Thomas Christensen

Counterpoint pedagogy in the Renaissance

How did Renaissance composers learn their craft? They could have learned much of their technique from treatises, especially from those portions devoted to counterpoint. Today, we often think of counterpoint as consisting primarily of rules of voice leading. Such rules, which are found in virtually every music treatise of the time, teach the student to regulate the melodic motions of lines in relation to the simultaneous intervals between them (e.g., conditions for approaching perfect consonances or for preparing and resolving dissonances). They were learned by young singers for the purpose of improvisation, and following them would have been as natural as speaking in grammatically correct sentences. Yet just as the art of oratory consists of more than correct grammar, so musical composition goes far beyond mere voice leading. Composers had to choose between many large-scale contrapuntal techniques involving texture, motivic and structural repetition, and variation. While there has been extensive study by scholars of the rules of voice leading in Renaissance music, less consideration has been given to these more advanced compositional techniques. In this chapter, then, we will examine some of these compositional techniques by reviewing some two dozen treatises written between the mid-fifteenth and mid-seventeenth centuries. We will see that the real challenge for a Renaissance composer consisted not of employing “correct” contrapuntal voice leading but rather of elaborating primary musical material – sometimes called a soggetto – by varying it or combining it with some other melodic material. (As we will see, a soggetto need not be simply a melodic subject in equal or mixed rhythmic values; it could also be a duo, or, in the case of parody technique, even an entire polyphonic composition.)

Musica practica: music theory as pedagogy

One of the most consequential developments in the long history of music theory has been its gradual integration with the discipline of musica practica , a discipline that until at least the eighteenth century was considered largely distinct from the rarefied concerns of classical musica theorica . In the present chapter, we will attempt to look at some traditions of “practical” music theory in more detail. We will begin first with a brief discussion of the difficulties in defining “practical” theory and assessing its relation to functions of music pedagogy. We will then proceed to a broad survey of some of the major contributions to practical music instruction from the Middle Ages to the present day. Needless to say, this constitutes a vast quantity of writings that cannot be analyzed comprehensively here. But by focusing upon a few selected examples at historically significant moments, we hope to illustrate the principal parameters – structural, stylistic and institutional – which have together helped shape the discipline of “practical” music theory. Praxis and pedagogy The notions of “pedagogy” and “practice” have historically been closely linked, although they are by no means synonymous. In ancient Greece, the pedagogue was the “leader” or “teacher” of boys (usually the slave assigned to transport the boys from one schoolmaster to another). Today, the term “pedagogue” often carries with it negative connotations of pedantry and dogmatism, although in music, the term has perhaps a somewhat more benign association related to the teaching of basic skills. As pointed out in the Introduction to this volume, the origin of the dialectical juxtaposition of theory with practice may be traced to Aristotle (see p. 2).

Greek music theory

Introduction In the history of Western music theory, technical works written in Greek on the general subjects of “music” ( μoνσική ) and “harmonics” ( αρμoνικά ) play an anomalous role. On the one hand, they are not “Western,” especially in the linguistic and geographic senses reinforced in the Middle Ages by the gradual schism between Eastern and Western Christendom. On the other hand, the tradition never ceased to exert an influence during this period, not only because some parts of it were carried over into the West by authors writing in Latin, but also because the early church readily acknowledged and accepted – though not without reservations – the ancient power of music and its centrality to human existence. This combination of causes was sufficient to sustain an interest in early writings on music, especially those in Greek, throughout the Middle Ages. Thus, unlike other early Eastern traditions, the tradition represented by Greek works on music and harmonics assumed a prominence in the West even as it acquired a sense of the esoteric and foreign, a duality of character it retains in the modern conception of “ancient Greek music theory.” Prior to the Middle Ages, the tradition of writing technical works in Greek “on music” ( πeρi μoνσικηs ), on the subject of “harmonics” ( άρμoνικά ), or as a general introduction ( eiσαγωγη ) to one or both subjects was extraordinarily resilient, extending easily over eight centuries. But by the collapse of Rome in the fifth century CE, the tradition had become moribund, though certainly not entirely forgotten.

Tonal organization in seventeenth-century music theory

Otherwise known as the Age of Reason for the scientific undertakings and discoveries it witnessed, the seventeenth century is frequently seen as a period of uncertainty and confusion in the context of musical thought. From our own present perspective, musical thought of this period is overshadowed both by what precedes it and by what follows: the opposing poles of Renaissance modal theory and eighteenth-century harmonic tonality each seem more intelligible than that which we perceive as the transition between the two. Indeed, much of what we find in seventeenth-century theory appears to present a puzzling mix: ideas that accord strikingly well with the precepts of major-minor tonality appear to mingle freely with the teachings and terminology of modal theory. Added to this seeming paradox is the darkened mood of the theorists themselves, many of whom lament the confusion of their age. And yet, seventeenth-century music treatises paint a clear, if richly detailed, picture of musical thought if we keep in mind that they address widely different purposes: to serve the church singer in a long-standing and relatively stable practice of liturgical chant; to train keyboardists and other chord-playing instrumentalists in the art of extemporizing harmonies over a bass; to educate the well-rounded musician in the established traditions of counterpoint and modal theory; to instruct the rational mind in the scientific bases of tuning systems; and finally, to enlighten the curious on more speculative and imaginative musical topics, such as Pythagoras’s fabled discovery of harmony in the sound of hammers at the forge, the Boethian harmonic strata, and the legendary origins of music itself.

Music theory and mathematics

In Chapter 6 of The Manual of Harmonics (early second century CE), Nicomachus of Gerasa narrates the legendary story of Pythagoras passing by the blacksmith’s shop, during which in an epiphany of sonorous revelation, he discovered the correlation of sounding intervals and their numerical ratios. According to Nicomachus, Pythagoras perceived from the striking of the hammers on the anvils the consonant intervals of the octave, fifth, and fourth, and the dissonant interval of the whole tone separating the fifth and fourth. Experimenting in the smithy with various factors that might have influenced the interval differences he heard (force of the hammer blows, shape of the hammer, material being cast), he concluded that it was the relative weight of the hammers that engendered the differences in the sounding intervals, and he attempted to verify his conclusion by comparing the sounds of plucked strings of equal tension and lengths, proportionally weighted according to the ratios of the intervals. Physical and logical incongruities or misrepresentations in Nicomachus’s narrative aside, the parable became a fixture of neo-Pythagorean discourse because of its metaphoric resonance: it encapsulated the essence of Pythagorean understanding of number as material or corporeal, and it venerated Pythagoras as the discoverer of the mathematical ratios underlying the science of harmonics. The parable also established a frame of reference in music-theoretical thought in the association between music and number, or more accurately, music theory and mathematical models, since it is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one can best begin to apprehend the broad range of interrelationships between music theory and mathematics.

The Spanish Baroque Guitar and Seventeenth-Century Triadic Theory

In searching for the origins of harmonic tonality, the historian must be careful not to fall into the trap of fallacious geneticism by anachronistically interpreting some musical event or theoretical formulation in the light of later tonal theory. For example, because a harmonic progression in a 16th-century madrigal might look like a tonal authentic cadence, it does not follow that the progression actually fulfills such a tonal function in context. Likewise, when a theorist from the same time notes that in practice the fourth scale degree of a Lydian mode is often lowered through musica ficta, it would be Procrustean for us today to interpret his observation only as an adumbration of the modern major/minor key system. Any sophisticated theory of tonality, as Carl Dahlhaus has shown us, must be a dynamic one comprising a nexus of features that cannot be facilely reduced to a composite of individual constituents, however critical any one of these constituents may be to the theory.1 In this regard, the question of triadic theory in the 17th century stands as a paradigmatic test case. It is easy for us to project back-

Renaissance modal theory: theoretical, compositional, and editorial perspectives

The word “mode” is one of the most richly textured and problematic terms of Renaissance discourse about music. The difficulties associated with interpreting modal theory are hardly recent creations: sixteenth-century writers frequently framed their discussion of modal theory with assertions that they intended to clarity a difficult concept that earlier writers had not fully grasped. Thus Pietro Aaron claimed in 1525: And knowing [an explanation of the modes in polyphony] to be exacting and strange, I judge that it was abandoned by the celebrated musicians… not through ignorance but merely because it proved otherwise troublesome and exacting at the time. For it is clear that no writers of our age have explained how the many different modes are to be recognized … I show briefly what I know to be necessary, for I see that many are deceived about the true understanding, and regarding this I hope in some measure to satisfy them. The central problem - a problem that Aaron appears to have been the first to articulate explicitly - hinges on the nature of the relationship to polyphony of a theory intimately tied to monophonic music in its origins. If anything, twentieth-century attempts to recover, explain, and in some cases extend, modal theory have become even more subject to contention than their sixteenth-century antecedents. Several facets of the tradition, history, and reception of the body of writings generally known as “modal theory” contribute to these difficulties. Discussions of mode appear in wide-ranging contexts that reflect antecedents in classical and ecclesiastical traditions as well as a complex, and at times contradictory, network of concepts associated with the term.

Rhythm in twentieth-century theory

We have seen in the previous chapter how Riemann attempted to consolidate various trends in nineteenth-century rhythmic theory, synthesizing rhythm, meter, agogics, and phrase structure within his overarching theory of harmonic functionality. As was perhaps inevitable with such a comprehensive project, various tensions and problems remained in Riemann’s mature theory. Many early twentieth-century theorists such as August Halm, Ernst Kurth, and Hans Mersmann were critical of Riemann’s accentual theory, and so in part the history of rhythmic theory, at least at the beginning of the century, can be characterized as “a reaction to Riemann.” In addition, there were other musical and intellectual developments which shaped twentieth-century rhythmic theory, among which can be mentioned: 1. There were new ideas of motion and time, from physics, philosophy, and psychology, that led a number of theorists to place musical energetics and motion at the center of their approach to rhythm. 2. Schenker’s theory of tonality and tonal dynamics influenced a number of approaches to rhythm and the temporal unfolding of musical events, especially amongst North American theorists in the second half of the century. 3. Developments in linguistics and gestalt psychology influenced “architectonic” approaches to rhythm, engendering structuralist theories that emphasize the hierarchical aspects of rhythm and form. 4. Radical changes in musical style, especially the rise of atonality and serialism, demanded new conceptions of rhythm and meter. This led to various prescriptive theories of rhythm that were often developed (and commented upon) by the composers themselves. This chapter will selectively review the work of theorists and musicians from each of these four areas.

Mapping the terrain

Every musical culture possesses its own representation of what constitutes its music theory, a “map” of the domains of inquiry or precept and the relations between these domains, thus providing a degree of completeness and coherence to the discipline. The works of some theorists contain explicit and comprehensive mappings: this is particularly the case in music theory of the sixteenth and early seventeenth centuries. Yet there also are implicit mappings of music theory, which are to some degree recoverable by the historian. Often, the appearance of new theoretical constructs is symptomatic of an underlying remapping of the realm of theory. Particularly after the middle of the seventeenth century explicit mappings of theory come to have a metatheoretical and disciplinary function, often seeming to be attempts to stabilize a discourse perceived as being on the verge of fragmentation. For the purposes of this exposition, we will distinguish three broad historical cartographies, the first governing music theory through the sixteenth century, the second governing seventeenth- and eighteenth-century theory, and the last governing theory after the turn of the nineteenth century. Architectures and harmonizations The most basic representation of music theory is the schema , an attempt to analyze or systematize a body of knowledge through its division into idealized categories. As such, it is akin to classification, with the distinction that the latter pertains to autonomous data (harmonies, pieces, styles, etc.). A simple phenomenal schema of music might hypothetically distinguish the attributes of pitch and temporality, and indeed, theoretical schemas often do involve such binary discriminations.

Theories of musical rhythm in the eighteenth and nineteenth centuries

Everyone agrees: it is difficult to talk about rhythm in music, or, for that matter, the temporal experience in general. Compared with spatial relations, which appear to us as fixed and graspable, temporal ones seem fleeting and intangible. As a result, the language of time and rhythm is complex, contentious, and highly metaphorical. Considering that theorists today continue to have difficulty dealing with the metrical and durational organization of music from the eighteenth and nineteenth centuries – our most familiar music – it should come as no surprise that theoretical writings from those centuries often present themselves as perplexing and in need of explication. Though their manner of formulation may at times seem odd or convoluted, these theorists nonetheless ask many of the same questions about musical rhythm that underlie current concerns: What is a metrical accent? How do the profusion of time signatures relate to each other? Do the groupings of measures create a sense of larger-scale rhythm? Can various durational patterns be organized according to some scheme or another? How does our understanding of musical rhythm affect performance, especially tempo, phrasing, and articulation? Like many other domains of music theory, rhythmic theories are largely formulated in relation to a distinct compositional practice. Thus when compositional styles change, theorists respond by modifying their conceptions and formulating new ones in order better to reflect such transformations in practice. The high Baroque style, with its motoric pulses, regularized accentuations, and dance-derived rhythms, induced early eighteenth-century theorists to focus in detail on the classification of various metrical and durational patterns and to begin accounting for that most elusive concept – metrical accent.