Volume 15, Number 1, March 2009
Copyright © 2009 Society for Music Theory
Stephanie Lind*An Interactive Trichord Space Based on Measures 18–23 of Clermont Pépin’s Toccate no. 3 

KEYWORDS: Pépin, transformational theory, motivic repetition and development, transformational theory, Québécois composers, Canada
ABSTRACT: Clermont Pépin’s Toccate no. 3, a work for piano composed in 1961, features extensive repetition and sequencing through the development of smaller atonal intervallic motives. Measures 18–23, in particular, manifest several elements that return repeatedly in the course of the work, and consequently a space defined by the objects and transformations characteristic of this passage can provide a framework for interpreting transformational relations throughout the piece. This paper provides such a space, in an interactive format that allows the reader to explore grouping structure within the passage. The space is then expanded to accommodate later passages involving similar motivic material.
Received November 2008
Introduction
[1.1] Clermont Pépin (1926–2006), a prolific and influential twentiethcentury Canadian composer, incorporates elements from serialism, jazz, and neoclassicism, but like many Québécois composers of his generation also employs the repetition and development of smaller atonal intervallic motives. By interpreting these intervallic motives transformationally (that is, understanding them as a series of transformations rather than as a grouping of pitch classes), motivic repetition can be depicted as a series of moves through a space. The structure of the space, its objects and transformations, and the path through the space all express characteristic elements of the associated musical passage.
[1.2] The analysis presented in this paper will examine one such case of motivic development in the Toccate no. 3, a work for piano composed in 1961. In this piece, the right and left hands of the pianist present two independent but interactive melodic voices. Recurring intervallic motives are commonly heard, as are exact and nearsequences. Measures 18–23 of the work, in particular, present all of these features, and are also interesting because the musical objects contained therein, SC 027 trichords repeated within each sixnote motivic statement, are transformed in similar ways at multiple levels. In this article, the objects and transformations of this passage will be conceptualized in the context of a particular geometrical figure, which the reader can explore through an interactive animation with sound. To motivate and explain this choice of space, I will present a detailed analysis of measures 18–23. I will then explain how the organization of the passage’s trichords is reflected by the transformational structure of the geometric space, and how the motion between trichords mapped within the space outline hypertransformations that help to understand structural similarities between different pitchclass groupings. Last, I will use the chosen space as a model for other related passages, demonstrating that the composer has organized elements similarly throughout the work.
A Suitable Transformational Space
[2.1] Because trichords are emphasized throughout the Toccate via rhythm (in particular, they are isolated with rests), phrase markings, and repetition, they will form the objects of the present transformational analysis. This focus on trichords is established early on, as heard in measures 8–12 (given in Figure 1). Choosing T_{n}/T_{n}I types (i.e. set classes) as the analytical objects allows one to hear an alternation between two trichord types, set classes 016 and 027, within this passage. Figure 1 shows that these two set classes are heard both in alternation and in overlap; green boxes isolate instances of SC 027 and orange boxes SC 016. These two trichords also saturate the music later on, for instance in measures 18–23, given in Figure 2. In this excerpt, instances of SC 016 are heard linearly starting on the downbeat, while instances of SC 027 are heard linearly starting on the second sixteenth of each measure. The latter grouping will be used in this analysis since it more clearly isolates the two alternating voices by emphasizing the downbeat rest, and reinforces T_{5} from one pitch class to the next through repetition. In addition, T_{5} occurs between pitch classes in both trichord types and is influential in structuring the work.
Figure 1. Pépin, Toccate no. 3,
measures 8–12 
Figure 2. Pépin, Toccate no. 3,
measures 18–23 
[2.2] Identifying these particular objects within measures 18–23 helps us to focus our attention on just the few transpositions that relate them, illustrated on Figure 2. T_{11} generates the second trichord of each measure from the first, T_{10} generates the upper voice from the lower, and T_{5} generates the second twomeasure segment from the first. Some of these transformations also act on a larger scale in this passage: for example, T_{5} is heard from one pitch class to the next, but is also heard between twomeasure segments generating a longer instance of SC 027 over measures 18–23 (indicated with dotted lines below the score on the figure). In fact, two of the characteristic transformations mentioned above, T_{5} and T_{10}, are the same transformations we hear between pitch classes of the 027 trichord, shown in Figure 3. 
Figure 3. Transformations characteristic
of SC 027 trichords 
[2.3] These are not the only transformations conceivable between the given trichords. Because the manifestations of SC 027 in
measures 18–23 all occur symmetrically in pitchspace, one might instead use inversions to demonstrate these relationships, along the lines of Figure 4a. If pitchclass relationships are expressed as standard inversions, the higherlevel relationships among trichords must be expressed as hypertransposition, as depicted in Figure 4a. If pitchclass relationships are instead illustrated with contextual inversions, as shown in Figure 4c (which expresses the idea that ordered instances of SC 027 are generated by the inversion of the first pitch class about the second to produce the third; Figure 4b is included to show the derivation of this contextual inversion), the relationships among the trichords cannot be expressed via the same contextual inversion since this transformation acts only upon individual pitch classes, not SC 027 
Figure 4. Other possible
analyses of pitch and pitchclass relations
among instances of SC 027 
[2.4] Figure 5 presents one view of a cubic lattice incorporating the main objects and transformations of measures 18–23. The vertices of the cube represent the objects of the space, the ordered 027 trichords. These are arranged such that the cube’s three dimensions each involve one of the three main transformations of the passage, T_{5}, T_{11}, and T_{10}. Thus each face of the cube represents four trichordal objects and two characteristic transformations. 
Figure 5. A cubic lattice for Pépin’s Toccate no. 3 
Table 1. Cubic lattice relationships 
[2.6] Animation 1 presents this cube in an interactive application that allows the user to examine its every facet and to relate it to the music. It shows three faces of the cube. Clicking on any one of these will play the succession of trichords in the passage associated with it, and traces a zigzag path on the cube corresponding to the order of trichords. The boxes on the score, indicating these trichords, change in accordance with the clicks. The cube can be rotated in any way by clicking on the arrows at the lower left of the frame.^{(3)} 
Animation 1.
(click to view the animation) 
[2.7] A trichord path is given abstractly in Figure 6a: the first trichord generates the second via an axial transposition, the second generates the third by a different (nonaxial) transposition, and the third generates the fourth via the original transposition. Figure 6b presents a specific example, the path between the consecutive trichords of Face 1. On this face, the trichords are transformed by <T_{11}, T_{6}, T_{11}>. The similar motionpath on each face of the cube reflects the similar structures present in different levels of sequence within the passage (from trichord to trichord, from measure to measure, and from twomeasuregroup to twomeasuregroup).
Figure 6. Motion paths and transformations between trichords, where T_{x} and T_{y} are axial transpositions 
Figure 7. Still shots from the animations on three faces of the cube 
[2.8] Figure 7 and its associated realizations in Animation 1 provide further instances of this shared structure. Part A of the figure shows the motion path between ordered trichords on Face 6, representing the trichordtotrichord transformations within the passage. Part B shows the motion path on Face 2, representing the measuretomeasure transformations. Part C of the figure shows the motion path on Face 1, representing the twomeasure sequence. The remaining three faces of the cube depict similar trichord groupings, and can be explored in the animation; by selecting the “play all” feature, the motion paths on the three visible adjacent faces are displayed simultaneously.
Extended Applications, 1: Hyperrelations
[3.1] As demonstrated in Figure 8, each face of the cube can be interpreted as a transformational network, incorporating the ordered trichords of the passage as objects, and the characteristic transpositions and their compositions as the transformations. For the sake of visual clarity I have only indicated transpositions in a single direction on the figure; however, the inverse operations may be substituted by changing the direction of the arrow. Thus the six networks can be interpreted as sharing a single nodearrow system, suggesting that the trichords progress similarly within each grouping (and thus may follow similar motion paths within the geometric space).
[3.3] Musically, these Mrelations support several observations made in conjunction with previous figures. The principal transformations that structure the cubic lattice are T_{5}, T_{10}, T_{11} and their inverses (T_{7}, T_{2}, and T_{1}); the diagonal transformations are produced by the composition of these axial transpositions. <M_{5}> and <M_{7}> can be applied to each of the axial transformations to produce one of the same six transpositions, resulting in an exchange of the T_{5/7} and T_{11/1} axes for one another while keeping the T_{10/2} axis constant. This exchange of axes at a hyperlevel implies that a T_{5} sequence is (abstractly) analogous to a T_{11} sequence via <M_{7}>, an observation already alluded to by the zigzag motion path on each face. Thus the sequences are structured in the same manner, even though they occur at different structural levels.
Extended Applications, 2: Measures 18–23 as a model for other passages
[4.1] Although the transformational space derived from the events of measures 18–23 is interesting from a theoretical perspective, one might wonder about the value of an analysis focusing on such a short passage. In fact, the transformations introduced in measures 18–23 recur throughout the work, and thus the cubic lattice can be understood as a model for several passages. The following discussion will analyze excerpts clearly associated with the musical materials of measures 18–23 through rhythm, intervals, contour, texture, and other features.
[4.2] Measures 111–119 (given in Figure 11) present music that is very similar to that of measures 18–23; both employ the same rhythm (a series of sixteenth notes starting one sixteenth after the downbeat), same dynamic level (pp), same expression (leggiero), and both feature an alternation between the right and left hands from one measure to the next. In addition to these similarities, the upper voices of measures 18 and 111 are identical, further linking the two passages. In measures 111–118, the arpeggiated trichords from measures 18–23 are heard in the upper voice, but with several distortions: while the trichords of measures 18–23 were all members of SC 027, several trichords in measures 111–118 are not. Figure 12 gives the hypothetical upper voice resulting from the pattern established in measures 18–23 (in other words, incorporating only SC 027 trichords). By comparing this with the actual score of measures 111–119, one can see that the chain of 027 trichords is only disrupted in a few locations, indicated with open note heads on Figure 12. Figure 13 presents a network that illustrates the transformations between the “expected” trichords formed from the pitch classes of Figure 12. As in measures 18–23, SC 027 trichords are transformed by T_{11}; however, this new version reverses the melodic motion halfway through at measure 115. Thus in the second half of measures 111–118, T_{1} rather than T_{11} generates consecutive trichords. Because of this reversal, measures 115–118 can be understood as a distorted retrograde of measures 111–114. The notes indicated with open noteheads on Figure 12 are shown in bold within the network; they occur as the upper pitch class of the middle four trichords, at corresponding locations within the first and second fourmeasure phrases. [4.3] A second network, given in Figure 14a, interprets measures 111–118 in a slightly different manner. Whereas the previous network assumed a grouping structure based on that of measures 18–23, with instances of SC 027 occurring at the same metric location, the new network instead suggests that the “distorted” pitch classes establish a new grouping structure in measures 115–116 beginning on the third sixteenthbeat of the measure. Trichords affected by this change are highlighted in red on the figure. While this interpretation has a significant weakness (the third and fourth trichords are not analyzed in the same manner as the fifth and sixth, even though they have been asserted as analogous trichords within the retrograde progression), it more clearly establishes the retrograde pattern by immediately repeating pitchclass trichords {927} and {816}. It also establishes a largerscale repeated transformation, T_{3}. To more clearly illustrate the repeated transformations within this passage, Figure 14b and c interpret the trichords of measures 111–119 with a product network and a networkofnetworks, respectively. The former emphasizes the repetition of transformations within the passage, whereas the latter emphasizes the trichordal grouping structure (that is, each iteration of the sixnote motive consists of two trichords related by T_{11}). More importantly, the combination of transformations, and the motionpath they form within a threedimensional space whose characteristic transformations are those of the cubic lattice, will become characteristic of analogous passages later in the work, to be discussed shortly. 
Figure 11. P้pin, Toccate no. 3, measures 111119
Figure 12. Expected upper voice in P้pin,
Figure 13. A network depicting
transformations between
Figure 14. A second interpretation of the
uppervoice 
[4.4] Since the trichords of measures 111–119 do not all appear within the original cubic lattice, a complete threedimensional representation requires a larger structure. Figure 15 therefore gives a segment of a threedimensional lattice based on the previouslydefined cube (shaded on the figure for reference), where the vertices represent nodes containing instances of SC 027, and the three dimensions represent the transformations T_{5}, T_{10}, and T_{11}.
[4.5] The motion path among trichords in measures 111–118 can also be mapped within a single tier of the expanded threedimensional lattice given in the previous figure; this is demonstrated in Figure 16. Part (a) of the figure presents a threedimensional view of the entire tier, and part (b) shows a twodimensional view of the bottom horizontal plane. As in Figure 15, the shaded portion represents the original cubic lattice. The transformations T_{11} and T_{3}, the main transpositions of the product network, are represented via horizontal and diagonal motion, respectively. Since T_{3} results from the combination of the axial transformations T_{11} and T_{10} (in fact, their inverse operations: T_{1}T_{2} = T_{3}), every trichord move within the passage can be represented along a single plane, the bottom face of the figure. The motion path repeats the same combination of transformations twice, representing the repetitive nature of the passage, and its combination of axial and nonaxial transformations is reminiscent of the zigzag pattern examined in connection with the sequences of measures 18–23.
Figure 15. A segment of a 3D lattice based on the established cubic lattice 
Figure 16. One plane of the
expanded threedimensional lattice 
Conclusion
[5.1] The preceding examples have demonstrated how the cubic lattice and its associated twodimensional spaces can provide a framework for interpreting transformational relationships in various passages. Indeed, its broad applicability results from Pépin’s replication of the characteristic transformations and objects of measures 18–23 throughout the work. While the three passages analyzed above, measures 111–119, 194–198, and 300–302, clearly use motives and rhythms similar to those of measures 18–23, their related transformational structures are not immediately apparent. The space of Animation 1 helps us to perceive commonalities among these objects and transformations via repeated visual motifs such as the shared zigzag motion path. Furthermore, the repetition within the motion paths of measures 111–119, 194–198, and 300–302 is reminiscent of the sequences outlined in measures 18–23, providing another link to the original passage (these passages do not involve multiple overlapping sequences as heard in measures 18–23, but instead develop the motive at a single level).
[5.2] The hypertransformations discussed in conjunction with measures 18–23 transformed the graphs of different cube faces into one another. These relationships support the idea that the various sequences of the passage, occurring at different structural and transpositional levels, are analogous. Conceptually, <M_{5}> and <M_{7}> swap T_{5} and T_{11} axes/operations for one another; these two transformations correspond to the longerscale and local processes, respectively, in Pépin’s original sequence.
[5.2] Other spaces are possible for the characteristic transformations presented here. The analysis of Berg’s op. 2, no. 2 presented in this issue of Music Theory Online presents a toroidal space incorporating the same transformations. That choice of space emphasizes the congruence of melodic, metric, and textual elements within the Berg Lied, whereas the cube space presented in this paper emphasizes the similar structure of sequences involving different transposition levels. The animation can be explored further to hear different subgroups, sequences, and combinations of groupings. While this type of analysis may not be practical for every example, it does provide interesting insights into structural similarities, recurring motivic relationships, and the congruence of a group of twodimensional spaces.
Stephanie Lind
Queen’s University School of Music
HarrisonLeCaine Hall
39 Bader Lane
Kingston, ON K7L 3N6
Canada
linds@queensu.ca
Acknowledgments
The author wishes to thank the Canadian Music Centre/Centre de musique canadienne (and in particular Mireille Gagné, director of the Quebec branch) for permission to reproduce excerpts from Clermont Pépin’s Toccate no. 3 in this publication.
Works Cited
Brown, Stephen C. 2003. “Dual Interval Space in TwentiethCentury Music.” Music Theory Spectrum 25/1: 35–57.
Gollin, Edward. 1998. “Some Aspects of ThreeDimensional ‘Tonnetze’.” Journal of Music Theory 42/2: 195–206.
Klumpenhouwer, Henry. 1998a. “The Inner and Outer Automorphisms of PitchClass Inversion and Transposition: Some Implications for Analysis with Klumpenhouwer Networks.” Intégral 12: 81–93.
———. 1998b. “Network Analysis and Webern’s Opus 27/III.” Tijdschrift voor Muziektheorie 3/1: 24–37.
Kochavi, Jonathan. 1998. “Some Structural Features of ContextuallyDefined Inversion Operators.” Journal of Music Theory 42/2: 307–320
Lambert, Philip. 2000. “On Contextual Transformations.” Perspectives of New Music 38/1: 45–76.
Lewin, David. 1982–83. “Transformational Techniques in Atonal and Other Music Theories.” Perspectives of New Music XXI/1–2: 312–71.
———. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University.
———. 1990. “Klumpenhouwer Networks and Some Isographies That Involve Them.” Music Theory Spectrum 12: 83–120.
———. 1993. Musical Form and Transformation. New Haven: Yale University Press.
———. 2008. “Transformational Considerations in Schoenberg’s Opus 23, Number 3.” In Music Theory and Mathematics: Chords, Collections, and Transformations, ed. Jack Douthett, Martha M. Hyde, and Charles J. Smith, 197–221. Rochester: University of Rochester Press.
Morris, Robert. 1987. Composition with Pitch Classes: A Theory of Compositional Design. New Haven: Yale University Press.
Peck, Robert. 2004–05. “Aspects of Recursion in MInclusive Networks.” Intégral 18–19: 25–70.
Footnotes
1. These contextual inversions are modeled on those of
Lewin, as discussed in Lewin 2008. Earlier references to contextual inversions can be found in
Lewin 1993,
Kochavi 1998, and Lambert 2000.
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2. Technically, the arrows and transformations among different object types are not “the same” since one type of arrow may represent a change of pitch class, and the other type represent may represent a change from one trichord to another, one network to another, and so forth. However, they function analogously.
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3. This interactive animation was designed by Stephanie Lind and animated by Reuben Heredia.
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4. Isography via hyperM_{5} and M_{7} is
discussed in Lewin 1990, 88; note that the automorphisms of hyperM_{5} and M_{7} are notated in the form F<5, j> and F<7, j>, respectively, in Lewin’s article.
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5. Ibid.
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6. The previous two pitch classes can be understood as part of an incomplete 027 trichord.
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