VARIETIES OF FORCE IN THE PRINCIPIA

Newton's physics is based on two fundamental concepts: mass and force. In the Principia Newton explores the properties of several types of force. The most important of these are the forces that produce accelerations or changes in the state of motion or of rest in bodies. In Definition 4 of the Principia, Newton separates these into three principal categories: impact or percussion, pressure, and centripetal force. In the Principia, Newton mentions other types of forces, including (in Book 2) the forces with which fluids resist motions through them. Of a different sort is Newton's “force of inertia,” which is neither an accelerative force nor a static force and is not, properly speaking in the context of dynamics, a force at all.

The structure of Newton’s Principia follows a classical pattern: definitions and axioms, followed by the statement of propositions and their demonstrations. Newton’s treatise differs, however, from classical (or Greek) geometry in two respects. First, there is a constant appeal to the method of limits – Newton’s “first and ultimate ratios,” as set forth in Book 1, Section 1. Second, the validity of propositions is tied to evidence of experiment and critical observation.

In the demonstrations in the Principia, Newton generally proceeds by establishing a series of proportions from a geometric configuration. He then allows one or more of the parameters to be diminished without limit, thereby obtaining a limiting (“ultimate”) value of the geometric ratio. It is in the limit that Newton’s proofs are valid.

After his first optical publications in 1672 Newton was identified by his contemporaries and later generations as a supporter of the corpuscular or emission theory of light, in which light is assumed to consist of corpuscles, or atoms, emitted from a luminous source such as the Sun. While it is true that Newton believed in a corpuscular theory, utilized it in developing many of his optical experiments and theories, and argued vigorously against the wave theory of light, he never believed that it was a demonstrated scientific truth and considered it to be only a probable hypothesis. This distinction explains why, for example, he never set forth a synthetic account of the emission theory and eschewed it in his public accounts of his scientific theories. In order to understand Newton's advocacy and use of atomism in his optics it is necessary to understand his views on hypotheses and certainty in science.

HYPOTHESES IN NEWTON’S SCIENCE

From the beginning of his scientific career Newton was concerned with establishing a new, more certain science to replace contemporary science, which he felt was rife with “conjectures and probabilities.” He believed that he could establish a more certain science both by developing mathematical theories and by basing his theories on experimentally discovered properties. To establish a more certain science, Newton insisted that one must “not mingle conjectures with certainties.”

“Anyone who aspires to understand contrapunctus should write down the following matters.” So begins a short manual on counterpoint from the early fourteenth century that circulated widely under the authority of Jehan des Murs but is best identified (anonymously) by its incipit “Quilibet affectans.” The theorist’s remarks are straightforward indeed. Contrapunctus observes a strictly note-against-note texture. Only select intervals – some perfect in nature (unisons, fifths, octaves), others imperfect (minor third, major third, and major sixth) – are admitted between the voices, as are their octave compounds. The natural sequel to any authorized interval is that interval from the opposite category closest in size: minor third (imperfect) after unison (perfect); octave (perfect) after major sixth (imperfect). Although characterized as “natural,” these contiguous progressions are by no means mandatory. The motion of the cantus, the pre-existent melody to which another line is joined in counterpoint, may well prompt other intervallic successions. Within the latitude this affords, no perfect interval may be reiterated in direct succession, but any imperfect interval may be followed by another, or by several, of the same kind. Aside from such parallels, the two voices ought usually to proceed in contrary directions, so that when the cantus ascends, the new line descends, and vice versa. All contrapunctus must begin and end with perfect consonance.

The compact array of precepts set forth in “Quilibet affectans” hardly seems to qualify as theory. It reads as a set of rudimentary guidelines for production of correct note-against-note polyphony in two parts. Yet the opening declaration that the path to understanding contrapunctus is to write these rules (rather than, say, to sing or to compose model progressions or phrases) does appear to claim some theoretical status for what follows (as does one stray reference to Boethius).

THE INTELLECTUAL BACKGROUND OF A NATURAL PHILOSOPHER OF THE SEVENTEENTH CENTURY

Newton's theological manuscripts are concerned principally with two subjects: the interpretation of the prophecies of the Apocalypse and Daniel, and the history of the early Church. These two subjects are linked, but it was as a consequence of his interpretation of the Apocalypse that Newton undertook his study of the history of the Church. The study of prophetic literature was firmly rooted in Cambridge, where this subject was taught by Joseph Mede, author of a Clavis Apocalyptica (or Key to the Apocalypse), a work much used by Newton.

Newton’s interest in the prophecies is already documented in the “Quaestiones” of the Trinity Notebook (1664–5). In “Of Earth” (c. 1664) Newton made deductions about physics, “in rerum natura,” directly from the Scriptures: the final conflagration of the earth, and the probable succession of worlds. This last affirmation was supported by a passage of the Book of Revelation which referred to days and nights after the Last Judgment, which would have made no sense had the world finished for ever. In “Of the Creation” (c. 1664) Newton made use of a passage from Genesis to prove that God had created time. From these entries it is evident that Newton used biblical texts to determine the truth of a philosophical proposition. Strange or ingenuous as this approach of Newton’s might seem, given that the trial and condemnation of Galileo had shown the difficulty of reconciling philosophy and religion, it was one he maintained in subsequent years.

Today most scholars agree that Rameau was the founder of modern harmonic theory. As we have seen in the previous chapter, Rameau attempted to synthesize in his many writings a multiplicity of ideas – both old and new, speculative and practical – into a unified theory of tonal harmony grounded upon a single underlying principle, the corps sonore (see Chapter 24, pp. 759–71). If he did not succeed in creating a truly systematic and stable theory of harmony owing to his many differing and often contradictory theoretical arguments and intellectual borrowings, he nonetheless bequeathed to the nineteenth century a number of compelling and richly suggestive ideas that would inspire theorists in their own efforts.

In this chapter, we will look at the evolution of harmonic theory in the nineteenth century. This is of course a vast and complex topic. Given the profound changes in harmonic language between 1800 and 1900 – a period covering the end of Viennese Classicism and closing with Schoenberg’s first tentative steps beyond the tonal system – it is not surprising that theorists expended extraordinary energy and efforts in their attempts to rationalize this shifting practice. As it is impossible here to describe all of these efforts in detail, we will limit our scope to the development of Austro-German harmonic theory, a tradition which arguably encompasses some of the most innovative and influential writings on this topic for the entire century. Within this tradition, three individual trajectories can be traced back by differing routes to Rameau’s own theory: scale-degree (Stufen) theory, fundamental-bass theory, and function theory.

As scholars begin to gain a sense of historical perspective on art music in the twentieth century, it seems clear that the introduction and development of twelve-tone compositional procedures will remain one of the cardinal markers of musical modernism. The careers of Schoenberg, Berg, Webern, Boulez, Stockhausen, Babbitt, and even Stravinsky (among many others) are all at some point intimately bound up with dodecaphonic concerns, as is the course of avant-garde music generally. No matter what one may think of the twelve-tone idea – and it has been the source of considerable controversy almost from the start – understanding dodecaphony and its appeal to several generations of composers in Europe and America will continue to play a central role in understanding twentieth-century music and culture.

The twelve-tone idea has also played a pivotal role in the development of music theory as a professional discipline, especially in the United States during the post-World-War-II period. Indeed, twelve-tone theory and composition are deeply interdependent, and this is in no small measure attributable to the fact that in many cases the theorists involved were also composers. Unlike Schenkerian theory – which along with twelve-tone theory has played an important role in the professional growth of music theory in the second half of this century – twelve-tone theory often seems more prescriptive than descriptive; rather than explicating the structural features of works already established within the canon of Western art music, dodecaphonic theory is frequently speculative, suggesting structural possibilities for pieces yet to be written (or in some cases, pieces just finished by the composer himself).

The psychology of music is a subfield of psychology that addresses questions of how the mind responds to, imagines, controls the performance of, and evaluates music. The history of this subfield has been greatly influenced by the major trends and developments in the parent discipline, and the organization of this chapter follows the traditional rubrics of that history. Earlier in the twentieth century there was a frequent distinction made between Tonpsychologie (the study of vibration, the ear, and the sensation of sound) and Musikpsychologie (the study of music as a form of cognition). Though the distinction seems less clear-cut today, this chapter recognizes its historical force and focuses on the latter category, with the former receiving extended treatment in Chapter 9, passim.

Since at least the seventeenth century, proponents of one or another theory of music have frequently used the psychology of music as a touchstone. They assert propositions in the general form of “musical relationship has a valuation because there exists a relevant phenomenon or principle in the psychology of music.” For example, one might view Rameau as having asserted that “the fifth and third, as progressions of the fundamental bass have the qualities of being good, natural, and fitting because Sauveur and other acousticians have shown these intervals to be present in every musical tone, as a macrocosm within a microcosm.” Or Riemann could be viewed as having asserted that “harmonic relationships based on progressions of a major third or perfect fifth between the chordal ‘roots’ have the qualities of being directly intelligible and foundational because Helmholtz and other physiologists have shown that the frequency analysis of the inner ear privileges these intervals.”

The “canon” is the monochord, a single-stringed instrument suited for the production of musical pitches and the comparative measurement of the lengths of the string segments that produce them. In Plate 6.1, from Lodovico Fogliano’s Musica theorica of 1529, the monochordist has placed two movable bridges “about three fingers apart” at points marked A and B (the letters do not indicate the names of pitches, but designate points as in a geometric diagram); he has marked equal segments AC, CD, DE, EF, and BG, and placed bridges under points F and G. By moving the bridge he holds in his right hand, the monochordist can demonstrate that string segment DF, twice the length of BG, produces a pitch an octave (diapason) below that of BG; that CF, three times the length of BG, produces a pitch a twelfth (diapasondiapente) below that of BG; that AF, four times the length of BG, produces a pitch two octaves (bisdiapason) below that of BG.

In a systematic division of the monochord, a musician defines a number of pitches successively, at each step specifying the ratio between the length of the string segment that produces one pitch and that of the string segment that produces some other. The end results of such a monochord division are an array of pitches (which can be arranged in a scale) and a set of intervallic relationships between them specifically defined by numeric ratios (a tuning system). Canonics is the study of such pitch arrays and intervals and the ratios through which they are defined.

INTRODUCTION: PHILOSOPHICAL CONTROVERSY OVER NEWTON’S IDEAS OF SPACE, TIME, AND MOTION

Newton's concepts of “absolute space,”“absolute time,” and “absolute motion” met with serious objections from such philosophical contemporaries as Huygens, Leibniz, and Berkeley. Among philosophers of the early twentieth century, after the advent of Special and General Relativity, the objections bordered on scorn: Newton's concepts were not only lately outmoded, but they were also epistemologically inherently defective, empirically unfounded - concepts not scientific at all, but “metaphysical,” in so far as science is concerned precisely with “sensible measures” rather than obscure notions of what is “absolute.”The prevailing idea was that Einstein had established not only a new theory of space and time, but a deeper philosophical viewpoint on space and time in general. From this viewpoint, space, time, and motion are essentially relative, and to call them absolute was an elementary philosophical error. As Einstein put it, General Relativity had taken from space and time “the last remnant of physical objectivity.”

The philosophical motivation for this viewpoint seems obvious. Space cannot be observed; all that we can observe is the relative displacement of observable things. Therefore, if we observe two bodies in relative motion, to say that one of them is “really” moving, or that it is moving “relative to absolute space,” is to pass beyond the bounds of empirical science. If we wish to decide which bodies are moving, we have to construct a frame of reference – that is, we must designate some reference-points to be fixed, and compare the motions of other bodies to these.

Defending “Berenice,” in a letter to Thomas White, Poe confesses that the tale's “subject is by far too horrible” yet asserts that the “history of all Magazines shows plainly that those which have attained celebrity were indebted for it to articles similar in nature - to Berenice.” This nature consists of “the ludicrous heightened into the grotesque: the fearful coloured into the horrible: the witty exaggerated into the burlesque: the singular wrought out into the strange and mystical.” Poe justifies such work: to “be appreciated you must be read, and these things are invariably sought after with avidity.” While Poe couches his economic interest within assertions about originality and careful style and also by noting that “some very high names valued themselves principally upon this species of literature,” he concludes the letter by stating that the marketplace rather than the critic is the final arbiter: “The effect - if any -” he writes, “will be estimated better by the circulation of the Magazine than by any comments upon its contents.”Elevating economic success over good taste, Poe demonstrates early in his career his desire to master the literary marketplace and his awareness that manipulating its conventions offered the key to such mastery.

The aspect of Newton's Principia that has provoked the most controversy within the philosophy of science, other than his invocation of absolute space, time, and motion, has been his claim to have “deduced” the law of universal gravity from phenomena of orbital motion. In particular, a tradition that began with Pierre Duhem and continued with Karl Popper and then Imre Lakatos has argued that this claim is at best misleading (Duhem) and at worst a subterfuge (Lakatos). Among other reasons they have advanced against any such deduction is the objection that no deduction from consistent premises can yield a conclusion that entails one or more of these premises is false; yet one consequence of the law of universal gravity is that all the orbital phenomena from which Newton proceeds in his supposed deduction are, strictly, false. Duhem, Popper, and Lakatos insist, to the contrary, that only a hypothetico-deductive construal of Newton's evidence for universal gravity makes sense, Newton's outspoken objections to hypothetico-deductive evidence notwithstanding. More recently, Clark Glymour has offered a “bootstrapping” construal of Newton's evidence, proposing that it captures the logical force of the reasoning for universal gravitation in the Principia better than a straight-forward hypothetico-deductive construal can. Glymour too, however, sees no way around concluding that some of what Newton seems to think he is doing cannot be correct.

One issue this raises is understanding the reasoning Newton offers in arriving at the law of universal gravity and describes as a “deduction” from phenomena. Another is the extent to which such reasoning is cogent and illuminates scientific method. The simplest way to respond to these questions is to proceed step-by-step through Newton’s reasoning.

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).

As a rubric for music-theoretical literature focused on music’s dynamic qualities, “energetics” is unrestrictively broad in scope on the one hand, and restrictively narrow on the other. It is broad because ever since ancient times authors have identified motion as a fundamental aspect of music, and narrow because specific references to “energy” in music, or analogies with force, power, or similar concepts from the domain of physics, are historically limited, appearing first with regularity in the decades straddling 1900. In fact, the term energetics was first coined in 1934 by an historian of aesthetics, Rudolf Schäfke, who proposed it as a way of characterizing the work of several theorists active in the early twentieth century, primarily Heinrich Schenker, August Halm, and Ernst Kurth, although the nature and language of certain contemporaries, likewise German-speaking (Arnold Schering, Hans Mersmann, Kurt Westphal), associate them with energetics. As Schäfke points out (p. 395), if authors had long recognized the primacy of motion and tonal flux in music, they did not thematize motion to the same degree as did the energeticists, or isolate it from music’s affects.

2.

Many poets who have commented on Poe's verse have expressed amazement regarding the relative paucity of his poetic output in relation to his status as a great poet. William Carlos Williams observed that though Poe is known as a poet, “there are but five poems, possibly three.” In his essay, “From Poe to Valèry,” T. S. Eliot noted, “He wrote very few poems, and of those few only half a dozen have had a great success: but those few are as well known to as large a number of people, are as well remembered by everybody, as any poems ever written.” Daniel Hoffman has called Poe's poetic oeuvre “one of the teeniest bodies of verse of any poet the world has applauded for over a century.” In his sonnet, “For a Copy of Poe's Poems,” Edward Arlington Robinson eloquently characterized Poe's poetic output as “wonder-songs, fantastically few.”Some have their special favorites - H.D., who named Poe her “favorite among American writers,” preferred “To Helen”;William Carlos Williams, “To One in Paradise”; Robert Pinsky, “Fairy-Land”- yet many agree that Poe's two finest poems are “The Raven” and “Ulalume.”

Few would hazard a challenge to long-standing opinions that Poe was a master of the Gothic horror tale, although many might not as readily be aware that he did not invent Gothic fiction. When he began to attract widespread attention by publishing several macabre tales in the Southern Literary Messenger in early 1835, critics sounded negative notes concerning his “Germanism,” a synonym for Gothicism, just as they deplored his wasting talents on what they deemed had become an outmoded type of fiction. Such caveats, as well as many offered over the course of the century succeeding his death, notwithstanding Poe's Gothic tales, are what have typically attracted greatest numbers of readers, and that allurement is wholly understandable. A descent from such British milestones in literary Gothicism as Horace Walpole's The Castle of Otranto (1764), William Beckford's Vathek (1786), W. H. Ireland's The Abbess (1798), or Sir Walter Scott's The Bride of Lammermoor (1819) is evident in Poe's writings. In his own day the brief tale of terror, familiarly known to the Anglo-American readership as the signature for fiction in the popular Blackwood's Edinburgh Magazine, served as Poe's, and other Americans', model, time and again, although his accomplishments in the short story far surpassed what now often reads like so much dross in the pages of the celebrated Scottish and other contemporaneous literary magazines from the first half of the nineteenth century.

“The epistemological underpinnings of Schenker’s theory,” writes Leslie Blasius, “are far from obvious.” Such a statement might well give his readers pause. After all, Blasius is talking about what must be the most widespread approach to the advanced analysis of the common-practice repertory today, and the doubt he is expressing goes to the heart of what Schenkerian analysis tells us: what sort of knowledge of music it gives us, what sort of truth it aspires to. And this of a theorist who devoted considerable attention to the underpinnings of his theory, for instance by carefully distinguishing those elements of music that he saw as given in nature from those that resulted from artifice, and thereby demarcating the province of the scientist from that of the music theorist. Most music-theoretical writing betrays few of Schenker’s epistemological qualms; Allen Forte’s The Structure of Atonal Music, to cite an example more or less at random, plunges straight into its topic in the same spirit of epistemological self-evidence that characterized the contemporary scientific writing on which Forte modeled both his literary and his theoretical approach. Like scientists, perhaps, music theorists address epistemological issues only when the truth-value of their work no longer seems self-evident to them. And if this is the case – if music-theoretical concern with epistemology is at root an expression of anxiety – then we have a fundamental problem in trying to unravel the epistemological underpinnings of music theory: when theorists are confident of the epistemological status of their work they will say nothing about it, whereas when they do talk about it we can deduce they are not quite sure about what they are saying.

Edgar Allan Poe, poet, short story writer, and critic, was a controversial figure in the publishing world of antebellum America. His ability to spark controversy stemmed not only from an image concocted by his contemporary detractors but from the sharp tone and pointed content of the critical articles he wrote during his lifetime. He worked as an editor and contributor to magazines in several American publishing venues, including Richmond, New York, and Philadelphia. His continuing ambition was to found and edit his own magazine, an outlet that would have granted him financial security and artistic control in what he deemed an antagonistic literary marketplace. Poe's challenge to moralistic strictures against literature, his confrontations with the New England literary establishment, and his caustic and satirical critical style won him many enemies.

In the Preface to the first edition (1687) Newton informs the reader straight off that he intends the Principia to illustrate a new way of doing what we now call empirical science:

Surprisingly, however, the main body of the first edition contains only two further comments about methodology: (1) a cryptic remark at the end of the opening discussion of space and time, announcing that the purpose of the work is to explain “how to determine the true motions from their causes, effects, and apparent differences, and, conversely, how to determine from motions, whether true or apparent, their causes and effects”; and (2) a scholium buried at the end of Book 1, Section 11 in which Newton proposes that his distinctive approach will make it possible to argue more securely in natural philosophy.