Notes on Prosody: numerical classification of rhythms

November 27th, 2013 § 0 comments

I will be posting more of these “notes” from time to time. They are mostly drawn from old drafts of writings on prosody, but some will be new. This second posting is a continuation of the first. I hope readers will not be too put off by this overgrown thicket of definitions. The aim was to find a more useful and consistent terminology for describing rhythms in the abstract. A lot is borrowed, naturally, from music, but oriented towards a more general application that would encompass verse as well. 

First of all, we will need terms to refer to a rhythm’s basic “count.” Standard terms like binary or duple for rhythms of two, ternary or triple for rhythms of three, quaternary or quadruple for rhythms of four, etc., will be used here, along with the simpler expressions rhythms in or of two, three, four, etc. To these we may add unary to denote rhythms in, or of, one.

Every rhythm, in a sense, has a unary component to it, since this merely denotes the fact of repetition as such. So the term is perhaps of limited usefulness; nevertheless, we will occasionally have need of it to refer to rhythms that achieve their effect by conspicuous repetition of a single element throughout, e.g. a chant based entirely on a refrain.

The number that defines a rhythm’s basic count is called its numerator, on the analogy of a time signature in music, where the number of beats in a measure is given as the upper of a pair of numbers, the lower one (denominator) giving the unit of measurement. We can dispense with the latter: it is a result of the convention of infinite divisibility of time in Western musical practice. In poetry, a lower limit of divisibility is already set, for metrical purposes, by the syllable, and any larger divisions (foot, line, stanza) are more clearly referred to by name.

Rhythms metrically organized on more than one level are said to be nested. The number of levels at which a rhythm is metrically organized is its degree or level of nesting. This is counted inclusively, e.g. a rhythm of four overlying a rhythm of two is nested to two levels.

Nesting may be either multiplicative or additive. In multiplicative procedures, a numerator is multiplied, either by itself or by another, to yield larger metric units.

The ability of a numerator to compose larger units in multiples of itself, as, for instance, quatrains out of couplets or eight-bar phrases out of measures in duple time, gives the degree of multiplicativity of rhythms based on that numerator. .

Unary rhythm is the most multiplicative of all, but in a fairly trivial way, for the same reason that it is trivial mathematics to say that all whole numbers are divisible by one. The most significantly multiplicative rhythms are binary ones, because of the special place two has among whole numbers, as the one that makes all the others even or odd. It has a special relation to the idea of symmetry, being indeed at some level identical with it. But perhaps at least as important, as applied to rhythm, is the fact that multiples of two can yield a maximum of nested levels of rhythm within the narrow range of tempos at which rhythm as such is perceptible to the human organism. Multiples of three, or any higher number, more quickly reach the threshold at which repetitions, being too far apart, fail to be perceived as rhythms.

Three, however, is the next most important number rhythmically, and in combination with two and multiples of two, it is perhaps even more important than it is on its own. The special qualities ascribed to three, aesthetic and spiritual as well as mathematical, need not be pointed out to anyone who has grown up in the Western tradition. It is not symmetrical in the same sense as two, but another kind of symmetry, represented by the image of a triangle, can be felt in it.

If a numerator is not multiplied, or multiplied only by itself, to yield larger levels of rhythm, then the rhythm as a whole is said to be simple. Two, and, to a much lesser degree, three, are really the only numbers usable as numerators for nested simple rhythms. It is quite possible, though, to have simple rhythms that use a higher numerator, and are not nested.

Rhythms that multiply different numerators we will refer to as compound. Examples of compound rhythms in poetry are the French trimètre, which divides the twelve syllables of the alexandrine into three groups of four, and the tétramètre, which divides them into four groups of three.

The terms simple and compound are used somewhat differently in musical metrics. There the first refers to rhythms that are not multiplied at all, and the second to rhythms that are multiplied by three, whatever their numerator. Both terms apply only within the musical bar, ignoring any larger units, as well as any that are smaller than the bar’s “denominator.” In practice, the majority of compound musical rhythms are binary rhythms multiplied by three; my use of the term can be seen as a generalization from that category.

Rhythms may also be nested by being mixed. Mixing is an additive procedure. Different numerators are applied not simultaneously, but in succession. Now, if numerators are mixed randomly, in no particular sequence, they cease by that token to be significant numerators; therefore rhythms that are mixed in a strictly metrical way have different numerators that occur in a regular, predictable sequence. Examples from poetry would include “regular” classical hexameters, consisting of five dactyls plus a spondee, and many stanzaic forms, such as the ottava rima, whose abababcc rhyme scheme gives a line grouping that could be expressed in terms of numerators as (3 X 2) + 2. We will count any mixed sequence as being nested to at least two levels, one for the sequence as a whole, another for its components. These, in turn, may individually contain further levels, as in the compound part (3 X 2) of the last example.

Four has a special importance as a numerator because of its “dual” nature, in several senses. Though it is always more or less felt as the multiplication of a duple rhythm, it can often be perceived as a simple numerator in its own right. As the first power of two, it is unique in this regard. Six is the only other even number that can even be perceived, generally speaking, as a simple numerator, and clearly much less so than four, especially in English-language poetry. Eight will almost always be perceived as two groups of four, or some other nested sequence, whether multiplicative (4 X 2) or additive ( 3 + 2 + 3).

Of course these generalizations are relative, not absolute, and how well they hold will depend on many factors. We can’t deny, for instance, that in languages that base their versification on syllable count rather than feet or accents, higher numerators may often be perceived as such. Since the basic unit is smaller, multiplications of it will stay within a more perceptible range, and even twelve- or fourteen-syllable sequences may sometimes be perceived without being broken down into multiplicative or additive components. Mostly, of course, verse written in such languages does so break them down, and often in a consistent pattern from line to line.

Five and seven are the last numbers we need mention here. Being odd, they obviously can only be broken down additively, and this gives them an asymmetry that none of the lower numbers has. Of course three can be broken down this way as well, but it is most often perceived as simple and, in a certain sense, symmetric. Five is important in English, naturally, because of the prevalence of the pentameter. Seven occurs mostly as a regular additive sequence of 4 + 3.

Instead of or in addition to being nested, rhythms may overlap. In this case there is no overall numerical coordination between them, though there may be moments at which they coincide. An example would be a poem with one rhythm given by its metrical lines, and another by its syntax and phrasing. These two rhythms, unless they happen to coincide perfectly, would overlap.

The properties of numbers can tell us a good deal about the properties of the rhythms they describe. But there is a sense in which, the more completely we can define a rhythm numerically, the less complex it really is. Complexities may in fact be too subtle for analysis, though a person speaking or reading a poem, hearing music, or watching a dance, can readily feel them. From this point of view, we could say that the possibilities of even the simplest unary rhythm are far from having been exhausted.

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