Hierarchical set theory

Abstract

Musicians often operate with a hierarchy of scale-like collections, each embedded within the next, and with transposition and inversion available at every level. A particularly common technique is to counteract a transformation at one level with an analogous transformation in the intrinsic scale consisting of a chord’s own notes.

Keywords:

• neo-Riemannian theory
• contextual inversion
• voice leading
• twentieth-century theory
• set theory
• hierarchy

Acknowledgements

Thanks to Richard Cohn, Nick DiBerardino, Fred Lerdahl, Jacob Shulman, and Jason Yust.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In the words of Robert Morris, “the transformational voice-leading frequently ‘contradicts’ the proximate voice-leading” (1998, 187). Among later writers, Fiore and Satyendra (2005) and Straus (2011) consider neo-Riemannian transformations to be contextual inversions, while I have interpreted them as voice leadings (Tymoczko 2011 and 2020).

2 This inversion is doubly contextual in that both the index number a + b and the scale in which it operates are determined by the collection itself.

3 In these voice leadings, the voices will never cross no matter how they are rearranged in register. The connection to efficient voice leading is discussed in Tymoczko (2006) and (2011), and Hall and Tymoczko (2012).

4 It is also possible to invert around a single note, or combine multiple inversions to obtain neo-Riemannian voice leadings with no common tones at all, as in Figure 5.

5 Previous theorists have often focused on the special case in which the two parallel voices stay exactly fixed.

6 Without octave equivalence, the pitch set (C4, E4, G4) is unrelated to (G3, C4, E4). However, the two are related by one-step intrinsic transposition, and hence belong to the same intrinsic set class.

7 The spaces of voice-leading geometry arise as quotients of ordered pitch space by global symmetries such as octave shift, permutation, transposition, and inversion. With the exception of transposition these global symmetries do not directly correspond to kinds of voice leading, or collections of paths in pitch-class space. For example, since inversion around C4 sends C5 to C3 and C6 to C2, there is no single path it attaches to the pitch class C; this means that pitch-space inversion cannot be represented as a kind of pitch-class voice leading. It is rather remarkable that two pitch-space inversions do define a kind of pitch-class voice leading, an element of inversional set-class space’s fundamental group. These neo-Riemannian voice leadings can be understood as the analogues of inversion-as-global-symmetry, just as voice exchanges are the analogues of permutation (Tymoczko 2020).

8 We can obtain all the bijective voice leadings between unrelated chords by adding one crossing-free voice leading X→Y to the collection X→X; nonbijective voice leadings can be reinterpreted as bijective voice leadings with fixed doublings (Tymoczko 2020).

9 The perspective I am describing brings together two research traditions. The first originates with the cognitive scientist Herbert Simon (Simon and Kotovsky 1963; Simon and Sumner 1968) and develops with Deutsch and Feroe (1981) and Lerdahl (2001). This work focuses on the hierarchical nesting of musical “alphabets.” Several of these writers postulate that the triadic and scalar alphabets are mobile with respect to their enclosing alphabets, while the upper levels of the hierarchy – e.g. the root or the root-fifth dyad – are stationary with respect to the triad. The second tradition begins with John Clough (1979) and continues with David Lewin (1987), generalizing musical set theory to arbitrary collections. This work emphasizes generality, aspiring to comprehend all the possible “alphabets” contained in a given scale. Combined, the two approaches allow us to consider arbitrary nestings of arbitrary alphabets, and to consider how operations at multiple levels can counteract one another.

10 See http://www.madmusicalscience.com/quadruple.html.

11 The intrinsic scale thus offers a slightly more general understanding of musical symmetry:{C5, D5, E5, G♯5} is not invariant under any single pitch-space inversion, but is invariant under a combination of intrinsic and chromatic inversions; similarly, {C4, E4, G♯4} is not invariant under any single pitch-space transposition but is invariant under the combination of intrinsic and chromatic transposition. Once again, our perspective lies between pitch and pitch class, with intrinsic pitch-space operations manifesting what traditional theory would consider to be pitch-class symmetries.

12 In nonbijective voice leadings, notes often change status from octave doublings, or redundant representations of a single scalar voice, to representations of independent scale degrees. This provides a way to interpret Callender’s “split” and “merge” transformations, the latter involving the converse process (Callender 1998).