An Analysis of the Compositional Techniques in John Chowning’s Stria

This article describes an analytic study of the process used by John Chowning for the composition of Stria. This article is intended to complete the description of the compositional process given in a previous work (Meneghini 2003), largely restated by Bossis in a subsequent paper (Bossis 2005). Stria was composed in 1977 and was fully generated by means of computer algorithms (Chowning 1977c) and the corresponding input files (Chowning 1977a, b): all the parameters of the sounds generated in Stria are determined and calculated by these algorithms on the basis of specific mathematical rules and of numerical parameters chosen by Mr. Chowning as input for the programs. For this reason, an accurate analysis of these algorithms is fundamental for a complete comprehension of the compositional process.

This analysis has three degrees of interest: at first, it enables a study of the composer's intentions regarding the piece's characteristics. Secondly, it complements the comprehension of Stria obtainable from listening with analytic data on the properties of the sounds. Finally, it provides information on the characteristics of the piece that can be used for the re-writing of the algorithms and for the analysis and re-synthesis of the piece (Baudouin 2007; Dahan 2007).

The work described in this article began in 2002, and it started with the study of the programming language (SAIL) used for the generation of the algorithms (Smith 1976; see also www.xidak.com/mainsail/documentation_set_1630_html/docset.html), which was necessary for the understanding of the programs. The algorithms (Chowning 1977c) have been then carefully analyzed, line by line, to achieve a complete understanding on how the parameters of each sound generated in Stria are determined. Personal communication with the composer was indispensible for the interpretation of some of the procedures used, and for the understanding of Mr. Chowning's intentions related to programming choices.

General Properties

This section explains the general properties that characterize Stria. The piece was fully composed by means of a computer program: The properties of each sound were generated by specific subroutines, starting from the input data chosen by the composer, and taking into account a set of mathematical relationships which will be explained shortly. Most of the information described here was extracted from the source code of the programs used by Mr. Chowning himself.

Starting from some basic definitions about the Golden Mean, this section gives a description of the pitch space and spectrum division, of the characteristics of the sounds, and of the temporal structure of the piece.

The Golden Mean

Stria is based on the mathematical properties of the Golden Mean: The basic properties of this important ratio are summarized in this section. Consider a segment of length one, and look for a subset of length x such as the ratio between the segment's overall length and x is equal to the ratio between x and the remaining part of the segment itself. To do this, we consider the equality

Solving this equation for x, we find that

We can then extend this result to the continuous proportion

which numerically corresponds to

In ancient and current times, the important ratio we have obtained this way (and its reciprocal Φ, usually referred to as "Golden Mean" or "Golden Section") has been considered a rule of physical perfection. Many authors have indicated that it is easily recognizable in many human works (e.g., in architecture) and in nature as well (e.g., Runion 1990; Markowsky 1992).

In music, the Golden Section represents (to a good approximation) the interval of a minor sixth in Western notation. Traditionally, the minor sixth was expressed as the ratio 8/5 = 1.6, which is close to the Golden Mean. In twelve-tone equal temperament, the minor sixth is eight semitones or

Another important property of the Golden Mean is related to the Fibonacci series: Each of the terms of this series, starting with {0, 1, 2}, is the sum of the two immediately preceding terms. It can be proved that the ratio between two consecutive terms of this succession quickly approaches to the Golden Section. From this, we can easily derive that...