A Mathematical Theory of Communication. His paper in turn bears a debt to Alan Turing, whom Shannon met in 1942. Norbert Wiener also published his work on Cybernetics the same year as Shannon. All of these ideas were bubbling up together out of the minds of the leading intellectuals of the day. Ideas about computable numbers, the transmission of information, communication, and thinking in systems, all of which would give artists practical tools for connecting one field to another as Hesse showed was possible in the fictional world of Castalia.
Robin Maconie again had the insight to see the connection between the way Alan Turing visualized “a universal computing machine as an endless tape on which calculations were expressed as a sequence of filled or vacant spaces, not unlike beads on a string”.
As the Common Practice era of western music came to an end at the close of the 19th century, the mathematically inclined serialism came into its own, and as the decades wore on games of chance made a resurgence, defining much of the music of the 20th century. With the advent of computers the paper and pencil method have taken a temporary backseat in favor of methods that introduce programmed chance operations.
Composers like John Cage took to the I Ching with as much tenacity as the character Elder Brother did in Hesse’s book. Karlheinz Stockhausen meanwhile used his music as means to make connections between myriad subjects and to create his own unique ‘Magic Theater’. Cybernetics and Information Theory each contributed to thinking of these and other composers.
Dice Music in the Eighteenth Century, pp. 184–185, Music and Letters 59: 180–87.
Conceptualizing music: cognitive structure, theory and analysis, by Lawrence M. Zbikowski, Oxford, 2002
The New Common Practice by Alvin Curran
Other planets: the complete works of Karlheinz Stockhausen 1950–2007, Rowman & Littlefield Publishers, 2016
A set of musicians dice have been made that offer up numerous possibilities for the practicing musician. Using random process doesn't just have to be for avant-garde composers anymore!
"The Musician’s Dice are patented, glossy black 12-sided dice, engraved in silver with the chromatic scale. They can be used in any number of ways – they bring the element of chance into the musical process. They're great for composing Aleatory and 12 tone-music, and as a basis for improvisation – they’re really fun in a jam session. They also make an effective study tool: they can be used as “musical flash cards” when learning harmony, and their randomness makes for fresh and challenging exercise in sight-singing and ear training. Plus, they look really cool on the coffee table, and give you a chance to throw around words like "aleatory.""
Below two musicians play around with using these dice.
Read the rest of the Radiophonic Laboratory series.
Justin Patrick Moore
Husband. Father/Grandfather. Writer. Green wizard. Ham radio operator (KE8COY). Electronic musician. Library cataloger.