I

Analytic reduction, with its implicit recognition of music as a multi-leveled structure in which surface complexities conceal more basic underlying patterns, forms one of the cornerstones of Heinrich Schenker's theory of tonality. Schenker's work is well known today, and there are few theorists active who have not been influenced to some degree by his ideas. Moreover, it is precisely Schenker's insistence on reducing complex musical phenomena to simpler foundations that has had the most widespread effect upon recent theoretical conceptions. Thus reduction technique, both in its specifically Schenkerian form as well as in other less orthodox (though Schenker-derived) versions, is widely current and represents a topic of general familiarity among musicians.

Indeed, Schenker's current prominence is such that musical reduction is often viewed as an exclusively Schenkerian phenomenon, or even as a purely Schenkerian invention. This stems partly from Schenker himself, who was inclined to emphasize differences between his own ideas and those of his predecessors and contemporaries. The reasons are not difficult to find: increasingly attacked by all but a small group of devoted followers, he preferred to stress his conceptual isolation and to emphasize the innovative nature of his work. In addition, Schenker was not himself overly concerned with the historical aspects of music theory: aside from an occasional remark,¹ there is little indication that he felt any desire to uncover historical precedents for his own work.²

One unfortunate result of this can be seen in a tendency to separate Schenker from the larger course of Western music theory. He is too often considered a musical thinker sui generis, as if his ideas developed solely as the result of a miraculous and entirely personal (and thus ultimately idiosyncratic) conceptual leap. Although it would be absurd to deny the extraordinary originality of Schenker's achievement (which is patently clear to anyone with even a superficial knowledge of his work), overemphasis on his uniqueness has tended to obscure his position in the larger historical context. What is more, it has alienated many musicians who, less immediately drawn to his approach, are led to believe that his ideas are completely without precedent and thus totally removed from the terra firma of everyday practical musical discourse. There is no denying that, for many, Schenker remains even today a sort of musical aberration, an outsider to be looked upon with suspicion if not fear.

The present article proposes to show that Schenker's ideas are in fact firmly rooted in the historical tradition. The point, again, is not to distract from his importance or originality. On the contrary, the aim is to reveal that his theory represents a remarkable synthesis of some of the main currents of Western musical thought.

The roots of reduction technique, the foundation of the Schenkerian system, extend far back into Western music history and encompass a wide range of theorists and theoretical ideas. Although a complete account would exceed the scope of this paper, an effort will be made to trace the most important lines of development preparatory to and, in an historical sense, leading toward Schenker. These can be grouped into three principal areas: 1) diminution technique; 2) musica poetica, or the theory of musical figures; and 3) functional harmonic theory. It should be borne in mind, however, that all of these areas, and particularly the first two, are closely interrelated and to some extent overlap with one another.

 

II

Diminution—the division of a note into several notes of shorter value through various kinds of elaboration, such as repetition, circling around, or filling up the space between tones—is, considered from the point of view of reduction technique, the other side of the coin: an augmentation of musical content through addition, and thus a process of compositional synthesis, rather than an analytic process involving musical subtraction.³

As a musical practice, diminution no doubt dates back to the origins of music, as it seems reasonable to assume that ornamental processes of some sort have always been associated with the act of singing or playing an instrument. A characteristic feature of some of the oldest musical cultures about which we have knowledge is the existence of a repository of standard melodic formulae—e.g., the Greek nomoi or the Byzantine echoi—that serve as models for the invention of new melodic material through variation, elaboration and rearrangement. In the European tradition, early examples of diminution are found in melismatic forms of Gregorian chant (visually reflected in neumatic notation), which often represent amplified and more ornamental versions of simpler chant types.⁴ Similar florid melodic structures continue to appear in early polyphony: in the Duplum diminutiones of melismatic organum, as well as in various later forms of polyphonic composition in which one or more faster-moving voices occur in combination with a more sustained tenor.

With the evolution of written polyphony toward an integrated system of fixed durations, enabling rhythmically independent voices to move simultaneously in mutually coordinated values, diminution becomes a central topic of theoretical concern. In the thirteenth century theorists begin discussing the introduction of shorter note-values into the rigid proportions of modal rhythm. The Anonymous IV treatise De Mensuris et Discantu, for example, contains a section "on the diminishing and breaking up of the modes" (De minutione et fractione modorum), in which the author states that a long may be split up into as many as eight shorter values.⁵ Significantly, these shorter notes are initially considered to be "beyond measure" (ultra mensuram), for they owe their existence solely to the ornamentation of the larger durations properly belonging to the mode.⁶ Only later do they gain independence and, consequently, self-sufficiency. One such ornament, the plica, which receives particular theoretical attention, is described by Franco of Cologne as "a note in which the same sound is divided into low or high,"⁷ a definition clearly indicating that the decoration represents—and is thus in some sense reducible to—a single underlying note.

The gradual assigning of definite durational values to these shorter notes introduces a new stage of theoretical development, in which the various possibilities of subdividing larger rhythmic units are subjected to intensive investigation. One of the most interesting of the treatises dealing with this problem is the Ars Contrapuncti secundum Johannem de Muris, which dates from the fourteenth century. Here, in an extended section entitled De diminutione contrapuncti, a series of examples illustrates systematically, through progressive divisions of a discant, the combination and rhythmic coordination of an increasing number of shorter notes with each note of a tenor. Divisions from two to nine are given, of which the first three and last are as follows:⁸

 

Ex. I.

 

Here, then, the theoretical conception of diminution as a layered succession of increasingly complex elaborations is already clearly established.

The practice of diminution continues throughout the Renaissance and Baroque periods in both written and improvisatory form. In written composition it produces the type of florid polyphony designated by Tinctoris in the fifteenth century as Contrapunctus diminutus in distinction to Contrapunctus simplex (i.e., note-against-note counterpoint in equal values). The determination of the relationship between these two types of counterpoint becomes one of the principal concerns of the theory of musica poetica and will be discussed in the next section. For the moment we shall focus upon diminution as an art of improvised performance, a time-honored practice that eventually emerges in the sixteenth century as a subject of widespread and thorough theoretical treatment, principally in instruction books for singers and instrumentalists.

The earliest manual on performance practice to encompass diminution is Conrad Paumann's Fundamentum organisandi, a mid-fifteenth century book of organ instruction consisting almost entirely of musical examples presumably intended as models for improvisation.⁹ The Fundamentum illustrates how a faster moving, ornamental discant can be combined with an underlying cantus, the complexity of which is gradually increased from simple rising and falling scales in regular durations (the ascensus simplex and descensus simplex) to more differentiated melodic and rhythmic forms.

Although Paumann's examples already demonstrate how standard ornamental figures can be joined in various melodic combinations, the diminutions are not yet grouped according to underlying intervallic structure. This next step, leading to a fully developed theory of improvised ornamentation, is taken by Sylvestro Ganassi in his Fontegara, a recorder method published in 1535.¹⁰ Ganassi's work contains, among other things, a series of basic models of intervallic progression, followed by a number of different diminutions. The following, drawn from the diminutions for a rising second, are typical:

 

Ex. II.

 

It is clear from Ganassi's examples that the underlying second may be radically altered by the surface elaborations. Particularly suggestive from Schenker's point of view is the transformation of the rising second into a descending seventh or rising ninth through octave transference. In summary, Ganassi offers, even at this early stage, a clearly formed conception of musical elaboration as a process that preserves the structural meaning and integrity of the foundation upon which it is based.

Similar examples of diminution can be found in numerous treatises devoted wholly or partly to performance practice and improvisation published during the following one hundred years or so, among them works by Coclico, Ortiz, Finck, Zacconi, Diruta, Bovicelli and Mersenne. The diminutions are not necessarily limited to a single voice, as is illustrated in the following elaboration of a three-voice choral (with certain voice-leading peculiarities) from a manual on organ playing by Hans Buchner, a composer and organist active during the first half of the sixteenth century:¹¹

 

Ex. III.

 

Illustrations of diminutions for entire pieces are also commonly found (e.g., in Zacconi, Bovicelli and Mersenne). Often these consist of elaborations of already existing compositions by such masters as Palestrina and Cipriano de Rore, but original compositions also appear. The opening portion of a complete one-voice motet from Zacconi's Prattica di Musica, on which the author comments that he has "not only presented the ornamental voice but also the part in its natural form," may serve as an example:¹²

 

Ex. IV.

 

Evident in all of these treatises is a tendency toward schematization. Conforto's manual Breve et facile maniera d'essercitarsi (1593?), for example, contains as many as thirty variants for a single two-note figure, while there are as many as forty-three in Bovicelli's Regole, Passaggi di Musica (1594). The diminutions thus take on the character of set musical formulae, and as such they become part of the standard vocabulary of written, as well as improvised, composition. In both of these forms diminution technique continues as a vital part of musical instruction and practice until well into the late eighteenth century, as is evident from the importance accorded it in the performance treatises of C.P.E. Bach, Leopold Mozart, Tartini and Quantz, as well as in more general theoretical works such as Kirnberger's Die Kunst des reinen Satzes. Indeed, the central position of diminution in both the practical and theoretical areas of musical life accounts in significant measure for the unusually close correspondence between theory and practice notable during this entire period.¹³

 

III

Whereas diminution technique, viewed as an historical phenomenon, has received some consideration within a specifically Schenkerian perspective,¹⁴ the theory of musical figures has attracted little attention in reference to Schenker's ideas. Yet the theory of figures—or more generally, of musica poetica or the rhetoric of music—is perhaps an even more important precedent in terms of the overall focus of his analytic method. Moreover, musica poetica itself encompasses diminution, treating such elaborations as one of a number of possible types of rhetorical embellishment.

The idea that music is organized according to principles analogous to those of the verbal art of rhetoric, or poetics, was in general acceptance from the sixteenth through the eighteenth century, and there is scarcely a theorist during this period who fails to betray its influence.¹⁵ Two basic assumptions underlie the notion of musica poetica, both of which are important for our considerations: first, that there are certain standard types of musical elaboration, called figures, through which both literal meaning and affect are communicated; and second, that these figures belong to a heightened, "poetic" style of communication, employed for rhetorical or expressive purposes, that represents a departure from the simpler, "normal" forms of musical discourse.

Diminutions are among the first figures to be considered within a musico-rhetorical framework. Thus Coclico, whose Compendium Musices (1552) is one of the earlier works to discuss the use of diminutions in performance, introduces the subject in a section entitled De musica figurali and remarks that music "is taught in the same way as rhetoric." Moreover, his designations simplex and elegans, distinguishing the basic model from its elaboration, are taken directly from rhetorical terminology.

The first theorist to develop a more or less complete system of musical rhetoric, however, is Joachim Burmeister, whose highly influential Musica Poetica appeared in 1606. Burmeister, who wished to develop a method of compositional instruction based upon actual practice, took the highly evolved organizational structure already existing in rhetoric as a model for the classification of different types of musical phenomena. In his treatise Musica autoschediastike, published in 1601, he already distinguishes levels of rhythmic and contrapuntal complexity by differentiating between simple and ornamental forms of musical "syntax," commenting explicitly on the analogy with language: "In my opinion . . . there is a difference between what the ars musica prescribes for the (simple) syntax of consonances, and what is added to this syntax when the work of art is to shine forth in full brilliance. And if we reduce music to a more exact conceptual scale, we will be able to show unequivocally that there is only a slight difference between the nature of music and that of speech. For the great power of the art of speech does not lie in the simple succession of simple words, nor in the correct articulation of sentences, nor in simply combining sentences in always similar ways, but in the fact that speech reveals its power of expression through its pleasing ornamentation and through important words, and because emphatic words are included in the sentences. In the same way music offers the listener more than just a combination of pure consonances. It offers a text capable of moving the heart, made up of a mixture of perfect and imperfect consonances as well as dissonances."¹⁶

But it is in his Musica Poetica that Burmeister first arrives at a systematic classification of compositional types according to four generic categories (the genera styli) borrowed directly from rhetorical studies: 1) the genus humile (the rhetorician's "everyday language"), in which "the intervals follow one another in stepwise progression and the composition consists solely of a pure combination of consonances; 2) the genus grande (including in rhetoric, in addition to normal expressions, an abundance of metaphors, elaborations, etc.), in which "more distant intervals appear and many concealed dissonances are mixed among the consonances"; 3) the genus mediocre, which "assumes a middle position between the humile and the grande"; and 4) the genus mixtum, which "encompasses the previous three in a mixture, although not in the same moment but alternately according to the nature of the text."¹⁷ Burmeister notes that every composer develops his own style, and he gives names of representative musicians for each of the four types. The student of composition, however, is instructed "to begin with the imitation of those composers who have written all of their works in the genus humile and then progress by step [per gradus!] to the higher genera."¹⁸

Burmeister's distinction between compositional genera according to levels of technical complexity and expressive power already suggests certain parallels with Schenker's theory of Schichten, or compositional layers. Yet although Burmeister notes, in turning to a discussion of the different figures, that musical figures, like those of rhetoric, depart from the simple forms of composition, he does not posit a specific musical correlation between the two, so that one is shown to be logically derived from the other, thus offering a true analogue to Schenker's notions of compositional foreground and background.

The first theorist to deal with this correlation is Christoph Bernhard, whose theoretical works appeared some fifty years later.¹⁹ Although Bernhard does not mention Burmeister, he would seem to have been greatly influenced by the earlier theorist. This is apparent first in his emphasis on parallels between music and language, as when he remarks that "the art of music has attained such a height in our day, due to the large number of figures, as seen especially in the recently founded and, up to now, ever more embellished recitative style, that it may be compared to a rhetoric."²⁰

Bernhard's classification of various types of counterpoint reveals a more specific relationship with Burmeister. In his principal theoretical work, the Tractatus compositionis augmentatus, he makes several distinctions, the first and most basic of which is between contrapunctus aequalis (or simplex), in which "all notes sound together with the same durations" and "which consists solely of consonances," and contrapunctus inaequalis (or diminutus), in which "one voice has slower notes while the other has faster" and "which consists of both consonances and dissonances." This latter type is then further divided into contrapunctus gravis (or stylus antiquus), which "consists of relatively slow notes and makes use of only a few types of dissonances," and contrapunctus luxurians (or stylus modernus), which "consists of relatively fast notes, strange leaps to move the affects and more types of dissonances (or more figurae melopoeticae, which others call licentiae), and whose melodies are more in agreement with the text." Finally, contrapunctus luxurians is itself divided into contrapunctus communis and contrapunctus theatralis, "the first of which is used everywhere, while the second is used mostly in the theater."²¹

The significance of these distinctions, which are made near the opening of the treatise, becomes apparent when Bernhard discusses musical figures, which are now—significantly—defined as a purely musical phenomenon: "a certain way of using dissonances, so that they become not only inoffensive but quite agreeable."²² Bernhard notes that only certain types of figures are appropriate for a given style (although each more complex style encompasses all those belonging to simpler ones); and he introduces them according to the contrapuntal category to which they belong, starting with the simplest and leading only gradually to the most complex.

What is more important from the present point of view, however, is that Bernhard no longer justifies examples of freer treatment of dissonance merely by assigning them to a particular stylistic category. He shows that a logical musical connection exists between even the most complex figures and more basic contrapuntal forms underlying them. Thus, except for those figures belonging to the stylus gravis (i.e., the most basic type), he not only gives musical examples for particular figures (as did Burmeister) but follows these with a simpler version introduced by a comment to the effect that "it would be like this in the simpler style."

In respect to Schenker, the most interesting figures discussed by Bernhard are those of the stylus theatralis, as it is here that the departures from "simple composition" are most pronounced. The Heterolepsis, which—again significantly—does not appear in rhetoric but is a purely musical figure of Bernhard's own invention, may serve as an example. Bernhard's definition, with two of his five examples, reads: "Heterolepsis is the taking up of another voice and exists in two forms. First, if after a consonance I leap to a dissonance, so that a passing motion can be formed with another voice:

 

Ex. V.

The other voices would thus be:

 

Second, if in conjunction with a syncopated bass the top voice, having formed a fourth [i.e., the above the bass E in m. 2] falls a third instead of rising a second":²³

 

Ex. VI.

should be thus:

 

Bernhard's reductions for these examples reveal a pattern of "strict" voice leading, which underlies the surface elaborations and assures their comprehensibility. As he says elsewhere, in discussing the most complex figures: "Such figures . . . have the old masters as their basis, and what cannot be excused through them should be weeded out of the composition as an abomination."²⁴ Here, then, we find a conception of background and foreground in all respects except name.²⁵ And in his last theoretical work, the Ausführlicher Bericht vom Gebrauche der Con- und Dissonantien, Bernhard even approaches the latter when he classifies all figures according to two basic types: the figurae fundamentales (all of which belong to the simplest style, the stylus gravis) and the figurae superficiales (belonging to the more complex styles).²⁶ With the notions of surface and foundation we are very close to Schenker indeed.

The influence of Bernhard, and of the rhetorical tradition in general, is evident in fundamental bass theory throughout the first half of the eighteenth century, where the problem of resolving the apparent contradictions between the strict contrapuntal procedures of the stile antico and the voice-leading liberties introduced within the harmonic framework of the stile moderno remains a central issue. Thus Johann Gottfried Walther, in his Praecepta der Musicalischen Composition (1708), retains Bernhard's distinction between fundamental and superficial figures and offers simple versions of the latter in strict style; and Johann David Heinichen, who devotes an extended section of his Der Generalbass in der Composition (1728) to dissonances in the "theatrical style," follows his examples of these freer dissonances with reductions consisting of what he calls the "fundamental notes," which "show the correctness of the passage":²⁷

 

Ex. VII.

Fundamental notes:

 

Even more suggestive is the following example from Johann Adolph Scheibe's Compendium Musices (c. 1730), in which the polyphonic implications of an ornamental top voice are made explicit in the reduction:²⁸

 

Ex. VIII.

 

The most systematic of these treatises is the Handleitung zur Variation (1706) of Friedrich Erhardt Niedt, which undertakes to illustrate how a complex and extended compositional surface can be derived from a simple fundamental bass line. Niedt begins with basic two-note intervallic models and their elaborations, similar to those found in sixteenth century manuals on diminution, and then proceeds to combine these with one another in order to form florid bass lines for complete phrases. Finally, he illustrates how various short pieces (preludes, allemandes, etc.) can be developed through variational procedures from a single fundamental bass. In the introduction to his revised version of Niedt's Handleitung, Johann Mattheson, himself one of the most important theorists in the rhetorical tradition, remarks that this process involves the "transformation of certain slow bass notes into shorter notes in various ways while maintaining their linear progression, so that the phrase retains its fundamental essence yet is subjected to diminution and is divided and articulated so as to receive more life, strength, spirit and ornament."²⁹

Two additional theorists in the musica poetica tradition should be discussed at least briefly. The first is Moritz Johann Vogt, whose Conclave thesauris magnae artis musicae appeared in 1719.³⁰ In a section devoted to musical figures, Vogt discusses the derivation of a particular figure—the so-called fuga, or imitative passage—from a simpler underlying structure, the phantasia simplex, and offers the following example:

 

Ex. IX.

 

In summary the author notes: "For fugues, the best method is to derive the theme from a phantasia simplex. . . . And therefore, all fugal themes can be reduced to such regulated progressions of phantasias."³¹ The implicit reciprocal relationship between elaboration—or in Schenker's terms, Auskomponierung—and reduction is here explicitly stated for the first time.

Finally, at the end of the period during which the musical-rhetorical relationship played a dominant role in music theory, Johann Mikolaus Forkel devotes an extended section of the long introduction to his Allgemeine Geschichte der Musik to the rhetorical foundations of music. In speaking of the "arrangement of musical ideas with regard to the total scope of the composition," Forkel introduces the concept of Zergliederung, the "dissection" of a principal idea in order "to show it from all of its various sides and points of view":

 

Ex. X.

 

Commenting on this example, he remarks that "the two measures are exactly the same in their basic meaning; yet they so modify the idea that the ear is more inclined to understand the second as a new idea rather than the same one."³²

Another technique discussed by Forkel is Umschreibung, whereby a short phrase "acquires through paraphrase" a more extended form, which "gathers together its most important characteristics and thereby tries to make it clearer":³³

 

Ex. XI.

 

Finally, Forkel examines what he calls the "individualization of more general ideas": "Almost every chord can serve to express a general impression; but one rarely notes the differences lying within it if one is not practiced in the language of music, because taken together they all make up only a single totality." Commenting on the first part of the following example, he remarks that "every practiced ear experiences its meaning just as one who knows a language understands abstract expressions; but the liveliness and beauty of the expression gain enormously when it is also occasionally dissolved and individualized."³⁴

 

Ex. XII.

 

Although the examples offered by Forkel are admittedly very simple, they represent striking conceptual prototypes for the transformational techniques later introduced by Schenker.

We can conclude by summarizing those aspects of musica poetica and Figurenlehre that anticipate Schenker: the distinction between a simpler, ordinary manner of writing and a more elaborate, poetic one; the notion that the ordinary way is "fundamental" while the differentiated one, which is based on the former, is "superficial" (in the sense of forming the surface of a more basic foundation); that the surface is individualized while the foundation is abstract; that composition in some sense involves a derivation of the surface from the foundation; and finally, that the surface is analytically reducible to the foundation.³⁵

 

IV

Despite these correspondences, it is nevertheless notable that no attempt was made by any of the musica poetica theorists to deal with matters of large-scale pitch organization. One of the most remarkable aspects of Schenker, on the other hand, is the way he regards pitches within an all-encompassing framework: his reductions are based upon, and form part of, a synoptic view of tonality through which the individual pitch events are interpreted as ramifications and expressions of a single underlying source.

This brings us to the third general area of historical importance for Schenker, the theory of functional tonality. At first thought the differences between functional theory, in whichever of its various guises, and Schenker seem more apparent than the similarities; and Schenker himself considered his approach to be in direct opposition to functional thought. Yet on further reflection it appears likely that Schenker's ideas could not have developed at all—would, indeed, have been literally "unthinkable"—without the prior development of certain basic concepts of functionalism. There are many important parallels between the two approaches, not so much in regard to specific solutions for theoretical problems, however, as to the identification and formulation of these problems and, above all, to the whole question of what properly constitutes the requirements of a unified theory of pitch structure.

We may start at the beginning, with Rameau, who was the first theorist to attempt to reduce the entire range of tonal pitch possibilities to a single, all-embracing principle, which he considered to be harmonic in nature and to reside in the mathematical characteristics of the individual tone. The belief that music and all of its "laws" are reducible to the properties of a single tone not only supplies the conceptual foundation for all later functional theory, it represents the basic assumption underlying Schenker's concept of the Ursatz. Consider the following quotations from Rameau's Traité de l'harmonie (whose title, we should remember, continues: réduite à ses principes naturels), which, with only minor modifications, might well stem from Schenker himself: "The principle of harmony does not subsist only in the perfect chord, from which is formed the chord of the seventh, but more precisely in the lowest tone of these two chords, which is, so to speak, the harmonic center to which all the other tones must be related."³⁶ "How marvelous is this principle in its simplicity! However many chords, many different melodies, this infinite diversity, these expressions so beautiful and correct, such well-formed sentiments, all this arises from two or three intervals disposed by thirds, the principle of which is founded in one tone. . . ."³⁷ Moreover, Rameau's insistence on the fundamental nature of the dominant progression clearly anticipates Schenker: his remark that when the dominant progresses to the tonic, "it seems as if the fifth returns to its source,"³⁸ is, if one be permitted a chronological inversion, a profoundly Schenkerian observation.

One of the most interesting concepts introduced by Rameau to explain the diversity of actual music in terms of a single abstract principle is that of Sous-entendre—the notion that a tone or chord can be "understood" to be present solely through implication.³⁹ The assumption that one chord may be functionally represented by another, and thus be functionally present even when it is not actually sounding, occurs repeatedly after Rameau and ultimately finds its most elaborate expression in Riemann's concept of Klangvertretung. It then reappears, radically altered to fit its new surroundings, as one of the fundamental tenets of Schenker's system. The need for such an assumption resides in the belief, shared by all of these musical thinkers, that a theoretically consistent justification must be found for all apparent compositional "deviations" from the basic principle of the underlying theoretical system. As Rameau asserts, exceptions—or "licenses"—derive their meaning "only from the perfect chord, to which [they] are added, and without which [they] cannot subsist."⁴⁰ Indeed, Rameau explicitly defines such licenses as "everything that deviates from this natural principle," and then adds the all-important proviso: "we have no license to propose that is not derived from this source (this 'principle')."⁴¹

Rameau's principle—that the complex surface of any actual musical composition is reducible to (and thus explainable in terms of) a simple triadic structure in root position expressed by and contained within the basse fondamentale⁴²—has its roots in Zarlino and is already at least implicit in figured bass theory, which posits a latent structure of triadic voice-leading established by the harmonic implications of the bass line and notated (in an appropriately "idealized"—i.e., generalized and abstract—form) by numerical figures. Only after Rameau's more explicit formulation, however, is a systematic codification of harmonic reduction made possible, enabling it to become a standard part of functional theory.

A particularly clear exposition is found in Georg Joseph Vogler's Handbuch zur Harmonielehre. Vogler is the first theorist to speak of a Reduktionsystem, which he defines as "that system of instruction in which one breaks down all structures to the one simple form of the triad and assures oneself that in the case of every harmony that occurs, no matter how complicated it may appear, a principal chord with a third and fifth can be discovered, and that the third and fifth form a fundamental chord [i.e., a chord in root position ]."⁴³ Vogler also adds an important contribution of his own to functional theory in introducing a new method of indicating harmonic relationships within the key system by means of Roman numerals. In his words, these numerals are employed "in order to distinguish the principal chords; for example, if in a given case I declare C to be the principal chord, and want to indicate which scale degree C is—whether, for example, C is the first and principal tone, or the fourth of G, or the fifth of F, etc."⁴⁴

This means that Vogler designates each chord not only in terms of its own internal structure but as a representation of a particular functional relationship, and thus in a certain sense as a representative of the tonic triad itself. In Rameau's terms (for this can be considered an extension of his concept of Sousentendre), the tonic triad is "understood" or "implied" by all the chords related to it through the key system. This idea, so important for Schenker, is further developed after Vogler (for example, in Gottfried Weber and François-Joseph Fétis) and eventually reaches its most systematic formulation in the theoretical works of Hugo Riemann.

Before turning to Riemann, the nineteenth century Austrian theorist Simon Sechter should be considered. Sechter, one of the most influential theorists of his time, is of particular interest in that he taught Anton Bruckner, with whom Schenker studied and through whom he acquired a faithful transmission of the earlier theorist's ideas.⁴⁵ Although Schenker later came to regard Bruckner (and thus presumably Sechter) with decided reservations, a study of Sechter's Grundsätze der musikalischen Komposition, published in three volumes in 1853-54 and certainly well-known to Schenker, indicates that he supplied at least the basic outlines for Schenker's general theoretical orientation.

In the Grundsätze Sechter adheres to many of Rameau's concepts, such as the fifth as sole fundamental progression and the idea of "implied" harmonies (which he calls Stellvertreter, anticipating Riemann). But he contributes an important innovation in expanding the meaning of the term Stufe. Vogler, for example, defined Stufen in his Handbuch as the "rising and falling tones that form a scale,"⁴⁶ a definition followed by Sechter at the beginning of his treatise when he uses the term simply to designate scale degrees. Later, however, in discussing the chromatic elaboration of a simple harmonic progression, he remarks: "The meaning of chromaticism acquires a great extension if, with the exception of the seventh degree, one considers every scale degree as a tonic, letting the fundamentals occur in such an order that each one can be independent."⁴⁷ This means that for Sechter any given major or minor chord can temporarily take on the meaning of a tonic without relinquishing its secondary position within the principal key.

In order to illustrate this Sechter takes a simple succession in four-part harmony (referred to as the Thema) and shows how its diatonic quality can be altered through various stages of "transformation." Example XIII presents the original succession and Sechter's four transformations.

 

Ex. XIII.

 

Of the first elaboration he notes that "each chord becomes a tonic for the duration of a measure, during which its own dominant is heard, which as a secondary harmony, however, is not to be considered as part of the actual fundamental progression in the C major scale."⁴⁸ After similar descriptions for the second and third transformations, Sechter notes of the last: "Without first presenting all possible transformations, one can finally introduce in each measure all the fundamentals of the original Thema as secondary fundamentals . . . Thus in this phrase there are secondary scales, all of which are related to the principal scale of C major."⁴⁹

This enlarged concept of Stufe, taken up and developed some fifty years later by Schenker, radically redefines the sense of scale degree. The latter no longer represents merely a tone, or even the chord whose fundamental that tone forms. It becomes "idealized" as a Source of compositional expansion, a temporary nucleus for musical prolongation.

The third volume of Sechter's Grundsätze contains a section on "strict Composition with a brief consideration of free composition." The German title—Vom strengen Satz, mit kurzer Andeutungen des freien Satzes—immediately recalls Schenker. Moreover, like Schenker, Sechter conceives of strict composition as species counterpoint confined to the major and minor scales. Free composition is touched upon only briefly, and for the most part Sechter limits himself to giving examples of short passages in both strict and free styles (the latter being little more than simple variations of the former). He does make clear, however, that he considers the "deviations" (Abweichungen) of free composition to be grounded in strict composition: "those who are accustomed to free composition and thus believe that they are able to turn up their noses at the rules of strict composition will see, after having thoroughly examined these side-by-side comparisons of simple passages with their elaborations, that strict composition must supply the foundation if free composition is to amount to anything at all."⁵⁰

In concluding his remarks on free composition, Sechter's language again recalls Schenker: "For those who have read the preceding it would now be well to study the compositions of Sebastian Bach, of Friedrich Händel, of Mozart and of Joseph and Michael Haydn, after having learned the foundation on which these masters constructed. In recent times one can find many worthy examples of compositions in which strict and free composition are so mixed that free composition gets the upper hand and strict composition only glimmers through. But there are also many compositions in which free composition is so dominating that strict composition is forced into the background. For an understanding of this final type it would be unnecessary to give instructions, for on the one hand there is little change in the fundamental and on the other there is no appropriate order in the succession of the fundamentals themselves."⁵¹ The parallels with Schenker are remarkable, certainly too close to be coincidental. Only Sechter's final sentence (and that only in part) indicates how far we still are from the world of Der freie Satz.

This brings us to Riemann, the last of the important theorists in the functional tradition and the final one to be considered in this survey. In Riemann one finds the beginnings of a view of large-scale linear structure bearing distinct resemblances to that of Schenker. His concept of "dead intervals" provides an example. These are intervals Riemann considers to be only "apparent" (as opposed to "real"), since the notes involved, though they appear in the same instrumental voice, do not belong to the same structural lines (or as he phrases it, to the same "motivic unit"). Concerning the opening of the Trio of the second movement of Beethoven's Piano Sonata, Op. 26, he observes: "We can . . . assume, to be sure, that the individual rising intervals [i.e., A-flat—f, A-flat—E-flat, etc.] can be understood as motives in the narrower sense, between which dead intervals are inserted (between the higher note and the lower one that begins the next motive); but a motivic structure of a higher order occurs, which follows the melodic connections of the boundary tones F—E-flat, G-flat—F:

 

Ex. XIV.

 

"Here we clearly follow the motion from F to E-flat while the punctuated second voice remains on A-flat; and since we also understand the continuation from G-flat to F as corresponding to this, we again eliminate the interior interval E-flat—G-flat as dead."⁵²

Similarly, in discussing the opening measures of the E major Prelude from the Second Book of the Well-Tempered Clavier, Riemann notes that the piece should be understood "not so much as a combination of the rhythms of the three individual voices as the animation of a rigid three-voice structure through a single voice moving in similar note values."⁵³

 

Ex. XV.

 

A particularly suggestive example is found in Riemann's linear analysis of Bach's E-flat minor Prelude from the First Book. After speaking of the "imposing correspondences between the general character of the key and the specific content of Bach's conception" and the "great and noble feeling that is expressed through the long spans of the melody," Riemann presents "the structure of the piece in its fundamental spans":⁵⁴

 

Ex. XVI.

 

This last reduction moved Schenker to comment: "Do I need to show that these basic spans have nothing to do with my Urlinie?"⁵⁵ Schenker's rhetorical question is perhaps justified in view of the particular form in which the spans appear, which, one must grant, has little in common with the elegant structures revealed by his own highly developed system of linear reduction. On the other hand, the very notion of mapping the large-scale linear progression of a composition in order to show how the melody manifests "the general character of the key" marks a crucial step in the direction of that system. For up to a certain point, at least, Riemann is doing something closely analogous to Schenker: isolating particular notes from a larger melodic continuum for the purpose of revealing a more fundamental underlying linear structure characterized by stepwise progression. (Riemann's term for these notes is Melodiespitzen, or "melodic peaks"; but significantly, they do not necessarily represent the highest pitches in the "foreground.") Admittedly, no more than four notes participate in any given linear progression (e.g., the F-flat to C-flat span in mm. 5-8); yet the general concept of span (for which Riemann uses the same word as Schenker: Zug) is clearly adumbrated.

Riemann, moreover, introduces the concept for a similar reason: the belief that all significant features of a tonal composition must express the content of a single underlying key. It is, finally, this generalized conception of tonality, rather than any particular analytical technique ensuing from it, that forms his most important contribution to the theoretical atmosphere in which Schenker developed—an atmosphere which Riemann shaped, at least in the German-speaking countries, to a remarkable extent.

Already in a relatively early lecture entitled "The Nature of Harmony,"⁵⁶ which he delivered in 1882, Riemann observed:

Musical consonance in the word's strictest sense—that is, a chord capable of closure, requiring no further motion—is solely and uniquely the tonic chord. . . . If I think of the G major chord in relation to the C major tonality, then I think of it as the upper fifth of C major; that is, the C major chord itself is encompassed by the conception as that chord that determines the meaning of the G major chord as something that is a departure from it. . . . And exactly the same holds for the F major chord and for every other chord of the key. This modern concept of key, or as one says to distinguish it from the old concept of key, of tonality, is not, however, tied to the scale; other chords that have notes foreign to the scale can also be understood in relation to the tonic and through this acquire their own particular meaning. . . .

The relationship of tone connections even allows for a further extension, namely that of the relationship of keys to one another. Just as the principal tone of a chord is related to its secondary tones (the fifth and third and more distantly related tones), so is the principal chord related to the secondary chords (the fifth-related chords, third-related chords, etc.), and the principal key to the secondary keys (the fifth-related keys, third-related keys, etc.). In a piece in C major, the key of G major plays the same or similar role as does the G major chord in a cadence in C major, or as the tone G in a C major arpeggio, or the tones B and D in the C major scale—that is, it has the effect of a dissonance, does not exist independently; rather, its justification is conditional and has only temporary validity.⁵⁷

Riemann, even at this early date, views tonality as a hierarchical configuration in which the same relationships are preserved at different structural levels: given the C major tonality, the key of G, the chord on G and the tone G all have the same functional meaning; and this meaning is acquired solely through their relationship to the tonic chord. Put in Schenker's terms, this means that all tonal motion is derived from, and thus analytically reducible to, a single background triad. Of course the particular formal model developed by Schenker to elucidate precisely how this derivation—or, conversely, reduction—is accomplished is not even hinted at in Riemann's remarks; yet its philosophical and conceptual foundation could hardly be more succinctly stated.

 

V

The question of the degree to which Schenker was aware of these prior developments is not without interest; and as has been noted at several points above, there is some indication that he was at least partially cognizant of them. It would be mistaken, however, to contend that he in any way consciously adapted these earlier conceptions to fit his own more general theoretical framework. They are too completely integrated within, and transformed by, the particularities of his own personal theoretical vision for that to appear even remotely plausible.

Yet if the relationship between Schenker and these historical precedents is, on the one hand, less specific than such a view would indicate, it is on the other much more fundamental. The material presented in the present paper indicates that reduction and prolongation are musical concepts deeply embedded within the Western musical tradition, and thus within the consciousness of its composers and theorists. The real point to be drawn from these investigations, then, is that Schenker's contribution is both more and less original than is generally assumed: less in that it does not offer a totally unprecedented way of looking at musical structure; but more in that it radically transforms inherited theoretical ideas, lending them a totally unforeseen new meaning and unexpected new life.

Schenker, then, has maintained the tradition in the very moment of altering it. (He thus conforms to a pattern already established by its most important composers and theorists.) It would be surprising if this were otherwise: however limited Schenker's view of the Western musical tradition might have been, there has surely been no theorist more devoted than he to its integrity and continuity.

In the closing remark of his 1882 lecture (see above), Riemann observes: "If it is possible to develop the theory of harmony that I have sketched here into a complete system, then the theory of harmony will become a true exercise in musical thinking that is directed both to the simplest matters and to the most complex. . . ."⁵⁸ Although there are probably few musicians today who would maintain that Riemann himself succeeded in realizing this goal, many may feel tempted to read his statement as a prophetic reference to the theoretical system later developed by Schenker. By extending not only the concept of tonality outlined in Riemann's lecture, but also those other techniques of reduction traced historically in this article, Schenker has provided us with an extraordinarily suggestive model for the nature of musical thought as manifested in common practice tonality.

---

¹See, for example, "Rameau oder Beethoven," Das Meisterwerk in der Musik, III (Munich, 1930), p. 17, where Schenker speaks of thorough bass theory ("the first victim of Rameau's theory") as the true basis for a "theory of voice leading." A comment in the introduction to Kontrapunkt, I (Stuttgart and Berlin, 1910), p. xxix, also indicates that Schenker had some awareness of the influence of earlier theoretical developments on his own approach: "When we see how Fux on the one hand and Rameau on the other almost simultaneously presented the world with their theories, the one a theory of voice leading and the other a theory of scale degree [Stufe], one recognizes a wink from fate that these two disciplines should be understood and treated independently of one another!"

²This is in contrast to Riemann, for example, that most historically oriented of theorists, who seems to have taken considerable pride in noting anticipations of his own ideas in earlier theoretical works.

³It is thus closely related to the Schenkerian concepts of prolongation and compositional unfolding (Auskomponierung).

⁴Cf. Willi Apel, Gregorian Chant (Bloomington, 1958), p. 246: "Some of the chants have melodies which, in spite of their seemingly free design, may be elaborate versions of a recitative. Such melodies, then, would be 'free' only at the level of variation or decoration technique, not in their thematic substance." Apel also speaks of the basic design of the psalmodic recitative as "the prototype of the Gregorian phrase, from the analytical and probably also from the historical point of view" (Ibid., pp. 249-50). For an example of the melody of a gradual verse that, "though florid, retains the essential form of simple psalmody," see The New Oxford History of Music, ed. Don Anselm Hughes (London, 1954), p. 120. Gustave Reese, Music in the Middle Ages (New York, 1940), contains an example of a gradual and an antiphon representing, respectively, florid and simple versions of the same "root melody" (p. 165). See also Thomas Forest Kelly, "Melodic Elaboration in Responsory Melismas," Journal of the American Musicological Society, XXVII (1974), pp. 461-74.

⁵Edmond de Coussemaker, Scriptorum de musica medii aevi (Paris, 1864-67; repr. Hildesheim, 1963), I, p. 336ff. Cf. Reese, op. cit., p. 281.

⁶For example, the anonymous Discantus Positio Vulgaris, the earliest treatise (c. 1230-40) to deal with mensural notation, defines all values of shorter duration than a long (as well as those greater than two longs) as ultra mensuram (Coussemaker, Scriptorum, I, p. 94). Cf. Reese, op. cit., p. 283, and Johannes Wolf, Geschichte der Mensural-Notation von 1250-1460 (Leipzig, 1904), p. 3.

⁷Coussemaker, Scriptorum, I, p. 123. See also Apel, The Notation of Polyphonic Music (Cambridge, 1942), p. 234ff.

⁸Coussemaker, Scriptorum, III, p. 62ff. This example offers perhaps the earliest instance of the pedagogical practice of ordering contrapuntal "species" according to degree of rhythmic complexity, initiating a development that then progresses through Paumann, Diruta, Banchieri and Zacconi (all of whom are important diminution theorists: see below) until it reaches its most comprehensive formulation in Fux's Gradus ad Parnassum. Thus the roots of diminution technique and species counterpoint (which Schenker felt provided the only adequate method of teaching voice leading: see Footnote 1 above) would seem to have a common conceptual, as well as historical, source.

⁹Das Locheimer Liederbuch nebst der Ars Organisandi von Conrad Paumann, ed. Friedrich Wilhelm Arnold (Leipzig, 1926; repr. Wiesbaden, 1969), pp. 177-224.

¹⁰Sylvestro Ganassi, Opera Intitulata Fontegara (Venice, 1535). There is a modern English edition, translated by Dorothy Swainson and edited by Hildemarie Peter (Berlin-Lichterfelde, 1959).

¹¹Hans Praesler, "Fundamentbuch von Hans von Constanz," Vierteljahrschrift für Musikwissenschaft, V (1899; facs. ed. Hildesheim, 1966), pp. 46-47. Buchner's treatise, which exists in a manuscript dated 1551, is given in its entirety (in the original Latin) within this article.

¹²Lodovico Zacconi, Prattica di Musica, I (Venice, 1592; facs. ed. Bologna, 1967), p. 63. Zacconi's use of the word "natural" in reference to the basic structure underlying a diminution (la parte come sta naturale) recurs in later theorists (e.g., Bernhard: see below) and anticipates Schenker's view of the triad as a Naturidee (in distinction to its unfolding, which is a Kunstidee).

¹³On diminution in general, see Max Kuhn, Die Verzierungs-Kunst in der Gesangs-Musik des 16.-17. Jahrhunderts (Leipzig, 1902; repr. Wiesbaden, 1969) and Ernst Ferand, Die Improvisation in der Musik (Zurich, 1938). The latter is especially valuable, as it offers considerable evidence for the close relationship and continuous mutual influence of improvised and written forms of diminution in Western music.

¹⁴A brief discussion of sixteenth century diminution theory in a Schenkerian context appears in Allen Forte, The Compositional Matrix (New York, 1961), pp. 16-17. Schenker himself frequently employs the word "diminution" and devotes a lengthy section of Der freie Satz (2nd. ed., Vienna, 1956), pp. 145-65, to the subject. The close ties between diminution and improvisation are examined in his article "Die Kunst der Improvisation," Das Meisterwerk in der Musik, I (Munich, 1925), pp. 9-40. Here Schenker's view of music theory as a "practical" discipline closely aligned to actual music making is stated especially clearly. It is above all in this light that Schenker's reaction against the "abstract" harmonic and contrapuntal theories of the nineteenth and twentieth centuries should be understood.

¹⁵For a useful survey, see Hans-Heinrich Unger, Die Beziehungen zwischen Musik und Rhetorik im. 16.-18. Jahrhundert (Würzburg, 1941; repr. Hildeshaven, 1969).

¹⁶Quoted in Martin Ruhnke, Joachim Burmeister (Kassel and Basel, 1955), pp. 144-45.

¹⁷Burmeister, Musica Poetica (Rostock, 1606; facs. ed. Kassel, 1955), p. 75. Quoted in Ruhnke, p. 106. Burmeister also distinguishes three contrapuntal genera, which have certain correspondences with these stylistic differences: the genus simplex (note against note), genus fractum (notes with different values) and genus coloratum (a predominance of short durations). (Musica Poetica, p. 71; Ruhnke, p. 162.) Burmeister's debt to rhetorical studies, especially those of Lucas Lossius (1510-82), is discussed in Ruhnke, passim. See also Heinz Brandes, Studien zur musikalischen Figurenlehre im 16. Jahrhundert (Berlin, 1935), pp. 24-29.

¹⁸Musica Poetica, p. 75; Ruhnke, p. 107.

¹⁹All of Bernhard's theoretical works are available in a modern edition in Die Kompositionslehre Heinrich Schützens in der Fassung seines Schülers Christoph Bernhard, ed. Joseph Müller-Blattau (2nd. ed., Kassel, 1963); and they appear in English (trans. by Walter Hilse) in Music Forum, III (1973), pp. 13-179. References will be given to both the German and English editions, although translations are by the present author. See also Helmut Federhofer, "Der strenge und freie Satz und sein Verhältnis zur Kompositionslehre von Heinrich Schütz in der Fassung seines Schülers Christoph Bernhard," Beiträge zur musikalischen Gestaltanalyse (Graz, Innsbruck, Vienna, 1950), pp. 61-77.

²⁰Müller-Blattau, p. 147; Music Forum, pp. 90-91.

²¹Müller-Blattau, pp. 42-43; Music Forum, pp. 34-35.

²²Müller-Blattau, p. 63; Music Forum, p. 77.

²³Müller-Blattau, pp. 87-88; Music Forum, pp. 118-19. Bernhard's word for "taking up" or "grasping" the inner voice is Ergreiffung, the root of which appears with a similar meaning in Schenker's Übergreifen and Untergreifen (Cf. Schenker, Der freie Satz, pp. 85-87.)

²⁴Müller-Blattau, p. 147; Music Forum, p. 91.

²⁵Although not, to be sure, in a specifically Schenkerian form: In the reduction to Example VI, for instance, Schenker would presumably wish to show, in addition to the -B middle-voice progression stressed by Bernhard, the composing out of the top voice E by means of an octave transfer, resulting in an 8-7 exchange with the bass and a resolution to in the final measure. This might be expressed graphically as follows (in a form suggested by Professor David W. Beach of the Eastman School of Music in a letter to the author):

 

 

²⁶Müller-Blattau, p. 144; Music Forum, p. 77.

²⁷Heinichen, Der Generalbass in der Komposition (Dresden, 1728; facs. ed. Hildesheim and New York, 1969), pp. 591-92.

²⁸This work, though not published in Scheibe's lifetime, appears as an appendix to Peter Benary, Die deutsche Kompositionslehre des 18. Jahrhunderts (Leipzig, 1961).

²⁹Niedt, Handleitung zur Variation, 2nd. ed., rev. and ed. by Johann Mattheson (Hamburg, 1721), p. 3.

³⁰Vogt has received some notoriety of late for his suggestion that, when inspiration fails, one can compose by bending horseshoe nails in different shapes so as to make them distinguishable from one another. After associating a conventional musical figure with each shape, one tosses the nails in the air so that the order in which they fall can determine the order of the figures in a composition. Cf. Arnold Schering, "Die Lehre von der musikalischen Findekunst 'ars inveniendi,'" Das Symbol in der Musik (Leipzig, 1941), pp. 10-11; Helmut Kirchmeyer, "Vom historischen Wesen einer rationalistischen Musik," Die Reihe VIII (1962), pp. 20-21; and Konrad Liebe-Boehmer, Zur Theorie der offenen Form in der neuen Musik (Cologne, 1966), p. 42. Vogt also recommends alcohol as a possible source of creative motivation.

³¹Vogt, Conclave thesauri magnae artis musicae (Vetero-Prague, 1719), pp. 155-56.

³²Forkel, Allgemeine Geschichte der Musik, I (Leipzig, 1788; facs. ed. Graz, 1967), pp. 51-52.

³³Ibid., p. 52.

³⁴Ibid., pp. 52-53.

³⁵That Schenker had some awareness of the musica poetica tradition, or at least was inclined to think of music in similar terms, is suggested by his comment on Beethoven's departure from normal practice in omitting the dominant before the recapitulation of his Op. 109: "Since tonal language has a completely analogous syntax to verbal language . . . one could accordingly characterize the analogy here with 'Anakoluth' (literally: a failure to proceed—namely, with an expected continuation)." Erläuterungsausgabe der letzten Sonaten von Beethoven, Op. 109, rev. and ed. by Oswald Jonas (Vienna, 1971), p. 33.

³⁶Jean-Philippe Rameau, Traité de l'harmonie (Paris, 1722), p. 127. English version: Treatise on Harmony, trans. Philip Gossett (New York, 1971), p. 141. References will be given to both of these editions, although the translations are by the present author.

³⁷Traité, p, 128; Treatise, p. 142.

³⁸Traité, p. 129; Treatise, p. 143. See also Rameau, Nouveau Système de musique théorique (Paris, 1726), p. 30.

³⁹Traité, p. 9; Treatise, p. 12.

⁴⁰Traité, p. 109; Treatise, p. 123.

⁴¹Traité, p. 111; Treatise, p. 124.

⁴²Rameau actually uses the word réduire in a specifically harmonic context in his Génération harmonique (Paris, 1737), p. 37, to refer to the placing of the notes of a harmony in the closest possible position—i.e., within a single octave.

⁴³Vogler, Handbuch zur Harmonielehre (Prague, 1802), p. 6.

⁴⁴Ibid., p. 12.

⁴⁵On Bruckner's regard for Sechter, with whom he studied for over five years and whose "instruction formed the basis for all his future works," see Anton Bruckner, Vorlesungen über Harmonielehre und Kontrapunkt an der Universität Wien, ed. Ernst Schwanzara (Vienna, 1950), p. 21 et al. See also Leopold Nowak, "Ein Doppelautograph Sechter-Bruchner" in Friedrich Wilhelm Riedel and Hubert Unverricht, eds., Symbolae Historiae Musicae. Hellmut Federhofer zum 60. Geburtstag (Mainz, 1971), pp. 252-59.

⁴⁶Vogler, Handbuch, p. 7.

⁴⁷Sechter, Grundzüge der musikalischen Komposition (Vienna, 1853-54), I, p. 157.

⁴⁸Ibid., p. 158. The musical quality of Example XIII, which is in any event not germane to our present concern, can be passed over in silence. It is nevertheless worth mentioning that Sechter was himself a composer, whose practical interest in such "techniques of transformation" is attested to by his having written a set of 104 (!) variations on a tune consisting of 104 bars.

⁴⁹Ibid., pp. 159-60. Looking at this process from the opposite side, Sechter observes (in a sentence that might have been written by Schenker): "A diatonic structure must lie at the foundation of every chromatic structure."

⁵⁰Ibid., III, p. 155.

⁵¹Ibid., p. 160. Sechter's word for "background" is Hintergrund, the same as Schenker's. Another linguistic correspondence is found in Sechter's Veränderung (for "transformation"), a close relative of Schenker's Verwandlung (as in Verwandlungschichten).

⁵²Riemann, System der musikalischen Rhythmik und Metrik (Leipzig, 1903), p. 23. See also idem, Die Elemente der musikalischen Aesthetik (Berlin and Stuttgart, 1900), pp. 40-42.

⁵³Riemann, System, pp. 178-79.

⁵⁴Riemann, Katechismus der Fugen-Komposition (Leipzig, 1890-1904), I, p. 58. Riemann's harmonic analysis, presented symbolically under the staff containing his "basic spans," has been omitted in the example.

⁵⁵Schenker, "Joh. Seb. Bach: Wohltemperiertes Klavier, Band I. Präludium Es-Moll," Der Tonwille, I (1921), p. 45.

⁵⁶Published in Sammlung musikalischer Vorträge, IV, ed. Paul Graf Waldersee (Leipzig, 1882), pp. 159-90.

⁵⁷Ibid., pp. 187-88.

⁵⁸Ibid., p. 190.