Prolongation of Harmonies Related to the Harmonic Series in Early Post-Tonal Music

Yale University Department of Music

Prolongation of Harmonies Related to the Harmonic Series in Early Post-Tonal MusicAuthor(s): Olli VäisäläSource: Journal of Music Theory, Vol. 46, No. 1/2 (Spring - Autumn, 2002), pp. 207-283Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/4147681Accessed: 23/06/2009 16:13

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PROLONGATION OF HARMONIES

RELATED TO THE HARMONIC SERIES

IN EARLY POST-TONAL MUSIC

Olli Vaisal*

1. Introduction

In the history of music theory, the notion that a significant relationship exists between musical harmony and the harmonic, or overtone, series enjoyed more or less general acceptance from the time of Rameau to the first half of the twentieth century.' Theoretical thinkers of the latter period who drew on this notion include such pre-eminent figures as Schenker, Schoenberg, and Hindemith-whose aesthetic conclusions were, of course, quite divergent.2 In more recent times, attitudes towards this notion have become much more cautious or indifferent. While Schenkerian theory has established a leading position in the analysis of tonal music, present-day Schenkerians seldom appeal to the derivation of the (major) triad from the overtone series.3 And while the standard approach to post-tonal music-pitch-class set theory-is in many respects indebted to Schoenberg,4 its conception of intervals and harmonies bears no trace of Schoenberg's contention that "all musical phenomena can be referred to the overtone series, so that all things appear to be the application of the more simple and more complex relationships of that series" (Schoenberg 1975, 271).

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The present paper includes an argument for a qualified rehabilitation of the harmonic series as an entity relevant to music theory and analysis.5 More specifically, I shall focus on such early post-tonal music which, while having abandoned the triad as its basis, would seem to be illuminated by allowing for properties that derive from a relationship between harmony and the harmonic series.6 I shall analyze four works from around 1910, Scriabin's Vers la flamme, Berg's song op. 2, no. 2, Debussy's Voiles, and Webern's song op. 3, no. 1, suggesting that a central position in their organization is granted to harmonies that consist of pitches ap- proximating those in the harmonic series up to the eleventh harmonic; see chord A and its subsets P, Q, and U in Example 1 below. I shall argue that in these works (or in parts thereof) such harmonies not only occur as foreground sonorities but also become referential for prolongational organization more or less analogous to the Schenkerian concept of triadic prolongation. To lay groundwork for the analyses, I consider first the correspondence between musical harmony and the harmonic series from a perceptual viewpoint and then set out the theoretical principles applied in the analyses. The principles of prolongation will be discussed in detail, on the basis of the four conditions of prolongation postulated by Joseph Straus (1987); this discussion is several respects related to my earlier study of post-tonal prolongation, Vaisalk 1999.7

By the analyses I attempt to demonstrate that, at the time when the four works were composed, the structural principles discussed were utilized by composers with a remarkably wide scope of stylistic and expressive ends. The way in which the subsequent structural readings illuminate musical experience and expression is largely comparable with conventional Schenkerian analysis. The hierarchy of structural levels is, rather than an end in itself, a frame of reference for considerations of various musical aspects, such as large-scale articulation, less obvious motivic relationships or "concealed repetitions," and the expressive significance of primary and secondary motions. As in conventional Schenkerian readings, the moments of greatest structural weight by no means always coin- cide with those of greatest expressive intensity. Structural weight is basically connected with the impression of stability, whereas expressive high points often involve instability or tension. One noteworthy aspect of musical experience not so evident in the analytical graphs is the role of shift- ing temporal perspectives.8 In the structural readings, goals are prior to the motions that proceed to them. In listening, motions are, of course, prior to goals and there may be uncertainty and delusive impressions regarding a goal before it is reached. For the musical experience, such temporary impressions are as important as the retrospective clarification of the goal. At a few instances, I expressly point out such momentary impressions. I also present some ideas about the connection between structure and expression in the two songs. For reasons of space, however, the com-

208

parison between the structural readings and musical experience and expression is mostly left to the reader.

1.1. Perceptual Background: Harmonic Root and Virtual Pitch For a perceptual explanation that sheds most light on present consid-

erations, I shall refer to the notion, originally suggested by Ernst Terhardt (1974), of virtual pitch as the psychoacoustical correlate of the harmonic root. Virtual pitch (or residue pitch) is the pitch which corresponds to the fundamental frequency of a complex tone (a tone with many harmonics) and which is perceived, in the "synthetic" mode of hearing, as the pitch of the tone. Virtual pitch may be heard even when the fundamental is physically absent, on the basis of just a few harmonics.9 According to Terhardt's suggestion, the pattern-recognition faculty employed in virtual-pitch perception also affects the "tonal meaning" of musical harmonies (ibid., 1067).10 If a bass tone forms similar intervals with the upper tones as those between a fundamental and its harmonics, it will have a tendency to be perceived as a root of the harmony-that is, governing the overall pitch pattern in the manner of a virtual pitch.

This explanation has, with respect to its musical implications, several appealing features. Unlike some earlier ideas about the relationship between the harmonic series and music, it does not regard the harmonic series as a universal source of musical material. Rather, it delimits the relevance of reproducing the intervallic relationships in the series to a spe- cific property, the "rootedness" of harmonies; the significance of this property varies, of course, according to stylistic and aesthetic circumstances. Moreover, since virtual-pitch perception allows mistunings of harmonics up to at least a quarter-tone,"1 the musical relevance of this phenomenon is in no way negated by the use of equal temperament. As discussed by Richard Parncutt (1988), whose considerations are based on a revised version of Terhardt's model, this explanation enables us to pre- dict the roots of conventional harmonies in accordance with musical practice. This also applies to a chord such as the minor triad, an often- mentioned counterexample to theories attempting to derive harmonies from the harmonic series. In Parncutt's model, bass-related intervals bear root-supporting weights that correlate with the closeness of the approxi- mated harmonic (ibid., 73-75). In the case of the minor triad, the root- supporting weights of the unison/octave and the fifth are sufficient to support its conventional root, even though the minor third is not a root support. (However, the "root ambiguity" of the minor triad is greater than that of the major.)12

Despite the cogency of this explanation, it should be borne in mind that, whereas virtual pitch is a purely psychoacoustical concept, harmonic root is culturally influenced, and the two cannot be totally equated. The root concept involves cognitive extensions and generalizations related to

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aspects that are central to musical organization but play little or no role in virtual-pitch perception. Such aspects include the nonsimultaneous presentation of pitches, i.e., arpeggiation, and octave equivalence. Arpeg- giated harmonies are associated with the corresponding block chords and roots thereof, although an arpeggiation (usually)13 evokes no virtual pitch. As for octave equivalence, it may be noted, for example, that the (main) virtual pitch evoked by the triad C4-E4-G4 is C2, but the root is not usually understood as lying specifically at octave 2. Roots, in normal musical parlance, are pitch classes rather than pitches, and in order for a harmony to be in root position it is generally sufficient that the bass is of the correct pitch class.

This kind of octave generalization is allowed for in Parncutt's root model. In this model, intervals are regarded as root supports if they cor- respond to those between the fundamental and its harmonics, irrespective of octave enlargements or reductions. The range of harmonics allowed for in this model includes the first ten harmonics, which yields the following list of root supports, from the strongest to the weakest: octave/unison, P5, M3, m7, M2, or, in semitone notation, 0, 7, 4, 10, 2 (Parncutt 1988, 73-75). This range conforms roughly with the range of harmonics most relevant for virtual-pitch perception.14 The eleventh harmonic is rejected from Parncutt's model on the grounds that it "is hard to categorize in the chromatic scale (it lies roughly midway between a fourth and a tritone) and only rarely influences tone perception anyway, due to masking by neighboring harmonics" (ibid., 70). There are, however, experiments implying that the eleventh harmonic does affect virtual-pitch perception in circumstances in which it is audible, as when the spectrum of a tone consists of only odd harmonics so that the masking by neighboring harmonics is absent (Houtsma 1975, Gerson and Goldstein 1978). As regards the equal-tempered "categorization" of the eleventh harmonic-which lies 2.6 hundredths of a semitone closer to the tritone (6) than to the fourth (5)-it may at least be noted that the fourth is made a less feasible option by its association with its inversion, the fifth (7), which is a strong root support. The interval of a fourth strongly points to the pitch class of its upper tone as the root, a feature probably linked with its conventional status as a dissonance (cf. Parncutt 1996, 72) and likely to override any slight root-supporting effect deriving from its connection with the eleventh harmonic.'5

All in all, it seems a plausible hypothesis, at least, that the tritone (6) may be perceived as a supplement to the pattern of root supports, especially when registrated above the other root supports in accordance with the harmonic series (as in the chords in Example l a). For other kinds of spacings, the root-supporting status of the tritone has less direct perceptual support-but might, to some extent, apply through the cognitive octave generalization discussed above.'6 In the following, I shall regard

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the tritone as the weakest root support, observing, however, the depen- dance of its root-supporting effect on registration.

Without considering in any more detail the virtual-pitch-root connection and the historical and psychological processes involved, I shall hold the following general twofold assumption in the subsequent considerations. First, the use of root supports as bass-related intervals increases rootedness, or the extent to which a harmony tends to be understood as an entity governed by its bass. Second, spacings resembling the harmonic series produce especially direct psychoacoustic backing for this property.17 My special aim is to suggest that, in addition to conventional tonality, this property continued to be significant during the advent of post- tonal harmony. For following the analyses, this perceptual assumption is, in principle, no more necessary than in conventional Schenkerian analysis it is necessary to accept Schenker's derivation of the major triad from the overtone series; one can simply take the referential harmonies as "given." However, the pertinence of this assumption for the present repertoire is supported by the correlation between the property of rootedness and the syntactic properties of harmonies, and, in particular, by the way in which the assumption illuminates the function and effect of certain bass tones. The analyses may thus be viewed as inductive evidence for the perceptual assumption; to some extent such evidence may also concern the significance of the eleventh harmonic, as I suggest specifically in the discussion of Voiles.

1.2. Conception of Harmony; Conventions of Terminology and Notation

As argued in Vdiisili 1999, considerations of prolongational organization (tonal or post-tonal) require, in general, a conception of harmony based not only on pitch-class sets but that allows, in an appropriate degree, for registral disposition. Under the present perceptual assumptions, the most important registral issue concerns the identity of the bass-as in conventional tonality-and harmonies will thus be conceived primarily as sets of bass-related registrally ordered (mod 12) intervals. In Robert Mor- ris's (1995) terminology for different pitch-set equivalences, this kind of conception is called "FB," for "figured bass." Morris's other two equivalences, "PCINT," based on registral ordering, and "PSC," based on pitch intervals, are also, however, not without significance. The perceptual connection with the harmonic series is made stronger by spacings that reproduce its registral ordering and stronger still by those that reproduce the actual pitch intervals. In general, the registration of upper voices is somewhat more significant for the character of harmonies in the present examples than in triadic music. Spacings close to the harmonic series often assume important positions in compositional strategy, as is pointed out in the analyses. However, since various musical purposes call for other kinds

211

(a) (b) (c) (d) (e) (f) (g) r

• l22-•1 1H

2 L I

!6

2 9

lr l ) I

Il

S2. . 2 10 -

. 910_ 1•0_I

i (or 4)

5 4 _

4 J 4 4 4 4 4 L f 4(or 4 O77 0 7

" " "

0 0 mo 0o 0 o o0

, 0 0 0 0

T06A P Q TU T6A ToU(6) A+9 A9) oA+1(10) T6A T+9 T6P (or T)

=To(PUQ)

Example 1. Conventions of analytical notation

of registrations, no formal rules are given for restricting the spacing of upper voices.

Bass-related intervals are called fb-intervals and identified by the numbers 0-11, which show the intervals in semitones, after having sub- tracted octaves (12s). Registrally ordered intervals, in general, are called ro-intervals. In the musical examples, as in Example la, fb-intervals are indicated by ordinary numerals, whereas italics are used to indicate numbers of corresponding harmonics. Ro-intervals and pitch intervals between upper voices may be shown using brackets that connect the notes in question (see, for example, Example 2c below).

The harmony formed by all the root-supporting fb-intervals, that is, approximations of the first 11 harmonics, is called chord A; see Example 1a. Chords P and Q are arbitrary names for two subsets of A. Chord U is the union of P and Q. In order to specify transpositional levels, Tn markings are employed; see Example la-b. In the Introduction's examples, as in the Berg and Debussy analyses, To has Bb as its bass; in the Scriabin and Webern analyses the bass of To is E. Incomplete occurrences of harmonies may be indicated by showing the missing fb-interval in parentheses, as in Example ic. On the other hand, harmonies may include additional fb-intervals, i.e., such that are not root supports. These are denoted by + signs. As discussed by Parncutt (1988, 87 ff.), such added tones- "root detractors"-tend to have the least effect on rootedness when registrated above the root supports, as in Example id. Example le combines + signs with parentheses. The notation +11 (10) conveys the idea that fb- interval 10 is replaced by 11.

While the present conception of harmony is primarily based on bass- related intervals, there may be contextual justification for conceiving a harmony as inverted instead of in terms of literal fb-intervals. Such justification is roughly comparable to that required in conventional tonality for interpreting the 6 chord as an inverted triad. It is often offered by a bass motion that connects the inverted harmony with one in root position. Chord inversion is indicated by a number beneath the letter symbol showing the FB-interval of the new bass in the original harmony, as in Exam- ple lf. In inverted chords, all interval markings refer to the root instead of the actual bass. Example 1g demonstrates a special property of chord P that stems from its pitch-class invariance under tritone transposition: FB-interval 6 may be viewed as an alternative root in this harmony.

Two spacings are given special labels. The spacing employed in Ex- amples l a and 2a corresponds to the harmonic series without octave duplications, in other words, it only comprises approximations of odd harmonics. It is called the OH spacing. The term OH ordering refers to spacings that reproduce the registral ordering of the OH spacing but not necessarily its pitch intervals. The spacing with correspondents of all harmonics is shown in Example 2b. Example 2c shows the LT spacing, for

213

(a) (b) (c) (d)

ICf. _ __

- _~

1 -

1J

6 6 7

To 5 5 0 4 5 40

4 5

_ 3 7

7 1CO 7 1

Harmony: 0T0A TA TOU TOA

Spacing: OH all harmonics LT proximate = odd =low harmonics tritone

Example 2. Examples of spacing

low tritone. This spacing, in which fb-interval 6 occurs next to the bass, is used for chords P and U at the beginning of Vers laflamme (and is generally characteristic of Scriabin). In this spacing, the registration of fb- interval 6 contrasts sharply with the harmonic series, which makes its root-supporting status questionable or, at least, entirely dependant on the association with spacings closer to the harmonic series. A perhaps more significant perceptual aspect of an LT-spaced chord P is that it consists of two superimposed tritones and the lower tritone is formed by alternative roots of the upper tritone; see the illustration beside the chord in Exam- ple 2c.

One principle of spacing with more or less significance for all the example compositions is the tendency to avoid small pitch intervals, semitones and whole-tones (and especially clusters thereof). This principle will be called the proximity principle of spacing. (The proximity principle of spacing thus means avoiding proximity but is so named because its connection with the proximity principle of voice leading, to be discussed below.) A "proximate realization" of chord A, as shown in Example 2d, would thus be uncharacteristic of the present repertoire. This principle also implies that for the harmonics 7-11 the OH spacing (Example 2a) is usually employed rather than the spacing containing all harmonics (Example 2b). The correspondents of the odd harmonics 7-9-11, fb-intervals 10-2-6, form an "augmented triad," which is a special characteristic of chord Q.

While the chords discussed assume a structural position in the present works that deviates from conventional tonality, some aspects of their pre- history in tonal music are worth observing. Chord A may be formed by adding a sharp eleventh to a major dominant ninth chord. Chord P, on the other hand, occurs in tonal music as a "French sixth" chord and, also, as a dominant seventh chord with a flat fifth.18 By and large, we may out- line a psychoacoustically influenced historical process, in which such chords first emerge in poetically significant but structurally secondary functions, and are then more or less gradually "emancipated" from the need for a triadic resolution until becoming referential themselves.19 Such a process is particularly evident in the personal stylistic evolution of Scri- abin.20 As described by several analysts,21 Scriabin's tonal works typically employ chord P as a "French sixth" on bII and as a V7 -chords with identical pitch-class content-thus utilizing the alternative roots of chord P as bass tones. The tritone spans between bII and V foreshadow those in Scri- abin's later works, such as Vers la flamme, in which no resolution to a tonic in the conventional sense (A or Eb in this case) is realized or suggested. An interesting point of comparison is given by Berg's op. 2, no. 2 ("Schlafend triigt man mich"), which stands just on the threshold between "tonal" and "atonal" in this transitional process. It does contain allusions

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to an Eb tonic but, as I argue in the subsequent analysis, these allusions remain structurally subordinate to chord U on Bb.

Another noteworthy aspect of chords P, Q, and U is their whole-tone content. The position of the whole-tone set varies, however, greatly in the four works. Debussy's Voiles presents an example of pervasive whole- tone organization based on chords Q and U.

1.3. Theoretical Principles: Conditions of Prolongation I shall now discuss Straus's four conditions of prolongation and the

principles of their application and, to some extent, adaptation in the present analyses. These principles are, of course, partly determined by the musical context but partly they are also supported by conformance with more general principles of auditory perception. Hence, in the first place, the main principle of functional consonance is closely related with the property of being root-supporting, whose perceptual underpinnings were discussed above. Moreover, the proximity principle of spacing, which sometimes functions as a supplementary principle of consonance, contributes to the reduction of effects produced when two frequencies come closer to each other than the critical bandwidth. Such effects include masking and roughness, the latter of which is often identified as sensory dissonance.22 The critical bandwidth is, from the middle register upwards, about three semitones (in lower registers it is wider); hence, a violation of the proximity principle of spacing between two tones causes their fun- damentals and all corresponding harmonics to lie within the critical bandwidth.23 Finally, the basic principle of classifying melodic intervals as voice-leading intervals or arpeggiations, the proximity principle of voice leading, conforms with the well-established relevance of pitch proximity towards auditory streaming (Bregman 1990).24

Condition 1 is the consonance-dissonance condition. It requires a distinction between functional consonances and dissonances, harmonies or intervals that can support tones of greater structural weight and those that cannot. In the present analyses, the main principle of consonance is that the referential harmonies-chord A, its subsets, and, in one case, a super- set (A+9)-are consonant. Under this principle, consonant tones form root-supporting fb-intervals, unless added tones or chord inversions are used. As observed above, the proximity principle of spacing may sometimes be viewed as a supplementary consonance rule, according to which semitones and whole-tones between any two voices are dissonances. Generally speaking, this supplementary principle has a less consistent and less essential impact than the main principle.

Condition 2 is the scale-degree condition and requires a hierarchy among the consonant harmonies to determine their mutual structural weights. In tonality, the basis of such a hierarchy is the relationship between I and V, the bass arpeggiation that connects the root and the fifth

216

of the tonic triad. In three out of four of the analyses (all except the Debussy), we shall find a "scale-degree" system based on an analogous bass arpeggiation within the referential harmony. It should be borne in mind that such an analogy with the tonic-dominant relationship only concerns functions in the prolongational structure and not, for example, cadential effects.

In tonality, the consonance-dissonance and the scale-degree condition, as described by Straus, form a two-stage stability system. The factors underlying these conditions are chord construction and chord transposition, respectively. On the small scale, chord construction differentiates consonances from dissonances; on the larger scale, transpositional levels of the triad determine the scale-degree hierarchy.25 In post-tonal circumstances there is, however, reason to allow room for a wider spectrum of possibilities in which these factors cooperate in determining harmonic stability. The relationship between chord construction and structural functions may take more nuanced forms than the two-alternative consonance- dissonance system and substitute, in part, for a transpositional scale- degree system.26 Moreover, for the occurrence of prolongational systems, it is not necessary to have invariable consonance systems for entire compositions. The roles of chord construction and chord transposition on different scales of organization may also be the reverse to those in tonality. In Vers la flamme, the large-scale organization is based on relationships between harmonies of different, though related, construction. While one of these harmonies (A+9) is primary, or most stable, other harmonies assume locally referential status for considerable stretches of time, giv- ing rise to small-scale "scale-degree" systems based on transposition. (On a yet smaller scale, a more or less ordinary consonance-dissonance distinction becomes active; see Example 8e below.)

Condition 4-to be treated here prior to condition 3-is the harmony/voice-leading condition. It calls for "a clear distinction between the vertical and the horizontal dimensions" (Straus 1987, 5). The essential requirement is the classification of melodic intervals either as arpeggiations, i.e., those that stand for a "vertical" interval, or as voice-leading intervals, i.e., those that function purely as "horizontal" connectives. As discussed by Straus, such a distinction in tonality is based on the one between "steps" and "leaps." "Melodic motion by step takes place within a single voice; motion by an interval larger than a step goes from voice to voice and arpeggiates some harmony" (ibid.). The present analyses employ a related principle, the proximity principle of voice leading, in the interpretation of melodic intervals: small pitch intervals function as voice- leading intervals and larger ones as arpeggiations. In post-tonal circumstances, this distinction has, in general, to be based on the absolute size of the interval, instead of "steps" and "leaps" in a referential overall collection, such as the diatonic set.27 The "default" assumption in the pre-

217

(a) (b) (c) (d)

.

.4 1"

,N NH " ,, " "rp"

Q P Q Q Q PQ U U PUU A U U A Q U U A

Example 3. Illustration of the proximity principle of voice leading

sent analyses is that the borderline between small and large intervals lies between 2 and 3 semitones, in agreement with the proximity principle of spacing, with the critical bandwidth,28 and also with conventional tonality.

The proximity principle of voice leading, as presently understood, allows little octave generalization. Whereas the harmonic conception is primarily based on fb-intervals, the interpretation of melodic motions depends crucially on actual pitch intervals. As a rule, simple melodic semitones and whole-tones are regarded as voice-leading intervals, but expanded realizations of interval classes 1 and 2 ("sevenths" and "ninths") rank as arpeggiations. Example 3 illustrates. The "stepwise" C5-D5-C5 line in Example 3a is a neighboring motion with no effect on the governing Q harmony, whereas the "leaping" C5-D4-C5 motion in Example 3b is an arpeggiation and enlarges the harmony from Q to U.29 In Examples 3c and 3d, it may seem paradoxical that the whole-tones and the semitone are not taken as arpeggiations although they connect fb-intervals that belong to the prevailing harmony. However, since these melodic intervals are not alone sufficient to establish the presence of the fb-interval of the embellishing tone in a harmony, they do not actually stand for vertical ones. Neighboring and passing tones in an octave (or unison)30 relationship with a harmonic tone will be called h-neighbors and h-passing tones and denoted N(H) and PT(H) in the graphs; dotted lines are used to indicate concomitant octave (or unison) relationships, as in Example 3c-d. (To simplify notation, these markings are sometimes omitted.)

Despite the crucial significance of the proximity principle of voice leading in the present approach, its application in different musical contexts requires some flexibility. This concerns, first of all, the "default" borderline between the small and the large intervals. For example, there are no grounds to categorically rule out the possibility that simple whole- tones could function as arpeggiations. In the present study, this issue is relevant for the Webern analysis, in which whole steps are regarded vari- ably as harmonic or voice-leading intervals according to broader melodic context; melodic semitones are, however, always taken as voice-leading intervals. On the other hand, there may be special contextual justification for interpreting larger realizations of interval classes 1 and 2 as substitu- tions for the small voice-leading intervals, against the main rule described above. It seems natural to assume that ro-intervals 1 and 2 ("ninths") are more liable to function in this way than 10 and 11 ("sevenths").

There is an evident relationship between the proximity principle of spacing and the proximity principle of voice leading. The size of pitch intervals is crucial for both principles. The smallest intervals do not occur in structurally supportive harmonies, but function as dissonances when presented simultaneously, and as voice-leading intervals when presented successively. The correspondence between the two principles is, however, not perfect but more or less rough. As already observed, the appli-

219

(a) IN(D) (b) (a) 4IN(D)

IN(D) IN(D)

(b) IN(D)

2 IN(D) 9 6 6 N(D) IN(D) IN(D)

4 13

Example 4. The dimming tone

cation of both aspects of the proximity principle in different contexts requires flexibility. Moreover, the two aspects are not entirely dependent on each other. Even when the proximity principle of spacing is not in force (i.e., block chords containing small intervals are regarded as valid occurrences of consonant harmonies, as in parts of Vers laflamme), it is possible to postulate a syntactical rule excluding the arpeggiation of such intervals.31 In such a case the "distinction between the vertical and the horizontal dimensions" is made for certain intervals literally according to their vertical or horizontal presentation. In general, the proximity principle of voice-leading is more crucial for the analyses than the proximity principle of spacing, and proximity principle, without qualification, will hereafter refer to the former aspect.

A note relevant to both aspects of the proximity principle is that these principles and the attendant perceptual effects tend to lose force when different timbres are involved.32 In the songs analyzed, registral connections between the piano and vocal part are sometimes, under clarifying circumstances, utilized in the voice-leading structure. In other instances, however, the two parts are registrated more or less independently of each other, and small intervals may occur between them with no structural consequences.33

Finally, it is important to note that the harmony/voice-leading distinction of melodic intervals, as outlined above, often cannot be applied to all voices in a contrapuntal setting. "Leaping" intervals are not always arpeggiations of a harmonic interval but may have a function in counterpointing another voice with a more essential structural role. Such leaps frequently occur in the bass line, both in tonal music (see Example 5a below) and in some of the present analyses.

Condition 3, the embellishment condition, requires a consistent set of (melodic) relationships between structural and non-structural tones. In the present analyses, the types of embellishment are basically the same as in tonality: arpeggiations on the one hand, and, on the other, those involving voice-leading intervals: neighboring and passing tones and suspensions. These are denoted in the graphs by conventional abbrevia- tions. (Passing tones are denoted as PT in order to avoid confusion with chord P.)

One special type of embellishment that recurs in the analyses is the incomplete neighbor that follows a main note a semitone lower; see Example 4a. Characteristically, such incomplete neighbors involve a motion from ro-interval 4 to 3, or from the major to the minor third, in relation to another voice. If this relationship comes to the fore in the music, as is often the case, the effect of this embellishment is, in a sense, closer to mixture than to an actual incomplete neighbor in tonality. I shall call such incomplete neighbors dimming tones and denote them IN(D). The typical compositional function of a dimming tone is to articulate the end-

221

point of the temporal presence of the main note while prolonging it in a more abstract sense, a point to be elaborated in connection with Voiles. A dimming tone may also serve to trigger a linear descent, as illustrated in Example 4b.

The concepts of h-neighbors and h-passing tones most often cause no special complications for analysis. Whether the flexibility and "paradox- icalness" involved by them are liable to jeopardize the clarity of structural relationships may be put into perspective by considering certain tonal practices and their relationship to Straus's conditions. For example, h-neighbors might be objected to on the grounds that the distinction between structural and non-structural tones is not supported by the consonance-dissonance condition. In tonality, however, one not only finds consonant neighbors but even consonances prolonging dissonances and "tonics" prolonging dominants, as in Example 5a-b (cf. also Cinnamon 1993, Fig. 1; Larson 1997, Exs. 4-5). This illustrates the important point that the clarity of prolongation does not always require the concurrent ful- fillment of all four conditions. In the present examples, condition 3 overrides, so to speak, conditions 1 and 2: the melodic and metric circumstances of the embellishing figure are sufficient to clarify the hierarchical relationships to the extent that makes the criterion of harmonic stability locally irrelevant.34 Moreover, while h-neighbors are not possible in the prolongation of triads they occur frequently in the figuration of seventh chords, as in Example 5b. The organization of tonal music is also not entirely devoid of ambiguity and contextual influence concerning the harmony/voice-leading distinction. While it is natural to interpret G "hori- zontally" as a neighbor of F in the fragment shown in Example 5b, im- planting this fragment into the context of Example 5c makes us more inclined to understand it "vertically" as an arpeggiation.35 Incidentally, this example also illustrates that a rigid harmony/voice-leading distinction is not always necessary for the clarity of the broader prolongational hierarchy: whether we choose to emphasize the "horizontal" or "vertical" relationship between F and G, the primary top-voice motion is E-F-E.36

While in most cases the octave relationships involved by h-neighbors and h-passing tones are thus relatively inconsequential, there are cases in which both the "stepwise" and the octave relationship are more significant for the organization, so that there is reason to speak of an actual double function.37 Example 6a shows a relatively simple instance of this. Four upper voices are transposed down an octave; Bh4 and Eb5 concurrently anticipate the tones one octave lower and function as incomplete neighbors to Bb4 and D5. (Later, Eb4 is "resolved" to Ei4, which belongs to the primary harmony, chord A).38

That a double function may sometimes be connected with more radi- cally unconventional features of organization is illustrated by Example

222

(a) (b) (c) arp. arp- arp-

r1 . I . -I

C: V Iv I V7 I I V7 I I I V- V7 I

Example 5. Some special cases in tonal music

6b, which shows the overall structure of Voiles in a simplified form. The middle-register D in the middle part of the piece functions concurrently as a neighbor of E-paralleled by F# as a neighbor of Ab-and as a registral transfer of the structural D7 established in m. 22. The latter function is vital for the prolongation of the D7 (hence the solid line), which is not repeated or embellished in its initial high register until m. 43. The role of D in Voiles may be explained in terms of an unconventional combination of structural and temporal aspects. The D7 is a structural tone, but the elements prolonging it are confined to the middle part of the temporal organization; hence its temporal position resembles that of non-structural, unstable elements in conventional organization. Such unconventional circumstances link with the unconventional way in which register affects the structural implications of pitch relationships. D7 is established structurally by registral separation but, at the same time, it is a source of lower-register Ds, whose proximity with structural tones C and E creates a kind of instability. Eliminating this instability-which is underlined, for example, by the violations of the proximity principle in mm. 33-35-is central in the temporal plot of the piece. A distinction between structural and temporal aspects is also crucial for the function of the dimming tone Db7 in m. 43. While prolonging D7 in the structural sense, it articulates the temporal endpoint of D's impact in a most dramatic fashion.

2. Four Analyses

2.1. Scriabin, Vers laflamme, op. 72 (1914) As observed above, the structure of Vers laflamme may be interpreted

on the basis of several interrelated harmonies with locally referential status. Such status is assumed by chord U, which characteristically occurs in the LT-spacing, and by chord A+9 in both 7-inversion and root position.

223

(a) Webern, Op. 3/1 (b) Debussy, Voiles IN(D)

-1 2] 2 3

I .O IT r- I I re g tr

antic. 0

2 IN o 8(diss.) -

7

A Q U-

Example 6. Double functions

The prolongation of the first of these harmonies in the opening section (mm. 1-26) serves particularly well to illustrate several important aspects of the present approach and will be treated in detail. At first, however, let us briefly consider the three above-mentioned harmonies and their mutual relationships in Vers laflamme.

Example 7a-c shows how the three harmonies are thematically highlighted by three harmonizations of the opening material. (Example 7d shows a fourth harmony, which appears at the conclusion and is discussed later.) This Example also illustrates the close correspondence between the registration in these statements and the harmonic series of El, the bass note with primary status in the overall organization. The low tritone, or fb-interval 6 (Bb), in the opening U chord stands out as the only element that deviates from the series (Example 7a). In the final statement (Exam- ple 7c) this deviation is "corrected" by the substitution of fb-interval 7 (BM). A special feature of the second statement (Example 7b) is the excep- tionally blatant violation of the proximity principle of spacing; it incorporates approximations of even harmonics in addition to odd ones (cf. Example 2b).

The large-scale relationships between these harmonies are illustrated by graphs (a) and (b) in Example 8. The LT-spaced ToU functions as an "appoggiatura" chord in relation to the ToA+9, which may be regarded as the primary harmony in the overall organization. The impression of its superior stability is supported by the increase of rootedness effected by the substitution of fb-interval 7 (BM) for 6 (A#/Bb) in the second lowest voice (Example 8a).39 The Bb-BM relationship has several important ramifications in the organization. Example 8b shows bass arpeggiations that highlight this relationship by transferring it to the bass line. The 7-inverted ToA+9 (m. 27) results from the large descending arpeggiation E2-B 1-E 1.

While both the T0U and the 7-inverted ToA+9 are thus ultimately subordinate to the root-position ToA+9, during the spans governed by them they become locally referential, setting the norm of consonance for subordinate local harmonies. Graphs (c)-(e) in Example 8 depict the opening prolongation of ToU (1-26) and the beginning of the subsequent section based on the 7-inverted ToA+9. The borderline between the sections governed by these harmonies is shown by a vertical line.

The piece begins by an emphatic repetition of a block-chord P, a subset of U (P c U c A c A+9). By virtue of the proximity principle, however, the "leaping" top-voice interval D4-F#4 (mm. 1-3) forms an arpeggiation, which incorporates fb-interval 2 to the harmony and enlarges it to U; see Example 8d-e. Two sequential repetitions of the initial pattern follow in rising 3s (minor thirds). As beamed in Example 8c-e, this produces a framework in mm. 1-11 which horizontalizes, in parallel 10s, the two tritones superimposed in the initial LT-spaced ToP. Such parallel 10s

225

(a) (b) O 9 @

2 "' - *" -

ToU T0

+9

(d) ( 9 ...

0 * 15 -

AV I I I

I' I -

" 40 I

l

1

1' _

I I,6 -

1 :;6

,-. " -- • ' ,

9 - 10 -0 ** ***

W5.0

I iT A,0

TOA•9) TOA +9+l~O (2)

Example 7. Scriabin, Vers laflamme, reharmonizations of the opening subject and correspondence with the harmonic series

(minor sevenths) are, alongside 2s (major ninths), reminiscent of the cus- tomary tenths and sixths in conventional tonality and are generally characteristic of the present repertoire.

The interpretation of fb-interval 2 as a harmonic element is confirmed in m. 19, in which the U harmony materializes for the first time as a block chord, at T6. As the top voice of this chord, fb-interval 2, C, also assumes structural significance. The registral transfer in m. 23 brings the C to the proximity of the initial D, enabling it to operate as a middleground passing tone; see Example 8c. The function of the T6U in the opening section is analogous to the dominant in a tonal Ursatz. The bass Bb of T6U stems from the tritone arpeggiation within the locally referential ToU and supports the passing C by fb-interval 2 (major ninth), which, in the present context, is a consonance unlike the 8 (minor sixth) that would be formed against a stationary bass E (Example 8c). The structure is not, however, completed by a return of the opening ToU. Instead of it arrives the 7- inverted ToA+9 (m. 27).

The "scale-degree" system in the opening section is supplemented by the T3U in m. 5. Citing Schenker's terminology (1979, ?53), it functions as a "space-filling" between the two primary harmonies, T0U and T6U, and is thus comparable to tonal pre-dominants, II, III, and IV. However, the equal division of the tritone E-BL neither participates in the arpeggiation of the referential harmony ToU, in the manner of III in tonality, nor forms a "step" with the framing points, in the manner of II and IV. This raises the question of the status of the bass-line 3s in relation to the harmony/voice-leading condition. Since interval class 3 is absent from the locally referential U harmony, a simple solution to this problem is to interpret these 3s as special kinds of voice-leading intervals, by adjusting the "default" borderline between small and large intervals in the application of the proximity principle; a similar interpretation may be applied to the counterpointing top-voice 3s. According to this interpretation, the T3U is a passing chord.

Example 8e is a detailed foreground graph of mm. 45. Points worth observing in mm. 1-26 include several h-neighbors and h-passing tones, and the effective use of dissonant fb-intervals 3, 5, 9, and 11 as suspensions, accented passing tones, and a double neighbor.

The 7-inverted ToA+9 is established in mm. 27-29 in a subtle way so as to point to E as the root, on the one hand, and to B as the locally governing bass, on the other. Justification for viewing this harmony as an inversion is given, in the first place, by the large-scale arpeggiation E2-B 1-El (Example 8b). Significantly, however, there is also a local reference to the root at the introduction of the harmony, made by the bass motion B 1-E1-B 1 (Example 8e, mm. 27-29; interval numbers refer here to the root E). Despite the significance of this reference, several features, such as the metric strength of mm. 27 and 29 and the clarity of the accompa-

227

(a) (b) (c) 008 (LTinstead of

coPs.PT ")dis -.

FI IN(H) 6, 7

1, TU- T3 I

TOU j

dm+9 T 3 TA9 wTUm9 T 6U- T6 A+9 (LT spacing) TU TU7 T7

--- 0( T - e

x (N cons.)

(d) T y

P9 A T3A T ) T3A T6A6 0A TO)

u 0

eq.div.

TOP T3P T6P T6U

ToU- T3U- T6U ToA+-9

(e) (D - @

D,•• I."

PT(H)

TOA -

T6U A (4) T6U TOA+9

TOP T3P T6P 7(0) 7(o)

T0U T3U

T6U0 TOA'

667,6 7

x L

A susp. 1? ( (su

To A4 ToP+

+9 d

d+9

~(4) ToP''9 TTo4) 1T 4) To T9, ) T • T6A6) To (49) T A+9

TOA+9 7 . @

7 6 7

-• ,.

. _. '

Example 8. Vers laflamme, mm. 1-45: voice-leading structure

nying inner-voice neighboring figure F#3-G#3-F#3 (in circumstances roughly comparable to Example 5a), support the local predominance of B.40 Moreover, it is interesting to note that while the main principle of consonance would support the primacy of the root-position chord on E (m. 28), this is, to some extent, balanced by the proximity principle of spacing, since that chord includes a whole-tone, G#3-A#3, which "re- solves" to F#3-A#3 above B.41 Once the relationship of the 7-inverted A+9 to its root has been clarified in mm. 27-29, the harmony is, so to speak, temporarily emancipated from this relationship and assumes an indepen- dent referential status. For example, the chord on D in m. 30, a transposition of that in the first beat of m. 27, functions as consonant support for inner-voice neighbors F? and AM, with no reference to G, its "theoretical" root (Example 8e).42

Returning to the relationship between the opening T0U (m. 1-26) and the subsequent 7-inverted ToA+9 (m. 27 ff.), it may be observed that in terms of pitch classes the former is a subset of the latter, but systematic differences in their initial registrations enhance their contrast; see Exam- ple 8b. The most important voice-leading motions are the semitones that lead to the new pitch classes of the latter harmony, D-C# and A#-B. Both of them are highlighted by outer-voice presentation. D-C# is the structural top voice, and the inner-voice motion A#-B is, as already observed, transferred to the bass by arpeggiation. A remarkable feature in the voice- leading structure in mm. 1-27 is the way in which it prepares ground for subsequent thematic material. Top-voice progressions D-C# and D-F-Ab are adopted as prominent foreground motives in m. 27 ff., as indicated by x and y in Example 8d. Example 8e shows, moreover, a "concealed repetition" of motive y in mm. 33-45, which goes over a change of texture, creating an artful transition to a section with a developmental character.43 The A#/BK-B? relationship is also manifest on different levels, as clarified by the lowest stave of Example 8e. Premonitions of the B?, and the harmony above it, occur as neighboring chords in mm. 19 and 23, and the sforzato harmonies in mm. 33-34 and 37-38, in turn, hark back to Bk.44

The overall organization is sketched in a cursory fashion in Example 9a. It shows the main points in the outer voices, some significant inner voices, and some of the most important harmonies in actual registration. The main sources of the bass-line's drama are those already indicated in Example 8a-b: the Bb-B? motion and the downward arpeggiation towards El; for clarification, see the lower stave of Example 9b. The span governed by the 7-inverted ToA+9 is concluded by the reharmonization of the opening subject shown in Example 7b and by a new statement of the B-E-B bass motion (mm. 77-80). This is followed by a powerful renewal of Bb (m. 81: Eclatant, lumineux), prior to the arrival at the root-position ToA+9 (m. 95). The top voice consists of the registral ascent of C#, fb-interval 9, embellished with incomplete neighbors and arpeggiations.

229

IN(H) IN(H)2

(b)[

TOA+

(b) 1• 6 ]6

- I

6], ]7 IT- ~---~~

Example 9. Vers laflamme, overall sketch

Such a special position of fb-interval 9 is in keeping with its status as the only member of A+9 that is not a root support. As noted in the Introduction, rootedness is enhanced by voicing such tones above the root supports.

On a smaller scale, a pervasive bass-line technique is the chaining of equal intervals, 3 and 4 (Example 9a). In most cases such intervals pose a much greater problem with respect to the harmony/voice-leading condition than in the opening prolongation of ToU. The span governed by the 7-inverted ToA+9 (mm. 27-80) includes equal divisions of the octave in 4s and 3s. Both interval classes 3 and 4 are present in that harmony but the bass-line intervals do not compose an arpeggiation of it. To cite Howard Cinnamon's (1986) expression, this is the kind of equal-division technique that manifests a "dichotomy between the harmonic and linear dimensions."45 Such technique is by no means unique to Vers la flamme, or to Scriabin; whether or not we should regard it as a prolongational technique is a complex issue that cannot be discussed thoroughly here.46 In brief, two viewpoints seem essential. On the one hand, the lack of significant relationship between the governing harmony and the linear framework certainly weakens the sense in which the former is "composed out," or informs all aspects of organization. On the other hand, if the technique is carried out with such clarity as in the present case, the primary status of the governing harmony is not actually jeopardized. In terms of Straus's conditions, we might say that even though the technique may fail the harmony/voice-leading condition it clearly fulfills the scale-degree condition, i.e., creates no ambiguity with respect to chord hierarchy.

The conclusion of Vers laflamme is marked by a new kind of harmony, characterized by the substitution of fb-interval 11 (D#) for 10 (D?); see Example 9 and Example 7c above. As illustrated by the two staves of Example 9b, one may see a certain kind of parallelism between the 10-11 (D?-D#) change and the earlier 6-7 (Bb-B?) motion. The D?-D# change effects a 6-7 motion for the upper of the tritones superimposed in the opening P chord; hence it completes an overall sonorous process "from tritones to fifths."47 The two motions differ, however, with respect to structural significance. We have viewed the Bb-B? motion as a resolution of a large-scale "appoggiatura," but it does not seem appropriate to regard the D? as structurally subordinate to D#. The function of the final harmony might be compared with that of the Picardy third in conventional tonality: it highlights the conclusion but is otherwise of little structural conse- quence. The 6-7 and 10-11 motions also differ with respect to their effect on rootedness. The former increases rootedness, but the latter decreases it, since fb-interval 11 is not, under the present assumptions, a root support. (The comparison with the Picardy third thus does not apply to this perceptual aspect since substituting a major third for minor does increase rootedness.) However, it is not without interest that the registration of the D#5 corresponds with the fifteenth harmonic of El. While the direct root-

231

2 (antic.)

6 4 -

12

6 1 4 1012 4

10 0 0

0 N0 0

ToP T4U T3U ToU ToU

Example 10. Berg, op. 2/2, prominent chords with OH ordering

supporting effect of such a remote harmonic is negligible, it might perhaps have some indirect significance as the third harmonic of the fifth harmonic or vice versa, that is, supporting strong root supports as secondary roots; see brackets in Example 7d.48

2.2. Berg: op. 2, no. 2 (1909) Berg's op. 2, no. 2, "Schlafend trdgt man mich," is an oft-cited exam-

ple in analytical literature. Several authors have discussed the elements of Eb tonality emerging in mm. 9-12 as well as the pervasive use of the "French sixth" chord-chord P in the present terminology-elsewhere in the song.49 A previous interpretation of its voice-leading structure has been presented by Craig Ayrey (1982), whose discussion focuses expressly on the relationship between "tonal" and "atonal" aspects. Ayrey regards the "atonal aspect" as "predominant" (ibid., 192) but, nevertheless, shows the bass of an Eb "triad" in m. 9 as belonging to the "fundamental bass progression" (ibid., 197; "Level 4" in Ayrey's Example 4 is reproduced in Example 12c below); according to Ayrey, the function of this "triad" is to articulate "the high-level atonal progression" (ibid., 192) "by the allusion to tonality" (ibid., 200).

In the subsequent analysis, I shall view the structure more purely from an "atonal" perspective, demonstrating how it may be interpreted on the basis of the referential status of chord U-which, as in the preceding Scriabin example, is produced by the incorporation of fb-interval 2 into P-with no invocation of the tonal allusions as structural determinants.50

Viewing the tonal elements as non-structural allusions seems reasonable because of their temporary nature and because even in mm. 9-12 they are obscured to the extent that leaves the establishment of conventional tonality far from unequivocal. No triad or seventh chord actually materializes-whereas the other parts of the song manifest the prevalence of chord U in a straightforward way.51 This structural view in no way ne- gates the expressive significance of the tonal allusions. In my reading, the passage of mm. 9-12 appears as a complex and unstable, "dreamlike," collision of elements, whereas the ensuing chord on Db (m. 13, T3U) restores relative stability and offers retroactive clarification for the preceding passage; such a picture would seem to be well in accordance with the musical impression.

While several structural features in the Berg song are reminiscent of the opening section of Vers la flamme, an important distinctive aspect is spacing. In contrast to the Scriabin, in which the LT spacing prevails, Berg's op. 2, no. 2 features salient occurrences of chords P and U in the OH ordering (0, 4, 10, 2, 6), as illustrated in Example 10. These occurrences culminate in the precise realization of the OH spacing by the T0U (on Bb) in m. 15. This Example also illustrates that, despite all the com- plexity and the allusions to tonality in mm. 9-12, the melodic frameworks

233

(a) (b) (c) (d)0) ( 00 Op.2/2 Op.2/3 0 0 00 IN(D)T PT(H)IN 0

1( IN(D) IN

6 2 10 6 0 2 0 6 0 2 2 6 0 Piano: 6 0 (2] Voice: [6] 2 2 2 6 6 0

5 4eq. eq.

4arp. d---

PT

div.PT

(OH) T0U T6P ToU TOUT6P T3U ToU T5P T0U T6P T3U TIU ToU T5P

(e) ) 000(0( 0 00@( N(D)

atPT(H)

IN

ino 6 4 0 10!

Voice: [0] 2 2! 2 2 2 6 6 0

asPT 5s desc 5s descv. IsPT

ToP T OU T6P T4U T3U TIU TOU T6P TsP

ToU 6 -- - - - - - - - - - - - -) -• • -- "•"------------------------

asc. 5s desc. 5s desc. Is

TOP T5P TI TP T8P

TIPp T7P T2PT9P T4U TIiU T6U (ToP TIiU)

T5P T4U

%u T6P (T6U) (continued)

Example 11. Berg, op. 2/2, voice-leading structure

234

In2T4U: 2

antic.?-

^ L •T4 antic.? 7W

y x I y x y N(H)

PT/

T4U desc. Is

T3U (T2P) T3U T2P T1P ToU

asc. 5s

ToU TsP(6)T1QPT3P T8P TIP T6P T5P

Example 11. (continued)

in this passage hold a simple relationship with the surrounding T4U and T3U chords, by horizontalizing their contents. This relationship, rein- forced by the inherent capability of the bass tones D and DM to function as roots of the frameworks, gives a most significant clue to the structure.

What I shall be viewing as the main part of the structure is the progression framed by the To harmonies in mm. 1 and 15. At the end of the song there is another E1 that alludes to but fails to establish a tonal resolution of the harmony on Bb. I will treat the end of the song-shown after a vertical line in the graphs of Examples 11 and 12-as transitional, leading to the next piece in the opus. The graphs extend their scope to the beginning of op. 2, no. 3, viewing the organization in terms of the referential status of the ToU, although the latter song, in general, is based on different structural principles. A more complete picture of the relationship between the two songs calls for considerations from complementary

235

IN(D) PT PT(H) IN

1 2Piano: 6 4 0,

10!

(a) Voice: [0] 2 2! 2 2 2 6 6 0 eq.

PT div. PT

ToP TloU

T6P T4U T3U T1U ToU T6P T5P

T0U -

IN(D) PT Or: PT(H) IN

Piano: 6 4 0 10 (b) Voice: [0] 2 2 2 2 2 6 6 0

- r

_

PT IN(D) PT

ToP T 10U T6P T4U T3U T1U T0U T6P T5P

ToU

Cf. Ayrey's reading:

(C) pc sets: 4-25 4-25 4-25 4-25 ! 4-25 4-21 5-33 5-33 4-25 4-25 4-25

Example 12. Berg, op. 2/2, alternative interpretations

236

angles; some provisional ideas concerning this relationship are presented at the end of the analysis.52

As in Vers laflamme (Example 8d), the opening sonority in this Berg song is not chord U but its subset P, which is then completed by the integration of fb-interval 2 (major ninth) to form U. Such integration is first manifest, in a twofold sense, in the chord on Ab, TloU, in m. 2 (for a detailed foreground graph, see Example 1 If). Firstly, this harmony includes a local 2, Bb. Secondly, by virtue of the proximity principle, its top-voice C6 may be taken as forming an arpeggiation with the initial Fb6 and thus implying the integration of fb-interval 2 into the governing harmony on Bb.53 Such integration becomes more explicit in the overall structure. The large-scale top voice reproduces the opening E-C motion in enlargement, supporting the C (FB-interval 2) by the T0U in m. 15, as indicated in Example 1 la. On both levels, the motion is embellished by an Eb to become E-Eb-C; see Example 1 ic and small notes in Example 1 le. Eb functions as a dimming tone, a type of incomplete neighbor discussed in the Introduction. After m. 15, the top voice continues to Ab, extending, as it were, the downward arpeggiation of the OH-spaced T0U to the beginning of the next song (Example 1 lb). This continuation is elaborated by another incomplete neighbor, AM (the last note in the voice part of op. 2/2); see Example 1 ic.

The main top-voice events are counterpointed by the bass line Bb- E-Db-Bb; see Example 1 lb-c. As in Vers la flamme, the scale degree hierarchy is based on a tritone arpeggiation and its equal division, although the arpeggiation goes now both up and down and it is the downward tritone that is divided. After m. 15 the bass presents an ascending 10 divided to 5+5 (Bbl-Eb2-Ab2), an enlargement of the opening foreground bass motion (Example 11lie, small notes). Under the proximity principle, these bass-line 5s are not voice-leading intervals. Their function may be explained on the basis of their contrapuntal relationship with the semitonal motions in upper voices. The same applies to the 5s and 7s chained at the foreground at the beginning and the conclusion.54

Examples 11lid and lie show further elaborations of the frameworks discussed. The structural descents in both outer voices are equipped with passing tones. Prior to its descent, the bass line presents two parallel figures, Bb-Ab-(Gb)-E and E-D-(C)-Bb (see brackets), which, taken together, horizontalize the opening verticality ToP; the small notes

D3-C3-Bb2 (m. 4) in Example lie indicate a detail which adumbrates the latter figure at its start. A key role in clarifying the bass-line structure is played by register. In terms of pitch classes, the bass line in mm. 1-11 is a whole-tone scale descending from Bb to Bb, but the registration helps to determine that Gb (m. 3) and C (m. 10) are passing tones (Example 1 le).ss Moreover, register is one of the factors that suggests viewing the events in mm. 9-11 as structurally subordinate. The Bb2 in m. 11 is pri-

237

marily the upper tritone of E2 and thus does not establish the structural return of Bb; however, since it represents a retained inner voice (Exam- ple 11 Ic), the Bb2 may be understood to make a secondary reference to the beginning, as suggested by the beamed upward stems in Example 11d.56 The lower registral placement of Eb2 in m. 12, in turn, signals a return to a more fundamental structural level. It directs the focus to the Db2 in m. 13 and may be understood as a passing tone coming from E2.

Except for measures 9-12, this reading accords with the referential status of the U chord in an unproblematic way. Examples 11 d and 11 e model these exceptional measures in terms of temporal displacement. Example 1ld shows a hypothetical alignment between the bass Bb2 in m. 11 and upper voices C6 and Fb5; the Bbl-Bb2 connection (mm. 1-11) would serve as appropriate support for the Fb6-C6 arpeggiation in the piano part, which foreshadows the main descent in the voice. In the actual music, however, the C6 and F65 occur two measures earlier, above the D in m. 9, producing the salient T4U at the second entry of the voice part; see exclamation marks in Example 1 le. Above the Bb bass, in turn, is an anticipation of the dimming tone Eb--and, in fact, of the entire T3U (see the box in Example I If). This anticipation produces dissonances in m. 11, which may, by virtue of the Bbl -Bb2 connection, be understood as foreground manifestations of the dissonance of the dimming tone against the background Bb, and which heighten the impression of instability at the moment when the outer voices are just about the begin their structural descent.

What may seem unnecessarily complicated in this reading is the formation of the salient T4U in m. 9 as a combination of elements of different structural levels. According to Example 1 Id, its outer-voice fb-interval 2 (D3-Fb5) is not structurally connected to the chain of 2s which it apparently initiates. One might try to improve the reading by assigning a higher structural rank to the D in m. 9, as sketched in Example 12b, in alignment with a reproduction of the original interpretation (Example I le) in Example 12a. This alternative reading shows a direct voice-leading connection between the D and the Db in m. 13, the bass notes of T4U and T3U. There are problems with this interpretation, however. It breaks down the lucid relationship between structure and register as well as the parallelism between the bracketed bass figures. This parallelism, incidentally, also applies to the introduction of fb-interval 2: in both the TIoU (m. 2) and T4U (m. 9), this interval is formed by sustaining or delaying the octave of the preceding, structurally superior bass. All in all, while some kind of an associational connection between the T4U and T3U is clearly relevant for the music, I find the original reading (Example 12a) preferable as a description of prolongational relationships.57 The peculiar formation of the T4U as a coincidence of levels is, I would suggest, in

238

keeping with the expressive content of the passage: the dreamlike, rather than real, arrival at the "homeland" ("Heimatland").58

An important factor that reinforces the coherence of the present reading is the regularity in the structural interpretation of motives. In general, such regularity may be regarded as a prima facie criterion in prolongational considerations: while different occurrences of a motive do not always require identical structural interpretations, deviant interpretations presuppose clear justification by other criteria, such as harmonic stability.59 In particular, when harmonic organization is less transparent, as in mm. 9-12, motivic organization is all the more important for structural clarification. In terms of Straus's conditions, the invariance in reading a melodic figure produces special consistency in the realm of the embellishment condition, which, in m. 11, for example, offers compensation for the local failure of the consonance-dissonance condition.

As observed by several analysts, the main motives of this song are introduced at the outset by the piano and voice; see Example 13, a and d. In the present reading, the unaccented second and fourth notes of motive x are consistently regarded as non-structural. In mm. 9-12, this produces the frameworks that relate with the T4U and T3U harmonies, as illustrated in Examples 10, 1 if, and 13b. The special significance of the T4U stems from its capability to incorporate the untransposed x motive, to offer a fleeting "home" to it, so to speak. The overall top-voice line, on the other hand, might be viewed as a version of motive x adapted to fit to ToU (and to the harmonic series of Bb), as suggested by Example 13c.

The prevalence of T4U and T3U in mm. 9-12 is also supported by the use of motive y. Examples 13d and 13e, incidentally, illustrate a case in which two motivically related passages call for different interpretation; this difference is clearly supported by harmonic as well as rhythmic and metric circumstances. In the occurrences in m. 9-13, however, motive y preserves its original metric guise, which emphasizes its initial three-note figure. As indicated in Example 1 If, in mm. 9 and 11 this figure, moving in parallel 4s, outlines fb-intervals 2, 6, and 10 of T4U and T3U, respectively. Moreover, the connection between the upper voices of m. 11 and the subsequent T3U is underlined by the reappearance of the same transposition of motive y an octave lower in m. 13.

This reading of mm. 9-12 contrasts with Ayrey's, who regards the second note of motive x, EL5 in m. 9 and D5 in m. 11, as structural in this passage, presumably on the basis of allusions to tonal resolutions (Exam- ple 12c). Whereas the harmonic foreground in this passage is more or less obscure both in terms of chord U and in tonal terms, motivic considerations unequivocally support the present reading.

Another prima facie criterion that supports the present analysis is the correspondence between metric and structural weight. (This criterion is,

239

(a) (b) (c) variant of x?

Cf. x Cf. also

IN' ,

10 2 4 6 2 10

Within TgU: 1 [0] Within TIOU: 2

IN IN

Cf.

Within T3U:

4 N N(H) PT

PT N N(H) PT (e)(H)

N(H)

Example 13. Berg, op. 2/2, basic motives

of course, often overridden in tonal music by suspensions and appog- giaturas, usually on the basis of the consonance-dissonance condition.) The above readings of foreground motives manifest such correspondence. On a somewhat larger scale, the passing-tone functions in mm. 10 and 12 (Example 11 d-f) are in keeping with the hypermetrical weakness of these measures. In particular, hypermeter contributes to the impression of the Db bass in m. 13 as a relatively stable element, which overrides the suggestion of a tonal V-I relationship between the preceding Bb and Eb.60

While all these factors speak for the view that the "tonal" allusions in op. 2, no. 2 yield to an "atonal" background, the tension between these aspects is not over, but continues in op. 2, no. 3. Example 14a-b focuses on the relationship between the tritone (6) E/Fb-B1b and the fifth (7) Eb-B6b, intervals which may be taken as epitomes of the "atonal" and "tonal"

(a) (b)

Op. 2/2 Op. 2/3

Interpretation in present terms: TOU TP T3UT62) Tonal Tonal allusions: E

k":

V II(Fr.) V I interpretation: '(IIGer.+Fr.)(V9) I

(c) Op. 2/2 Op. 2/3: Provisional sketch (d)

(I. 6 9 1 ( Cf. also antic.

--l -- 5K. I -:4 F?,, 09 21P

•: ' aI Ger.576

V

T0U T6A(2)

V b 'II (Ger. Ger.+Fr.) I

Example 14. Berg, relationships between op. 2/2 and 2/3

241

aspects, respectively. From the tonal viewpoint, the tritone suggests a

blI-V relationship. In the former song, Eb flirts with but fails to establish a tonal resolution, but at the end of the latter song, tonality takes an indis- putable vengeance. Roughly speaking, the pair of songs offers first an "atonal" and then a "tonal" interpretation for similar harmonic material, with chord P-a "French sixth" on blI or a V75 from the tonal viewpoint- as a common denominator. While a proper treatment of the structural principles in op. 2/3 is beyond the scope of the present discussion, I would suggest that the tritone Bb-E/Fb and its resolution also underlies the entire bass line of the pair of songs, as illustrated in the provisional sketch of Example 14c. According to this reading, the structure of op. 2/3 is based on the composing-out of an augmented-sixth harmony on Fb/E and its resolution to the E' major triad. The tonal perspectives in op. 2/3 are complicated by the fact that its basic harmonic progression also evokes the key of Ab minor (in accordance with the key signature), as indicated in Example 14d. Despite this evocation, there are features-such as the low register of Fb (m. 4) and the consistency with which the bass-line events in both songs prior to the eventual Eb would seem to spring from the ToP or T6P-that speak for the high structural status assigned to the Fb.61

2.3. Debussy: Voiles (1908)62 The piano prelude Voiles, based on the juxtaposition of the whole-tone

and the pentatonic set, is often cited as an example of Debussy's non-traditional tendencies.63 However, two central analytical questions have remained without a fully satisfactory answer. First, how is the whole-tone material-which prevails most of the time, mm. 1-41 and 48-64-inter- nally organized? Second, how does the intervening pentatonic episode relate with the rest of the work? In the subsequent analysis, I deal with the first question on the basis of the prolongation of chord U. While such a basis is shared with the preceding examples, the structure of Voiles otherwise shows very different features.64 There are no transpositions of the U chord: the bass is constantly Bb (Tn markings will be omitted as unnec- essary). Instead, a central role in the organization is played by an internal tension within chord U, which, as preliminarily discussed in the Intro- duction, derives from the registral treatment of D (fb-interval 4). The special role of D is also relevant for the second question, the function of the pentatonic section, since the pentatonicism is triggered by semitonal voice-leading motions, which serve to articulate the temporal endpoint of D's impact.

Unlike the preceding examples, Voiles introduces no block-chord material at the outset. The opening harmonic framework may, however, be traced by interpreting the melodic motions according to the proximity principle, allowing for rhythmic and motivic features. In the whole-tone environment, the proximity principle may be based on a step-leap dis-

242

tinction within the whole-tone scale; hence the whole step is the only (and unequivocal) voice-leading interval.

In Example 15a-b, this analytical strategy is applied to the two subjects introduced at the opening section, the introductory parallel-third subject (mm. 1-5) and the middle-register melody in mm. 9-13. The latter subject, hereafter the A-melody, is the leading thematic idea in Voiles. It plays a central role in the formal design and also in introducing and highlighting some of the most important features in the overall pitch relationships.

According to Example 15a-b, both subjects are based on the horizontalization of the "augmented triad" Ab/G#-C-E. A similar interpretation has already been presented by Adele T. Katz (1945, 284-88)-whose reading of the beginning of the prelude is perhaps the most illuminating of the previous analyses of Voiles. While Katz's analysis may be adopted as a starting point, it remains to be supplemented in several important respects. A primary issue left unexplained in Katz's approach is the relationship between the almost ubiquitous pedal Bb and the upper-voice material. In the present terminology, the combination of the Ab-C-E triad and the Bb bass constitutes chord Q. The choice of BE as a pedal stems thus from its suitability as a root.

Another feature overlooked by Katz is the special role of D and its striving to be integrated into harmony.65 In the present terms, this means that chord Q tends to be enlarged to U by adding fb-interval 4-whereas in the preceding Scriabin and Berg examples chord P was enlarged to U by adding fb-interval 2. This tendency also corroborates the status of Bb as an ideal root: for only Ab, C, and E as upper voices, D and F# would be alternative bass tones completing chord Q.

While the structural significance of D is fully evident only in the large- scale organization, there is a distinct sense in which the D in the A-melody (m. 10) functions as a "seed" of the large-scale D. According to Example 15b, the D in m. 10 is non-structural, a neighbor to C (or a passing tone between C and E; in Example 17e below, occurrences of the A-melody are shown in a simplified form by beaming together Ab, C, and E). Sev- eral factors emphasize, however, the arrival at D: its relatively long dura- tion, crescendo and tenuto markings, and expectations created by the reit- eration of the Ab-Bb-C figure in mm. 7-9. Moreover, this arrival is highlighted by the concurrent reappearance of the parallel-third subject, which completes the local harmony to form chord U, in the OH spacing (with doublings of D and A6/G#); see Example 16, a comprehensive graph of the opening. For all these reasons, there may arise a momentary impression of D as the goal of the scalar ascent, as sketched below Example 15b. According to this impression, the framework would be the tritone Ab-D (10-4 in fb-intervals) instead of the "augmented triad" Ab-C-E (10-2-6).

Despite the salience of the D and the U chord, the impression of their

243

Parallel-third subject

(a)

S - P pia p

A-melody C E N ------- A

(b)..

PT or IN

A possible momentary impression:

A6 D

(c) -O ,,,, / "t J l ..J, -_U-

Example 15. Debussy, Voiles, thematic subjects

"4

(con 8va) 0 8 (diss)

Q (U) QU Q

b) 5

--4 5 _14

Example 16. Debussy, Voiles, mm. 1-26

structural primacy is overridden by subsequent events. The priority of the Ab-C-E framework is suggested, first of all, by the motivic parallelism between Ab-Bb-C and E-D-C, as bracketed in Example 15b. As illustrated by vertical alignment in Example 15a-b, there is also a subtle wider parallelism of phrase rhythm between the two subjects, which conforms with the indicated structural relationships. In each subject, the first and the third measure are rhythmically and motivically similar, and out- line a harmonic interval (see brackets), whereas the second beat of the second measure introduces a salient neighbor; both melodies reach a clo- sure in their fifth measure.66

The most decisive confirmation for the primacy of C4 and E4 over D4 is given by the registral connection between the A-melody and the conclusion of the parallel-third subject; see the dotted slurs in Example 16a. In mm. 13-14, the endpoints of the two subjects combine to form the first vertical occurrence of chord Q, in the OH ordering (0, 10, 2, 6), as a kind of cadential chord. The parallel harmonization of the Ab-Bb-C motive in m. 15 ff. (which foreshadows the second statement of the A-melody in mm. 33-37) further corroborates the status of the "augmented triad" and chord Q as primary harmonic entities.

All in all, the A-melody elegantly embodies two features with central significance for upper-voice relationships in Voiles: the primary significance of the Ab-C-E framework and the role of D as a "rival" of this framework and of C and E in particular. While D remains subordinate within the A-melody, the momentary suggestion of its structural significance is actualized in the events to come. The way in which the large- scale D grows out of the "seed" in the A-melody is illustrated by the alignment of Examples 15b and 15c and by the brackets between the staves of Example 16a. The harmonized Ab-Bb-C motive in mm. 15, 17, and 19 is followed in mm. 21-22 by D octaves, which echo the introduction of the original D, mark D7 as a registral extreme, and establish D structurally.

The overall structure was preliminarily discussed in reference to Ex- ample 6b and is more comprehensively sketched in Example 17. Graphs a-d limit their scope to mm. 1-54. The conclusion, sketched in Example 17e and, in more detail, in Example 17f, shows the emergence of elements whose relationships with the otherwise prevalent organizational principles are less transparent. I discuss the conclusion briefly after the main part of the analysis. Graphs c and d also include symbols for the formal design, which proceeds in close coordination with the voice-leading structure. Sections A, (mm. 1-221), A2 (33-37), and A3 (48-64) contain a statement of the A-melody; the two latter statements of this melody occur in a higher register (Ab5-E6; see Example 17e). Section B (222-32) has the character of a developing episode (for motivic connections between A, and B, see x and y in Example 16a). The passage labeled "transition"

246

(38-41) leads to section C (mm. 42-47), which is singled out by the shift to pentatonicism.67

The "rivalry" of D with the Ab-C-E framework is also manifest in the overall organization. The D7 (m. 22) initiates the constant presence of D as a salient element in lower-register accompaniment figures, which lasts until the pentatonic section C; see Example 17c-e. On the other hand, the opening G#5 initiates a large-scale projection of the Ab-C-E framework, whose goal, E6, is expressed by the two latter statements of the A- melody, as beamed in Example 17e. The "rivalry" of D is especially relevant for the relationship between these statements. In section A2, D6, the highest note of an ostinato figure, "disturbs" C6 and E6 by proximity (see especially m. 35),68 whereas the "undisturbed" statement in section A3 offers compensation.

The endpoint of the temporal presence of D is articulated by semitonal voice-leading motions, D-Db and D-Eb, which lead to the new pitch classes of the pentatonic section C (Example 17c-e). These new pitch- classes feature as the framing points of the quasi-glissando gestures that introduce the pentatonic set (mm. 42-43).69 Example 18a illustrates foreground connections between these gestures and previous events. The dimming-tone relationship D?7-Db7 in the registral extreme is underlined by a most remarkable parallelism that connects the externally con- trasting gestures at the framing points of D's temporal presence: the pianissimo octaves (D6-D7) in mm. 21-22 and the crescendo quasi-glis- sandi (Db6-Db7).70

Example 18b shows another token of Debussy's sensitivity to corre- spondences between small and large-scale organization. The form-defining D?-Db relationship is reflected in the sixty-fourth notes of m. 31, the only instance of foreground semitones. At this point, the large-scale Ab5-C6-E6 motion has just arrived at C and D reminds us, so to speak, of its existence through this exceptional figure. D6 functions here as an h-neighbor of C6, but a subtle connection with the D6 of the subsequent ostinato is suggested by the way in which m. 31 prepares for the ostinato, as illustrated by circles.

As observed in the Introduction, the role of D in the organization combines structural and temporal aspects in an unconventional way. Struc- turally, D belongs to the sonority (Example 17a) that is prolonged, or elaborated, on later levels (Example 17b-e). Temporally, however, the D7, the main manifestation of the structural D, is not "prolonged" after the dimming tone Db7; the temporal confinement of D to the middle parts of the form links with its role as a source of unstable lower-register Ds.71 Both aspects are essential for the description of the organization. For example, one might attempt to strip D of its structural status by reading the D7 as a transferred neighbor, as sketched beside Example 17c, but such a reading is less satisfactory because it does not do justice to the

247

4 4 IN(D) (a) (b) (c)

10 ratherthan: reg. tr 2_6 1

Section: Al B A2 C A

,6 N .4N(. (H 44(diss.)

NI N o N- N - N U Q U Q U Q Q

U- U

@ ?? ?? 0 0 10 4 IN(D) 10

(d) antic.?--------- -

ection: Al B/ A2 trans- CTI A' ition"

'teams------ Schmitz's

r, teams": "t" ' d" (pentatonic) ("d')"t"

? -o , Ii r[- / .VL-

6

- -- - -

------------i llf-o I•

(e) 2 T I _

r, o-rL II• IN(H)?

A-melody A-melody A-melody (disturbed / (undisturbed ("t")

6 2statement) statement) L 2 s2

se

U (df See (f)

---antticI

L;N ?;U N 0 Uf

Example 17. Debussy, Voiles, overall structure

D6 D7

D66 D?7

(a) cresc. molto

D Ep

D Dk

(b) *j

......

Example 18. Debussy, Voiles, the Dt-DM relationship

form-defining significance of the D-Db relationship. D is more than a non-structural tone, even though its temporal treatment resembles that of non-structural tones in conventional organization. On the other hand, the dimming tone Db and the accompanying Eb3 are structurally incomplete neighbors of D, but such an identification does not capture the essential function they have in breaking off the temporal impact of D. To quote Steve Larson's expression, the Db7 "displaces the trace" of the D7, the source of instability, without canceling the structural independence of D.72 (If D were led to one of its adjacent tones within the whole-tone collection, C or E, this could be understood as structural resolution.)

One element in the whole-tone organization remains to be discussed: F#, the only pitch class in the prevalent whole-tone set that is not a root support and that lies outside chord U.73 F# has its most significant function in the large-scale Ab3-F#3-Ab3 neighboring motion, which parallels the E-D-E motion above it, as indicated in Example 17c. This parallelism clarifies the non-structural status of the middle-register D. While D, fb-interval 4, is consonant under the main principle of consonance, it participates in a middle-register element which includes the dissonant F#, fb- interval 8. Example 17d shows in more detail the position of the "minor sixth" intervals Ab-E and F# -D in the overall organization; the linear relationship between these intervals is made explicit by their foreground juxtaposition at the beginning of section A3 (m. 48 ff.). Symbols "t" and "d" beneath the graph indicate one way to conceive these intervals. They refer to E. Robert Schmitz's notions of "tonic team" and "dominant team," as the basis of pitch organization in Voiles (Schmitz 1966, 133-36). Schmitz, like Katz, has recognized the primary significance of the Ab-C-E triad in this prelude. It constitutes the "tonic team," whose members are employed at "points of rest," whereas the remaining tones of the whole-tone set, F#-Bb-D, form the "dominant team" and occur at "points of tension" (ibid.: 133). Such descriptions would seem to apply well for several events on both the small and large scale.74 The F#-D dyad may be seen as the main large-scale agent of the "dominant team." Its role as an element of tension becomes especially evident in the "transition" of mm. 38-41, where it is alone prevalent, and where there seems indeed to be a con- centration of tension (see especially m. 40, Serrez) just before the out- burst of pentatonicism.75

In the concluding section of Voiles, the final statement of the A- melody (m. 50 ff.) is followed by the appearance of G#6 in m. 54, which corroborates the status of G# as the primary member within the G#-C-E triad.76 The ensuing G#6-Bb6-G#6 neighboring motion recalls the opening parallel-third subject, an association confirmed by the reappearance of the subject itself in m. 58 ff.; see brackets in Example 17e. While these top-voice features are readily understood in terms of the general principles of pitch relationships, the interpretation of the inner-voice material

250

in m. 54 ff. is more complicated." Example 17e-f shows a possible reading, which includes the revival of D in a less salient register. (The D3 is not structurally connected with the previous D7 but is rather a new manifestation of fb-interval 4.) The very end is also problematic. The last sounding interval C4-E4 represents the "tonic," but the "dominant" interval F#3-D5, which reappears in m. 62, is left hanging without leading to proximate pitches (G#3-E5 or E3-C5).78 The effect of the conclusion is, I would suggest, one of deliberate, "impressionistic" vagueness, which does not call into question the structural relationships in general.79

This analysis of Voiles helps to bring some analytical and perceptual premises into particularly sharp focus. For an experiment, let us view its pitch relationships in terms of an intervallic conception of a more standard approach to post-tonal analysis, i.e., pitch-class set theory. Two premises underlie such a conception. First is octave equivalence. Second is the treatment of other intervals in terms of the equal division of the octave.80 Under these premises, octave is the only interval whose special (psycho)acoustical properties are reflected in the conception. Such a conception does, in fact, shed considerable light on the organization of Voiles. A fundamental feature, introduced by the A-melody and with crucial ramifications for the overall organization, is the "rivalry" between the "augmented triad" Ab-C-E and the tritone Ab-D (Example 15b-c). One may readily account for these frameworks on the basis of the equal division of the octave into three or two parts. Moreover, these frameworks are the only options in the whole-tone set for satisfying an octave-generalized version of the proximity principle, i.e., avoiding interval class 2 alto- gether.81 Against the "rivalry" of D with C and E, the pentatonic set may be understood as an escape from this "rivalry" by the replacement of all these pitch classes by the intervening ones, D6 and Eb. However, an element which seems to lie beyond this line of explanation is the pedal Bb. Neither octave equivalence nor equal divisions of the octave illuminate its relationships with the upper-voices frameworks, whereas both frameworks are readily found from the harmonic series of B6 (see brackets in Example 17a).

As a particularly interesting listening experiment I would suggest comparing the two chords formed by the combination of Schmitz's "tonic team" and "dominant team" with the pedal Bb; see chords Q and R in Example 19. In Voiles, chord R functions as a dissonance in relation to Q. From the point of view of octave equivalence, this order of stability seems strange: in terms of set classes R (3-12) is a subset of Q (4-24). The situation becomes even more strange if we view the chords in terms of the intervallic conception of conventional tonality-the conception which presumably has the strongest cultural effect on our way of hearing. In conventional terms, all fb-intervals in chord Q-minor seventh, major ninth, and tritone-are dissonances, whereas those in chord R-minor

251

Q R

Example 19. Debussy, Voiles, comparison of two chords

sixth (ignoring enharmonic notation), octave, and major tenth-are consonances! Nevertheless, I would suggest that there is, in accordance with the chord functions in Voiles, a perceptual aspect in which Q sounds more "stable," "satisfied," or "harmonious" than R-or, more cautiously, at least not so much more "dissonant" as the above considerations would imply. If, as I think, this can be explained by the better conformity of Q with the harmonic series, this supports the hypothesis that fb-interval 6 has perceptual relevance as an approximation of the eleventh harmonic, since otherwise the explanation would be much weaker.82 Ultimately, however, this hypothesis is not indispensable to the root status of Bb in Voiles, since in any case it is supported by D (fb-interval 4) in addition to Ab (10) and C (2), an aspect set aside in this experiment.

At this point there is need to consider the issue of registration in Voiles. To understand this issue, we have to allow for aspects of both structural clarification and foreground sonority. The structural prominence of the strongest root supports, D and Ab, is clarified by their high registral placement in the overall structure (Example 17c), which, of course, contrasts with the registration in the harmonic series. However, spacings closer to the harmonic series occur in foreground chords at strategically important moments. As discussed above, the "seed" of the structural D7 is the D4 in m. 10, which is accompanied by the OH spacing. The connection between D7 and D4 (the fifth harmonic) is confirmed by the reappearance of a similar spacing immediately after the D7 (m. 22; see Example 16 above). Instances of the OH ordering (4, 10, 2, 6) in other registers are indicated by numbers in Example 17e. Moreover, the effect of F# as a dissonant "root detractor" (Parncutt 1988, 87 ff.) is enhanced by its location next to the bass in octave 3.

All in all, Voiles appears as an ingenious exploration of the structural potential of the whole-tone set, utilizing aspects that derive from both equal divisions of the octave and the harmonic series. This raises the question whether similar principles of organization are to be found elsewhere

252

in Debussy's music. The connection between harmony and the harmonic series and the attendant root phenomenon certainly have a more general relevance for Debussy (often in a much more obvious way than in Voiles).83 As regards Debussy's use of the whole-tone set, however, organization similar to the present one cannot always be traced. The predominance of this set in Voiles is exceptional. In general, the focus tends to be more on its relationships with other kind of material and less on its internal struc- turing. Nevertheless, whole-tone organization comparable to that in Voiles does occur, if not at the highest structural level. Example 20b-c, presents two examples. The introductory cadenza of L'Isle joyeuse shows particularly remarkable convergences with the opening of Voiles, as indicated by the alignment of Examples 20a and 20b. The main difference between these cases concerns structural order. In L'Islejoyeuse the tritone 4-10 is structurally superior to the "augmented triad" 10-2-6, in reverse to the circumstances at the opening of Voiles. Moreover, after the cadenza, fb- interval 7 (the fifth) is incorporated into the harmony; hence on the whole L'Isle joyeuse focuses on stronger root supports (closer harmonics) than Voiles.84

2.4. Webern, op. 3, no. 1 (1908-9) In my final example, Webern's song op. 3, no. 1, "Dies ist ein Lied,"

the primary harmonies, chord A and its subsets, are somewhat less apparent as surface sonorities; in fact, none of them materializes as a root-position verticality. The clearest clues to their significance are found by ana- lyzing the melodic frameworks and their intervallic relationships with low bass tones. The recognition of the primacy of these harmonies leads to a coherent view of the voice-leading structure but does not alone provide a fully satisfactory picture of the harmonic organization. In terms of the primary harmonies, several prominent-looking foreground chords appear as opaque conglomerates of pitches representing different harmonies and of different structural rank. It is, however, possible to identify other kinds of regularities in chord formation; hence the Webern song exemplifies the possibility to combine the main principles of the present approach with other, subsidiary principles to create a more varied sonorous surface.

Prior to any prolongational considerations, it is worthwhile to survey the melodic material in purely motivic terms; the relevance of motives for prolongation has already been observed in the previous Berg example. The remarkably dense motivic organization of the Webern song is illustrated in Example 21.85 The connections at the "middleground" and "background" levels in this multileveled reading are not based on the conditions of prolongation but on factors such as rhythmic emphasis, registral and phrasal location, and, very significantly, similar positions in lower- level motives; hence it exemplifies the kind of associational approach that was advocated by Straus (1987) as a more viable method for dealing with

253

Voiles o

(a),

L'Isle joyeuse o

Ce qu'a vu le Vent d'Ouest 46

(c) I

P U U

Example 20. Debussy, Voiles and two comparable cases.

post-tonal large-scale organization.86 The melodic line shown in the Ex- ample comprises the entire vocal material and the top-voice notes of the piano interpolations that occur during the pauses in the voice part in mm. 5 and 7-8. The integration of these interpolations into the line is assisted by semitone and whole-tone relationships between the parts, as indicated by straight lines.87

Occurrences of two four-note motives, x and y, are shown. A "minus" sign indicates the absence of the last note in a motive. At the "fore-

254

Section: Ai B

(Rhythm) ;TRx

"Foreground" ToY

-1

lY T1 y Piano - -

"Middleground"

(Rhythm) o.T0

x x

"Background" ToIx-

A2 ToY

"Fgd'

Piano- T

••T1yTO

"Mgd." (T Ix--) - T

"Bgd."

**

Retransition chord

Example 21. Webern, op. 3/1, motivic organization

ground," motive y occurs at the beginning of each phrase in the framing sections, A1 (mm. 1-5) and A2 (mm. 8-12); moreover, motive x is em- bedded in the framing occurrences of Toy (mm. 1-2 and 9). The con- trasting middle section, B (mm. 6-7), features a registrally transformed retrograde of x. The "middleground" and "background" consist of versions of motive x and its shortened forms.

An important nodal position in the organization, matched by formal and rhetoric significance, is occupied by the piano chord leading from section B to A2 (mm. 7-8), hereafter the retransition chord. At the "middleground" of the framing sections, the first three notes of motive x (= x-), at To (D-Db-Eb) and T11 (C(-C0-D) determine the transpositional levels of "foreground" occurrences of y. The top-voice C# of the retransition chord is an exception to this scheme, since it does not initiate a "foreground" statement of y. As indicated by the boxes in the Example, the chord itself compensates for this absence, as it verticalizes the pitch classes of the second vocal phrase (m. 3)---except for E, which is saved for the subsequent phrase (m. 8).88 Hence the retransition chord helps to create a cogent association between the Db/C#s in the "middleground" and "background" Tox(-), without a literal repetition of the T1iy.

While such a motivic reading sheds considerable light on the organization, significant structural aspects remain to be revealed by allowing for the conditions of prolongation. As a first step in this direction, let us interpret the first vocal phrase according to the proximity principle, allowing for rhythmic relationships. A similar strategy was applied to the opening subjects of Voiles (Example 15a-b); however, whereas in the whole-tone environment of Voiles it was natural to regard the whole-tone as an unequivocal voice-leading interval, in the Webern semitones abound, which gives more reason to consider the possibility that whole-tones might stand for arpeggiations. Example 22 shows three alternative readings of the phrase. Reading (a) regards whole-tones strictly as voice-leading intervals and reading (b) as harmonic. Reading (c) combines elements of (a) and (b), viewing Gb-Ab as harmonic but D-E as voice leading. I shall later formulate pitch-based principles for this reading; at present we may ob- serve that it displays the most complete correspondence between rhythmic and structural weight.

As the next step towards a prolongational analysis, let us relate the opening melodic framework with the bass E2. This yields an important clue to a consonance-dissonance system: the fb-intervals are those of chord U (10-2-4-0-6). According to readings (b) and (c), the phrase arpeggiates that harmony; according to reading (a), it progresses from Q to P.

Example 23 provides preliminary illustration of salient elements in the piano part that support the notion that ToU (with E as the root), it subsets, and also ToA, are harmonies with central significance for the organ-

256

A r-3-- r3--

10 2 4 0 6

(a) IN(D) IN IN PT(H) or or PT PT'

ToQ ToP 10 2 4 0 6 ToP--

(b) IN(D) IN IN or or PT PT

Sr r TOU 10 2 4 0 6

CIN(D) IN IN PT(H) or or PT PT

ToU---

Example 22. Webern, op. 3/1, alternative readings of the opening phrase

ization. At the end of m. 3, ToA occurs as a block chord.89 This chord is a 4-inversion, but hearing it as connected with the opening E2 is assisted by the absence of bass-register material between them. Incidentally, there is also evidence that the connection between E2 and G#2 was an essential part of the composer's own conception of the music, as a preliminary version of the song included a slur going from a repetition of the E2 (m. 2) to the G#2.90 The piano material in mm. 1-2 may also be interpreted as subordinate to the ToA, albeit in a rather complex way (for details, see

257

(a) (b)

TOP horizontalized

Cf.

, chod r6 ~

-

[T0- ToP [o I] 8 bassa

ToA To- ToA

Example 23. Webern, op. 3/1, manifestations of ToA: preliminary observations

Example 27 below). In the concluding section, the 4-inverted ToA makes a return and prevails up to the end with less complex elaboration. As illustrated by Example 23b, the spacing of the five lowest tones adheres here to the harmonic series-although the implied fundamental (EO!) is too low to have musical utility. As suggested by beams in Example 23a, the concluding G# bass may be understood as produced by a large-scale bass arpeggiation, E-G#, whose association with the small-scale arpeggiation in mm. 1-4 is enhanced by the participation of D in both cases (presuming that the whole-tone E2-D2 may be regarded as an arpeggiation). Register is an important clarifying factor also for the large-scale organization: certain registers are reserved for certain structurally connected elements. All material in the register between D2 and F#3 participates in the arpeggiation within ToA on different levels, except for the upper voices of the retransition chord, which would appear as lower neighbors to ToA; see lines in Example 23.

All the material in Examples 22 and 23a display a consistent internal order of structural importance within ToA. The subset with the greatest prominence is ToP ({ E, G#, D, Bb }). Fb-irterval 2 (Gb) plays a less prominent role and 7 (B?) still less. This order is clearly manifest at the conclusion of the song in which ToA shrinks, via ToU, to ToP.

In extending the interpretation to the remaining material in the framing sections, the simplest way to exploit motivic regularities is to transfer the reading of motive y in the first phrase to its other occurrences. Such an interpretation is carried out in Example 24a. The fb-interval numbers above this Example relate the melodic frameworks with the bass tones shown on the lowest stave (other piano material is left out of consideration for the time being). This interpretation involves complex tem-

258

A reading suggested by the use of motive y Y Y Cf. Voiles (Ex. 15b)

y y y I - I I I

10 2 4 (o)

6 10 2 4 0 4 6 In relation to D: 10 2 4 0 4 6 10 2 4 In relation to E: 10 2 4 6

(a)

ToA T, U T10UT

Better, in retrospect 7 9 (10) (8) 11 In relation to D: 10 2 4 0 6 2 10 2 4 6

10 2 4 (0) 6 10 2 (4) (0) 6 2 In relation to E: 0 2 10 4 0 6 IN D)

PT

(b) b. ..- liar[", L- --- -" V/ A

ToA T,1Q TIo

1

A

SIN(D) 8_....

Example 24. Webern, op. 3/1, readings of phrases in the framing sections

poral relationships. The El1 bass coincides with the apex point of the framework in mm. 3-4, and the D2-the bass of the retransition chord- precedes the framework to which is connected. However, the reading is supported by the striking regularity of fb-intervals it reveals. In relation to both Eb (T11) and D (TIo), the fb-intervals of chord U are produced, corroborating the earlier clues to a consonance-dissonance system, and even repeating the original succession, 10-2-4-0-6. The end of the vocal line is interpreted as belonging to the 4-inverted ToA discussed above (the interval numbers relate to the root E rather than the actual bass G#). The point of the change between the last two harmonies (m. 8 ff.) seems ambiguous, owing to four pitch classes shared by TIoU and ToA (D, E, F#, G#). This ambiguity is reflected in the crossing slurs and in the over- lapping sets of numbers. In any case, the main determinants of the change, C (v ToA) and Bb (0 Tl0U), occur conspicuously at the framing points of the concluding A2 section.91

While Example 24a shows a simple relationship with the motivic organization, it may be refined in the way shown in Example 24b. The difference between these readings in the structural significance of tones is indicated by the size of the interval numbers above the staves. In reading (a), the fb-intervals of chord P, 10-4-(0)-6, predominate in each framework, since 2 only occurs in the midst of larger 10-4 arpeggiations (mm. 1-2, 3 and 8). In reading (b), on the other hand, the apex tone of y in mm. 3 and 8 and the headtone in the subsequent measures are understood as participating in a "stepwise" motion going over a motivic and phrasal border. This means that the predominant fb-intervals in the T11 and T0o harmonies are those of chord Q, 10-2-6, in the OH ordering, instead of P. In the

T•IQ (mm. 3-5), the framework Db4-F4-A4 occurs

in the register of harmonics 7-9-11 of Eb 1, which, under the present perceptual assumptions, reinforces the connection between the framework and the bass. (If Eb4 and G4 are included in the framework, they supplement the pattern by harmonics 8 and 10.) Incidentally, this connection and the primacy of the Db4-F4-A4 framework is also underlined by a motivic feature: the triplet F-E-C# from the beginning of the melodic framework reappears in the piano concurrently with the E 1 and the A4 (m. 3, beat 2); see asterisks in Example 28c below.

Although the reading of Example 24b ultimately accounts better for the voice-leading motions, this does not totally cancel the relevance of Example 24a. The present situation invokes temporal perspectives that are reminiscent of the A-melody in Voiles; see the right edge of the Exam- ple (cf. Example 15b). Just as in the Voiles excerpt, the melodic motion in the T,I and T0o harmonies starts from fb-interval 10, and fb-interval 4 makes, at its appearance, the momentary impression of a goal tone. Only subsequent events point to the primacy of the "augmented triad" 10-2-6 over the tritone 10-4.

260

At this point, the status of melodic whole-tones needs to be taken up. From the perceptual standpoint, the most appropriate model in this issue might be one that allows for a gradation between purely harmonic and purely voice-leading functions, depending on the combined amount of proximity in a melodic figure (cf. endnote 36). For example, the Gb in m. 2, which occurs within the tritone arpeggiation D-Ab, has a whole-tone relationship only with Ab, whereas the G0 in mm. 3-4 is surrounded by two whole-tone relationships, with both F and A (Example 24b). Hence we could regard the G6 as "more harmonic" than the G0, but even the latter could be understood as having some degree of presence in the harmony. For the sake of simplicity, however, I will abide by the two-alternative model, which is adequate for the present purposes if the following rule is observed:

Let P1-P2-P3 be a three-note succession, in which P2 is subordinate to pi and P3. If there are two or three "stepwise" intervals (= smaller than 3) between any two of the pitches, then P2 is a voice-leading connec- tive. If, however, there exist one whole-tone and two larger intervals between pl, P2, and P3, then P2 participates in an arpeggiation.

Under this rule, fb-intervals 4 and 0 in the T11 and T10 harmonies are passing tones, as indicated in Example 24b; hence these frameworks are based on Q instead of U. In fact, the only instances of arpeggiating whole- tones under this rule are the GL4-AL4 in the first phrase and the E2-D2 in the large-scale bass arpeggiation (Example 23). Both participate in the composing-out of the untransposed referential harmony, ToA, which might be taken as an additional factor supporting their harmonic function.

All in all, considerations of the melodic frameworks and their relationships with the low bass tones suggests that the material in the framing sections is based on four harmonies: T0U and T11Q in the opening section, and T10Q and 4-inverted ToU in the concluding section. A similar examination of the middle section (m. 6-7) is undertaken in Example 25b, which also graphs the piano material. Most significantly, this con- trasting section is not based on a subset of A, but on a harmony that includes fb-intervals 3 and 11 as well as 2 and 6 (Example 25a).92

On the basis of these preliminary considerations, we are ready to deal with the structure in a comprehensive manner, as illustrated in Example 26. As already observed, register plays a key role in the clarification of the structure. The correspondence between register and structure is not, however, as straightforward as in the bass line of the previous Berg example (Example 11), which displays a direct correspondence between lower registral location and greater structural weight. In the Webern, registral expansion is characteristic of the middle phases of the composition, which nevertheless have to be regarded as structurally subordinate to the framing events. In the following, the "ninths" that connect the two lowest bass

261

IN and antic ?

P PT

L IN and

antic'

2 PT

IN(H)

---------------.----

q: -

-PT(H)

8oTIo

TT0Q

Example 25. Webern, op. 3/1, middle section: foreground

(b) (c) 0( IN IN(D) PT Cf cons. Cf. diss.Cf. PTPTIN(D)IN

10 8 Ix 3 11

TA TA TA T IQ TA 8N(D) A ToA T Q T TA

IN(D)

I-11 *

8-8------------

ToA TTQQ T0A

IN(D) 4 ToA TI Q ToQ Toe

aanfic., -tc

antic. cantic.

ToA TlQ

TloQ To4

Example 26. Webern, op. 3/1, overall structure

263

tones, Ebl (m. 4) and Cl (m. 3) with the structural E?2 (m. 1) and D2 (mm. 7-8), respectively, are interpreted as enlarged voice-leading intervals (1 and 2). The same applies to the interval between the initial D4 and the Eb5 (m. 6) in the vocal line; this is also clarified by the actual semitone D6-Eb6 in the piano (Example 26d).93

Example 26a presents the large-scale arpeggiation that connects the root-position and the 4-inverted ToA; the outer voices project ToP. Exam- ple 26b supplements the bass arpeggiation by D, the bass tone of T10Q. This harmony supports the upper-voice passing tone C and is thus analogous to the dominant in a tonal Ursatz (cf. also the T6U in the opening section of Vers laflamme; Example 8c).

Example 26c adds two prominent incomplete neighbors to the upper voice, Eb and C#, beaming them together with the structural descent. The D-Eb-C# figure is an enlargement of the initial foreground motive, the "background" Ix- of Example 21; the voice-leading analysis shows that the large-scale Ix is completed by the final Bb. The bass C, which coun- terpoints the treble Eb in the middle section, is also a kind of incomplete neighbor. As suggested by the S-formed slur, its function is comparable to that of the subdominant in tonality: it leads to the "dominant" D by an ascending whole step.

The principal top voice progression from D4 to Bb3 is carried out in the vocal part. The graphs from (d) onward indicate vocal and piano material separately, and add inner voices when relevant. Graph (d) illustrates the symmetrical clarification of the double-incomplete-neighbor figure D-Eb-C# by motions in parallel 4s (major thirds) in characteristic registers above and below the vocal part. The inner-voice motion F#3-F?3 continues to E3 and D3 and thus parallels (though not particularly saliently) the entire top-voice progression in lower 8s.94

Example 26e shows two more incomplete neighbors: the Eb bass in m. 4, counterpointed by the Db in the treble. The harmony on Eb contains, apart from the T1'1Q, a "pedal" of D3 and F#3, which is retained from the opening harmony up to the end of the middle section. The fb-numbers in Example 26c-e illustrate a most noteworthy feature in the harmonic organization, a manifestation of a more nuanced correspondence between structural function and chord construction than the two-alternative consonance-dissonance system. Prominent incomplete neighbors are consistently associated with fb-intervals 3 and 11, which are thus better viewed as constituting a special "incomplete-neighbor sonority" than as arbitrary dissonances. This observation offers additional illumination for the choice of C as the bass tone against Eb and B. The harmonies on Eb (m. 4) and C are (m. 6), in fact, similar in terms of fb-intervals, except that 10 does not occur on C.95

Example 26f shows further elaborations. The harmonic connections in both framing sections are characterized by complex temporal relation-

264

ships, which make the point of harmonic change more or less ambiguous. I have indicated these points at the last possible moment, in which the melodic frameworks of T11Q and T10Q reach their apex. Consequently, both the T11Q and the 4-inverted ToA are preceded by extensive anticipations. In the piano chord on the strong beat of m. 4, these anticipations produce another example of the "incomplete-neighbor sonority" (bass D + upper voices C# and F); this is a common subset with the retransition chord, which, as suggested by asterisks, enhances the association between small-scale and large-scale events for both the Db/C0 dimming tones and the Ds in the beamed bass arpeggiations.

A significant upper-voice progression in the opening section (mm. 1-5) is D-E-F-Gb. While it may be traced from the vocal part, as indicated by lines in Example 26f, it is brought to the fore an octave higher in the piano, as shown by a beam. The lines also indicate how the motion goes on to Ab so that one may see an enlargement of the opening D-Gb-A6 framework in the overall structure (a concealed one, to be sure). The

D5-E5-F5-Gb5 progression is paralleled in lower 4s, Bb4-C5-C#5-D5, in the piano. An interesting feature of these parallel progressions is their retrograde relationship with the large-scale progressions shown in Exam- ple 26d. The small-scale upper voice retrogrades the large-scale inner voice and vice versa (see also Example 28b below).

As noted in the Introduction, structural analogies with tonal organization do not necessarily apply to cadential effects. The last harmonic change, from the T10Q to the 4-inverted ToA, demonstrates this in a rather extreme manner. The structural function of the T10Q is analogous to the dominant in the Ursatz, but its progression to the 4-inverted ToA has a character that could hardly differ more from the decisiveness of a perfect authentic cadence. This character stems from the creeping temporal effect and also from the outer-voice elements themselves. The opening harmony returns only as an inversion (more akin to 16 than to I) and the top-voice is directed to Bb, the weakest root support (fb-interval 6 of E), which is a goal but by no means a tonal center like the 1 in tonal music.96

As observed in the beginning, the issue of harmony warrants considerations that go beyond the main rule of consonance and the recognition of chord A and its subsets as primary harmonies. Significant supplementary principles regulate the formation of secondary chords whose relationships with the primary harmonies are complicated. One such principle, the "incomplete-neighbor sonority," has already been identified. Still another principle is especially evident in the opening section. This principle is not based on fb-intervals, but comprises the use of ro-interval 11 ("major seventh") between elements of relative structural importance and of ro-interval 1 ("minor ninth") as a "dissonance" relative to 11. As I suggested in Viisali 1999, such a principle assumes primary significance in some other music of the Schoenberg school. Example 27, a foreground

265

antic.

acc. FT IN(H) IN(D)

IN(H) a

a . N)IN(D)

1 11IN(D)

11

.

113 1

[•.

13 11 antic.-

------- ---

IN(D)

8- -

ToA Tl 1Q

Example 27. Webern, op. 3/1, opening section: foreground

graph of the opening section, shows the is as 13s, in accordance with their actual width.97 The importance of Its is especially evident in the parallelism between the high-register chords in mm. 1 and 3 and their middle- register correspondents in the subsequent measures (see brackets)- whereas in terms of the primary harmonies there is no analogy between these chords. Only one of the 1 Is shown in the Example, B?3-Bb4, belongs to a primary harmony (ToA), as others combine elements of primary and secondary structural rank. The relative dissonance of 13, on the other hand, clarifies, for example, that the F5 in m. 3 bears a passing function despite its metric strength.

Another issue worth taking up is the relationship between the prolongational and motivic-associational aspects of organization. As in tonal music, the two aspects are best viewed as autonomous but subtly inter- acting. Motivic regularities (Example 21) have a role to play in tracing the voice-leading structure, but only some of the motives, such as the "background" Ix, have been "confirmed" by correspondence with voice- leading relationships. The lack of such confirmation does not, however, invalidate a motivic relationship. For example, even if we perceive the

Eb4 in m. 4 and the D4 in m. 9 retrospectively as passing tones in the voice-leading structure (Example 24b), this does not cancel their significance in the "middleground" motives, Tox and T11x (Example 21), but rather adds to the interest of the situation. Moreover, the D4 in m. 9 has obvious associational significance-not evident in Example 21-in ini- tiating the recapitulation of the opening phrase, in circumstances reminiscent of non-structural tonics in tonal recapitulations.98

Voice-leading considerations also give feedback to the view of the motivic organization, by providing a rationale for revealing more concealed relationships. From the perspective of voice leading, a particularly significant family of motives grows out of the dimming-tone figure com- bining 4 and 3 ("major" to "minor" third), introduced by the first two high-register chords; see Example 28a. Linearizations of this element, including retrogrades and inversions, play an important role at all structural levels, as exemplified by Example 28b-c. (This element is shared by motives x and y of the original motivic interpretation; see Example 28d.)

While the almost incredible wealth of purely musical relationships in this song is by no means exhausted, I will conclude the analysis with a few remarks on the relationship between musical organization and text expression. The musical setting seems to reflect and amplify a dichotomy suggested by the poem. This dichotomy, introduced in the first two verses, is one between the song ("This is a song...) and the one to whom it is directed (...for you alone."); see Example 29a. This dichotomy is reproduced in the whole of the poem. Up to the point where the retransition chord occurs in the music, it describes the song, the feeling it contains,

267

(a) IN(D) (b) (C) *4 (d) 33

4

4

f "%

4 •

Piano

..• *,*

8

4

Example 28. Webern, op. 3/1, motivic ramifications of the dimming-tone figure

31 .-

3 L

(a)

Dies ist ein Lied/ fUir dich al - lein:

song direction

retr.

8.--------

registral width, rhythmic activity,

(C) amount of elaboration

Example 29. Webern, op. 3/1, elements of text expression

and its sounding through "morning gardens." After that, it returns to its longed for direction. In the music, this is the point in which the upper voice sets off on its structural descent towards Bb; see Example 29b. In general, the song is connected with the musically elaborative and the direction is connected with the structural. By structural I refer to the quasi- Ursatz of Example 26b, to the primary consonances, and, within ToA, to its most prominent subset ToP. Both in the first vocal phrase and in the overall organization, elements foreign to this subset leave off towards the end. The elaborative includes the incomplete neighbors, the supplementary principles of harmony, and the ascent towards fb-interval 2, D-Gb, which contrasts with the structural descent, D-Bb, and links with the song on both scales of organization. As illustrated in Example 29c, the overall upper-voice events are accompanied by other organizational aspects,

269

registral width and rhythmic activity, which realize a roughly similar curve, first rising, and then falling below the starting point.

All in all, the song/elaborative part (mm. 1-7) is coupled with increasing activity and liveliness, whereas the direction/structural part (mm. 8-12) is coupled with the opposite tendency and also by the lack of decisiveness that characterizes the final harmonic change. If we take the structure as standing for reality, the basic dichotomy appears as a contrast between imaginations and reality. The song is enlivened by wishful imaginations ("Wihnen"), which, as implied by the voice-leading structure, are touchingly removed from the real, insecure prospects of its reaching the one it yearns to move.99

3. Conclusions

On the basis of these analyses, one could almost speak of the emergence of a new kind of "common practice" at the time when the four works were composed. As in the old triadic practice, there is basic correlation between harmony and syntax, utilized by different composer per- sonalities for different stylistic and expressive ends. One can think of several reasons why the new "common practice" remained an "almost," that is to say, never yielded a corpus of works quite comparable to the triadic practice, or crystallized into compositional norms with equal historical hegemony. In general, modernist composers seem to have striven for a more comprehensive "emancipation of dissonance" than that enabled by the utilization of the weaker root supports. In particular, "rootedness"-the conception of harmony as a unified pattern governed by the bass-may have been sensed as antithetical to the linear independence of voices, a central striving in the Schoenberg school, for example.

Despite its limits, the scope of music illuminated by principles more or less similar to the present ones is not negligible. Of the four composers, organization comparable to the present cases is most readily found in the music of Debussy and Scriabin, although the details of structural principles certainly vary. Both composers also influenced the musical language of their time, and the chords in Example 1 are by no means uncharacteristic of the harmonic vocabulary of that period in general. For Berg and Webern, the works analyzed may be regarded as representing an experimental, transitory phase in evolutionary paths that were soon to lead in new directions.'00 However, as illustrated by the Webern song, with its combination of different harmonic principles, such harmonies may play a key role in organization even when they are less apparent as surface sonorities. The limits of the relevance of root supports thus cannot be drawn lightly on the basis of surface sound. In principle at least, root supports might enjoy some kind of special status in structural articulation

270

in music whose sound world is still more removed from my first three examples.

As for analytical approaches, the present cases demonstrate the significance of aspects not covered by the more established methods of Schenkerian and set theory. While the methodology is otherwise "Schen- kerian," the conception of harmony and consonance is non-traditional. The recognition of non-traditional options such as the use of 10 (minor seventh) and 2 (major ninth) as functional consonances would appear crucial for the analysis of music by composers such as Scriabin and De- bussy.101 And regarding post-tonal prolongation in general, some of the present principles, such as the proximity principle, are applicable also in cases in which rootedness and the correspondence with the harmonic series have little or no relevance for harmony.

The present conception of harmonies and intervals also includes aspects that deviate from pitch-class set theory. The premises underlying the intervallic conception of set theory-octave equivalence and the treatment of other intervals in terms of the equal division of the octave-were already considered in connection with Voiles. These premises lend a simplification to the theory and facilitate a systematic command of pitch collections. However, they also reduce information that is perceptually significant and potentially determinative of structure. In Vaisdli 1999, I argued that the first premise, octave equivalence, has to be restricted in considerations of (post-tonal) prolongation. In the present paper, I have focused on the second premise. While this premise is adequate for gen- erating the equal-tempered pitch space, it is less adequate for the description of the perceptually and structurally significant aspects that emerge in music employing that collection. Discussing some of the most significant music from the advent of post-tonal harmony, I have attempted to show that its organization is illuminated by allowing for the root-supporting status of intervals, a property that does not stem from their width in fractions of the octave but from their correspondence with the intervals in the harmonic series.102 While the precise limits of this aspect are best here to be left open, the awareness of its potential would appear essential for our analytic arsenal.

271

NOTES I am indebted to Lauri Suurphi for several detailed comments on this paper, to Richard Parncutt for discussion about the psychoacoustical arguments relevant to both this paper and Viisili 1999, and to Joseph Straus for his comments on an earlier version of the Voiles analysis.

1. On Rameau's notion of corps sonore, see, for example, Christensen 1993. 2. See, for example, Schenker 1954 (German original 1906), ? 9 ff.; Schoenberg,

"Problems of Harmony" (German original 1934), in Schoenberg 1975, 268-87; and Hindemith 1942.

3. However, in his commentary on Schenker's Free Composition, Carl Schachter suggests that while "one can reformulate Schenker's theory so that the overtone series does not figure in it at all," this would entail that "something important- even essential-would be lost: the connection between the simplest musical element, the single tone, and the masterpieces of the great composers in all their com- plexity" (Schachter 1999, 187-88).

4. On the conceptual history leading to pitch-class set theory, see especially Bernard 1997.

5. As regards compositional practice, the rehabilitation of the harmonic series is hardly in need of argument, in view of the recent "spectral composition" move- ment.

6. By "post-tonal," I refer to 20th-century music not based on the norms of conventional triadic tonality, even if it displays some other kind of tonal centricity.

7. Questions of intentionality-i.e., the extent to which the composers were aware of the connection with the harmonic series-are left outside the present discussion. While some sketch material of Scriabin, for example, suggests that he was con- scious of this connection, I think it is safe to maintain that the primary motivation for the use of harmonies derives in all four cases from the produced musical effects rather than the outward resemblance with the acoustic phenomenon.

8. For an extensive argument for the importance of temporal perspectives in music perception, see Lewin 1986.

9. Virtual pitch can be heard on the basis of just two harmonics, even when these are presented separately to two ears (Houtsma and Goldstein 1972). It should be noted that virtual pitch is a different phenomenon from combination tones, on which Hindemith (1942, 68-74) based his explanation of roots. For the superiority of the present explanation, see Parncutt 1988, 66-67.

10. In addition to pattern-recognition models, there are explanations of virtual-pitch or residue-pitch perception based on the timing of neural impulses. There is, in fact, experimental evidence for both explanations; experiments speaking for a pattern recognition model include those reported in Houtsma and Goldstein 1972 and Hall and Peters 1981 (see notes 9 and 13). For an overview, see, for example, Moore 1997, 188-208. In any case, though the pattern-recognition theory is especially suggestive of musical ramifications, the explanation of roots is not really dependent on it.

11. This estimate, which corresponds to a 3% shift in frequency, is based on the results of Moore et al. 1985. See also Parncutt 1988, 70-71.

12. An alternative root of the minor triad, sometimes utilized in early twentieth-century music, is the lower fifth of its conventional root. The notes of the triad corre- spond to harmonics 6, 7, and 9 of this alternative root.

272

13. In the experiments of Hall and Peters (1981), virtual pitches were evoked by nonsimultaneous harmonics, but the tempo of presentation was very rapid: three (sine- wave) tones in 0.14 seconds.

14. Houtsma and Smurzynski (1990, 309) divide harmonics into "those of low order and those of high order, with the separation somewhere between the 10th and 13th harmonic," on the basis of the more distinct virtual pitch produced by the former.

15. The results of Moore et al. (1985) on the effect of mistuned harmonics do not, however, reach harmonics as high as the eleventh. I am not aware of studies that would confirm the relevance of the equal-tempered approximation of the eleventh harmonic for virtual-pitch perception (though my own, very informal experiments support it).

16. It would seem that the significance of registration for sonorous character is greater for weaker root supports (especially fb-intervals 2 and 6, i.e., "major ninth" and "augmented eleventh").

17. Parncutt's root algorithm is based on octave equivalence, but he discusses the impact of registration verbally (1987, 87-89).

18. There is, of course, an enharmonic difference between these chords. Enharmonic differences play no role in the present conception of harmony and are left outside consideration in the four analyses.

19. For one example in conventional tonality, in which a dominant ninth chord, or a combination of V and II, carries an expressive function far from the usual conno- tations of "dissonance," consider the opening of Schumann's Mondlicht, op. 39, no. 5. Such a combination adumbrates the 20th-century usage mentioned in note 12.

20. James Baker (1986, 99) criticizes authors who "have persisted in explaining [Scri- abin's 'mystic'] chord as a derivation of the upper partials of the overtone series [...] in spite of the evidence that the harmony originated in the context of tonality." I can see, however, no contradiction between explanations concerning the (psycho)acoustical relevance of a harmony, on the one hand, and its historical origins, on the other.

21. E.g., Varvara Dernova (Guenther 1983) and Baker (1986, 28 ff.). 22. Terhardt speaks of two components of musical consonance: "sensory consonance

and harmony" (1984, 282). The former derives from roughness and the latter from virtual-pitch perception.

23. In more general terms, overall critical-band effects for complex tones tend to be minimized by simple frequency ratios (Plomp-Levelt 1965). This impact of frequency ratios is left outside the present discussion.

24. The perceptual factors discussed here--virtual-pitch perception, critical-band effects, and streaming-are also included in Fred Lerdahl's (1999) list of perceptual principles relevant for atonal prolongation. A difference between Lerdahl's approach and mine is that according to Lerdahl "for atonal music the chief measure of tension is psychoacoustic at all levels," (ibid., 23) but in the present approach psychoacoustic principles have more indirect relevance as background factors of syntactic rules.

25. Straus does not actually specify that a "scale-degree" system should be based on transposition but speaks of "some kind of hierarchy among the consonant harmonies" (1987, 7).

26. Even conventional tonality is not actually devoid of such nuances. 5/3 and 6/3

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chords are both consonances, but the former is more stable. On the other hand, the structural significance of chords such as the V7 and the cadential 6/4 clearly deviates from that of arbitrary dissonances.

27. A similar point is made by Joel Lester (1970, 13-14). In special cases, however, there are referential overall sets, such as the whole-tone set in Voiles, which enables a more literal distinction between steps and leaps.

28. However, a direct connection between the critical band and streaming is, as far as I am aware, somewhat conjectural; cf. Lerdahl 1999, 19-20.

29. The effects of "steps" and "leaps" in these examples can be aptly described by quoting Steve Larson (1997, 105): "In a melodic step, the second note tends to dis- place the trace of the first, leaving one trace in musical memory; in a melodic leap, the second note tends to support the trace of the first, leaving two traces in musical memory."

30. A unison relationship between an h-neighbor and a chord tone presupposes that the proximity principle of spacing is violated.

31. The status of the fourth in conventional tonality presents a converse case. If the fourth occurs as a simultaneity against the bass it is dissonant, but bass-line fourths often function as arpeggiations of the triad.

32. As regards voice leading, the effect of timbre links with its effect on streaming. As regards spacing, Bregman (1990, 509 ff.) has suggested that stream segregation also affects the perception of roughness-based dissonance; hence timbral difference may also alleviate this perceptual effect. Moreover, tones with different spec- tra have different amplitudes for corresponding harmonics, which also may reduce the roughness caused by violations of the proximity principle of spacing.

33. For example, in the Berg song (Example I If), the piano chord in m. 13 includes

Db3-F3-Cb4-Eb4 whereas the vocal part employs these notes an octave higher in mm. 11-13. The whole-tone relationships between the Cb4 and Eb4 in the piano and the Db4 and F4 in the voice have no structural impact.

34. For a discussion on melodic embellishment versus harmonic stability as prolongational determinants, see Larson 1997 and Straus 1997.

35. For a real musical example, see Bach's Prelude in C minor from The Well-Tem- pered Clavier I. Compare the turn-like figures in m. 12, beat 1, r. h. (Ab-G-F-G); m. 14, beat 2, 1. h. (Ab-G-F-G); m. 19, beat 2, r. h. (F-Eb-D-Eb). In the first case, the semitone Ab-G is an arpeggiation within a 6/5 chord; the second case is a gen- uine turn: Ab is a neighbor; in the third case, F is an h-neighbor of Eb in a 6/5 chord.

36. As in the case of the consonance-dissonance condition, some musical circumstances might call for more nuanced models than the two-alternative model also in the realm of the harmony/voice-leading condition. For the sake of simplicity, however, such models will not be systematically elaborated in this study.

37. In general, there is no reason to rule out multiple functions for embellishing tones if such multiplicity does not jeopardize the clarity of the basic structural order. This agrees with Schachter's (1999, 11) "liberal" attitude reflected in the following comment on Schenker: "In a melodic line D-E-C over V-I in C major, he would call the E an anticipation, not a neighbor-note. But why not both?"

38. I discussed such double functions also in Viiisdilii 1999, in reference to Ex. 13. 39. The substitution of a fifth for a tritone also diminishes roughness. 40. Expressed in J. J. Bharucha's terms (1984, 421), E is superior in the "tonal hierar-

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chy" and in the overall "event hierarchy," but B is superior in the local "event hierarchy" of mm. 27-29. The local position of the El in m. 28 is comparable to the upper-voice notes F#4 in m. 3 and G#4 (in m. 31). All these notes occur in a registral position beyond the corresponding structural outer-voice notes, to which they are connected by arpeggiation.

41. Even the rootedness aspect is not so unambiguously in favor of E, since the subset of the 7-inverted ToA+9 in mm. 27 and 29 contains salient occurrences of those members of the harmony that are root-supports of B, i.e., F# and C#. As regards the proximity principle of spacing, it is not in force throughout the span governed by the 7-inverted A+9, as noted in reference to Example 7b. While this principle cannot be taken as a syntactic dissonance rule in mm. 77-80, its violation serves to underline the restlessness of the harmony at the end of its prevalence.

42. However, in the transpositions of the 7-inverted A+9 in the developmental section of mm. 41-76 (see mm. 45, 53, 59, and 65 ff.), the root is characteristically set next to the bass and the two are connected by arpeggiation.

43. In these occurrences of motive y, structural circumstances differ from those in the opening section, but this does not invalidate the motivic relationship.

44. The harmony above these Bbs momentarily observes the consonance principle based on actual root-supporting fb-intervals, an exception to the surrounding norm based on the 7-inversion. The slurring in Example 8e indicates the Bis in mm. 19 and 23, and the Bbs in mm. 33 and 37 as incomplete neighbors in order to convey the characteristic way in which the musical circumstances (slurs in mm. 17-27 and the dynamic markings in mm. 33-39) suggest a motion directed from B6 to B?, rather than vice versa.

45. Cinnamon describes an "evolutionary progression from correlation, to dichotomy, to correlation between the harmonic and linear dimensions" in Liszt's music (Cin- namon 1986, 23). In the former correlation, both dimensions derive from (major or minor) triads. In the latter, they derive from equal divisions of the octave- characteristically the augmented triad.

46. To some extent, this technique could be explained on the basis of underlying or accompanying "stepwise" progressions. In mm. 27-65 of Vers laflamme, the bass line supports a "stepwise" motion from G#4 to C#5. Moreover, the tritones B 1- F2-B2 in mm. 66-77 link with semitonal relationships; see lines in Example 9a.

47. This process is coupled by overall source sets or scales, whose more or less conventional names are the "mystic" (Example 7a), the "acoustic" (Example 7b-c), and the diatonic (Lydian) scale. The name of the "acoustic" scale (= A+9) refers, of course, to the harmonic series; fb-interval 9 (major sixth) is sometimes taken as the representative of the thirteenth harmonic. While spacings like that in Example 7b might tempt one to make such a connection, there may not be sufficient perceptual justification for it. First, the thirteenth harmonic has still less relevance for virtual-pitch perception than the eleventh. Second, the thirteenth harmonic is actually 19 hundredths of a semitone closer to fb-interval 8. Third, whereas the significance of the first eleven harmonics for the evolution of harmony is supported by the fact that the order of the root-supporting weights of fb-intervals 0, 7, 5, 10, 2, and 6 agree with the historical order of their integration to harmony, fb-interval 9 (major sixth) has a long history as a consonance prior to 10, 2, and 6. All in all, it seems more prudent to regard fb-interval 9 as a non-root-support frequently added to harmonies otherwise formed by root supports.

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48. Cf. also Vincent Persichetti's (1961, 24-25) considerations of "resonance" and brilliance": "Resonant harmony is not formed by seeking higher and higher overtones but by using overtones of overtones. [...] For maximum brilliance, let the lower tones of the chord be accompanied by their own overtones." In these terms, the D-D# motion decreases rootedness but adds brilliance, a property that accords with the work's title: "Towards the Flame."

49. For considerations of this song, see Perle 1977, 3; Ayrey 1982; Headlam 1996, 33 ff.; Pople 1997.

50. While the importance of chord P in the song has been widely discussed, the significance of fb-interval 2 (major ninth) seems to have gained less notice.

51. In my view, Headlam's (1996, 41) characterization of mm. 9-12 as "the only clearly triadic bars in the song" is an overstatement.

52. The connection between the two songs is discussed, e.g., by Ayrey (1982, 192) and Pople (1997, 67 ff.).

53. As indicated in Example I If, fb-interval 2, Gb, is also present in the implicit T6U harmony in mm. 5-7. The proximity principle implies, however, that the interval

G#4-Gb4 in mm. 4-5 is a voice-leading interval. The Gb could perhaps be understood as a passing tone on its way to the Fb4 in the T4U (m. 9).

54. The 3s in the equal division of the tritone may be interpreted as voice-leading intervals on the same grounds as in the opening section of Vers la flamme. How- ever, not all melodic 3s in this work are voice-leading intervals. For example, intervals Eb3-C and Gb3-EL3 in m. 10 are arpeggiations within a dissonant passing chord C-Eb-Gb-At-Db (Example I1 f).

55. For the upper voices of the piano part, however, registral motions do not always serve to clarify the structure, but in mm. 9-14, for example, they freely parallel those in the bass. This produces points in the upper voices in which "ninths" and even "sevenths" substitute for small intervals as voice-leading intervals; see especially the bracketed notes in Example I lf, mm. 11 and 13.

56. The present situation derives from the special position of the tritone as half of the octave and lacks precise correspondents in triadic organization. However, retained tones may lead to other kinds of parenthetical references to the tonic scale degree; see, for example, Schachter 1999, 128-30.

57. One aspect in the connection between the T4U and the T3U is that the latter chord continues the semitonally descending chain of transpositions interrupted after the former (as observed in Pople 1997, 70-71).

58. The present explanation of the text expression in mm. 9-12 is not intended to con- tradict those based on other aspects, such as the tonal allusions (Ayrey 1982, 192-93).

59. For a polemical argument for the inevitability of the motivic aspect as an autonomous criterion of prolongational considerations in tonal analysis, see Cohn 1992.

60. The Bb and Eb in mm. 1 1-12 are viewed in terms of a tonal V-I relationship, for example, by Headlam (1996, Example 1.16b). On the other hand, the role of Db is emphasized in Pople's graph of the "tonal framework" of the song (1997, Exam- ple 4.5), in which the bass line reads Bb-Db-Bb-Eb.

61. A connection between F,/EO

and Eb is also suggested by the "rhyme" between the cadential melodic gestures in mm. 4-5 and 10-1 1. The structural weight of Fb in my analysis deviates from several previous readings (Headlam 1996, Example

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1.17; Pople 1997, Example 4.5; Gauldin 1999, Example 3), but is in accord with Edward Pearsall's (1999) view of Fb and D as neighbors of Eb.

62. Vaisala 1993 is an earlier version of this analysis. 63. For example, Perle (1962) regards Voiles as a forerunner of serialism. 64. Despite the differences, there are some striking points of resemblance between

Voiles and Berg's op. 2, no. 2. The referential harmonies are identical (with Bb1 as the bass), and both pieces also contain some kind of an allusion to Eb as a conventional tonic; in Voiles this is effected by the pentatonic section (m. 42 ff.). In both cases, moreover, Eb tonality becomes predominant in the next item in a series of compositions: in Berg's op. 2, no. 3 and in Debussy's prelude Le vent dans la plaine. Since there can hardly be any causal relationship between these two more or less concurrently composed works, these resemblances appear at least as "mys- terious" as the one between foreground chord progressions that was discussed in Stuckenschmidt 1965.

65. Although Katz's graph (1945, Example 93) covers the first fourteen measures, verbally she asserts that "the first thirty-eight [probably means thirty-seven] measures of the prelude are provided solely by this augmented chord, with or without embellishments, over a pedal point on Bb" (ibid., 286).

66. This rhythmic parallelism, occurring at a somewhat "deeper" level than the very surface, may be viewed as complementing Forte's (1983) observations of "foreground rhythm" in Voiles.

67. Despite the contrast between pitch sets, sections B and C are materially related. Compare m. 29 (with upbeat) with Emporte' in mm. 43-44. Also compare m. 31 with mm. 45-46.

68. The simultaneous whole-tone D-E also occurs in the B section, in mm. 26-27. 69. A precise description of the internal structural relationships within the pentatonic

material is left outside the present analysis. For example, Example 17e does not specify the structural order between the bracketed Db5 and Eb5 in mm. 45-46 but only draws attention to their association with the subsequent D?5 and E?5.

70. The precision of temporal proportions in Voiles is noteworthy. The D?7 (m. 22) and Db7 (m. 43) segment the composition into three parts in proportions of 21 : 21 1/2 : 21 1/2. On the other hand, the borderline between sections B and A2, at which the horizontalization of the Ab-C-E triad halts on its midpoint C, occurs at the temporal midpoint, dividing the piece into two groups of 32 measures each. Such a combination of bisection and trisection contrasts with or complements Roy Howat's (1983) extensive observations on the golden section in Debussy.

71. The concept of "prolongation" as commonly used nowadays deviates from Schenker's usage. On the structural aspect of "prolongation," see Schenker 1979, ? 45 ff. Another sense of the word refers to the prolongation of rules rather than musical entities, as discussed, e.g., by Joseph Dubiel (1990, 292 ff.).

72. For the whole Larson quote, see note 29 above. 73. All in all, the present analysis demonstrates that the functions of each member in

the whole-tone set are sharply individualized. This contrasts with Arnold Whit- tall's contention that "the persistence of a pedal Bb enables the ear to distinguish between a fundamental element and its subsidiaries, even if there is little sense in which any of the other five notes is less equal than the others." (Whittall 1975, 261.)

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74. Although Schmitz makes no use of linear analysis, and his description of Voiles is far from comprehensive, he makes observations relevant for both the small and large scale, concerning, for example, the opening parallel-third subject and the F#-D ostinato in m. 23 ff.

75. Since chord U is not transposed in Voiles, there is no scale-degree system similar to the preceding examples. However, Schmitz's "tonic" and "dominant teams" might be understood as forming a two-element scale-degree system, if the pedal Bb and the D7 are excluded from consideration. In contrast to the other examples, this hierarchy does not stem from bass arpeggiation but from stepwise relationships (whereas the tonal I-V relationship combines both factors: bass arpeggiation with stepwise upper-voice motions). The issue may, however, be transposed to the realm of the consonance-dissonance condition by viewing, instead of the "teams" themselves, their combinations with the pedal, shown as Q and R in Example 19.

76. G# governs the top voice both at the opening and at the conclusion. The primacy of G# is also reflected in the character of the G#-C-E motion. Both in the parallel-third subject and in the overall structure, E is the goal but the motion towards E has a "floating" character, which totally deviates, say, from the decisiveness of the paradigmatic tonal progression, the Urlinie. It would seem that such effects correlate with root-supporting weights: the G#-C-E motion proceeds towards the weakest root support.

77. This reading is complicated especially by the temporal alignment of structural and non-structural elements. In mm. 54-55, the top voice is structural but the middle- register material non-structural. Mm. 56-57 show the reverse circumstances. The tendency to such alignment is already evident in the preceding statement of the A melody (mm. 50-54): the filled-in "tonic" intervals Ab-C and E-C (mm. 50 and 52) are accompanied by the "dominant" (F#-D) and the "dominant" D and Bb (mm. 51 and 53) by the "tonic" (Ab-E).

78. In the middle-register parallel chords of mm. 56 and 59, F#3 returns to Ab3 (see Example 17f), but in m. 71 the last chord is omitted. This allows the upper voices of these chords to stay on the structural C4 and E4 but leaves F# unresolved.

79. Comparable "vagueness," created by the presence of non-structural elements at the end of a piece, also occurs in such Debussy's works whose organization is closer to triadic tonality. See, for example, the D-major triad in Le vent dans la plaine and the B-diminished in Brouillards.

80. The two premises are lucidly manifest in Morris's (1987, 34 ff.) notation for the equal-tempered pitch space, [2(1/12)], in which 2 stands for the frequency ratio of the octave and 1/12 for its division in twelve parts. As discussed by Morris, such a pitch space is, from a mathematical viewpoint, "fundamentally different from p[itch]-spaces where intervals have been traditionally produced from ratios of integers such as 1:2, 2:3, 8:9, and the like" (ibid., 35). From the perceptual viewpoint, however, the difference is less fundamental; virtual-pitch perception, in particular, does not presuppose purity of frequency ratios.

81. Straus (1987, 5) justifies the special status of the (major or minor) triad in the diatonic system on the basis of its being "the maximal subset of the [...] collection consisting entirely of non-adjacent elements." An analogous reasoning applies to the "augmented triad" in the whole-tone collection.

82. The perceptual aspect under question involves factors that are not allowed for in Parncutt's (1988) root algorithm, even if we supplement it by assigning a small

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root-supporting weight for FB-interval 6. Such factors include registral ordering and perhaps also a more holistic sense in which chord Q fits to the pattern of harmonics whereas chord R includes one element, F#, that deviates from such a pattern. Moreover, as suggested in Parncutt 1996, fb-interval 8 (minor sixth) may, alongside with 5 (fourth), have special impact in increasing root ambiguity as an inversion of a strong root support.

83. I discuss some examples in Viisildi 1997 and in my forthcoming paper. For another recent study on overtone structures in Debussy, see Don 2001.

84. The overall structure of L'Isle joyeuse is a lucidly articulated conventional Ursatz, as I described in Viisdildi 1997. The structure of Ce qu 'a vu le Vent d'Ouest, as analyzed in my forthcoming paper, is unconventional, but the referential harmony is not a subset of chord A. Chord U is a local subordinate harmony in the occurrences shown in Example 20c.

85. The motivic organization in this song has previously-if less comprehensively- been discussed by Budde (1971, 44-46) and Wason (1996, 111-17). Wason focuses on the motive of "encircled D," which corresponds to Tox- and ToIx- in the present notation.

86. The present criteria of interpretation are similar to those listed by Straus (1987, 2-4, 13), as criteria of associational middlegrounds, and by Lerdahl (1989, 73- 74), as salience conditions for "atonal prolongation." However, whereas Straus mentions the similarity of register as an associational factor, one of Lerdahl's salience conditions is based on the extremity of registral position. Both aspects are relevant for the present motivic reading: the "middleground" connections are consistently supported by registral closeness, whereas the inclusion of the Eb5 in the "background" is based on the salience given by its registral height.

87. The melodic line is thus held together by the cooperation of two primary factors of auditory streaming, timbral similarity and pitch proximity (see Bregman 1990).

88. Another verticalization of the pitch classes of the second phrase occurs in m. 3, beat 2. This chord omits G, the apex point of the phrase, which occurs just after the chord.

89. The chord at the end of m. 3 and the retransition chord (mm. 7-8) exemplify two different kinds of verticalization, relevant for two different aspects of organization. The former chord verticalizes the structurally superior tones of the first vocal phrase, clarifying the prolongational organization. The retransition chord, by contrast, verticalizes both structural and non-structural tones of the second phrase, creating a motivic-associational connection.

90. For a discussion of the different versions of the song, see Budde 1971. 91. Also B?, fb-interval 7 in the ToA, is outside T10U, but, as observed, is much less

prominent in the organization than Bb. 92. The phrase in m. 7 consists of elements subordinate to those of m. 6, a reading that

accords with its "lightly winged" expression. The piano material, based on the varied imitation of the vocal phrases in a higher and lower octave, offers structural clarification. All the incomplete neighbors of the vocal part occur as complete neighbors in the piano figuration. The priority of fb-interval 3, Eb, in relation to 4, E?, which reverses the norms prevalent in the rest of the music, is confirmed by the return of Eb6 at the end of the B section.

93. While D4-Eb5 functions as a voice-leading interval in the overall structure, locally, in m. 6, the interval D4-E65 is an arpeggiated dissonance.

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94. The C#4 and the F3 in the retransition chord are regarded here primarily as dimming tones and thus as related to the preceding D4 and F04, respectively. In a certain sense, however, we may view them also as participating in the large-scale

descents and bearing double functions similar to those shown on the small scale in Example 22 (IN or PT). Furthermore, as illustrated in Example 23, it is also possible to view the upper voices of the retransition chord as complete neighbors. The relationships indicated in Example 26 appear primary, however.

95. The harmony on Eb thus prepares for that on C with respect to chord construction as well as the register of the bass note. The Eb-C bass line also forms an anticipatory arpeggiation of the harmony on C. Pursuing the analogy with tonal harmony, this connection might be compared to that between VI and IV in a I-VI-IV-V-I progression.

96. As observed in note 76, the use of FB-interval 6 as a goal creates a less decisive impression also in Voiles.

97. The middle section (mm. 6-7) contrasts with the opening also with respect to this principle. The D and Eb form an ro-interval 1, which may, in some sense, be heard as "resolving" to the 11, D-C#, in the retransition chord.

98. For a discussion on this topic and the temporal perspectives involved, see Smith 1995.

99. In both songs discussed, I have interpreted elements with salient appearance but far removed from the basic structural level as representing dreams or imagina- tion. For a parallel case in tonal music, see Schachter's (1999, 216-19) discussion of a G major chord based on a chromatic passing tone in Schubert's Nacht und Trdume.

100. However, a striking foreground occurrence of chord A in more "mature" Berg is the final chord of Wozzeck: G-D-B-F-A-C#. (For an earlier occurrence, see Sonata op. 1, m. 12, beats 1-2.)

101. Hence, for Scriabin and Debussy, the perspectives offered by conventional Schenkerian and set-theoretical methods, as in Baker 1986 and Parks 1989, have considerable limitations.

102. There are also other potentially relevant psychoacoustical aspects of intervals, such as roughness, which have a connection with approximate frequency ratios (see note 23).

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Documents

Prolongation of Harmonies Related to the Harmonic Series in Early Post-Tonal Music