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Rhythm, §I: Fundamental concepts & terminology

I. Fundamental concepts and terminology

The elements of rhythm are necessarily intertwined in their origin and effect. Thus, for example,

while durations have certain essential properties, one's experience of duration will vary according tomusical context (i.e. given the presence or absence of a regular metre). Nonetheless, this discussion

begins with the duration of single notes, then proceeds to the organization of successive durations

into coherent groups, the emergence of metre and metric listening, and so forth. In addition, since

rhythm and metre are coherent phenomena only for a listener who can capture and remember the

music as it unfolds, the following discussion engages psychological theory and research to a

significant degree.

1. The distinction between rhythm and metre.

A series of events (whether musical notes or the blows of a hammer) is commonly characterized as

‘rhythmic’ if some or all of those events occur at regular time intervals. But being ‘rhythmic’ is not the

same thing as being ‘a rhythm’. For a musician or musicologist ‘rhythm’ signifies a wide variety of

possible patterns of musical duration, both regular and irregular. A musician is apt to observe that a

regular rhythm exhibits metric properties – or, to put it directly, regular rhythms involve metre. While

all music involves some type or rhythm, not all music involves metre. Thus in common usage the

adjective ‘rhythmic’ often signifies what might more precisely be described as a ‘metrically regular

series of events’. However, irregular rhythms can occur in the context of a regular metre (e.g.

syncopated figures and asymmetrical phrase structures), and not all metres require regular or even

patterns of duration (e.g. Bartók's ‘Bulgarian’ rhythms). Thus there is more to the distinction between

metre and rhythm than regularity versus non-regularity.

Broadly stated, rhythm involves the pattern of durations that is phenomenally present in the music,

while metre involves our perception and anticipation of such patterns. In psychological terms, rhythm

involves the structure of the ‘temporal stimulus’, while metre involves our perception and cognition of

such stimuli. Perhaps the earliest recognition of this distinction was made by Butler (1636; see

Houle, A1987). Butler's illustration is given as ex.1; he remarked that in the top staff the minims go

‘jumping by threes’ whereas in the bottom staff they go by twos, based on the metric context

established in the previous bars. That is, even if the second and third bars are performed in the

same way, we will hear them differently; different perceptual attitudes give rise to different metres.

As Gjerdingen (I1989) has aptly put it, ‘metre [is] a mode of attending’, while rhythm is that to which

we attend.

Ex.1

Butler (1636) on metric context and its perceptual effects; repr. in Houle (A1987), 31

Many of the salient differences between rhythm and metre are summarized in Table 1.

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While the majority of contemporary music theorists embrace a ‘strong separation’ of rhythm and

metre into separate ontological and analytical domains, not all do so. Schachter (G1987) and

Rothstein (E1989) explicitly de-emphasize the rhythm–metre distinction, preferring instead to

address a more general concept of ‘phrase rhythm’ (see §III, 3). Hasty (G1997) gives an especially

thorough critique of this separation, drawing on the work of Friedrich Neumann (G1959). Hasty notes

that the ‘strong separation’ between metre and rhythm can be traced to 19th-century theorists such

as Lussy (D1885), and he also questions the basis on which rhythm and metre can interact if such a

strong separation exists between them.

2. The perception of duration and succession.

Not all durations are perceived alike, as there are a number of psycho-physical limits on our ability to

perceive durations and durational succession. Hirsh (I1959) demonstrated that two onsets must be

separated by at least two milliseconds (ms) in order to be distinctly perceived, and that at least 15–

20ms are required to determine which onset came first. There also seems to be about a 50ms

perceptual decay time – a minimum interval needed to hear one element follow another without

overlap. Hirsh and others (I1990) found that 100ms seems to be the threshold for reliable judgments

of length, and Roederer (I1995) noted that 100ms seems to be a threshold for processing in the

cerebral cortex, and is thus the minimum duration that engages a ‘musical’ (i.e. learnt) understanding

of sound and constructs such as scale degrees and rhythmic archetypes. The maximum interval forreliable estimates of the length of single durations, as well as for the connection of successive

articulations, is usually 1·5–2·0 seconds. This limit is related to the limitations of short-term memory

and the perceptual present (Fraisse, I1978; Handel, I1989). Musicians and musicologists have long

been aware of these upper and lower bounds, as they appear in the context of discussions of tempo

and performance limits (see, for example, Westergaard, E1975).

Just as there are perceptual and cognitive biases and constraints on our understanding of duration,

there are also similar constraints on our apprehension of musical texture, some of which impinge on

our understanding of rhythm. In perceptual psychology texture is investigated under the rubric of

‘auditory streams’ which are the ‘perceptual grouping(s) of the parts of the neural spectrogram that

go together’ (Bregman, I1990). It is through the process of auditory streaming that we are able to

pick out some sounds in our environment and hear them as connected and coherent, whether theyare a single voice in a crowded room or a single part in a complex musical texture. Research in

auditory streaming has shown that pitch, tempo, timbre and loudness are all factors that affect our

ability to segregate sounds into separate streams. Some streaming effects interact with our

perception of duration; for example, Van Noorden (I1975) has shown that when two isochronous

streams cross, an uneven (‘galloping’) rhythm is nonetheless perceived. It is through streaming that

we are able to hear compound melodies, and hence perceive a series of different durations within a

musical surface consisting of even articulations, as in ex.2.

Ex.2 (a) compound melody (b) auditory effect



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Along with streaming effects, other factors can influence our perception of duration. As summarized

by Handel (I1989), intensity and/or pitch differences between evenly spaced notes tend to be heard

as durational differences (with a longer duration perceived from the onset of the unstressed or lower

note to stressed or higher note, and vice versa). Similarly, durational differences may be perceived

as differences in intensity or loudness, and the durational and/or intensity difference(s) of one note

may interfere with durational judgments of other notes. Repp (I1995–6) has reported that in metric

contexts listeners expect slight variations in duration, such that actual durational differences may gounnoticed if they occur where they are expected, or may seem exaggerated if they occur in

unexpected metric positions.

Musical durations (and hence rhythmic groups) are almost exclusively recognized from note onset to

onset. Ex.3 contains three different musical figures: (a ) a series of staccato notes; (b ) a series of

legato notes; and (c ) an arpeggio. While 3a and 3b differ with respect to the absolute value of their

component notes, and while 3c is texturally different from 3a and 3b , all three express the ‘same’

rhythmic pattern – a three-element series with a sense of accent on the first element (prosodically, a

dactyl). While rests between sounds may be salient, most often they inform the perceived quality of

articulation (i.e. staccato versus legato) rather than being heard as musical objects in themselves

(except that in established metric contexts some rests can be heard as having definite duration: see

London, G1993).

Ex.3 Onset patterns

See also HEARING AND PSYCHOACOUSTICS and PSYCHOLOGY OF MUSIC, §II, 2 .

3. Durational patterns and rhythmic groups.As James noted, ‘To be conscious of a time interval at all is one thing; to tell whether it be shorter or

longer than another interval is a different thing’ (I1890, p.615). Metre obviously plays a crucial role in

the determination of relative duration. Nonetheless, we are also aware of relative durations in non-

metric contexts. The first judgment to be made regarding two successive durations is whether they

are the same or different. This distinction is not as trivial as one might suppose. Within the range of

rhythmic acuity described above, the perception of duration follows a modified form of Weber's law,

in which the just-noticeable difference between two successive durations is proportional to their

absolute length (plus a constant of minimal discrimination: see Allan, I1979). There are also effects

of ordering (whether the longer note comes first or last), pitch proximity and differences in loudness,

all of which influence durational comparisons. If two successive durations are judged as different,

then their difference can be conceived and represented in different ways. One may simply comparethe first interval with the second, using the first as a sort of temporal yardstick. This strategy may be

used when the two durations are close to each other in length. If the two durations differ to a greater

degree, rather than comparing the second directly with the first, one may relate each to previously

established perceptual categories of duration. These categories may simply be ‘long’ versus ‘short’,

or they may be more nuanced (e.g. ‘very long’, ‘long’, ‘medium’, ‘short’, ‘very short’). Such

categorical perception of duration has been documented in psychological studies, and it does not

require precise (i.e. proportional) definition of the respective categories (see Clarke, I1987).

Two or more musical durations may cohere into a larger unit, termed a ‘rhythmic group’. The creation

of coherent, well-articulated rhythmic groups is one of the principal tasks the performer faces in

realizing a musical score: to project a sense that some notes go with other notes, and that these

groups themselves form larger units. From this process the basic musical shapes of a piece may bediscerned.



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In the absence of phenomenal cues for group boundaries we will arbitrarily impose a sense of group

structure on a series of events. 18th-century theorists such as Koch (1787) noted this propensity

(see discussion in Hasty, G1997), and in 20th-century psychological studies this is known as

‘subjective rhythmization’ (Fraisse, in Action and Perception , I1985; Handel, I1989). The two

principal factors that influence group boundaries are proximity and similarity (see Meyer, I1956;

Lerdahl and Jackendoff, E1983; Handel, I1989; and Bregman, I1990). In ex.4a proximity (as well as

placement on a common horizontal line or pitch plane) gives the sense that the series of Xs or notesis organized into three groups of three. Composers, copyists and engravers have long understood

the importance of note-spacing in projecting a sense of group boundaries. It can be seen that this

proximity is marked from note onset to note onset. In an analogous case which employed a

continuous series of durations (e.g. two crotchets plus a semibreve) we would find the same

grouping structure. Small variations in onset timing may have an impact on proximity judgments (and

hence on grouping structure); the common performing practice of taking extra time for bowing or

breathing following longer notes exploits (and underscores) our innate proclivity to hear a long note

as the end of a group (see Gabrielsson, H1988).

Ex.4b illustrates the effect of similarity on the perception of group boundaries. Even though the

pitches are equally spaced in this example, one again perceives a series of triplets. In ex.4c the

effects of similarity and proximity are combined. As can be seen, proximity is of greater salience than

similarity, for despite the sharp differentiation of pitch, the differentiation of note-onset proximities

creates a series of four-element groups. Group boundaries can be marked by changes in any

musical parameter, including dynamics, timbre and texture. However, grouping is primarily marked

by patterns of duration and timing, with pitch playing an important, though secondary role (see §III,

6).

Ex.4 (a) proximity (b) similarity (c) proximity versus similarity

Once the boundaries of a group have been established one may describe its internal structure,

though often these two issues are interdependent. Cooper and Meyer (E1960), whose prosodic

approach to musical rhythm has its antecedents in 17th- and 18th-century discussions of

rhythmopoeia such as those of Mersenne (MersenneHU ) and Mattheson (C1739), begin with a list of

two or three-element archetypes for musical groups; these involve the relationship between one

accented and one or two unaccented elements in each group (iamb, trochee, dactyl, anapest and

amphibrach). Cooper and Meyer then consider each archetype in various metric contexts. Lerdahl

and Jackendoff (E1983) do not employ durational archetypes in their analysis of rhythmic grouping;

rather, they focus on the determination of group boundaries and on the hierarchical nesting of

subgroups. In Lerdahl and Jackendoff's theory the internal structure of each group is determined by

the interaction of metrical and tonal components (see §III, 3).

The same pattern of pitches and/or durations may allow for more than one grouping interpretation

(ex.5). These various groupings may be differentiated compositionally (for example, by the

patterning of the accompaniment) or by articulative and dynamic cues in performance. The fact that

the same pattern of pitches and durations may give rise to different grouping (as well as metric)

structures has implications for the historical study of rhythmic notation and performance. Rhythmic

notation must somehow inform the performer how to make rhythmic nuances in performance, either

with explicit markings, with hints from spacing and orthography, or through shared conventions of

score interpretation (e.g. metre as an indication of bowing and hence grouping, characteristic styles

of rhythmic performance for various dance genres, and so forth).

Ex.5 Same pitch-durational pattern with different group boundaries



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Rhythmic groups may be nested hierarchically, and thus smaller groups may function as composite

elements within a larger group. Ex.6a consists of a pair of simple dactylic groups, while in ex.6b the

dactyls are themselves composed of trochaic subgroups. One reason we are able to hear ex.6b as a

variant of ex.6a is that we treat the grouping structures as commensurate; the elaborations in ex.6b ,

while adding extra depth to the rhythmic hierarchy, do not alter the basic rhythmic pattern on the

crotchet level.

Ex.6 Nested rhythmic groups

4. Metre: beats, metric cycles and tempo.

Metre is a structured attending to time which allows the listener to have precise expectations as to

when subsequent musical events are going to occur. Dowling, Lung and Herrbold (I1987) have

studied and described such directed attention in terms of ‘expectancy windows’; Jones and her

colleagues (I1981, 1989, 1990, 1995, 1997) have given considerable attention to the process of

entrainment, whereby one synchronizes one's attention to regular patterns of information present inthe environment. Musical metre, then, would seem to be a particular utilization of our more general

capacities for temporal perception. Thus, while knowing that waltzes are in triple metre tells us

something (at least in broad terms) about their rhythmic structure, it also tells us even more about

the way we listen to and/or perform them.

First and foremost, metre requires an awareness of a beat or pulse. A series of very rapid notes

(such as a sustained tremolo) does not give rise to a sense of metre, nor does a series of very long

or widely spaced notes. Only when we hear a series of regular articulations in a certain range (from

100ms to 2 seconds apart) does a sense of pulse arise. More familiarly, a 2-second duration is a

semibreve at a tempo of crotchet = 120 (the upper limit), while a 100ms duration is a semiquaver at

a tempo of crotchet = 150 (the lower limit). While these very short and rather long durations may

constitute individual levels of a metric hierarchy, they are at the ends of the metric spectrum.Psychological research has validated what musicians have long known: we have a preference for

tempos within a narrower subrange of these extremes (see also PSYCHOLOGY OF MUSIC, §II ). Parncutt

(I1993–4) has examined a range of ‘maximal pulse salience’ (from 60 to 150 beats per minute,

anchored at approximately 100 beats per minute) wherein pulses tend to be most strongly felt. Our

sense of tempo is not simply based on the shortest (or longest) durations present in the musical

surface, though such durations contribute to the particular quality of motion present, such as a

‘tense’ adagio or a ‘serene’ allegro. As Epstein (H1995) notes, tempo is an aggregate effect of the

total metric hierarchy – its relative depth, the continuity of its various levels and so on.

The perception of a beat or pulse is not only necessary for a sense of ‘connectedness’ among

successive events; it may also be necessary for a sense of motion, a temporal/auditory analogue of

the ‘phi phenomenon’ in visual perception (Wertheimer, I1912; Bregman, I1990). The specialsalience of the pulse level has been acknowledged in various theoretical models of metre, from

Cooper and Meyer's ‘primary rhythmic level’ (E1960, p.2) to Lerdahl and Jackendoff's

‘tactus’ (E1983, p.71). If the beat level should drop out, we immediately feel suspended in time and

await its restoration.

A sense of beat, while necessary, is not sufficient to engender a sense of metre. Another layer of

organization is also required, giving rise to a metric hierarchy which contains two or more

coordinated levels of motion. Koch (D1782–93, ii) recognized beats (Taktteile ) as primary

components of the measure (Takt ), while various subdivisions (Taktglieder and Taktnoten ) are

produced by analogous partitions of the beat. Weber (D1817–21), with an emphasis on symmetry as

an organizing principle, noted that

“Just as beats together form small groups, several groups can also appear bound

together as beats of a larger group, of a larger or higher rhythm, a rhythm of a

higher order. One can go even further and place such a rhythm of a higher order



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with a similar one, or a third, so that these two or three together form yet a higher

rhythm. (1824, pp.102–3; trans. from Morgan, E1978, p.437)”

Thus Weber extended metric relationships both upward and downward, as opposed to earlier

theorists who had begun with the measure and then had proceeded by division. Similar descriptions

can be found in Hauptmann (D1853), who spoke of ‘metric construction inwards and outwards’, as

well as in Riemann (D1903–4), who articulated the principle of Achttaktigkeit to describe an eight-

measure metric schema (see also Klages, E1934; Leichtentritt, E1951).

Given the emphasis on symmetry and the pervasiveness of the systole/diastole metaphor for

musical motion, 19th-century accounts of metre were strongly biassed towards binary principles of

metric organization. Present-day theorists continue the hierarchic approach to metric structure but

have relaxed the strictures on its organization. Cooper and Meyer define metre as ‘the measurement

of the number of pulses between more or less regularly recurring accents’ (E1960, p.4). Yeston

describes metre as the interaction between two adjacent levels of motion (E1976, p.67).

Lerdahl and Jackendoff note that ‘fundamental to the idea of meter is the notion of periodic

alternation of strong and weak beats’ and that ‘for beats to be strong or weak there must exist a

metrical hierarchy – two or more levels of beats’ (E1983, p.19). They go on to specify how the metric

hierarchy must be organized through a series of metric well-formedness rules. These rules work in abottom-up fashion, prescribing that beats come in twos or threes and that beats and measures be

isochronously spaced. Kramer (K1988) relaxes these constraints and allows for the non-isochronous

spacing of beats and hyperbeats (though they still must occur in cycles of twos or threes).

While two levels of structure will give rise to a sense of metre, most metric hierarchies are more

richly organized. Time signatures used in Western notation since the 17th century specify four basic

metric types, based not only on the organization of each measure into two or three beats, but also on

the organization of beat subdivisions into duplets or triplets (see Table 1). These time signatures

specify three levels of structure and thus define the pacing of rapid, moderate and slower events in a

coordinated fashion. Additional subordinate and superordinate levels of organization are possible.

Subdivisions are constrained by our abilities to discriminate among rapid stimuli, as individual note

onsets become a perceptual blur (as in rapid sweeps in Chopin). Limits on superordinate levels aremore controversial. Clearly there may be a sense of attending above the measure (see Caplin,

D1981, on ‘notated versus expressed’ metres), but the extent of such ‘hypermetres’ remains a topic

of considerable debate (see §III, 1).

The beat level of the metric hierarchy serves as the temporal anchor for the other levels. Thus

neither a ‘top-down’ approach (where one starts with the duration of an entire measure or larger unit

and then proceeds to partition that span into beats, and those beats into subdivisions) nor a ‘bottom-

up’ approach (where one starts with the shortest durations present on the musical surface, combines

them into beats, and those beats into measures and so forth) is quite right. Rather, a sort of ‘middle-

out’ perspective on metre seems most consonant with the way we attend to (as well as represent)

rhythmic events. For higher and lower levels may come and go over the course of a piece without

breaking its temporal thread; a layer of subdivision may be absent entirely, or freely change fromduplets to triplets without perturbing our sense of temporal continuity. Ex.7 illustrates how the

number of metric levels may fluctuate over the course of the piece, from two or three above and

below the beat in the opening bars (ex.7a ) to its distillation down to simple pairs of beats (ex.7b ).

(Here and elsewhere, layers of dots under the staff correspond to levels of the metric hierarchy. A

metric event is more or less accented depending upon the number of levels on which it occurs,

indicated by the greater number of vertically aligned dots under that event.) In a very real sense

these fluctuations are changes of metre, as they represent changes in the degree and depth to

which the listener is entrained to the music.



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Ex.7 Fluctuation in metric depth over the course of Beethoven’s Fifth Symphony, first movement

Finally, if metre is a complex mode of musical attending, in characterizing a piece or passage as

‘metric’ one usually means that composer, performer and listener all share the same temporal

perception(s) of the music. If a performer maintains a complex counting framework in the execution

of a musical work, but this framework is aurally inaccessible to the listener, then that musical work is

not metric in the ordinary sense. If the performer maintains one sense of metre while the listener

infers another (which may be possible in a wide variety of contexts), then the music is metric but in amultivalent sense. If on the other hand the performer maintains one sense of metre but the listener is

unable to infer any sense of metric organization at all, then the music is in some sense non-metric.

Thus the presence of metric notation (whether in Babbitt or Baude Cordier) does not guarantee the

presence of a metre (see §II, 4).

5. Rhythmic and metric accent.

Perhaps the thorniest problem in discussing rhythm and metre involves their respective accents. In

its broadest sense, ‘accent’ is a means of differentiating events and thus giving them a sense of

shape or organization. Many sources distinguish between ‘qualitative’ and ‘quantitative’ accent.

These terms originated in discussions of poetry and linguistic accent, and were subsequently applied

to music. Originally, qualitative accent referred to poetic rhythms whose elements were differentiated

by dynamic or intonation stress, as distinct from those differentiated by length. This opposition in turn

informs the distinction that has been made between stress-timed and syllable-timed languages. Yet,

as Handel (I1989) notes, all languages employ both qualitative and quantitative accents. Moreover,

from a perceptual point of view quantitative differences result in qualitative distinctions, and

conversely qualitative distinctions are often perceived not simply as, for example, difference in

dynamic emphasis, but also as differences in duration.

A metric accent marks one beat in a series as especially strong or salient, such that it functions as a

downbeat, while a rhythmic accent makes one element in a series of durations the focal member of

the rhythmic group. Cooper and Meyer, in recognizing the difficulty of pinning down rhythmic and

metric accent, simply said that an accent event is ‘marked for consciousness in some way’ (E1960,

p.8). They distinguished accent from stress, which is the ‘dynamic intensification of a beat, whether

accented or unaccented’. Thus, for example, the dynamic and textural emphasis that occurs on the

second beat of each bar in a mazurka does not displace the downbeat (and hence the sense of

metric accent).

The relationship between tonal motion and rhythmic and metric accent(s) has generated a

considerable amount of discussion, much of it focussing on the accentual status of cadences and

other components of phrase structure (for a summary see Kramer, K1988). Analyses of rhythm and

metre by Schenkerian theorists, most notably in the work of Schachter (E1976, E1980, G1987) and

Rothstein (E1989), engage analogous concerns. Not surprisingly, from a Schenkerian perspective

rhythmic accent is generated via top-down linear processes, as it derives from tonal motions which

serve to articulate various structural levels. Schachter (G1987) also notes that metric accent accrues

to the boundary points of various time spans, and that such spans may be articulated by both

durational and tonal processes. Rothstein, drawing on the work of Koch (D1782–93) and Schachter,

posits that middleground tonal motions are inherently regular, based on symmetrical archetypes

which may be modified or transformed in the foreground. Rothstein also emphasizes the ways in

which rhythmic groups and metric structures tend to be commingled, and so approaches questions

of metre and accent through a consideration of ‘phrase rhythm’ which includes both metrical and

rhythmic components (see also §III, 2).

Other theorists consider accent by focussing on musical motion and on those points which serve as

crucial moments in its ebb and flow. This perspective naturally leads to considerations of metre and

metric accent. Momigny (D1803–6) spoke of an upbeat-downbeat motion (from levé to frappé ) which

inheres at both foreground and higher levels. A similar account of arsis and thesis was given byRiemann (D1903–4), who integrated a recursively applied concept of upbeat-to-downbeat into an

eight-bar metric/rhythmic schema. Zuckerkandl (G1956) developed a wave model of metre, linking

successive upbeat and downbeat motions:



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“With every measure we go through the succession of phases characteristic of

wave motion: subsidence from the wave crest, reversal of motion in the wave

trough, ascent toward a new crest, attainment of a summit, which immediately

turns into a new subsidence. … Our sympathetic oscillation with the metre is a

sympathetic oscillation with this wave. (p.168)”

The crest of the metric wave is a point of metric accent, a moment of beginning. The correlation

between a point of beginning and metric accent has been explored by Berry (G1976), who speaks of

‘reactive’ and ‘anticipative’ impulses and their corresponding accents; this parallels Zuckerkandl's

account of motion from trough to crest (see also Brower, E1993). Kramer similarly defines metric

accent as ‘a point of initiation’ (K1988, p.86). Benjamin (G1984), drawing on the work of Berry,

attempts to quantify and tabulate various factors (harmonic change, relative stability, relative

dissonance, contour, textural density and so forth) which mark such points of initiation and thus

define accent algorithmically; Lester (E1986) espouses a similar approach.

Other theorists have refined Cooper and Meyer's distinction between accent and stress. Epstein

distinguishes stress, rhythmic accent and metric accent. He places metre and rhythm into separate

temporal domains, a ‘chronometric time’ consisting of beats and metric accents, and an ‘integraltime’ which contains pulses and rhythmic groups (E1979, pp.58–62; see also Souvtchinsky, K1939).

Lerdahl and Jackendoff distinguish three varieties of accent: phenomenal accents which ‘give

emphasis or stress to a moment in the musical flow … such as sforzandi, sudden changes in

dynamics or timbre, long notes, leaps … and so forth’, structural accents which are ‘caused by

melodic/harmonic points of gravity in a phrase or section’, and metrical accents which accrue to a

‘beat that is relatively strong in its metrical context’ (E1983, p.17). Their metric accent is hierarchic in

nature, in that a metrically accented event on one level is also present on higher levels; these

relationships are indicated by their dot notation which marks the emergent metric grid. In Lerdahl and

Jackendoff's approach metre is read in relation to grouping structure, though it is restricted to a few

levels in the musical foreground (see §III, 2). This hierarchic approach to metric accent has its

precursors in Komar (E1971) and Yeston (E1976) and is also adopted by Kramer (K1988).

6. Interactions of rhythm and metre.

The first question to consider in discussions of rhythm-metre interaction is, ‘Which comes first,

rhythm or metre?’. Hauptmann (D1853), for example, viewed metre and metric processes as prior to

rhythm, while Neumann (G1959) takes the opposite view. Metric priority implies that durational

patterns are understood only within the context of a metric framework (and, on some views, are

generated from that framework; see Johnson-Laird, I1991). The pulse train which is the substrate of

metre may be internal to the listener, or it may be given in the music, but in either case rhythmic

shapes gain their identity in relation to it. Conversely, from the point of view of rhythmic priority, the

metric framework is inferred from the unfolding durations; Lerdahl and Jackendoff (E1983) give aformalized treatment of such a metric discovery process. On this view it also follows that metre is not

inherently regular, given that it flows from the fluctuating stream of durations. Rather, metricity lies in

the listener's sense of temporal comparison or measurement. At the root of the question of rhythmic

or metric priority are fundamentally different conceptions of time and temporal consciousness (for a

discussion of the historical and philosophical aspects of this issue, see Hasty, G1997). One may

steer a middle course between these two perspectives. When a piece begins, its metre and tempo

are usually not known to the listener, and so the listener must make metric inferences (usually

quickly and without difficulty) from the pattern of durations and stresses that are given. Once the

metre is established it takes on a life of its own; the listener may then project a sense of accent on to

an event even if it is not otherwise marked by duration, dynamics, contour or harmonic change. Thus

at some times a sense of accent flows from the musical surface to the emerging metre, and at other

times from the metre to the unfolding musical surface. Metre is not constant over the course of apiece, and should the metre falter or collapse later on, the listener must again seek an appropriate

metric framework.



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We often speak of conflicts between rhythm and metre. As Hasty has rightly observed, ‘to enter into

conflict metre and rhythm must share some common ground’ (G1997, p.17). In hearing metrically, a

listener also generates a series of rhythmic groups, a pattern of durations and rhythmic accents.

Thus two grouping patterns are usually present in the musical experience: that expressed by the

musical surface, and that generated by the metrically entrained listener. While often these patterns

are perfectly congruent (ex.8a ), quite often they are not (ex.8b ). In Beethoven's ‘Ode to Joy’ there is

a one-to-one correspondence between rhythmic and metric events, and both measures and groupshave their boundaries in the same locations. In the Haydn example, however (ex.8b ), the rhythmic

groups in the first violin part have a different number of elements from the metre (which is clearly

expressed by the other members of the quartet, who play a foursquare accompaniment in quavers),

and these groups are offset from the metric boundaries because of anacrustic figures. Nonetheless,

in ex.8b these groups are coordinated in a number of important ways. Both the ‘rhythmic group’ and

the ‘metric group’ express the same length. It may also be observed that in both examples the

rhythmic and metric accents coincide – the downbeats of each bar are also the location for the note

which functions as the focal note in each group. In other words, though groups and measures are

non-congruent in ex.8b , the result is not a polyrhythm or polymetre.

Ex.8 Congruent and non-congruent metre grouping boundaries (a) Beethoven, ‘Ode to Joy’; (b) Haydn, String Quartet

op.33 no.2, first movement, bars 1–3

More serious conflicts occur in the cases of syncopation and hemiola (ex.9). Yeston (E1976) and

Krebs (G1987), building on the work of Sachs (A1953), have described these conflicts in terms of

‘metric dissonance’, and their approach has been taken up by theorists such as Rothstein (E1989),

Cohn (G1992, G1992–3), Kamien (G1993) and Grave (G1995). Metric dissonances occur when

secondary accents and/or group lengths undermine the established metre to the point that a

secondary metric framework may emerge; in ex.9a Holst's melody refuses to settle into the 2/4

framework, but neither can it quite establish a new metre. In ex.9b , however, there is a local shift of

metre, from compound duple to simple triple in the second and fourth bars. Note in this case that the

quaver and dotted minim levels of metre remain the same; the shift occurs only on the beat level(from dotted crotchets to crotchets and vice versa).

Polyrhythms entail even greater complexity, as they involve the simultaneous (as opposed to

successive) presence of two different rhythmic or metric streams. Thus, for example, if the bass line

in ex.9b maintained the 6/8 metre while the upper part shifted to 3/4, the result would be a true

polyrhythm. Polyrhythms are often described in terms of the presence of two (or more) concurrent

metres (thus, more properly, polymetres); descriptions of metric dissonance often make similar

assumptions. However, work in perception of polyrhythms (Lashley, I1951; Handel and Oshinsky,

I1981; Handel, I1989; Klapp and others, I1985; Grieshaber, I1990; and Jones and others, I1995)

suggests that we are unable to hear two metric frameworks at the same time, but either hear

polyrhythms in terms of a dominant metre, or construct a composite metre to accommodate both

rhythmic streams.

Ex.9 Syncopation and hemiola (a) syncopation in Holst, The Planets, ‘Jupiter’, bars 6–12 (b) hemiola in Gaspar Sanz,

Canarios, bars 9–12

A number of theorists (Cone, E1968; Westergaard, E1975; Benjamin, G1984; Schachter, E1976;

Berry, G1976; Komar, E1971; and Lerdahl and Jackendoff, E1983, summarized in Kramer, K1988)

have concerned themselves with the relationship between phrase structure and metre or, more

precisely, hypermetre. Here the chief concern is the alignment between the tonal structure of a

phrase and the accentual organization of the hypermeasure, especially with respect to the

interaction(s) between cadences and hypermetric accent. A related concern is whether the tonal

structure of the phrase determines the higher-level metric accent or vice versa (see §III, 1).



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7. Complex rhythms and complex metres.

While complexities such as hemiola and syncopation may arise from the interactions between

rhythm and metre, rhythmic and metric structures may also exhibit considerable complexity in their

own right. In describing either a measure or a rhythmic group one may note (a ) its overall size, in

terms of both its absolute duration and the number of elements it contains; (b ) the number of

structural levels it comprises; (c ) the variety of its elements (e.g. the range of durational values within

it); and (d ) the degree of redundancy in its organization. These factors must be considered together,

for a large number or variety of elements does not in and of itself entail rhythmic or metric

complexity. For instance, ex.10 contains many notes and many levels of metric structure, yet it is not

rhythmically or metrically complex, given the high degree of redundancy in its grouping and metric

structure.

Ex.10 Multi-level, simple metre in Bach, Prelude in C major (Das wohltemperirte Clavier, bk 1)

More complex rhythms involve a variety of contrasting durational values. In ex.11 there is alternation

between a series of short durations and then an extremely long inter-onset interval between groups.

As the metric subdivisions (and indeed almost the beats themselves) lapse during the grand pauses,

the result is both an indeterminacy with respect to the ‘end’ of each group and a sense of

discontinuity between successive metric units. Ambiguity of group organization and/or boundaries

adds to rhythmic complexity. In ex.12 the sense of closure and group articulation that occurs on the

second beat of bar 4 is undermined by the sequential repetition of the motif in bars 5–6; the result is

a sense that bars 3–6 form a group in their own right, blurring the cadence in bar 4.

Ex.11 Collapse of metric subdivision in Haydn, String Quartet op.33 no.2, finale, bars 153–9

Ex.12 Indeterminate phrase/group boundary owing to motivic continuation in Haydn, Symphony no.92, finale, bars 1–8

Complex metres involve irregular relationships among elements on a single metric level as well as

between adjacent levels (Jones and Boltz, I1989). The hallmark of complex metres is that some

levels are non-isochronous. In ex.13a , showing a metre commonly referred to as mixed or

alternating, the downbeats are unevenly spaced, but regularity occurs every five beats. In ex.13b the

quavers are too rapid to be felt as beats, and the result is a series of uneven beats with a ratio of

2:2:2:3. Such 2:3 ratios for the duration of successive beats are characteristic of metres with

complex beat patterns and give the music its ‘limping’ quality (see Brăiloiu, J1951, on ‘aksak’

rhythm). Complex metres typically are indicated with a fractional time signature such as 2+2+2+3/8.

Complex metres also tend to be more fragile, in that they readily devolve into metres with

isochronous beat or measure periods, and in that only a limited range of durational patterns is

possible in any given complex metre. Subdivision is also usually explicitly present to ensure the

intelligibility of the metre.

Ex.13 (a) ‘mixed metres’ in Bartók’s Mikrokosmos, bk 4, no.102, bars 33–7 (b) non-isochronous beats in Dave Brubeck,

Blue Rondo à la Turk, bars 1–3

On a larger scale, the use of constantly changing patterns of rhythm and/or shifting metres adds

another level of structural complexity. Elliott Carter has developed and described the technique of

‘metric modulation’, which he uses in his percussion piece Canaries (ex.14). Of this excerpt Carterwrites that ‘to the listener, this passage should sound as if the left hand keeps up a steady beat

throughout the passage … while the right-hand part, made up of F-natural and C-sharp, goes



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through a series of metric modulations, increasing its speed a little at each change’ (F1977, p.349).

Canaries has its antecedents in the use of a series of proportional changes in mensuration in pre-

tonal music.

Ex.14 Metric modulation in Carter, Canaries

8. Additive versus divisive rhythm.

In discussions of rhythmic notation, practice or style, 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 × 2/8. In addition, additive is associated with asymmetrical

rhythms, while divisive rhythms are often assumed to be symmetrical. As a result, the first problems

that often arise occur in the case of triple metres, which can be regarded as divisive (e.g. 3/4 = 3 ×

1/4) but often involve the pairing of unequal durations (e.g. a minim plus a crotchet, creating a 2/4 +

1/4 figure, to put it in appropriately metric terms; see Rastall, A1982). While this problem stems from

a conflation of metric beats with rhythmic durations, deeper confusions regarding additive versus

divisive rhythm also occur.

These confusions stem from two misapprehensions. The first is a failure to distinguish between

systems of notation (which may have both additive and divisive aspects) and the music notated

under such a system. The second involves a failure to understand the divisive and additive aspects

of metre itself. Few notational systems are wholly additive or wholly divisive, given the practical

problems such systems create (see §II, 1). Any notational system that uses different orthographic

forms (or patterns of forms, in the case of modal rhythm) for different durations will have to define

their interrelationships in some way, and this is almost universally done in terms of proportional

differences between different shapes. Therefore, on a note-to-note level successive durations will be

expressed in terms of some multiplicative or divisive relationship (e.g. that a given note is half or

twice as long as the note that preceded it). Yet ‘additive’ and ‘divisive’ mean more than simple note-

to-note durational relationships, in that they imply the manner in which a series of durations coheres

into a larger figure. Additive rhythms are constructed and understood from the ‘bottom up’, while

divisive rhythms are constructed and understood from the ‘top down’. Additive and divisive are

therefore claims about the essential nature of the rhythmic hierarchy in a particular piece or style.

Notational systems that are metrically equivocal tend to give representations of durational

sequences that are neither specifically additive nor divisive. For example, while there is a divisive

aspect to modal rhythm, in that one must see the entire ordo in order to understand which mode is

present, this is not the same as understanding shorter durations in terms of their generation from

longer spans via some process of division. Later Franconian and Petronian refinements of modal

practice, while giving greater precision and flexibility, essentially remain systems representing a

linear series of durations. Such notational systems do not express any hierarchical structure of either

an additive or a divisive nature. This is not to say that the music written in modal notation cannot

contain hierarchical rhythm; one would be hard pressed to claim that pieces which employed

primarily 3rd- or 4th-mode rhythms do not project a sense of metre akin to compound duple time.

Conversely, just because a notational system is hierarchic (i.e. divisive), it does not follow that the

musical rhythms notated under such a system are (see ex.14).



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Copyright © Oxford University Press 2007 — 2011.

Metre itself contains additive and divisive components, and this suggests that our understanding of

rhythm in general involves both additive and divisive aspects. Psychological studies of metre indicate

that above the level of the beat all metres are additive: 2/4 (1+1), 3/4 (1+1+1) and so forth. On the

other hand, below the level of the beat metric and rhythmic relationships are usually divisive, in that

these shorter durations are given definition through the subdivision of the beat in simple metres (see

Shaffer, H1982). Standard metric pedagogy and practice reveal this basic distinction between beats

and subdivisions: one counts a basic frame additively (‘1 2 3, 1 2 3 …’) and then interpolatessubdivisions within it (‘1 and 2 and 3 e-and-uh, 1 2 and-uh 3 and …’). For this reason, metric

subdivision can come and go over the course of a passage, and even shift from duple to triple

without seriously disturbing the sense of metre. Complex metres differ from simple metres in that the

units of addition, rather than simple isochronous beats, are nested ‘packets’ of shorter durations

which themselves define long versus short beats. While this means that in complex metres the

listener's sense of the subdivision cannot come and go (which may account in large part for the

qualitative differences between simple and complex metres), the counting process in complex

metres remains analogous to the counting of rapid subdivision in simple metres.


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