God created the integers; the rest is the work of man. — Leopold Kronecker
What are we saying when we talk about music?
This question captures the paradox that lies at the heart of musical theory. Put in its most basic form, the problem that has dogged musical theory since Boethius has to do with the relationship between reason and the esthetic sense. The earliest theories show that the coexistence between the two was never an entirely easy one:
In the final analysis, it was to this that the Pythagoreans’ harmonic analysis of the universe led: the discovery of incommensurables. And no matter how they might juxtapose the numbers, no matter to what lengths they might extend their mathematical circumlocutions, one fact remained, a fact that has ever since proved resistant to mathematical rationalization: there is no fraction m/n that will divide the whole-tone into two equal parts.
The Pythagorean construction of music was an attempt at reconciling the rational and the beautiful — at showing that they are, indeed, one and the same. In this sense it was a corollary to the impulse behind the Parthenon: the Athenians believed that the golden ratio, applied to every dimension of a structure, would create something that was beautiful precisely because of its mathematical perfection.
In music, as mentioned in the previous quote, this dream was quickly shown to be illusory. While the Parthenon was constructed ex nihilo, and could perfectly mirror the rational dreams of its designers, the Pythagorean theorists of music were confronted from the beginning with a stubborn fact: there were pre-existing and deeply engrained notions of what constituted the proper and beautiful in music — the whole tone, the semi-tone, the modes, the tuning of the lyre — and, although they hovered tantalizingly close to the realm of reason, they ultimately eluded its grasp.
The early history of music theory was thus a reflection of fundamental debates about the nature of the beautiful. Boethius, through whom the most significant elements of ancient theory have come down to us, saw a basic opposition between the rationalism of the Pythagoreans and the empiricism of Aristoxenus, with Ptolemy mediating between the two. Boethius himself came down firmly on the side of the rationalists:
There are three classes of those who are engaged in the musical art. The first class consists of those who perform on instruments, the second of those who compose songs, and the third of those who judge instrumental performance and song… [the third] class, since it is totally grounded in reason, will [alone] rightly be esteemed as musical. That person is a musician who exhibits the faculty of forming judgments according to speculation or reason relative and appropriate to music. 
This rationalism, which defined an important part of the heritage of western theory, is deeply tied to its Hellenistic source. The association between beauty and proportion (and hence reason), coupled with the invocation of the Greeks, may be found, essentially unaltered, a millennium later in the writings of Zarlino:
It must be understood that nothing whose parts possess what the Greeks called [symmetry] can fail to please the senses, for the senses are delighted by proportioned objects. Conversely, it is impossible for the senses to be pleased by what is not proportioned. 
Set against Hellenic rationalism was an empiricism closely tied to medieval Christianity. The tenets underlying this empiricism may be observed most clearly in the Musica and Scolica Enchiriadis, a pair of medieval handbooks intended for practical instruction in music making. The emphasis of these works is entirely on what may be done with the empirical givens of music; the justification of these givens is not attempted, or even considered important. This is partly because the Musica Enchiriadis and the Scolica Enchiriadis were intended primarily as pedagogical, rather than as abstractly theoretical, treatises. But there is a deeper reason as well, which is intimately bound to a specific conception of faith:
[Music is] the science of regulating properly the movement of sound. [That is,] to control melody so that it sounds sweet. But this must be done in full conformance with the rules (ad artem.) It is clear to me that one who misuses the sweetness of this art for worthless purposes, just as one who does not know how to apply [the rules of] the discipline (ars) where it is necessary, does not regulate sound properly. Rather, only someone with a heart full of devotion sings sweetly to the Lord. 
Beauty, by this conception, can be reached only through the full and fruitful mastery of a system. But this alone is not enough: “only someone with a heart full of devotion sings sweetly to the Lord.” Moreover, this system is not an object of rational contemplation, but something given to us. Nowhere in the Enchiriadis is the possibility entertained that this system might be questioned; a heart full of devotion must be matched by a humble acceptance of the rules and traditions of the practice. These rules are themselves a gift of divine grace.
In modern terminology, then, the Musica Enchiriadis and its successors—such as the work of Guido D’Arezzo and the modern system of notation—constitute a tradition that treats music axiomatically. Their concern is with the system as it unfolds before us. But to the authors of the Enchiriadis, this axiomatic treatment of music could exist only as a manifestation of the idea of faith, in the full, theological sense in which the medieval Church understood this term. The axioms did not have meaning and force simply because they were necessary for the system to work. Rather, they were gifts of God, and they could be mastered only through a humble submission to God’s will. But this meant also that, for those who conceived of music in this tradition, the Hellenistic attempt to make beauty coeval with reason was not merely futile, but hubristic.
These, then, are the fundamental terms of the dialectic that has unfolded throughout music history: reason and faith, faith and reason. To the rationalists the passive acceptance of the rules must have seemed an unforgivable abandonment of intellectual responsibility, while the empiricists, as we have seen, saw an unseasonable pride in the attempts of reason to penetrate God’s mysteries.
It is a hoary truism that Western civilization was formed through the confluence of the Greek and Judaic cults. But in the case of music this applies in an immediate and technically quantifiable sense. The origins of the enlightenment and of Western civilization more generally, Leszek Kolakowski has argued, lie in the contradictions of the medieval Christian world, which necessitated the endless mediation of irreconcilable dualities:
In the Christian faith, as in the other great religions, an inevitable tension exists between the notion of a finite world through which the Creator is revealed, and the image of that same world as the negation of God; between a vision of nature as displaying the glory and goodness of God and one in which nature, through her corruption and her contingency is seen as a source of evil; between the biblical cuncta valde bona and the earth as a place of exile or even, in the extreme version, almost as the fruit of God’s sin. The Christian idea as it has been developed and expressed over the centuries has had to wage an unceasing battle with heretical tendencies which affirmed one of the elements of this tension while neglecting or forgetting about the other. 
It is precisely this process that gave Western music its marvelous, syncretic strength. With the infusion into the Christian world of the Platonic and Aristotelian concepts espoused in the works of the medieval scholastics philosophers, a dialectic between faith and reason began that can have no final resting point. This dialectic has controlled the debate over music and shaped the very essence of musical practice. The absolute triumph of either school of thought would spell the end of our musical tradition as we know it, but the tension between the two gives it life. It is in the space between these two extremes that western musical practice, in an act of unceasing mediation, has carved out its distinct territory, making the vivifying tensions of this dialectic its own. Sadly, it is also the very possibility of such mediation, and with it the survival of this tradition, that is now being called into question.
The history of music theory is thus a logical unfolding of one of the fundamental intellectual dualities that have defined western thought. To understand the consequences that this has had for theory since the Enlightenment, we may appeal to yet another duality, that between content and form. If we see the form of theory to be the historical dialectic in which it has been embedded, and the content to be the specific arguments of the various theoretical systems, we will see that theory since Rameau has tended toward a unity of form and content. In other words, being part and product of an inescapable dialectic, it has itself become dialectical.
A dualistic turn of mind shaped Rameau’s theoretical writings at the most elemental level. As Joel Lester has noted, Rameau embodied in his own work the dialectical conflict that characterizes Western theory more generally:
Because Rameau both searched for the basic principles underlying music and also addressed practical concerns, his works often present two seemingly separate explanations for a given phenomenon. As speculative theorist he will offer an explanation, while as practical theorist he will concentrate on the results. When a speculative perspective is prominent in one work and a practical perspective predominates in others, it has seemed to many scholars that Rameau changed his opinions on the issue. But often, Rameau held similar views in both works, although he may have spoken speculatively in one and practically in the other. 
Lester might also have noted that the dialectic between practical and speculative theorist was itself a microcosm of a more fundamental dialectic between composer and theorist, and that the work of the practical theorist was principally geared toward answering the needs of the composer, while the work of the speculative theorist was ostensibly in the interest of pure understanding.
These dualities of form correspond to a single, overriding duality of content; and it is this duality of content that marks Rameau’s greatest achievement. Combing through the vast, undifferentiated apparatus of thoroughbass theory, he refined it down to the simple opposition of “dominant” and “tonic,” consonant and dissonant chords. This opposition was the most significant element of his theoretical system, and has had farther-reaching consequences even than the famous theory of the “fundamental bass.” To fully grasp Rameau’s significance, we must consider both the strengths and limitations of this viewpoint.
Among the paradoxical results of Rameau’s rigorous dualism is that he was, in fact, the first theorist of dissonance. This may seem an extraordinarily strange statement; the question of dissonance had occupied theorists from the beginning, and Boethius and Zarlino had both devoted extended passages to the codification and enumeration of dissonances. But they had, for the most part, classified dissonance only in order to exclude it. In ancient theory, the dissonance had no part at all, and this remained true for much of the Renaissance. Zarlino’s writings show the first significant departure from this tradition: in his dialectic between “fuller” and “more pleasing” sonorities, he imagined a scale of increasing and decreasing tension which mediated between the harshest contrapuntal dissonances and the barest consonances, down to the unison itself. What prevented Zarlino’s dialectic from constituting a true theory of dissonance was its local and ornamental conception of dissonance: a dissonance in Zarlino’s writing (as, indeed, in the music it was intended to elucidate) was a purely contrapuntal phenomenon, which was understood in the context of the relation between two voices, and which had no effect on the fundamental structure of a work.
Set against this background, we may understand that the fundamental revolution of Rameau’s thinking was not the division between consonance and dissonance per se, but the division between essential and non-essential dissonances, which was taken over by Kirnberger and Riemann after him. The idea of a distinction between essential and non-essential dissonance seems insignificant to us only if we fail to understand the new category it is introducing. If this new category were the non-essential dissonance, it would be comparatively insignificant. But, in fact, the new category that Rameau is introducing is the essential dissonance. That is to say, all dissonance in theory prior to Rameau was non-essential; all earlier theories of dissonance, including Zarlino’s, treated it as a phenomenon ultimately reducible to ornamentation and contrapuntal practice, and hence as ultimately insignificant on a structural level. Rameau, by comparison, posited the simple but epochal notion that a dissonant chord exists as a fundamental structural element of a tonal composition, and that the dialectical interplay between consonance and dissonance, occurring not at the contrapuntal but at the harmonic level, is the structural engine of a work.
The weaknesses of Rameau’s dualism are less readily apparent than its strengths, but they are significant enough that Rameau himself, toward the end of his life, became increasingly aware of them, and went to great lengths to abrogate them. The fundamental difficulty arose from Rameau’s desire to tie every element of his system into one overriding opposition. Hence oppositional pairs whose equivalence to each other was not beyond question—structural dissonance and consonance, dominant and tonic functions, seventh chord and triad—were yoked together to form a single dialectic. This equivocation had problematic consequences, not least for Rameau’s own music, which tended to reach full consonances only at cadential points, as a result of his inability to separate the formal category of structural dissonance from the locally harmonic category of the seventh chord.
Perhaps more significantly for the history of theory, the simple duality of “tonic” and “dominant” chords was ultimately felt by Rameau to be insufficient. By introducing the concept of subdominant in his late works, and thus expanding the range of basic tonal structures to three, Rameau abandoned the consistent, unifying dualism that had marked is earlier theoretical writings. But the result of this abandonment was an understanding of tonality more comprehensive and accurate than that which had characterized his earlier writings.
But the high-water mark of explicit dualism in music theory—and, indeed, in Western thought—was the nineteenth century. In music theory, this development was epitomized by the theories of Hauptmann and Riemann. Hegel’s dialectic, in Hauptmann’s eyes, had opened new paths to an understanding of the problem of major and minor, which he characterized in terms of the Hegelian distinction between “having” and “being”—opposing the C major triad, in which C has a fifth (G), to the F minor triad, in which C is the fifth of F. Hence “having”, an active principal, is opposed to “being”, a passive principal.
As abstract as this notion was, it remained the strongest rationale for the theories of harmonic dualism that were developed later by von Oettingen and Riemann. From this distance, the dialectic of Riemann’s thought can ultimately be justified only by an idealistic dualism, such as that which Hauptmann developed, rather than by the acoustically dualistic constructions, such as that between overtone and undertone series, that Riemann and von Oettingen advanced in support of their constructions. This is for the simple but uncomfortable reason that acoustics, unlike music, shows no sign of behaving dialectically, as Riemann himself was ultimately forced to admit, discarding the notion of the undertone series as a source of the minor triad.
If acoustics proved to have a tenuous relationship to the dialectical constructs through which Rameau and his successors viewed music, then Riemann was left with a choice between abandoning the acoustic interpretation of music, and abandoning dialectic. As Carl Dahlhaus writes, he ultimately chose the former.  The principle project of late 19th-century theory, then, was a re-imagining of Rameau’s elaborate symmetries in the terms of Hegelian dialectic. This becomes apparent when we compare the form of Riemann’s theories with that of Rameau’s. What is striking is that, although their content is in many ways very different, the shape in which this content is expressed is identical: that of an elaborate set of dualities, all of which are reducible to a single, fundamental opposition. Moreover the fundamental terms of Riemann’s opposition are designed to account for precisely that harmonic discovery that upset the symmetry of Rameau’s system—the subtonic.
In Riemann’s harmonic dualism, the fundamental opposition is not between tonic and dominant, consonance and dissonance, but dominant and subdominant, major and minor. The “over”- and “under-dominants,” as Riemann termed them, are the basic harmonic manifestations of this opposition, depicting two distinct but equal aspects of the tonic.
This reflects the fundamental feature that distinguished Hegelian dialectic from the dualism of the enlightenment. The basic terms of Kant’s duality—the empirical and noumenal worlds—were irreconcilable; this was, indeed the entire point of his philosophy, whose goal, as he said, was to “limit reason to make room for faith.” Thus sense experience could never give us access to the transcendental. The terms of the duality were fixed.
Hegel’s critique of Kant, as we will see, had important consequences for the future of music theory (as, indeed, for Western civilization as a whole.) The essence of this critique, as set forth in The Phenomenology of Mind, was that Kant and his immediate successors saw logic and the mind and as fixed, immutable categories, whereas they were in fact processes unfolding in time. This is what distinguishes dialectic from simple duality. The basic function of Hegel’s logic was that of exploring the ultimate consequences of this logical unfolding. This can be seen, for example, in the famous master-slave dialectic, where the master’s ever-greater reliance on the slave leads ultimately to his own subservience, and thus to the slave becoming the master.
Thus the fundamental terms of Hegel’s dialectic are negation: dualities ultimately lead to their opposite, and the pendulum swings forward and backward from thesis to antithesis, ultimately coming to rest at a point of synthesis. But Hegel saw this synthesis as existing not merely in individual dialectical relationships; rather, he believed that all the smaller dialectics of the age were part of one grand unfolding, which was the ultimate realization, in the human world, of the spirit of Freedom. Ultimately, all dualities would be dissolved in a grand unity, which was itself the endpoint and culmination of History. And since this dialectical process was an unfolding taking place within Reason itself, this meant that the territory of Reason had no bounds. Hegel thus felt that he had found a way to know the noumenal, which Kant had tried to show was fundamentally unknowable.
Given the derivation of Riemann’s system from this school of thought, this background puts us in a position to understand several things of fundamental importance about his thought. The first thing we realize is that the dialectical constructs of 19th-century musical theory were in fact the continuation of the rationalistic school of theory that descended from the Pythagoreans, but centered around the unfolding of logic itself, rather than around mathematical processes. Hegel’s dialectic was ultimately a project of rationalistic system-building, fundamentally opposed to Kant’s project of “limiting reason to make room for faith.” Thus Riemann’s musical logic, which was the ultimate expression of Hegelian thinking in music, was also an attempt to find a way by which reason could encompass the entirety of music, without having to accept the limitations to musical knowledge advocated by empiricists such as the author of the Enchiriadis.
The analogy with Hegelian logic will also be able to show us why the purity and consistency of Riemann’s dualism have problematic implications for his theory as a whole. The subdominant and the minor mode are derived from the same process of reflection—what in Riemann’s earlier, more acoustically derived theories was called the undertone series—and are thus intimately coupled. This means, however, that in the dichotomy between dominant and subdominant, the subdominant in its most basic form—as the “underclang” of the tonic pitch—is literally the reflection of the dominant, extending downward from the tonic pitch just as the tonic itself extends upward from the fifth of the tonic sonority. This fact is more accurately represented in Riemann’s original terminology, in which dominant and subdominant are referred to as “over-” and “under-” dominants. Thus the rigorously dualistic nature of the theoretical system is reproduced in the terminology, which treats dominant and subdominant as being of equivalent structural importance—the equal and opposite manifestations of two aspects of the tonic itself.
From this we may derive the entirely unproblematic proposition that the prime form of the dominant is always major, regardless of the prevailing mode. Thus Riemann elegantly explains the actual behavior of the dominant in all tonal music. But the necessary corollary of this is the argument that the “prime” form of the subdominant is that which is formed directly from the “underclang” of the tonic pitch—in other words, the minor. The major subdominant, which is formed from the “overclang” of the under-fifth of the tonic, is a mathematically and syntactically secondary sonority. And this, like the major quality of the dominant, obtains regardless of mode. Riemann himself seems largely oblivious of this particular implication of his harmonic thinking, and in Harmony Simplified places greater emphasis on the major subdominant, despite the fact that its derivation from his system is tenuous by comparison with that of the minor subdominant.  But the logic of this system entails something quite different: that the “natural” form of both the major and the minor mode is mixed—the minor with the leading-tone, and the major with the flattened sixth—and therefore partially chromatic.
Riemann’s adherence to a rigid dualism in his derivation of tonic and dominant results in a fatal undermining of precisely the duality he initially sets out to explain, that of major and minor. In turn, his system served as something he would never have wanted: one of the most potent theoretical justifications of the chromatic musical logic of Romanticism and atonality. The tonal system, as Riemann envisions it, is thus pregnant with its own dissolution, precisely by virtue of the burgeoning chromaticism that its dialectic necessarily implies. But this is, after all, just another manifestation of the Hegelian logic on which Riemann’s system is based: the dynamic force of an autonomous dialectic leads inevitably to its own negation. This is succinctly expressed in Peter Heller’s maxim that “the history of dialectic is dialectical to the point of paradox:” 
Lessing conceived “pure striving” as a dialectic sustained in its direction by a divine and infinitely distant goal. The increasing emancipation of dynamic striving from the faith in this or in any objective goal led to Nietzsche’s frantic attempt to embrace a boundless and autonomous dialectic. Nietzsche’s endeavor established the context for the pervasive ambivalence and perennial oscillations of Thomas Mann. Finally, the levelling and neutralizing despair of Kafka demonstrated that dialectical striving, without a goal, is futile. 
Heller’s principle concern was with the expression of dialectical logic in literature. But this logical progression can be seen as manifesting itself in many other areas, including philosophy and music. If there is one lesson that the history of the twentieth century teaches us, it is that dialectical logic is like the Golem of the old Yiddish tale: once summoned, it has purposes of its own, and leads in directions which may not be congenial to the men who thought they were its masters. Its logic is ultimately that of a self-negation that leads not toward the unity and perfection Hegel expected, but toward the negation of this goal itself: thus Hegel’s dialectic of Freedom gave birth, through Feuerbach and the young Hegelians, to Marxism, in the name of which millions were enslaved; and Riemann’s system, formulated with an eye toward preserving the system of classical tonality, acted instead as the handmaiden of its dissolution. Riemann, like Hegel, believed that reason would lead him to the promised land. But, to invert Pascal’s phrase, reason has its reasons that the heart knows not of.
After Riemann’s system revealed itself to be inherently unstable, what remained to his successors? The predominant theoretical methods of modernity turned away from dualism, and, in a sense, from rationalism altogether: Schenker’s primal tonal shapes, like the row in Schoenberg’s compositional practice, existed not as rational explanations for the way music is, but as primal, generative phenomena. This is why the essence of Schenker’s system was expressed graphically: it is not intended as a rational explanation of why music is the way it is; it is, simply, a picture of what it is. Schenker explicitly associated the Viennese classicism around which his theories were built with the Kantian project of establishing the limits of reason:
Just as Kant established these limits for human thought as a whole, so, too, did the great masters of Germanic composition establish the limits of specifically musical thought. 
This conception of the role of theory applies also to the twelve-tone system: it simply is, existing as a practice rather than an explanation. And, as Robert Watson relates, Schoenberg also conceived of his harmonic theory in terms, not of a unified system in the Riemannian sense, but as what Wittgenstein called a “perspicuous representation” of facts:
The dislike of absolute systems (and Riemann’s identification of ‘speculative’ theory with aesthetics) would lead Schoenberg to maintain that harmony instruction should not aspire to ‘theory’ or ‘aesthetics,’ but merely to a clear and practical system of presentation. 
In this respect Schoenberg and Schenker, despite their intense antipathy toward each other, were engaged in the same basic enterprise, which was expressed nearly simultaneously in music, philosophy, history, and the physical sciences. This was the attempt to construct a morphological vision of their field, which in turn derived from Goethe’s study of the morphology of plants, which he embarked on as a corrective to what he perceived to be the mechanistic nature of Newtonian science. The morphological vision of theory can thus be seen as fundamentally opposed to the rationalist one promoted by Rameau and Riemann: Schenker wanted, not to explain music, but simply to show us its shape. This is parallel to the theories of statistical mechanics pioneered by Hertz and Boltzmann, who invented a purely graphical method for the representation of the state of a particle. It is also parallel to the early philosophy of Wittgenstein, who attempted to limit the metaphysical speculations of the post-Hegelian idealists by applying the methods of Hertz and Boltzmann to language. The characteristic position of Wittgenstein’s philosophy, that one should not attempt to say what can only be shown, is also the unspoken principle underlying Schenker’s graphical method of analysis. It is thus not surprising that Wittgenstein felt there to be a deep affinity between his thinking and Schenker’s.  Similarly, a number of scholars of Viennese philosophy have postulated a link between the methodology of Wittgenstein’s Tractatus and twelve-tone technique. 
What, then, of Hegel and Riemann? As we have seen, it is precisely the triumphant rationality of their systems that foiled their goals. The history of dialectical logic thus returns us, paradoxically, to our starting point, for it is an object lesson in the dangers of unfettered reason, and the necessity of tempering reason with faith. This is why the logic of Hegel was seen by many of his critics as a kind of hubris of the intellect. Kierkegaard expressed this view succinctly:
If Hegel had written his whole ‘Logic’ and in the Preface had disclosed the fact that it was merely a thought-experiment, he would have been the greatest thinker that has ever lived. Now he is a comic. 
If this was hubris, what follows? The old morality, which we in our enlightened age believe ourselves to have overcome, held that hubris invites nemesis. The history of the twentieth century showed that nemesis was not long in coming.
 Flora D. Levin, trans.: The Manual of Harmonics of Nicomachus the Pythagorean [Newburyport: Phanes Press, 1993], p. 136.
 Boethius: The Fundamentals of Music, trans. Calvin Bower and ed. Claude Palisca [New Haven: Yale University Press, 1989.] p. 51.
 Gioseffo Zarlino: The Art of Counterpoint, trans. Guy Marco and Claude V. Palisca [New Haven: Yale University Press, 1968], p. 283.
 Claude V. Palisca, trans.: Musica Enchiriadis and Scolica Enchiriadis [New Haven: Yale University Press, 1995], p. 33.
 Leszek Kolakowski: Modernity on Endless Trial [Chicago: University of Chicago Press, 1997], p. 27.
 Joel Lester: Compositional Theory in the 18th Century [Cambridge: Harvard University Press, 1992], p. 92.
 Dahlhaus, Carl. “Harmony,” [in Grove Music Online, accessed March 2009], section 4.
 Hugo Riemann, Harmony Simplified [London: Augener, 1895], 8.
 Peter Heller, Dialectics and Nihilism [University of Massachusetts Press, 1966], 92.
 Ibid., vii.
 (Cited in Nicholas Cook: The Schenker project: Culture, Race, and Music Theory in Fin-de-siècle Vienna [Oxford: Oxford University Press, 2007], p. 31.)
 Robert W. Watson, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg [Rochester, University of Rochester Press, 1985], 119.
 See Eran Guter: “Where Languages End: Ludwig Wittgenstein at the Crossroads of Music, Language, and the World” [PhD diss., Boston University, 2004.]
 See Alan Janik and Stephen Toulmin: Wittgenstein’s Vienna [New York: Simon and Schuster, 1973.]
 Walter Lowrie: A Short Life of Kierkegaard [Princeton: Princeton University Press, 1942], p. 116.