Operation Attribution: The seeming comprehensiveness of the LPR group for describing triadic progressions—as though any chord motion could be automatically and robotically

“explained” by a mix of the operators—is an illusion. Even relatively simply moves such as Cg admit multiple transformational narratives (LRP or PRL? These are not the same!). As soon as non- parsimonious logic begins to suggest itself on the musical surface, the integrity of those transformations and compounds comes to into question. The most basic decision the transformational analyst must be confronted with is thus the attribution of operations to the progression under their scrutiny. Specific transformations are selected from specific families of transformations. How this family is chosen, and what operators it admits, is contingent upon its musical relevancy to the music—the degree to which its elements can illuminate analytically worthwhile features and represent intuited musical distances satisfactorily. Those distances, and the pathways they generate between other musical objects (both sounding and implied), in turn generate conceptual spaces through which normative and exceptional motions alike are captured. In Edward Gollin’s formulation, “music-transformational spaces provide or describe contexts, from which otherwise indifferent transformations derive meaning. Such ‘meanings’ within the spatial metaphor take the form of pathways through those spaces—pathways which reflect ways of experience or conceiving a given transformation within the context of that space.”5 _{This is the more profound of the}

two senses of “contextuality” often invoked in transformation theory. (The other being more strictly

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algebraic: that a given transformation is sensitive to the quality of its input.) Gollin’s sense of contextuality denotes the extreme flexibility one has in selecting what counts as a transformation, how transformations combine, and where they are situated within a tonal space. The vast number of possible transformational readings for a given piece are reduced and managed by accounting for the influence of local and global context, which may range from immediately flanking chords to broader harmonic-motivic trends. In our special repertoire, this process is additionally and crucially informed by the information provided and influenced by the rest of the film.

To the film music analyst the selection of transformation is a thoroughly interpretive act. We have seen the hermeneutic complexity attendant with operation selection in several examples, most notably in the Dg# motion in “Cadillac of the Skies” of Chapter 1. In any possible case, the precise transformation chosen to represent passage from one object to another represents a peculiar (not comprehensive) feature of the motion. The chosen descriptor will often imply a larger family of operations that come with their own implied geometries and attendant measures of proximity, as well as the baggage of ingrained topics or associations.

Consider a decontextualized fifth motion between major chords, G to C. The relationship that arises between these two Klangs can be formulated in a number of ways, not all of which involve the hierarchies of diatonic tonality. Figure 4.1 tabulates an abundance of such potential transformational accounts of GC. Despite offering sometimes wildly incompatible or incongruent accounts of the progression, they nevertheless form a loose equivalence class of transformations, all achieving the same effect in the end.6 This list is not exhaustive—infinitely more operations, formal

and informal, can be devised to describe the relationship between G and C major. While each of the nine categories of transformation describes the same chordal motion, each captures some

6 _{This is a loose equivalence class because of the radically different definitions of several of these transformations (nos. 3}

and 4) involve fundamentally different collections of objects (independent voice-leading streams, and scales, respectively) compared to the Klang-manipulation definitions of the others.

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idiosyncratic feature of it. Some are at odds (the frequency, transpositional, and clausula operations imply sharply differing voice-leading procedures) while other complement each other (DOM and clausula operations). Some require a single step or motion (T5, DOM). Others a dictate series of

moves (clausula, RL), literal or elided. Others a dictate series of moves (clausula, RL), literal or elided. One does not expect the tenor voice to proceed through C# en route to C from D; neither does one expect G major first to go to E minor (R), then C major (L), though compound relations by nature assume that the constitutive operations are relevant in some way, otherwise one would use a unary operation. Two transformations, the compounds HS and P·R·P·R·S·H·DOM·P·MED, entail tonal pathways space so foreign from our everyday conception of pitch relations that it is questionable whether the label “authentic cadence” could hold any meaning for what is now cast as an impeccably chromatic transformation. This abundance of options is one of the great boons of the transformational stance. For any parameter specified, there exists at least one—and in all likelihood, many—transformation(s) that can be formulated to describe it.

Figure 4.1: Differing Transformational Accounts of GC motion.

1) Frequency Ratio Shift: 4/3 – Move all pitches by multiplying frequency by 4/3 to arrive at chord all of whose members are a just-intoned perfect fourth higher.

2) Transposition/Inversion or Mode Change: T5 – Transpose up five semitones, or a perfect fourth.

3) Voice-Leading/Clausulae: BASizans·TENozirans·SOPrizans. E.g., lead bass voice (5) down fifth (to 1), tenor (2) down M2 (to 1), soprano (7) up semitone (1).

4) Scale Alteration: Tmode5 – Move from Mode V (Mixolydian) on G to Mode I (Ionian) on C [as in jazz

improvisation].

5) Functional/Dominant: DOM – Become dominant of key. Alternatively, (G)DOM=C, meaning “move G so that it becomes the dominant [of C].

6) Functional/Subdominant: SUBD’ – Go to subdominant of.

7) Neo-Riemannian Group: RL – Invert about major third (I ), then minor third (I )

8) HEX/SLIDE Group: HS – Move to hexatonic pole (Cab), then invert about third (abG).

9) Functional-NRT-HS Group Combined: T1·T11·T11·T1·P·R·P·R·S·H·DOM·P·MED – Oscillate about semitonal

neighbors, then initiate octatonic cycle on G, then SLIDE after reaching octatonic-pole, then take to hexatonic pole, then become dominant, then shift mode, then become mediant.

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Distance between musical objects is implicitly conveyed by the length of the transformation, in terms of the accumulation of component moves within it. Compounds of long word length, such as the extravagantly convoluted, but perfectly realizable T1·T11·T11·T1·P·R·P·R·S·H·DOM·P·MED,

entail a huge conceptual journey to arrive at G from C, with the listener in this case being asked to reconstruct a 13-stage path. Unary operations, on the other hand, are intelligible entirely on their own, and represent short and direct paths through pitch space. Distance, as we have already demonstrated in Chapter 3, figures heavily into the generation of harmonic associativity. Because so many absolute progressions host resilient affective connotations across styles, those associations undergo even greater development, mixing with and interpenetrating one another, when we begin to attribute combinations of APs to sounding harmonic motions, either within a single cue or across an entire score. When considered in tandem, then, the algebraic and affective properties of tonal relations can take on an unparalleled hermeneutic fecundity.

2. Network Spatio-Temporal Design: Selection of operations from the unlimited