Lessons in Counterpoint : Observe the Bagel

Virgil Thomson usefully suggested that one describe a contrapuntal setting in terms of the number of voices, the characteristic intervals between the voices, and whether the voices were were differentiated or undifferentiated in character. Since the basic idea of counterpoint is that two or more melodies are chugging along simultaneously and that those melodies are distinct, a degree of differentiation is basic, and the various sets of rules for counterpoint that emerged from either observation or speculation are designed to insure some degree of distinction, for example through the exclusion of parallel unisons, octaves, and fifths and limits on the number of consecutive parallel thirds or sixths (which are more distinct in melodic character than the "perfect" unisons, octaves, and fifths through their admixtures of major and minor intervals).

But this differentiation can be greater or lesser. Sometimes a composer would like an ensemble in which the individual voices are relatively indistinct and it is less important to perceive the individual lines as distinctive melodies. This can be achieved by using uniform rhythms, a similar and narrow tessitura, compact voice leading (which is not quite the same as a parsimonious voice leading as such a setting may also include voice crossing), etc..

Often, however, a composer chooses instead to increase the differentiation, moving on an axis of differentiation from homophonic to heterophonic, while remaining in the same place in the monophonic/polyphonic axis, as they say in the systematic musicology biz. While this may be done by brute force, increasing the variation from line to line in contour, rhythmic content, etc., it may also be desirable to figure out how to increase differentiation without necessarily decreasing the audibility of the individual lines.

Observe, for a moment, the bagel, or its toroid, sweet, and goyischer sibling-by-topology, the common fried doughnut. These are two, as far as we know, independently developed, examples of finding the same solution to the problem of an optimal form for a differentiated texture in a bread-like good. With a torus* of relatively uniform dimension, the ratio of crust to sponge, or surface to mass, is optimized, allowing a more even distribution of heat during the boiling and baking or frying processes, and -- in cooperation with a cool rising in which yeast and bacteria are encouraged to get to work as well -- maximizing the distinction between smooth and relatively dense, if not hard or crispy, crust and the airy sponge. (The presence of the whole also assists in draining a doughnut of oil or a bagel of water). The success of both baked goods is largely due to this spatial optimization, and from that we can perhaps intuit a bit about making the lines in our contrapuntal ensembles more distinctive.

Space is a common metaphor for the pitch-height continuum among musicians, and if one wants to differentiation voices it is certainly useful to separate them in space (sometimes, in an ensemble, it is useful to have subgroups of voices separated from one another, for example, in a choir, separating male and female voices, or the lower voices from a soaring treble line, or in basso continuo, the bass line from the upper voices). But allow me now to push the spatial metaphor a bit further, and suggest that it is useful to consider that each individual line possesses a number of spatial elements -- and those concern both tessitura (in height, and extent, and whether its pitch space is a sole possession or shared or overlapping with other voices) and, more critically, contour, the melody's crust. We might even describe contour in terms cannily like those used to describe a crust -- is it smooth? is it variegated? does it cover the range of a melody in a compact way or does it wander? In a contrapuntal ensemble, the contrast between and -- frequently -- the audibility of the individual voices can be increased by using alternative tessituras and surfaces.
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* The torus, is of course, familar to all students of tuning theory as the toroid projection is ideal for describing a tuning system in which two generating intervals (fifths and major thirds, for example) are so tempered as to form closed cycles of pitches upon a single lattice. I recall once showing a toroid tuning lattice in a dissertation by Scott Mackeig to the theorist Ervin Wilson, who asked the inevitable: "But does it sound like a doughnut?"