Volume 13, Number 3, September 2007 Copyright 2007 Society for Music Theory

Transcription

1 of Volume 3, Number 3, September Copright Societ for Music Theor * Dmitri Tmoczko REFERENCE: ABSTRACT: In this article I propose a new notation for hper-transpositions and hper-inversions, in which the usual subscripts are divided b. This new notation allows us to recast the hper operations as ordinar transpositions and inversions operating on equivalence classes in 4-tone equal temperament. This suggests a response to Buchler and Losada, who both criticize the conceptual foundations of standard K-net analsis. Received August I. The Problem [] In a recent article, Michael Buchler observes that K-nets such as those in Figure, which I will notate as {C, E} + {G} and {G, B} + {D}, are related b <T > ( hper-t ). (), () He asks, in effect, what s T -like about the relation between C major and G major triads? [] This question is worth taking seriousl. Although hper-transposition is different from ordinar transposition, being a function over functions rather than a function over pitch classes, comparisons between these two tpes of transposition are intrinsic to the practice of K-net analsis. And although the primar analtical use of K-nets is to relate sets belonging to different set classes, the technolog applies equall well to chords such as C major and G major. Buchler has uncovered an eample that seems to demonstrate that there is onl a tenuous analog between the two sorts of transposition. It will not do simpl to reiterate that the are different. For Buchler s challenge is, given that the disagree so dramaticall about such a simple case, what s the musical value of comparing them? [3] It is possible, however, that a simple change in notation might help meet his objection. For suppose we used the label <T > to refer to the hper-transposition linking {C, E} + {G} to {G, B} + {D}. In that case, the force of Buchler s worr would be significantl ameliorated, since there is obviousl something T -like about the relationship. We might therefore ask whether it is possible to label the members of the hper-ti group <TI>, such that, if T or I transforms the pitch classes in one K-net K into those of another K, with arrows being updated accordingl, <T > or <I > transforms the arrow-labels in K into the arrow-labels in K? (See Figure.) In other words, can we label the elements of the hper-ti group in a wa that is consistent with the TI group? II. The Solution [4] Yes. Simpl divide the current hper-t and hper-i labels b. The onl complication is that division is not uniquel

2 of defined in circular pitch-class space. [5] As shown in Figure, {C, E} + {G} relates to {G, B} + {D} b <T > in ordinar K-net notation. I propose instead that we label this hper-transposition <T >. This is because + = + = (mod ). In other words, divided b can be either or in pitch-class space. Note that the label <T > does not refer to a dual transposition : it does not mean that one part of the K-net moves b semitone while the other moves b semitones. Instead, it sas that the two parts of the K-net move b a total distance that is equal to both + and + (mod ). Intuitivel, the parts move b an average distance of or (mod ): thus the might move b and, and, and, 6 and 8, and so on. [6] Now what s T -ish about the relation between C major and G major? The T relationship is obvious. What about T? Well, {C, E} + {G} is strongl isographic (<T >) to {G, B } + {D } and the G major chord has to be transposed up one semitone to become G major. This is illustrated in Figure 3. The hper-transpositional labels thus reflect actual transpositions: C major is related to G major b <T > because C major is strongl isographic to chords that are T - and T -related to G major (G major and C major respectivel). For the sake of clarit, I m going to drop the clums <T or T > notation and use whichever of the pair is most appropriate to the contet. (3) [] Q: In the new sstem, what hper-transposition relates {C, E} + {G} and {C, E} + {A }? A: <T > ( hper-t / one-half ). What could that mean? Here s an informal wa to think about it: in 4-tone equal-temperament, {C, E} + {G} and {C, E } + {G } (C and E quarter-tone flat, G quarter-tone sharp) are strongl isographic, and hence related b <T >. Transposing the second chord up b half a semitone ields {C, E} + {A }. Again, in the new sstem, the hper-t labels reflect actual transpositional relationships: if A is related to B b <T > then there is a chord strongl isographic to A that can be transposed b semitones to get chord B. Note however that this chord need not live in -tone equal temperament in fact for half-integer hper-t, it will live in 4-tone equal-temperament. Happil, once we pa attention to these quarter-tone chords, then positivel isographic to A simpl means transpositionall related to a chord that is strongl isographic to A. Likewise, negativel isographic to A means inversionall related to a chord that is strongl isographic to A. [8] The new notation suggests a response to those, like Buchler, who worr that hper-t and hper-i are too abstract. Ever K-net defines an equivalence class of strongl isographic chords in 4-tone equal-temperament, as shown in Figure 4. The statement that two (-tone equal-tempered) K-nets are related b <T > or <I > is logicall equivalent to the statement that these equivalence classes relate b T or I (Figure 4). Thus we need not think of hper-transposition and hperinversion as being etremel abstruse functions over functions : instead, we can understand them as ordinar transpositions and inversions of equivalence classes. (4) In principle, there is nothing problematic about using transposition and inversion to relate equivalence classes; we do this ever time we talk about transpositionall related pitch-class sets. The onl question is whether equivalence classes of strongl isographic chords are analticall or perceptuall meaningful an issue we will return to shortl. [9] I know that man music theorists dislike quarter tones. So here s an argument for m new notation that relies on -tone equal temperament eclusivel: in twelve-tone equal-temperament, no chord that is strongl isographic to {C, E} + {G} is transpositionall related to {C, E} + {A }. Hence we need to identif the hper-t operation that takes {C, E} + {G} to {C, E} + {A } using a label not used b an equal-tempered transposition. (Remember, we want to ensure consistenc between labels for the TI group and the <TI> group.) Furthermore, we want to label this hper-transposition with a number such that <T > + <T > = <T > = <T >. (This is because doing <T > twice produces what we have decided to call + <T >.) This strongl suggests taking to be /. (5) [] Since there are several was to notate ordinar inversions, there are several was to notate hper-inversions. Alternative. If ou label inversions with inde numbers, ou should divide the current hper-i labels b. In this new sstem, <T >I + and <I >I. Some hper-inversions will be labeled with half-integers, just like hper-transpositions. Alternative. If ou use Lewin-tpe labels, such as I, then <T >I will transpose a and b b an average of semitones. Thus <T >I, since here we transpose E b and E b, for an average transposition / of ( + )/ = /. In this sstem, some hper-inversions will be labeled with quarter tones. Alternative 3. Perhaps the best solution is to divide ordinar inversion labels b, labeling inversions b their fied points, or. In this sstem, I is represented b I. (Informall, this is an inversion that 3.5

3 3 of maps pitch class 3.5, or E, to itself.) This allows us to keep the traditional rule <T >I. However, in + this sstem, some hper-inversions will have quarter-integer labels, such as <I >. /4 [] Table compares these alternatives. Note the consistenc, in Alternative 3, between the wa the TI operations operate on pitches, collections, and operations: in each case, T adds to something (a pitch or an inversion subscript), while I subtracts something from (a pitch or an inversion subscript). In standard notation, there is an inconsistenc between the wa TI operate on collections and operations, while in Alternative, there is an inconsistenc between the wa TI operate on pitches and on collections and operations. M goal here is to ensure consistenc between collections and operations. (6) III. Problems for recursivit [] What happens to eisting K-net analses if we adopt m proposal? Well, ordinar T labels sta the same while hper-t labels are divided b two. This means that in the new notation, eisting recursive K-net analses purporting to demonstrate isographies between K-nets and hper K-nets () will no longer seem to be recursive. It follows that their apparent recursivit depends on a particular (and possibl ad hoc) wa of labelling hper-operations. I therefore agree with Buchler that we should be concerned about the significance of these analses. [3] One can save these analses b asserting that the require onl a group isomorphism between the hper-ti group and the ordinar TI group. M rejoinder is threefold:. Group isomorphism is actuall a weak relationship. (8). The use of labels like <T > and T misleads people b making the isomorphism look stronger than it actuall is: if all we care about is group isomorphism, then we can associate <T > with either T, T, T,. From the 5 standpoint of group theor, there s no principled reason for choosing among these. 3. In the new notation, we can have recursive K-net analses that are underwritten b something stronger than mere group-theoretical isomorphism. M <T > reall can be described as transpositions: the transpose equivalence classes of strongl isographic chords b semitones. (Similar points appl to <I >, which invert these equivalence classes.) Consequentl, there is a canonical wa to associate each ordinar transposition T to a particular hper-transposition <T > something that group theor alone does not provide. Furthermore, in the new sstem, there is a clear analog between the techniques of K-net analsis and those of standard set theor the main difference being that K-net analsis manipulates equivalence classes of strongl isographic chords, rather than equivalence classes of transpositionall or inversionall related chords. All of this, in m view, argues strongl in favor of the proposal. IV. Going up the hierarch [4] Q. Can we do this again at the net level? That is, can we label hper-hper-transpositions <<T >> consistentl with our labeling of the transpositions and hper-transpositions? [5] A. Yes, as Figure 5 demonstrates. There are no new difficulties here and no need for anthing other than half-integer labels. (In particular, there is no cascade to ever-smaller fractions, as the hper-operations get more hper.) I ll leave the details up to the reader. The resulting sstem has the nice feature that if ou take a big hper-hper-hper -...-hper-network of pitch classes, transpose all the pitch classes b semitones, update all our arrow-labels accordingl, doing the same for our hper-arrows, our hper-hper-arrows, etc., our two networks are related b T, <T >, <<T >>, and so on. (Similar points appl to inversion and hper-hper-...-hper-inversional labels.) In other words, the ordinar transpositional and inversion labels propagate up the sstem. This is not the case in standard notation, a fact that can lead to philosophical puzzles. (9) V. Hper-transposition and dual transpositions [6] Buchler perceptivel remarks that K-net analses tr to pack too much information into a single number. He suggests that we use O Donnell s dual transpositions and dual inversions rather than hper-transposition and hper-inversion, identifing the particular TI operations that appl to each of the K-net s two parts. () This is illustrated in Figure 6. I think this is a reasonable suggestion. However, it should be noted that there are two independent questions here. First, should we describe the relation between K-nets using one number or two? And second, should we use dual transpositions or hpertranspositions?

4 4 of [] Buchler labels the relation between K-nets {C, E} + {G} and {C, E} + {A } using the dual transposition T, indicating that the major third stas fied while the singleton moves b one semitone. But we can also label this relation <X / / >. (Here X stands for expand. ) The subscript of <X > is calculated b taking half of the difference between the two components of Buchler s dual transposition, while <T > is equal to their average: mathematicall, the dual transposition T a b is equivalent to the pair of transformations <X (b-a)/ (a+b)/ >. Thus, since {C, E} + {G} and {C, E} + {A } relate b T, the also relate b <X ( )/ (+)/ >= <X / / >. [8] What do these numbers mean? Figure presents a two-dimensional graph listing all the K-nets, in 4-tone equaltemperament, that are positivel isographic to {C, E} + {G}, and that belong to set classes found in -tone equal temperament. Strongl isographic chords lie on the same vertical line. Transpositionall related chords lie on the same horizontal line. The graph should be interpreted as a -torus, with its right edge glued to its left, and the top edge glued to the bottom. (As in earl video-games such as Pac-Man or Asteroids, one can move off of an edge to reappear on the opposite side of the figure. () ) What I am calling the <X > transform moves the K-net vertical steps on this figure, while hpertransposition <T > moves steps horizontall. The first transformation represents motion within the equivalence class of strongl isographic chords, and describes how the structure of the set class is altered: under <X > each part of the K-net moves b semitones in contrar motion, possibl changing the set class in the process. () The second transformation represents motion from one equivalence class to another, moving all the pitch-classes in the K-net upward b semitones. As can be seen from the figure, dual transpositions represent an alternative wa of describing relationships on this two-dimensional surface: the dual transposition T a b moves a chord a positions diagonall southeast, and b positions diagonall northeast. The two notational sstems are fundamentall equivalent, and are related b what phsicists would call a coordinate transformation. (3) [9] Let's now return to the issue of K-nets abstractness. Buchler is right to observe that standard K-net analses use one number where it is possible to use two. In fact, we can restate his observation more precisel: K-net analses use onl the -coordinate to refer to relationships between objects situated on a two-dimensional surface. One possible response is to adopt dual transposition labels. A slightl more conservative alternative, from the standpoint of traditional K-net theor, is to use the <X > labels described here. These simpl etend the techniques of K-net analsis b adding the missing coordinate, representing motion within equivalence classes of strongl isographic chords. [] To m mind, the deepest question raised b Buchler s article is this: what coordinate sstem should we use when navigating Figure? I share his feeling that dual transformations are somewhat more general than hper-transpositions and inversions. (In particular, I think we should be reluctant to use standard K-net technolog in cases where the musical surface does not clearl project eact contrar motion.) Furthermore, I am smpathetic with Buchler s complaint that K-net analsis throws awa too much information. I see no reason wh we should have developed an analtical tradition that pas attention to onl one of the two dimensions of Figure. Buchler s worr, which I share, is that our analtical practices ma derive not from deep conceptual reflection, or from underling musical necessit, but simpl from the force of institutional habit: we disregard the second coordinate because we have alwas done so, perhaps without even noticing that it could be incorporated into our K-net analses. VI. Philosophical issues [] The proposal in this paper is at bottom notational, a suggestion that we use new words to describe familiar relationships. It might therefore seem that I do not go to the musical heart of the matter. M response is that notation is not at all trivial, but is rather something that shapes thought. The importance of good notation is well understood b phsicists and mathematicians, and is an issue that deserves more music-theoretical scrutin especiall since some of our basic notational conventions are conceptuall quite confusing. [] Among these is the practice of referring to inversions b inde number. Ultimatel, the notational question I have been pursuing is this: what should we call the function F(I ) = T (I )T - +? M suggestion is that we should call it <T >, or hper-t, because it shifts the ais of inversional smmetr up b semitone: C is mapped to C under I, and C is mapped to C under T (I )T -. This fact is obscured b inde numbers, which represent the one-semitone shift of the ais of inversional smmetr b a two-unit change to the inversion s subscript. Standard notation might therefore mislead the unwar theorist into thinking that, when the inversional ais of smmetr shifts up b one semitone, something else the inde number, whatever that is has shifted b two. [3] Familiar K-net terminolog inherits this problem, using the label <T > to refer to the operation F(I ) = T (I )T - =

5 5 of I. This ma be a case of bad notation leading to confused thought. For though it can feel like we are transposing b two + when we increase the inde numbers b two, and though thinking in this wa ma help us calculate, this is not at all what is happening musicall: transposition b changes inde numbers b, as both Figure and Table demonstrate. I worr that David Lewin ma have been mislead b this simple but pernicious feature of our ordinar notation when he invented the now-standard labels for hper-transpositions and hper-inversions. [4] However, it ma be that Lewin was motivated not b conceptual confusion, but b the desire to identif T, a generator of the T group, with <T >, a generator of the hper-t group. Presumabl, this desire was in turn motivated b Lewin s goal of eploiting the group isomorphism between the <TI> hper-operations and the TI operations. But is it so clear that this particular group isomorphism is, musicall speaking, the most important one? M notation emphasizes a different isomorphism: that between the quotient group TI 6 (the TI group for tritone-smmetrical objects, such as equivalence classes of strongl isographic chords) and a particular subgroup of <TI> those that relate K-net arrows in networks whose pitch classes are related b ordinar transposition and inversion. (In Lewin s notation, these are the hper-operations with even-numbered subscripts; in m notation, the have integer subscripts.) This relationship is more than a mere group isomorphism: an action of TI 6 on K-net nodes induces a corresponding action of the <TI> subgroup on arrows; conversel, an action of the <TI> subgroup on arrows can be realized b networks whose PCs are related b a corresponding action of TI 6 on nodes. In man circumstances, the two perspectives provide alternate descriptions of the same musical process. [5] The interesting point is that Lewin chose to emphasize a relativel weak relationship (group isomorphism) at the epense of this stronger relationship and that almost all subsequent users of K-nets have followed him. In doing so, the have asked us to overlook the fact that <T > (in m notation) actuall transposes something b one semitone and focus instead on the ver abstract fact that m <T > generates onl half of the <T> operations. But it should be understood that this approach is rooted in a discrete, group-theoretical perspective. I would argue that it s not necessar and perhaps not even productive to think about K-nets in this wa. In fact, there s a ver beautiful (and in m view much more natural) geometrical interpretation of K-nets and their significance. (4) [6] What rests on the choice between standard notation and m own? First, convenience and conceptual clarit: the sstem I advocate is (I claim) simpler and more logical, once ou get used to it. Second, generalizabilit: K-nets define wedge voice leadings in which the two parts of the K-net move in eact contrar motion; the sstem I propose can be etended to generalized K-nets whose parts move along arbitrar voice leadings and not simpl those involving eact contrar motion. These arbitrar voice leadings define generalized analogues to strong isograph and give rise to equivalence classes related b generalized analogues to the <TI> operations. (Unfortunatel, describing this generalization further is beond the scope of the current paper. (5) ) Third, our understanding of the relationship between <T > and T. Under m proposal the re etremel close, both transposing something b chords in one case, equivalence classes of strongl isographic chords in the other. [] How should we understand Buchler s criticisms in light of m proposed notational reforms? As I have indicated, I believe that some of his complaints can be ameliorated b a simple change of notation. There is indeed a close relationship between hper-transposition and ordinar transposition, though it is obscured b standard K-net terminolog. At the same time, even in m new notational sstem, it is clear that some of Buchler s criticisms remain intact. K-nets are woven with a ver coarse mesh, allowing a lot of useful musical information to wriggle free. (Indeed, relative to Figure, K-nets are not nets at all, but rather threads using one dimension where two are needed!) I applaud Michael for having the courage to point this out, and for challenging us to think about how the practice of K-net analsis might be improved. Dmitri Tmoczko Princeton Universit dmitri@princeton.edu Works Cited Buchler, Michael.. Reconsidering Klumpenhouwer Networks. Music Theor Online 3.: 69. Callender, Clifton, Ian Quinn, and Dmitri Tmoczko.. Generalized Voice Leading Spaces. Unpublished draft.

6 6 of Klumpenhouwer, Henr.. Aspects of Depth in K-net Analsis with special reference to Webern s opus 6, 4. Forthcoming in the Journal of Music Theor. Lewin, David. 99. Klumpenhouwer Networks and Some Isographies That Involve Them. Music Theor Spectrum : 83. Losada, Catherine.. K-nets and Hierarchical Structural Recurrence: Further Considerations. Music Theor Online 3.3. O Donnell, Shaugn J. 99. Transformational Voice Leading in Atonal Music. Ph.D. dissertation, Cit Universit of New York. Footnotes * Thanks to Michael Buchler, Norman Care, Henr Klumpenhouwer, and Steven Rings for helpful conversations and suggestions. Return to tet. As Buchler points out, a K-net partitions a chord into two parts. I will therefore use the notation { } + { } to refer to K-nets. Pairs of notes within a single set are connected b T-arrows, while pairs of notes from different sets are connected b I-arrows. Return to tet. Buchler actuall uses different sets, but the essential point is the same. See Figures 4 and 5 in Buchler. All subsequent references to Buchler s work refer to this paper. Return to tet 3. This is mere notational shorthand: <T > should alwas be taken to abbreviate <T +6 (mod ) >. Return to tet 4. Of course, the <TI> operations can still be understood as functions over functions. The point is that an statement about these functions over functions can be translated into an equivalent statement about ordinar transposition and inversion of equivalence classes. Return to tet 5. Or 3/. In some sense, what s happening here resembles what happened when our teacher started talking about negative and zero eponents in junior high-school algebra. We etend our sstem b introducing new notation, at first a little paradoical, but in a wa that is consistent, and that increases the formal sstem s power. Return to tet 6. In her response to Buchler, Catherine Losada shows that when we relabel pitch-class space b adding c to ever pitch-class label, we add c to ever inversion label and 4c to to ever hper-inversion label. (See Eamples 8 and 9 in Losada, where c = 8.) It follows that associations between ordinar inversions and hper-inversions will depend on which pitch class we arbitraril choose to label. Under the alternative notation described here, this problem does not arise: inversion and hper-inversion labels transform in the same wa under relabelings of pitch class. Return to tet. The nodes of a hper-k-net are themselves K-nets, connected b arrows indicating hper-transpositional and hperinversional relationships. See Figure 5. Return to tet 8. See m Lewin, intervals, and transformations, forthcoming in Music Theor Spectrum. Return to tet 9. In a forthcoming article in the Journal of Music Theor, Henr Klumpenhouwer sorts through these puzzles, engaging in a careful investigation of the ontological difference between transposition, hper-transposition, hper-hper-transposition, and so on. B contrast, I propose an alternative notation in which the inconsistencies between these levels simpl evaporate, along with the appearance of a philosophical puzzle.

7 of Return to tet. See O Donnell 99. Return to tet. Here, however, the left edge needs to be glued to the right with in a diagonal fashion as shown b the lines A and B. Note that, for reasons of space, chords on the top edge also appear on the bottom edge, but chords on the left do not also appear on the right. Return to tet. Given a K-net {a } + {a }, the <X > transformation transposes {a } up b semitones and {a } down b semitones. Thus the notation requires us to identif which of the K-net s two parts is the first and which is the second. Return to tet 3. Indeed, phsicists have a specific name for the coordinate transformation involved: the would sa that m labels use the center-of-mass reference frame to relate K-nets. In elementar phsics, one often represents the motion of a group of particles as having an internal component defined relative to a coordinate sstem in which there is zero total momentum, and an eternal component representing the motion of this coordinate sstem relative to some other reference frame. If we imagine the two parts of the K-net to have equal mass, then motion between strongl isographic chords occurs in the centerof-mass reference frame. Return to tet 4. Clifton Callender, Ian Quinn, and I describe this interpretation in a collaborative paper which we hope to finish (and publish) soon. A preliminar draft can be found at Return to tet 5. See the paper cited in the previous footnote. Interested readers can probabl work out the details themselves: the idea is to consider all the equal-tempered set classes produced b an arbitrar voice leading even if the sets themselves do not lie in -tone equal-temperament. These sets can be understood as equivalence classes, can be related b transposition and inversion, and can be used to construct two-dimensional spaces eactl analogous to Figure. Return to tet Copright Statement Copright b the Societ for Music Theor. All rights reserved. [] Coprights for individual items published in Music Theor Online (MTO) are held b their authors. Items appearing in MTO ma be saved and stored in electronic or paper form, and ma be shared among individuals for purposes of scholarl research or discussion, but ma not be republished in an form, electronic or print, without prior, written permission from the author(s), and advance notification of the editors of MTO. [] An redistributed form of items published in MTO must include the following information in a form appropriate to the medium in which the items are to appear: This item appeared in Music Theor Online in [VOLUME #, ISSUE #] on [DAY/MONTH/YEAR]. It was authored b [FULL NAME, ADDRESS], with whose written permission it is reprinted here. [3] Libraries ma archive issues of MTO in electronic or paper form for public access so long as each issue is stored in its entiret, and no access fee is charged. Eceptions to these requirements must be approved in writing b the editors of MTO, who will act in accordance with the decisions of the Societ for Music Theor. This document and all portions thereof are protected b U.S. and international copright laws. Material contained herein ma be copied and/or distributed for research purposes onl. Prepared b Brent Yorgason, Managing Editor and Stefanie Acevedo, Editorial Assistant