Concepts of harmony and prolongation in Schoenberg's op. 19/2

Music Theory Spectrum

Volumn 21, Issue 2, 1999, Pages 230-259

  Väisälä, Olli ^a  

^a NONE

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EID: 64949089797     PISSN: 01956167     EISSN: None     Source Type: Journal    
DOI: 10.1525/mts.1999.21.2.02a00040     Document Type: Review

References (110)

1. Arnold Schoenberg, Theory of Harmony, trans. Roy E. Carter (London: Faber Music, 1978), 421
2. Roy Travis, "Directed Motion in Schoenberg and Webern," Perspectives of New Music 4/2 (1966): 84-89
3. Joseph Straus, "The Problem of Prolongation in Post-Tonal Music," Journal of Music Theory 31/1 (1987): 1-22
4. Fred Lerdahl, "Atonal Prolongational Structure," Contemporary Music Review 4 (1989): 65-87
5. See also Deborah Stein. "Schoenberg's Opus 19 No. 2: Voice Leading and Overall Structure," In Theory Only 2/7 (1976): 27-43
6. For an extensive bibliography of studies of post-tonal music applying Schenkerian principles, see Joseph N. Straus, "Voice Leading in Atonal Music," in Music Theory in Concept and Practice, ed. James M. Baker, David W. Beach, and Jonathan W. Bernard (Rochester: University of Rochester Press, 1997): 237-74, note 1
7. Straus, "The Problem of Prolongation," 6-8, 17-19
8. Unlike Straus, who suggests that the four conditions could in principle be met by means other than tonality, Lerdahl rejects the role of stability conditions in "atonal" music on a priori grounds: "Atonal music almost by definition does not have stability conditions" (Lerdahl, "Atonal Prolongational Structure," 73)
9. See Straus, "The Problem of Prolongation," 2-7
10. 2|/ 12
11. see also Jack Boss, "Schoenberg on Ornamentation and Structural Levels," Journal of Music Theory 38/2 (1994): 187-216 [193]
12. Travis, "Directed Motion," Examples 4 and 5c
13. In the present article the ordering is always calculated from the lowest to the highest note. For previous articles discussing registrally ordered sets from various analytical and theoretical viewpoints, see, for example, William E. Benjamin, "Ideas of Order in Motivic Music," Music Theory Spectrum 1 (1979): 23-34
14. Alan Chapman, "Some Intervallic Aspects of Pitch-Class Set Relations," Journal of Music Theory 25/2 (1981): 275-90
15. John Roeder, "A Geometric Representation of Pitch-Class Series," Perspectives of New Music 25/1-2 (1987): 362-409
16. idem, "Harmonic Implications of Schoenberg's Observations of Atonal Voice Leading," Journal of Music Theory 33/1 (1989): 27-62
17. Robert Morris, "Equivalence and Similarity in Pitch and their Interaction with Pcset Theory," Journal of Music Theory 39/2 (1995): 207-44
18. Marcus Castrén, "Joukkoluokitukseen perustuva sointuluokitus: perusperiaatteet ja esimerkkejä sovellusmahdollisuuksista" [Chord Classification Based on Set Classification: Basic Principles and Examples of Applicational Possibilities], Sävellys ja musiikinteoria 7/1 (1997): 6-25
19. In the Examples, ro-interval 1 has been denoted as 13 whenever this is the actual width of the interval
20. Stein, "Schoenberg's Opus 19 No. 2," Sketch 1, etc.; Lerdahl, "Atonal Prolongational Structure," Figure 11
21. For Schoenberg's presentation of this notion, see "Composition with Twelve Tones (1)" (1941), in Arnold Schoenberg, Style and Idea, ed. Leonard Stein (London: Faber and Faber, 1975), 214-43 [220]
22. On the history leading to the concept of pitch-class set analysis, see Jonathan W. Bernard, "Chord, Collection, and Set in Twentieth-Century Theory," in Music Theory in Concept and Practice, ed. James M. Baker, David W. Beach, and Jonathan W. Bernard (Rochester: University of Rochester Press, 1997), 11-52
23. Joseph N. Straus, Remaking the Past: Musical Modernism and the Influence of the Tonal Tradition (Cambridge: Harvard University Press, 1990), 24
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
25. 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 Spectrum 14/2 (1992): 150-70
26. 6 chords with different functional status, see Peter H. Smith, "Brahms and Motivic 6/3 Chords," Music Analysis 16/2 (1997): 175-218
27. For a review of the history of the concepts of consonance and dissonance, see, for example, James Tenney, A History of "Consonance" and "Dissonance" (New York: Excelsior Music Publishing Company, 1988)
28. By 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 of the Acoustical Society of America 38/4 [1965]: 548-60)
29. and virtual-pitch perception, on the other (Ernst Terhardt, "Pitch, Consonance, and Harmony," The Journal of the Acoustical Society of America 55/5 [1974]: 1061-69
30. and idem, "The Concept of Musical Consonance: A Link Between Music and Psychoacoustics," Music Perception 1/3 [1984]: 276-95)
31. 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 Theory 41/1 (1997): 101-36 [110-11]
32. and Fred Lerdahl, "Issues in Prolongational Theory: A Response to Larson," Journal of Music Theory 41/1 (1997): 141-55 [151-52]
33. 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 compositions and passages by Liszt in which, according to analyses of Robert P. Morgan and of Howard Cinnamon, the augmented triad assumes the role of stable referential harmony to which other chords, including positions of minor triads, resolve. See Robert P. Morgan, "Dissonant Prolongation: Theoretical and Compositional Precedents," Journal of Music Theory 20/1 (1970): 46-91
34. and Howard Cinnamon, "Third-Relations as Structural Elements in Book II of Liszt's 'Années de Pèlerinage' 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 Faust symphony
35. On the concept of "auditory stream," see especially Albert S. Bregman, Auditory Scene Analysis (Cambridge, Mass.: MIT Press, 1990)
36. Straus, "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)
37. 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
38. Schoenberg himself also referred to "the tendency to soften the dissonance through wide spacing of the individual chord tones" (Theory of Harmony, 417-18)
39. Plomp and Levelt, "Tonal Consonance." For an overview of the multifarious impacts of the critical band for pitch perception and its width in different registers, see, for example, William M. Hartmann, Signals, Sound, and Sensation (Woodbury, N.Y.: AIP Press, 1997), 249ff
40. The first to explain dissonance as the beating between partials (though with no notion of critical band) was Hermann L. F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music, second English edition, trans. Alexander J. Ellis (New York: Dover Publications 1954 [1885]), Chapter VIII
41. For more recent confirmation, see Plomp and Levelt, "Tonal Consonance," and Akio Kameoka and Mamoru Kuriyagawa, "Consonance Theory Part II Consonance of Complex Tones and its Calculation Method," The Journal of the Acoustical Society of America 45/6 ( 1969): 1460-69
42. That 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. 11), and Kameoka and Kuriyagawa ("Consonance Theory Part II," Figs. 7 and 8)
43. For overviews of experimental confirmation and musical implications, see Bregman, Auditory Scene Analysis, chapters 2 and 5
44. Straus also discusses voice-leading in terms of the structure of the diatonic 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 principle 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)
45. One 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 Symphony in Three Movements, might be a candidate for showing this kind of organization), or simply on a syntactic rule excluding the arpeggiation of semitones
46. Paul Hindemith, The Craft of Musical Composition, Book 1: Theoretical Part, fourth ed., trans. Artur Mendel (New York: Schott, 1942), 68-74
47. See Terhardt, "Pitch, Consonance, and Harmony," and idem, "The Concept of Musical Consonance." For a historical overview of virtual-pitch research, see, for example, Bertram Scharf and Adrianus J. M. Houtsma, "Audition II: Loudness, Pitch, Localization, Aural Distortion, Pathology," in Handbook of Perception and Human Performance, 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, i.e., based on each listener's actual perception, and to what extent indirect, i.e., based on the adoption of musical conventions originally affected by virtual-pitch perception, is outside the discussion of the present paper
48. 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 Perception 6/1 (1988): 65-94
49. 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 problems for acoustical explanations ever since Rameau. On the dissonance of the fourth, see Richard Parncutt, "Praxis, Lehre, Wahrnehmung. Kritische Bernerkungen zu Roland Eberlein: 'Die Entstehung der tonalen Klangsyntax,'" Musiktheorie 11/1 (1996): 67-79
50. It may also be noted that Schoenberg himself repeatedly referred to the idea of relating harmonic material to the overtone series (Theory of Harmony, 18-22, and Style and Idea, 258-64), an idea for which virtual-pitch theory provides a rationale
51. Of these, only ro-interval 8 appears in the chord at all; ro-intervals 1, 5, and 9 are not formed
52. See Parncutt, "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 orderings 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
53. Enhancing 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: "Kokosävelisyys Debussyllä - Voilesin sävelfunktioista" [Whole-Tone Organization in Debussy: On Pitch Functions in "Voiles"], in Sibelius-Akatemian Aikakauskirja Sic, vol. 4, ed. Veijo Murtomäki (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
54. Robert Morris, "Equivalence and Similarity," 213ff. The intervallic 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 Spectrum 20/1 (1998): 72-96
55. The notion of an "initially ordered pitch-class set" comes from Heinemann (ibid.)
56. Extending the equivalence to inversionally related pitch-class sets is a further, and by no means self-evident, step of abstraction
57. In 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 Pathétique Sonata). For a different argument that registral disposition tends to play a more important role for atonal than tonal harmonies, see Benjamin, "Ideas of Order," 24-25
58. Castrén, "Joukkoluokitukseen perustuva sointuluokitus," 10
59. However, one should refrain from postulating an identical ro-interval vector 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 Castrén, "Joukkoluokitukseen perustuva sointuluokitus," Example 16
60. More precisely, the set-theoretical contrast between chords A and B may be expressed, for example, in terms of Messiaen's modes of limited transposition, 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 Theory Review 15/2 (1994): 17-40
61. "The Problem of Prolongation," 8
62. j + 1 (mod 12) against rule (iii). Consequently, neither assumption can be true
63. n set-class labels come from Marcus Castrén, Joukkoteorian peruskysymyksiä (Helsinki: Sibelius-Akatemia, 1989), 33-39
64. Straus, "The Problem of Prolongation," 7
65. Larson, "The Problem of Prolongation."
66. Lerdahl, "Atonal Prolongational Structure," 67-68. The charge of circular reasoning is also made by Roy Travis, Communication, Journal of Music Theory 34/2 (1990): 380
67. and repeated by Howard Cinnamon, "Tonal Elements and Unfolding Nontriadic Harmonies in the Second of Schoenberg's Drei Klavierstücke, Op. 11," Theory and Practice 18 (1993): 127-70 [128-29]
68. The examples are adapted from Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 2nd ed. (New York: Harcourt Brace Jovanovich, 1989), Examples 8-15, 16-11, and 13-9
69. Larson, "The Problem of Prolongation," Examples 4 and 5
70. "Tonal Elements and Unfolding," Figure 1
71. 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
72. Joseph N. Straus, "A Response to Larson," Journal of Music Theory 41/1 (1997): 137-40
73. For 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 Symphony in Three Movements ("The Problem of Prolongation," 10-13) or Boss's discussion of a passage in Schoenberg's song Seraphita ("Schoenberg on Ornamentation," 190-91)
74. Cf. Straus, "The Problem of Prolongation," 2-5
75. See, for example, Roy Travis, "Toward a New Concept of Tonality?," Journal of Music Theory 3/2 (1959): 257-84
76. idem, "Directed Motion"; Morgan, "Dissonant Prolongation"; Cinnamon, "Third-Relations as Structural Elements"; Edward Laufer, "Schoenberg's Klavierstuck, 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
77. Timothy Jackson, "'Your Songs Proclaim God's Return': Arnold Schoenberg, the Composer and His Jewish Faith," International Journal of Musicology 6 (1997): 281-317. It should perhaps be mentioned that Jackson's article came to my attention only after writing the present paper
78. Cf. 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)
79. "The Problem of Prolongation," 7; "A Response to Larson," 138
80. The reader may recognize a similarity between this view and some of Schoenberg's ideas on gradated consonance (Theory of Harmony, 18-22, and Style and Idea, 260-61)
81. Schoenberg, however, discusses this notion from a perceptual rather than functional perspective and seems to have doubted whether the functional potentialities of harmonies are likely to be based on "the quality of sharpness or mildness of the dissonances." See Arnold Schoenberg, Structural Functions of Harmony (London: Williams and Norgate, 1954), 194-95
82. In addition to Travis, "Directed Motion," these include Steve Larson, "A Tonal Model of an 'Atonal' Piece: Schoenberg's Opus 15, Number 2, "Perspectives of New Music 25/1-2 (1987): 418-33
83. Cinnamon, "Tonal Elements"; and James Baker, "Voice-Leading in Post-Tonal Music: Suggestions for Extending Schenker's Theory," Music Analysis 9/2 ( 1990): 177-200
84. Travis, "Directed Motion," Example 1, etc
85. Cf. the related discussion by Straus, "The Problem of Prolongation," 19
86. See, for example, Aldwell and Schachter, "Harmony and Voice-Leading," Example 19-27
87. That 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 improbable that in a well-constructed organism, such as a work of art, anything will happen that exerts absolutely no influence anywhere in the organism" (Theory of Harmony, 311)
88. Example 14c even contains an ro-interval 10, Amusic sharp sign3-C3, in a voice-leading function. This is clarified by anticipating the C in octave 4
89. "The Problem of Prolongation," 19
90. Cf. Straus's discussion on this distinction in "The Problem of Prolongation," 6
91. Such a progression occurs in Chopin's Mazurka op. 30, no. 2. According to Schenker, "the uncertainty which rises about the tonality ... almost prevents us from calling this Mazurka a completed composition." Heinrich Schenker, Free Composition, trans. Ernst Oster (New York: Schirmer Books, 1979), §307. For less dogmatic listeners, however, this very " uncertainty" is an integral part of the fascination of the Mazurka
92. In fact, the distinction based on registral ordering has had decisive analytical significance only with respect to interval classes 1 and 2
93. See Travis, "Directed Motion," Examples 3 and 4, and Stein, "Schoenberg's Opus 19 No. 2," Sketch 1, etc
94. Travis, "Directed Motion," Example 4
95. Lerdahl, "Atonal Prolongational Structure," Figure 11
96. For Lerdahl's formulation of these, see ibid., 73-74
97. This conclusion is corroborated by other studies of the present author, including those mentioned in note 36
98. It 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 Musikalische Formenlehre, 3rd ed. (Leipzig: Breitkopf & Härtel, 1927), 437ff
99. This kind of "voice exchange" (with no connotations of consonance and dissonance) has also been discussed, among others, by Benjamin ("Ideas of Order," Example 1)
100. The prolongation of this harmony extends, I believe, also through those parts of mm. 23-25 that are not shown in Example 21b
101. The 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
102. For 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 of Music Theory 38/1 (1994): 79-101
103. In 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
104. This view is not shared by Cinnamon ("Tonal Elements," 148-50), who sees a D-Bmusic flat sign-Gmusic flat sign-Emusic flat sign progression at the beginning of the piece (see the present Example 22b) but not at the conclusion
105. This "voice exchange" is also observed by Cinnamon, whose view of the structural order of the harmonies in mm. 39 and 40, is, however, the reverse of mine. According to Cinnamon, the harmony in m. 39 is a "cadential" tonic and the subsequent harmony a music flat signII (ibid., 150ff.)
106. Since this is not a complete analysis of op. 11, no. 2, it remains to be specified to what extent the bass line shown in Example 22a is a genuine bass arpeggiation and to what extent its elements constitute a "middleground pun," to cite Straus's expression ("The Problem of Prolongation," 15). However, I think the present arguments suffice to strongly suggest, if not to prove, that more than just a pun is involved
107. 1 in The Structure of Atonal Music (New Haven: Yale University Press, 1973), 46ff
108. Cinnamon, "Tonal Elements," 137
109. Example 21c reproduces the upper stave of the graph in Cinnamon's Figure 8. Example 22b reproduces the bass line shown in the lower graph of Cinnamon's Figure 13 and a part of the upper (more detailed) graph. Cinnamon also discusses unfoldings of non-triadic harmonies, which are interpreted as pitch-class sets derived from the "transformed" tonic and dominant harmonies (including the set classes of X and Y, 4-18 and 4-19; see "Tonal Elements," Figure 23). However, the parsing of the bass line is essentially the same in both discussions
110. According to Cinnamon, in Schoenberg's style "the tendency to disperse traditional stepwise voice-leading between two or more different registers "often obscures the underlying voice-leading structures" ("Tonal Elements," 143). Although instances of registrally transferred voice-leading also occur in the present analysis (see especially Example 21d and 21g), this is constrained by the important distinction between ro-intervals 1 and 11. Voice-leading semitones may be expressed as "compound" versions of 1 but not usually as 11s, which is a harmonic interval of central importance. Technically speaking, the "double double neighbor" figure in m. 7, shown in Example 21d, includes rointerval 11s, Gmusic sharp sign4-A3 and C4-Dmusic flat sign3, as voice-leading intervals. However, the situation is clarified by the (more salient) 1s, Cmusic sharp sign5-C4 and A3-Amusic flat sign3, and, in addition, by the contextual justification provided by the connection to the original double neighbor in the first phrase (Example 21a). Perceptual justification for using "compound" versions of ro-intervals 1 and 2 as voice-leading intervals (cf. also Example 14c) might, perhaps, be based on the proximity between the nth harmonics of the upper tone and the 2nth of the lower (cf. Bregman's tentative explanation of a minor ninth trill in Stravinsky, Auditory Scene Analysis, 477)