1. The progression in Example 1b:i does not literally fulfill Jackson's definition of "tonic" auxiliary cadence, as cited here, since the initial C is devalued as the dominant rather than as a pre-cadential chord; however, it would seem reasonable to apply the term also for this case. One example of this progression occurs at the opening of Beethoven's "Kreutzer" Sonata, according to Lauri Suurpää's analysis, presented at the Third International Schenker Conference in New York, March 1999.

2. To depict temporally changing impressions of structural relationships, it is probably less advisable to mingle them in a single graph than to provide separate graphs, as is done, for example, in Peter Smith, "Structural Tonic or Apparent Tonic?: Parametric Conflict, Temporal Perspective, and a Continuum of Articulative Possibilities," Journal of Music Theory 39/2 (1995): 245-84.

3. Fred Lerdahl, "Atonal Prolongational Structure," Contemporary Music Review 4 (1989): 65-87; Joseph Straus, "The Problem of Prolongation in Post-Tonal Music," Journal of Music Theory 31 (1987): 1-22.

4. In Jackson's graph, the location of the parentheses in m. 61 is unclear. I have "diplomatically" corrected their location in my reproduction. Similar corrections are tacitly made elsewhere.

5. In Nicholas Marston's analysis (which Jackson quotes in his note 34) this C assumes an even greater structural weight (Nicholas Marston, Schumann: Fantasie, Op. 17 [Cambridge: Cambridge University Press, 1992], Ex. 4.7). The only--but certainly not sufficient--reason for such a reading I can think of is that the inner-voice theme introduced in m. 34 ff. is later, in different harmonic and metric circumstances, employed to establish the C minor (m. 133 ff.).

6. By a "psychoacoustical basis," I refer to the model of harmonic roots based on virtual-pitch perception; see, for example, Richard Parncutt, "Revision of Terhardt's Psychoacoustical Model of the Root(s) of a Musical Chord," Music Perception 6 (1988): 65-94.

7. This correspondence between large and small scale is observed by Marston (Schumann: Fantasie, Op. 17; 54, Ex. 4.7). According to Marston the connection goes on to the ascending notes D-E-F, for which I find less justification.

8. Jackson's quotation, taken from Robert Layton's translation of Erik Tawastsjerna's biography, reads actually "the crystallization of ideas," in plural. This seems to be a translation mistake since both the Finnish and Swedish versions of the biography use singular.

9. Edward Laufer, "On the First Movement of Sibelius's Fourth Symphony. A Schenkerian View," in Schenker Studies 2, ed. Carl Schachter and Hedi Siegel (Cambridge: Cambridge University Press, 1999), 136-7.

10. Heinrich Schenker, Free Composition [1935], trans. Ernst Oster (New York: Schirmer Books, 1979), 27.

11. For discussion on multikey schemes, see, for example, William Kinderman and Harald Krebs (ed.), The Second Practice of Nineteenth-Century Tonality (Lincoln and London: University of Nebraska Press, 1996).

12. For Schenker's graph of this prelude, see Free Composition, Fig. 110a:3.

13. Carl Schachter, Unfoldings (New York and Oxford: Oxford University Press, 1999), 260-88; see especially Ex. 11.4.

14. Free Composition, Fig. 13.

15. Jackson goes so far as to call The Wood Nymph "a remarkable example of super-sonata form" (215). I find it difficult to trace any sonata-form features with respect to either thematic disposition or harmonic structure.

16. Jackson shows E as the governing bass tone from m. 82 (I6) to m. 495 (III) (Ex. 8.30). As regards the significance of E, his intuitions might be credited with being on the right track, even if the rationalization of these intuitions in terms of prolongational superiority is totally misguided.

17. This small-scale C-Eb-F-G bass line is identical with the structural bass line that accompanies the Urlinie descent from ^5 to ^2, according to Laufer's analysis (Ex. 12.27, mm. 343-449). The connection is enhanced by the use of a Neapolitan sixth above F in both cases. Laufer points out how the same harmony in m. 500 "sums up what was heard before, in 446-448," i.e., above the structural F (Ex. 12.25), although the chord progression in m. 500 is actually familiar from mm. 88, 148, 172, and 195.

18. Such techniques and their historical precedents are discussed in greater detail in Laufer's previous essay "On the First Movement of Sibelius's Fourth Symphony."

19. In the Seventh Symphony, too, there is an important "rotational" relationship that partly involves an inner voice, between mm. 34-59 and 64-91. In mm. 34-45 and 64-75 the relationship becomes evident by comparing the voices of second violin and first horn, respectively. From mm. 46 and 76 onwards it concerns the top or most prominent voice.

20. For another representative example of recent Sibelius research one may consult Sibelius Forum: Proceedings from the Second International Jean Sibelius Conference, Helsinki November 25-29, 1995, ed. Veijo Murtomäki, Kari Kilpeläinen, and Risto Väisänen (Helsinki: Sibelius Academy, 1998). As regards discussion of the formal language in Sibelius's symphonies, an indispensable source is Veijo Murtomäki, Symphonic Unity: The Development of Formal Thinking in the Symphonies of Sibelius, Studia Musicologica Universitatis Helsingiensis 5 (Helsinki: University of Helsinki, 1993).

End of footnotes