Abstract
In Sect. 13.1 the (implied) key-distance theories of Heinichen, Kellner, and Weber are contrasted with the (incomplete) theory of Riepel. For Riepel, the parallel key, the tonic of which is chromatic relative to the home key, is more distant from the home key than a key (for example, the relative) the tonic of which is diatonic. Section 13.2 examines the prevalent belief that Weber’s theory has been empirically validated by Krumhansl and her associates. Finally, Sect. 13.3 posits a “neo-Riepelian” theory of key distance. Unlike other theories, by neo-Riepelian theory a distance between two keys does not necessarily exist; as a result, except for diatonic key relations, key distances in general do not conform to what is known in algebra as “metric space.”
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Notes
- 1.
Every connected graph on which a similar distance is defined is a metric space. Heinichen (1728, p. 837) announces the chapter devoted to his “musical circle” with a caption that refers to the “relationship… of all keys” (Verwandschaft… aller Modorum Musicorum). Lester (1989, pp. 108–109) notes that the musical circle “…is presented [by Heinichen] not as a revolutionary method of establishing a new tonal system, but as a practical convenience to aid in modulation from one key to another…. Heinichen allows modulations not only from one key to another in order in either direction, but also by using alternate keys (for instance, C major to F major). Skipping keys in this manner works for a single modulation, but, as Heinichen notes, is hardly possible for continued use.” In Fig. 13.1b and subsequent diagrams upper- and lower-case letters represent major and minor triads/keys, respectively.
- 2.
Undoubtedly referring to Heinichen’s circle, Mattheson (1735) offers an “Improved Musical Circle, which leads around through all keys better than the ones previously invented” (see Werts 1985, p. 97). However, Mattheson’s circle does not contribute to reducing the overall number of degrees of key-relatedness, and its first- and second-degree relationships do not seem intuitively more satisfying than those offered by Heinichen.
- 3.
The Arbre Généalogique de l’harmonie (ca. 1767) of François-Guillaume Vial anticipates Weber’s table. See Lester (1992), pp. 229–230.
- 4.
- 5.
Bernstein ’s (2002, p. 784) assessment that “Weber’s tonal grid exhaustively measures all key relationships according to their proximity to any tonic key, and thus supplants the more limited conceptual mapping of key relations afforded by the eighteenth-century music circle…” is misleading. The eighteenth-century musical circle, we have seen, implies an equally exhaustive (though different) key metric.
- 6.
- 7.
See also Aldwell and Schachter (2003), p. 448 ff.
- 8.
Note that the claims challenged here are positivistic: Krumhansl’s results prove Weber’s theory. Cohn ’s (2007, pp. 110–111) critique of Lerdahl (2001), though bearing some noteworthy points of contact with the argument that follows, is “negativistic”: the theory of “tonal pitch space” is inconsistent with, and therefore falsified by, (some of) Krumhansl’s data.
- 9.
More precisely, classes of enharmonic key classes in the sense of Definition 10.7. However, for present purposes we may assume that keys have no scores (Definition 10.4), such that “key class” reduces to “key.”
- 10.
Such an assumption is implicit in circular key metric s in general, including Heinichen’s and Kellner’s. Note in Fig. 13.6 that in order to close the circles of fifths and major thirds, Krumhansl and Kessler write D
/C
and e/d. This move implies that every “key” in their representation is in fact an equivalence class of enharmonic keys. - 11.
Apparently because Schoenberg highlights the first-degree relations to C major or C minor on his chart, Krumhansl and Kessler also state (ibid.) that “the [Krumhansl/Kessler] torus has the advantage [over Schoenberg’s chart of regions] of simultaneously depicting all interkey relations, not just those immediately surrounding a single major or minor key” (i.e., C major or C minor). Clearly, however, like Weber’s chart, Schoenberg’s has no fixed center of reference.
- 12.
In their discussion on pp. 345–346 Krumhansl and Kessler note some of these fine distinctions. For example, they note “the closer distance between C major and A minor than between C major and C minor.” However, although they note that “C major and E minor are closer than are C major and D minor,” they conveniently fail to mention that E minor is closer to C major than is C minor.
- 13.
Emphasis added. Krumhansl goes on to claim that “the convergence between the two maps of key distance derived from the tonal and harmonic hierarchies is of considerable interest for a number of reasons” (p. 187), listing no fewer than three.
- 14.
In Fig. 13.8 keys K in bold typeface indicate that the distance from K to HK is different from the distance from HK to K.
- 15.
The key-scheme of the Waldstein Sonata is anticipated by Beethoven in the G-major Piano Sonata, Op. 31 No. 1, where B major in the exposition (m. 66 ff.) is answered in the recapitulation by E major (mm. 218−225). The apparent symmetry by which III
and VI are equidistant from the tonic is also confirmed by Brahms
’s Symphony No. 3 in F major, Op. 90, where in the first movement A major in the exposition (m. 36 ff.) is answered in the recapitulation by D major (mm. 149−153). See also Beethoven
’s “Pastoral” Symphony, Op. 68, where a move from B major to D major in the development of the first movement (mm. 151–190) is answered by a move from G major to E major (mm. 197–236). - 16.
/C
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Agmon, E. (2013). A Neo-Riepelian Key-Distance Theory. In: The Languages of Western Tonality. Computational Music Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39587-1_13
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