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M U S I C T H E O R Y O N L I N E
A Publication of the
Society for Music Theory
Copyright (c) 1993 Society for Music Theory
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| Volume 0, Number 2 April, 1993 ISSN: 1067-3040 |
+-------------------------------------------------------------+
General Editor Lee Rothfarb
Co-Editors David Butler
Justin London
Elizabeth West Marvin
David Neumeyer
Gregory Proctor
Reviews Editor Claire Boge
Consulting Editors
Bo Alphonce Thomas Mathiesen
Jonathan Bernard Benito Rivera
John Clough John Rothgeb
Nicholas Cook Arvid Vollsnes
Allen Forte Robert Wason
Stephen Hinton Gary Wittlich
Editorial Assistants Natalie Boisvert
Cynthia Gonzales
All queries to: mto-editor@husc.harvard.edu
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AUTHOR: London, Justin M.
TITLE: Loud Rests and Other Strange Metric Phenomena (or, Meter as Heard)
KEYWORDS: meter, rhythm, accent, parameter, perception, cognition, entrainment, ontology, dynamic model of meter
REFERENCE:
Justin M. London
Carleton College
Department of Music
Northfield, MN 55057
jlondon@carleton.edu
ABSTRACT: This is an excerpt from a work-in-progress, portions of
which will be read at the 1993 meeting of the Society for Music
Perception and Cognition. In order to give an adequate account of
"loud" rests, metric articulations, and meter's propulsive character
(as noted by theorists such as Berry and Zuckerkandl), a dynamic model
of meter is proposed. In framing this model a brief overview of
theoretical strategies for metric models is given. The dynamic model
regards meter as a listener-generated framework for the understanding
musical time. The theoretical implications of the dynamic model are
discussed, as it challenges traditional notions of meter as "part of
the music," that is, as musical parameter with the same ontological
status as pitch, loudness, timbre, texture, and rhythm.
ACCOMPANYING FILES: mto.93.0.2.london.gif
INTRODUCTION
[0] NOTE: The three musical examples in the .gif file, while helpful,
are not absolutely necessary for comprehension of this article; verbal
descriptions of the salient features of each example are given in the
text. Also, there are several citations to the author's dissertation
in the following article which are the natural outcome of the close
relationship between the essay below and the dissertation, the former
being an extension of some ideas already explored in the latter.
[1] Cooper and Meyer have noted that the downbeat of measure 280 in
the first movement of Beethoven's "Eroica" symphony "must be the
loudest silence in musical literature."(1) This moment, where triple
meter has at last been re-established following an extended
duple-vs.-triple conflict (in mm. 250-75) is unequivocally striking,
a moment whose poignancy and power is immediately felt.(2) But what
exactly happens on the downbeat of measure 280? First and foremost,
does something happen, or is it the absence of something which strikes
us so profoundly? But if something does happen, what kind of thing is
it?
=======================================
1. Cooper, Grosvenor, and Meyer, Leonard. *The Rhythmic Structure of
Music*, Chicago: Phoenix Press (1960): 139.
2. Cooper and Meyer (op. cit.) view this moment as the culmination of
an extended anacrusis, the moment of thesis following an 8 mm. arsis.
While I would not describe mm. 272-79 as an extended anacrusis, as I
feel it is an inappropriate application of a phrase-level phenomenon
to higher structural levels, nonetheless I agree with Cooper and
Meyer's characterization of the essential rhythmic feel of the passage.
=======================================
[2] The downbeat of measure 280 in Beethoven occurs in an incredibly
rich musical context--as there are not only metric intrigues here,
but also a nexus of motivic, phrasing, tonal, and formal events--and
thus it may be helpful to consider the metric issues in a simpler
context, and so I have composed the following two (and admittedly banal)
examples. [EX. 1 and EX. 2]. In example 1 we have another "loud"
downbeat, though one not as markedly loud as m. 280 in the Beethoven
example. Following a clear antecedent phrase in 3/8 time which ends
with a melodic half-cadence, the consequent phrase leads us to expect a
tonic arrival on the downbeat of the eighth measure. But in example 1,
nothing happens. Instead, we have an 8th rest and then a shift in
motivic pattern, with the melody coming in on the mediant (i.e. on the
third of the tonic chord rather than the expected root/do which would
complete the "sol-la-ti" melodic line). While nothing happens on the
downbeat of measure 8 in the music, what happens at that moment in
the listener's mind? Similarly, in example 2, a dactylic pattern
(strong-weak-weak) of rhythmic grouping is established in the first
two measures, a pattern than is congruent with the 3/4 metric context.
In the third measure, however, instead of three quarter notes we have
a half note and a quarter. Yet is there not some sense in which the
dactylic pattern persists in this measure? And if so, why?
[3] The quick answer to such questions is, of course, "because one
hears a beat or a downbeat in these cases." But what exactly is meant
by "hearing a beat?" I am quick to add that to come up with adequate
definitions for "beat" and "downbeat" is no easy task, as the work of
theorists past and present will attest. Since beats and downbeats are
"primitives" in most metric theories, these questions ultimately drive
us down the slippery slope to ask "what is meter?"
THE PHYSICIST AND THE MUSICIAN LOOK AT MUSICAL PARAMETERS
[4] Meter is often regarded as one of the parameters of music, an
aspect of musical structure (or to put it another way, a dimension
of musical description) that taken together with pitch, duration,
timbre, articulation, dynamics, and texture allows for a thorough
account of a musical event. Kramer puts it most boldly: "Meter is
not separate from music, since music itself determines the pattern
of accents we interpret as meter. . . .Music not only establishes,
but also reinforces and sometimes redefines meter."(3) But meter is
not cut of the same cloth as the other parameters. Let us consider
how a musician and a physicist would describe these different aspects
of musical sound:
The MUSICIAN The PHYSICIST
Pitch Frequency of waveform
Rhythm Duration in microseconds of event
Timbre Shape of waveform
Articulation Envelope of waveform
Loudness Amplitude of waveform
Meter (????)
While these pairs of terms are not merely synonymous, they do show to
serve how the physical attributes of sound inhere in the various
musical parameters (for example, the physicist's understanding of
frequency is stated in terms of cycles per second of periodic
vibration, and even if she takes octaves into account by mapping the
frequencies onto a logarithmic scale, this is still quite different
from the notion of "pitch" which defines tones relative to some scale
or tuning context). And I hasten to add that these features are
interdependent, especially those of timbre, articulation, and loudness.
But where is meter for the physicist?
===================================
(3) Jonathan Kramer, *The Time of Music,* New York: Schirmer Books,
(1988): 82.
===================================
[5] Well, meter has something to do with musical time, so one might
place meter under rhythm, as the term "rhythm" is admittedly vague.
For clearly "rhythm" is more than just the duration of single
notes/events; it also involves patterns of durations (i.e rhythmic
groups), and patterns of patterns, and so forth. Perhaps meter can
be subsumed under rhythmic grouping, that is, a special kind of
temporal patterning.(4) We will allow our physicist, with help from
a friendly, nearby music theorist, to consider meter to be a special
kind of periodicity present in the music, one which is based on
hierarchically regular isochronous durations.
=======================================
4. The author is well aware of the fact that rhythmic groups consist of
patterns of duration, that is, of time spans, versus metric patterns
which may be viewed as an ordering of time-points. I do not wish to
imply here that I regard meter as a time-span phenomenon (as will be
made clear below). It has also been posited that metric patterns need
not be isochronous, especially on higher metric levels (see Kramer, op.
cit., Ch. 4). For the moment, let us allow our physicist to struggle
toward meter as best she can, with an admittedly over-simplified metric
model.
======================================
[6] We give our physicist the following example [example 3], where the
duration of each pitch is an 8th note, and each rest two eighth notes
(rests are indicated by (r)), at a tempo of a quarter-note = 100:
c-d-e-f-g-a (r) d-e-f-g-a-bb (r) b-c-b-c-b-c (r) d-c-bb-a-g-f
(N.B. "b"= b-natural and "bb"= b-flat)The physicist notes that there
are clear periodicities for every 8th-note duration as well as every
8 eighth-notes, the latter marked by the rests. Furthermore, with a
nudge from her theorist companion, the physicist is able to interpolate
two intervening levels of duple organization. But of course, she has
not yet found the meter--for where is the downbeat (that is, where
does the metric pattern properly begin)? Well, with another nudge
from the theorist, the physicist learns that there are two possibilities
in this passage, either (a) the downbeat occurs on the very first
note/musical event, or (b) it doesn't. In the latter case, one must
then look for other information (tonal cues, dynamic stress, textural
changes, etc.) which would mark another note/event as the downbeat.
Luckily, our nearby theorist tells the physicist that if the downbeat
is not on the first note, one then usually has a conventionalized
anacrustic pattern, the most common of which goes "sol-la-ti-DO" (with
the tonic pitch on the downbeat). And of course that is what we have
here. So our physicist, with a little help, has found periodic
patterns of duration within the musical signal. She can then assign
the following index to these spans:
c-d-e-f-g-a (r) d-e-f-g-a-bb (r) b-c-b-c-b-c (r) d-c-bb-a-g-f
2-1-2-1-2-1-2-1-2-1-2-1-2-1-2-1--2-1-2-1-2-1-2-1-2-1-2--1-2-1-
2---1---2---1---2---1---2---1----2---1---2---1---2----1---2-
1-------2-------1-------2--------1-------2--------1-----
1---------------2----------------3----------------4-----
[7] So, the physicist can assign a unique label each note/event based
on its position within the hierarchy of periodicities (for example,
the "a" which starts the second rhythmic/motivic group can be labelled
as occurring at "1.2.1.2," that is, Measure #1, 2nd half-note span of
Measure #1, first quarter-note span of 2nd half note, 2nd
eighth-note-span of first quarter-note). But (again, leaving the
time-span versus time-point question aside for the moment) what does
the physicist have? Is it meter? More to the point, what can the
physicist *do* with her periodic recognition? Well, along with a
particular pitch, duration, waveform, articulation, and dynamic, she
can now give each musical event a location (e.g. 1.2.1.2) relative
to the other events in this passage.(5) Our physicist now has an
acoustical correlate for meter (though we admittedly have fudged
quite a bit on the problem of finding the downbeat). While our
physicist is now happy, our theorist is not. Clearly something is
lacking in this meter-as-periodicity account, and that something is
the listener, or rather, the listener who can use meter as something
more than a temporal yardstick.
=========================================
5. The physicist's metric labels function as a temporal index for each
event--but one could also give an equally-useful index (at least for a
physicist) by simply labelling each event at the 8th-note level as
1,2,3,4,5,6,7 . . etc. What we have here is an index that at least
appears to be hierarchically organized--but is it? The physicist is
counting in modulo 2 up to the level of the downbeat--but does this
counting method create a true set of subordinate and super-ordinate
relationships? The answer to this question depends, I think, on how
the counting system is applied *to* the events, which is rather
different than reading the counting system *off of* the events. So
for the moment, all we can say is that we have a method of locating
musical events relative to others--a sophisticated measurement of a
note/event's being "before X and yet after Y." I would also add that
we should not be troubled by the fact that this system is relativistic
(i.e. dependent on the knowledge of surrounding events)--for after all,
the assignation of scale degree and harmonic function in the pitch
domain is similarly relativistic, as they are not context-free
attributes of a musical sound the way that frequency measurements are
context-free descriptions of sonic events.
=========================================
[8] Our unhappy theorist has read a bit of psychology; he knows that
when we listen to a periodically regular stimulus (to speak in
psychological terms for a moment), psychological experiment has shown
that we tend to respond by "entraining" our perceptions; that is, we
tune our rates of attending to the rhythms present in our environment.
As a result of this entrainment, we anticipate the occurrence of future
events. We also seem to have particular range of sensitivity (whether
learned or innate is another question) for attending to periodicities,
what psychologists have termed "natural pace" or "preferred tempo"
which falls in the range of 60-120 beats/minute.(6) Given these
cognitive proclivities, we would expect that not only is meter used to
give a location to previous events, but also to anticipate the location
of future musical events. Furthermore, the presence of this
projected/anticipatory framework can and at times does affect our
interpretation of the ensuing musical events. Thus, to assuage his
displeasure, our unhappy theorist turns to the current literature to
see if his colleagues have provided an account of meter which can
accommodate both the physicist's periodicity and the psychologists
entrainment.
=========================================
6. For a detailed overview of psychological research in these areas
see Justin London, "The Interaction Between Meter and Phrase Beginnings
and Endings in the Mature Instrumental Music of Haydn and Mozart," Ph.D.
Diss., U. of Pennsylvania (1990), Ch. 1, and David Butler, *The
Musician's Guide to Perception and Cognition* New York: Schirmer (1992).
=========================================
THEORETICAL STRATEGIES FOR DEFINING METER
[9] There are three broad strategies for defining meter and metric
accent: (a) one may divide time spans into smaller chunks, and then
sub-divide the chunks, and so forth, with meter as the fallout of this
segmentation process; (b) one may have an emergent hierarchy of time
points independent of (though still interdependent with) concomitant
durations; or (c) one may have an ordered series of time points (that
is, counting patterns) whose accent is not hierarchically determined by
"external" factors, but rather whose generative process itself gives
rise to a modest time-point hierarchy.(7) In the previous paragraphs
our physicist used the first strategy to determine the meter in example
3. The end product of all three strategies is a set of temporal
locations for musical events--either the "edges" of real durations,
or time-points apart from durational phenomena. But if we assume
that meter is crucially linked to our cognitive process, then we must
ask which of these three strategies is best suited to the way(s) we
actually deal with musical structures in our real-time listening
experience--in other words, meter-as-heard.
======================================
7. Given the scope of this essay I wish to avoid specific critiques
of other recent work in metric theory. Also, several fine overviews
of current work in metric theory are available elsewhere; see Fred
Lerdahl and Ray Jackendoff, *A Generative Theory of Tonal Music,*
Cambridge, MA: MIT Press (1983), esp. Chs. 2 and 4; Jonathan Kramer,
op. cit., Ch 4; and Justin London, op. cit., Chs 1 and 5.
======================================
[10] The first observation one might make regarding meter-as-heard is
that we can differentiate two distinct phases of metric cognition.
The first phase involves the initial recognition/discovery of the
metric context, as happens either (a) at the very beginning of the
piece, or (b) when we find ourselves thrown into a piece *in medias
res* (as when we turn on the radio to the middle of a symphony or
blues song). The second phase involves the continuation of an
established context. The cognitive tasks are very different in these
two phases. The first involves a rather high processing load, as
every event an equal amount of metric significance (or potential
significance). At the same time the listener is searching to find
the most salient parameter(s) for metric information. Fortunately,
in this first phase normally we are not trying to re-invent the metric
wheel, as it were, but rather simply trying to match the initial
series of musical events to a small number of metric archetypes.(8)
Once the meter has been recognized the cognitive load drops considerably.
Now the listener is entrained and needs relatively little information
to maintain the metric pattern. Indeed, as is well known, we will
continue to maintain the chosen pattern even when confronted with a
fair amount of contradictory information (e.g. an extended passage of
syncopation, or a series of stressed weak beats, etc.). In order to
break or shift an established metric pattern we must be presented with
a strong and continuing series of cues in order to achieve a metric
reconfiguration.
======================================
8. The number of metric archetypes is quite small: binary or ternary
patterns of beats, and simple versus compound subdivisions of those
beats. Moreover, I would posit that listeners have a store of
durational and pitch/durational templates which fit into the metric
archetypes. And so, for example, our anacrustic figure in example 3
involves the matching of a pitch-durational sequence (sol-la-ti-do
in even durations) into a limited number of metric possibilities. If
the notes are assumed to be subdivisions of the beat (a reasonable
assumption given the notion of natural pace) then the metric
recognition task boils down to simple versus compound subdivision--that
is, if do is the downbeat, then are the beats quarters or dotted
quarters? Furthermore, in actual performance (that is, by a human
player as opposed to a "deadpan" realization on a synthesizer) it is
likely that timing and dynamic cues within the sol-la-ti anacrusis
would indicate simple or compound time; see, for example, Eric F.
Clarke, "Categorical Rhythmic Perception: An Ecological Perspective"
in *Action and Perception in Rhythm and Music,* ed. Alf Gabrielsson,
Stockholm: Royal Swedish Academy of Music (1987): 19-33.
======================================
[11] While all three metric strategies listed above may be used as means
for metric recognition/ discovery--that is, during the first phase of
metric cognition--one realizes that the first two strategies create
problems in the second phase of metric cognition in that they allow
only for the retrospective hearing of metric patterns. In the case of
time-span segmentation this limitation is readily apparent, for one
cannot begin sub-dividing a time span until its duration is complete.
For the hierarchically-minded time-span segmenter this becomes an
especially acute problem, for if the determination of the beat is
determined by the partitioning of the measure, one must first have the
location of the downbeats. But if the downbeats are determined by the
partitioning of the next-larger span, then one must wait--and so on,
and so on--and thus one does not know the location of the first beat
(if a top-down partitioning plan is rigorously followed) until the
piece is over (!). The problem is alleviated somewhat if we employ
the second strategy and consider meter to be a hierarchy of time
points built from the bottom up. Here we can (usually) read the
lowest level of subdivision "right off the surface," as it were.
As soon as periodicities emerge we can retrospectively (but relatively
quickly) tag particular moments at higher levels, such as the beat.
Downbeats remain a potential problem, however, in that one is always
looking backward for cues which mark a higher-level metric
articulation.(9)
======================================
9. Downbeats are, by definition, metrically-accented beats. Yet there
are a considerable number of problems with the notion of an accented
beat, for how do phenomenal properties, such as length, loudness,
contour salience, tonal emphasis, etc. inhere in a time-point? While
it is clear that these parametric differences can and do give rise to
rhythmic accent, how can one connect these to a nearly durationless
instant? Well aware of these problems Wallace Berry has spoken of
downbeats as "iterative impulses" (*Structural Functions in Music*,
2nd ed., Mineola: Dover Publications (1987): 327) and Kramer has
approached metric accent as an "accent of initiation" (op. cit.,
p.86). The problem here is that in order to recognize that something
has been initiated, it must endure for a while (and the higher the
structural level, the longer one must wait), and thus metric
accents--if they are accents of initiation--can only be tagged
retrospectively.
======================================
[12]. Fortunately, there is a fairly simply solution to the problem of
retrospective metric hearing, and that is to combine a model of meter as
hierarchic patterning of time-points with a knowledge of metric
templates and our proclivities for entrainment--in other words, let's
tap our feet and count along. In counting along, we not only mark
locations for events as they occur--we also anticipate the locations
(and musical salience) of future events. This is a *dynamic* model of
meter which assumes that meter is an active part of the listening
process. It is the listener who, once the meter has been recognized,
creates the "generative process that gives rise to a modest
time-point hierarchy." This assumption is admittedly restrictive, in
that meter, for the most part, requires known archetypes. Similarly,
since metric hearing is assumed to be a form of temporal entrainment,
it demands that metric patterns be largely isochronous.(10) However,
the dynamic model accords nicely with the simulations of metric
attention proposed by Gjerdingen, where events at metrically important
locations are assumed to be of greater structural importance, as well
as Clarke's experimental studies of metric perception, which use known
metric patterns along with two basic durational categories (long vs.
short) to account for a wide variety of rhythmic phenomena.(11)
======================================
10. Two comments regarding these restrictions: first, I stress/repeat
that the known archetypes are not "2/4," "3/4," "6/8," "12/8" but rather
a matrix of "duple or triple" orderings for three to four layers of
time points. Thus one could "build" a template for a new or unusual
metric pattern even if one had not experienced it before. Second,
and following from the first, by "largely isochronous" I mean that
most (but not necessarily *all*) layers of the metric pattern be
isochronous. One can entrain to a pattern which encompasses some
irregularity. For example, Dave Brubeck's jazz standard "Blue Rondo
al a Turk" is based on a complex meter of 2+2+2+3/8, where the
8th-note level is isochronous, the downbeat level is also isochronous,
but the intermediate "beat" level is not. Nonetheless, once the
pattern is recognized, one can tap along to the "limping" meter quite
nicely.
11. Robert O. Gjerdingen, "Meter as a Mode of Attending: A Network
Simulation of Attentional Rhythmicity in Music," *Integral* 3 (1989):
67-91; Eric F. Clarke, op. cit.
======================================
THE ONTOLOGICAL STATUS OF METER
[13] If we adopt the meter-as-counting-time-point-patterns model, we
have made some rather far-reaching commitments regarding meter's
ontological status. Under this framework meter is a listener-generated
construct that is intertwined with the musical surface. Meter is not
"part of the music" in the same way that pitch, timbre, and duration
*are*. This commitment may be more troubling for some theorists than
others, and to explain (at least in part) this uneasiness I will
arbitrarily divide my colleagues into two groups, the "structuralists"
and the "phenomenologists."
[14] The structuralist regards music as existing "out there," apart
from the listener, and thus treats our listening and cognition
experiences as our efforts to understand these external sound
objects. Given this assumption, meter as I have defined it is a
particular kind of response to a particular kind of sound stimulus.
As such, meter would then seem to be in the same basket as our other
responses to sonic stimuli, such as feelings of sadness, surprise, or
pain (if the music is unbearably loud), evoked remembrances, and so
forth. This stimulus-response approach to meter, with its behaviorist
overtones, is justifiably suspect. By contrast, the phenomenologist
regards musical structure(s) as the product of the interaction between
a sound object and our cognitive faculties; she disdains the notion
that music qua *music* is only an external sound object, separate from
the listener. For her the meter-as-counting model is more plausible.
While meter is *not* part of the sound object, it nonetheless may still
be regarded as "part of the music."(12) Meter is neither a parameter
like pitch or timbre, nor is it part of a nested measuring of
durational patterns and/or periodicities. It is something that is
heard and felt. And this is of course why the physicist has so much
trouble with meter, for physics is not phenomenology. The physicist's
job is to describe the structure of physical objects in the world.
Understanding our interaction with those objects is beyond the scope
of the physicist's mission--at least if we stay above the quantum level.
======================================
12. Of course, she now has another problem, and that is to confront
the different kinds of "phenomenological fallout" that the interaction
between the sound object and the listener may generate. For clearly
one would not want to put meter and remembrances of things past (as
triggered by hearing "our song") in the same cognitive/phenomenological
basket.
======================================
[15] The dynamism of the meter-as-actively-counting-time-point-patterns
explains how and why we hear loud rests and metric articulations, as
well as meter's propulsive character. In most cases our self-supplied
metric articulations go unnoticed because most of the time they are
redundant: metric articulations at the levels of the downbeat, beat,
and beat-subdivision(s) tend to be phenomenally present somewhere in
the musical texture. What makes the *Eroica* example so striking is
the absence of that redundancy just where we expect it the most. For
at the very moment where we expect the culmination of a tissue of
musical processes, all we get is the "default" articulation of the
downbeat as we count along. With so much riding on that moment, the
little metric "click" we hear/create in our heads is deafeningly loud
indeed. In other cases, such as example 2, the metric clicks are not
so loud, but they nonetheless may be heard. A few theorists, most
notably Berry and Zuckerkandl, have at length described meter's
propulsive character.(13) Here is Zuckerkandl's aptly-worded account:
A measure, then, is a whole made up, not of equal fractions of time,
but of differently directed and mutually complementary cyclical
phases.. . . With every measure we got 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 ofthe summit, which immediately turns into a new
subsidence--a new wave has begun. (168)
He goes on to comment that:
Now we see the wrong-headedness of the doctrine that musical time,
that is, the grouping of beats into measures, springs from
differentiation of accents. There is no need for externally derived
accents in order to distinguish weak and strong beats from one
another and thus establish the metrical pattern. It is the wave
released by the regular succession of marks in the time flux that
in each case emphasizes the beat which falls on "one"; brings all
the beats between "one" and "one" into a group. The theory that
the metrical pattern depends upon accentual differences confuses
cause and effect. It is not a differentiation of accents which
produces meter, it is meter which produces a differentiation of
accents. (168-69)
If meter were a partitioning of time-spans or a hierarchy of time points
it would be difficult to see why meter should have such propulsive
properties, but these properties are the natural fallout of a dynamic
model. Indeed, under such a model it seems difficult to avoid such
properties.
======================================
13. Wallace Berry, op. cit., and Victor Zuckerkandl, *Sound and
Symbol,* translated by Willard R. Trask, New York: Pantheon Books,
1956.
======================================
[16] Embracing a dynamic model of meter is not without theoretical cost.
First and foremost, one must confront the ontological considerations of
meter noted above. If meter is still "part of the music," it is no
longer phenomenally part of musical sounds and structures in the same
way as pitch, timbre, dynamics, articulation, and duration. Since
meter is based upon known archetypes, it is a facet of musical
listening that is acquired, rather than innate (though metric hearing
probably does not depend on formal training)--and so the theorist
becomes interested in how we acquire such skills. As part of our
cognitive matrix for musical experience, our metric sensibilities
would also appear to be bound up with our other kinesthetic activities,
and thus that too becomes an area of interest. One is also perhaps
ruling out a number of structures that are often listed under the
rubric of meter as non-metric phenomena, i.e."mixed meters" (where
there is no substantially continuing metric pattern, but only a
succession of ever-changing metric notations) and thoroughly irregular
meters (as contrasted from the modestly irregular meter noted above).
And of course, the dynamic approach to meter creates large (and perhaps
insoluble) problems for hypermeter--but that is another paper.
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