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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) 1995 Society for Music Theory
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| Volume 1, Number 6 November, 1995 ISSN: 1067-3040 |
+-------------------------------------------------------------+
All queries to: mto-editor@boethius.music.ucsb.edu or to
mto-manager@boethius.music.ucsb.edu
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AUTHOR: Royal, Matthew S.
TITLE: Review of *The Analysis and Cognition of Basic Melodic
Structures* and *The Analysis and Cognition of Melodic
Complexity* by Eugene Narmour
KEYWORDS: Narmour, Implication-Realization, melody, analysis,
cognition, auditory streaming
Matthew S. Royal
University of Western Ontario
Faculty of Music
Talbot College
London, Ontario, Canada, N6A 3K7
mroyal@julian.uwo.ca
ABSTRACT: This article summarizes and offers appraisal of
Narmour's Implication-Realization model as presented in the
following volumes: Eugene Narmour, *The Analysis and Cognition of
Basic Melodic Structures: The Implication-Realization Model*, and
Eugene Narmour, *The Analysis and Cognition of Melodic
Complexity: The Implication-Realization Model*, both published by
the University of Chicago Press.
*Introduction*
[1] "There is at present virtually no viable conceptual framework
for the analytical criticism of melody."(1) Thus wrote Leonard
B. Meyer in 1973 before setting out to expound his theory of
implication in tonal melody in his volume *Explaining Music*. A
paragraph later Meyer went on to add,
Schenker's theories are primarily concerned to explain the
middleground and background organization of tonal music--the
large-scale structure. My concern will be to explain the
foreground and its adjacent levels.(2)
This quotation might also serve as a manifesto for the two recent
volumes by Meyer's colleague and one-time student, Eugene
Narmour, which build on the foundation laid in *Explaining
Music*.(3) The model presented in Narmour's books, the
implication-realization model, constructs a quite radical
conceptual framework for melody analysis, and, following in
Meyer's rather than Schenker's footsteps, takes as its central
focus the foreground note-to-note relationships between tones.
===============================================================
1. Leonard B. Meyer, *Explaining Music* (Berkeley, LA: University
of California, 1973), 109.
2. Meyer, *Explaining Music*, 109-110.
3. Eugene Narmour, *The Analysis and Cognition of Basic Melodic
Structures: The Implication-Realization Model* (Chicago and
London: University of Chicago, 1990); Eugene Narmour, *The
Analysis and Cognition of Melodic Complexity: The
Implication-Realization Model* (Chicago and London: University of
Chicago, 1992).
===============================================================
[2] The two volumes reviewed here complement each other to a
large extent, although, taken sequentially, the second provides
some minor modifications of the first. Volume 1, *The Analysis
and Cognition of Basic Melodic Structures: the
Implication-Realization Model* (hereafter called "Vol
1") explicates the basic building blocks of the theory, and
concentrates on note-to-note relations in melody cognition and
analysis. Volume 2, *The Analysis and Cognition of Melodic
Complexity: the Implication-Realization Model* (hereafter called
Vol 2"), explores relations within larger groups of notes and
relations between non-contiguous tones. Given that
Narmour's theory, at least in its present manifestation, is
relatively new, this review will attempt to provide a precis of
it before offering any critical appraisal.
[3] Taking the theory as a whole, several overarching goals may
be identified. Firstly and most generally, Narmour aims to
explain the "'genetic code' of melody" (Vol 1, p. xiv),
to provide a taxonomy of archetypal melodic shapes. The
implication-realization model thus strives for parsimony in that
it tries to classify the infinite number of possible melodies
according to a short list of melodic archetypes.
[4] Secondly, Narmour's approach is bottom-up. Although, in the
analysis of melody, considerations of style, culture, and the
listener's previous experience (various types of memory) come
into play, the theory largely parses melodies from note to note
on the basis of the raw parameters of musical sound. These
parameters include interval size, direction, rhythmic duration,
dynamic accent and so on. Thus, even quite low-level culturally
specific cognitive constructs such as scale degree, are regarded
as secondary in importance to the theory.
[5] Thirdly, the word "cognition" alongside the word "analysis"
in the titles of both volumes signals the theory's strong
perceptual and cognitive leanings. That is to say, although
Narmour develops a sophisticated symbology in order to perform
analyses on the score, his ultimate aim is to describe how
listeners hear melody. To this end the implication-realization
model rests on hypotheses substantiated or suggested by the
research literature of music perception and cognition. In this
respect at least, Narmour's orientation is similar to that of
Lerdahl and Jackendoff.(4) In line with this psychological bias,
at the end of the first volume reviewed here, Narmour proposes a
number of experimental studies intended empirically to test the
validity of his theory.
===============================================================
4. Fred Lerdahl and Ray Jackendoff, *A Generative Theory of Tonal
Music* (Cambridge, MA: MIT Press, 1983).
===============================================================
[6] Fourthly, and perhaps most importantly, the words
"implication-realization" betoken the theory's concern with
tracing the listener's changing expectancies over time, and the
extent to which those expectancies are realized or denied. Thus,
based on bottom-up implications in the various parameters of
musical sound, the theory measures listeners' ongoing levels of
surprise as an aesthetic response to music as well as listeners'
ongoing perceived structural closure.
*Basic Archetypes*
[7] As intimated above, the most far-reaching aspect of Narmour's
theory is that it postulates a set of principles of melodic
analysis and cognition that work on individual parameters of
musical sound from the bottom up. These principles are wholly
divorced from any influences of style, culture or musical
ability/training on the part of the listener. As such they are
presumed to be universal to all listeners and therefore, from an
analytical point of view, universally applicable to all types of
melodies, regardless of historical or geographical provenance.
[8] Narmour's primary hypothesis is that any two successive
pitches (i.e. one melodic interval) imply a third pitch (a second
interval). Thus, at the most basic level, melodies can be
divided up into elemental building blocks of three pitches, with
pitches one and two of each block forming the antecedent interval
of "implication," and pitches two and three forming the
consequent interval of "realization." Whether the antecedent
implicative interval is completely realized, only partially
realized or denied depends on size and registral direction of the
consequent interval. These two parameters, interval size and
direction, lie at the heart of Narmour's theory, yet are
envisaged as working independently in melody cognition.
[9] The parameter "interval size," designated in Narmour's
symbology by the abbreviation "I," is for most purposes
dichotomized into small and large intervals. Small intervals
range from the unison to the perfect fourth, and large intervals
comprise the perfect fifth and wider. The tritone is an
ambiguous interval that can be counted as either small or large,
depending on the context.
[10] In addition, the relationship between successive intervals
is characterized by calculating the difference between them (for
example a major 3rd followed by a perfect 5th have a difference
of three semitones or a minor 3rd). Depending on their
intervallic difference, antecedent and consequent intervals are
"equal" (difference of a perfect unison between two intervals),
"similar" (difference of less than a minor or major 3rd) or
"different" (difference of more than a minor or major 3rd).
Whether the difference limen between two intervals is a minor or
a major third depends on the shape of the three-pitch structure
in question (see Vol 1, pp. 85-88 for the reasoning behind
choosing particular difference limens).
[11] The parameter "registral direction," designated by the
abbreviation "V" (from the word "vector"), has three categories:
up, down and lateral (or a repeating pitch). Up followed by up,
down followed by down or lateral followed by lateral are counted
as continuations of registral direction, and any other
combinations are counted as changes of registral direction.
[12] By applying the Gestalt principles of good continuation,
similarity and proximity to the parameters of interval size and
registral direction, Narmour posits two basic three-pitch melodic
archetypes. Narmour hypothesizes that if a listener hears an
initial small interval (i.e. narrower than a tritone), that
interval will imply a second similarly small interval continuing
in the same registral direction. If this implication is realized
by the third pitch, giving an up-up or down-down motion comprised
of two small intervals, then the resultant melodic shape is a
"Process" (designated by the symbol "P"). Examples of Processes
would be C4-D4-Eb4 (assuming now and hereafter that C4 is "middle
C"), or A#3-F#3-D3. If the interval of implication is a unison
(giving lateral motion between the two pitches) then the listener
will expect a second interval in the lateral direction, giving a
shape of three repeated pitches. When this implication is fully
realized the result is what Narmour calls a "Duplication"
(designated by the symbol "D"). An example of a Duplication is
D#2-D#2-D#2.
[13] The structures P and D arise, therefore, from the
realization of melodic implications generated by Gestalt laws.
However, Narmour also posits the operation of an opposite,
counterbalancing law, namely reversal. It should be stressed
that no "law" of traditional Gestalt psychology embodies
reversal, and thus it is not so solidly grounded in accepted
perceptual principles as are P and D. Rather, evidently
influenced by Meyer's notion of "gap-fill melodies," the author
seems to have included reversal for theoretical completeness.(5)
===============================================================
5. Meyer, *Explaining Music*.
===============================================================
[14] In contrast to the implication of a small antecedent
interval, when an initial large interval (i.e. larger than a
tritone) is heard, it implies a small consequent interval
changing direction. In other words a large antecedent interval
implies both a change in interval size and a change in direction.
If the implications in both the parameters of interval size and
registral direction are realized, then a third basic melodic
archetype is generated, namely a "Reversal" (designated by the
symbol "R"). Examples of Reversals are A3-F4-Eb4, C#5-F#4-A4 or
D4-C#5-C#5. (Note that the last example involves lateral motion
between the second and third pitches, up-lateral being a change
in direction.)
[15] The three-pitch structures P, D and R, the primal structures
of Narmour's theory, are generated when melodic implications in
both parameters of interval size and registral direction are
realized. However, a further group of three-pitch structures are
possible when an implication in only one or the other of the two
parameters is realized. As mentioned above, a small antecedent
interval (other than a unison) implies a similarly small interval
continuing in the same direction. If the interval-size parameter
implication alone is realized, i.e. small-small but involving a
change of direction, then the resultant three-pitch structure is
called an "Intervallic Process", or "IP." Examples of IPs are
C4-E4-D#4 or Bb3-F3-G3. If only the direction parameter
implication is realized, i.e. up-up or down-down but moving from
small to large, then the resultant structure is called a
"Registral Process," or "VP." Examples of VPs are D4-E4-C#5 or
G3-E3-A2.
[16] Corresponding types of partial realizations are also
possible with large antecedent intervals. Thus, as mentioned
above, a large antecedent interval implies a change of interval
size and registral direction. If, however, just the interval-
size parameter implication is realized, i.e. large-small but
evincing no change in direction, then the resultant structure is
an "Intervallic Reversal," or "IR." Examples of IRs are C3-B3-C4
or D#5-G#4-F#4. If, on the other hand, just the direction
parameter implication is realized, i.e. a change in direction but
the second interval is larger than the first, the resultant
structure is a "Registral Reversal," or "VR." Examples of VRs
are Eb3-Bb3-C3 or D5-E4-G#5.
[17] The last type of partial realizations to be considered are
those derived from the Duplication. Firstly in this regard,
Narmour argues that a "Registral Duplication" [VD] does not
exist, since one cannot have realization of a lateral registral
direction (unison-unison) without also repeating the antecedent
interval and therefore realizing the intervallic implication.
Therefore the only remaining derivative of the Duplication is the
"Intervallic Duplication," or "ID."
[18] In the opinion of this reviewer, Narmour's use of term
Intervallic Duplication is inconsistent given the logic of the
other structures so far identified. Narmour identifies any pitch
structure consisting of any two identical small intervals moving
in opposite directions as an ID. A defining characteristic of an
ID is therefore that the first and third pitches are identical.
Thus, examples of IDs would be C4-D4-C4 or Bb3-F3-Bb3. The
reason given by Narmour for calling this type of structure an
Intervallic Duplication seems to be that the antecedent interval
size is replicated exactly in the consequent interval, but the
registral direction of the antecedent is denied by the registral
direction of the consequent. However, this argument treats the
structure as a complete whole, in retrospect, whereas all other
structures hitherto have been derived by considering the
realization prospectively. For example, with the C4-D4-C4
structure, when the listener hears the antecedent interval
between C4-D4, according to Narmour's theory he/she will expect a
small consequent interval that continues upward (with a third
pitch of E4 or F4, for example). When the third pitch, C4,
occurs, admittedly the implication is partially realized
(interval size is realized and direction is denied), but why is
it a Duplication that has been partially realized? Why, on the
basis of the first two pitches, would the listener expect a
Duplication rather than a Process? This quirk is made more
confusing by the fact that structures such as C4-C4-D4, whose
antecedent interval *is* in fact a unison, are labelled as
Intervallic Processes or IPs (see Narmour Vol 1, p.358, Ex.
21.7). In this latter case, with the antecedent unison (C4-C4),
the listener might reasonably expect the lateral registral
direction to be repeated in the consequent interval (resulting in
C4-C4-C4), so that when the D4 sounds the prospectively implied
lateral motion is denied. This type of structure seems to me to
be a more logical candidate for the title "ID," than does a
structure such as C4-D4-C4. My argument is one of nomenclature
and ultimately, if accepted, does no great damage to Narmour's
theory. However, the reader interested in pursuing Narmour's
theory further may find it helpful to remember the ID as a
specific subset of IP, an ID being an IP where the first and
last pitches are identical.
[19] The structures so far summarized, in which the implication
in at least one parameter is realized, Narmour calls
"prospective" because at least one of the listener's unconscious
expectations is fulfilled. In contrast, a third group of
structures in which both interval-size and registral-direction
implications are denied, Narmour dubs "Retrospective". They are
thus called because it is only on the completion of the three-
pitch structure that the true contour of the structure becomes
apparent; in prospect, on the basis of the antecedent interval,
the listener expects something quite different. For example, if
the listener hears a large antecedent interval, this implies a
small consequent interval in the opposite direction. If,
however, what actually follows is a similarly large interval in
the same direction, then implications in both parameters have
been denied. The resultant structure is here a "Retrospective
Process" since, like a Process, the structure contains two
similarly sized intervals in the same direction, but, unlike a
Process, the initial interval is large. An example of a
Retrospective Process would be C4-A4-F#5.
[20] In Narmour's theory, every prospective structure has a
retrospective counterpart, with retrospective structures being
designated by the symbol for their prospective counterpart in
parentheses. Thus, for example, a Retrospective Process is
symbolized by "(P)." Generally, retrospective structures have
the same directional characteristics and the same relationship
between the antecedent and consequent intervals (same, similar,
or different) as their prospective counterparts. Retrospective
structures differ from their prospective counterparts, however,
in the size of their antecedent interval. Therefore, if a
prospective structure has a small antecedent interval, then its
retrospective counterpart will have a large antecedent interval,
and vice versa. An exhaustive account of the logic behind every
retrospective structure is beyond the scope of this review.
Those interested readers should consult Vol 1, pp. 195-198,
and ch. 14.
[21] For the sake of convenience, Table I below summarizes all
the prospective and retrospective structures with their
intervallic and directional characteristics. For a summary of
these structures in a different order with additional
information, see Narmour Vol 1, Appendix 2, pp. 435-436.
Table I
Category Structure Intervals Direction
Ant Con Ant-Con
----------------------------------------------------------
Processes P sm sm sim (/= M3) contin
(VP) la v.la dif (>/= M3) contin
Dupli- D uni uni = (= P1) contin
cations (D) uni uni = (= P1) contin
ID sm sm = (= P1) change
(ID) la la = (= P1) change
Reversals R la sm dif (>/= m3) change
(R) sm v.sm dif (>/= m3) change
IR la sm dif (>/= M3) contin
(IR) sm v.sm dif (>/= M3) contin
VR la v.la dif (>/= m3) change
(VR) sm la dif (>/= m3) change
----------------------------------------------------------
[22] The "Category" column divides the structures into Processes,
Duplications and Reversals. The abbreviations for all the
structures are given in the second column. Intervallic
information is divided into three columns. The "Ant" column
describes the size of the antecedent interval, where "sm" =
"small", "la" = large and "uni" = "unison" according to the
criteria outlined earlier in this review. The "Con" column
describes the size of the consequent interval using the same
abbreviations as the "Ant" column, as well as "v.sm" and "v.la"
which stand for "very small" and "very large" respectively. The
adverb "very" indicates an interval that is even smaller or even
larger than the small or large antecedent interval. The column
"Ant-Con" describes a comparison between the antecedent and
consequent intervals, and provides two pieces of information.
The first piece of information compares the size of the
antecedent and consequent intervals as "=" (exactly equal) "sim"
(similar) or "dif" different. The second piece of information in
the Ant-Con column gives the requisite absolute difference
between the antecedent and consequent intervals for these
intervals to be classified as a particular structure. Here "m" =
"minor," "M" = "major" and "P" = "perfect." Thus, for example,
for the antecedent and consequent intervals to be counted as
similar in a Process structure, the difference between these
intervals must be less than or equal to a minor third. The final
column, "Direction" compares the registral direction of the
antecedent and consequent intervals, with "contin" signifying a
directional continuation and "change" signifying a change of
direction.
*Structures arising from Parametric Interaction*
[23] All of the structures summarized in Table I are derived
purely from implications and realizations in the pitch domain.
However, one of the great strengths of Narmour's theory is that
it also incorporates other parameters of musical sound as
centrally important determinants of analysis. Thus, in naming
particular melodic structures, the analyst must also take into
account note durations, dynamic accent, consonance and so on. In
the parameter of duration, for instance, Narmour identifies three
types of relationships that may inhere between successive note
values, namely "additive" (isochronous repetition of the same
note duration), "countercumulative" (longer to shorter) and
"cumulative" (shorter to longer). Cumulative rhythms may serve
to suppress the melodic implication between two pitches if the
second note is greater than or equal to 1.5 times the duration of
the first note. If this durational relationship is present, for
example in the rhythm quarter->dotted-quarter or quarter->half,
then the interval between the pitches having these rhythmic
values is simply identified as a dyad. A dyad is thus a
disconnected interval whose implication fades because the
listener has to wait too long for the realization. Note that
this type of suppression of an implication is to be distinguished
from a denial; in a denial the implication is still pending when
the unexpected tone of realization sounds. In Narmour's
symbology a dyad is simply labelled with a numeral indicating the
width of the interval, e.g. "6" for a major or minor sixth, "4"
for a diminished, perfect or augmented fourth and so on. (Vol 1,
pp. 391-410).
[24] Rests (silences) can also function to suppress implications
and therefore perceptually isolate particular pitches. For
example the rhythm quarter->quarter->quarter-rest will isolate
the pitches of the first two quarters as a dyad. Again, whatever
consequent interval may have been implied by the interval between
the first two pitches is suppressed by the silence that follows
this interval. Rests may also serve to isolate single notes so
that no continuation is even implied. These single-note
structures are called "Monads" and symbolized on the score by the
letter "M" (Vol 1, pp. 410-414).
[25] Returning to Table I, all the structures listed here consist
of three pitches in their most basic format. Thus if one
imagines the rather artificial situation of a perfectly
isochronous monophonic series of pitches, undifferentiated as to
dynamic and timbre, and with no meter suggested, then this melody
would be analyzed as a series of linked three-pitch structures
where the third and final pitch of one structure becomes the
first pitch of the next structure. Such a sequence is analyzed
after Narmour in Example 1 below:
Example 1
___P_____P______R____ID_____IP____VP_____VR___(R)__
! ! ! ! ! ! ! ! !
C4-D4-E4-D4-C#4-A4-A4-G4-A4-F4-Bb4-A4-D4-B4-C4-F4-E4
______
! ! = a horizontal square bracket
However, if the series of pitches in Example 1 were also marked
by durational, metric, dynamic, harmonic and articulative
accents, then, insofar as these accents coincided, Narmour's
analytical methodology would attempt to reflect the grouping
suggested by these accents. I have already noted how durational
cumulation or use of rests can cause individual pitches or pairs
of pitches to be isolated and thus form Monads or Dyads. By the
same token, if metric accent, change of harmony, articulation or
whatever suggests a grouping of more than three pitches, then
Narmour makes use of two further structure types: "combinations"
and "chains."
[26] A combination is simply the interlinking of two basic
melodic structures so that the consequent interval of realization
of the first structure is also the antecedent interval of
implication of the second structure. A combination therefore
consists of three intervals (four pitches) that cohere owing to
the operation of parameters other than interval size or
direction. Consider Example 2:
Example 2
Analysis: ____IPP___
! !
___P___
! !
___IP__
! !
Melody: C4-B3-D4-F4
X X
Harmony: C+: I IV6
In this example the three pitches C4-B3-D4 form an Intervallic
Process [IP], while the pitches B3-D4-F4 form a Process [P].
However, because B3 and D4 are dissonant with the underlying
harmony (Narmour uses an X to indicate dissonance), and because
of the change of harmony on the F4, the most stable and salient
pitches in this pattern are C4 and F4. the boundary pitches.
To reflect this four-pitch structure the analysis shows the
combination IPP rather than two separate structures. Notice
also that, whereas in Example 1 adjacent structures share only
one pitch in common, the two structures that make up a combination
share two pitches (one interval) in common. Combinations can
occur in numerous situations where four pitches are naturally
grouped together, and those readers interested in extensive
examples should consult Vol 2, chs. 2-7. Narmour identifies
217 possible archetypal combinations, all of which are listed
and exemplified in Vol 2, Appendix 5, pp. 391-396.
[27] Having briefly described combinations, chains take the
process of expanding the basic melodic structures one stage
further. A chain is the interlinking of *three or more* basic
melodic archetypes. As is the case with combinations, each
archetype within a chain shares two pitches (one interval) with
its immediate neighbours. Example 3 shows a modification of
Example 2 to give the chain IPIPIDP:
Example 3
Analysis: ____IPIPIDP_______
! !
___IP_____P___
! ! !
___IP____ID___
! ! !
Melody: C4-B3-D4-C#4-D4-F4
X X X X
Harmony: C+: I IV6
Since chains can be any number of pitches greater than or equal
to four in length, the number of possible chains is theoretically
limitless. However, the vast majority of the chains that Narmour
identifies when he expands on the subject at length in chapters 8
through 10 of Volume 2, are no longer than four or five pitches.
Again, as was the case with combinations, it should be stressed
that chains frequently arise from the operation of parameters
other than interval size or direction.
[28] It will be recalled that one of the purposes of Narmour's
theory is to record the ebb and flow of melodic closure as a
series of notes is unfolded in time. All of the structures
introduced thus far embody different, and quantifiable degrees of
melodic closure according to Narmour's theory. In fact, Narmour
proposes that closure can occur in one or more of six
parameters (see Vol 1, pp. 11-12):
1. when a rest or repetition interrupts and therefore
suppresses an implication, e.g. a dyad;
2. on a metrically strong beat;
3. on a harmonically consonant tone, following a harmonically
dissonant tone (a resolution);
4. on a long duration following a short duration (cumulative
rhythm);
5. when a large melodic interval moves to a small melodic
interval (the pitch concluding the small interval is
closural--if the reverse occurs, namely a small interval
followed by a large one, then non-closure is the result);
6. when registral direction changes (the pitch concluding the
change is closural--lateral to lateral is neutral so far as
closure is concerned, and up to up or down to down are non-
closural).
As might be expected, several of these parameters allow multiple
gradations of closure. Harmonically, the greater the difference
in consonance between a dissonance and its resolution, the
stronger the resolution (Vol 1, pp. 288-289). In his analyses,
Narmour makes use of three gradations of dissonance, and three
gradations of consonance or resolution, "weak," "moderate" and
"strong" in each case (see Vol 2, Appendix 2, p. 375 for
definitions and symbology of each). Incidentally, although this
is regrettably not made explicit, Narmour seems to treat
dissonance and consonance as psycho-acoustic phenomena, and thus
maintains that they act from the bottom-up rather than being
stylistically determined. Durationally, the greater the ratio of
the long duration to the short duration, the greater the closure
of the cumulation (Vol 1, pp. 292-293 & p. 314). Intervallically,
the greater the difference between the large and the small
intervals, the greater the closure (Vol 1, pp. 283-284 & p. 308).
[29] In addition to the multiple gradations of closure along
several of the parameters, it is also important to realize that
points of closure in different parameters may or may not coincide
with each other (what Narmour calls "parametric congruence" and
"parametric non-congruence" respectively). The greater the
number of parameters that evince closure on a particular note,
the greater the overall closure of that note. Taking the
gradations of closure on one parameter together with the varying
degrees of parametric non-congruence, one can see that Narmour's
theory posits a wide range of types and degrees of closure,
allowing for many subtle gradations of closure on a particular
tone.
[30] For the purposes of analysis, however, Narmour divides
degrees of closure into three categories: "articulation,"
"formation" and "transformation." Articulation occurs when
closure among the parameters is completely non-congruent.
Formation is a stronger form of closure than articulation, and it
occurs when closure among the parameters is partially non-
congruent. Transformation occurs when closure among the
parameters is sufficiently congruent for a note to be labelled as
the beginning or end of a structure, that structure being a basic
archetype [P, D, R, IR, VP etc.], a combination or a chain. As
was intimated earlier with the C4 and F4 in Example 2, for
instance, another way of conceptualizing transformational tones
is as the stable boundary tones of a coherent melodic grouping.
[31] As a theory of melodic perceptual grouping, I think the
implication-realization model is very successful. Previous
theories, those of Cooper and Meyer and Lerdahl and Jackendoff
being the most eminent, have viewed groups as temporally
contiguous yet mostly self-contained chunks;(6) only in the case
of elision do successive groups share a tone or chord. However,
one problem with such self-contained groups is the often
arbitrary placement of an accented tone as either a group-
initiating or group-ending accent. To be sure, some perceptual
research indicates that, for instance, dynamic accents tend to
initiate groups whereas durational accents tend to end groups.(7)
That being said, the experimental jury is still out on the sort
of grouping likely to result from parametric interaction. What
happens, say, when dynamic and durational accent coincide, as in
Example 4?
Example 4
Dynamic > > >
Rhythm half->quarter->half->quarter->half->quarter etc.
Grouping: a !___________! !___________! !____________!
b !___________! !___________! !__________
c !______________!______________!______________
Are the dynamically accented half notes group-initiating
(analysis a) or group-ending (analysis b)? Narmour's solution
(what would be analysis c) felicitously circumvents the problem.
By simply labelling the longer and louder tones as the group
boundaries he recognizes their greater salience and stability
while remaining agnostic as to their position within a group.
Also, to the extent that a graphic analysis on paper can ever be
analogous to the aural experience of sounds in time, analysis c
better captures the balance between continuity and segmentation
that is surely so central to melodic perception. Analyses a and
b visually chop up the flow of the rhythm in a manner that does
not accord with perception. Narmour's conception of grouping,
then, is a novel and welcome departure from previous theories.
===============================================================
6. Grosvenor W. Cooper and Leonard B. Meyer, *The Rhythmic
Structure of Music* (Chicago and London: University of Chicago,
1960); Lerdahl and Jackendoff, *Generative Theory*.
7. Herbert Woodrow, "The role of pitch in rhythm,"
*Psychological Review* 18 (1911): 54-77.
===============================================================
[32] Correct recognition of the transformational boundary tones
is necessary in Narmour's model because it allows the analyst to
begin to build a hierarchical analysis of a melody. The pitches
of transformational tones are the pitches that "survive" the
winnowing out process of analysis, and are consequently those
that are left at the next level up in a reductional hierarchy.
Example 5 reproduces the rather simple series of pitches from
Example 1 giving higher level analyses.
Example 5
_________________________3__________________________
! !
C4 E4
_________________________R__________________________
! !
C4 A4 E4
____________(VR)______________________R_____________
! ! !
C4 C#4 A4 D4 E4
______IP___________R__________(VR)__________IP______
! ! ! ! !
C4 E4 C#4 A4 A4 Bb4 D4 C4 E4
___P_____P______R____ID_____IP____VP_____VR___(R)__
! ! ! ! ! ! ! ! !
C4-D4-E4-D4-C#4-A4-A4-G4-A4-F4-Bb4-A4-D4-B4-C4-F4-E4
Lest, on the basis of Example 5, any reader should draw the
erroneous conclusion that tracing transformations from one
hierarchical level to the next simply consists of selecting every
second note, remember that the melody here exhibits no
combinations or chains. Since only those pitches at the
beginning or end of a square bracket would be found in the
analysis at the next level up, in combinations or chains the
reductional process would remove all the pitches that lie within
a bracket (i.e. more than every second pitch). For a richer
example involving varying degrees of durational, harmonic and
metrical closure see Vol 2, p. 32, Ex. 1.8.
[33] With regard to this Example 1.8, Narmour states that the
ability to extrapolate higher hierarchical levels from
note-to-note structures is desirable primarily for the sake of
analytical completeness:
The extent to which listeners actually transform
structural tones to such levels remains an open question.
The main point for the reader to observe is that, despite
the simplicity of the theory and the apparent paucity of
available structures, the analysis generates from the bottom
up a sophisticated hierarchical accounting of every melodic
tone.(8)
Despite the hierarchical component in the implication-realization
model, then, Narmour's primary interest in these two volumes is
still note-to-note relations. This predilection is reflected in
the fact that the vast majority of musical examples in both
volumes show solely surface-level analyses.
===============================================================
8. Narmour, Vol 2, 32.
===============================================================
[34] All of the structures, whether archetypes, combinations or
chains, and whether considered on the note-to-note level or on a
higher hierarchical level, are hypothesized as resulting from
bottom-up processes. However, Narmour is also ready to admit the
influence of top-down (learned or style-specific) processes on
melodic implication. One example that immediately springs to
mind is the strong pull that scale-step implications will have
for enculturated listeners of Western tonal music (e.g. the
leading-tone tending towards the tonic). Symbologically Narmour
accounts for such learned or style-specific influences by
appending the abbreviation "(xs)" (for "extra-opus style")
whenever these are deemed to affect melodic implication.
Similarly, when motivic references or simple repetitions within a
piece may be assumed to mold a listener's expectations, Narmour
uses the abbreviation "(os)" (for "intra-opus style").
[35] Recognition of the influence of tonality (xs) is
particularly necessary in these volumes since many of Narmour's
examples are of tonal, common-practice period melodies. The
author's reasons for relegating style-specific influences to
secondary variables in the theory are laid out lucidly and at
some length in Chapters 2 and 3 of Volume 1. However, given that
the theory does try to model perception of an "educated"
listener, one has to wonder whether the top-down influences might
not be more powerful than the bottom-up processes in many of
Narmour's examples. It is quite possible that the effect of
bottom-up implications would be more tangible in music for which
the average listener has not developed sophisticated schemata.
For this reason, not only is the implication-realization model
pan-stylistic, but it may actually be *better* suited for
analyzing music outside the (Western) common-practice period.
Interestingly, Narmour offers analyses of excerpts of pre-tonal
music in Appendix 3 of Volume 2, and analyses of non-Western
melodies in Appendix 4 (presumably as heard by a naive cultural
outsider). One other repertoire where listeners' top-down
schemata are likely less developed is 20th-century post-tonal
music. Therefore, given its ability to generate reductional
hierarchies using only bottom-up principles, Narmour's theory may
offer another useful perspective on the thorny problem of
post-tonal prolongation.
*Streaming and Tempo*
[36] One type of melodic texture that requires special attention
is pseudopolyphony, or what Berry calls a "compound line."(9) As
the reader will recall, pseudopolyphony occurs when an ostensibly
monophonic line gives the illusion of tracing two or more
polyphonic voices through rapid change of register, or rapid
intervallic leaps. This type of writing is the hallmark of
Bach's solo 'cello suites, for instance.
===============================================================
9. Wallace Berry, *Structural Functions of Music* (New York:
Dover, 1987; reprint, Englewood Cliffs, NJ: Prentice Hall, 1976).
===============================================================
[37] In fact, pseudopolyphony is one of the most studied types of
pitch pattern in experimental perceptual research, coming under
the general heading of "Auditory Stream Segregation" or simply
"streaming."(10) A number of converging experimental studies
suggest that the occurrence of streaming in the pitch domain is
dependent on two parameters: interval size and rate of occurrence
(tempo). In a now classic experiment, van Noorden showed that
the faster the tempo, the narrower must be the interval between
two pitches for those two pitches to form one coherent
stream.(11) If the tempo is faster than about ten tones per
second and the interval between these tones is a minor third or
wider then the two pitches will always form separate streams. At
a tempo of five tones per second stream segregation always occurs
at an interval of about a major ninth or wider. At the other
extreme, two tones separated by the interval of a major second or
smaller always cohere as one stream regardless of the tempo.
Between these extremes, van Noorden identified a perceptually
ambiguous region where most musical melodic patterns would lie.
Despite this ambiguity, however, it is possible to derive
measures of streaming for melodic patterns that estimate how many
perceptual streams are likely to occur at a given point.(12)
===============================================================
10. Albert S. Bregman, *Auditory Scene Analysis: The Perceptual
Organization of Sound* (Cambridge, MA: MIT, 1990); Stephen
McAdams and Albert S. Bregman, "Hearing Musical Streams,"
*Computer Music Journal* 3 (1979): 26-43, 60, 63.
11. Leon P.A.S. van Noorden, "Temporal Coherence in the
Perception of Tone Sequences" (Ph.D. diss., The Institute of
Perception Research, Eindhoven, The Netherlands, 1975).
12. David Huron, "Voice Segregation in Selected Polyphonic
Keyboard Works of Johann Sebastian Bach" (Ph.D. diss., University
of Nottingham, UK, 1989), ch 14.
===============================================================
[38] What is important to emphasize with regard to Narmour's
theory is that streaming is an automatic perceptual process that
occurs regardless of the style of music in which it is
embedded.(13) Therefore streaming is exactly the sort of
phenomenon that Narmour's bottom-up perceptual model should
capture. To be fair, Narmour is evidently aware of much of the
literature on streaming (see Vol 1, p. 352, footnote 5); however,
his recognition of the importance of streaming in melodic
analyses is somewhat inconsistent. For instance, Narmour's
Example 14.18 (Vol 1, p. 276), an extract from Handel's violin
sonata in G minor, HWV 368, contains the following pitches (all
eighth notes in groups of three) analyzed as combinations:
Example 6
___________P_________________
! !
(os)
___(P)VR_______PVR_______PVR______PVR___
! ! ! ! !
D4-Bb4-F5-Eb4-Bb4-G5-E4-C5-G5-F4-C5-A5-F#4 etc.
To be sure, the upper-level analysis captures the Process, P,
that is present between the Eb4, E4 and F4 that sound on the
beat. Unfortunately the lines that also emerge in the "middle"
and "upper voices" (Bb4-C5-C5 and G5-G5-A5 respectively) are not
reflected at all. In my opinion, in this case, Narmour's concern
with the note-to-note implications and realizations overshadows
the perceptually more salient stream segregation that is likely
to occur at this point.
===============================================================
13. William M. Hartmann and Douglas Johnson, "Stream segregation
and peripheral channeling," *Music Perception* 9 (1991): 155-84.
===============================================================
[39] By Volume 2, Narmour's treatment of streaming is more finely
honed. With Example 3.5 (Vol 2, p. 77) Narmour introduces the
idea of concurrent overlapping structures caused by streaming.
Here, he allows multiple analyses of a compound line:
one analysis traces the note-to-note structures while other
analyses trace the structures that arise from non-contiguous yet
registrally proximate pitches. This, I feel, captures quite
neatly the perceptual ambiguity that is inherent in many passages
of pseudopolyphony. However, Narmour is not always so vigilant
in this regard, there being a number of examples throughout the
two volumes where the stream segregation that would likely result
from abrupt changes of register (and return) is left by the
analytical wayside in favor of note-to-note implications and
realizations (Volume 1: Examples 8.17a p. 175; 10.10, p. 204;
14.17, p. 275; 21.3, Vol 1, p. 353. Volume 2: Examples 8.8,
p.183; 9.9, p. 195; 10.11, p. 227).
[40] My contention that streaming should figure more centrally in
a bottom-up theory of melody perception and cognition,
foreshadows a more general refinement that I shall offer, namely
consideration of tempo. The extent to which intervallic
relationships between contiguous pitches are registered by the
human auditory apparatus, and therefore the extent to which
implications based on interval size and direction can be generated,
will depend on the absolute rate at which those pitches sound.
[41] At one extreme, as the streaming research cited above
suggests, if notes sound too rapidly the intervallic and temporal
relationship between them becomes blurred. At the other extreme,
if notes sound too slowly, owing to the limitations of short-term
auditory store the brain is not able to integrate the separate
pitches into one coherent whole.(14) Between these two extremes,
tones are most readily grouped together when separated between
about 250 and 900 milliseconds, that is between metronome
markings 240 and c. 70.(15) A reasonable hypothesis might then be
that melodic implications are more compelling within this range
than outside it. Of course, for an analytical system that takes
the musical score as its source, tempo does not immediately
present itself as a vital parameter. Tempo is one of the least
well defined parameters in a score and probably the one that
varies the most within one performance and between different
performances. However, this problem is far from insurmountable--
one might take the tempi from "classic" or "benchmark"
recordings, for example--and some consideration of the range of
tempi at which a passage might be presented to a listener would
be a welcome addition to the model.
===============================================================
14. For a review of the literature on the low-level auditory
store see Nelson Cowan, "On short and long auditory stores,"
*Psychological Bulletin* 96 (1984): 341-70.
15. John A. Michon, "The making of the present: A tutorial
review," in *Attention and Performance VII*, ed. Jean Requin
(Hillsdale, NJ: Lawrence Erlbaum, 1979).
===============================================================
*Conclusion*
[42] Despite some minor inconsistencies, Narmour's ideas merit
wide dissemination. Unfortunately Narmour is not always his own
best evangelist; his presentation and ordering of concepts often
obfuscates rather than elucidates, with many of the principles of
the model being introduced piecemeal via numerous musical
examples throughout the two volumes. Also, possibly in an effort
to reveal the multifaceted nature of the theory as quickly as
possible, Narmour sometimes presents concepts in musical examples
before these concepts have been given more than a very cursory
introduction in the text. In this regard, Chapter 1 of Volume 1,
for example, makes a rather discouraging beginning to what is
ultimately a rewarding read.
[43] On a more positive note, the model offers a wealth of
experimental hypotheses for researchers in music perception and
cognition. Indeed, if the 1995 Society for Music Perception and
Cognition annual conference in Berkeley is any indication,
Narmour's ideas are currently the object of favorable attention
among many music psychologists. Also, partly in response to
Narmour's own list of experimental questions posed at the end of
his first volume, there is now a small but ever-burgeoning
literature of empirical studies that have tested aspects of the
model.(16) While in the interests of internal validity these
experiments have generally used short and simple melodic stimuli,
their converging results tentatively support many of Narmour's
principles.(17)
===============================================================
16. For a good list of recent empirical investigations of
Narmour's theory see William Forde Thompson, review of *The
Analysis and Cognition of Basic Melodic Structure* and *The
Analysis and Cognition of Melodic Complexity*, by Eugene Narmour,
in *Journal of the American Musicological Society* (1995): in
press.
17. For example Carol L. Krumhansl, "Music psychology and music
theory: Problems and prospects," *Music Theory Spectrum* 17
(1995): 53-80.
===============================================================
[44] Overall, my complaints notwithstanding, the implication-
realization model is surely a highly impressive undertaking. In
drawing together the numerous parameters of musical sound it
fastidiously reflects much of the richness of the note-to-note
experience of melody. Yet, despite its attention to detail, the
theory provides a parsimonious taxonomy of melodic shapes. This
classification system could well prove invaluable in such
analytic enterprises as tracing a composer's stylistic evolution
or comparing melodic styles among two or more different
composers. Some may feel that style-specific top-down processes
are given rather short shrift; however, the author promises
further coverage of such matters as well as the treatment of
counterpoint and large-scale tonal structures in future volumes.
In the meantime, Narmour should be credited for going much of the
way towards addressing the complaint by Meyer that was quoted at
the beginning of this review--with the implication-realization
model, Narmour has offered an extremely viable conceptual
framework for the analytical criticism of melody.
*References*
Berry, Wallace. *Structural Functions in Music*. New York:
Dover, 1987; reprint, Englewood Cliffs, NJ: Prentice Hall,
1976.
Bregman, Albert S. *Auditory Scene Analysis: The Perceptual
Organization of Sound*. Cambridge, MA: MIT, 1990.
Cooper, Grosvenor W., and Leonard B. Meyer. *The Rhythmic
Structure of Music*. Chicago: University of Chicago, 1960.
Cowan, Nelson. "On short and long auditory stores."
*Psychological Bulletin* 96 (1984): 341-70.
Hartmann, William M., and Douglas Johnson. "Stream segregation
and peripheral channeling." *Music Perception* 9 (1991):
155-84.
Huron, David. "Voice Segregation in Selected Polyphonic Keyboard
Works of Johann Sebastian Bach." Ph.D. diss., University of
Nottingham, U.K., 1989.
Krumhansl, Carol L. "Music psychology and music theory: Problems
and prospects." *Music Theory Spectrum* 17 (1995): 53-80.
Lerdahl, Fred, and Ray Jackendoff. *A Generative Theory of Tonal
Music*. Cambridge, MA: MIT, 1983.
McAdams, Stephen, and Albert S. Bregman. "Hearing musical
streams." *Computer Music Journal* 3 (1979): 26-43, 60, 63.
Meyer, Leonard B. *Explaining Music* Berkeley, CA: University of
California, 1973.
Michon, John A. "The making of the present: A tutorial review."
In *Attention and Performance VII*, ed. Jean Requin.
Hillsdale, NJ: Lawrence Erlbaum, 1979.
Thompson, William F. Review of *The Analysis and Cognition of
Basic Melodic Structures* and *The Analysis and Cognition of
Melodic Complexity*, by Eugene Narmour. *Journal of the
American Musicological Society*, in press.
van Noorden, Leon P.A.S. "Temporal Coherence in the Perception
of Tone Sequences." Ph.D. diss., The Institute of
Perception Research, Eindhoven, The Netherlands, 1975.
Woodrow, Herbert. "The role of pitch in rhythm." *Pychological
Review* 18: (1911): 54-77.
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