Maximally smooth cycles, hexatonic systems, and the analysis of late-Romantic triadic progressions

Music Analysis

Volumn 15, Issue 1, 1996, Pages 9-40

  Cohn, Richard ^a  

^a NONE

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EID: 60949266789     PISSN: 02625245     EISSN: None     Source Type: Journal    
DOI: 10.2307/854168     Document Type: Review

References (73)

1. Stanley Sadie (ed.), The New Grove Dictionary of Music and Musicians (London: Macmillan, 1980), s.v. 'Wagner, Richard' (Vol. 20, p. 123).
2. Felix Salzer and Carl Schachter, Counterpoint in Composition (New York: McGraw-Hill, 1969), p. 215.
3. For expressions of similar judgement by earlier theorists, see Rudolf Louis & Ludwig Thuille, Harmonielehre, 7th edn (Stuttgart: Carl Grüninger Verlag, 1920), p. 345;
4. Ernst Kurth, Romantische Harmonik und ihre Krise in Wagners 'Tristan', (Bern: Paul Haupt, 1920), p. 248;
5. Donald Francis Tovey, 'Tonality in Schubert', in Tovey, The Mainstream of Music and Other Essays (London: Oxford University Press, 1949), pp. 154-5
6. (originally published as 'Tonality', Music & Letters, 9 (1928), pp. 341-63);
7. Adele Katz, Challenge to Musical Tradition (New York: Alfred A. Knopf, 1945), p. 213.
8. David Lewin, 'Amfortas's Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic Cmusic flat sign/B', 19th-Century Music, 7/iii (1984), pp. 345-9 and passim.
9. Lewin affiliates these paradoxes with the magical qualities of the opera, and by implication with the entire nineteenth-century tradition of associating equal divisions with magic, a tradition that goes back to Schubert's Die Zauberharfe and Weber's Der Freischütz, and forward to Stravinsky's Scherzo Fantastique and beyond. For more on this tradition, see Richard Taruskin, 'Chernomor to Kashchei: Harmonic Sorcery; or, Stravinsky's "Angle"', Journal of the American Musicological Society, 38 (1985), pp. 72-142.
10. For evidence suggesting that the affiliation of enharmonicism with altered states of consciousness may have deeper historical roots, see Edward Lowinsky, 'Adrian Willaert's Chromatic "Duo" Re-examined', Tijdschrift voor Muziekwetenschap, 17 (1956), pp. 1-36
11. reprinted in Lowinsky, Music in the Culture of the Renaissance and Other Essays, ed. Bonnie J. Blackburn (Chicago: University of Chicago Press, 1989), pp. 681-98.
12. For a penetrating discussion of the problems encountered in assigning diatonically based interpretations to enharmonically ambiguous events, see Gregory Proctor, 'Technical Bases of Nineteenth-Century Chromatic Tonality' (PhD diss., Princeton University, 1978), esp. pp. 140-43, 177-8.
13. Lewin, 'Amfortas's Prayer', p. 344.
14. Brian Hyer, 'Reimag(in)ing Riemann', Journal of Music Theory, 39/i (1995), pp. 101-38, discusses Riemann's views on intonation, and their implications for enharmonic problems.
15. See also Robert W. Wason and Elizabeth West Marvin, 'Riemann's "Ideen zu einer Lehre von den Tonvorstellungen": An Annotated Translation', Journal of Music Theory, 36/i (1992), pp. 69-117.
16. For a general discussion of the impact of just intonation on nineteenth-century harmonic theory, see Wason, 'Progressive Harmonic Theory in the Mid-Nineteenth Century', Journal of Musicological Research, 8 (1988), pp. 55-90.
17. Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987), pp. 175-80;
18. 'Some Notes on Analyzing Wagner', 19th-Century Music, 16/i (1992), pp. 49-58.
19. Riemann's table took several forms. The one that is reproduced in Wason and Marvin, 'Riemann's "Ideen"', p. 102
20. On the development of the table, see Martin Vogel, On the Sensations of Tone, tr. Vincent Jean Kisselbach (Bonn, 1993), pp. 108-10.
21. On its potential for interpretation as a group structure, see Hyer, 'Tonal Intuitions in Tristan und Isolde' (PhD diss., Yale University, 1989), pp. 214-15 and passim; and 'Reimag(in)ing Riemann'.
22. For an overtly neo-Riemannian consideration of the dispensability of dualism, see Hyer, 'Reimag(in)ing Riemann'. For a recent defence of Riemann's dualism, see Daniel Harrison, Harmonic Function in Chromatic Music (Chicago: University of Chicago Press, 1994), Chap. 1.
23. Lewin notes that a transformational perspective is implicit in Riemann's conception of functions (Generalized Musical Intervals, p. 177).
24. In a review of Generalized Musical Intervals (Intégral, 2 (1988), pp. 161-77) Bo Alphonce has suggested that such a perspective is already present in much post-Riemannian work in Northern Europe, and perhaps even explicit (though not fully formalised) in Riemann's own writings. David Kopp has called attention to a passage in Riemann's writings where he explicitly 'argues for an essential identity of chord progression types existing independently of the character which they take on in the context of a key'.
25. See Kopp, 'The Function of Function', Music Theory Online, 1/iii (1995), paragraph 9.
26. Hyer, 'Tonal Intuitions', pp. 175-226.
27. These transformations are introduced in Lewin, Generalized Musical Intervals, p. 178, using the labels PAR, REL, and LT respectively. The single-letter labels are from Hyer, 'Tonal Intuitions'.
28. Some readers may be struck, as I have been, by the kinship this caveat bears with similar formulations by scholars of 'pre-tonal' music. See, for example, Don Randel, 'Emerging Triadic Tonality in the Fifteenth Century', Musical Quarterly, 57 (1971), pp. 73-86.
29. This re-frames an observation of Gerald Balzano, 'The Group Theoretic Description of 12-Fold and Microtonal Pitch Systems', Computer Music Journal, 4/iv (Winter 1980), pp. 66-84.
30. The name was suggested to me by Easley Blackwood. Set-class 6-20 has borne many other rubrics in the music theoretic literature, including 'Wunderreihe' (sometimes translated as 'Miracle hexachord'), 'Ode to Napoleon' (due to its prominence in Schoenberg's piece of that name), 'source-set E', 'augmented scale' and 'Liszt mode'. See Donald J. Martino, 'The Source Set and its Aggregate Formations', Journal of Music Theory, 5 (1961), pp. 224-73;
31. Lewin, 'Intervallic Content of a Collection of Notes and its Complement: an Application to Schoenberg's Hexachordal Pieces', Journal of Music Theory, 4 (1960), pp. 98-101;
32. Jeff Pressing, 'Pitch Class Set Structures in Contemporary Jazz', Jazzforschung, 14 (1982), pp. 133-72;
33. Henri Pousseur, 'Stravinsky by Way of Webern: The Consistency of a Syntax', Perspectives of New Music, 10/ii (1972), p. 33.
34. Although such representations are associated with Ernst Krenek's name, they originate in nineteenth-century writings. See Wason, 'Progressive Harmonic Theory'.
35. Ernö Lendvai, Béla Bartók: An Analysis of his Music (London: Kahn & Averill, 1971), pp. 51-4
36. and The Workshop of Bartók and Kodály (Budapest: Editio Musica, 1983), pp. 370-81.
37. For further discussion of the properties of 6-20, see Wason, 'Tonality and Atonality in Frederic Rzewski's Variations on "The People United Will Never be Defeated"', Perspectives of New Music, 26 (1988), pp. 108-43;
38. John Schuster-Craig, 'An Eighth Mode of Limited Transposition', The Music Review, 51 (1990), pp. 296-306;
39. Richard Cohn, 'Bartók's Octatonic Strategies: A Motivic Approach', Journal of the American Musicological Society, 44 (1991), pp. 262-300.
40. The tri-partition of third-relations, on the basis of common-tone retention and modal correlation, is suggested in Louis & Thuille, Harmonielehre, 7th edn, p. 347.
41. Richard Isidore Schwartz's dissertation, 'An Annotated English Translation of "Harmonielehre" of Rudolf Louis and Ludwig Thuille' (Washington University, 1982), p. 413, indicates that the passage was included in the 4th edn of 1913.
42. He does not indicate whether it was passed on from earlier editions, which are unavailable to me for examination. For a similar taxonomy, see Hans Tischler, 'Chromatic Mediants: A Facet of Musical Romanticism', Journal of Music Theory, 2 (1958), p. 95.
43. In an article that focuses on polar relations between large-scale tonics in sonata form movements, Rey M. Longyear and Kate R. Covington suggest measuring 'degrees of remoteness among third relationships' on the basis of 'the presence or absence of common tones' ('Liszt, Mahler and a Remote Tonal Relationship in Sonata Form', in Anke Bingmann, Klaus Hortschansky and Winfried Kirsch (eds.), Studien zur Instrumentalmusik: Lothar Hoffman-Erbrecht zum 60. Geburtstag (Tutzing: Hans Schneider, 1988), pp. 457-69).
44. This is the sense, for example, in which transposition by seven semitones generates the chromatic universe via the 'cycle of fifths'. For an introduction to generators in music theory, see Balzano, 'The Group Theoretic Description of 12-Fold and Microtonal Pitch Systems'.
45. Salzer and Schachter observe (Counterpoint in Composition, p. 215) that the major cycle is more common than the minor. The explanation for this remains an open question.
46. Lendvai, Workshop, pp. 199, 235-8 and 377-81. Lendvai's term for this triadic pairing is 'complementary', i.e. with respect to the '1:3 model'. Lendvai uses illustrations from music by Beethoven, Wagner, Verdi, Bartók and Kodály.
47. See Lewin, 'Music Theory, Phenomenology, and Modes of Perception', Music Perception, 3 (1986), pp. 327-92.
48. The passage is discussed in somewhat similar terms in Longyear & Covington, 'Liszt, Mahler', p. 466
49. and in Leonard Ratner, Romantic Music: Sound and Syntax (New York: Schirmer, 1992), p. 289.
50. Carolyn Abbate, Unsung Voices (Princeton: Princeton University Press, 1991), p. 152.
51. Kurth, Romantische Harmonie, p. 246.
52. Translated as 'supernatural strangeness' in Lee Rothfarb (ed.), Ernst Kurth: Selected Writings (Cambridge: Cambridge University Press, 1991), p. 124. Further support for the interpretative claim concerning the 'ethical' qualities of hexatonic poles will be provided in 'Uncanny Resemblances', a chapter of a book (as yet untitled) on which I am working.
53. The relations between members of 6-20 are discussed in detail in Wason, 'Tonality and Atonality', p. 122.
54. The most familiar citation of this 'law' is found in Arnold Schoenberg, Theory of Harmony, trans. Roy E. Carter (Berkeley and Los Angeles: University of California Press, 1978), p. 39: 'Each voice will move only when it must; each voice will take the smallest possible step or leap, and then, moreover, just that smallest step which will allow the other voices also to take small steps'.
55. See Proctor, 'Technical Bases of Nineteenth-Century Chromaticism', p. 159.
56. See also Rothfarb (ed.), Ernst Kurth: Selected Writings, p. 113.
57. 2 relation with its predecessor. I do not mean by this to suggest that Liszt was thinking about pc relations in the same way as Schoenberg was in the 1940s, but rather that Schoenberg would have had some interest in the way that Liszt was thinking about pc relations in the 1870s. For some music by a contemporary of Schoenberg that follows a plan very similar to Ex. 7, see the refrain of Hugo Distler's 'Fürwahr, er trug unsere Krankheit', his Motet No. 9 for mixed four-voice chorus (1934-6).
58. Liszt's strategies for inter-relating diatonic and chromatic modes of organisation are an important concern of Ramon Satyendra, in 'Chromatic Tonality and Semitonal Relationships in Liszt's Late Style' (PhD diss., University of Chicago, 1992). Satyendra's discussion of the Polonaise (pp. 80-81) serves as a point of departure for my treatment of Ex. 7.
59. and Lewin ('Some Notes', p. 57).
60. The need to constrain the application of super-potent descriptive systems has been a theme of much meta-theoretic writing, including William Benjamin, 'Ideas of Order in Motivic Music', Music Theory Spectrum, 1 (1979), pp. 23-4;
61. Richard Taruskin, 'Reply to van den Toorn', In Theory Only, 10/iii (1987), pp. 47-57;
62. and Patrick McCreless's review of Warren Darcy, Wagner's 'Das Rheingold', 19th-Century Music, 18 (1995), pp. 289-90.
63. 'Fumeux fume' is transcribed in Peter M. Lefferts, 'Subtilitas in the Tonal Language of "Fumeux fume"', Early Music, 16 (1988), pp. 176-83.
64. See bars 27-9 and 32-3 (other published transcriptions agree with Lefferts for the passages in question). For a discussion of the problems associated with analysis of chromatic third relations in Gesualdo's music, with special attention to hexatonic poles, see Carl Dahlhaus, 'Zur chromatischen Technik Carlo Gesualdos', Analecta Musicologica, 4 (Köln: Bühlau Verlag, 1967), pp. 77-96.
65. Dahlhaus calls attention to this passage in 'Zur chromatischen Technik Carlo Gesualdos', p. 87.
66. These passages are discussed respectively in Mario LeBlanc, 'Franz Schubert: un pas vers l'atonalité', Canadian University Music Review, 9/ii (1989), pp. 84-115
67. and Paula Jean Telesco, 'Enharmonicism in Theory and Practice in the Eighteenth Century' (PhD diss., Ohio State University, 1993), p. 157.
68. I have discussed this passage in 'As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert', unpublished paper presented at a Conference on Schubert's Piano Music, Washington DC, 1995.
69. See especially 'Music Theory, Phenomenology, and Modes of Perception', pp. 357ff.
70. Generic triads and sevenths are discussed in John Clough and Gerald Myerson, 'Variety and Multiplicity in Diatonic Systems', Journal of Music Theory, 29 (1985), pp. 249-70.
71. Insight into their maximally smooth potential may be gained from Eytan Agmon, 'Linear Transformations Between Cyclically Generated Chords', Musikometrica, 3 (1991), pp. 15-40.
72. Robert Cook discusses the hexatonic potential of the diatonic Grail in 'Alternative Transformational Aspects of the "Grail" in Wagner's Parsifal', unpublished paper presented at the annual meeting of Music Theory Midwest, 1994.
73. This theme is developed in different ways in the dissertations of Hyer ('Tonal Intuitions') and Satyendra ('Chromatic Tonality').