Of Renaissance Canons and Computer Music
An Essay on the Representation of Algorithims in Music
by Jaren Feeley
Computers and Renaissance musicians have at least one thing in common: they can run simple algorithmic processes on fragments of music to create vastly more complex and nuanced musical compositions. This algorithmic compositional technique was very popular with Renaissance composers, who could generate contrapuntal polyphony from a simple melody and a few basic rules. This technique is today known as canon. Canons are the link that will allow us to bridge the worlds of Renaissance music and computation. It will be considered that those qualities that made canons compelling to Renaissance musicians also allow canons to be easily produced using DSP (digital signal processing) techniques. Through a comparative analysis of how musicians and computers execute canons, this essay will demonstrate some of the ways that compositional and performance practices are intertwined with the way we represent our music.
The 15th century composer and music theorist Tinctoris famously described the canon as “a rule showing the composer’s intention behind a certain obscurity.” (Melson 1) Computers don’t care much for intention or obscurity, but they are extremely adept with rules, or what we could more accurately term algorithms. An algorithm is a self-contained step-by-step set of operations to be performed. When a piano accompanist transposes a piece of music, they are performing a kind of mental algorithm: they identify a written note, and then they mentally process the note with the interval it needs to be transposed to (up a semitone, or a major third perhaps), and then they play the “output” of the little mental algorithm—the newly transposed note.
The algorithm is the secret behind the musical canon; it is the elegant and concise explanation for how to create a rich polyphonic music out of a simple melody. Canons are generated through imitation: different versions of the same melody are juxtaposed and interwoven to create (at least in Renaissance times) musical counterpoint. The initial melody is often overlaid with an identical version of itself that begins after a certain time interval, and often beginning at a different pitch interval. There are many such techniques for altering an imitated melody in order to artfully juxtapose it against an original version. These techniques are the “operations that are to be performed” upon our melody, which give an algorithmic quality to this technique of composition.
What accounts for the popularity of the canon in Renaissance times, through to the present day? It can largely be explained by the satisfaction that is drawn from recognizing and predicting imitation in music, a quality that is very common in music from many cultures. The canon can be thought of as a particularly rigorous—and “globally” consistent, in computing parlance—application of imitative musical techniques, which can be useful as a large-scale framework for both composers and listeners. Canons also ensure that all voices have melodic integrity and are of relatively equal importance, which is in line with the polyphonic sensibilities of much Renaissance music. (Davidian 117)
Of particular interest to this investigation is the sheer efficiency of the canon. Both human memory and written records can more easily store and transmit data that is represented in an efficient manner. For example: 88 is clearly a more memorable and efficient representation of the number 16777216. Canons allow composers to create pieces that not only have the aforementioned positive attributes of imitative writing, a consistent formal framework, and polyphonic integrity—but canonic pieces are also strikingly efficient to write down and pass on. However, just as with the example of 8^8, which relies on an understanding of the concept and notation of exponents to “unpack” 16777216, canonic music can only be efficiency conveyed in writing with an appropriate musical system of notation in place.
One of the most obvious differences between the way computers and Renaissance musicians interact with music (besides circuits and warm blood) is this notation, or “input”: computers work upon audio files, which describe a sonic waveform with millions of 0s and 1s, while Renaissance musicians primarily worked with mensural notation, which is made up of interdependent symbols (such as notes, rests, mensuration signs, staff lines, etc.).
In the 1502 publication of Missa L'homme armé super voces musicales by Josquin des Prez, the 68-note-long melody of the Agnus Dei II is represented with 68 note head symbols. Here is the elegance: two mensuration symbols signal to the performers that this piece is a canon, and from these two little symbols, a distinct 47-note-long secondary melody and a 23-note-long tertiary voice can be quickly deduced and overlaid upon the primary melody, outlining a highly complex and very beautiful polyphony—all this extra music from just two additional mensuration signs!
By comparison, computer music “notation” (.wav audio files) would require about 120,600,000 zeros and ones to represent the unique melodic material of these three voices in three 60-second recordings. The algorithmic nature of canons, in conjunction with clever symbolic systems (as evidenced in mensural notation) clearly offers a massive increase in representational efficiency over the millisecond-by-millisecond sound record captured by the .wav audio file on my computer.
This example does not aim to suggest that computers are inherently unable to gain representation efficiency from the algorithmic nature of musical canons—merely that a proper symbolic system of representation must be created and utilized. We can begin by asking ourselves how a computer might efficiently represent the aforementioned canon in Missa L'homme armé super voces musicales. The key is finding an equivalent to the mensuration signs, which signaled to the Renaissance performers that this was a canon and the initial melody could be overlaid with “copies” which are sung at proportionally slower speeds (1.5x slower and 2x slower). In this way, we would only need a sound record of one of our three voices, and thus only 1/3 of our 120,600,000 zeros and ones—the other two voices could be generated by our digital mensuration signs.
As it turns out, computers are very good at producing copies of data (there is nothing ineffable about strings zeros and ones), and so a computer could easily be directed to create two copies of an initial melody. A computer could also easily make these copies proportionally slower by taking tiny chunks of the millisecond-by-millisecond audio data and duplicating them, such that every millisecond of recorded audio data would be reiterated before moving onto the next millisecond of recorded audio data. (To visualize this: the binary string 01010101 could be broken into its constituent numeral parts, each of which could be duplicated, to produce 0011001100110011.) Through this process of duplicating data at the macroscopic and microscopic scales, a computer could easily execute the algorithmic processes behind a mensuration canon.
Note: a presentation was later given to UC Berkeley's "Early Music" course, in which I performed Josquin des Prez's "Missa" mensuration canon live, using my solo voice and a custom digital signal processing instrument.
What is the essence of this similarity between Renaissance music notation and computing that allows a mensuration canon to so easily be represented in both spheres? As we’ve seen, clearly the ability to create copies of a piece of data is essential for the computer—and this finds its human equivalent with the separate bodies of Renaissance musicians, who are able to intonate musical “copies” from a single sheet of music notation by virtue of multiple sets of vocal chords. A subtler underlying connector is tactus, which in Renaissance music is the basic rhythmic pulse that is shared in common by performing musicians. In a mensuration canon, musicians are singing the same melody but at proportionally different rates of speed, and in order to stay in sync with one another they must be grounded by this common tactus. The computer version of tactus is the sample rate, which is the number of times per second that audio is represented with a piece of data. This sample rate (generally 44,100 Hz, about 44x faster than the “millisecond-by-millisecond” vernacular used previously in this essay to describe the rate of recording) is held in common among all audio files being played on a computer system at the same time, and is the common denominator among sound files that keeps them in sync, even while data is being manipulated to create the effect of precisely proportional speed changes.
Though computers and Renaissance musicians share a rigorous commitment to tactus, the same cannot be said about pitch classification. The pitched note head is one of the most salient features of Western music notation, but pitch designation does not feature at all in the way audio is represented for computers. Computers of course can represent pitched sounds, but with the ability to sample audio at 44,100 times per second there comes the ability to represent the entire range of perceivable pitches, which makes a comprehensive system of classification as unpractical as creating a system for classifying grains of sand on a beach. The Renaissance creation of scale degrees and hexachord pitch classification systems is in many ways a tremendous pruning back of the wilderness of pitches that the human mind is able to perceive. There are advantages to creating a limited alphabet of pitches: this allows musicians to develop a shared tonal language. This tonal language finds its symbolic representation in the staff lines and note heads of music notation. And as was previously stated, the algorithmic nature of musical canons can afford great representational efficiency, but only in the presence of an appropriate system of symbolic representation. Canons which rely on an understanding of things such as hexachordal relationships (or diatonic interval relationships) are not able to be recognized or generated easily by computers, as these qualities are not inherent in way computers represent audio, as it was with tactus. Thus there is currently to my knowledge no computer system on earth that can take a piece of audio and apply algorithmic processes to produce an inversion canon—whereas this process would be relatively straightforward for a Renaissance musician.
Canons, as an algorithmic musical technique that can be executed by musicians and computers, is not only a compelling and efficient musical form, but can also be a vehicle for exploring the way we understand and represent music. Through a comparison of two systems of representation—digital and notational—it has been demonstrated that aspects of our compositional and performance practices are intertwined with the way we represent our music in symbolic systems.
Bibliography
Davidian, Teresa. Tonal Counterpoint for the 21st-Century Musician. Rowman & Littlefield Publishers, 2015.
DeFord, Ruth I. Tactus, Mensuration and Rhythm in Renaissance Music. Cambridge University Press, 2015.
Elders, Willem. Josquin Des Prez and His Musical Legacy: An Introductory Guide. Rev. and Translated ed. Leuven University Press.
Melson, Edward. “Compositional Strategies in Mensuration and Proportion Canons, ca. 1400 - ca. 1600.” Master’s thesis, McGill Universtiy, 2007.
Newes, Virginia. “Mensural Virtuosity in Non-Fugal Canons c.1350-1450.” In Canons and Canonic Techniques, 14th-16th Centuries: Theory, Practice, and Reception History ; Proceedings of the International Conference, Leuven, 4-6 October 2005. Leuven: Peeters, 2007.
Urquhart, Peter. "The Persistence of Exact Canon throughout the 16th Century." In Canons and Canonic Techniques, 14th-16th Centuries: Theory, Practice, and Reception History ; Proceedings of the International Conference, Leuven, 4-6 October 2005. Leuven: Peeters, 2007.