UFPr Arts Department Electronic Musicological Review Vol. 2.1/October 1997 Home
SCHOENBERG ON TONAL FUNCTION
This essay discusses the notion of tonal function in schoenbergian theory. Related to this subject are the concepts of transference of function by imitation of a tonal model, transformation of chords, and a special category of chords, the vagrant chords, which are the best examples of multiple meaning, i. e., multiple function.
Tonal Function - General Concepts
Rarely found in harmony text books, the entry "tonal function" is subject of many discussions. The texts by Arnold Schoenberg are no exception. Barely treated in his "theoretical oeuvre", the notion of tonal function for Schoenberg, is the subject of this essay. Before we proceed to Schoenberg's concepts on this subject, it is necessary to make some general considerations about tonal function.
The term "tonal function," normally employed in the sense of "harmonic function," is far from being clearly defined and its use has been vague. Basically, function means harmonic meaning or action, and both terms heve been used differently (1). For example, harmonic meaning or tonal function might be used as a scalar degree and its variations, used as a root of different chords (2); or even it can be associated to tendencies of individual pitches of a chord (3).
The most frequent use of the term function has been that of relating the harmonic meaning of an element capable of expressing a tonality with a tonal center, a tonic. The main question here concerns the identification of these elements and the way that they express a tonality.
Generally, two distinct but complementary theories have been dealing with the subject of tonal function. The first refers to the traditional theory of the fundamental bass, an heritage of the XVIIIth and XIXth centuries theories, which considers chords reductions to its root position. These chords roots are marked with Roman numerals and they are related in this way to a tonic. The second, Hugo Riemann's "functional theory," tries to reduce all chords functions of a tonality to three main functions: T, S, D. The distinction between both theories has been presented in the following manner: the first, being melodic, for it considers degrees of a scale to define tonal function; the second, harmonic by considering chords to define it (4).
Schoenberg's notion of tonal function, as discussed below, comprises both trends, and can be considered a complete theory on the subject. Firstly, he considers melodic elements as capable of expressing a tonal function; secondly, he introduces the concept of transference of function of a chord to another chord, implying that a chord has a tonal function, not depending on any specific pitch, such as the root, to express a function; and thirdly, introducing the concept of multiple meaning, the function of a chord is always established depending on its context.
Tonal Function for Schoenberg
In schoenbergian theory, tonal function depends on the relationship among the elements which express a tonality. For example, there is no function in a succession of chords that does not express a tonality. In this way, Schoenberg distinguishes a progression of chords from a succession of chords. The former expresses a tonality, either by establishing it or by contradicting it, while a succession, does not express a tonality satisfactorily:
A succession is aimless; a progression aims for a definite goal...A progression has the function of establishing or contradicting a tonality. The combination of harmonies of which a progression consists depends on its purpose - whether it is establishment, modulation, transition, contrast, or reaffirmation. A succession of chords may be functionless, neither expressing an unmistakable tonality nor requiring a definite continuation. (5)
In a general sense, tonal function for Schoenberg presents two aspects: one of a specific character and the other of a general character. The first is concerned with specific elements which express a tonality, that is, individual pitches and chords. These specific elements do not acquire the functional character by themselves, but they depend on the context where they are used and they will have its specific tonal function established by its relationship with other elements present in the same context, thus, this function is indeterminate. Schoenberg explains that "any given chord can have diverse functions, corresponding to its various tendencies, hence, that it is not unequivocal, and that its meaning is established only by its environment"(6).
The general function refers to tonal functions that express a tonality by affirming or contradicting it through the specific elements. These are: the centripetal function, which establishes a tonality; and a centrifugal function, which contradicts a tonality: "the centripetal function of progressions is exerted by stopping centrifugal tendencies, i. e., by establishing a tonality through the conquest of its contradictory elements" (7).
The concept of tonal function for Schoenberg is related to the expression of a tonality. Many theoreticians have not considered in depth the complexity of this concept. Schoenberg's idea of tonal function is based on the principle that a tonal work presents its tonal function, specific and general, related to a central tonality of the whole work represented by the principle of monotonality. According to this principle "every digression from the tonic is considered to be still within the tonality whether directly or indirectly, closely or remotely related...There is only one tonality in a piece" (8). Therefore, tonality is defined as "a network of functions, defined by scale degrees, each related to a single tonal center in a specific way. That relation gives to each pitch, chord, key area, to each event, its particular function" (9).
Basically, tonality and tonal function are well summarized by Patricia Carpenter:
Tonality for Schoenberg is not merely a certain collection of pitches of a scale, but more importantly, a kind of centricity. All pitches of a key-collection are related to a single tonal center, each in a specific way. The function of a single tone is signified by the degree of the scale it represents. The function of a chord depends upon its root, which is, in turn, the scalar degree upon which the chord is constructed. Tonality, then, is a set of functions of scalar degrees. (10)
The specific tonal function occurs in two elements: pitches and chords, each of them being related to a tonal center. The function of a pitch is defined by being a scalar degree related to a tonic of a tonality, or of a tonal region. Chords have their function expressed through their root, which is also related to a tonic. This relationship includes the notion of region which, in its turn, relates a fragment of a tonality to a tonal center, as if it were a scalar degree which includes the functions of chords and pitches. With these two elements, Schoenberg relates any segment of tonal music to a tonic, regardless of the degree of chromaticism or its degree of distance from the tonal center.
Schoenberg's most significant contribution to the function theory is his concept of monotonality and regions, expressed by the procedures of substitution and neutralization, which express a tonal unity in a piece of chromatic music. Through the principle of monotonality and the concept of regions, one can relate to a tonal center what was formerly understood as an independent tonality or others tonalities within a work.
The tonal relationship in monotonality is expressed in the Chart of Regions, a well designed schema which represents all tonal relationships within a tonality. Traditional relations are presented in the Chart, 1) vertically, the circle of fifths; and 2) horizontally, the relative and parallel minor relations. Schoenberg's principle of monotonality determines distances from the tonic. This classification is based on a common-note principle in which the regions with more common-notes with the tonic are considered "Direct and Close," and those with less "Indirect and Remote" or "Distant" (11).
Thus, diatonic or chromatic pitches can be assimilated within a tonality that ultimately will be considered as an extended tonality.
The possibility of expressing a tonality through characteristic pitches in major or minor makes those specific pitches acquire its functional character in schoenbergian theory. This functional character is accentuated when discussing the role and the inclusion of artificial leading tones through substitution and by the laws of the pivot tones (Wendepunktgesetze) which can be applied wherever false-relation occurs. Thus, one can deduce that individual pitches acquire the capacity to express a tonality through specific functions, acquiring, in this way, a functional character.
The characteristic pitches of a tonality have the primary function of expressing a tonality by distinguishing it from those tonalities that most resemble it. The function of degrees 4th and 7th of a scale is of extreme importance, and prevents a possible false interpretation of a tonality with its closest neighbors, the tonalities on both sides of the circle of fifths. These scale degrees, fourth and seventh, act preventing the expression of other tonal regions:
The 4th tone represents the subdominant region (IV and II), and thus prevents the interpretation of a segment as expressing a tonality a fifth above. The 7th tone represents the dominant region (V), and thus prevents the interpretation of a segment as expressing a tonality a fifth below. To achieve a cadential effect these two tones, or their substitutes, must appear immediately before the final tonic. (12)
These characteristic notes are not related to a specific chord, they present the same function in several chords and tonalities independently of their root. They must be considered as having a specific function by themselves.
The minor mode gives us another example of pitch function. Through the procedure of substitution and neutralization, artificial dominants can be created by transferring the functions of altered 6th and 7th degrees, raised or lowered. The laws of the pivot tones can be applied independently of a specific region, for they can convert temporarily any degree of a scale into a 6th or 7th degree with the purpose of neutralization (13). These pivot tones and the pitches in which these are neutralized constitute a model of pitch functions which is applied wherever false-relation occurs: "This means that every non-diatonic tone will be regarded either as the sixth or the seventh tone of an ascending or a descending minor scale" (14). Those pivot tones help to express a tonality. Neutralization, in its turn, or the lack of it, determines the function of a substitute note, either as a "chromatic substitute" or as a "quasi-diatonic substitute," the latter presenting a strong centrifugal function (15).
Example 1 illustrates the application of the functional character of 6th and 7th degrees of the minor mode (here they are marked with Arabic numerals, whereas crossed numerals indicate the introduction of substitute tones). In the last three measures of section a, Schoenberg indicates the pitches with their respectives numbers: g (7), f (6), e (5), and the substitute notes 6 and 7 which acquire a distinct functional character by expressing unequivocally the tonality of A minor through the 6th and 7th raised degrees. In the second measure of section b, Schoenberg indicates a change of the function of the 7th degree of B minor by introducing the substitute a#, 7, which acquires the function of a leading-tone.
example 1; from Structural Functions, example 36 a, c
The function of a chord is represented by its root and defined by its relationship to a tonal center. Schoenberg emphasizes the importance of the root of a chord over its other tones, affirming that "from the standpoint of structural functions only the root of the progression is decisive" (16).
The scalar degree of a root is indicated by Roman numerals and by traditional names of function, i. e., dominant, subdominant, etc. This terminology shows how a root of a chord relates to a tonic and how it expresses a tonality (17).
For instance, in C major, the major triad on the IV acts as a subdominant of the tonic region. Schoenberg explains that "degrees are marked by Roman numerals, and the first six of them also bear names: I, tonic; II, supertonic; III, mediant; IV, subdominant; V, dominant; VI, submediant; VII has not been given a name. These numbers refer to the place within the scale and determine the functional relations of the triads (or seventh or ninth chords, etc.) built on them" (18).
The establishment of a determined function also depends on its context: "A Triad standing alone is entirely indefinite in its harmonic meaning; it may be the tonic of one tonality or one degree of several others. The addition of one or more other triads can restrict its meaning to a lesser number of tonalities. A certain order promotes such a succession of chords to the function of a progression" (19).
The notion of region is also applied to define the specific function of a chord. Each region represents a segment of a tonality which involves individual pitches and chords. These specific elements acquire their function through the relationship established at the Chart of the Regions where any pitch or chord can be related to a tonic (20).
Transference of function by imitation of a model
When referring to the enrichment of the tonal system by creating secondary dominants, Schoenberg mentions the principle of transference of function by imitation of a model, a prototype:
The principle of analogy, of imitation, which tentatively transfers the characteristics of one object to a different object, producing, for example, the raised seventh tone in minor. We shall adhere to this principle when in the course of our study we repeatedly transfer to other degrees that which is possible, for example, on the IInd degree. (21)
The principle of imitation, of analogy, is used to describe the historical development of harmony, the development of a diatonic scale and the incorporation of chromatic notes into the tonal system. Schoenberg refers to the observations made by Robert Neumann(22) regarding his system of presentation of harmony. This refers to the development of the harmonic resources which is explained mainly "through the conscious or unconscious imitation of a prototype; every imitation so produced can then itself become a prototype that can in turn be imitated" (23).
Diatonic chords that are chromatically altered through substitution normally imitate the characteristics and the function of other diatonic chords. Thus, the function of a "prototype," a model, of a diatonic chord, is transferred to another degree of the scale. Transferring the function of a triad or a dominant seventh degree to other degrees of a scale has the advantage "that it allows the transfer (imitation) of all functions manifested by the basic triad to the new secondary dominants" (24). In this way, the principle of transference of function by imitation of a model is present in schoenbergian theory.
The characteristics of a chord can be transferred to any other chord. For that purpose, it is established a rule which observes the tendency of chords with substitutes in imitate a common prototype, such as the cadence or the progression II - V - I:
Every chord will (if not hindered by its environment) require a continuation like that of [some prototype] chord (consisting of entirely different tones) which has identical intervals. Thus, the constitution of the seventh chords on II, III, and VI in major is precisely the same. Now, since the one on II has a definite, familiar function (II - V - I, II - I6/4 - V), the ear consequently expects the same continuation from III and VI, whose structure is the same. (25)
The transference of function of II7 to seventh chords on III and VI supposes a correspondence of possibilities with the functions of tonic and dominant characteristic of the I and V. For example, in C major, if the III7 imitate the function of II7, the VI, with c# substituting c natural, will imitate the function of V, and II, with f# substituting f natural, will imitate the function of I. Thus, the cadence which originally reads II7 - V - I, will read III7 - V - I, with the same functions.
Through this association with the functions of tonic and dominant, transference acquires a centrifugal character (function) for it supposes a new tonic chord, or another chord with the function of tonic.
Transformation of a chord is originated through substitution. When discussing the transformations on the II, Schoenberg writes that these "result from the influence of D, SD, and sd. Under the influence of D, the [minor] third of II is substituted for,..., by a major third" (26). Transformation does not change the root function expressed as a scalar degree of a chord and its relationship to a tonal center. Thus, a chord can be transformed, for example, from a major triad to a minor, a diminished, or an augmented triad, even though it will keep its diatonic root. In this way, Schoenberg keeps the principle of monotonality still valid referring to a "monotonal root" which is kept despite the substitution introduced (27).
Schoenberg illustrates the transformations of the II as follows:
example 2, from Structural Functions, example 50
These transformations produce a certain number of chords on II, and some of them combine substitutes from two distinct regions. For instance, in sections a and b is found the substitute f#, from D, substituting the third f natural; and ab, from sd, substituting the fifth a natural. In this way, chords of secondary dominants, diminished triads, and dominant sevenths are produced. In section d, the II is transformed with substitutes f# and eb, in a dominant ninth chord (root omitted), i.e., the diminished seventh chord. In this case, Schoenberg emphasizes root progressions considering the II as the root; if f# were considered as the root would occur the substitution of roots (f natural for f#), which for Schoenberg is "an assumption which must be rejected as nonsensical" (28). Considering the II as the root, the model of strong or ascending root progressions (V-I or II-V) is emphasized (29).
Section g illustrates the Neapolitan sixth chord, which is justified by Schoenberg as being borrowed in toto from subdominant minor region as is not considered as a transformed chord on II. Schoenberg conception about the Neapolitan sixth chord acquires a character more important than of a single chord which substitutes the subdominant in the same function. According to the principle of transference, the notion of Neapolitan sixth is transferred to other degrees of the scale acquiring the "status" of an important tonal region (30). Indeed, even though the Neapolitan sixth chord be derived from subdominant minor region, this harmony had its potential recognized by composers who saw in it a tonal region in its own right. Example 3, a harmonic reduction from Schubert's Quintet in C major, Op. 163, illustrates the "possibility of using harmonies in a manner different from their original derivations", (31) as Schoenberg recognizes the changing of harmonic possibilities through transformation of chords. In this example, a small fragment of the development section, mm. 181-97, the Neapolitan sixth region is emphasized by the context (32).
Applying the transformations of the II to other degrees of the scale, Schoenberg creates a great expansion on the harmonic vocabulary. Some of these transformations may seem too distant from the original tonal center and they can endanger the supremacy of the tonic. To avoid the endangerment of the tonal center, Schoenberg directs the transformations on I, III, IV, V, VI, and VII to an application of the models of V-I, V-VI, and V-IV, according to his model of root progressions. Example 4 illustrates the transformations on the other scale degrees:
example 4, from Structural Functions example 55
For the minor mode Schoenberg applies the procedures done in major, and transformations on the II are applied to other degrees (see example 5):
example 5, from Structural Functions, example 63
The concepts of transference and transformation seem to contradict each other, but actually they are complementary. Transference suggests a movable function, while transformation does not admit the substitution of a root and is based on the function of diatonic degrees of a scale. Both represent complementary general functions: the first is centrifugal by alluding to other tonics, the second is centripetal by affirming the central power of one tonic. Transference promotes modulation among regions. The new region is established by transferring the function of the original scalar degrees to a new group of scalar degrees. Transformation simply enriches the harmony with the introduction of substitute notes and keeps the original root as a fixed point where the tonal relationships are measured.
Multiple Meaning and Vagrant Chords
Schoenberg considers every chord as belonging to two or more tonalities. The principle of transference of function implies a multiple meaning of a chord and consequently its functional ambiguity (33).
To illustrate this principle we can consider two cases. In the first, a pitch collection is fixed and the degree of the scale changes. For instance, the pitch collection c, e, g; can be interpreted, in C major, as I; or IV, in F major; or III, in A minor (34). The second case is concerned with different pitch collections related to the same scalar degree: I, in C major, has c, e, g as its pitches; I, in A minor, has a, c, e as its pitches; or I, in G minor, has g, bb, d as its pitch collection.
The principle of multiple meaning is best exemplified by a special category of chords: the vagrant chords. "Such chords belong to no key exclusively, rather, it can belong to many, to practically all keys without changing its shape"(35). These chords are derived by transformation and have its multiple meaning by presenting a specific form. The most evident cases are the diminished seventh chord, the augmented triad, and the augmented sixth chord.
These chords have an incapacity to express a tonal function, a tonal meaning, mainly due to its inability to define their root. The definition of a root is due in great part to an asymmetric form of a chord. For example, a major triad is formed by a minor third above a major. The root is recognized by this asymmetric form and by the definition of a perfect fifth. Vagrant chords are symmetric and due to its specific shape the root is not defined. Generally, they present a repetition of one interval once or more times and often the interval of a perfect fifth is not present to define a root (36).
The diminished seventh chord is one of the best examples of this kind of symmetrical form. It is composed of superimposed minor thirds that do not allow an identification (in this terms) of a specific root. For Schoenberg, the relation of its root to other chords depends exclusively on the context in which it is found. Even when inverted, there is no new form emerging and allowing an identification of its root:
Each of its tones can thus be the root, consequently each can be the third, diminished fifth, and diminished seventh. If we invert the chord, no new structural patterns emerges, unlike the inversion of a major or minor chord; we shall still always have minor thirds (augmented seconds). It will thus be unclear to which key it belongs whenever the diminished seventh chord appears out of context or in an ambiguous one.(37)
For Schoenberg, the root of a diminished seventh chord is revealed only when occurs its resolution according to root progressions. The root of this chord is understood as being omitted and located a major third below one of its tones, consequently, it is considered as a dominant ninth chord (V9) in minor. The multiple meaning of this chord is emphasized by the fact that it belongs to, at least, eight tonalities or regions. Each of its tones being considered as a potential leading-tone, four in major and four in minor.
Example 6 illustrates the derivation of the diminished seventh chord from a transformed II. Schoenberg begins his explanation with a diminished triad on II in C minor; this chord is transformed to a secondary dominant, then to a secondary dominant seventh chord, and then to a secondary dominant ninth chord. The diminished seventh chord appears by omitting its root:
example 6, from Theory of Harmony, example 183.
The augmented triad is also considered as a vagrant chord. It is composed of two equal intervals: two major thirds. In the same manner as the diminished seventh chord, the augmented triad belongs to several distinct tonalities, three in major and three in minor, and its pitches are potential leading-tones. Schoenberg defines its application explaining that "it can be introduced, because of its ambiguity, after almost any chord"(38). It can be used in any region as an altered chord of V or as a secondary dominant.
Augmented triads may be used to produce a tonic, and that, to this end, they may be introduced artificially on the V of the major key in question, following the idea of the secondary dominants. They are most simply introduced through chromatic alteration upward of the fifth.(39)
Example 7 illustrates the alteration of a fifth of a chord in the progression V-I:
example 7, from Theory of Harmony, example 176.
The augmented sixth chord has its derivation from a minor ninth chord on II with root omitted. In example 6b, the diminished seventh chord has its fifth altered to ab and its seventh (eb) enharmonically reinterpreted to d#. The augmented sixth appears between ab and f# in the first inversion of the chord (II6/5) (40).
Example 8 illustrates the derivation of a minor ninth chord with the diminished fifth and root omitted in its second inversion (4/3), the augmented sixth is located between ab and f#. Commonly, this chord is presented as having its root altered from f natural to f# (41). In Schoenberg's explanation, the root of the chord is not altered, even though it is omitted. He recognizes the augmented sixth between the fifth - ab - and the third of the chord - f natural.
example 8, from Theory of Harmony, example 184a, b, c, d
Enharmonic reinterpretation serves as an example of the quality of vagrant chords. In example 8c, the diminished seventh chord (f#, ab, c, eb) is reinterpreted as V7 in third inversion of Db major - gb, ab, c, eb. For Schoenberg "the fact that the sound of an augmented six-five (four-three, two, or sixth) chord is identical with the sound of a dominant seventh chord can now be easily exploited by treating (introducing and continuing) the one as if it were the other"(42). Such enharmonic reinterpretations implies a change of the root of the chord from d to ab, and a change of the region, in C major, from d natural (S/T) to db (Np).
These vagrant chords, without a defined tonal function, easily fluctuate between two or more regions and tonalities, and can have several functional interpretations (43). These chords endanger the establishment and the concept of tonality. They also characterize what Schoenberg called "Fluctuating Tonality," defined by Carl Dahlhaus in the New Grove as a "vacillation between two or more keys, though not in the sense of modulation but of ambiguity: the capacity of being related simultaneously to different centres" (44).
Tonal function for Schoenberg is based on tonal expression. His notion of tonality as a network of functions of specific elements which always refer to a tonal center comprises individual pitches, chords, and tonal regions. Pitches act as a melodic element capable of expressing a tonality, acquiring in this way their tonal function, whereas chords have their tonal function expressed by their root. Both elements, pitches and chords, are included in the notion of tonal region which considers scalar segments to establish a relationship between two or more tonalities.
Introducing the concept of transference of function, Schoenberg considers the harmonic function of a chord as present, understanding that a chord presents specific characteristics that are transferred to other chords. This procedure is based on the imitation of a model of tonal expression: the cadence IV(II)-V-I. Thus, the notion of chord function is expressed not only by its root but also by its characteristic sound, i.e., the sound of the whole chord.
The category of vagrant chords is defined as not having any specific function. Instead, they have multiple meaning. They act as chords which can be introduced anywhere and after any chord. Again, Schoenberg points to the direction of defining the function of a chord by relating it to a model of tonal expression, the cadence. The roots of these chords are defined only by the context in which they are present.
Wason's observation of two different trends on tonal function is achieved on Schoenberg's theory of tonal function. Reinforcing this opinion is the concept of monotonality which presents a complex schema of tonal relationships expressed in the Chart of Regions. The first sketches of the Chart of Regions based on Riemann's function notation points in this direction. Even though the function of chords for Schoenberg is expressed by their root, it can be inferred from his texts that a whole chord also expresses a function.
Norton Dudeque teaches at the Federal University of Paraná and has developed research on schoenbergian theory. (back)
(1) Cf. in KOPP, David. "On the Function of Function." Music Theory Online. Volume 1, no 3, May 1995. (back)
(2) "Each scale degree has its part in the scheme of tonality, its tonal function." Cf. in PISTON, Walter and DeVOTO, Mark. Harmony. p. 53. (back)
(3) This approach is used by Daniel Harrison in his Harmonic Function in Chromatic Music. p. 43-72. (back)
(4) Cf. in WASON, Robert. "Fundamental Bass Theory in Nineteenth Century Vienna." p. 256-7. (back)
(5) See Structural Functions of Harmony, p. 1. (back)
(6) See Theory of Harmony, p. 191-2. (back)
(7) See Structural Functions of Harmony, p. 2. (back)
(8) Ibid. p. 19. (back)
(9) Cf. in CARPENTER, Patricia and NEFF, Severine. In the commentary to The Musical Idea. p. 61. (back)
(10) Cf. in CARPENTER, Patricia. "Grundgestalt as Tonal Function." p. 16-7. (back)
(11) Dunsby and Whittall explain that "Schoenberg is concerned with tonal regions, rather than common-note relationship between triads, which select only three notes from the scale of a region." Cf. in DUNSBY, Jonathan and WHITTALL, Arnold. Music Analysis in Theory and Practice. London: Faber & Faber, 1988. p. 78. Schoenberg's conception is unique in this sense. Even when compared with Riemann's "functional theory" Schoenberg's is more complete in the manner that it can relate all diatonic or chromatic digressions from the tonic. Schoenberg recognizes an influence and "a certain similarity to Riemann's function notation, but it avoids his error, revealed mainly in the drastic reduction to three functions...Perhaps Riemann himself felt this, because he states in the "Lexicon" that all modulations within a movement stay under the influence of the main key." Cf. in The Musical Idea. p. 331. Schoenberg's also draw an initial sketch of the Chart of the Regions based on the main functions of Riemann's theory. See The Musical Idea. p. 338-41. (back)
(12) See Preliminary Exercises in Counterpoint, p. 73. Schoenberg considers the cadence I - IV (II) - V - I, as a model of tonal expression. The characteristics pitches act in the cadence as follows: the IV (4th) after I acts as an element that contradicts the tonic and may establish a new tonic. The introduction of V with the 7th, contradicts the IV and its possible tonality, confirming the tonic, I. In The Musical Idea Schoenberg illustrates the cadence as: I=assertion (of a tonality); IV(II)=challenge; V=refutation of IV (self-assertion of V); and I=confirmation (of the tonality). See The Musical Idea, p. 311. (back)
(13) The laws of the pivot tones (Wendepunktgesetze), here they are presented in a minor, of the minor scale are:
First pivot tone, g #: g# must go to a; for g # is used only for the sake of the leading tone progression. Under no circumstances may g or f follow g #, nor may g # go to f # (at least for now).
Second pivot tone, f#: f# must go to g #; for it appears only for the sake of the g #. Under no circumstances may g follow, nor, of course, f. Nor e, d, a, etc. (for now, at least).
Third pivot tone, g: g must go to f, because it belongs to the descending form of the scale. Neither f# nor g # may follow it.
Fourth pivot tone, f: f must go to e, because it belongs to the descending form of the scale. F# may not follow it.
See Theory of Harmony, p. 98. (back)
(14) See Theory of Harmony, p. 178. (back)
(15) The distinction on the introduction of substitute notes depends on the application of the laws of the pivot tones. The "quasi-diatonic" introduction is characterized by its application and promotes "modulation" between regions; "chromatic" introduction acts mainly as an enrichment of harmony and is unable to produce a change of region, or tonality. (back)
(16) See Structural Functions of Harmony, p. 46. This concept is derived from Simon Sechter's theories about root progressions. For him the root defines the progression. As Phipps puts it: "Sechter applies Rameau's theories of triple and quintuple progression as the fundamental principle of all harmonic successions. The most basic of all successions is the diatonic circle of fifths - I-IV-VII-III-VI-II-V-I- a progression illustrating Rameau's concept of triple progression. Sechter recognizes that there is one imperfect fifth in this progression - a diminished fifth between IV and VII in the major mode or between VI and II in the minor mode." In this respect he proposes that [the fifth of the vii° chord] "became the octave [of the IV chord], then the resolution of the false fifth would be in danger of being forgotten..." The movement by third are described by Sechter in the progressions I-VI-IV-II-VII-V-III-I. All the progressions described by Sechter were taken from these and presented isolated or not, in direct or retrograde form. The progression by steps, up or down, for example IV-V, presents an omitted root (Zwischenfundament) which causes the progression to consist of a combination of root movement by third down and by fifth, IV-(II)-V. Cf. in PHIPPS, Graham. "A Response to Schenker's Analysis of Chopin's Etude, Opus 10, nº 12, Using Schoenberg's Grundgestalt Concept." p. 548. See also CHENEVERT, James. "Simon Sechter's "The Principles of Musical Composition": A Translation of and Commentary on Selected Chapters." p. 34-5. Schoenberg's root progressions are defined as: 1) Strong or ascending (a fourth up, third down); 2) Descending (a fourth down, a third up); 3) superstrong progressions (one step up or down). See Structural Functions of Harmony, p. 6-8. (back)
(17) According to Robert Wason, Georg Vogler was the first to use this system of notation: "Vogler's major theoretical achievement - his invention of Roman numeral chord notation - was obviously an important contribution to "chord identity" theorizing." See WASON, Robert. "Fundamental Bass Theory in Nineteenth Century Vienna." p. 21. (back)
(18) See Models for Beginners in Composition, p. 54. (back)
(19) See Structural Functions of Harmony, p. 1. (back)
(20) For the complete Chart of Regions see Structural Functions of Harmony, p. 20 and 30. (back)
(21) See Theory of Harmony, p. 176. (back)
(22) Dr. Robert Neumann is identified as a young philosopher in Theory of Harmony. There is no reference about him in Schoenberg's correspondence. (back)
(23) Transference of function is yet derived from the principle that "affirms that a bass tone strives to impose its own overtones, thus has the tendency to become the root of a major triad..." See Theory of Harmony, p. 385. (back)
(24) Ibid. p. 385. (back)
(25) Ibid. p. 192. (back)
(26) See Structural Functions of Harmony, p. 35. (back)
(27) Schoenberg considers the root of a chord as an immutable and fixed point. Transformation occurs in the other tones of a chord. Thus, only the third, the fifth, the seventh, or the ninth of a chord can be altered. Besides, Schoenberg considers the root as "fixed points from which relationships are measured. The unity of all measurements we have found is guaranteed by the immobility of those points." See Theory of Harmony, 234. (back)
(28) Ibid. p. 35. (back)
(29) Schoenberg explains that the diminished seventh chords "were formerly considered seventh chords on natural or artificial leading tones. Accordingly a diminished seventh chord in c minor (b-d-f-ab) would be considered to be on VII and, worse, [50d] section d would be considered as based on a "substitute" root (f#)." See Structural Functions of Harmony, p. 35. It is important to remember that Schoenberg follows the Viennese tradition in this subject. Formerly, Bruckner, Sechter, and Weber, considered the diminished seventh chord in the same way. See DINEEN, Phillip Murray. "Problems of Tonality." p. 164, footnote 5. (back)
(30) Schoenberg's conception on the Neapolitan sixth chord is unique and distinct from that of other authors. For example, Riemann considers this chord originated from the substitution of the fifth of a chord by a minor sixth, and the major third by its minor, in C major: f-a-c to f-ab - db. Cf. RIEMANN, Hugo. Armonia y Modulacion. p. 122, 182 and 194. See also in DAHLHAUS, Carl. Studies on the Origin of Harmonic Tonality. p. 50. For Schenker this chord results from the Phrygian II in which the II is descendently altered and explained in terms of voice leading. See SCHENKER, Heinrich. Harmony. p. 109-10. More recently other authors consider this chord in a different manner. Piston considers that "the major triad whose root is the chromatically lowered second degree of the scale is known as the Neapolitan Sixth." See PISTON, Walter. Harmony. p. 407. And for Aldwell and Schachter it "is a chromatic variant of II6 with 2 lowered to b2; the alteration produces a major triad that replaces the normal diminished triad where the latter might give an unsatisfactory effect." See ALDWELL, Edward and SCHACHTER, Carl. Harmony and Voice Leading. p. 457. (back)
(31) See Structural Functions of Harmony. p. 35. (back)
(32) Other examples of this use can be found in Beethoven's Third Symphony, first movement, m. 284 and ff., where a change to e minor region (fb minor) represents the Np minor region (np). See also the second movement of Schubert's Quintet in C major, the tonal relationship between the sections A-B-A are: section A in E major (T) and section B in f minor (np). The structural relation is T - np - T. Certainly, this two examples does not match to the common usage of the Neapolitan sixth chord as a substitute chord in the function of subdominant. See also in PHIPPS, Graham. "Comprehending Twelve tone Music." p. 39-42. (back)
(33) Even when considering root progressions Schoenberg implies the notion of transference of function and multiple meaning. In his classification of root progressions specific tones of a chord are promoted or degraded in its importance in a triad. For example, in ascending progressions a fourth up, the root of a chord is degraded to become the fifth of another chord. So, the same pitch has two functions in this progression, it is the root of the first chord and the fifth of the second. See Structural Functions of Harmony, p. 6-9; and Theory of Harmony, p. 115-21. (back)
(34) See Models for Beginners in Composition, p. 54. (back)
(35) See Theory of Harmony, p. 195. (back)
(36) See DINEEN, Phillip Murray. "Problems of Tonality." p. 182-3. (back)
(37) See Theory of Harmony, p. 194. (back)
(38) Ibid, p. 243. (back)
(39) Ibid, p. 242. (back)
(40) Schoenberg's derivation of this chord is very similar to Sechter's. According to Phipps, Sechter derives from Rameau, "the diminished-seventh chord is a substitute for a dominant-ninth chord whose real root is a major third below the root of the diminished-seventh chord. He extends this principle to account for the augmented-sixth chords. Sechter describes the chord as an illegitimate chord (Zwitterakkord), which is built on the second scale degree with a major third and a diminished fifth." See PHIPPS, Graham. "The Tritone as an Equivalency: A Contextual Perspective for Approaching Schoenberg's Music." p. 55. (back)
(41) This is the conception of Aldwell and Schachter who explains that "if the augmented sixth follows a strong IV, 4 will carry over as root (altered, of course, to #4). See ALDWELL, Edward and SCHACHTER, Carl. Harmony and Voice Leading. p. 484. (back)
(42) See Theory of Harmony, p. 254. (back)
(43) One of the best examples is certainly the "Tristan Chord", which was subject to several analysis of its meaning or function. These interpretations range from a diminished seventh chord on VII by Kistler (1879); a VII°7 in f# minor by Jadassohn (1899); to SVII by Riemann (1909). Schoenberg considers the "Tristan Chord" as a II in the Structural Functions (see example 85). On the other hand, the discussion on the "Tristan Chord" as a vagrant chord is defined when Schoenberg proposes many interpretations for the same chord. One considers g# as a suspension going towards a natural; another considers a natural as a passing (through a#) to b natural; and the other considers the chord derived in eb minor key [?]. All these interpretations imply that the "Tristan Chord" is vagrant and has multiple meaning. See Theory of Harmony, p. 257. For other interpretations of this chord; see NATTIEZ, Jean Jacques. "Harmonia." In Enciclopédia Einaudi. v. 3. p. 245-71. (back)
(44) See The New Grove Dictionary of Music and Musicians, v. "Tonality" by Carl Dahlhaus, vol. 19, p. 54. (back)
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