1. How to picture meter

In trying to understand musical phenomena, and in trying to understand how we understand those phenomena, our musical descriptions are centrally important. For the words we choose when we talk about music, and the pictures we draw of it, frame and direct our understanding. To the extent to which these verbal and pictorial representations are "true" or "veridical"--whatever true or veridical may really mean--we may come closer to understanding the nature of our mental representations of musical phenomena. We may also examine similarities and differences amongst these representations in order to get a better grasp of similarities and differences in the music itself.

Traditional western musical notation is a continuous graph of pitch and time, with pitch on the vertical axis, and time on the horizontal. Thus each musical score represents these primary musical parameters as a long ribbon, with recurring patterns of pitch and duration rendered in similar orthographies. What is implicit, but never actually notated in our familiar system of staves, rests, noteheads, stems, and beams, is meter. Now to be sure, there is usually a time-signature which gives the performer an indication of how durational patterns are to be interpreted (at least initially). But what is indicated within (and sometimes across) each bar is a pattern of durations, not the hierarchical arrangement of temporal locations that is the essence of meter.

Among music theorists a number of representations for meter have been developed. Perhaps most well known to researchers in music perception and cognition, is the "dot notation" for meter developed by Lerdahl and Jackendoff (1983; their notation is drawn from the work of Arthur Komar, 1971), which aligns a grid of time points below the staff, such that each and every musical articulation aligns with at least one dot. Higher levels of metric structure align with lower levels, and metrical accent is product of hierarchical relationships between dots on one level and their alignment with a dot or dots on higher level(s). Like other metric representations, Lerdahl and Jackendoff's metric grid is yoked to the traditional notation for pitch and time, and so it too takes the form of a long ribbon which represents the unfolding of a hierarchy of metric time points.

I like Lerdahl and Jackendoff's representation, and one can learn much about music and our musical understanding from it. But it seems to miss a crucial aspect of meter, and that is its cyclicity. Meter, by just about anyone's definition, is a recurring pattern of time, one we infer from the musical surface and then project forward in time, a hierarchically-structured anticipation of future musical events. One important aspect of meter is its function as a timekeeper, a clock, and one of the best ways to represent clock-like processes is with a circle.

2. Cyclical Representations of Meter

Here is a very simple meter, a cycle of four isochronous beats:

One reads these cycles by moving through them in a clockwise fashion, starting from the "12 O'clock" position. Most metric patterns are more interesting than this, in that they involve a hierarchy of coordinated time-cycles created by temporal connections between "non-adjacent" events on the "outer rim" of metric diagram:

The outer rim represents the basic cycle of a meter; it represents the lowest/shortest/fastest level of the metric hierarchy--not beats, typically, but beat subdivisions. The interior line-segments represent higher levels of metric structure. The meter in this example is based on an 8-cycle, and the various pathways within the cycle correspond to different levels of the metric hierarchy: the outer level defines the cycle, the next defines the beats, the next the half-bar level, and the red loop the measure itself.

Metric well-formedness may be expressed in terms of the following rules for constructing cyclical representations:

Given a basic cycle of N elements, additional levels may be constructed, provided:

(a) each line segment connects non-adjacent time-points on the cycle (with the exception of (d) noted below);

(b) each and every series of segments that represents a metric level must start and end at the same location (for convenience, notated here as the "12 O'clock" position), forming a sub-cycle;

(c) no crossing of line segments is permitted;

(d) the highest level of metric structure is represented by a loop to and from the cardinal metric position.

Note in the previous example each level follows these rules recursively--what were non-adjacent points on the basic cycle become adjacent on the "first interior sub-cycle" of the diagram. N.B., Since my main interest is the relationship between the basic cycle and the beat level of the measure, I will omit higher levels of metric structure in my subsequent graphs. Also, to avoid un-necessary clutter, I will omit the directional arrows, as one may assume all basic cycles and sub-cycles involve clockwise directed motion.

These local constraints on metric well-formedness capture some basic aspects of metric structure. One is that higher levels of meter are usually comprised of two or three elements from the level underneath. Another is that one point in the measure--the downbeat--is of cardinal importance in the alignment and coordination of metric processes. What they do not capture is a global constraint on the spacing of higher-level articulations relative to the basic cycle, the principal of maximal evenness. Maximal evenness is a concept developed in the study of pitch-class sets (Clough and Douthett, 1991). A maximally-even pattern is one in which a subset of M-elements are spaced "as far apart as possible" on the circle that represents their N-element superset. We may therefore note:

(a) The basic cycle itself is, by definition, maximally even

(b) Regular meters are, by definition, maximally even, since each beat is comprised of the same number and kind of sub-division units.

(c) Complex meters are also maximally even

While the first two points are not remarkable, why should complex meters tend toward maximal evenness? On a cyclical representation of a metric pattern, the fundamental constraint on the formation of its metric hierarchy is the number of time-points in the basic cycle. This number determines the various configurations that are possible within it, and so one can represent cyclically-defined set of meters in terms of the different interior patterns it may contain.

Consider a 9-cycle. According to the metric well-formedness rules given above, it may contain the following three and four-beat sub-cycles: (a) a pattern of three evenly-spaced beats (familiarly, 9/8), or (b) a complex four-beat pattern, short-short-short-long (2+2+2+3):

As an aside, note in the case of the complex meter, that the location of the long relative to the downbeat may shift, but the SSSL series remains unaffected, since the L always loops back to the first S. Why not, however, have a sub-cycle of 2+3+4 (?):

A common argument against this configuration is that the segment which spans four articulations of the basic cycle "naturally" devolves to 2+2, since the "duple" unit tends to persist in the listener's perception and anticipation. This is a level-specific rationale. Maximal evenness provides a global rationale, one that assumes that the listener will gravitate towards the most parsimonious attending strategy. The simplest attentional frameworks are comprised of categorically-equivalent spans on each and every metric level. As a metric pattern, the 2-3-4 sub-cycle involves three categorically different time intervals, a short, a medium, and a long. However, this pattern of time intervals does align with the maximally-even 2-3-(2-2) sub-cycle, and this pattern involves only two categorically distinct time-intervals. Thus I would conjecture that while a four-beat pattern is nominally more complex than a three-beat pattern, because the 2-3-2-2 pattern is both maximally even and involves fewer distinct durational categories, it is in fact the preferable attending strategy. This is an instance of a general problem: in inferring a meter from a durational surface, when are long surface durations "split" into shorter sub-articulations when metrically interpreted? An answer may be: whenever splitting a long duration preserves maximal evenness.

This brief examination of the metric possibilities of the 9-cycle is but one instance of how cyclical representations allow one to examine the formal properties of various metric systems. One may also, for instance, consider the differences in N-element basic cycles when N is a prime versus non-prime number. Similarly, an examination of the 12-cycle shows that contains a great number of sub-cyclic configurations, from symmetrical 3, 4, and 6 beat patterns to a wide variety of complex meters. This perhaps explains its cross-cultural ubiquity, as it is so rife with metric possibilities.

3. Cyclical Representations of Meter and Cognitive Constraints

The cyclical representations of meter presented thus far involve both elements in a metric system as well as connections or pathways between them. In addition to well-formedness rules for metric cycles given above, one may also add some perceptual and cognitive constraints. Each line segment which connects a pair of metric elements can be assigned a time value. Given what we know about the maximum and minimum time-intervals between metric elements (i.e., the shortest "countable" time interval is about 100ms, and the longest is about 5 seconds), and that we have a strong preference for events in the 500 to 900ms range, we can thus specify the following requirements on the "length" of the segments which comprise metric cycles and sub-cycles:

(a) a minimum length requirement: elements of the basic cycle must be spaced at least Å100ms apart (Roederer 1995; Hirsh et. al. 1990);

(b) a maximum length constrain: the cumulative time span for basic cycle is Å5 sec. (Fraisse 1982; Berz 1995).

These two requirements work in tandem, since, for example, as the basic interval on an 8-cycle approaches 650ms, the cumulative time-span begins to exceed the 5 second limit. Conversely, if a metric cycle has, for example, a cumulative time-span of about 3 seconds, and the shortest possible interval is Å100ms, then there is a maximum of 30-32 elements which might appear as part of a metric cycle (N.B. even if these maxima and minima change as the result of further experimental research, the reciprocal relationships between minimum time intervals on the basic cycle, the maximum possible number of elements in a basic cycle, and the cumulative duration should still hold). We may also stipulate that:

(c) if possible, a metric system should contain a sub-cycle comprised of segments which fall in the 500-900ms range (Parncutt 1994; Duke, et. al 1991).

Consider a complex 3-beat pattern within an 8-cycle:

Here the beats follow a 3-3-2 pattern of basic cycle elements; the ratio of the long beat to the short beat is, as is typical, 3:2. Since we prefer a meter in which all beat-level periodicities fall in the range of maximal pulse salience, we can note the following effect of tempo changes on this pattern:

Long Beat Interval	Short Beat Interval	Basic Cycle Interval
600ms	400ms	200ms
750ms	500ms	250ms
825ms	550ms	275ms
900ms	600ms	300ms
1050ms	700ms	350ms
1200ms	800ms	400ms

Only when the interval of the basic cycle falls within the 250-300ms range do both the long and short beats fall within or near the range of maximal pulse salience. This suggests that complex meters may be more sensitive to tempo constraints than simple meters, and that tempo constraints may play a role in limiting the range of possible sub-cycles when the basic cycle itself is made up of many (that is, more than 16) elements.

4. Expressive Variations and Cyclical Representations of Meter

Since a metric cycle can represent not just the formal relationship between metric events and levels, but also actual timing values, we may incorporate expressive variation into our specifications for timing relationships, either in the form of decomposable variations from an isochronous norm, or in terms of absolute timing values from empirical data. For real-world musical performances (and the metric attending behaviors they engender) do not involve isochronous durations. Here is an example based on timing data collected by Repp (1998b):

This diagram is for the second measure of Chopin's E-major etude, taken from the grand average of timings from 27 performances by nine different pianists (see Repp 1998a, p. 268). As can be seen, neither the basic cycle nor the 4-beat sub-cycle involves isochronous time intervals. Indeed, the 4 beat sub-cycle bears more than a little resemblance to the 4-in-9 sub-cycle given above, though of course here the last beat is considerably shorter than the others, whereas in the 4-in-9 sub-cycle, the last beat is considerably longer. Notice also the range of time intervals that occurs here on the basic cycle itself, from a minimum 436ms to a maximum of 617ms.

What is going on in this measure? In a word, rubato. The story goes something like this: In the first part of the measure we had a slight bit of expressive timing variation, with the 2nd half of each beat being stretched by about 40-50ms; in this fashion, the first two beats follow a predictable pattern (more on this in a moment). On the third beat, however, we have a more dramatic bit of rubato (corresponding with the onset of a sustained tone in the melody), as the first half of beat 3 is Å100ms longer than the first half of beats 1 and 2. Now the pianist must regain the time s/he has taken, and the remaining time-intervals make up the "stolen time"; as such they are correspondingly short. Notice, however, that even under this constraint the second half of the last beat has a discernible stretch relative to its first half.

Here is where a bit of mathematical graph theory may be of use. Let us suppose that the rubato on the third beat of this measure had not occurred. In that case, the timing data might have looked something like this:

Here we see a pattern on the basic cycle of a regular alternation of slightly shorter--slightly longer time intervals, what I have labeled T1 and T2. Notice that each "odd" location (the filled dots) on the graph is symmetrically positioned, as each sub-cycle re-integrates to a constant timing value. In graph theory, cycles with such symmetrical properties are reducible to more compact graphic representations, what are referred to as voltage graphs (a term obviously borrowed from their usage in electrical engineering). The right-hand panel above shows the voltage graph for an 8-cycle comprised of alternating T1 and T2 values, values for the different "voltages" between the on-beat and off-beat timepoints. The voltage graph above will generate various basic cycles when "counted" according to a particular arithmatic modulus. Thus the 8-cycle is generated by the given voltage graph, modulo 4. In the case of the E-major prelude, we can specify that the ratio between T1 and T2 should be Å48:52. Other timing ratios may be specified, such as a shift from "straight" to "swung" 8th notes in a jazz performance style.

Because the rubato performance reported by Repp lacks the symmetry of the simplified version given above, one cannot reduce it to a simple voltage graph (nor can one use a voltage graph to generate its complete metric cycle). Similarly, one cannot reduce the 4-in-9 sub-cycle in terms of a corresponding voltage graph, since it too lacks the requisite symmetry. Let me add that it may be possible to make alternative graphic representations of the "rubato 8" and 4-in-9 patterns, using what are known as permutation voltages, and so it would not be correct to infer that such graphs are irreducible. But the geometric similarities between the "rubato 8" and the "4-in-9" are highly suggestive.

To the extent that a meter can be reduced to a simple voltage graph, one need not include higher-level timings as part of a structural representation, which is to say, as part of the listener's temporal attending strategy. In these instances, the higher levels "take care of themselves" as a byproduct of the cyclical generation from the underlying voltage graph. By contrast, complex meters (such as the 4-in-9) and simple meters with multi-leveled expressive variation (the rubato 8) require more levels of structure in their representation(s); the rubato 8 example shows how low-level timing changes can "trickle up" to affect higher levels of attending/anticipation. This implies that as attentional strategies, these meters require a greater interplay of top-down and bottom-up information--indeed, one cannot build up higher levels from lower levels, but must instantiate the metric hierarchy in toto (see London 1995, p. 73).

Complex meters (such as the 4-in-9) and simple meters performed with a high degree of expressive variation (such as the rubato 8) have a number of formal and cognitive similarities, from maximal evenness of events on each level to timing constraints on local and global levels. These meters are thus in many ways more alike than they are different. Given that most human musical performance involves multi-leveled expressive variation, one is led to question the validity of the simple-complex metric distinction. While one may draw this distinction in theory, in practice, metric attending is almost always complex.

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