A system (or systems) of graphic devices in musical notation, by the application of which the durations ostensibly prescribed for notes conveyed by pre-orthochronic notations were modified and made greater or less in accordance with a specified proportion.
ROGER BOWERS
Such graphic devices fall into two principal classes. First is coloration (or blackening): somewhat simplified, this may be described as the blackening of notation otherwise void, or prior to the adoption of void notation the execution in red full (also, rarely, red void or black void, even blue full) of notation otherwise black full. Second is the graphic modification of any of the four standard mensuration signs, either through its inscription in mirror-image or through the addition of a stroke and/or a number (or pair of numbers). The apparent note values could be either reduced in value or augmented; diminution was by far the more common. Resort to these devices is found in notation from the early 14th century to the early 18th, but enjoyed three particular periods of favour among composers: c13801420, c14651525 and c15801650. The reasons for their adoption were diverse and legion, and differed somewhat from period to period, albeit within fixed boundaries. Overall, to the extent that musicians who composed were familiar with the mathematics of the Boethian proportions by which intervals were identified and explained, it was probably natural that when seeking to invent notational neologisms to convey rhythmic patterns unnotatable by conventional means, or to render difficult notation in a more digestible form, they should resort to devices based similarly on proportional concepts.
The device that was both the simplest and the most stable and durable was that known as coloratio. In principle, any note or group of notes subjected to coloration or blackening was reduced to two-thirds of the value that it would have enjoyed in its pristine state. In respect of any note in mensural notation that was equal in duration to two of that next smaller in value, the coloration of three in succession caused each to undergo reduction to two-thirds of its erstwhile value, so creating a triplet (ex.1a). In the case of any note that was equal in duration to three of that next smaller, the coloration of three together likewise effected a proportional reduction in the value of each to two-thirds, so reducing perfect value to imperfect and commonly creating the effect called hemiola (ex.1b). As an aid to comprehension by the reader, coloration was usually applied to notes in groups that added up to conventional and readily recognizable ensembles of three units to be reduced overall to the value of two; this device might commonly entail the coloration of smaller notes whose values individually were not thereby affected (ex.1c). On occasions coloured notes could appear singly to denote imperfect value, especially to inhibit unwanted perfection and alteration.
In comparison with coloration, proportional usage generated by graphic modification of the standard mensuration signs was far less stable in its signification. It was practised for over 300 years; usage evolved and changed over time, and at any given moment a single manifestation might convey in one cultural region a meaning somewhat different from that prevailing in another. Secure interpretation by modern editors of such notational phenomena therefore relies on the perception and identification of fine distinctions both geographical and chronological. Because much of this work yet remains to be undertaken few aspects of the subject are uncontroversial, and no real justice can be done to it in a brief article.
To original performer and modern editor alike, resort by the composer to a proportional signature introduced the need to resolve two principal issues: firstly, the mensural relationships prevailing among the principal components of the hierarchy of note values (primarily long, breve, semibreve, minim); and secondly, the temporal equivalence between note values of proportioned notation and those of the adjacent unproportioned notation (integer valor). The former issue can usually be determined by inspection; essentially, original performer and modern editor alike use their eyes and inner ears to determine which values are arriving in twos and which (if any) in threes. In the latter case, all the usual rules of perfection, imperfection and alteration apply (see Notation, §III, 3). For the second issue, the appropriate resolution can usually be calculated readily when the proportion occurs in one part contemporaneously with integer valor in another. However, when all the constituent voices of a polyphonic composition progress simultaneously from integer valor to proportion (or vice versa), the temporal relationship cannot be resolved by inspection and resort must be made to contemporary theory. At this stage, distinction must always be made between those writers who were faithfully and objectively describing current usages, and those who took up the pen to advocate novel systems and approaches consciously divergent from the contemporary practice, revised and reformed in ways commendable to the author but not necessarily to anyone else. Both are illuminating in academic terms, only the former in practical terms.
Three basic practices informed the system. (1) The inscription of a stroke through any signature conveyed diminutio dupla, namely execution of the notes concerned in values half those of integer valor. Execution of the signature in mirror image could likewise convey diminutio dupla, or occasionally some more irregular proportion. (2) The appending to a signature of two numbers written as a fraction (or, rarely, in succession) indicated the inauguration of performance of the upper number of notes in the time previously taken to perform the lower; whether the note value concerned was breve, semibreve or minim is usually evident on inspection, and largely may be determined by chronology. Diminutio sesquialtera (3 in the time of 2) was inaugurated by such a sequence as C→C3/2 (the numbers written as in the modern time signature), diminutio dupla (2 in the time of 1) by C→C2/1, and diminutio tripla (3 in the time of 1) by C→C3/1. Many other fractional diminutions were possible in theory, but were engaged in practice only rarely. (3) The appending of a single number to a signature originally conveyed diminution by a proportion indicated by the number, plus the transfer of the mensural relationships of integer valor to the degree next higher (modus cum tempore). However, following the extinction of modus cum tempore by the later 15th century, C→C2 could be used as an abbreviation for C→C2/1 (dupla), and C→C3 commonly for C→C3/2 (sesquialtera) or, more rarely, for C→C3/1 (tripla). (When not necessary for comprehension, repetition of the base-line signature could in practice be omitted.) Only one form of augmentation was ever common. C occurring simultaneously with conveyed duple augmentation; it could be engaged, for example, to preserve the original major-prolation mensuration of the L'homme armι melody when used as the tenor of a mass otherwise notated in .
Each resort to proportional notation was no isolated intellectual game, but was part of the standard notation practice of its particular time and location. Only chronologically, therefore, not analytically, can proportional usages be genuinely understood; some good beginnings have been made on such work, but much remains to be done. Many of the earliest manifestations of proportional usage arise in music in the style dubbed Ars Subtilior (c13801420). A notational principle fundamental at this period was that of the constancy of the value of the minim from one mensuration to another succeeding it; proportional usages were developed as a means of subverting this principle. Particularly challenging in their intricacy and not intimidating in aggregate number, the interpretation of these early instances is now largely resolved.
Beyond about 1420 occurrences of proportional usage fall into two areas. The simplest coloration/blackening and sesquialtera are encountered continuously well into the 17th century, and were part of any composer's stock-in-trade, for use in any kind of composition. More complex instances occurred predominantly in sacred music, in which they could make a contribution to the composer's offering of profundity and learning. Prior to about 1450 such examples occurred commonly in motets that engaged isorhythmic diminution of the tenor, enabling that voice to be notated in a manner at once erudite and concise. By the middle of the century, however, some fluidity was entering the system, as rules yielded (temporarily) to conventions. In particular, while the juxtaposition of with simultaneously (and C with C) continued to specify diminutio dupla, the occurrence of in succession to (and C in succession to C) appears to have been intended to prescribe a tempo that was faster than and C respectively, but less than twice as fast. There are grounds for interpreting the sequence →C, frequently encountered in music of the mid-century, to convey a proportion of four semibreves of C to three of .
Towards the end of the 15th century the emerging practice of choral performance for church polyphony instigated formalization of the concept of the Tactus, entailing in its turn some radical rethinking of the practice of proportional notation. In particular, equivalence of the semibreve (rather than of the minim) was confirmed as the principle now primarily subvertible by resort to proportional notation. Some examples of cumulative proportion became common, especially C→C3/2 serving as a means of denoting diminutio tripla (sesquialtera [3:2] and dupla [2:1] applied simultaneously produce 3:1). For so long as the principle of an all-pervasive uniform tactus subsisted, commonly measured by its similarity to the human pulse, it provided a mental anchor upon whose stability theorists such as Tinctoris and Gaffurius could hang elaborately all-inclusive systems of proportional usage (almost as impractical as they were erudite and encyclopedic), composers could create elaborate mensuration- and other canons and incorporate similar offerings of elegant learning in their church composition, and controversialists among the theorists could engage in learned disputations on finer points.
Interest in proportions other than the simplest rather waned after about 1525. By this time C was established as an initiating mensuration signature in its own right. It was used in conjunction with note values that were longer than those employed under C, performed in a faster tempo so that the tactus fell on the breve (that is, alla breve). Theorists of the preciser kind proved unable to rationalize this essentially irrational usage (as a diminution ostensibly dupla but in practice imprecisely related to C); those more pragmatic simply accepted it, and presently recognized C as a standard usage for church music, C for madrigals and other secular genres.
Towards the end of the 16th century, and especially with the inception of the Baroque style, proportional usages were revived and extended, and were applied increasingly to secular as well as to sacred genres. By about 1615 even a diminutio sextupla (six semibreves in the time of one) had been invented; initiated by the signature sequence C→3/1, it was created to permit the notation of a triple time sufficiently quick for it to be conveyable only in tactu aequali. Certain composers now extended their resort to the hooked minim as an unambiguous alternative to the blackened minim; elsewhere, however, the potential for confusion between the blackened minim and the crotchet was already beginning to undermine the whole practice of coloration. In terms of the sheer quantity of music affected, this is the most significant period of proportional usage. However, the performers and scholars who first disinterred this repertory in the early 20th century found that the application of interpretations ostensibly correct for these proportions delivered results inconsistent with their aesthetic preconceptions for it. Consequently, the notation was conveniently dismissed as being in a state of chaotic confusion, and the evidence of the sources, both theoretical and musical, was disregarded.
It is true that some publications of the period manifest a potentially somewhat confusing proliferation and diversity of symbols to convey proportional usage, to which their composers, untutored in the correct application of a system that had been believed obsolete at the period when they were under training, were now making totally unnecessary (and sometimes incorrect or irrational) resort. Moreover, certain German practices differed from Italian; particular idiosyncrasies could be found (e.g. Giovanni Gabrieli's consistent application of the number 3 alone to denote diminutio tripla); and local peculiarities occurred (e.g. the use in some early 17th-century Dutch and north German sources of a practice endorsed by certain local theorists whereby three blackened minims may denote crotchet-crotchet-minim). Nevertheless, to theorists of the time, including Morley, Zacconi, Banchieri and Michael Praetorius, the primary system inherited from the past was still in good working order. Coloration, sesquialtera, tripla (and now sextupla) meant what they had always meant, and Praetorius in particular advocated a severe pruning of the symbols used in his day so as to clarify the basic simplicity underlying the system. Modern performance of late Renaissance and early Baroque music may well start to sound somewhat different when these principles have become further assimilated into editorial practice.
The very beginnings of the dissolution of the proportional system, and of its orderly evolution onward, can be traced to about the 1620s, with the inception of a progressive evaporation of the concept of the tactus on which it had come to depend. Prior to the early years of the 17th century it was a principle too pervasively fundamental to require statement that, once established at the beginning of the performance of any piece of music, the chosen tactus did not change; on this certitude hung all calculation of proportional usage. In avant-garde circles the principle was being questioned by as early as 1609, when the singer Aquilino Coppini observed that the emotionalism of certain of Monteverdi's most recent madrigals was best served by a degree of local flexibility in the tactus. From a conservative theorist such as Agostino Pisa (1611) such thoughts provoked a clear statement of the traditional immutability of the tactus; presently, however, that principle proved equally ungrateful to such a composer as Girolamo Frescobaldi, who risked a conscious break with received wisdom and practice by prefacing his Primo libro di capricci (1624) with verbal instructions for relative tempo, modifying and refining the message conveyed by his array of proportion signs (resourceful and elegant though that already was). Only slowly did the proportional system dissolve and mutate, however, and odd items in the output of Purcell, Handel and J.S. Bach show that it was still being taught to young composers far into the 17th century. Nevertheless, by then its evolution into the modern system of time signatures was well under way, though the detail of this transition still awaits its elucidation through research.
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