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Volume 31 , Issue 2. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Music Analysis Volume 31, Issue 2. Matthew Riley Search for more papers by this author. Read the full text. Tools Request permission Export citation Add to favorites Track citation.

Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Citing Literature. Volume 31 , Issue 2 July Pages Related Information. Close Figure Viewer. For present purposes, a set may simply be understood as a collection of distinct elements, fnite or infnite.

Te elements are distinct in that none of them occurs more than once in the set. Te space S may consist of pitches, or pitch classes, or harmonies of a particular kind, or time points, or contrapuntal confgurations, or timbral spectraand so on. GISes thus extend the idea of interval to a whole host of musical phenomena, not just pitches; this is one of the senses in which they are generalized.

Note that the elements s and t are given in parentheses in the formula, sepa- rated by a comma. Tis indicates that they form an ordered pair: s, t means s then t. Te ordered pair s, t is distinct from t, s. GISes thus measure directed intervalsthe interval from s to t, not simply the undirected interval between s and t. Tis difers from some everyday uses of the word interval, in which we might say, for example, Te interval between C4 and D4 is a diatonic step.

GISes do not model such statements, but instead statements of the form Te interval from C4 to D4 is one diatonic step up i. Te element i to the right of the equals sign is a member of a group. A group is a set that is, a collection of distinct elements, fnite or infnite plus an additional structuring feature: an inner law or rule of composition that states how any two elements in the set can be combined to yield another element in the set. Lewin calls this inner rule a binary composition, and we will follow that usage here.

For example, take the set of all integers, positive, negative, and zero. But once we introduce the concept of addition as our binary composition, the set of integers coheres into a group, which we call the integers under addition. Tis is called the group property of closure: the composition of any two elements under the binary compo- sition always yields another element in the same set.

Groups have three other properties. First, they contain an identity element, labeled e for the German word Einheit. Te composition of e with any other group element g yields g itself. In our group of integers under addition, the iden- tity element is 0: 0 added to any integer x yields x itself. Further, for every element 7. A set in which elements appear more than once is called a multiset. Multisets have music-theoretical applications, but we will not explore them in this study.

Most mathematicians call Lewins binary composition a group operation or binary operation.

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Lewin, however, somewhat idiosyncratically reserves the word operation for a diferent formal con- cept, as we will see, thus making binary composition preferable in this context. Te element g 1 is called the inverse of g g 1 is read g -inverse.

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In the group of integers under addition, the inverse of any integer x is x e. Finally, composition within any group is associative. Mathematicians study groups primarily for their abstract structure, a structure that is suggested in its most basic terms by the four conditions outlined above clo- sure, existence of an identity, existence of inverses, associativity.

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Te GIS formula- tion rests on the idea that intervals, at a very general level, have this same abstract structurethey are group-like. Tat is, the combination of any two intervals will yield another interval closure. Any musical element lies the identity interval from itself existence of identities.

Given an interval i from s to t, there exists an interval from t to s that is the reverse of ithat is, i 1 existence of inverses. For instance, there is nothing in the four group conditions that says anything about direction or distance two attributes ofen attributed to intervals. Tis is one area in which the GIS concept has recently been criticized. Tat is, the group itself, qua abstract algebraic structure, knows nothing of one step up or fve steps down.

Instead, it knows only about the ways in which its elements combine with one another according to the properties of closure, existence of an identity and inverses, and associativity. We can conclude two things from this: 1 GISes are formally quite abstract, and may not capture everything we might mean by interval in a given context; and 2 not all of the intervals modeled by GISes need to be bound up with the metaphor of distance. Te symbol here is a generic symbol for the binary composition in any group. When the idea of group composition is understood, such symbols are sometimes eliminated.

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  5. See GMIT, Tymoczko and As Rachel Hall puts it, GISes can express notions about distance, but are not forced to do so , Hall , We are nevertheless free to add notions of distance and direction to our inter- pretations of GIS statements, if so desired. Dmitri Tymoczko , Lewins main critic on this front, has indicated how distance may be reintroduced into a GIS by adding a metric that formally ranks the distances between all pairs of elements in the space of the GIS.

    We will not employ Tymoczkos distance metric explicitly in our formal work in this study, but we will ofen rely on the idea implicitly, whenever we wish to interpret intervals as representing various distances. Tis is because each group has an underlying abstract structureor, we might say fguratively, a certain shape.

    Tis shape is determined by the number of elements in the group and the various ways they combine with one another and with themselves. A group, for example, may be fnite or infnite. It may contain certain patterns of smaller groups called subgroups that articulate its structure in various ways. One way to think of this is that a given intervallic statement in Lewins model inhabits a certain conceptual topographya sort of landscape of intervallic relationships given shape by the structure of the group IVLS.

    Diferent types of interval may thus inhabit considerably diferent conceptual topographies, based on the structure of their respective groups e. Tis suggests that the intervallic experiences corresponding to such intervals have Tat GIS, which calculates intervals between qualitative tonal scale degrees, involves a group of intervals that might better be understood as comprising familiar intervallic qualities rather than distances, such as the quality of a minor third, as opposed to that of an augmented second. Te distance metaphor is especially inapt in connection with certain exotic interval types that we will explore in sections 2.

    Edward Gollin explores another model for distances in a GIS, measuring word lengths in the elements of the intervallic group. In the group of neo-Riemannian operations, for example, the word PLP, of length 3, is longer than the word RL, of length 2. A given group admits of multiple distance-based interpretations, based on which group elements are chosen as unitary words of length 1 via the formalism of group presentation, as Gollin demonstrates.

    Two familiar noncommutative groups in music theory are the group of transpositions and inver- sions from atonal theory, and the group of neo-Riemannian operations. In the former group, it is not generally true that T m followed by I n is the same as I n followed by T m. Such diferences are ofen interpretively productivethe formalism encourages us to attend to them carefully, as we pursue and extend any given intervallic statement within a particular analytical context. Tus far in our survey of GIS structure, we have two separate collections that are as yet entirely independent: the space S of musical elements and the group IVLS of intervals.

    We have not yet shown how various intervals in IVLS can be understood to span pairs of elements in S. Te lefmost element in the GIS for- mula, int, provides that connection. As indicated in Figure 1. A function from a set X to a set Y sends each element x in X to some element y in Y. Te set X of all arguments is called the domain of the function, while the set of all images in Y is called the range.

    Te domain for our function int in a GIS is not simply the space of musical elements S itself, but the set of all ordered pairs of elements from S. Our arguments are thus not single elements from S, but ordered pairs of the form s, t. Te function int sends each ordered pair to an element in IVLS.

    Te group IVLS is the inte- gers under addition, our familiar group discussed above. Te mapping int sends every ordered pair of diatonic pitches to some element in the group of integers. Tat is, the interval from Te locutions one diatonic step up and one diatonic step down evoke ideas of distance and direction, suggesting the pertinence of Tymoczkos distance metric to this particular GIS.

    Within the set of all diatonic white-note pitches, there is of course only one pitch that lies one diatonic step up from C4, that is, D4. Tese two conditions lend a certain logical tightness to GIS structure, providing only one interval between any two musical elements within a GIS.

    For example, in the GIS corresponding to pitch classes in tone equal temperament, there are 12 elements in S the 12 pitch classes and 12 intervals in IVLS the integers mod We now turn to some philosophical and methodological matters raised by GISes. Note that the main action modeled in a GIS is the action car- ried out by the mapping int. Te active nature of int is especially evident if we use an arrow notation to rewrite the function.

    Tis notation makes visually vivid the fact that int is the primary action involved in a GIS statement, capturing the act of pairing two musical elements with an interval. Te relevant thought process might be verbal- ized thus: I just heard a C4 and now I hear a D4; the interval from the former to the latter is one diatonic step up. Lewin characterizes this attitude as Cartesian because it is the attitude of someone passively calculating relationships between entities as points in some external space. Te action of passively measuring is embodied by the mapping int itself.

    One might think of int as analogous to pulling out some calculating device and applying it to two musical entities out there to discern their intervallic rela- tionship. Te action in question is not one of imaginatively traversing the space from C4 to D4 in time, construing and experiencing a musical relationship along To be clear, there is only one interval between two musical entities within a single GIS.

    As dis- cussed in section 1. Tat multiplicity arises not within a single GIS, but via the multiple potential GIS structures that may embed the two elements in question. Te GIS formula is further like the Cartesian mindset in that it exhibits a certain fracturing of experience, a conceptual split between musical elements the space S , musical intervals the group IVLS , and the conceptual action int that relates the two. Te cumbersome nature of the GIS formalismwith its three com- ponents S, IVLS, int , all of which need to be coordinated, and with the action int placing the musical perceiver in an explicit subject-object relationship vis--vis the music being perceivedthus encodes aspects of the Cartesian split between res cogitans and res extensa , a familiar trope in Lewins writings.

    In short, despite the fact that the GIS formalism enacts the Cartesian problematic that Lewin so eloquently criticized, it is still a produc- tive and suggestive technology in many theoretical and analytical contexts. It will be valuable for us to spend a little time here thinking about the matter, as the questions that it raises bear directly on the relationship between transformational technology and musical experience.

    Tough Lewin gives us no clear defnition of what he means by intuitions, we can infer two crucial characteristics of the term as he uses it in his writings: 1 His intuitions are culturally conditioned. Lewin states 1 explicitly: Personally, I am convinced that our intuitions are highly conditioned by cultural factors GMIT, By cultural factors, Lewin Te relevant philosophical matters are penetratingly treated in Klumpenhouwer On the prob- lematics of the subject-object relationship in passive musical perception , see Lewins well-known phenomenology essay Lewin , Chapter 4.

    My ideas on these matters were clarifed through conversation with Henry Klumpenhouwer. My view difers slightly from Klumpenhouwers published comments, in which he states that Lewin wants us to replace intervallic thinking with transformational thinking , I feel that Lewins ethical directive in GMIT is not quite this strongthat he wants us not to replace inter- vallic thinking but to become more aware of its Cartesian bias, and to be self-conscious about that bias whenever thinking intervallically in some analytical context.

    For example, a sixteenth-century musician conditioned by ideas about modes, hexachordal mutation, mi-contra-fa prohi- bitions, and so forth would have diferent intuitions about a given musical pas- sagesay in a motet by Palestrinathan would a modern musician conditioned by ideas about keys, diatonic scales, tonal modulation, and so forth. Te general pertinence of characteristic 2 to Lewins thought is manifest throughout his writings, as theoretical structures of various kinds are brought to bear on var- ious musical experiences, sharpening, extending, or altering those experiences in diverse ways.

    Indeed, Lewins entire analytical project can be understood as a process of digging into musical experience and building it up through analyt- ical refection. Stanley Cavell, paraphrasing Emerson, provides a very suggestive wording that we can borrow for the idea: such work involves a reciprocal play of intuition and tuition, or, even more suggestively, it is a project of providing the tuition for intuition.

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    Te word intuition, by contrast, runs the risk of naturalizing GIS and transformational statements, treating them as unmediated reports on prerefective or at least minimally refective experience. Tis risk is especially Cavell , a play of intuition and tuition and Cavell , section 4 providing the tuition for intuition.

    For Emersons original quote see Emerson , As Henry Klumpenhouwer notes, In distinction to other uses of the term, Lewins intuitions have some conceptual content , n3. In this book I will understand apperceptions loosely in William Jamess sense, as experiences col- ored by the previous contents of the mind , Such apperceptions may involve conscious refection, or they may not. For example, ones past experiences with a certain musical idiom will strongly color ones current and future musical experiences with music in that idiom, whether one has consciously refected on the idiom or not.

    Apperceptions, thus conceived, are simply current experiences under the infuence of past experience, and open to present refection. Tis departs from certain philosophical understandings, in which conscious refection is a necessary compo- nent of all apperceptions. By hewing to the word apperception, I instead hope to make clear that the sorts of experiences explored in this book are by no means universal, and will be strongly shaped not only by ones cultural background and historical context, but also by the concrete particulars of present analytical engagement.

    Yet, despite this abstraction, GISes are not as general as they might at frst appear, nor are they applicable to all musical situations. GISes, for example, cannot model intervals in musical spaces that have a boundary or limit. Consider an example that Lewin himself raises: S is the space of all musical durations measured by some uniform unit. Tis space has a natural limit: the shortest duration lasts no time at allthere is no duration shorter than it. Now imagine that we choose to measure the interval from duration s to dura- tion t in this space by subtracting s from t IVLS would then be the integers under addition.

    Such a t would be 2 units shorter than no time at all. Tus, the given musical space of durations under addition, though it is musically straightforward, can not be mod- eled by a GIS. Similar problems arise with any musical space that has a boundary beyond which no interval can be measured. Tis relates to a more general limitation. Given Lewins defnition, any interval in a GIS must be applicable at all points in the space: if one can proceed the interval i from s, one must also be able to proceed the interval i from t, no matter what i, s, and t one selects.

    Tis limits GISes to only those spaces whose elements all have uniform intervallic environments. Te vast majority of familiar musical spaces do have this property. Tis is so even when the result would be too high or too low to hearthe space is still in principle unbounded. Neo-Riemannian spaces are also On theoretically unbounded spaces that can be perceived only in part, see GMIT, Nevertheless, there do exist spaces that do not have this uni- form quality, such as the durational space outlined above, or any number of voice- leading spaces that are better modeled geometrically as discussed in Tymoczko Tus, despite their generalized qualities, GISes are not as broad in scope as they might initially appear to be: they only apply to uniform intervallic spaces.

    We will return to this important criticism in section 2. Such an interval could obtain between any other pair of major-third-related elements in the space, say F4D. Tymoczkos formulation provides a difer- ent and useful perspective, focusing more attention on the concrete endpoints of a specifc interval s and t , and less on the ways in which a listener or analyst might construe the relationship between those endpoints as some general interval-type i. But an attractive aspect of GIS theory is lost in the process, to which we now turn. Tere is thus no single interval from G4 to E.

    In the Beethoven, the pitch topography is diatonic, and the GIS might be any one of a number of diatonic GISes pitch-based or pitch- class-based. What I have been calling uniform , Tymoczko calls homogenous and parallelized. Te latter term means that one can move a given interval from point to point, applying it anywhere in the space. Te GIS introduced in Chapter 2 would be especially well suited to modeling the interval in question. It would further distinguish the G4E.

    Tis is a rather obvious instance of what we might call apperceptive multiplicity in intervallic experience. Less obvious, perhaps, is GIS theorys insistence on apper- ceptive multiplicity when confronting a single interval in a single musical passage. Tis suggests that the interval in question can inhabit multiple musical spaces at once. Lewin puts it somewhat more strongly than I would: we do not really have one intu- ition of something called musical space.

    Instead, we intuit several or many musical spaces at once GMIT, Per the discussion in section 1. Tose diferent conceptions can subtly change our experience of the interval, leading to new musical apperceptions. An arrow labeled i extends from the cellos opening G2 to the B3 at the apex of its initial arpeggio. Figures 1. Figure 1. Tis suggests the context shown on the staf: B3 is nine steps up the G-major diatonic scale from G2. Te fgure shows this by placing in parentheses the elements that i skips over in the space S of the relevant GIS.

    While my rewording focuses on listening experiences stimulated by analytical refection, it is not clear from his comment what sorts of listening contexts Lewin has in mind. He may indeed have meant that multiplicity is a fact of everyday musical experience: when we hear music in any con- text, we intuit multiple musical spaces at once and thus hear intervallic relationships in manifold wayseven when we are not in an analytically refective mode. Tis may be true, but I am not sure how one could test the idea, nor do I know what exactly is meant by intuit and intuition in Lewins passage.

    Is the listener consciously aware of these manifold intuitions? Or are they perhaps instead a congeries of more or less inchoate sensations that one has when listening, which can be brought into focus through analytical refection? I am more comfortable with the latter position, which moves toward my rewording. Lewin GMIT, discusses the discrepancy between the familiar ordinal intervallic names of tonal theory tenths, ffhs, thirds, etc. Te GIS introduced in Chapter 2 employs the familiar ordinal names for intervals between scale degrees.

    If we say that i is a tenth , we are likely not thinking primarily about a number of steps up a scale, but about a privileged harmonic interval. G2 is adjacent to B2 in this space, as is B2 to D3, and so on. Adjacent pitches here correspond to the steps in Fred Lerdahls triadic space , or to steps in the chordal scale of William Rothsteins imaginary continuo , Note, however, that the integers now represent acoustically larger inter- vals than did the same group elements in the GIS of Figure 1.

    For example, in the GIS of 1.

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    See also Tymoczko and Te intervals available to the style bris lutenist within any given harmony are exactly those of the present GIS. Tis model is suggestive, given the spacing of Bachs opening arpeggio: the G2D3B3 succession corresponds to partials 2, 3, and 5 in the over- tone series of G1. In a more historical-theoretic vein, we might say that the notes project elements 2, 3, and 5 of a Zarlinian senario. We can sense that diference in topography when we recognize that, in the arpeggio GIS of 1.

    In the just ratio GIS of 1. Te diference registers the acoustic distinction between a just perfect ffh and a just major sixth. Te resonant, partial-rich open strings of G2 and D3 with which the arpeggio begins strengthen the relevance of the just-ratio GIS here. While both j and k were skips in 1. Tis calls attention to the missing G3, partial 4 in the overtone series note the question-marked dotted arrows on the examples right side. As Figure 1. Te G3 bears a considerable tonal accent as a pitch that completes several processes set in motion in the works opening measures.

    Note that the interval from D3 to G3labeled l in the exampleis flled in by step. Tis stepwise motion is the Te cellist can emphasize the partial series by placing a slight agogic accent on the opening G2, a gesture that makes good musical sense anyway, given the works upcoming stream of constant sixteenth notes.

    Te partials activated by the G2 are an octave higher than those in Figure 1. I have worded this carefully: interval k in 1. In the GIS underlying 1. Te group IVLS in this GIS consists of all of the positive rational numbersnot just the low-integer ratios explored in the fgure , , and so on , but also higher integer ratios like a major semitone in Pythagorean theory , and even enormous integer ratios such as ,, the acoustically tiny Pythagorean comma.

    For example, it includes the pitch residing a Pythagorean comma above G2. Tis makes clear that the GIS structuring Figure 1. In fact, by the end of m. One could invoke other GIS contexts for i as well. One could model i as span- ning the interval from 1. Many other intervallic contexts for i are possible as well, but not all of them are relevant to the opening bar of Bachs prelude.

    For example, one could conceive i to extend up 16 semitones in a chromatic pitch gamut. Tis is a somewhat strained understanding within the context of m. Such a space is invoked, however, at the works climax in mm. Here it is very easy to hear the interval spanned from D3 in m. Such an analysis could continue, modeling other notable intervallic phe- nomena in the prelude and exploring their interactions.

    For present purposes, it is important merely to note the style of the analysis, particularly its focus on multiple intervallic interpretations of single musical gesture. As Lewin puts it in one of his most frequently quoted passages, instead of regarding the i-arrow on fgure 0. GMIT, xxxi Lewin elsewhere dubs this the transformational attitude, and it has become a familiar part of the interpretive tradition of transformational theory. It is a subtle and somewhat elusive concept; I will ofer my own gloss on the idea and its relevance to certain acts of tonal hearing in section 3.

    For now, the reader may simply con- ceive of transformational arrows as goads to a frst-person experience of various gestural actions in a musical passage, actions that move musical entities or confg- urations along, or that transform them into other, related entities or confgurations. A transformational network is a confguration of nodes and arrows whose nodes contain elements from some set S of musical elements analogous to the set S of elements in a GIS and whose arrows are labeled with various transfor- mations on S.

    A transformational graph resembles a transformational network in all respects but one: its nodes are empty. We have already encoun- tered functions in the GIS discussion above, with the function int. Te transfor- mations and operations in a transformational graph or network are also functions, but rather than mapping pairs of elements to intervals as int does in a GIS , they act directly on single musical entities, transforming them into each other.

    Before exploring how this works in practice, it will be valuable to distinguish between a transformation and an operation. We now defne a transformation on s that we will call resolve to C, abbreviated ResC. Tis transformation sends every element in S to the element C. ResC is indeed a function from S to S itself: it takes as input each element of S, and returns as output an element of S.

    We can represent it by a mapping table, like that shown in Figure 1. Both of these transformations can be conceived as idealized musical actions. But it is only when we consider the entire mapping table that we get a full sense of just what these actions are.

    To see this, consider the fact that both transforma- tions have the same efect on the note B: they both map it to C. At this local level, the transformations appear to be indistinguishable. But if we perform the same actions elsewhere in the space, their diferences emerge. It is only in this broader context that we can see that Step raises pitches by one step, while ResC resolves notes to C. Tese two actions have the same efect when applied to B, but the specifc kinetics they imply are diferentResC suggests a gravitational centering on C, or an action that yields to such gravitation, while Step suggests a more neutral, uniform motion of single-step ascent anywhere in the space.

    Step also difers from ResC in a more formal way. Every element from S appears on the right-hand side of the table for Step Fig. While both ResC and Step are transformations, Step is a special kind of transformation that we will call afer Lewin an operation: an operation is a transformation that is one-to-one and onto. Operations thus have inverses: one can undo any operation simply by reversing the arrows in its mapping table. Tus, we can defne Step 1 , as shown in Figure 1.

    We cannot, how- ever, defne an inverse function ResC 1. As shown in Figure 1. Functions must be defned on all elements of their domain, but ResC 1 is not defned on all of the notes in S; for example, it is not defned on D, as D does not appear anywhere at the beginning of an arrow on the table for ResC 1. Furthermore, ResC 1 is not even well defned on C, as it seems to send that note to seven diferent places; a function must send each element it acts on to only one element.

    Tus, ResC has no inverseit is a transformation, but not an operation. Tis process is illustrated in Figures 1. Functions that are one-to-one and onto are also called bijections. For further discussion, see the entry for Function in the Glossary. Most of the familiar transformations in transformational theory are operations that is, they are one-to-one and onto, and thus have inverses.

    Te neo-Riemannian transformations, for example, are all operations, as are the familiar T n and I n operations on pitch classes from atonal theory. Te new operation is defned by combining the mapping tables in 1. Trough processes like this, sets of transformations and operations can be combined to yield groups, or group-like entities called semigroups, in which composition of mappings serves as the inner law, or binary composition.

    Since operations have inverses, they can combine into groups. We remember that each element in a group must have an inverse. We speak in this case about a group of operations. Transformations that are not operations, however, cannot combine into groups, as they do not have inverses. Tey can instead combine into a more general structure called a semigroup. A semigroup is a set of elements with a binary composition like a group, but one that needs only to satisfy two of the four group properties: closure and associativity. A semigroup need not contain an identity element, nor does each element in a semigroup need to have an inverse.

    We thus speak of a semigroup of transformations. Te structure underlying transformational systems is thus what mathemati- cians would refer to as a semigroup action on a set or a group action on a set if a group of operations is involved. Te set in question is a set S of musical elements notes, harmonies, rhythmic confgurations, etc.

    Te semigroup of transformations then acts on this set, modeling certain musical behaviors that are performed directly on entities in S, transforming them one into another along the arrows of the net- work. Te action modeled by our formalism has thus changed in a subtle way from the GIS perspective. Tere, the action was one of calculating, modeled by the function int, which matched an ordered pair of elements with an intervallic distance.

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    Te action enshrined by a transformational arrow, by contrast, is the active performance of some characteristic musical gesture, which transforms one musical element into another. Tere is no equivalent to int here: semigroup or group elements act directly on the musical entities themselves. Specifcally, any GIS statement can be refashioned into a transformational statement; such transformational statements can also be turned back into GISes. Tis process of translation, taking one from an intervallic perspective to a transformational perspective, is a central theme in GMIT, as Klumpenhouwer persuasively argues.

    A striking aspect of Lewins project is the way in which the conceptual transition from intervallic to transformational thinking is mediated by the technology of his theorythe technological transformation from a GIS perspective to a transformational perspective enacts formally the conceptual transformation that Lewin wishes us to undergo as we switch from an intervallic Cartesian, observational mode of thought to a transformational frst-person, active one.

    In the process, the action of measuring int disappears and is replaced by an imaginative musical gesture, which the analyst is urged to perform in his or her re-creative hearing. Only certain kinds of transformation statements are GIS-able, or conceivable in intervallic terms. Tere are two requirements for such a conceptual shif from transformations back to intervals.

    First, the transformations in question must be operations. To see this, let us return to Figures 1. Note that we can conceive of the operation Step as interval-like. First, there is an interval that we can associate with the dis- tance traversed by Step, namely up one step in diatonic pc space. Second, Step is has an inverse, and is thus reversible, as we expect all intervals to be. We can thus reframe any transformational statement that we make using Step as a GIS statement, and vice versa.

    We cannot, however, develop an intervallic interpre- tation of ResC. First of all, it is very dif cult to see how we could conceive of a single interval that would correspond to the action traced by all of the arrows in Figure 1. In such a case, the interval from some white note to C would be the same as the interval from any other white note to C! Even if we could wrap our heads around such a curious idea, this putative interval would lack an inverse, thus failing the basic requirement that all intervals should be reversible.

    In short, there are certain musical actions we can conceive of performing that cannot be interpreted intervallically; these are the actions modeled by transformations that are not operations. Second, in addition to the requirement that a transformational graph or net- work must include only operations to be interpreted in GIS terms, that group of operations must act on the elements in the space S in a particular way, which is called simply transitive an idea that already arose in our discussion of GISes. A group acts simply transitively on a set if, given any two elements a and b in the set, only one element g in the group takes a to b.

    Simple transitivity will not be a prop- erty of all transformation graphs or networks, even if they include only operations. Consider, for example, a transformation network with node contents drawn from the set of 12 chromatic pcs, and arrow labels bearing a mixture of atonal transpo- sitions T n and inversions I n. Klumpenhouwer networks are familiar instances of this kind of network. Ramon Satyendra clearly explains why such a non-simply-transitive situation conceptually resists translation into intervallic terms: When reckoning intervallic distances we intuitively expect unique answers.

    It is counterintuitive to describe the straight-line distance between the chair and the table as both two feet and three feet. By requiring that a musical system satisfy the simple transitivity condition we are assured that the interval formed between any two points in a musical space may be uniquely determined.

    If a system is not simply transitive, it becomes counterintuitive to shif between transformational and inter- vallic perspectives. For instance it is intuitive to say that both T 3 and I 3 transform C to E. Te translation from such a transformational system into a GIS is formally rather involved, and I will not run through the details here.

    But the basic idea is simple. One merely keeps the space S the same from the transformational network to the GIS, reinterprets the group of operations in the transformational network as the group IVLS in the GIS, and applies int so that pairs of elements and intervals match up in agreement with the original transformational actions. X is the pieces Hauptmotiv a falling fgure frst heard in mm. Altered forms of both X and Y appear in the passage: X changes the intervallic structure of X slightly, and T Y is a transposition of Y.

    At the right-hand side of the example, two cadential gestures are identifed, one in the soprano Cad and one in the bass BassCad. Te phrase concludes in m. We will be interested in the way these gestures are internally structured, as well as in the ways in which they are transformed into one another. Te network of Figure 1. Both X and X begin with B4 -1 A4. Schuberts articulation makes these gestures vivid. Te slurred appoggiatura from B4 to A4 underlies the 1 motion, pulling B4 forward to A4, and the three gently rebounding eighth notes that follow on A4 staccato, and slurred together lead forward to the motives concluding pitches.

    Te calm repetition comes to seem unhealthily obsessive by the time of the climax in m. Note that, despite the evident alteration of Xs internal structure in X, the ges- tural motives of 1 and 3 are retained in the latter. Yet 3 now acts as an internal Satyendra ofers a lucid account of this process of translation, which Lewin defnes formally at the beginning of Chapter 7 in GMIT. We are thus dealing with a group of operationsthe integers under addition. Te operations act simply transitively on the infnite set of diatonic pitches, from which our node contents are drawn.

    Te networks in Figure 1. Peter Smith , 6 makes suggestive observations along these lines, especially involving the way A4s evident structural status is undercut by the articulation, which causes it to lead ahead to F. For its part, 1 remains in its original initiating position, linking the BA appog- giatura that plays such a prominent role in the movement. Indeed, 1 is the most persistent melodic fgure in the piece, initiating nearly every one of its thematic and motivic units.

    Te dashed arrow in 1. Te initiating 1 from X and X remains. In Y it joins B4 and A4, as in the X-forms. In T Y , however, it joins F. Te sense of a change of direction in the Y-forms is refected in other parameters as well, as we will see presently. Te resulting confguration would then need to be transposed by 1 to produce X, once again demonstrating the thematic role of 1 in the music. Like 1, which took X to X, 3 is also present locally in both X and X. Te 3 arrow in X connects the same two elements connected by the dashed 3 arrow in 1.

    Rather than merely afecting one note, it serves to transpose all of Y into T Y. Tus, while 1 acts as an internal transformation in X and as a single-note transformation between X and X, 3 acts as a spanning transformation in X and as an agent of wholesale transposition between Y and T Y. For the frst time, 1 does not initiate the gesture, but terminates it. Te reversal is appro- priate for a cadence.

    Te sense of reversal is heightened by the presence of x the mini form of X at the end of the phrase, turning the movements initiating gesture into an agent of closure. Te diatonic pitch-space opera- tion that takes T Y to Cad is retrograde-inversion-about-F. Cad is the frst gesture to begin with an ascent, as well as the frst to depart from the articulative pattern of X: it is fully legato, covered by a single slur, and linked by ornamental connectives between its nodal points. Of the three transformations in X, only 1 does not appear in positive that is, ascendingform in the melodic gestures of Figures 1.

    It does appear in ascending form in the bass, however, as shown in 1. Until this point, B has always proceeded to A via 1. Te sense of cadential ret- rograde interacts nicely with the comments just made about Cads various reversals. Minus signs in X are replaced by pluses in BassCad, as Xs falling, initiating gesture is transformed into a rising, cadential bass fgure.

    Tese pitch and contour relationships interact compellingly with durational aspects of the music. Some of these interactions are shown in Figure 1. Te contents of the nodes in Figures 1. In the examples in the lef column 1. For example, the proportional transformation 2 in 1. In the exam- ples in the right column 1. In X, for example, the successive durations increase Note that in Figure 1.

    Te reading corresponds Peter Smiths understanding of C. See also the Schenkerian sketch in Figure 1. On the role of reversals as closural, see Narmour For example, the F. I employ it here to show its musical intuitiveness, and to show that it can work as a transformational system, though it is formally awkward: one must posit an element in the space S of durations that corresponds to a duration-less instant. Any duration x is transformed to if it is acted on by a negative duration whose absolute value is greater than x s. An elegant way around this problem is to treat the system in question not as a transformational system at all, but as a system based on a tangent space, as explored in Tymoczko Such spaces admit of bounded dead ends beyond which no transformations or intervals may be conceived.

    Our duration-less instant is one such dead end. Te conceptual and experiential diferences between the two species of rhythmic transformation are also refected in their difering group structures: the integers under addition in the additive rhythmic transformations vs. Tese dual transformation systems reveal interesting correspondences bet- ween pitch and rhythm in the passage. Note frst that for all descending pitch motions, durations increase; for all ascending pitch motions, durations decrease.

    Te connection is suggestive of a metaphorical correspondence between durations and weight, with the longer, heavier durations at the bottoms of the gestures. Durational transformations in a , c , and e are proportional, while those in b , d , and f are additive. X, by contrast, constantly sinks, as note values gradually increase in length and hef. Te vertical arrows show the alteration of each successive element in the three-note gestures. Pitch one is not altered at all; it remains a quarter in both X and Y durational proportion 1. Te rightmost events in each gesture are thus at the durational extremes of the network.

    Te proportional relationship of X to Y is palpable to both performer and listener; it corresponds to the increase in harmonic rhythm in mm. For example, the 3 gesture that links B4 and F. Te arrow labels in 1. Tis observation interacts suggestively with BassCads role as a textural inverse of X bass rather than melody as well as a syntactic inverse a cadential rather than initiating gesture.

    Further, those insights in no way call into doubt For a compelling discussion of the thematic role of gesture in Schubert, see Hatten X and X correspond to the basic idea b. Te increase in harmonic rhythm and surface activity in mm. Tis increase in activity is visibly evident in the piling up of gestures shown on the right-hand side of Figure 1.

    In making this statement, we rely on the fact that the group of trans- formations in question is a direct product group, as discussed in the Glossary. Yet, while tonal aspects of the two works were dis- cussed in informal ways in the analyses through references to things like tonics, dominants, and cadences , the formal apparatus of the analyses did not model those ideas in any direct way.

    Tis was especially evident in the Schubert analysis, which made no attempt to explore the subtle interpenetration of D major and B minor that characterizes the movements harmony. As Peter Smith has noted, the relationship between the pitches B and A is especially striking in this regardtheir status relative to one another, as either stable or decorative pitches, depends heavily on tonal concepts. In the context of the opening bar, it functions as a tonic D chord subjected to a contrapuntal 56 displacement.

    But, as Smith notes, it also carries hints of B-minor in frst inversionhints that connect both to the concluding moments of the previous movement, and to later events in the Andante such as the root position B-minor chord in mm. Te subtle shif of hearing that Smith notes in regard to the opening six-three is a characteristically tonal efect, but one that our transformational methodology, in its current state, cannot capture. Te development of the apparatuss tonal sensitivity is the work of Chapters 2 and 3.

    Te need to do this with transformational approaches is perhaps more pressing than usual, as developments in neo-Riemannian theory have generated a degree of antagonism between adherents of the two methods. In this section I will briefy compare the methodological characteristics of transformational and Schenkerian approaches with the aim of demonstrating that they difer in important ways in terms of analytical technique, theoretical content, and methodological goals.

    I will ultimately propose that any tension or competition between the two methodologies is misplaced and unnecessary. Such a tension suggests that Schenkerian and transformational theories represent two versions of the same kind of music theorythat their claims are equivalent and competing. I will instead argue that they are not competing forms of the same kind of music theory, but represent distinctly diferent styles of music-analytical thought.

    Te sketch shows a three-voice contrapuntal structure underlying mm. Te Kopfon 3. B3 is decorated with a complete neighbor C4, while the inner voice horizontalizes the fourth bet- ween D3 and G3 through stepwise motion. Tere are some evident visual parallels here with Figure 1. But the two analyses diverge notably in their content and in the nature of their analytical claims. Instead, it sketches multiple intervallic contexts in which the opening G2B3 interval may be experienced.

    Its various analytical representations do not concern the immanent stuf of the piece per se, but rather model a handful of intervallic apperceptions that one might have when hearing the preludes opening. One may even argue that Figure 1. It instead proposes diverse ways in which one might conceive of the G2B3 interval, and, in so conceiving, potentially recon- fgure ones experience of the suites opening. John Rahn has drawn a useful heuristic distinction between two kinds of the- ories: on the one hand is the theory of experience, on the other, the theory of piece , 51, Rahns categories recall two of the three levels of the tri- partition of Jean Molino and Jean-Jacques Nattiez Nattiez : Rahns theory of experience concerns the esthesic level, while his theory of piece concerns the neutral or immanent level however problematic the latter is from a philosophical standpoint.

    Te distinc- tion is crude because it oversimplifes: the GIS analysis still takes a piece of music as the object for its apperceptual musings, while the Schenkerian sketch of 1. Yet, in addition to its esthesic aspects, the Schenkerian analysis of Figure 1. Te sketch purports to reveal work-immanent relation- ships, in which the work is conceived as a structural entity composed of inter- acting harmonic and linear components, hierarchical levels, and so on.

    Such an immanent perspective is evident in familiar locutions in Schenkerian discourse, such as Te work is a 3. For critiques of Nattiezs immanent level, or niveau neutre, see Kof Agawu , , and Nicholas Cook , Te third level of the tripartition is the poietic, which is composer-focused. In practice, this statement may of course mean Te work may be heard as a 3. It is nevertheless ofen hard to tell the extent to which the familiar immanent wording is meant in these esthesic terms, or is instead intended as a propositional statement about the piece as an entity separable from the observer.

    Most important for our purposes is the fact that, as a discursive formation, such a statement is framed in immanent terms. Te conficts discussed in this section arise as a result of such discursive formations. Nattiez , provides a pertinent discussion of the admix- ture of work-immanent, esthesic, and poietic claims in Schenkerian discourse. For a strongly poietic account of Schenkerian theory as a model of expert monotonal composition, see Brown A look back at GMIT is helpful here.

    Lewin does not introduce his project by saying Music is made up of intervals, or even Music consists of a countless mul- tiplicity of intervals. Instead, he frames the issue in decidedly esthesic terms: In conceptualizing a particular musical space, it ofen happens that we conceptu- alize along with it, as one of its characteristic textural features, a family of directed measurements, distances, or motions of some sort.

    Contemplating elements s and t of such a musical space, we are characteristically aware of the particular directed measurement, distance, or motion that proceeds from s to t. GMIT, 16 Note what is being characterized: not the music as a Ding an sich , but our encounter with that music, and our concomitant conceptualizing or contemplating of it as a Ding an mich, as it were. Te statement introduces intervals and transforma- tions not as things in the world, but as relationships actively construed and apper- ceived by an individual in contact with some musical phenomenon.

    As one consequence of this fuid interpretive activity, the sounding entities of the music became defni- tionally mobile: the G2 and B3 in the Bach were variously construed as elements within a diatonic pitch space, elements of an infnite arpeggio, just overtones of a conceptual G1, 1. Other construals were possible as well none of these GISes, for example, construed G and B as pitch classes, either diatonic or chromatic. To the extent that such analyses reveal structures at all, they are esthesic structures rather than immanent structures.