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Categories and Timing: On the Perception of MeterAuthor(s): Tellef KvifteSource: Ethnomusicology, Vol. 51, No. 1 (Winter, 2007), pp. 64-84Published by: University of Illinois Press on behalf of Society for EthnomusicologyStable URL: http://www.jstor.org/stable/20174502 .Accessed: 07/10/2014 02:34

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Vol. 51, No. 1 Ethnomusicology Winter 2007

Categories and Timing: On the Perception of Meter

Tellef Kvifte / University of Oslo

Introduction

In

his paper "Meter and Grouping in African Music," Temperley asks the

following questions: "How well can African rhythm be reconciled with the prevailing music-theoretical view of rhythm? What similarities and differ ences emerge between African rhythm and Western rhythm, as the latter is viewed by contemporary music theory?" (2000:65). My point of departure in this article is only slightly different. Like Temperley, I want to discuss implica tions of some recent theories of rhythm and meter in the light of evidence from outside so-called Western (art) music. However, I will not primarily address African music. I will instead draw on experiences from other types of music, such as jazz and traditional Scandinavian dance music. My conclu sions will also differ from Temperley's. In the introduction to his paper on

meter and grouping inAfrica,Temperley states "To anticipate my conclusions, I will argue that, at a fundamental level, African rhythm as described by eth

nomusicologists is similar to Western rhythm and can be accommodated in the same basic model" (ibid.). I am not going to argue against this specific conclusion, but I will point to theories and empirical evidence that seem not to fit well with the models of meter referred to by Temperley.

More specifically, this paper addresses what I perceive as a fundamental conflict between aspects of what might be called "the microrhythmic para digm," and models of meter that take for granted a low(est) metric level with isochronous pulses. I will argue that the latter models are, at best, limited in scope, and of little value to describe and explain certain quite common observable rhythmic behaviors. The idea of a lowest isochronous metric level is more or less explicit in concepts such as additive rhythms and den

sity referent, as well as in more modern models, like those of Justin London

(1995, 2002, 2004). With reference to Kauffman (1980:396), I will refer to

? 2007 by the Society for Ethnomusicology

64


Kvifte: Perception of Meter 65

such theories as theories of a Common Fast Pulse (CFP), using "pulse" instead of "beat" to avoid confusing them with the B(eat) level in the metric models under discussion. Finally, I will suggest that it is convenient to distinguish more clearly than is usual between models of metric categories and models of metric timing.

The problem The basic premise for research into microrhythmic phenomena is the

belief that we do not play any rhythm exactly as it is, or could be, written in standard notation. Further, deviations from the mathematically simple and exact values of notation fall into two categories, namely random variations due to imperfections on the part of the performers or the equipment, and

systematic variations that are important as part of the style, and/or important for the rhythmic feeling.

In such research, reference may be made to different metric levels. Studies

may concern, for example, the level of subdivision of beats, (e.g., Pr?gler 1995 ; Benadon 2006) or the beat and bar or measure level in studies of so-called

asymmetric rhythms in Scandinavian traditional music done by Bengtsson (1974),Groven (1971), and Kvifte (1999). The patterns studied concern not

only expressive variations of rhythmic figures played on a background of a

regular metric framework, but also the metric framework itself. One hesitates to use terms such as "deviation" or "variation" to describe these patterns as the patterns that are described are not perceived as necessarily deviating from a norm. Rather it is the norm itself that one tries to describe. Bengtsson puts it like this: "In fact, we should avoid calling it'deviations'when dealing with rhythm without stating clearly that we just mean deviations from a mechanical norm that we use as a sort of temporal ruler. We have no other ruler, mainly because we know far too little about such micro-structures" (1987:78, emphasis in original).

In some cases, therefore, what is implied is a metric grid where the rela

tionship between units on different levels does not conform to simple ratios between integers. Such an idea of a metric grid that is not regular, however, is

contrary to some prominent positions within rhythm research, and I will outline two of these and discuss some implications of the different positions.

A Few Remarks on Terminology

This study requires a note of clarification concerning terminology. The two central terms, "meter" and "rhythm," are used in many different ways, and there are many definitions in the literature. One of the more recent that I agree with is London's version from 2002:


66 Ethnomusicology, Winter 2007

Meter is defined as a stable and recurring pattern of hierarchically structured

temporal expectations. Metrical patterns, although related to the pattern of in

teronset intervals1 present in the musical surface, are distinct from that pattern.

(2002:529)

I also subscribe to his description of rhythm as "a coordinated and con nected temporal pattern," and will add that such patterns are usually to be understood relative to a meter, as is implied in his observation that "meter is also viewed, at least by most music theorists and psychologists, as being distinct from rhythm, where rhythm involves the phenomenal pattern of durations (more precisely, interonset intervals or'IOIs') and dynamic accents. It is acknowledged that the same melodic pattern may be heard in a number of different metric contexts" (London 2002:531). Such a clear-cut distinction between meter and rhythm is not shared by everyone. Clayton discussed the

question at length, addressing the question of the possible universal useful ness of these concepts (2000:33-35).

Also important for the definition of meter is that it consists of at least two, and usually more, levels. We will normally assume the levels of beats, subdivisions, and measures (abbreviated B, SD and M), where the beat level

commonly has a privileged position and is sometimes called a referent level or tactus.

Common Fast Pulse Theories

The widely discussed concept pair, additive/divisive rhythms, may go back to Curt Sachs' book Rhythm and Tempo (1953). But, as London wrote for the Grove's online dictionary:

Few terms are as confusing or used as confusedly as "additive" and "divisive".

Additive rhythms are said to be produced by the concatenation of a series of

units, such as a rhythm in 5/8 which is produced by the regular alternation of

(2/8 + 3/8). Divisive (or, more often, multiplicative) rhythms are produced by

multiplying some integer unit such that a measure of 2/4 is equal to 2 x 2/8. In

addition, additive is associated with asymmetrical rhythms, while divisive rhythms are often assumed to be symmetrical. (London 2005)

Adding to the confusion is the use of the term "rhythm" in this context. The way I understand this (and several other) definitions and descriptions of the concepts of additive and divisive, is that what is implied is really "meter" as I use the term in this article, and not "rhythm."

Examples of additive meters frequently include 5/8?either as 2+3 or

3+2,7/8 as 2+2+3 or 3+2+2 and 9/8 as 2+2+2+3. Divisive meters are typically shown as 2/4,4/4 or 3/4. With such examples, the distinction additive/divisive

may look quite obvious. But the mathematical construction of the additive and divisive meters is not the whole matter. There also seems to be consensus in



the literature that the difference is a question of perception. To perceive a

rhythm as additive is fundamentally different from perceiving it as divisive. Berger, for instance, says explicitly, "It is important here to realize that odd versus even meters must not be conflated with additive versus divisive time"

(1997:485). The point is that it is possible to perceive a given musical sound in both ways, with distinctly different musical experiences. One example, that Sachs himself used, is a pattern of 3+3+2 (1953:90). Perceived in a divisive

context, this pattern may be described as a syncopated pattern in a four beat bar. But in an additive context, each of the pulses is to be perceived as the actual beat of a three-beat bar, where the beats have different duration, as shown in Figure 1. The basis of the difference may be that in a divisive context one perceives the beats as the basis, and then finds the subdivisions

by dividing the beat. In the additive context, the subdivisions are basic and added together to form beats.

Relevant in this context is the concept of "complex meters" as described

by London: "Describing a musical passage as 'metric' usually implies that one can hear in it an isochronous series of beats and that these beats are

hierarchically structured. In some cases, however, one cannot infer a wholly isochronous metric structure from the durations present on the musical surface. In particular, there may be some meters where the beat level of the metric hierarchy consists of a nonisochronous series of durations; these cases are referred to as complex meters"2 (1995:59). In plain language, in a

complex meter the beats do not have the same length. This definition does not in itself presuppose anything in the direction of how the meter is expe rienced, specifically not whether it is the bar, beat or subdivision level that is at the center of the metric experience.

London uses the following excerpt (Figure 2) from Bernstein's melody "America" as an example. Instead of interpreting the perceived meter as chang

Figure 1. A 3+3+2 pattern perceived in a divisive (upper) and additive (lower) context.

r r "r pr r

r r T r r

Divisive

r p

Additive

3+3+2 r



Figure 2. From "America" by Leonard Bernstein, used by London (1995) as an example of a complex meter.

ing between 6/8 and 3/4, he suggests that one should allow for beats not being isochronous, and describes the pattern as a "five-beat metric structure: two

long beats followed by three shorter beats (L-L-S-S-S)" (London 1995:66). But in London's context, such meters pose a problem: "A series of non

isochronous B[eat]s poses a special problem because I have posited that the B level functions as the reference level in the construction of the metric

hierarchy. How can a nonisochronous level function as the temporal 'anchor' for the construction of other levels of the metric hierarchy? The answer is that it can do so via the stabilizing presence of a level of isochronous

S[ub]D[ivision]s" (ibid.). London goes on to write that "in this musical con text the listener will make an extra effort to maintain the SD level in order to assist with her/his comprehension of the B level," or, in other words, the SDs "provide an isochronous baseline upon which nonisochronous levels of the metric hierarchy may be constructed" (ibid.:67).

A strikingly similar idea is found in African rhythm discussions in the 1970s and 1980s. One important focus was how to understand what was

diversely called multimeter, polyrhythms or, as in Kauffman's paper, "African

Rhythm?a Reassessment," multi-rhythms: "One of the more widely accepted theories of African music is that multi-rhythms can be reconciled by relating them to a common fast beat." Under the heading,"The Theory of a Common Fast Beat," he considers both Waterman's concept of "metronomic sense,"

and Hood's "density referent" (Kauffman 1980:396). Richard Waterman wrote that African multi-rhythm was "structured along a theoretical framework of beats regularly spaced in time and of cooperation in terms of overt or inhib ited motor behavior with the pulses of the metric pattern whether or not the beats are expressed in actual melodic or percussion tones" (Waterman 1952:211-12, as quoted in Kauffman 1980:396). Kauffman continues: "Wa terman believes that these beats provide a 'metronome sense' that operates behind the music" (ibid.).

A similar idea is represented in Mantle Hood's concept of "Density Ref erent": "What was the fastest pulse in the piece, discounting momentary doubling or tripling characteristic of rhythmic ornamentation? Although no one could say what the slowest pulse of a piece might be, everyone agreed that each piece has a fastest pulse. This measuring device was dubbed the

Density Referent" (1982:114).



Lerdahl and Jackendoff's theories are explicitly formulated to be relevant for tonal western music, and they point out that the rules they describe may not be universally applicable in all details. In connection with meter, they give four rules of well-formedness, and indicate that the fourth rule may be

subject to variations in other styles of music. According to this rule, "The tactus and immediately larger metrical levels must consist of beats equally spaced throughout the piece. At subtactus metrical levels, weak beats must be equally spaced between the surrounding strong beats... [This] makes the tactus the minimal metrical level that is required to be continuous (though there is nothing prohibiting smaller levels from being continuous too). It also permits the tactus to be subdivided into threes at one point and twos at another, as long as particular beats of the tactus are evenly subdivided"

(Lerdahl and Jackendoff 1983:72). As it stands, this rule asserts that beats must be evenly spaced. But the

rule does not imply that Lerdahl and Jackendoff are of the opinion that there should be an isochronous lowest level. They say explicitly that the beats may be subdivided in different ways at different points in the music. Further, they are also open to meters where the beats actually have different durations.

They use Singer's paper on "The Metrical Structure of Macedonian Dance"

(Singer 1974), as an example on what kind(s) of variation to the rule they have in mind, indicating how her rules for generation of possible meters

might be used to form meters where the beats are of two kinds with a 3:2 ratio. Singer's observation that "the fact that the absolute duration of the units often does not make a perfect 2:3 ratio" (ibid.:386) is also relevant in this context, and I will return to this later.

The Problem Specified Common to descriptions of additive and complex meters, is the view

point that London formulates by asking: "How can a nonisochronous level function as the temporal 'anchor' for the construction of other levels of the

metric hierarchy? The answer is that it can do so via the stabilizing presence of a level of isochronous SDs" (1995:66). In other words, to be able to per form?or entrain3 to?a pattern of beats that do not have the same length, we

must, according to this view, keep a clock going that ticks at regular intervals, and the beats must conform to multiples of the clock units. London goes so far as to formulate this as one of his general principles of meter: "If the B level is not isochronous, then the SD level must be isochronous. Furthermore, the listener will maintain this level even when it is not present in the musical

signal in order to stabilize and track the nonisochronous patterning of the B level" (ibid. :69).



This brings up several questions. How are we able to maintain consistent

pattern of nonisochronous beats? How will we as listeners be able to entrain to such a series? If the experience of tempo is tied to the beat, how do we

perceive tempo when the beats are not all the same? The solution offered

by many authors is deceptively simple and convincing. Given a yardstick of

small, equal units, all the levels above will be easy to explain. Such a yardstick is (relatively) easy to entrain to, and if you are entrained to this level in the

hierarchy, it is almost trivial to entrain to other levels. If beats are unequal, one could assume that tempo is tied to the isochronous lower level rather than to beats. Therefore, it is not surprising to find this idea in several versions, as described above.

Discussion

But there are, in my view, several reasons why the CFP theories should not be regarded as universal theories of meter. The most obvious observa

tion, and also perhaps the most difficult to defend by hard evidence, is my own experience playing complex meters. I fully agree that counting equally spaced subdivisions was necessary for me the first time I tried to play melo dies in 5/8 or 7/8?and also for a long time thereafter. But I am equally clear that it is not possible for me to play a 7/8 tune in a musically satisfying way if I have to count the subdivisions to be able to keep the meter. To play

music in a complex meter and feel comfortable, I have to be able to feel the nonisochronous beats as basic. Instead of 1-2,1-2,1-2-3,1 will feel 1,2, long 3, etc, or Quick-Quick-Slow using Singer's (1974:386) terminology. I may of course feel the subdivisions if need be, but I don't have to, as for instance,

when a melody line articulates the beats, or even longer units, but does not articulate the subdivisional units. When I talk to fellow musicians about this, they are quite clear in their opinions that the beat level is the primary focus, and that paying too close attention to the subdivisions is detrimental to get ting the groove right. A 2+2+3 meter will therefore be described in terms of a three-beat structure, rather than as a structure with seven subdivisions.

One should also point out, as Singer did in the quote above, that the beats are not always simple sums of an even underlying pulse. Further, Blom states:

A third procedure employed by some dancers, is to depict the measure as com

posed of short vs long, or quick vs slow units. In terms of relative duration the

short / fast units are 2/3 of a long / slow... This method is largely but not entirely

adequate as far as the segmentation of time is concerned. There are dances, for

example, in which the last metrical unit of the measure is regularly elongated

by approximately half a temporal unit. (Blom 1978:4)

Following this idea, we may look at measurements made of actual sub divisions of the beat in performed music and other genres. Given a metric framework with a lowest level of fast, isochronous units, one would expect


Kvif te: Perception of Meter 71

to find subdivisions of beats that fall into very few categories of duration, conforming to a subdivision of the beats into 2,3,4, or in rare cases, up to 5 and 6.1 present one set of data for subdivisions of beats in my article "Descrip tion of Grooves and Syntax/Process Dialectics" (2004). Therein, a number of measures of a Norwegian springar (a particular dance genre) that has three isochronous4 beats are measured. The beats are typically subdivided in two or three notes. Looking at only the beats with two notes, time values

for the note pairs of 36 beats are summarized in a graph in Figure 3. Each line represents a beat with two notes, where the notes are represented by the endpoints of the line connecting them. The vertical placement of the

endpoints represents the length of the note, measured in percentage of the beat. As the sum of the two notes will always add up to 100%, a high value for the first note will produce a low value for the second, and vice versa. A note pair that divides the beat in two equal parts will thus produce a hori zontal line in the graph. The graph may indicate that the note pair values form almost a continuum between the extreme values. Also, controlling for

the possible systematic differences between the three beats of the bar does not bring any order to the chaos (Kvifte 2004:70).

Figure 3. Relative length of notes in note-pairs in a springar tune.

90

80

70

+J 60 05 0) _Q M? O 4-1 50 C

72 Ethnomusicology Winter 2007

Drawing on data from a different genre, Benadon's measurements (2006) seem to confirm this absence of a simple and clear-cut subdivision of the beat. Part of Benadon's paper describes the beat-upbeat ratio5 (abbreviated BUR) for series of eighths in solos by a number of well-known jazz musicians. His data does not seem to indicate any evidence for clear categories of subdivi sions of the beats. Specifically, the widespread belief that jazz phrasing is built on a division of the beat into three, is refuted: "To be sure, exclusively tripleted phrases do occur... though seemingly not often enough to bolster the claim that jazz eighths are fundamentally triplets" (Benadon 2006:91). Figure 4 shows BUR distribution for solos by two different performers. For the sake of comparison, the measurements shown in Figure 3 are converted to BURs, distributed on Benadon's scale, and shown in Figure 5. Here the

figures refer to percentages rather than absolute numbers.

One should also note that the music measured in the two mentioned

papers is not especially "untidy" in the sense of a marked tendency to agogic performance with a great deal of tempo stretching and compression. On the

contrary, both papers draw on genres where strict tempo is important and

valued; the Norwegian example being dance music. It would, in any case, be difficult to argue that tempo variations should be an important factor on the SD level, as long as relative, and not absolute values, are measured.

Another type of evidence may be found in a number of Scandinavian

springar and polska dialects. In these meters, the three beats are of different

length and the relative lengths of the beats do not conform to any simple multiple of common isochronous subdivision. These genres have received

Figure 4. From Benadon (2006), showing distribution of BUR values for a

large number of note-pairs for two performers. BUR scale at the bottom of

the figure.

Bill Evans

3 CZL

6

n 1

20

fi ri. PI 39 43

52 35 Dexter Gordon

6 in

2.0



Figure 5. Data from springar tune (performer Leif Rygg) shown as BUR

distribution.

15 Leif Rygg

D_

6 6

nnnn

12

n -nnn 3 3 3

D ? ? g 2.0

much attention in the Scandinavian research literature for more than 100

years (e.g.,Groven 1971;Blom 1981;Ramsten 1982;Saeta 1992;Kvifte 1999). However there is still no general consensus on how these meters are to be understood, except for the practically unanimous view that the three beats as a rule will not have a common isochronous SD level. Also noted by many, is the great variation on the relative length of the three beats; not only between different dialects of the music, but also between performers and even within

single performances. One suggested descriptive model is shown in Figure 6 where the short first beat in some types "gives" time to the long second beat within certain limits. In this model, the medium beat is always one third of the measure. However, measurements of actual performances also indicate

varieties such as short-long-long, with the three beats taking approximately

Figure 6. Variations in relative length of beat in a bar in springar as shown

in Blom (1993). Most performances are supposed to fall in the area from

3:3:3 to 3:5:4. The lowest line included for illustration of the ratios only.

h h .h

_5,5___J_5?.

1.

P P "P



25,37 and 37 percent of the measure, not at all conforming to this model. Also complicating the matter is the rich variety of possible subdivisions, as the above example indicated. One should also note that there seems to be no doubt about tempo being perceived as constant, and not affected by the different lengths of the beats. The springar genre is an example of a metric

type where neither the entrainment of a listener or the tempo-keeping of a

performer may be easily explained by a subdivisional clock pulse. In fact, a Common Fast Pulse theory is more of an added complication than an expla nation in this case because a common fast pulse can not be used to describe the observed relative lengths of the beats.

A Question of Expressive Variation?

One possible way to save the theory of a common fast beat, may be to invoke the theoretically important distinction between categories of duration and performed durations. In this perspective, the units referred to in the con struction of metric hierarchies are to be understood as durational categories, that (like any other category of any other musical parameter) are subject to variation during performance.6 Such variations may then be construed as faulty performance, as random variations or, more interesting in this connection, as a

means of expression. London, for instance, is of course aware of the problems raised by such variations, and spends a lengthy footnote on this:

A note regarding the use of the term "isochronous" here and throughout the rest

of this paper: It is, of course, well known that in performance, metric timings are rarely isochronous, but in fact are subject to a degree of expressive varia

tion?see, for example Gabrielsson (1982),Sloboda (1983), Clarke (1985,1989), Shaffer, Clarke, andTodd (1985), andTodd (1985). On the other hand, it is both necessary and desirable to distinguish between those temporal patterns that

involve more-or-less even durations and those temporal patterns that involve

categorically different durations. Therefore, when I speak of "isochronous" in

the following examples, I am using the term as a shorthand for "an underlying

representation of an idealized series of isochronous durations that are subject to expressive variation in performance." (London 1995:60)

I will not argue against this. On the contrary, I will underscore the im

portant point that we have to distinguish clearly between "those temporal patterns that involve more-or-less even durations and those temporal pat terns that involve categorically different durations." This is of course in no

opposition to microrhythmic studies, and the distinction is in fact one of the main points in one of the classic papers on microrhythmic structure

(Bengtsson 1974). Also, what has been said above on the subdivisions in

springar and jazz performances, is not in opposition to this. In both cases beats are subdivided in two or three. The confusion starts when questions of timing and questions of categories are mixed together. The point where



I disagree with London, is where he states that a nonisochronous level must be upheld by an isochronous lower level. This is a statement regarding not

only metric structure, but timing practice, and I understand this to mean that a nonisochronous level must get its timing information from a lower level clock. A similar thought seems to me to be the rationale for the concepts of both metronome sense and density referent.

I will not argue for, or against, the notion that African meters and rhythms (whatever one would take such a generalization to mean) are to be under stood in terms of a CFP theory. But I will argue against generalizing such theories to the extent that ALL metrical music is to be understood as built from a lowest isochronous level, in the sense that such a level gives stability and timing to the whole structure. This is how I read London, and also how I understand the concepts of density referent and metronome sense. Lerdahl and Jackendoff's rules imply a lowest isochronous level, but they do not tie the lowest level to timing questions. On the contrary, they seem to indicate

what they call the "tactus" level to be central in questions of timing:

However, not all these levels of metrical structure are heard as equally prominent. The listener tends to focus primarily on one (or two) intermediate level(s) in

which the beats pass by at a moderate rate. This is the level at which the con

ductor waves his baton, the listener taps his foot, and the dancer completes a

shift in weight (see Singer 1974, p. 391). Adapting the Renaissance term, we call

such a level the tactus. The regularities of metrical structure are most stringent at this level. As the listener progresses away by level from the tactus in either

direction, the acuity of his metrical perception gradually fades; correspondingly, greater liberty in metrical structure becomes possible without disrupting his sense of musical flow. Thus at small levels triplets and duplets can easily alter nate or superimpose, and at very small levels?imagine, say, a cascade of 32nd notes?metrical distinctions become academic. At large levels the patterns of

phenomenal accentuation tend to become less distinctive, blurring any potential extrapolated metrical pattern. (Lerdahl and Jackendoff 1983:21)

Alternative Explanation

If we can't use the fast isochronous clock pulse or Common Fast Pulse to explain all meters, what other possible explanations are there? Well, there is of course the Common Slow Pulse. Instead of thinking about the pulse in the same way as a clock where all time units are built from one fast reference

pulse, such as a pendulum or the oscillations of a crystal, we might hold on to a tactus level as described in the citation above, regarding the levels be low as formed by dividing the beat and the levels above by adding the beats

together. The levels below are perceived as divisive and the levels above are

perceived as additive. The premise that puts the CFP paradigm into trouble is the idea that

divisions can only be composed of equal units. If one divides a beat into



two, the two parts are supposed to be equal. And of course this is correct if one takes for granted that to divide is using the mathematical operation of division. But there is no reason why it should be so. As my children know

very well, when I divide a chocolate bar into three pieces, there is absolutely no guarantee that those three pieces will be of the same size. The reason is

simple; I do not use a calculator to divide the chocolate; I use my body. In the same way, when we, as listeners or performers of music, subdivide beats,

we do not use calculators, we use our bodies.

Seen in this way, if we have a referent level above the beats, there is no reason the beats should have not only different lengths, but also lengths that are not simple multiples of shorter subdivisions. In fact, they may have any kind of mutual relationship, as long as they add up to a unit on the level above. But in light of the following discussion where meter is seen as connected to body movements, one might wonder what kinds of constraints possible

ways of moving the body might place on subdivisions of units. If one tries to understand metric perception as the passive undertaking of

a person sitting in a chair, the idea of a fast clock pulse, produced somewhere in the nervous system, may seem a tempting explanation. But if we consider a different, but still quite common context for performance of music, namely

music played for dance, we may think in another way. The all-important factor in this connection is that it is possible to entrain your body movements to the music; that you, in your body, can experience a common movement with

the music and other possible participants. Such kinds of movements are, in

my opinion, more likely candidates for referent level units than a fast clock

pulse without any clear location in the body. One argument in favor of this view is the empirical evidence of the close relationship between observable

body movement patterns (as in dance) and metrical patterns in music. A fellow musician once remarked to me that to get the members of a

band to play in time with each other, and be in the same groove, is a question of the musicians "walking in the same way." Similar, but empirically far better

grounded arguments, are found in the dance research of Jan-Petter Blom. In some genres of traditional Scandinavian dance, there are many ways a dance

couple may move. The following quote has to do with so-called Norwegian bygdedanser, but, if taken as a general description, may be used for many other dances: "Dances are composed as a continuous sequencing of interac tion motifs and themes comprising systematic adaptations of holds, turns, distances and relative positions" (Blom 1981:306).

The point is that whatever types of interaction motifs a dance couple performs, there is an important invariant in the performance, namely the

pattern of up-and-down movements of the dancers' center of gravity. Such movements, called "librations" in Blom's terminology, are a necessary part of normal walking, and may, for the purpose of illustration, be represented by a series of straight lines as shown in Figure 7a. The vertical position of the



body's center of gravity is represented on the y-axis, and time on the x-axis. The basic unit for analysis is one down-and-up movement, a "dance beat" in Blom's terminology. Figure 7b shows a typical librational pattern for one

type of waltz. Here two dance beats, the first twice as long as the second, form a unit that is repeated throughout the dance. This pattern is a) of the same duration as the fundamental unit of the musical meter of 3/4, and b) distinctive for this kind of dance. Other dances, with a three-beat musical meter structure, have different and characteristic librational patterns. Differ ent dialects of the Norwegian springar dance may, in fact, be distinguished both on the basis of their respective librational patterns and on the basis of their specific version of the meter; their characteristic groove. Figure 8 shows librational patterns for two different varieties of the springar dance.

I believe the specificity of these observations form a convincing argu ment for body movements as central to meter. This is not just another general observation such as that heartbeats or walking steps have approximately the same frequency as fundamental beats in many kinds of music. On the contrary, Blom shows that there are specific characteristics of body movement that

correspond to characteristics of meter in the associated music.

Figure 7. Librational patterns, walking (a) and waltz (b). From Blom (1981). The axes are vertical displacement of body's center of gravity ("vertical

Space" or line S) and time (t). L and R for Left and Right foot. (L) and (R) auxiliaries; the tie is for continuity of support.

a S -

* LJ LJ LJ LJ* L - R - L - R -

r r w r r is R (L) R - L (R) L -



Figure 8. Librational patterns for two different varieties of the Norwegian

springar dance, as shown in Blom (1981).

A further observation along the same line is pertinent also to the remarks above on my practical experiences with so-called complex meters. In some Balkan dances and a variety of Norwegian gangar (a dance, like springar, belonging to the above mentioned bygdedans type) one may find a pattern where pulses are grouped 2+2+3+2+3. In a discussion that Blom and I pub lished, Blom wrote in response to some of my arguments:

However, it would be more disturbing to me if he finds it possible to explain the following without taking dance rhythm into account: The bowing figure 2+2+3+2+(2+l) in gangar is not experienced or considered by Kvifte or any other scholar to represent a musical meter. Why is this so in spite of the fact that

it represents, due to the bowshifts, a patterned distribution of stresses, while the

same grouping very well might have been the content of a Balkan 12/8 meter?7

(Blom and Kvifte 1986:515)

What is important here is the observation of the correspondence of the

normally perceived metric structure and the preferred pattern of bodily mo tion. In this case, we might find that the Balkan pattern shown in Figure 9a,

with one dance beat for each of the groups 2+2+3+2+38 effectively forms a structure of 5 dance beats, corresponding to a perceived 5-nonisochronous



Figure 9. Librational patterns for dance corresponding to a 2+2+3+2+3 musi

cal pattern; Balkan-style (above) and Norwegian gangar style (below).

beat musical meter. In the gangar, however, the same pattern will be perceived in a 4-beat structure with 4 isochronous dance beats. These may be shaped as a 2:1 down-up time relationship or, alternatively, alternating with some 1:2 groups to bring out the accents in the pattern, as shown in Figure 9b.

This argument may also be seen in the wider context of motor-mimetic and sensori-motor theories. God0y argues from a ecological point of view, with references to diverse fields such as linguistics and neurology, that our

perception of all kinds of musical events may be better understood on the

background of our practical experiences of how movements and manipula tion of objects (including musical instruments) produce sound:

I believe this points in the direction of what I would like to call a motor-mimetic

element in music perception and cognition, meaning that we mentally imitate

sound-producing actions when we listen attentively to music, or that we may

imagine actively tracing or drawing the contours of the music as it unfolds.

Although there are many unexplored aspects of such a motor-mimetic theory, I

believe there is now enough material to support this as a hypothesis as well as a

hopefully productive explanation for the crossmodal workings of music, which

may be summarized as follows: Motor-mimesis translates from musical sound

to visual images by a simulation of sound-producing actions, both of singular sounds and of more complex musical phrases and textures, forming motor

programs that re-code and help store musical sound in our minds. (Godoy 2003:318, emphasis in original)

God0y references primarily what he calls musical "objects" and how

they are understood, and does not explicitly tie his arguments to the time



dimension of meter and rhythm. A more specific argument in that direction is found inTodd where a "sensory-motor theory of rhythm, time perception and beat induction" is described:

We account for these phenomena by the interaction of sensory systems, which

represent temporal information in terms of the power spectrum of the sensory

image, and the motor system, which has certain natural frequencies. The central

assumption is that beat induction is not a passive process but, rather, a form of

sensory-guided action involving all the sensory and motor components that that

entails, i.e., major portions of the nervous system and the musculoskeletal system which it controls. Even if the musculoskeletal system.. .is not activated, i.e., there

is no motor output, the higher supraspinal levels of the system ("the controller") are. To be more specific, the principal agent mediating beat induction is the in

ternal representation of the musculoskeletal system and its dynamic properties which the "controller" requires as a "feedforward" model. (Todd 1999:5)

Connecting body movements along the lines described above, we do not need isochronous units on the lowest level in the meter, and meters like those described for springar above do not need to be explained away as "atypical" or "irregular," and may not pose great theoretical problems. On the other hand, we lose the neat descriptions offered by the CFP theories in terms of categories on different levels of a metric hierarchy.

The idea put forward here is quite simple. One should distinguish be tween categorical models and timing pattern models. The organization of durational categories may need different descriptions than the mechanisms of timing that are at work. The categorical models may very well be like the ones described by London or Lerdahl and Jackendoff, but will not need to

point out primary, referent or tactus levels. A timing model, however, will have to define such a primary level that may, or may not, coincide with what is perceived as a tactus level in the sense used by Lerdahl and Jackendoff. From this level, one can imagine two distinct processes, one additive, where isochronous pulses are added together to form higher-level units, and one divisive that splits a given level into smaller units that do not have to conform to simple mathematical fractions of the unit above. As data from springar performances indicate, such divisive processes may continue at least two levels down (e.g., affect both B and SD levels).

The distinctive feature of this perspective is that we do not have to relegate all deviations from mathematical ratios to "expressiveness," but are free to in clude any kind of subdivisions into models of metrical patterns. The question of how to represent such patterns in standard notation, however, remains un

solved, as does the whole question of a suitable description of such patterns. The perspective of a timing model puts the question of durational cat

egories in a different light. Under the CFP paradigm, the data concerning the



SD level in the springar and jazz data referred to above are difficult to explain. They do not seem to indicate any clear categories on the SD levels as would be expected if the durations are to be constructed from building blocks of fixed length. Using the Common Slow Pulse model, any subdivision of the beat would be possible, and the data pose no problems here. This is not to

say that the question of durational categories below the B level is without interest. Even if the actual time values for units on this level vary significantly,

we still have no trouble in assigning durational categories such as, for example, "equal-equal","long-short" or "short-long" to musical events splitting a beat in two. But the possible variation in actual duration may be much larger than if the units would also have to serve as a base for timing on levels above.

Instead of postulating a number of possible categories from the available units on lower levels in a categorical model, a Common Slow Pulse model

will, to a larger degree than the CFP model, encourage empirical studies of actual performances, of live musicians'and listeners'perception of the music in question, to unveil the categories at work. To what extent such categories are possible to infer from measurements of the musical sound alone, is an

open question, and, in my view, not to be taken for granted. The Common Slow Pulse model also draws attention to another aspect of the models of London, and Lerdahl and Jackendoff. These models try to explain how metrical information may, on the basis of general rules, be computed from a musical signal. In other words, learning is not really a part of these models, possibly apart from somehow learning the rules described by the theories. The Common Slow Pulse model, allowing for arbitrary divisions of units, will be much harder to explain using computing algorithms where sound data is the only input. On the other hand, one would expect learning and experience to be of importance using this model, and see the process of entraining to a

meter more as a pattern-recognition task than a computational task, that is, more a matter of learning to recognize and discriminate a large number of

(musical) patterns than of learning to apply a small number of rules. It is not central to the argument here whether meters with a lowest iso

chronous level exist or not. What I have tried to show is only that there is evidence from some (common) musical performances that does not fit such a model very well, and that another model may explain the observations more

completely. I do not argue that the CSP model is a better universal model of meter than the fast common pulse model, but rather I would like to state the need for better typologies of meter than those offered by western notation.

One possibility may be that meters with a common fast beat form a class of their own, distinct from meters where the lowest level is not isochronous.

Alternatively, one could imagine a typology of meters based on "lowest iso chronous level."



Concluding Remarks

If the idea of an isochronous basic clock pulse as the universal basis for meter in music were found exclusively in theories concerning western classical art music, I would perhaps not be too surprised. But finding this

conception to be central also in ethnomusicological literature is strange because there are quite a number of obvious observations to contradict the idea. Therefore, I think that insisting that additive meters must be perceived in terms of adding subdivisional units is an ethnocentric fallacy, promoted by researchers with either too little practical experience with living music, or with too much respect for traditional Western European music theory.

It should be remarked that in the discussions in this paper, two different

ways of viewing meter have been mentioned, namely meter as produced and

performed by a performer, and meter as perceived by a listener, dancer or

performer. It may very well be that these two situations should be treated

separately in a more detailed discussion. One might also want to discriminate further. For example, one could assume that there are significant differences between the demands for precision and stability in meter production and

perception put on a dancer in a crowd vs. a musician in a jazz band vs. a solo classical piano performer.

Have I really explained anything at all by asserting that units on a certain metric level may be subdivided in arbitrary nonisochronous units? Is this

really an alternative explanation? Will we not still have to explain what we

actually do to perform and maintain irregular relationships between beats?

Yes, of course. I have not explained how we do that. But I have given some

empirical evidence that we actually are able to do this, and I have tried to show where to look for explanations. The crucial difference here is that the low-level hypotheses leads us to look for fast oscillators, most likely located in the brain. The higher-level hypotheses will lead our attention in the direction of bodily movements in the same frequency range as normal walking speed or, as I prefer to see it, in the range of typical dance movements. The fact that we are able to subdivide such movements in ways that are not easily described

by simple division by integers is amply documented in the study of dance movements. How it is done is another matter that is not very well studied in

musicological literature, but this question deserves more thought. Further, the CSP model hinted at here also requires a level of isochronous

units, just as the CFP model does. The difference however is that in the CSP model it is not necessarily the lowest level. But then, is it possible to imagine a meter where NO level has isochronous units?

Notes

1. Interonset interval (abbreviated IOI) is the time between the onset of two consecutive

events; i. e., two beats or two tones.



2. Later, London left the term "complex meter": "... it is problematic to claim that non-iso chronous meters are more complex than isochronous meters tout court. Thus, I am rejecting the use of the term "complex meters" that I previously employed ..." (2004:174). The main

argument concerning the necessity of an isochronous lower level remains, however.

3. See Clayton (2004) for a discussion of the concept of entrainment. 4.The springar type of dance belongs to a group of dances known as bygdedanser or

"countryside dances "Couples are organized in a circle on the floor, and each couple performs a series of dance motifs only loosely coordinated with the other couples. The dance rhythm, however, is tightly coordinated between fiddler and dance couples, and is visible in the up down movements of the bodies of the dancers. The music is conventionally said to be found in three main categories, one with three isochronous beats in the measure, one with three beats in a pattern of short-long-medium, and one with three beats in a long-medium-short pattern.

5."The Beat-Upbeat Ratio [BUR] calculates the proportion between [two successive eighth notes] by dividing the durational value of the first by that of the second. Thus, two equally long eighths yield a BUR of 1.0, whereas a BUR of 2.0 represents a triple configuration ..." (Benadon 2006:75).

6. The literature in this field is comprehensive; see e.g., the references in the following quote by London (1995). See also Kvifte (1989:94-96 ) or Kvifte (1992:43) for a discussion of the concepts analog/digital in this connection.

7.Twenty years after this discussion, I am still unable to explain this without reference to dance meter, and do, accordingly, now look more to motor and gestural evidence than I did at the time of the quoted paper.

8.The down-up movement relations will normally be 1:1 for the "2"-groups. For the "3"

groups both 1:2 and 2:1 are possible. In this particular case, the 1:2 version is chosen.

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Article Contentsp. 64p. 65p. 66p. 67p. 68p. 69p. 70p. 71p. 72p. 73p. 74p. 75p. 76p. 77p. 78p. 79p. 80p. 81p. 82p. 83p. 84

Issue Table of ContentsEthnomusicology, Vol. 51, No. 1 (Winter, 2007), pp. i-viii, 1-180Front MatterFrom the Editor [pp. v-vi]Powwows, Intertribalism, and the Value of Competition [pp. 1-29]Public and Intimate Sociability in First Nations and Mtis Fiddling [pp. 30-63]Categories and Timing: On the Perception of Meter [pp. 64-84]Local Bimusicality among London's Freelance Musicians [pp. 85-105]Memories of Empire, Mythologies of the Soul: Fado Performance and the Shaping of Saudade [pp. 106-130]Book ReviewsReview Essay: Getting beyond Java: New Studies in Indonesian Music [pp. 131-142]Review: untitled [pp. 142-146]Review: untitled [pp. 146-148]Review: untitled [pp. 149-151]Review: untitled [pp. 151-153]Review: untitled [pp. 153-155]

Recording ReviewsReview: untitled [pp. 156-157]Review: untitled [pp. 157-159]Review: untitled [pp. 159-162]Review: untitled [pp. 162-165]Review: untitled [pp. 165-168]Review: untitled [pp. 168-172]

Film, Video, and Multimedia ReviewsReview: untitled [pp. 173-175]Review: untitled [pp. 175-179]

Back Matter

time perception finnish sem

Documents

african rhythm

african music

western rhythm

models of meter

paper meter

types of music

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