Volume 13, Number 3, September 2007
Copyright � 2007 Society for Music Theory
Dmitri Tymoczko*Recasting Knets^{(1)} 

REFERENCE: Michael Buchler, “Reconsidering Klumpenhouwer Networks,” Music Theory Online 13.2
ABSTRACT: In this article I propose a new notation for hypertranspositions and hyperinversions, in which the usual subscripts are divided by 2. This new notation allows us to recast the hyper operations as ordinary transpositions and inversions operating on equivalence classes in 24tone equal temperament. This suggests a response to Buchler and Losada, who both criticize the conceptual foundations of standard Knet analysis.
Received August 2007
I. The Problem
[1] In a recent article, Michael Buchler observes that Knets such as those in
Figure 1, which I will notate as
[2] This question is worth taking seriously. Although hypertransposition is different from ordinary transposition, being a function over functions rather than a function over pitch classes, comparisons between these two types of “transposition” are intrinsic to the practice of Knet analysis. And although the primary analytical use of Knets is to relate sets belonging to different set classes, the technology applies equally well to chords such as C major and G major. Buchler has uncovered an example that seems to demonstrate that there is only a tenuous analogy between the two sorts of transposition. It will not do simply to reiterate that they are different. For Buchler’s challenge is, given that they disagree so dramatically 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_{7}> to refer to the hypertransposition linking {C, E} + {G} to {G, B} + {D}. In that case, the force of Buchler’s worry would be significantly ameliorated, since there is obviously something T_{7}like about the relationship. We might therefore ask whether it is possible to label the members of the hyperTI group <TI>, such that, if T_{x} or I_{y} transforms the pitch classes in one Knet K into those of another K′, with arrows being updated accordingly, <T_{x}> or <I_{y}> transforms the arrowlabels in K into the arrowlabels in K′? (See Figure 2.) In other words, can we label the elements of the hyperTI group in a way that is consistent with the TI group? 
(click to enlarge)
Figure 2. _{ (click to enlarge) } 
II. The Solution
[4] Yes. Simply divide the current hyperT and hyperI labels by 2. The only complication is that division is not uniquely defined in circular pitchclass space.
[5] As shown in Figure 1, {C, E} + {G} relates to {G, B} + {D} by <T_{2}> in ordinary Knet notation. I propose instead that we label this hypertransposition <T_{1} or T_{7}>. This is because 1 + 1 = 7 + 7 = 2 (mod 12). In other words, “2 divided by 2” can be either 1 or 7 in pitchclass space. Note that the label <T_{1} or T_{7}> does not refer to a “dual transposition”: it does not mean that one part of the Knet moves by 1 semitone while the other moves by 7 semitones. Instead, it says that the two parts of the Knet move by a total distance that is equal to both 1 + 1 and 7 + 7 (mod 12). Intuitively, the parts move by an average distance of 1 or 7 (mod 12): thus they might move by 1 and 1, 7 and 7, 0 and 2, 6 and 8, and so on.
[6] Now what’s T_{1} or T_{7}ish about the relation between C major and G major? The T_{7} relationship is obvious. What about T_{1}? Well, {C, E} + {G} is strongly isographic (<T_{0}>) 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 hypertranspositional labels thus reflect actual transpositions: C major is related to G major by <T_{1} or T_{7}> because C major is strongly isographic to chords that are T_{1} and T_{7}related to G major (G major and C major respectively). For the sake of clarity, I’m going to drop the clumsy <T_{1} or T_{7}> notation and use whichever of the pair is most appropriate to the context.^{(4)} [7] Q: In the new system, what hypertransposition relates {C, E} + {G} and {C, E} + {A}? A: <T_{1/2}> (“hyperT onehalf”). What could that mean? Here’s an informal way to think about it: in 24tone equaltemperament, {C, E} + {G} and {C , E} + {G} (C and E quartertone flat, G quartertone sharp) are strongly isographic, and hence related by <T_{0}>. Transposing the second chord up by half a semitone yields {C, E} + {A}. Again, in the new system, the hyperT labels reflect actual transpositional relationships: if A is related to B by <T_{x}> then there is a chord strongly isographic to A that can be transposed by x semitones to get chord B. Note however that this chord need not live in 12tone equal temperament—in fact for halfinteger hyperT, it will live in 24tone equaltemperament. Happily, once we pay attention to these quartertone chords, then “positively isographic to A” simply means “transpositionally related to a chord that is strongly isographic to A.” Likewise, “negatively isographic to A” means “inversionally related to a chord that is strongly isographic to A.” [8] The new notation suggests a response to those, like Buchler, who worry that hyperT and hyperI are too abstract. Every Knet defines an equivalence class of strongly isographic chords in 24tone equaltemperament, as shown in Figure 4. The statement that two (12tone equaltempered) Knets are related by <T_{x}> or <I_{y}> is logically equivalent to the statement that these equivalence classes relate by T_{x} or I_{y} (Figure 4). Thus we need not think of “hypertransposition” and “hyperinversion” as being extremely abstruse “functions over functions”: instead, we can understand them as ordinary transpositions and inversions of equivalence classes.^{(5)} In principle, there is nothing problematic about using transposition and inversion to relate equivalence classes; we do this every time we talk about transpositionally related pitchclass sets. The only question is whether equivalence classes of strongly isographic chords are analytically or perceptually meaningful—an issue we will return to shortly. [9] I know that many music theorists dislike quarter tones. So here’s an argument for my new notation that relies on 12tone equal temperament exclusively: in twelvetone equaltemperament, no chord that is strongly isographic to {C, E} + {G} is transpositionally related to {C, E} + {A}. Hence we need to identify the hyperT operation that takes {C, E} + {G} to {C, E} + {A} using a label not used by any equaltempered transposition. (Remember, we want to ensure consistency between labels for the TI group and the <TI> group.) Furthermore, we want to label this hypertransposition with a number x such that <T_{x}> + <T_{x}> = <T_{x+x}> = <T_{1}>. (This is because doing <T_{x}> twice produces what we have decided to call <T_{1}>.) This strongly suggests taking x to be 1/2.^{(6)} 
Figure 3. _{ } (click to enlarge) Figure 4. _{} (click to enlarge) 
[10] Since there are several ways to notate ordinary inversions, there are several ways to notate hyperinversions.
Alternative 1. If you label inversions with index numbers, you should divide the current hyperI labels by 2. In this new system, <T_{x}>I_{y} = I_{2x+y} and <I_{x}>I_{y} = I_{2x–y}. Some hyperinversions will be labeled with halfintegers, just like hypertranspositions.
Alternative 2. If you use Lewintype labels, such as
I , then<T_{x}>I will transpose a and b by an average of x semitones. Thus<T_{1/2}>I = I, since here we transpose E by 1 and E by 0, for an average transposition of (1 + 0)/2 = 1/2. In this system, some hyperinversions will be labeled with quarter tones.Alternative 3. Perhaps the best solution is to divide ordinary inversion labels by 2, labeling inversions by their fixed points, or . In this system,
I is represented by I_{3.5}. (Informally, this is an inversion that maps pitch class 3.5, or E, to itself.) This allows us to keep the traditional rule <T_{x}>I_{y} = I_{x+y}. However, in this system, some hyperinversions will have quarterinteger labels, such as <I_{1/4}>.
[11] Table 1 compares these alternatives. Note the consistency, in Alternative 3, between the way the TI operations operate on pitches, collections, and operations: in each case, T adds x to something (a pitch or an inversion subscript), while I subtracts something from 2x (a pitch or an inversion subscript). In standard notation, there is an inconsistency between the way TI operate on collections and operations, while in Alternative 1, there is an inconsistency between the way TI operate on pitches and on collections and operations. My goal here is to ensure consistency between collections and operations.^{(7)}
Table 1. _{}
(click to enlarge)
III. Problems for recursivity
[12] What happens to existing Knet analyses if we adopt my proposal? Well, ordinary T labels stay the same while hyperT labels are divided by two. This means that in the new notation, existing “recursive” Knet analyses—purporting to demonstrate isographies between Knets and “hyper Knets”^{(8)}—will no longer seem to be recursive. It follows that their apparent “recursivity” depends on a particular (and possibly ad hoc) way of labelling hyperoperations. I therefore agree with Buchler that we should be concerned about the significance of these analyses.
[13] One can save these analyses by asserting that they require only a group isomorphism between the hyperTI group and the ordinary TI group. My rejoinder is threefold:
 Group isomorphism is actually a weak relationship.^{(9)}
 The use of labels like <T_{1}> and T_{1} misleads people by making the isomorphism look stronger than it actually is: if all we care about is group isomorphism, then we can associate <T_{1}> with either T_{1}, T_{5}, T_{7}, or T_{11}. From the standpoint of group theory, there’s no principled reason for choosing among these.
 In the new notation, we can have recursive Knet analyses that are underwritten by something stronger than mere grouptheoretical isomorphism. My <T_{x}> really can be described as transpositions: they transpose equivalence classes of strongly isographic chords by x semitones. (Similar points apply to <I_{x}>, which invert these equivalence classes.) Consequently, there is a canonical way to associate each ordinary transposition T_{x} to a particular hypertransposition <T_{x}>—something that group theory alone does not provide. Furthermore, in the new system, there is a clear analogy between the techniques of Knet analysis and those of standard set theory—the main difference being that Knet analysis manipulates equivalence classes of strongly isographic chords, rather than equivalence classes of transpositionally or inversionally related chords.
All of this, in my view, argues strongly in favor of the proposal.
IV. Going up the hierarchy[14] Q. Can we do this again at the next level? That is, can we label hyperhypertranspositions <<T_{x}>> consistently with our labeling of the transpositions and hypertranspositions? [15] A. Yes, as Figure 5 demonstrates. There are no new difficulties here—and no need for anything other than halfinteger labels. (In particular, there is no cascade to eversmaller fractions, as the hyperoperations get more hyper.) I’ll leave the details up to the reader. The resulting system has the nice feature that if you take a big hyperhyperhyper ...hypernetwork of pitch classes, transpose all the pitch classes by x semitones, update all your arrowlabels accordingly, doing the same for your hyperarrows, your hyperhyperarrows, etc., your two networks are related by T_{x}, <T_{x}>, <<T_{x}>>, and so on. (Similar points apply to inversion and hyperhyper...hyperinversional labels.) In other words, the ordinary transpositional and inversion labels propagate up the system. This is not the case in standard notation, a fact that can lead to philosophical puzzles.^{(10)} 
Figure 5. _{ } (click to enlarge) 
V. Hypertransposition and dual transpositions
[16] Buchler perceptively remarks that Knet analyses try to pack too much information into a single number. He suggests that we use O’Donnell’s “dual transpositions” and “dual inversions” rather than hypertransposition and hyperinversion, identifying the particular TI operations that apply to each of the Knet’s two parts.^{(11)} 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 Knets using one number or two? And second, should we use “dual transpositions” or hypertranspositions?
Figure 6. _{}
(click to enlarge)
[17] Buchler labels the relation between Knets {C, E} + {G} and {C, E} + {A}
using the dual transposition
T_{0}/T_{1}, indicating that the major third stays fixed while the singleton moves by one semitone. But we can also label this relation
<X_{1/2}/T_{1/2}>. (Here “X” stands for “eXpand.”) The subscript of
<X_{x}> is calculated by taking half of the difference between the two components of Buchler’s
dual transposition, while <T_{x}> is equal to their average: mathematically, the
dual transposition
T_{a}/T_{b} is equivalent to the pair of transformations
<X_{(ba)/2}/T_{(a+b)/2}>. Thus, since {C, E} + {G} and {C, E} + {A} relate by
T_{0}/T_{1}, they also relate by
<X_{(1–0)/2}/T_{(1+0)/2}>=
<X_{1/2}/T_{1/2}>.
[18] What do these numbers mean? Figure 7 presents a twodimensional graph listing all the Knets, in 24tone equaltemperament, that are positively isographic to {C, E} + {G}, and that belong to set classes found in 12tone equal temperament. Strongly isographic chords lie on the same vertical line. Transpositionally related chords lie on the same horizontal line. The graph should be interpreted as a 2torus, with its right edge glued to its left, and the top edge glued to the bottom. (As in early videogames such as PacMan or Asteroids, one can move off of an edge to reappear on the opposite side of the figure.^{(12)}) What I am calling the <X_{x}> transform moves the Knet 2x vertical steps on this figure, while hypertransposition <T_{y}> moves 2y steps horizontally. The first transformation represents motion within the equivalence class of strongly isographic chords, and describes how the structure of the set class is altered: under <X_{x}> each part of the Knet moves by x semitones in contrary motion, possibly changing the set class in the process.^{(13)} The second transformation represents motion from one equivalence class to another, moving all the pitchclasses in the Knet upward by y semitones. As can be seen from the figure, dual transpositions represent an alternative way of describing relationships on this twodimensional surface: the dual transposition T_{a}/T_{b} moves a chord a positions diagonally southeast, and b positions diagonally northeast. The two notational systems are fundamentally equivalent, and are related by what physicists would call a “coordinate transformation.”^{(14)}
Figure 7. _{}
[19] Let's now return to the issue of Knets’ abstractness. Buchler is right to observe that standard Knet analyses use one number where it is possible to use two. In fact, we can restate his observation more precisely: Knet analyses use only the xcoordinate to refer to relationships between objects situated on a twodimensional surface. One possible response is to adopt “dual transposition” labels. A slightly more conservative alternative, from the standpoint of traditional Knet theory, is to use the <X_{x}/T_{y}> labels described here. These simply extend the techniques of Knet analysis by adding the missing coordinate, representing motion within equivalence classes of strongly isographic chords.
[20] To my mind, the deepest question raised by Buchler’s article is this: what coordinate system should we use when navigating Figure 7? I share his feeling that “dual transformations” are somewhat more general than hypertranspositions and inversions. (In particular, I think we should be reluctant to use standard Knet technology in cases where the musical surface does not clearly project exact contrary motion.) Furthermore, I am sympathetic with Buchler’s complaint that Knet analysis throws away too much information. I see no reason why we should have developed an analytical tradition that pays attention to only one of the two dimensions of Figure 7. Buchler’s worry, which I share, is that our analytical practices may derive not from deep conceptual reflection, or from underlying musical necessity, but simply from the force of institutional habit: we disregard the second coordinate because we have always done so, perhaps without even noticing that it could be incorporated into our Knet analyses.
VI. Philosophical issues
[21] 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. My response is that notation is not at all trivial, but is rather something that shapes thought. The importance of good notation is well understood by physicists and mathematicians, and is an issue that deserves more musictheoretical scrutiny—especially since some of our basic notational conventions are conceptually quite confusing.
[22] Among these is the practice of referring to inversions by index number. Ultimately, the notational question I have been pursuing is this: what should we call the function F(I_{x}) = T_{1}(I_{x})T_{1} = I_{x+2}? My suggestion is that we should call it <T_{1}> , or �hyperT_{1},� because it shifts the axis of inversional symmetry up by semitone: C is mapped to C under I_{0}, and C is mapped to C under T_{1}(I_{0})T_{1} = I_{2}. This fact is obscured by index numbers, which represent the onesemitone shift of the axis of inversional symmetry by a twounit change to the inversion�s subscript. Standard notation might therefore mislead the unwary theorist into thinking that, when the inversional axis of symmetry shifts up by one semitone, something else—the index number, whatever that is—has shifted by two.
[23] Familiar Knet terminology inherits this problem, using the label <T_{2}> to refer to the operation F(I_{x}) = T_{1}(I_{x})T_{1} = I_{x+2}. This may be a case of bad notation leading to confused thought. For though it can feel like we are transposing by two when we increase the index numbers by two, and though thinking in this way may help us calculate, this is not at all what is happening musically: transposition by x changes index numbers by 2x, as both Figure 2 and Table 1 demonstrate. I worry that David Lewin may have been mislead by this simple but pernicious feature of our ordinary notation when he invented the nowstandard labels for hypertranspositions and hyperinversions.
[24] However, it may be that Lewin was motivated not by conceptual confusion, but by the desire to identify T_{1}, a generator of the T group, with <T_{1}>, a generator of the hyperT group. Presumably, this desire was in turn motivated by Lewin’s goal of exploiting the group isomorphism between the <TI> hyperoperations and the TI operations. But is it so clear that this particular group isomorphism is, musically speaking, the most important one? My notation emphasizes a different isomorphism: that between the quotient group TI/T_{6} (the TI group for tritonesymmetrical objects, such as equivalence classes of strongly isographic chords) and a particular subgroup of <TI>—those that relate Knet arrows in networks whose pitch classes are related by ordinary transposition and inversion. (In Lewin’s notation, these are the hyperoperations with evennumbered subscripts; in my notation, they have integer subscripts.) This relationship is more than a mere group isomorphism: any action of TI/T_{6} on Knet nodes induces a corresponding action of the <TI> subgroup on arrows; conversely, any action of the <TI> subgroup on arrows can be realized by networks whose PCs are related by a corresponding action of TI/T_{6} on nodes. In many circumstances, the two perspectives provide alternate descriptions of the same musical process.
[25] The interesting point is that Lewin chose to emphasize a relatively weak relationship (group isomorphism) at the expense of this stronger relationship—and that almost all subsequent users of Knets have followed him. In doing so, they have asked us to overlook the fact that <T_{1}> (in my notation) actually transposes something by one semitone and focus instead on the very abstract fact that my <T_{1}> generates only half of the <T> operations. But it should be understood that this approach is rooted in a discrete, grouptheoretical perspective. I would argue that it’s not necessary—and perhaps not even productive—to think about Knets in this way. In fact, there’s a very beautiful (and in my view much more natural) geometrical interpretation of Knets and their significance.^{(15)}
[26] What rests on the choice between standard notation and my own? First, convenience and conceptual clarity: the system I advocate is (I claim) simpler and more logical, once you get used to it. Second, generalizability: Knets define “wedge” voice leadings in which the two parts of the Knet move in exact contrary motion;
the system I propose can be extended to “generalized Knets” whose parts move along arbitrary
voice leadings—and not simply those involving exact contrary motion. These arbitrary voice leadings
define generalized analogues to “strong isography” and give rise to
equivalence classes related by generalized analogues to the
[27] How should we understand Buchler’s criticisms in light of my proposed notational reforms? As I have indicated, I believe that some of his complaints can be ameliorated by a simple change of notation. There is indeed a close relationship between hypertransposition and ordinary transposition, though it is obscured by standard Knet terminology. At the same time, even in my new notational system, it is clear that some of Buchler’s criticisms remain intact. Knets are woven with a very coarse mesh, allowing a lot of useful musical information to wriggle free. (Indeed, relative to Figure 7, Knets 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 Knet analysis might be improved.
Dmitri Tymoczko
Princeton University
dmitri@princeton.edu
Bibliography
Buchler, Michael. 2007. “Reconsidering Klumpenhouwer Networks.” Music Theory Online 13.2.
Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. 2007. “Generalized Voice Leading Spaces.” Unpublished draft.
Klumpenhouwer, Henry. 2007. “Aspects of Depth in Knet Analysis with special reference to Webern's opus 16, 4.” Forthcoming in the Journal of Music Theory.
Lewin, David. 1990. “Klumpenhouwer Networks and Some Isographies That Involve Them.” Music Theory Spectrum 12: 83–120.
Losada, Catherine. 2007. “Knets and Hierarchical Structural Recursion: Further Considerations.” Music Theory Online 13.3.
O’Donnell, Shaugn. 1997. “Transformational Voice Leading in Atonal Music.” Ph.D. dissertation, City University of New York.
Tymoczko, Dmitri. 2007. “Lewin, Intervals, and Transformations: A Response to Hook.” forthcoming in Music Theory Spectrum.
References
1. Thanks to Michael Buchler, Norman Carey, Henry Klumpenhouwer, and Steven Rings for helpful conversations and suggestions.
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2. As Buchler points out, a Knet partitions a chord into two parts. I will therefore use the notation {x_{1}} + {x_{2}} to refer to Knets. Pairs of notes within a single set are connected by Tarrows, while pairs of notes from different sets are connected by Iarrows.
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3. Buchler actually uses different sets, but the essential point is the same. See Figures
14 and
15 in Buchler 2007. All subsequent references to Buchler’s work refer to this paper.
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4. This is mere notational shorthand: <T_{x}> should always be taken to abbreviate <T_{x}
or T_{x+6 (mod 12)}>.
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5. Of course, the <TI> operations can still be understood as “functions
over functions.” The point is that any statement about these functions over functions can be translated into an equivalent statement about ordinary transposition and inversion of equivalence classes.
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6. Or 13/2. In some sense, what’s happening here resembles what happened when your teacher started talking about negative and zero exponents in junior highschool algebra. We extend our system by introducing new notation, at first a little paradoxical, but in a way that is consistent, and that increases the formal system’s power.
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7.
In her response to Buchler, Catherine Losada shows that when we
relabel pitchclass space by adding c to every pitchclass label, we
add 2c to every inversion label and 4c to to every hyperinversion
label. (See Examples
8 and
9 in Losada 2007, where c = 8.) It
follows that associations between ordinary inversions and hyperinversions
will depend on which pitch class we arbitrarily choose to label 0.
Under the alternative notation described here, this problem does not
arise: inversion and hyperinversion labels transform in the same way
under relabelings of pitch class 0.
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8. The nodes of a “hyperKnet” are themselves Knets, connected by arrows indicating hypertranspositional and hyperinversional relationships. See
Figure 5.
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9. See my “Lewin, intervals, and transformations,” forthcoming in
Music Theory Spectrum.
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10. In a forthcoming article in the Journal of Music Theory, Henry Klumpenhouwer sorts through these puzzles, engaging in a careful investigation of the ontological difference between transposition, hypertransposition, hyperhypertransposition, and so on. By contrast, I propose an alternative notation in which the inconsistencies between these levels simply evaporate, along with the appearance of a philosophical puzzle.
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11. See O’Donnell 1997.
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12. Here, however, the left edge needs to be glued to the right with in a “diagonal” fashion as shown by 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.
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13. Given a Knet {a_{1}} + {a_{2}}, the <X_{x}> transformation transposes {a_{1}} up by
x semitones and {a_{2}} down by
x semitones. Thus the notation requires us to identify which of the Knet’s two parts is the first and which is the second.
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14. Indeed, physicists have a specific name for the coordinate transformation involved: they would say that my labels use the
centerofmass reference frame to relate Knets. In elementary physics, one often represents the motion of a group of particles as having an “internal component” defined relative to a coordinate system in which there is zero total momentum, and an “external component” representing the motion of this coordinate system relative to some other reference frame. If we imagine the two parts of the Knet to have equal mass, then motion between strongly isographic chords occurs in the centerofmass reference frame.
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15. Clifton Callender, Ian Quinn, and I describe this interpretation in a collaborative paper which we hope to finish (and publish) soon. A preliminary draft can be found at
http://music.princeton.edu/~dmitri/geometry.pdf.
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16. See the paper cited in the previous footnote. Interested readers can probably work out the details themselves: the idea is to consider all the equaltempered set classes produced by an arbitrary voice leading—even if the sets themselves do not lie in 12tone equaltemperament. These sets can be understood as equivalence classes, can be related by transposition and inversion, and can be used to construct twodimensional spaces exactly analogous to
Figure 7.
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