conceptos de armonia y prolongacion en el op.19 -2 de schoenberg

Concepts of Harmony and Prolongation in Schoenberg's Op. 19/2

Olli Visl

Music Theory Spectrum, Vol. 21, No. 2. (Autumn, 1999), pp. 230-259.

Stable URL:http://links.jstor.org/sici?sici=0195-6167%28199923%2921%3A2%3C230%3ACOHAPI%3E2.0.CO%3B2-N

Music Theory Spectrum is currently published by University of California Press.

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/ucal.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. Formore information regarding JSTOR, please contact [email protected].

http://www.jstor.orgThu Apr 12 21:56:40 2007

Concepts of Harmony and Prolongation in Schoenberg's Op. 1912 Olli Vaisala Even the spacing is obligatory; as soon as a tone is misplaced the meaning changes, the logic and utility is lost, coherence seems destroyed. Laws apparently prevail here.

Arnold Schoenberg, Theory of Harmony1

I . INTRODUCTORY EXAMPLES

Schoenberg's Klavierstiick op. 19 no. 2, shown in its entirety in Example 1, has been a favorite subject for analysts discussing the applicability of quasi-Schenkerian methods to post-tonal music. Focal articles treating its voice-leading structure and the attendant issue of atonal prolongation include those by Roy Travis, by Joseph N. Straus, and by Fred LerdahL2 Analysts d o not, however, agree on the nature and validity of such analysis. Travis's pioneering study is criticized by Straus on grounds that it

This article is based on the author's previous "SchBnbergrn op. 19111- prolongaatiota vai ei?," Savellys ja nrusiikinteoria 712 (1997): 41-82

'Amold Schoenberg, Theory of' Harnrony, trans. Roy E. Carter (London: Faber Music, 1978), 42 1 .

2Roy Travis, "Directed Motion in Schoenberg and Webem," Per,vpertive.sof New Muslc 412 (1966): 84-89; Joseph Straus, "The Problem of Prolongation in Post-Tonal Music," Journal of Music Theory 3111 (1987): 1-22; Fred Lerdahl, "Atonal Prolongational Structure," Contemporary Music Review 4 (1989): 65- 87. See also Deborah Stein, "Schoenberg's Opus 19 No. 2: Voice Leading and Overall Structure," In TIzeory Only 217 (1976): 27-43. For an extensrve bibliog- raphy of studies of post-tonal music applying Schenkerian principles, see Joseph N. Straus, "Voice Leading in Atonal Music," in Music. Theory in Conc.ept and Practice, ed. James M. Baker, Davrd W. Beach, and Jonathan W. Bernard (Rochester: University of Rochester Press, 1997): 237-74, note 1.

has no consistent pitch-based criteria for differentiating between structural and non-structural tones o r determining the relationships between the two.3 Straus postulates four essential conditions for prolongational structures and argues that such conditions are not satisfied in "the most significant post-tonal music," including Schoenberg's op. 1912 as analyzed by T r a v i ~ . ~ Lerdahl, in turn, agrees that Straus's "stability conditions" cannot be satisfied, but instead proposes an "atonal prolongational structure" based on "salience condition^."^

In this article I will contest the views of Straus and Lerdahl and present an analysis of Schoenberg's op. 1912 that is-at least, to a significant degree-supported by such pitch-based norms as are required by Straus's conditions of prolongation. The main difference between Straus's approach and mine lies in the interpretation of harmonies. Whereas Straus identifies harmonies with un-ordered pitch-class sets, I argue that the structural status of harmonies and intervals is crucially influenced by the registral distribution of pitches. Before trying to justify this view in more general terms, I shall first discuss two excerpts from the work to demonstrate my approach and examine its analytic consequences. Then, after exploring relevant perceptual and theoretical issues, I present a full analysis of op. 1912 in section 4 of the article.

'Straus, "The Problem of Prolongation," 6-8, 17-19. Itbid., 4. Straus's conditrons are presented in ibid., 2-7. Sunlike Straus, who suggests that the four conditions could in princrple be

met by means other than tonality, Lerdahl rejects the role of stability condrtions in "atonal" music on a priori grounds: "Atonal music almost by definition does not have stabilrty conditions" (Lerdahl, "Atonal Prolongational Structu~e," 73).

Concepts of Harmony and Prolongation in Schoenberg's Op. 1912 231

Example 1. Schoenberg, Kleines Klavierstilck op. 1912

ARNOLD SCHONBERC "SECHS KLEINE KLAVIERSTUCKE OP 1 9 I1 (C) 1913BY UNIVERSAL EDITION A G . WlEN (c)RENEWED 1940 BY A SCHONBERG

Of Straus's four conditions, I will address only the first and In his discussion of Travis's analysis of op. 1912, Straus uses fourth at this stage.6 Condition 1 is the consonance-dissonance Travis's "tonic sonority," the harmony formed by the entire con- condition. It requires a clear distinction between functional conso- tents of the last measure, to test his conditions. Given the "stabi- nance and dissonance, between harmonies that can support tones lizing" musical effect of the harmony, this seems a sensible strat- of greater structural weight and those that cannot. Condition 4 is egy and will be followed here. Straus explains that because the set the hannony/voice-leading condition. It calls for a clear distinc- class of the chord, 8-19 (01245689), has an interval vector with tion between harmonic and voice-leading intervals, so that ample representation of every interval class, (545752), it becomes melodic motions may be clearly interpreted as either arpeggiation "impossible to interpret the voice-leading motions."' In other or voice-leading. words, the interval-class content of the referential harmony

6SeeStraus, "The Problem of Prolongation," 2-7.

232 Music Theory Spectrum

provides no foundation for the distinction between harmonic and voice-leading intervals; condition 4 fails. A similar conclusion may be reached with respect to condition 1: taken as a set class, Travis's tonic does not establish a clear distinction between consonant and dissonant intervals.

If we consider the registral positioning of the pitches in that harmony, however, a different picture emerges. To see this, let us adopt the concept of the reg i s t r a l ly ordered in t e rvu l (yo- interval) instead of that of the interval class. Under this concept, inversionally equivalent intervals, such as semitones and major sevenths, are distinguished from each other. At the same time, "simple" and "compound" versions of an interval, such as the semitone and the "minor ninth," remain to be treated as equivalents. Thus, excluding unisons and octaves, eleven ro-intervals are possible, numbered 1-11.8 Example 2 shows the subdivision of each interval class in the concluding harmony into ro-intervals. With the exception of interval class 2 (and, of course, 6), the presentation of all interval classes is maximally biased in favor of one or the other registrally ordered alternative. In itself, this speaks for the view that complementary intervals are not treated as "equivalents" in the music. Analytical considerations corroborate this view.

A distinction of special analytical significance for the Schoen- berg is the one between ro-intervals I and 1 1 . In Straus's approach, interpreting semitones as voice-leading intervals poses a major problem in music that also treats interval class I as a harmonic i n t e r ~ a l . ~ But since interval class 1 is consistently ex-pressed as ro-interval 11 in the referential harmony, we need not assume that a melodic ro-interval 1 stands for an arpeggiation of the same interval. Let us first see how this affects the interpretation of m. 3. In Travis's analysis, the eighth-note A is shown as an

8An ro-interval i may be defined on the basis of either pitch classes or pitches. Let pc, = the pc of the lower pitch and pc2 = the pc of the higher pitch. Then, I = pc, - pc, ~f pc, < pc,; i = pc, - pc, + 12 if pc, > pc,. Directly for pitches pi and p2, i is the remainder of the division ( p i - pz / I 12

9See Straus, "The Problem of Prolongation," 5ff. and idem, "Volce Leading in Atonal Music," 239-41, see also Jack Boss, "Schoenberg on Ornamentation and Structural Levels," Journal qf'Music. Theory 3812 (1994): 187-216 [I931

Example 2. Presentation of the interval classes of the concluding harmony as ro-intervals

0 times as ro-interval I interval class 1 occurs 5 times as ro-interval 11

2 tlmes as ro-interval 2 ~nterval class 2 occurs 2 times as ro-intervals 10

5 times as ro-interval 3 interval class 3 occurs 0 tlmes as ro-interval 9

5 times as ro-interval 4 interval class 4 occurs 2 tlmes as ro-interval 8

0 times as ro-interval 5 Interval class 5 occurs 5 times as ro-interval 7 interval class 6 occurs 2 times as ro-lnterval 2

upper neighbor to the subsequent Ab.loThanks to the functional distinction between ro-intervals 1 and 11 , this reading, criticized by Straus, can be rehabilitated. Taking the A as a non-harmonic tone leads to the view of the overall harmony in m. 3 shown in Example 3a. The resultant harmony, interpreted as a r eg i s t r a l ly orderedp i t c h - c l a s s s e t ( ro-set ) ," is transpositionally equivalent to

'OTravls, "Directed Mot~on," Examples 4 and 5c I1ln the present article the ordering IS always calculated from the lowest to

the hlghest note. For prevlous artlcles discussing reglstrally ordered sets from various analytical and theoretical viewpoints, see, for example. William E Benjamin, "Ideas of Order in Motivic Music," Mu.>!( Theory Spe(.rr~lnr 1 (1979): 23-34; Alan Chapman. "Some Intervallic Aspects of Pitch-Class Set Relations," Journal of' Mu.,ic. Theory 2512 (1981). 275-90, John Roeder, "A Geometric Representation of Pitch-Class Series," Persprc~rive.\ c?f'Neus Mu.>i(. 2511-2 (1987): 362-409; idem, "Harmonic Implications of Schoenberg's Observations of Atonal Voice Leading," Journal of Mu\.!(, T l~eo r y 3311 (1989). 27-62; Robert Morris, "Equivalence and Similarity in Pitch and the11 Inter- action with Pcset Theory," Journal of'Mu.\ic. Theory 3912 (1995). 207-44. Marcus Castrtn, "Joukkoluokitukseen perustuva sointuluokitus: perusperlaat- tee1 ja esimerkkejx sovellusmahdollisuuksista" [Chord Classification Based on Set Classification: Basic Prlnclples and Examples of Appllcatlonal Poss~b~ l l - tles], Siivelly.\ ja nlu.\iikinreoria 711 (1997) 6-25,

Concepts of Harmony and Prolongation in Schoenberg's Op. 19/2 233

Example 3. Harmonies in mrn. 3, 9.

a) b)

the pentachord formed by the five lowest-and five highest- tones of the concluding harmony, as bracketed in Example 3b. Thls, clearly, is an instance of harmonic similarity which goes much further than one based on pitch-class sets only. Taking the concluding harmony as referential, this similarity provides good reason for regarding the harmony completed by the Ab as a consonance, and the interpretation of Example 3a appears to be supported by condition I as well as 4. (Actually, this is one of only a few convergences between the present approach and Travis's view of the piece in general, as will be clear below.)

A distinction between ro-intervals I and 11 is relevant also with respect to the consonance-dissonance condition, as is suggested by studying m. 5. The total pitch-class content of that measure plus D equals the pc content of the piece's final harmony; thus a comparison between m. 5 and m. 9 is particularly illustra- tive of the impact of registral disposition on harmonic structure. In contrast to the final harmony and to the harmonies of the sur- rounding measures, ro-interval 1s (expressed as pitch-interval 13s) burst forth in m. 5.12I shall interpret these Is as dissonances. In Example 4, the C5 at the beginning of the measure is shown as a neighbor to B4 and the resultant harmony is compared with the last verticality of the composition. The most salient of the ro-interval Is, Eb5-Fb6, is "resolved" to the ro-interval 11 Eb5-D6, while the other pitches remain invariant. This interpretation holds an evocative relationship with the large-scale D6-Fb6-D6 top-

IZIn the Examples, ro-interval 1 has been denoted as 13 whenever this is the actual width of the interval

Example 4. The "resolution" of the interval Eb5-Fb6

voice line shown by some previous analysts.13 On the one hand, i t establishes a consonance-dissonance relationship between the non-structural Fb6 and the structural D6, supporting the neighboring motion. On the other hand, the presence of ro-interval 2 in the referential harmony means that the whole step cannot be justified as a voice-leading interval on the basis of the ro-interval content alone. Later I will justify the large-scale neighboring through other means. At present, it may be pointed out that whereas the melodic interval in the top voice is a simple whole-tone, the ro- interval 2s in the referential harmony are expanded by octaves. This would suggest a further restriction on octave equivalence, distinguishing between "simple" and "compound" realizations of ro-interval 2, as the basis of meeting the harmonylvoice-leading condition.

These preliminary examples suggest that, in dealing with ques- tions relevant to Straus's conditions of prolongation, treating harmonies as pitch-class sets may not be adequate. One may say that the concept of pitch-class set by and large stems from the Schoen- bergian notion of presenting an "idea" in the "two-or-more- dimensional" musical space14 (presupposing the "unity" of which

"Stein. "Schoenberg's Opus 19 No. 2," Sketch 1 , etc.. Lerdahl, "Atonal Prolongat~onal Structure," Figure 11

I4For Schoenberg's presentation of t h ~ s notlon, see "Compos~t~on with Twelve Tones (1 1" (1941). in Arnold Schoenberg. Splr and idea, ed. Leonard Stein (London. Faber and Faber, 19751, 214-43 [220] On the history lead~ng to the concept of p~tch-class set analys~s, see Jonathan W Bernard. "Chord, Collect~on, and Set in Twentieth-Century Theory," in Music Tl~eo t ym Concxpr and Pracric.e, ed. James M . Baker, David W. Beach, and Jonathan W. Bernard (Rochester. University of Rochester Press, 1997), 1 1-52


inherently contradicts the harmonylvoice-leading condition) and is thus connected with the motivic-associational rather. than the prolongational aspect of organization. In Straus's words, "A pitch- class set involves generalizing and extending the traditional concept of mot i~e ."~s And, I would suggest, accounting for associational organization and prolongational syntax (or something comparable to it) may call for different approaches to harmony.'6 Consideration of tonal practice offers a useful point of comparison.I7

In the harmonic syntax of tonality, octave displacements may occur with little consequence in the upper voices, but a change of bass note--chord inversion-affects chord function. chords often stand for or expand the corresponding root-position triad, but they are not as stable. The points of greatest structural weight, the I-V-I of the Bassbrechung, require root-position chords. 46 chords, on the other hand, most often function as dissonances. A prolongational analysis of tonal music failing to distinguish between cadential :chords and tonic triads would produce quite defective results. At the same time, however, the common pitch- class content of these chords may play an important role in the motivic and thematic organization. For example, it is customary to present a statement of the main theme on a cadential ,6 at the outset of a cadenza. A more subtle example is presented in the Mozart passage shown in Example 5. A motivic association, marked with brackets, connects two instances of the pitch-class set {G,B, D}

15Joseph N. Straus, Remaking the Past: Musical Modernisnt and the In- fluenre of the Tonal Tradition (Cambridge: Harvard University Press, 1990). 24.

"To be sure, registral issues are not irrelevant for associational organization, either. However, registral rearrangements cannot cause an association to turn into something opposite in the manner of syntactic functions such as consonance and dissonance.

''About the "autonomy" of the motivic and prolongational aspects of organization in tonal music, see, for example, Richard Cohn, "The Autonomy of Motives in Schenkerian Accounts of Tonal Music," Music Theory Spectrunt 1412 (1992): 150-70.

without in any way affecting the functional distinction between the and ,6 chords in the prolongational syntax.lS 2. PERCEPTUAL AND THEORETICAL NOTIONS

The examples presented so far point to registral disposition as a crucial factor for the syntactical status of harmonies and intervals in both tonal and post-tonal music. This, I believe, is connected with perceptual qualities that are affected by registral rearrangements and that tend to influence the functional status of harmonies and intervals. Traditionally, the terms "consonance" and "dissonance," for example, have had a perceptual meaning apart from the functional one (which is the one meant by Straus).I9 While it is clear that functional norms cannot be deduced from perceptual properties-the main idea of the present paper is to discover contextually established norms for Schoenberg's op. 1912-neither are perceptions irrelevant to functional norms. That is, some kind of correspondence between perceptual and functional principles may greatly enhance the viability and com- prehensibility of a functional system. In a stronger sense this means a more or less rough correlation between perceptual and functional consonance,20 or between "auditory streams" and

lSOn the related issue of motivically connected 6, chords with different functional status, see Peter H.Smith, "Brahms and Motivic 613 Chords," Mu.\ic. Analysis 1612 (1997): 175-21 8.

19For a review of the history of the concepts of consonance and dissonance, see, for example, James Tenney, A History of "Consonance" and "Di.sso-nanre" (New York: Excelsior Music Publishing Company, 1988).

20By perceptual consonance I refer here primarily to two types of sensory effects suggested by psychoacousticians as having influenced musical consonance and harmony: roughness or beats, on the one hand (R. Plomp and W. J M. Levelt, "Tonal Consonance and Critical Bandwidth," The Journal o f thr Acoustical Sociery of'America 3814 [1965]: 548-60), and virtual-pitch perception, on the other (Emst Terhardt, "Pitch, Consonance, and Harmony," Thr Journal of the Acoustical Sociery nfAmeric.a 5515 [1974]: 1061-69. and idem. "The Concept of Musical Consonance: A Link Between Music and Psycho- acoustics," Music Prrcepnon 113 [1984]: 276-95). Recent comments to the much-debated issue of the relationship between perceptual and functional consonance are included in Steve Larson, "The Problem of Prolongation In Tonal Music: Terminology, Perception, and Expressive Meaning," Journal of'Music.


Example 5. Mozart, Sonata K. 545, first movement, subsidiary theme

voices.*' In a weaker sense it means simply that functionally equivalent harmonies or intervals should not "sound" too dissimi-

Theory 4111 (1997): 101-36 [110-111, and Fred Lerdahl, "Issues in Prolonga- tional Theory: A Response to Larson," Journal ?f Music Theory 4111 (1997): 141-55 [151-521. In this discussion my view basically agrees with Lerdahl's, although I doubt whether atonal music really "ignores" degrees of sensory consonance to the extent that Lerdahl supposes.

As an interesting example of music in which the order of functional consonance is opposite to that of perceptual consonance, one may mention those com- positions and passages by Liszt in which, according to analyses of Robert P. Morgan and of Howard Cinnamon, the augmented hiad assumes the role of stable referential harmony to which other chords, including positions of minor biads, re- solve. See Robert P. Morgan, "Dissonant Prolongation: Theoretical and Compo- sitional Precedents," Journal ?f Music Theory 2011 (1970): 46-91, and Howard Cinnamon, "Third-Relations as Structural Elements in Book I1 of Liszt's 'AnnCes de Phlerinage' and Three Later Works" (Ph.D. diss., The University of Michigan, 1984), Chapter 5. Creating this effect seems to require all the more explicit contextual means: consider, for example, the opening of the Faut symphony.

llOn the concept of "auditory stream," see especially Albert S. Bregman, Auditory Scmr Ana l y ~ i ~ (Cambridge, Mass.: MIT Press, 1990).

lar if they are to be recognized as functionally equivalent. Con- sider, for example, the seven different registrations of the final harmony of op. 1912 shown in Example 7 below. Although chords i and vii are certainly related to each other on the basis of identical pitch-class content, their pitch relationships will cause them to be perceived quite differently. To examine such circumstances, and to lay groundwork for discussing the functional norms of tonality and of op. 1912, I shall now discuss two notions and their perceptual premises, the proximity principle and harmonic and intervallic roots.

By proximity principle I mean the simple notion, relevant to both conditions 1 and 4, that the smallest, "stepwise," intervals tend to be avoided as consonant harmonic intervals, on the one hand, and favored as voice-leading intervals, on the other. This principle has been observed in one form or another by various previous aythors. Straus, for example, says of condition 4 with respect to tonal music: "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."22 Joel Lester discusses a kind of proximity principle, and the relevance of spacing, in Schoenberg's music: "the [simultaneous] second, particularly when minor, is presented as a seventh, ninth, or a compound of these in almost all cases in order to avoid any confusion between step and simultaneity interval~."~3

Both the vertical and the horizontal aspects of the proximity principle have a correspondence in well-established perceptual phenomena. A concept of paramount importance in pitch perception is the critical band. When two simultaneous pure tones are closer to each other than the critical band, their interference pro- duces the perceptual effects of masking and roughness (beats), the latter being equivalent to what is often called sensory disso- n a n ~ e . ~ ~To account for critical-band effects between complex tones, such as normal musical tones, one must allow for those between all partials; simple (approximate) ratios between the fundamental frequencies, traditionally associated with consonance, tend to minimize these effectszs In the present paper this kind of impact

22Straus, "The Problem of Prolongation," 5. Cf. Larson's observation: "In a melodic step, the second note tends to displace 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" (Larson, "The Problem of Prolongation," 105).

2?Joel Lester, "A Theory of Atonal Prolongations as Used in an Analysis of the Serenade, Op. 24 by Arnold Schoenberg" (Ph.D. diss., Princeton University, 1970), 14. Schoenberg himself also referred to "the tendency to soften the dissonance through wide spacing of the individual chord tones" (Theory of Har-mony, 417-18).

24Plomp and Levelt, "Tonal Consonance." For an overview of the multifari- ous impacts of the critical band for pitch perception and its width in different registers, see, for example, William M. Hartmann, Signals, Sound, and Sensa- tion (Woodbury, N.Y.: AIP Press, 1997), 249ff.

first to explain dissonance as the beating between partials (though with no notion of critical band) was Hermann L.F. Helmholtz, On the Sensa- tions qfTone as a Physiological Basis for tile Thmry qfMusic, second English edition, trans. Alexander J. Ellis (New York: Dover Publications 1954 [1885]), Chapter VIII. For more recent confirmation, see Plomp and Levelt, "Tonal Consonance," and Akio Kameoka and Mamoru Kuriyagawa, "Consonance

of frequency ratios will not be considered, since it does not seem to play a decisive role in the Schoenberg. With tones of similar spectra, maximal critical-band effects are produced by the proximity of the fundamental frequencies, since this entails the proximity of all respective part ia l~.~6 As regards consecutive tones, those that are more proximate in pitch are more likely to be perceived as belonging to the same auditory stream.27

In musical practice, the borderline between the small, "stepwise," and larger intervals is, naturally, flexible and may vary according to the context. However, insofar as the conception of the music is basically harmonic-deriving from simultaneities-some kind of a psychoacoustical clue to the borderline is provided by the critical bandwidth, which, from the middle register upwards, is about three semitones. Drawing the borderline between pitch- intervals 2 and 3 conforms with the usual distinction between a step and a skip in conventional tonality and is also adhered to in the Schoenberg examples. Pitch-interval 3 is the smallest interval between adjacent tones in the referential harmony, and seml- tones and whole-tones function as voice-leading ~ntervals (see Examples 3 and 4).28

Theory Part I1 Consonance of Complex Tones and its Calculation Method," T l~ r Journal o f the Acoustic,al Society ofAmerica 4516 (1969). 1460-69

26That the semitone causes the greatest and the whole-tone the second greatest roughness-based dissonance also for typical complex tones conforms with the consonance curves presented by Plomp and Levelt ("Tonal Consonance," Fig. I f) , and Kameoka and Kuriyagawa ("Consonance Theory Part 11," Figs. 7 and 8).

"For overviews of experimental confirmation and musical implications, see Bregman, Auditory Scene Analy.sis, chapters 2 and 5

ZgStraus also discusses voice-leading in terms of the structure of the diatom collection: "voice leading in tonal music proceeds from one pitch-class to another pitch-class adjacent within the diatonic collection" ("The Problem of Prolongation," 5). This differs from the proximity principle of the present approach in two respects. First, instead of pitch-classes, the proximity pr~nciple applies primarily to pitches. Second, instead of adjacency within a given collection, the principle is based on proximity as related to the intervals of the referential harmony (and, more indirectly, to the critical bandwidth).


The proximity principle implies that the impact of registral disposition is particularly important with respect to interval classes 1 and 2. It also has indirect bearing on the distinction between ro- intervals 1 and l l , which, as we have seen, is significant for conditions 1 and 4 in the Schoenberg. If interval class 1 is to be integrated into harmony while the semitone is maintained as the strongest voice-leading interval, this distinction is the clearest conceivable way (though not the only possible wayz9) to avoid confusion between harmonic and voice-leading intervals and to fulfill condition 4. The exclusion of ro-interval 1 from consonant harmonies then opens the possibility to use it as a contextual dissonance, even though the perceptual dissonance based on proximity (i.e., the roughness) of a "compound" ro-interval 1 is no greater than that of ro-interval 11.

Harmonic and intervallic roots are a central notion in the stability system of tonality. Hindemith sought a perceptual basis for this concept in combination tones.30 A more recent and much more plausible explanation derives from the concept of virtual pitch, the faculty of the auditory system to extract the pitch of the fundamental of a complex tone from the patterns formed by its harmonics.3' In comparison with combination tones, virtual pitch is a

290ne should not infer that ro-interval 1 can never be included in a referential sonority if semitones are used as voice-leading intervals. The distinction between voice-leading and arpeggiating intervals could be based on a distinction between semitones and its compounds (another example discussed by Straus in "The Problem of Prolongation," Stravinsky's Syn~phony in Three Movements, might be a candidate for showing this kind of organization), or simply on a syntactic rule excluding the arpeggiation of semitones.

'OPaul Hindemith, The Craji of Musical Composition, Book I: Theoretical Part, fourth ed., trans. Artur Mendel (New York: Schott, 1942), 68-74.

"See Terhardt, "Pitch, Consonance, and Harmony," and idem, "The Con- cept of Musical Consonance." For a historical overview of virtual-pitch re- search, see, for example, Bemam Scharf and Adrianus J. M. Houtsma, "Audi- tion 11: Loudness, Pitch, Localization, Aural Distortion, Pathology," in Handbook ofPerception and Huntan Perforn~ance, vol. I , ed. Kenneth R. Boff, Lloyd Kaufman, and James P. Thomas (New York: John Wiley and Sons, 1986). To what extent the impact of virtual-pitch perception for roots is direct,

more central aspect of pitch perception and is less affected by the mistuned frequency ratios of equal temperament. Ro-intervals such as 4 and 7 may be considered root-supporting since they evoke a virtual pitch of the same pitch-class as the lower tone of the interval. In tonality, the property of being root-supporting strongly correlates with the functional stability of intervals. The major third and the fifth are, with relation to the bass, more stable than their inversions, the minor sixth and the fourth. The most dramatic effect on functional status, turning a consonance into a dissonance, is caused by inverting the strongest root-supporting interval, the fifth.32

Harmonic roots have hardly been standard concepts of post- tonal theory since Hindemith's time. However, I see no a priori reason to rule out their possible relevance for post-tonal harmony. Indeed, the property of being root-supporting seems to have significant bearing on the registration of the concluding harmony of op. 1912.33 The strongest root-supporting ro-intervals, 4 and 7, are among the favored ones in its total interval content (see Ex- ample 2). The ro-intervals that occur in relation to the bass ("figured-bass" intervals) are 2, 3, 4, 6, 7, 10, and 11. Excluded from these are precisely those ro-intervals-1, 5, 8, and 9-that

i.e., based on each listener's actual perception, and to what extent indirect, 1.e . based on the adoption of mus~cal conventions originally affected by virtual- pitch perception, is outside the discussion of the present paper.

"For a discussion on the weighting of root-supporting intervals and its psychoacoustical justification, see Richard Parncutt, "Revision of Terhardt's Psychoacoustical Model of the Root(s) of a Musical Chord," Music. Pen.eption 611 (1988): 65-94. That the fifth is the strongest root-supporting interval also helps to account for the root of a minor triad, an issue which has caused prob- lems for acoustical explanations ever since Rameau. On the dissonance of the fourth, see Richard Parncutt, "Praxis, Lehre, Wahmehmung. Kritische Bemer- kungen zu Roland Eberlein: 'Die Entstehung der tonalen Klangsyntax,'" Mu.siktheorie 1111 (1996): 67-79.

"It may also be noted that Schoenberg himself repeatedly referred to the idea of relating harmonic material to the overtone series (Theory of' Harntony, 18-22, and Style and Idea, 258-64), an idea for which virtual-pitch theory provides a rationale.


are under-represented in the total interval c0ntent.3~ Bass-related ro-intervals 2, 4, 7, and 10 reflect distances between a fundamental and its first ten harmonics and therefore rank as directly root- supporting.35 Moreover, Schoenberg has arranged the spacing of the harmony in a way that enhances the root-supporting character of ro-intervals 10 and 2 by placing Bb5 and D6 in the correct octave of harmonics 7 and 9 of C3; also G4 lies in the due octave of the third harmonic. These relationships are graphically demonstrated in Example 6. Thus, a perceptually salient majority of tones supports C as the root, while the inversions of the strongest root-supporting ro-intervals, 5 (F) and 8 (Ab), are excluded. Ulti- mately, however, while I think these features do contribute to the perceptual stability of the concluding harmony, the role of roots proves to be less decisive for the consonance-dissonance system of the piece in general.j6

In seeking theoretical concepts that would restrict octave equivalence to an appropriate degree for the present analytical

"Of these, only ro-interval 8 appears in the chord at all; ro-intervals 1, 5, and 9 are not formed.

35See Pamcun, "Revision of Terhardt's Psychoacoustical Model." Parncutt also considers it possible to regard the minor third (ro-interval 3) as "indirectly" root-supporting, since its third and fifth harmonics are octave equivalent with the root's seventh and third harmonics (ibid., 74-75). If we ignore this-in any case, weak--effect, the property of being root-supporting makes a clear distinction between alternative registral ordering only with regard to interval classes 4 and 5. With interval classes 1 and 3, neither ordering is root-supporting (in the direct sense); with interval class 2, both are.

16Enhancing the rootedness of the concluding harmony is somewhat analogous to the Picardy third in tonal practice (ending a minor-mode piece with a major tonic triad). In two other studies I have argued that the property of being root-supporting plays a much more decisive role in some other post- triadic stability systems: "KokosLvelisyys DebussyllL-Voilesin siivelfunk-tioista" [Whole-Tone Organization in Debussy: On Pitch Functions in "Voiles"], in Sibclius-Akutentian Aikakuuskirja Sic, vol. 4, ed. Veijo MurtomLki (Helsinki: Sibelius-Akatemia, 1993), 176-96; and "Projections of Post-Triadic Harmonies in Debussy," paper presented at the Third International Schenker Symposium, Mannes College of Music, New York, 14 March 1999.

Example 6. The relationship of the concluding harmony to the harmonic series

purposes, a point of departure is offered by Robert Morris's three candidates for pitch-set equivalences, PSC, PCINT, and FB.j7 Under PSC, pitch sets are equivalent if they are transpositions of each other. Under PCINT, they are equivalent if the series of ro- intervals between adjacent tones (pitch-class INTs) are the same, or, equivalently, if the corresponding ro-sets are the same or transpositionally related. Under FB (for "figured bass"), pitch sets are equivalent if the sets of the bass-related ro-intervals are the same, or, equivalently, if the "initially ordered pitch-class sets" (ordered pairs formed by the lowest pitch-class and the unordered set of the upper pitch-classes) are the same or transpositionally related.)* Of the registral modifications of the concluding harmony shown in Example 7, none are equivalent under PSC, i and i i are equivalent under PCINT, and i-vi are equivalent under FB.

"Robert Morris, "Equivalence and S~milarity," 213ff. The intervall~c notions of PCINT and FB are the same as Chapman's VP and AB, respectively, in "Some Intervallic Aspects." Stephen Heinemann's ois (ordered pitch-class intervallic structure) also corresponds to AB and FB. Stephen Heinemann, "Pitch-Class Multiplication in Theory and Practice," Music. Theory Spec.trunr 2011 (1998): 72-96.

'%The notion of an "initially ordered pitch-class set" comes from Heine- mann (ibid.).


Example 7. Different spacings of the pitch-class set of the concluding harmony (8-19) . . . i 11 111 iv v vi vii

PSC, PCINT, FB, and, finally, pitch-class set equivalence under TI, (rather than T,,P9) present a continuum of possibilities in which the freedom of registral distribution gradually increases. This continuum could be complemented by other kinds of partial orderings in addition to FB. For example, one could conceive of a "polychordal" harmonic identity based on the differentiation of registral layers. The pitch-class content of each layer would be fixed, but the internal ordering of the pitch-classes within a layer would permit permutations.

The analytical adequacy of such concepts depends on the aspect of organization to which they are applied and on the kinds of harmonies being interpreted. In the harmonic syntax of triadic tonality, FB is sufficient; the main registral concern is the identity of the bass, as related to the identity of the root. For large post- tonal harmonies containing multiple occurrences of interval-classes 1 and 2, the intervals between adjacent tones become more important, and I think, generally speaking, one has to move in the direction of PCINT (without, of course, ruling out the possible rel-

19Extending the equivalence to inversionally related pitch-class sets is a further, and by no means self-evident, step of abstraction.

evance of other equivalence types for some issues). The perceptual difference between chords i and vi in Example 7 ,for instance, is more pronounced than that between major :chords of different spacings, in part because of critical-band effects.40 Therefore, a system in which all of chords i-vi are functionally equivalent is less likely to be comprehensible than the triadic system. On the other hand, it would be going too far to require strict adherence to PCINT and allow absolutely no permutations in the registral ordering. A rough informal evaluation of the perceptual characteristics of the chords in Example 7 would suggest that chords ii-iv are clearly close enough to the referential harmony i to function as its equivalents and chords vi-vii clearly dissimilar to it, whereas chord v presents a borderline case, bearing the characteristics of i-iv with some distortion caused by the semitone A#-B.

'OIn this connection we should perhaps be reminded that the correspondence between perceptual and functional properties IS more or less rough. Thus, for example, triad tones may also fall within a critical bandwidth if they are placed in a low register, where the bandwidth is much wider (e.g., the first and the last chords of Beethoven's Pathktiqut. Sonata). For a different argument that regs- tral disposition tends to play a more important role for atonal than tonal harmonies, see Benjamin, "Ideas of Order," 24-25.


Establishing rigid rules for permissible octave transfers is diffi- cult and may be impossible even in principle, since the role of context may also be decisive in the interpretation of harmonies. An example from tonality is provided by the chord, which may function as a consonance in contexts that clarify its relationship to the corresponding root-formed triad.

Although the ultimate solution to the interpretation of harmonies will not come from simply identifying ro-sets (PCINT), the relevance of registral ordering in the present case is demonstrated by my earlier observations regarding m. 3 (see Example 3) and compels further investigation of ro-sets in op. 1912. To sum- marize the interval content of ro-sets containing no octave dou- b l ing~ , we may use an eleven-entry ro-interval vector, shown beneath the chords in Example 7. In a notation stemming from Marcus Castren, the numbers of ro-intervals 1-5 are set in an upper row from left to right, the number of ro-interval 6 appears on its own on the right, and the numbers of ro-intervals 7-11 are placed in a lower row from right to left; hence complementary intervals are aligned vertically.4' The similarities among the first four chords here are reflected in their identical ro-interval vectors -a relationship analogous to the familiar Z relationship between pitch-class sets.l2 Another concept related to ro-sets is the series of the ro-intervals between adjacent pitches, Morris's INT, which in the case of Schoenberg's concluding harmony is .

These concepts yield further crucial information about the harmonic organization of Schoenberg's op. 1912 when applied to the chord in m. 6, the piece's other sustained harmony and point of momentary rhythmic inactivity. Example 8 compares this chord, labeled B, with the concluding chord, labeled A. It identifies sev- eral points of contrast. With respect to interval-class content, the

41Castr6n, "Joukkoluokitukseen perustuva sointuluokitus," 10. 42However, one should refrain from postulating an identical ro-interval vec-

tor as a sufficient or necessary condition of harmonic equivalence. Examples of chords with identical ro-interval vectors but less than obvious perceptual similarity are presented in Castrtn, "Joukkoluokitukseen perustuva sointuluokitus," Example 16.

Example 8. Different intervallic notions applied to chords A and B

A B

ro-interval 02550 02202 vector 5 2 0 2 5 ~ 302202 ic vector 545752 324222

FBintervals 2,3,4,6,7,10,11 5 ,6 ,8 ,9 ,11

most pronounced contrast is that chord A favors interval class 4 whereas chord B favors interval class 3.43 The favored ro-intervals and FB intervals likewise show different features, with, however, one important exception: in both cases interval class 1 is represented exclusively by ro-interval 11. Studying the INTs, shown to the right of the chords in the example, confirms the decisive importance of this aspect for the chords. Within each INT is a repeated cyclic generator, or , that has a total width of 11. In both cases the cycle just stops short of including ro-interval 1: the next notes upward in the pattern, F6 in A and E4 in B, would form that ro-interval with lower notes (cf. Example 9 below).

How should we regard the structural status of chord B'? Ac-cording to Straus, this chord cannot be a consonance, since, in terms of set classes, it "is not even a subset" of the referential har- m ~ n y . ~ ~However, having argued that similarity in terms of pitch-

"More precisely, the set-theoretical contrast between chords A and B may be expressed, for example, in terms of Messiaen's modes of l~mlted transposl- tion, as their being subsets of modes 3 (9-12) and 2 (8-28 = octatonic), respectively. For an analysis focused on the dialectics between interval-classes 3 and 4 in this piece, see Thomas DeLio, "Language and Form in an Early Atonal Composition: Schoenberg's Op. 19, No 2." Indiana Tlleorv Review 1512 (1994): 17-40.

""The Problem of Prolongation," 8.


class sets is not a sufficient condition for similarity in terms of functional (or perceptual) properties, I would argue further that it is also not a necessary condition. Despite the set-theoretical contrast between A and B, a kind of parallelism does exist between these points of relative repose. Characteristic of both chords is the superimposition, at the distance of ro-interval 11, of harmonic units that may be conceived as positions of triads of tertian construction. Both adhere to rules of "consonance" that may be for- malized as follows. We conceive of a harmony here as an ro-set (pc,, PC*, ... ,PC,,);for all i, j: pc, # pcj. Rule (i): For all i, j 5 3: pc, - pc, + 1 and pc, - pcj + 2 (mod 12). (No interval class 1 or 2 is allowed among the three lowest tones. This is equivalent to their comprising a triad of tertian construction and conforms with the proximity principle.) Rule (ii): For all i s 4: pc, = PC,-, + 11 (mod 12). (Tones from the fourth onwards are added to the preceding ones in ascending order at the distance of ro-interval I I .) Rule (iii): For all i, j: i pc, - pc, f 1 (mod 12). (No ro-interval 1 is allowed between any two tones in the harmony.) If the harmony contains at least six tones, rule (i) is not logically necessary but follows from rules (ii) and (iii).45

Example 9 shows all ten ways of satisfying the three rules starting on C4, in a maximally compact spacing (pitch-intervals between the first and the third tones, as well as those between all adjacent tones must be smaller than 12). The chords extend as far as possible without including the bracketed black noteheads, which are the points at which rule (iii) would be violated. Com- paring the sets of chords shown in Examples 7 and 9 illustrates that pitch-class set identity, on the one hand, and the consonance

'5If we assume that pci - pc, = 1 (mod 12), i, j 5 3, rule (ii) implies PC,+, = pci + 11 = pc, + 1 + 11 = pc, (mod 12), hence pc,,, = pcj against the basic assumption. Again, if we assume that pc, - pc, = 2 (mod 12), i, j 5 3, rule (ii) implies PC,,, - pc, + 11 - pc, + 2 + 11 = pc, + 1 (mod 12) against rule (iii). Consequently, neither assumption can be true.

rules, on the other, define two very different aspects of similarity. For the consonance-dissonance system of Schoenberg's op. 1912, the latter aspect is of primary significance. In general I would suggest that the similarity between the chords in Example 9-which stems from the approximate registral density and the permeating sound of ro-interval I I-is more closely tied to their concrete perceptual properties and, therefore, provides a more viable basis for syntactic functions like consonance and dissonance than the similarity between, say, chords i and vii in Example 7. However, while chords A and B are both consonances, according to rules (i)-(iii), their stability is not necessarily equal. Regarding A as the referential harmony, it may be assumed that chord B, with its many contrasting features (for example, the use of "root-opposing" FB in-tervals), is a consonance of a somewhat lesser rank than A.

The consonance rules imply a conception of harmony based on the superimposition of registral layers, each of which contains three successive tones. Rule (i) regulates intervals within each layer, and rules (ii) and (iii) determine the relationships between the layers. It should be observed, however, that in the case of chord A, the borderlines between the layers are essentially arbi- trary, since any three adjacent tones fulfill the condition required by rule (i). Chords iii and iv in Example 7 are obtained by cycli- cally permuting the "augmented triads" of PC,-pc, and PC,-pc,; consequently, the ro-interval content is invariant.

The choice of chords A and B (e and j) from the ten possibilities in Example 9 deserves some comments. The INTs of these two chords include 2, 3, 4, and 6. It may be that 5 (the "fourth"), with its special perceptual characteristics, would be out of place in this piece (though Schoenberg readily used quartal harmonies elsewhere). Thus, it would be reasonable to stipulate an additional rule forbidding 5 as an adjacent-tone interval, thereby ruling out chords a, c, g, h, and i. Of the remaining chords, b and f resemble e (A) in that their INT is based on the same cycle . Choosing A as a kind of representative of these may be explained on the basis of the predilection for root-supporting intervals; B, on the other hand provides the greatest contrast to A with respect to


Example 9. Chords adhering to rules (i)-(iii)

FB intervals (see Example 8). Finally, in terms of pitch-class sets the chords in Example 9 may be divided into four groups according to the conventional triad type-major, minor, augmented, di- minished-that lies at the bottom of the chord; the hexachords formed by the six lowest tones are, respectively, 6-Z19B, 6-Z19A, 6-20, and 6-Z13.46 As is shown in Example 10, chords A and B contain a representative selection of these hexachords.

The question remains as to what the conception of harmony inherent in the consonance rules implies with respect to the harmonylvoice-leading condition. The introductory examples (Examples 3 and 4) show the use of the semitone and the whole- tone as voice-leading intervals. The voice-leading status of the semitone was readily justified by the absence of ro-interval 1 from

T h e Toset-class labels come from Marcus Castrkn, Joukkorrorian pe-ruskysymybia (Helsinki: Sibelius-Akatemia, 1989), 33-39.

the referential harmony. In the case of the whole-tone, however, it was suggested that a distinction between "simple" and "compound" versions of ro-interval 2 might serve as the basis for satisfying the harmonylvoice-leading condition (thus moving in the direction of PSC here). Although this principle would be in line with the proximity principle, in the actual music it is not rigidly observed. One might say that the distinction between harmonic and voice-leading intervals is not necessarily categorical but may be taken as a guiding principle only (as I discuss in the next section of this paper). There is, however, additional justification, suggested by the multilayered conception of the referential harmony, for stating that the voice-leading status of the ro-interval 2 is not really jeopardized. The risk of confusion arises only if we pre- sume that all intervals within the referential harmony are capable of being arpeggiated; I do not make such a presumption. Ac- cording to rules (i) and (ii), two kinds of ro-intervals are determi-


Example 10. Adjacent-tone hexachords in chords A and B

native for the harmony, those within a layer (3, 4, and 7 in chord A) and the one governing the relationship between the layers (11). Correspondingly, arpeggiations occur in the music either between adjacent tones (as in m. 3, see Example 3a) or between adjacent- tone dyads related by ro-interval 11 (as between B5-D6 and C4- D#4 in mrn. 2-3, and between G4-B4 and Ab3-C4 in rnrn. 5-6). Since melodic ro-interval 2s, by comparison, are much less strongly associated with the harmony, we can justify postulating that they will not function as arpeggiations.

3. STRAUS'S CONDITIONS: SOME CONSIDERATIONS

In addition to the two conditions already discussed-(1) con-sonance-dissonance and (4) harmonylvoice-leading-Straus's conditions of prolongation comprise conditions 2, the scale-degree condition, and 3, the embellishment condition. Condition 2 complements condition 1 in its requirement of a system of harmonic stability. Tones with consonant support have greater structural weight than those without, and the mutual hierarchy of the consonances is in turn determined by the hierarchy of scale-de- gees. Condition 3, in turn, calls for a well-defined set of melodic relationships between the structural and the non-structural tones: "we need a consistent model of voice leading that will enable us, for example, to tell an arpeggiation from a passing note."" In the present analysis the basis for meeting condition 3 is the same as in

LIStraus, "The Problem of Prolongation," 7

tonality. The basic types of embellishment are neighboring and passing tones, arpeggiations, and suspensions.

Within the criticism directed at Straus's four conditions of prolongation, one may distinguish two-apparently contradictory-types of counterarguments. According to the first, presented, for example, in a recent article by Steve Larson, the conditions are not met even by tonal music and thus cannot be taken as a prereq- uisite of prol~ngat ion.~~ According to the second, presented by Lerdahl and others, Straus's reasoning is circular because he "con- strains the concept of prolongation to fit only classical tonal music and then demonstrates that other music does not fit it."49

With respect to the first counterargument, it is true that one often encounters situations in tonal music in which the correct prolongational interpretation seems to require violations of Straus's conditions. Textbook examples of such are compiled in Example 11.50 In Examples 1 l a and 1 lb, similar to those presented by Larsons' and by Howard C inna rn~n ,~~ consonances

4SLarson, "The Problem of Prolongation." 49Lerdahl, "Atonal Prolongational Structure," 67-68. The charge of circular

reasoning is also made by Roy Travis, Communication, Journal of' Mu.ric Theory 3 4 2 (1990): 380, and repeated by Howard Cinnamon, "Tonal Elements and Unfolding Nontriadic Harmonies in the Second of Schoenberg's Drei Klavierstllcke, Op. 1 I , " Theory and Prartire 18 (1993): 127-70 [128-291.

T h e examples are adapted from Edward Aldwell and Carl Schachter, Harmony and Voire Leading, 2nd ed. (New York: Harcourt Brace Jovanovich, 1989). Examples 8-15, 16-11, and 13-9.

5iLarson, "The Problem of Prolongation," Examples 4 and 5. 52"Tonal Elements and Unfolding," Figure 1 .


Example 11. Prolongational readings of tonal progressions not adhering to Straus's conditions

prolong dissonances and a tonic chord prolongs other scale degrees, contradicting Straus's conditions 1 and 2. In Example I lc , in turn, melodic motions cannot be interpreted on the basis of the distinction between a step and a leap invoked by Straus in connection with conditions 3 and 4. In spite of the leaping intervals, the A in the bass does not function as a part of an arpeggiated A minor harmony but as a tone with a predominantly contrapuntal function, a "leaping passing tone."(' A reverse case, a "stepwise" interval C4-D4 which does not function as a voice-leading interval but whose member pitches belong to two separate voices, may be found in Example 5 , beginning of m. 15.

To address such criticisms we need only cite Straus's own response to L a r ~ o n . ' ~ He makes two important points. First, even if the pitch-based criteria stipulated by Straus's conditions are not the sole criteria for prolongational interpretations, but may occasionally be overridden by the impact of other parameters, they still

S'Quite speculatively, I would suggest that a psychoacoustical explanation for this kind of use of leaping bass might be based on the fact that the critical bandwidth is considerably larger In the low register.

S4JosephN. Straus, "A Response to Larson," Journal r fMus~c . Tlieorv 4111 (1997). 137-40

remain criteria among others, to say the least. Second, and above all, instances such as those in Example 11 typically occur near the foreground. There will always be a higher level where the dissonance is subordinate to a consonance and where all scale degrees are subordinate to the tonic. (And the strongest linear connections, such as those in the Urlinie, or in the upper voice of Example I Ic, remain to be made by steps.) Ultimately the point is not that Straus's conditions should be rejected but, rather, that they cannot be mechanically applied to the musical surface with no consideration for broader context. This, to be sure, is a lesson important to bear in mind when evaluating purported post-tonal prolongational structure^.^^

The relevance of Straus's conditions to the deep structural aspects of prolongation is best illuminated by studying how they relate to the idea of a referential harmony to be p r ~ l o ng ed . ' ~ In

55For example, I do not think the existence of a consonance-dissonance system can be reliably decided by studying individually each vertical "slice" In a passage in the manner of Straus's treatment of Felix Salzer's analysis of Stravinsky's Synipl~ony in Tl~rrr Movenirnt.> (("The Problem of Prolongation." 10-13) or Boss's discuss~on of a passage in Schoenberg's song Srrnpllrm ("Schoenberg on Ornamentation,'' 190-9 1 ).

s6Cf Straus, "The Problem of Prolongation," 2-5.


tonality, the triad serves both as the norm of harmonic stability and as the ultimate source of linear progressions. The consonant intervals are those that can be formed by the tones of the triad, with the partial exception of the fourth (condition 1).The foundation of the scale-degree hierarchy is the relationship between the root and the fifth of the tonic (condition 2). The primary voice- leading intervals, the seconds, on the other hand, are absent from the triad (conditions 3 and 4). Since the crucial idea in most of the proposed prolongational analyses of post-tonal music has been the replacement of the triad by other kinds of referential harmonies,s7 these may be assumed to offer similar clues to the ways in which the four conditions are met. As the reader has undoubtedly noted, precisely this kind of assumption has served as the basis of the present discussion.

Relating the four conditions to the concept of referential harmony also reveals interdependencies between the conditions. For example, the distinctions between consonant and dissonant intervals (condition l),on the one hand, and harmonic and voice-leading intervals (condition 4), on the other, tend to be closely related. If harmonic intervals are present in, and the voice-leading intervals absent from the referential harmony, the former will tend to be consonances and the latter dissonances.58 In the Schoenberg, we have seen the relevance of this interdependency to the dissonant status of ro-interval 1. However, this correspondence is not

%ee, for example, Roy Travis, "Toward a New Concept of Tonality?," Journal of Music Theory 312 (1959): 257-84; idem, "Directed Motion"; Morgan, "Dissonant Prolongation"; Cinnamon, "Third-Relations as Structural Elements"; Edward Laufer, "Schoenberg's Klavierstllck, Op. 33a: A Linear Approach," paper presented at a joint meeting of Arnold Schoenberg Institute and the Music Theory Society of New York State, 4 October 1991; Timothy Jackson, " 'Your Songs Proclaim God's Return': Arnold Schoenberg, the Composer and His Jewish Faith," Inrernarional Journal of Musicology 6 (1997): 281-3 17. It should perhaps be mentioned that Jackson's article came to my attention only after writing the present paper.

58Cf. Lester's observation: "Consonance and dissonance are closely related to skip and step, which, In turn, are closely related to simultaneity interval and non-simultaneity interval" ("A Theory of Atonal Prolongations," 2).

completely straightforward. There may be syntactical rules excluding certain arpeggiated harmonic intervals from being presented as consonant simultaneities (as is exemplified by bass-line fourths in tonality). Besides, the connection will not work the other way round: the dissonance of an interval does not imply its suitability for voice-leading (as is exemplified by the tritone in tonality). Furthermore, dissonances may also be arpeggiated, and the interpretation of horizontal intervals partly involves distinguishing between voice-leading intervals and arpeggiated dissonances (which in the case of the C4-D4 in Example 5 is decided on purely contextual grounds).

As regards the second counterargument against Straus's reasoning, that the four conditions were constrained to fit only conventional tonality, the present article takes the opposite stand. Straus himself has repeatedly emphasized that there may, in principle, be other ways to fulfill the conditions than those of conventional tonality.s9 The aim of the present paper is to suggest-by adopting methods of interpreting harmonies and intervals that deviate from Straus's-that this possibility may also have been real- ized to a greater extent than Straus assumed. I would just express one minor reservation about the way in which the conditions determine harmonic stability. I do not believe that the two-stage system defined by conditions 1 and 2 is the only possibility for doing this. A system of consonance and dissonance does not necessarily require a distinction between only two alternatives; there may be finer gradations in harmonic stability, as is already evident in the status of :and chords in tonality. It is conceivable that a yet richer system of gradated consonance could, at least partially, sub- stitute for the system of scale degrees.60 In Schoenberg's op. 19/2,

59"The Problem of Prolongation," 7; "A Response to Larson," 138. "The reader may recognize a similarity between this view and some of

Schoenberg's ideas on gradated consonance (Theory nf Harmony, 18-22, and Style and Idea, 260-61). Schoenberg, however, discusses this notion from a perceptual rather than functional perspective and seems to have doubted whether the funct~onal potentialities of harmonies are likely to be based on "the quality of sharpness or mldness of the dissonances." See Arnold Schoenberg,


the notion of gradated consonance applies to the relationship between chords A and B, the latter of which is assumed to be a consonance of a lesser rank than the former.

Apart from this reservation, what I find especially appealing about Straus's conditions is that they point out ways in which pitch-based norms affect prolongational structures, as these are known from tonality, without tying the norms to the particulars of tonality. In my approach to op. 1912, the relevant norms are de- rived from or related to the referential harmony, chord A. This contrasts sharply with those analyses in which concepts specific to triadic tonality, such as scale-degree functions or Urlinie descent, are directly applied to music in which there is little evidence of the normative position of the triad.61 In Travis's analysis of op. 1912, for example, the "V-I" relationship between the opening G and the closing C functions as the very basis of the reading.G2 Without denying the possibility of hearing such an allusion to tonality, the present approach places this allusion in a framework concordant with the harmonic material specific to this composi- t i ~ n . ~ ~The fifth-progression G-C is well compatible with the normative position of chord A with its abundance of ro-interval 7s (see Example 2), but it is not an event of the highest structural rank comparable to a Bassbrechung (cf. Example 16a below). Possible common features in tonal and post-tonal prolongational structures thus have to be traced back to the common features in the referential harmonies in order to determine their structural significance. These common features, in turn, might perhaps be taken as vestiges of tonality in atonality, but another way to look at them

Structural Functions of Harmony (London: Williams and Norgate, 1954), 194-95.

6'ln addition to Travis, "Directed Motion," these include Steve Larson, "A Tonal Model of an 'Atonal' Piece: Schoenberg's Opus 15, Number 2, "Perspec-tive.\ o fNew Must(. 2511-2 (1987): 418-33, Cinnamon, "Tonal Elements"; and James Baker, "Voice-Leading in Post-Tonal Music: Suggestions for Extending Schenker's Theory," Mu~ic.Ana1y.si.s 912 (1 990): 177-200.

62Travis, "Directed Motion," Example I , etc. "Cf. the related discussion by Straus, "The Problem of Prolongation," 19.

is to relate them to more general principles with perceptual back- ing, like, in the case of chord A, the proximity principle and the predilection for root-supporting intervals.

4.THE ANALYSIS COMPLETED

Measures 1-5,. The transpositional relationship, shown in Example 3,between the beginning and the end of the piece, can be extended to include the entirety of chord A. Example 12 illustrates. The dyad B5-D6 on the upbeat to m. 3 continues the cyclic interval pattern in a regular fashion (Example 12c). By the beginning of m. 5, again, the registral disposition has been trans- formed as shown in Example 12a and 12b--bearing in mind the non-structural status of the Fb6 (see Example 4).This spacing is a transposition of chord iv in Example 7, a very close variant of chord A on grounds of the above discussion. An important reason for using this spacing is that it brings the six highest tones to the same octave as in the concluding harmony, enabling the connection shown in Example 4.

A more problematic pitch in the opening measures is the F#4 at the end of m. 2, whose salience is highlighted as the piece's dy- namic high point. In principle, there are two possible interpretations for the status of this F# (see Example 12a). If we ma~ntain the distinction between ro-intervals 1 and 11, the semitone formed by the F#4with the left hand's G4 is to be regarded a dissonance, malung the F#a non-harmonic tone, a lund of incomplete neighbor to G.Another possibility is to take the F# 4 as a harmonic tone whose initial misplacement is corrected at the beginning of m. 5 (Example 12a: dotted arrow), somewhat analogously to the tonal practice of clarifying the functional consonance of a ,b chord by presenting its root in the bass afterwards.'j4 This reading might be defended on the grounds that the resulting harmony (see Example

"See, for example, Aldwell and Schachter, "Harmony and Voice-Leading," Example 19-27.


Example 12. Emergence of T,A

a)

12b, "?")--akin to the "borderline case" chord v in Example 7-is still relatively close to chord A.

Whichever of these interpretations we choose to prefer, two conclusions may be made. First, the opening (mm. 1-5,) intro-duces a harmony, which is, in its entirety, transpositionally related to the concluding one and will thus be denoted T,A. (This notation is slightly inexact in that no formal definition of "A" is given; the registral disposition of " A permits much less octave equivalence than FB but somewhat more than PCINT.) The "unproblem- atic" part of the harmony of mrn. 1-3 will be labeled T,A-, as shown in Example 12c. Second, against this harmonic background, the sernitone F#4-G4 stands out as a prominent "conflict- ing" element, be it a dissonance or an anomalous consonance. If the F # 4 is taken as a non-harmonic tone (as I am inclined to do),

Example 13. Voice-leading In registral transfers of T,A-

a) b)

the connection shown by the dotted arrow in Example 12a is not a connection in the voice-leading structure, but it is an important associational connection, which, as will be seen, extends to the F#s in the remainder of the piece (to be demonstrated later, with refer- ence to Example 17).65

Measure 4. As is shown in Example 4, the C5 primarily functions as a neighbor to B4. It should be noted, however, that the C5-EL5 dyad also recalls those same pitch classes occurring an octave lower in the melody of the preceding measure. This implies the possibility that in relating a harmbny to its octave transposition, different layers of the two chords may be connected by voice-leading intervals (semitones). More complete realizations of this possibility are shown in Example 13. Here, and in subsequent examples, curved broken lines denote octave transfers, and straight solid lines denote semitones connecting notes in different layers. Although this kind of voice-leading entails features that deviate from conventional tonality in matters related to Straus's conditions 3 and 4-like a double function of a neighbor and an octave transfer-these are definable in quite precise terms and do not impair the clarity of the prolongational structure.

6SThat an initially non-harmonic tone has important consequences for subsequent events is, of course, well in line with Schoenberg's general view on non-harmonic tones, namely, that "it IS ~mprobable that in a well-constructed organism, such as a work of art, anything will happen that exerts absolutely no influence anywhere in the organism" (Theov ~f 'Hnrn lony,311)


Measures 5-6. As was observed at the beginning of this essay, m. 5 is characterized by a burst of ro-interval 1s (pitch-interval 13s). After the first half of m. 6, which strongly recalls m. 3, chord B amves in the low register. In this chord, ro-interval 1s are again replaced by 11s. As is shown in Example 14b-d, these 11s may be viewed as registrally transferred resolutions of the 1s (13s) of the preceding bar (cf. Examples 13b and 14c). The confirmation of the Fb-D resolution over Eb in the original register at the end of the piece is shown in Example 4. It would also be possible to interpret the F#3 and A#3 in m. 5 as mere neighbors to Ab 3 and C4 in the manner shown in Example 14e. In the lowest stave of Example 14d and 14e, the two alternative interpretations are mod- eled by means of strict counterpoint. I prefer the first reading because the first half of m. 6 does not create an effect of stable return and because the registral position and the emphatic expression of the low F# s enhance their tendency to be heard as connected.

The effect of the registral transfers on the clarity of the voice- leading structure deserves some discussion. In section 2, it was argued that "compound" realizations of melodic ro-interval 2-such as the G4-F3 in Example 14c-may function as voice-leading in- t e r v a l ~ . ~ ~Moreover, the neat relationship between chords T8A- and B provides contextual justification for making the connection indicated in Example 14a-c. Nevertheless, the octave transfers go against the proximity principle, and their effect on the voice-leading structure is disguising rather than clarifying. As with the F # 4 in m. 2, registral means are employed to make certain elements stand out against the structural frame. As will be presently exam- ined, these singled-out elements form a line of development of their own, a thread of associational organization superimposed on the prolongational structure.

Measures 7-9. Straus comments on the concluding bass-line descent-which, externally, is most reminiscent of tonal prolongation-as follows: "That descending fourth may look like a pro-

MExample 14c even contains an ro-interval 10,A#3-C3, in a voice-leading function. This is clarified by anticipating the C in octave 4.

longational span, but it doesn't function that way."67 The prolongational nature of this passage is, however, well justifiable by the principles set out in the present article. Taking, as before, intervals 1 and 2 as voice-leading intervals and intervals 3 and 4 as harmonic, the bass line may be read as a fifth-progression in the manner shown in Example 15. 15a and 15b show two alternative interpretations of the passage, one starting anew after chord B in m. 6, the other viewing the F3 in m. 7 as a direct continuation of the F3 in chord B. In any case, the analysis satisfies the consonance- dissonance condition. This may be shown by means of INTs, which at the points of greater structural weight are subsegments of the INT of the referential harmony (Example 15c).

We are now ready to discuss the overall prolongational structure of Schoenberg's op. 1912. It is based on three consonant harmonies. The first evolves in the first three measures and lasts until the beginning of m. 5. The second is the chord in m. 6, and the third is the total harmony of the concluding measure. The first and the third, T,A and A, are similar to each other, and they similarly contrast with the second, B. Ro-interval Is emerge in m. 5 as prominent dissonances; their resolution to 11s is shown in Examples 4 and 14.

On the basis of this view of the harmonic structure, the overall outer-voice structure may be interpreted as shown in Example 16a. The top voice is governed by the D6-Fb6-D6 neighboring motion-that is, by prolongation of D6. The interpretation of the lowest voice is more complicated. If we assume the stability of chord B to be secondary in relation to the framing harmonies T8A and A, we may call upon condition 2, the scale-degree condition, to determine the structural order between the two. In Example 16a, the opening harmony is denoted as more structural, since i t functions as the source of the low- (and middle-) voice events (cf. Example 16b). This may seem surprising in the light of our initial assumption, stemming from Travis, of the concluding harmony as the referential, or "tonic," sonority. However, these two views can

67"The Problem of Prolongation," 19.


Example 14.Approaching chord B

a) = b)

be reconciled by distinguishing concepts based on prolongation, on the one hand, and centricity, on the other.68 In tonality, the first degree normally functions both as the bass of the prolonged harmony and as the tonic in the centric sense. A situation analogous to that in Example 16a would emerge if the overall harmonic progression were IV-I (in the finalistic centric sense), or I-V (in

T f . Straus's discussion on this distinction in "The Problem of Pro-longation," 6.

the prolongational sense), in which the V eventually takes over the role of a center.69 In the Schoenberg, however, the relationships between the harmonies of the background level are not based on

6 9 S ~ ~ ha progression occurs In Chopin's Mazurka op. 30, no. 2 According to Schenker, "the uncertalnty which rises about the tonality . . . almost prevents us from calling this Mazurka a completed composition" He~nrich Schenker, Free Conipos~rion,trans. Ernst Oster (New York: Schirmer Books, 1979). $307 For less dogmatic listeners, however, this very "uncertalnty" is an ~ntegral part of the fascination of the Mazurka.


Example 15.The concluding descent

Example 16. The overall voice-leading structure and its connection to the opening harmony


unequivocal subordination, as seems fitting, given the "atonal" tendencies of the music. The tonic-like quality of the concluding harmony stems, to a large extent, from registral manipulation: the overall arpeggiation of Ab-C is presented as a descending ro- interval 8 instead of an ascending 4.70 One might say that the external "weightinessW-created by the more block-like presentation and lower registral placement--of the three harmonies correlates inversely with the structural weights shown in Example 16a.

Let us return to the two points in which the use of register was somewhat problematic for the prolongational interpretation, the semitone F#4-G4 at the beginning (Examples 12a-b, 16b) and the registral transfers in approaching chord B (Example 14). In both cases, registral factors help bring out pitch-class F# in a manner not reflected in its indicated structural position. Thus the connection between the opening harmony and the overall bass line shown in Example 16 also concerns the special role of F#. The tendency of the F# to be in "conflict" with the prolongational structure does not, in my view, diminish the significance of that structure. Rather, the prolongational structure provides the foil against which the special role of F# gets sharply outlined. The F# s in mm. 2 and 6 are not connected by direct voice-leading, but they are bound together by clear associational logic. This is characterized by an overall registral descent, as shown in Example 17, and by the renewal of the ro-interval 1 F#-G in m. 5. This is then resolved to the ro-interval 11 F#-E# of chord B (see Example 14d), triggering (according tq Example 15b, at least) the final descent from the persistent G.

Finally, it may be of interest to compare this analysis with some previous readings. It should be noted that, except for the harmonic relationships in the very background, the above interpretations on pitch-based criteria are in close accordance with other perceptual factors. Rhythmic and metric features support the interpretation of the details in mm. 3, 6, and 7-9 and, on a larger

?Oln fact, the distinction based on registral ordering has had decisive analytical significance only with respect to interval classes 1 and 2.

Example 17. The role of F#

- .- - - - -

voice-leading

scale, the three stable harmonies neatly coincide with the formal articulation. The harmonic instability of m. 5 also has a correspondence in the restlessness of the external gesture. Thus it is not surprising to find some common traits between the present analysis and its predecessors, in which less attention was paid to pitch- based norms and conditions of prolongation. Apart from the top- voice D-Fb-D neighboring motion, these common traits include the prominent role of the descending F#." However, the present approach makes a significant distinction between these two cases. Whereas the top-voice line is a valid example of prolongation, the F#s are, in part, connected only through association.

Although some details of Travis's analysis have been restored (see Example 3), in general it gains little support from the views presented here. Travis makes connections, for example, between the E6 in m. 5 and the F3 in m. 6, and between the D4 in m. 6 and the EL5 in m. 9.72 These connections are supported neither by any kind of prolongational system in the Strausian sense nor by external association. More relevant to the issues discussed in this paper is the analysis by Lerdahl. Example 18 reproduces the prolongational reduction shown in one of Lerdahl's two complementary analytical graphs of the music.73 His reading of the top voice is

llSee Travis, "Directed Motion," Examples 3 and 4, and Stein, "Schoen- berg's Opus 19 No. 2,"Sketch 1, etc.

12Travis, "Directed Motion," Example 4. ?'Lerdahl, "Atonal Prolongational Structure," Figure 11


similar to the present one, but he interprets the lowest voice quite differently. The most striking difference is the low status given to the Ab 3 of m. 3 (cf. Example 16a). In part, this reflects the fact that Lerdahl's analysis ignores pitch-based "stability conditions" and, consequently, overlooks the similarity between the opening and concluding harmonies (cf. note 4). However, in terms of "salience condition^,"^^ one would expect the Ab 3 to gain higher standing by virtue of its framing function, both temporally and registrally, in the first melodic gesture.

5.CONCLUSIONS AND ADDITIONAL EXAMPLES

This analysis of Schoenberg's op. 1912 shows a structure supported by such pitch-based norms as are required by Straus's conditions of prolongation. Crucial for discovering these norms was the rejection of unrestricted octave equivalence-recognition of the registral disposition of pitch classes in interpreting harmonies and intervals. Would a similar approach to other comparable pieces yield similar results? I would be cautious about claiming that large-scale structures of comparable clarity would be revealed in every case. Instead, I would suggest two conclusions of a more modest kind. First, insofar as syntactic norms relevant to prolongation are to be found in post-tonal music, they are not likely to be discovered purely on pitch-class set basis.7s Second, registral disposition does play a more important role in "the structure of atonal music" than is often acknowledged, participating in different kinds of functions, including-at least occasionally-those re-quired by Straus's conditions. The structural norms of Schoen- berg's op. 1912, as outlined in the above rules of consonance for that work, are, of course, piece-specific. However, some features, like the proximity principle and the distinction between ro-intervals 1 and l l , have more general significance for the early

74For Lerdahl's formulation of these, see ibid., 73-74. 7SThis conclusion is corroborated by other studies of the present author, in-

cluding those mentioned in note 36.

atonal repertoire. I will demonstrate as much in the following additional examples from the first two "atonal masterworks" for piano, op. 11, no. 1 and no. 2.

In op. 1111, the most obvious function of the distinction between ro-intervals 1 and 11 is the differentiation of formal sections. In the framing sections (mm. 1-10 and 52-63), ro-interval 11 is a prevalent harmonic interval, whereas ro-interval 1 behaves more or less clearly as a voice-leading interval (as simple semitones) or as a dissonance (usually as "compounds"), as in the "suspension" shown in Example 19a. In intervening sections, pitch-interval 13s emerge as a prominent contrasting element, to- tally permeating the music in rnm. 33-46.76 Here, in turn, l l s are used to create an effect of repose at the end of the first two phrases; see Example 19b. This kind of 13-11 "resolution" by "voice exchange" occurs frequently in the early atonal music; an additional example, from Schoenberg's op. 19 no. I , within an ev-ident neighboring figure is shown in Example 1 9 ~ . ~ ~ Although these examples show syntactic (or, at least, prosodic) functions not unlike consonance and dissonance, it is not easy to see whether these functions contribute to any kind of prolongational organization in op. 1111.

Op. 1112 contains more straightforward instances in which the consonance-dissonance relationship between ro-intervals 1 and 11 is at the service of the composing-out of harmonies. Example 20 illustrates. At the beginning of the piece, Example 20a, Eb is introduced as a neighbor of Db; Example 21a shows a harmonic reduction of the passage. Then, B4 is used as a non-harmonic tone in relation to the harmonic entity shown inside the boxes in Example 20b. Note especially the B4-A4 resolution in m. 22, in which the

761t is not always clear whether these 13s should be regarded as voice- leading intervals (enlarged semitones) or as arpeggiated dissonances. Hugo Leichtentritt was probably the first to present the former explanation, in Mu.,ik-alische Formenlehre, 3rd ed. (Leipzig: Breitkopf & Hitrtel, 1927), 437ff

llThis kind of "voice exchange" (with no connotations of consonance and dissonance) has also been discussed, among others, by Benjamin ("Ideas of Order," Example I).


Example 18. Reproduction of Lerdahl's graph


Example 19. Schoenberg, op. 1111: uses of ro-intervals 11 and 1

b, 13 13 13 13 13 I I 13 13 13 13 13 l l C ) Cf Op. 1911 1 1 1 1 1 1 1 1 1

structural primacy of A occurs in marked opposition to any kind of "salience c0nditions."~8 Example 20c is basically a transposition (T,) of 20a. E3 occurs first as a neighbor and then as a passing tone. The filled-in D3-F3 is carried over to the next harmony; a transpositional PSC relationship is shown by the boxes (although the structural status of the pitches remains to be specified). The symbols below the example refer to chord X, introduced in Example 21a, which is equipped with an additional FB interval 2. Finally, Example 20d shows how the conclusion of op. 1112 echoes, in terms of ro-sets (PCINT), an earlier gesture, spiced with a dissonant appoggiatura.

Example 21 contains voice-leading graphs for the beginning of the piece. Clause a indicates how the first phrase introduces two semitonally related harmonies, X and Y, using a double-neighbor figure similar to one occurring in m. 22 (Example 20b). Chords X and Y are conceived here in terms of FB intervals, although actually the most important constraint on the registral disposition of X and Y is that the ro-interval 11 is not to be inverted to 1. In the example, FB intervals are shown with relation to D; the F1 is con-

18The prolongation of this harmony extends, I believe, also through those parts of mm. 23-25 that are not shown in Example 21b.

sidered a pedal. Both X and Y are prominent harmonies in the composition and function as consonances. They are also more generally characteristic of S ~ h o e n b e r g ; ~ ~ Example 21b quotes an excerpt from op. 1111 in which the progression X-Y occurs without elaboration. In the first fifteen measures of op. 1112, chord Y is predominant; its composing-out is sketched in Example 21d. Note especially the salient dissonant "double appoggiatura" in m. 7 , the resolution of which vigorously vindicates the primacy of Db3-Ab3 over Dk3, called, in some sense, into question in the preceding measure (see "?"). At the beginning of m. 8, Ab is lifted into octave 5, marking the completion of the "initial ascent" (cf. Example 21g). In the top-voice arpeggiation of Y that follows, interpreting the middle voices is more complicated. Example 21d shows a possible partial explanation.

The present line of analysis also sheds light on the chord progression of mm. 11-13.80 Observing the dissonance of the D#5

79The set class of X, 4-19, is especially ubiquitous in set-theoretical analyses of Schoenberg's music. X and Y also correspond to the four lowest tones of chords b and a, respectively, in Example 9.

8oFor a very different recent analysis of this passage, see David Lewin, "A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg's Opus 11, No. 2," Journal qfMusir Theop 3811 (1994): 79-101


Example 20. Op. 1112:ro-interval 1 used as a dissonance


Example 21. Op. 1112: voice-leading graphs of the beginning

a) b) (Cf op 1111) C) Ct Cirulatnon's reading


and the G2 as shown in Example 21e clarifies that the outer voices of the four chords of greater structural weight linearize transpositions of X in parallel r n o ~ em e n t . ~ ~ The first chord (T,(XnY)+a) lends support to G#5; the rest of the progression is shown to be led by the lower-voice arpeggiation of T 8X The upper voice presents a counterpoint in parallel ro-interval l l s , which in this context play a role analogous to that of parallel tenths in tonality. Taken in this way, the bass line anticipates the T8X that emerges as a governing sonority in the next formal section, starting in m. 16 (Example 21f). T8X is employed to enhance the harmonic support of top-voice A. Although A is included in chord X, the outer-voice ro-interval 11 and the attack sonority T8(XnY) (see Example 21a,f) provide, in this context, a more unequivocal support. (An analogous tonal practice is supporting 8with IV instead of Is".) As is shown in Example 21f, latter box, the harmony in m. 16 is, in fact, a combination, T8(XuY), of both chords introduced by the opening phrase. This combination is made possible by set- ting FB intervals 6 and 7 at the distance of ro-interval 11 instead of 1 as in the beginning. A sketch of the first eighteen measures is presented in Example 21g.

While a complete analysis of op. 1112 is beyond the scope of the present paper, some provisional remarks about the overall bassline are in order. The sketch presented in Example 22a shows the bass line of rnm. 1-40 as consisting of the arpeggiation D2- B b 1-F# 1-Eb 1 (T,X) combined with the neighboring movement F1-Ebl. The concluding section (rnm. 55-66) presents a simpli- fied and condensed recapitulation of the arpeggiation. A focal point in the overall form is m. 40, which marks the endpoint of the first arpeggiation and reveals its source as a foreground harmony T,(XUY). If we are searching for a single "referential harmony" for this composition, T,(XuY) is, I think, the strongest candidate. Some tentative anticipations of this harmony are worth noticing;

811n their entirety, the four chords may be taken as transpositions of X with two modifications: in the first and last chord there is an F instead of E.

conceptos de armonia y prolongacion en el op.19 -2 de schoenberg

Documents

atonal music

journal of music theory

faber music

jstor archive

joseph straus

jstor transmission

conditions of use

studies of posttonal