1. Introduction

Today I am going to give a talk that on music perception and cognition, but it will be masquerading as a comparison between neo-Riemannian tonal spaces versus graphic representations of metric systems. Many composers and theorists, throughout the ages, have argued that musical pitch and musical time are, at some deep level, isomorphic--that they are two sides of the same coin. And at first blush, these arguments, which range from Pythagoras and Hauptmann to Stockhausen and Boulez, may seem reasonable. Individual note onsets take specific locations in both measured and unmeasured time, just as individual pitches take discrete locations within the pitch continuum and as scale steps. Likewise both relative durations and the intervals between pitches can be discussed in terms of ratios or proportions. Pitch-classes, beat-classes (in an established meter) and scaled durations are all amenable to set and/or group-theoretic treatment. When the same ratios and patterns crop up in both tonal and temporal domains our suspicions are naturally aroused--could the recurrence of a 2-2-1-2-2-2-1 durational pattern in various musical cultures be a sign of some deep parallel between scales and meters?

In a word, no. I will begin by presenting spatial representations--graphs--of pitch and metric systems, starting with familiar tonnetz representations of pitch and pitch-class space. After a very quick introduction to mathematical graph theory, I will show that pitch/pitch-class space and an analogous "meter/tempo space" are fundamentally non-isomorphic. This non-isomorphism stems from the fact that there are no temporal analogs to octave and enharmonic equivalence and that there are no tonal analogs to various limits on our temporal perception and acuity. These non-isomorphisms in the spatial representations for pitch and meter call into question the validity of broader claims about the unity of pitch-time relationships.

2. Mapping PC Space

Starting with the graphic arrays or tonnetze developed by Riemann and Oettingen, a number of theorists have developed spatial representations of pitches, triads, and tonal distance. Hyer (1995, 102) reproduces the following figure from Riemann (1914-15):

Of this diagram Hyer says "It is as if, using C as ground zero, Riemann has taken the combinatorial intervals of the Klang and strewn them in all imaginable directions, mapping out an abstract terrain of harmonic consonances" (101). Hyer also notes: "A crucial feature of the grid is its extreme chromaticism: fig. 1 represents an unbounded conceptual area containing an infinite number of different pitches, no two of which are identical. Because Riemann assumes just intonation, the lattice is infinitely extensible on all sides" (105). Hyer then re-imagines Riemann's space under the constraints of enharmonic and octave equivalence, and notes that in so doing Riemann's "tabular representation of tonal relations gives rise to remarkable algebraic and topological properties" (p. 106). The result is the following figure (Hyer's fig. 3, p. 119):

Hyer notes that when one connects the edges in the manner shown in the diagram, the result is a doughnut or torus. Two dimensions are not adequate for mapping the spatial relationships amongst the elements in a tonnetz; to capture the continuities generated by enharmonic and octave equivalence requires at least three dimensions.

More recently Cohn (1997, 10) has produced a generalized form of the tonnetz and then has noted the special properties of the triadic tonnetz in the context of 12-tone equal temperament.

Like Hyer, Cohn notes that "if x and y [Cohn's variables for horizontal and vertical relationships in the tonnetz] are assigned to acoustically pure intervals (as in Euler, etc..), or to intervals in pitch-space, then the structure implicitly projects into an infinite plane. The realizations [of the tonnetz] that will hold our focus are generated by equally tempered intervals in some modular system, where the modular congruence represents octave equivalence. In such interpretations, both x and y axes become cyclic rather than linear, and the plane . . . therefore projects into itself as a torus" (Cohn 1997, 11-12).

Music theorists are not alone in recognizing the torroidal shape of tonal space. Researchers in music perception and cognition have empirically measured the goodness-of-fit for notes and chords in a tonally-primed context, and they too have mapped tonal space onto the surface of a torus. Krumhansl and Kessler (1982) give the follow representation of inter-key distances:

Lerdahl (1988), strongly influenced by Krumhansl and Kessler, as well as the work of Deutsch and Feroe (1981), has also produced a mapping of chord space which is symmetrical and which is torroidal. From these examples we may make the following observations.

o In a tonnetz there is a uniformity of relationships from one location to another--all of the links form the same pattern, generating a uniform space.

o Topologically no location in the tonnetz is privileged, though in applying the tonnetz to real musical surfaces, a particular location assumes the role of "tonic" such that other locations are heard relative to its location (at least as long as that particular tonic holds sway).

o Given enharmonic and octave equivalence, the tonnetz requires at least 3 dimensions (i.e., a torroidal surface) for its representation which preserves all of the relationships between adjacent tonal elements.

3. A Graph Theory Interlude.

A tonnetz is a kind of graph, a finite set of one or more vertices connected (or not) to each other with set of zero or more edges. Here are a number of graphs:

Figure 5A is a graph, although not a continuous one. It consists of seven vertices and four edges. We can specify edge-vertex relations by giving the degree of each vertex in a graph, that is, the number of edges that meet at each vertex. So in figure 5A there are five vertices of degree 1, one vertex of degree 2, and one vertex of degree 3.

Figure 5B is a cyclical graph. In a cycle one has the same number of edges and vertices, and each vertex is of degree 2. Figure 5C is a bipartite graph K3, 3 (K indicates completeness, and 3, 3 indicates the number of elements in each half of a bipartition, as it shows how a set of one elements relates to another--in this case, each of the three vertices on top connect to all three vertices on the bottom. It is also known as the utility graph, as it illustrates a classic problem of whether three utility companies can hook up their services to three houses without crossing their pipes or wires (they cannot, as we shall see). Finally, figure 5D is a complete graph, here K5. A complete graph is defined as the graph of some set of vertices in which each vertex is connected to every other. Thus for a complete graph consisting of N vertices, each vertex will be of degree N-1.

Two graphs are isomorphic if they share the following properties:

(a) they have the same number of vertices,

(b) they have the same number of edges,

(c) the same distribution of degrees (for each vertex in each graph)

(d) the same number of "pieces" in each graph

The first two requirements should be obvious, since two isomorphic structures each need to be comprised of the same number of elements. The last two insure that those elements stand in the same relationship(s) to each other. It does not matter whether the edges are straight or curved, nor does the relative spacing of the vertices. Here are some more examples:

Figure 6A is the complete graph K4--a set of four vertices, each connected to each other. Figure 6B is also K4, with one edge drawn "outside the box." Note that both have the same number of vertices, the same number of edges, all vertices are of the same degree, and both are in one piece. These two graphs are isomorphic. In Figures 6C and 6D we have a pair of graphs that have the same number of vertices, the same number of edges, and all vertices are of the same degree, but they are not isomorphic. Notice that 6C is in one piece--it is possible to "walk" from one vertex to any other vertex via edge connections, while 6D is in two pieces--if you are on the upper triangle, you can walk to two other vertices, but you cannot follow an edge to the lower triangle. Another way to put it is that 6D is a graph that contains two subgraphs, here each a 3-cycle, but there are no common edges between each subgraph.

Another important aspect of graph theory is planarity. A graph is planar if it is possible to draw it in a two-dimensional plane without edge-crossings. Because we can draw K4 without crossing the diagonals, it is planar. K3, 3 and K5 are non-planar. If one graph is planar, and another is non-planar, they cannot be isomorphic (though subgraphs of each can be).

We have already noted the difference between 6C and 6D in terms of "pieces." Here is a more formal way to describe the connectedness amongst the elements in a graph, modeled after Trudeau (p. 97): A walk in a graph is a sequence A1, A2, A3 . . . An of not necessarily distinct vertices in which A1 is joined by an edge to A2, A2 by an edge to A3, and so on, through An. If every pair of vertices in a graph is joined by a walk, then that graph is said to be connected. Thus the in 7D, there is no walk which joins opposite points of the star.

We have now gone, though rather quickly, through enough graph theory to give a fairly thorough characterization of various tonnetz representations. Riemann's original conception of the tonnetz is connected and it is planar. When we include octave and enharmonic equivalence, the graph becomes non-planar. The tonal torus is also a connected graph. If we following the "fifth spiral" around the torus, it will take us through each and every vertex just once--this is known as a "Hamilton walk," as it is a single path through all the the vertices of a graph. Better known is the "Euler walk" through a connected graph: a path which goes through each and every edge just once. Because the tonal torus is a connected regular graph, in that every vertex is of the same degree, and since that degree is even (in our case, 6, since every vertex connects two major thirds, two minor thirds, and two perfect fifths), the tonal torus also has an Euler walk.

4. Mapping Metric Space

How then to map temporal relationships? An obvious way to start would be to try and construct a temporal analog of a Riemannian tonnetz, a "zeitnetz,":

This is a partial representation of such a network, in which each vertex represents a particular periodicity, and each connects to four other vertices. So, for example, the vertex labeled "12" connects to two larger periodicities (24 and 36) as well as two smaller periodicities (4 and 6). Duple relationships are mapped on the diagonals that extend downward from left to right, while triple relationships are mapped on the diagonals that extend upward from left to right. Rick Cohn has developed a similar representation, what he has called "ski hill diagrams" for relationships amongst metric periodicities.

As nifty as this zeitnetz is, I do not feel it is the correct representation of metrical relationships. For one thing, certain levels--the measure and especially the tactus--are more essential than others. If a layer of subdivision or hypermeter should drop out, we still have metric and temporal continuity, whereas if the beat disappears, there is a palpable lack of motion. Thus the beat or tactus serves as the fundamental substrate for any metric system. As a result, I have proposed the following mapping of metrical relationships.

Here we have a configuration of vertices and edges. Each vertex in this graph represents a potential metrical periodicity, with the beat level (for convenience, represented by a quarter-note) serving as the origin for the space. This reflects the centrality of the beat level. Each vertex in this graph represents a level of periodic articulations linked to higher and lower levels in terms of either 2:1 or 3:1 ratios. The horizontal edges represent binary relationships, while the vertical edges represent ternary relationships. Vertices which involve concatenations of beats and larger units are measures and hypermeasures (the upper two quadrants), and vertices which involve fractions of the central beat are subdivisions (the lower two quadrants). Notice that the upper two quadrants contain most of standard time signatures in western music.

In graphic terms, we have a rooted tree whose central vertex is of degree four, with all other vertices are (at least in principle) of degree three--the gray vertices are used to indicate overlapping hierarchic configurations (for example, 6/2 vs. 12/4) which have identical periodicities, a bit of fudging in order to preserve the "vertical = ternary" and "horizontal = binary" relationships for the edges. Nonetheless one can draw this tree without edge crossings, and hence it is planar.

While we may imagine a graph of metrical relationships that is topologically open, extending ad-infinitum (i.e., like PC space under just intonation), we are able to perceive only a small region of its extended terrain. Just as our perception of octave equivalence determines the shape of PC space, so too do our perceptual and cognitive capacities influence the shape and extent of this "M-space" of metric possibilities. We can only attend to durational events within a certain range, an envelope of about 100 milliseconds (for the shortest notes that can be counted) upwards to about six seconds (for the largest metric units with meaningful beat relationships).

At a tempo of 92 beats/minute (a 650ms duration) for the central beat, one can see that not all periodicities will lie the "metric envelope" of our perceptual and cognitive capacities. On this example I have written temporal contour lines which mark the edges of metric envelope. Since we are now considering a metric system with respect to a particular tempo, I have called this a graph of "meter-tempo space," or "M/T space." Notice how our perceptual limitations truncate various branches of the graph, so that we now have a graph with a central vertex of degree four, intermediate vertices of degree three, and terminal vertices of degree one.

This last figure shows a number of graphs of M/T space, with vertices outside of the 100ms and 6 sec. boundaries omitted. As one would expect, at slower tempos there are limits to the number of levels above the beat (as they simply become too long), while, conversely, at faster tempos there are limits on the extent of metric subdivision (as the subdivisions simply get too short). Each graph is yoked to a beat rate that is typical of a particular tempo, and so the six graphs illustrate the metric differences between these six distinct tempo categories.

What is perhaps most interesting is that the number of possible meters in each graph of M/T space is not constant, nor is there a simple linear relation between the number of possible meters in any graph of M/T space and tempo:

Beat Rate (MM/ms)	# of nodes within the metric envelope	MxSD-1
40 (1500)	12	3x10-1 = 29
50 (1200)	13	4x10-1 = 39
60 (1000)	11	4x8-1 = 31
72 (833)	11	6x6-1 = 35
80 (750)	11	6x6-1 = 35
86 (700)	11	6x6-1 = 35
92 (650)	10	5x6-1 = 20
100 (600)	12	7x6-1 = 41
108 (555)	11	8x4-1 = 31
120 (500)	11	8x4-1 = 31
140 (428)	13	10x4-1 = 39
160 (375)	13	10x4-1 = 39
180 (333)	12	10x3-1 = 29
200 (300)	12	10x3-1 = 29

The left-hand column gives the beat rate in the musically-familiar term of beats per minute. The next column lists number of vertices present, and this varies only between 13-15. In order to see how tempo changes really effect the extent of M/T space, one must take note of the multiplicative relationships between vertices above versus below the central beat, as indicated in the "M times SD minus ONE" column. Note here the wider variation, from 29 to 41 metric vertices at various tempos. At different tempos, there are different numbers of metric possibilities--and why this is noteworthy will become clear in a moment, if it is not clear already.

5. Conclusion: M/T Space and Various Tonnetze are Non-Isomorphic

It is hoped at this point that the principal argument of this paper should be fairly obvious to the reader: metric space is planar, tonal space is non-planar; therefore the two spaces are non-isomorphic. And if the two spaces are non-isomorphic, then there are fundamental problems in trying to map elements or relationships (i.e., functions which employ those elements) from one space to another.

Differences beyond planarity versus non-planarity may also be discerned: the torroidal tonnetz is regular, with all vertices of the same degree; the graph of M/T space involves vertices of different degrees. The many vertices of degree 1 in M/T space create a large number of "dead ends" in the graph, and so it is not possible to have either an Euler or a Hamilton walk through M/T space. M/T space is a rooted tree, and even if we acknowledge Lerdahl's concerns regarding the role of a tonic in generating a tonal space, the root of a rooted tree is not the same as a tonic in tonal space. That one particular vertex serves as the origin of a tonnetz does not change the uniformity of the structure of the network itself (and indeed, various marvelous effects of chromatic harmony depend on this to move smoothly to "distant" chords and keys). By contrast the choice of an initial tactus has a dramatic impact on the shape and extent of the M/T space.

Finally, changing the tempo changes both the shape and extent of M/T space. Changing one's initial tonic does not alter the number of vertices or edges in the graph of tonal space--wherever you start, there are always the same number of tonal possibilities, the same number and kind of pathways to other chords or pitch-class complexes (i.e., all edge-relationships remain constant). Similarly, changing one's initial tonic does not change the degree of any other vertex in the tonnetz (i.e., all vertex relationships remain constant). In contrast, changing tempos does change the number of edges and vertices in the graph of tonal space, as well as the relationships amongst them, for example, as higher levels of subdivision that are vertices of degree 3 at slower tempos lose edges (and hence change degree) as the tempo increases.

From the outset Hyer reminds us that tonnetze--both his and Riemann's--are tonal representations, in that they are translation from that which is heard and remembered to that which is seen. This rings true; my account of the differences between various representations of tonality and meter stands or falls on the extent to which they truthfully represent relationships between pitch and time. But the use of representations--whether in musical notation, pictures, or words--is unavoidable in music theory and analysis. What is gained in this exercise is that by trying to follow the same "rules" in constructing graphic representations of tonal and metric relationships, we are forced to confront the differences between them. We also are reminded of how the topologies of both M/T space and PC space--the metric tree and the tonal tonnetz--arise from the combination of formal relationships among their component elements as well as the way human beings hear and understand those relationships. As in all of our musical representations, what we can hear and what we can imagine are intertwined and interdependent.

Babbitt, Milton. [1962] 1972. "Twelve Tone Rhythmic Structure and the Electronic Medium." In Perspectives on Contemporary Music Theory, ed. B. Boretz and E. T. Cone. New York: W.W. Norton, 148-79.

Berz, William L. 1995. "Working Memory in Music: A Theoretical Model." Music Perception 12(3): 353-364.

Boulez, Pierre. 1971. Thoughts on Music Today, trans. S. Bradshaw and R. R. Bennett, Harvard University Press.

Bregman, Albert. 1990. Auditory Scene Analysis. Cambridge: MIT Press.

Brower, Candace. 1993. "Memory and the Perception of Rhythm." Music Theory Spectrum 15(1): 19-35.

Clarke, Eric F. 2000. "Categorical Rhythm Perception and Event Perception." In Proceedings of the Sixth International Conference on Music Perception and Cognition, ed. Woods, C., Luck, G.B., Brochard, R., O'Neill, S. A., and Sloboda, J. A. Keele, Staffordshire, UK: Department of Psychology. CD-ROM.

Clarke, Eric F. and Windsor, Luke W. 1992. "Dynamic Information for Metre." In Proceedings of the Fourth Workshop on Rhythm Perception and Production, Bourges: 105-110.

Cohn, Richard. 1997. "Neo-Riemannian Operators, Parsimonious Trichords, and Their Tonnetz Representations." The Journal of Music Theory, 41(1): 1-66.

Cohn, Richard. 1998. "Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective." The Journal of Music Theory, 42(2): 167-80.

Fraisse, Paul. 1987. "Rhythm and Tempo." in The Psychology of Music, ed. Diana Deutsch. New York: Academic Press, 149-180.

Gabrielsson, Alf, Bengtsson, Ingmar, and Gabrielsson, Barbro. 1983. "Performance of Musical Rhythm in 3/4 and 6/8 Meter." Scandinavian Journal of Psychology 24: 193-213.

Hirsh, Hirsh, Ira J., Monohan, Caroline B., Grant, Ken W., and Singh, Punita G. 1990. "Studies in Auditory Timing: 1. Simple Patterns." Perception and Psychophysics 47(3): 215-26

Hyer, Brian. 1995. "Re-Imagining Riemann." Journal of Music Theory 39(1): 101-38.

Jones, Mari Riess and Marilyn Boltz. 1989. "Dynamic Attending and Responses to Time." Psychological Review 96(3): 459-91.

Jones, Mari Riess. 1992. "Attending to Musical Events." In Cognitive Bases of Musical Communication, ed. Mari Riess Jones and Susan Holleran. Washington: American Psychological Association, 91-110.

Krebs, Harald. 1987. "Some Extensions of the Concepts of Metrical Consonance and Dissonance." Journal of Music Theory 31(1): 99-120.

Krebs, Harald. 1997. "Robert Schumann's Metrical Revisions." Music Theory Spectrum 19(1): 35-54.

Krumhansl, Carol L., and Kessler, E. J. 1982. "Tracing the Dynamic Changes in Perceived Tonal Organization in a Spatial Representation of Musical Keys." Psychological Review 89(4): 334-68.

Lerdahl, Fred. 1988. "Tonal Pitch Space." Music Perception 5(3): 315-50.

Lewin, David. 1981. "On harmony and meter in Brahms's op. 76, no. 8." 19th-Century Music 4(3): 261-65.

Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press.

Longuet-Higgins, Christopher. 1962. "Letters to a Musical Friend," The Music Review 23: 244-48 and 271-280.

Messiaen, Oliver. 1956. The Technique of my Musical Language, 2 vols., trans. J. Satterfield, Paris, Alphonse Leduc.

Miller, Benjamin O., Scarborough, Don L., and Jacqueline Jones. 1992. "On the Perception of Meter." in Understanding Music with AI: Perspectives on Music Cognition, ed. M. Balaban, K. Ebcioglu, and O. Laske. Cambridge: MIT Press, 428-47.

Parncutt, Richard. 1994. "A Perceptual Model of Pulse Salience and Metrical Accent in Musical Rhythms." Music Perception 11(4): 409-464.

Pressing, Jeff. 1983. "Cognitive Isomorphisms Between Pitch and Rhythm in World Musics: West Africa, the Balkans, and Western Tonality." Studies in Music (Australia): 38-61.

Roeder, John. 1995. "A Calculus of Accent." Journal of Music Theory 39(1): 1-46.

Saslaw, Janna. 1996. "Forces, Containers, and Paths: The Role of Body-Derived Image Schemas in the Conceptualization of Music." Journal of Music Theory 40(2): 217-243.

Stockhausen, Karlheinz. 1959. " . . . how time passes . . . " Die Reihe, vol. 3, "Musical Craftsmanship," trans. C. Cardew. Bryn Mawr: Theodore Presser Co.

Trudeau, Richard. 1993. Introduction to Graph Theory. New York: Dover Publications.

West, Douglas B. 1996. Introduction to Graph Theory. Upper Saddle River: Prentice-Hall.

Windsor, Luke W. 1993. "Dynamic Accents and the Categorical Perception of Metre," Psychology of Music 21: 127-140.

Yeston, Maury. 1976. The Stratification of Musical Rhythm. New Haven: Yale University Press.

Zbikowski, Larry. 1998. "Metaphor and Music Theory: Reflections from Cognitive Science." Music Theory Online 4(1).