Prolongational Structure in

Bartok's Pitch-Centric Music:

A Preliminary Study
Ted Buehrer
The question of whether or not the analytical concept we know as
prolongation may be applied to music of the twentieth-century is one that
has been argued by many theorists in recent years. 1 Many of these efforts
have attempted to address this question by investigating music clearly from
the "atonal" literature-that is, music primarily from the Second Viennese
School. Such studies have yielded results that fall on both sides of the
prolongational fence. Joseph Straus, for example, in his article titled "The
Problem of Prolongation in Post-Tonal Music," asserts that analyses that aim
to uncover middleground structures of post-tonal music using the traditional
notion of prolongation as their analytic tool generally miss their mark,
primarily because music of this genre lacks four necessary "stability
conditions" that are present in tonal music. He proceeds to propose a "less
ambitious, but theoretically more defensible" approach based on an
associational model, which sheds all of the baggage that accompanies the
term "prolongation" when it is applied to atonal music. 2 Fred Lerdahl, on the
other hand, makes a case for prolongational structures in atonal music by
focusing not on the stability conditions that are lacking, but rather on a set of

1 See, for example, James Baker, "Schenkerian Analysis and Post-Tonal Music," in
Aspects of Schenkerian Theory, ed. David Beach (New Haven: Yale University Press,
1983), 153-86; Fred Lerdahl, "Atonal Prolongational Structure," Contemporary Music
Review 4 (1989): 65-87; Charles D. Morrison, "Prolongation in the Final Movement of
Bartok's String Quartet No.4," Music Theory Spectrum 13, no. 2 (Fall 1991): 179-96;
Joseph Straus, "The Problem of Prolongation in Post-Tonal Music," Journal of Music
Theory 31 (1987): 1-22; Roy Travis, "Tonal Coherence in the First Movement of
Bartok's Fourth String Quartet," Music Forum 2 (1970): 298-371; Paul Wilson,
"Concepts of Prolongation and Bartok's Opus 30," Music Theory Spectrum 6 (1984): 7989; as well as parts of Felix Salzer's Structural Hearing (New York: Dover, 1952).
2 Straus, 1.

Indiana Theory Review Vol. 18/2

"salience conditions" that replace them. Indeed, with these salience

conditions, Lerdahl develops "an atonal prolongational theory [that can]
shed its Schenkerian origins. Such a theory can account for the important
intuitions of elaboration and linear connection that atonal music evokes.,,3
Lerdahl's rule-driven approach, derived from his and Ray Jackendoffs
perceptually based A Generative Theory of Tonal Music (1983), produces
prolongational tree diagrams that reveal deep structure through multilevel
4
.
re ductlOns.
Lerdahl's theory, designed specifically to reveal prolongation in music
from the atonal literature, has merit as far as it goes. He recognizes its
limited scope and suggests some extensions of the theory for other
twentieth-century music, such as music that does have tonal or some other
element of stability at work. The pitch-centric nature of much of Bartok's
music seems a perfect medium in which to test this extension of the theory.
In such music, tonal and atonal elements are combined in a way that requires
a coordination of stability and salience conditions. An analysis of a
representative example from this literature that attempts to demonstrate
prolongation using only Lerdahl' s salience conditions would ignore
important tonal features of the piece. In this paper I shall illustrate this point
by comparing and contrasting two prolongational analyses of the coda (mm.
126-61) of the first movement of Bartok's Fourth String Quartet. The first
set of analyses shows the prolongational structure of the excerpt based on
salience conditions alone. As we shall see, this analysis is problematic
because it fails to recognize many tonal implications that are present in the
passage: a motion to the dominant, a prolongation of a pseudo-subdominant
harmony as well as that of a pseudo-dominant harmony. The second set of
analyses, a modification of the first, incorporates traditional stability
conditions along with salience conditions to provide a more accurate picture
of the prolongational structure of the excerpt. This latter analysis is then
3 Lerdahl, 68. As a side note, it is worth mentioning that Nicola Dibben ran a perceptual
study to see if listeners do indeed hear atonal music in a hierarchical fashion as Lerdahl
suggests in his article. She reported her findings in an article titled "The Cognitive
Reality of Hierarchic Structure in Tonal and Atonal Music," Music Perception 12, no. 1
(Fall 1994): 1-25. In brief, Dibben, using Lerdahl's own examples as stimuli, found no
empirical support for his hypothesis. Bartok's pitch-centric music, which is under study
in this article, was not tested by Dibben in her study.
4 Fred Lerdahl and Ray lackendoff, A Generative Theory of Tonal Music (Cambridge:
MIT Press, 1983).

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

compared to Travis's quasi-Schenkerian analysis of the same passage,

excerpted from his complete analysis of the first movement, to suggest
problems in the Schenkerian approach to this excerpt and, more generally, to
music of this genre.
In Lerdahl and Jackendoffs A Generative Theory of Tonal Music
(hereafter referred to as GTTM) , the authors make the claim that listeners
perceive events within nested rhythmic units which they call time-spans. As
Lerdahl summarizes:
At local levels time-spans consist of the distances between beats; at global
levels, of groups; at intennediate levels, of a combination of meter and
grouping. Within these spans the listener compares events for their relative
stability. Less stable events are recursively "reduced out" at each level,
until one event remains for the entire piece. Such in brief is time-span
reduction. Its essential function is to link rhythmic and pitch structure ...
[which is] needed to develop a deeper stage of analysis, prolongational
reduction, ... [which] evaluates the prolongational importance of events. 5

Listeners do not take an ad hoc approach to determining stability, the

authors claim. Indeed, Lerdahl and Jackendoff posit several "wellformedness" and "preference" rules to determine the length of each timespan, the grouping of time-spans at higher levels, and the selection of the
"head" or most important event of a time-span, and claim that listeners
intuitively apply these same rules when comparing events for their relative
stability. These rules might collectively be referred to as stability conditions.
Prolongational reductions are modeled on Schenkerian reductions in that
both types of reduction describe linear continuity, departure, and return of
events within the context of a hierarchy. Prolongational structures are not,
however, dependent upon pre-existing schema such as Schenker's Ursatz.
Rather, they derive from the most global to the most local levels of a piece's
time-span reduction. On each successive level, motion can be described in
one of two ways: as a tensing motion (departure), indicated in Lerdahl and
Jackendoffs analytical notation by a right branch, or as a relaxing motion
(return), indicated by a left branch. Each of these types of motion may be
described more specifically according to their prolongational function: as a
strong prolongation, in which an event repeats, indicated by an open circle at
the node (the point where the two branches merge); as a weak prolongation,
5

Lerdahl, "Atonal Prolongational Structure," 71.

Indiana Theory Review Vol. 18/2

in which an event repeats in an altered form, indicated by a closed circle at

the node; or as a progression, in which an event connects to an entirely
different event, indicated simply by the merging of two branches without an
open or closed circle. As in time-span reduction, a set of stability conditions
guides the analyst to selecting the proper type of connection.
In tonal music, it is these stability conditions, based on the grammar of
tonal music, that guide the listener at every stage of the decision-making
process involved in an analysis. In adapting the theory to atonal music,
however, Lerdahl acknowledges that "atonal music almost by definition
does not have stability conditions. Its pitch space is flat; sensory consonance
and dissonance do not have any syntactic counterpart.,,6 In order to develop
criteria for pitch reduction in the atonal language, Lerdahl suggests a set of
salience conditions that replace the stability conditions of tonal music. We
begin with a review of these salience conditions, shown in figure I, followed
by a review of two preference rules essential for the proper construction of
the prolongational trees, shown in figure 2.
Figure 1. Salience conditions (numbers in [ ] refer to relative strength of application)
Of the possible choices for head of a time span, prefer an event that is:

Local levels - - -

Global levels

--i

a)
b)
c)
d)
e)
f)
g)

attacked within the region [3]

in a relatively strong metrical position [I]
relatively loud [2]
relatively prominent timbrally [2]
in an extreme (high or low) registral position [3]
relatively dense [2]
relatively long in duration [2]

h) relatively important motivically [2]

i) next to a relatively large grouping boundary [2]
j) parallel to a choice made elsewhere in the analysis [3]

The salience conditions interact according to their relative strength of

application to select the most salient event within each time-span of a
passage. Upon recursive reduction to more global levels, less salient events
disappear from the analysis. The two preference rules are needed in order to
6lbid., 73.

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

determine the most important event within a prolongational region ("timespan") and for making the most stable connection within a given time-span.
These prolongational rules interact at each level, sometimes mutually
supportive of each other but at other times in conflict, forcing the analyst to
seek the most stable connections from the next lowest time-span level or to
assign less stable connections from the more global level.
Figure 2. Preference Rules
A. Preference Rule for Prolongational Importance:
In choosing the prolongationally most important event ek within the
prolongational region (ei - ek), prefer an event that appears in the two most
important levels of the corresponding time-span reduction.
B. Preference Rule for Prolongational Connection:
a. Stability of Connection: Choose a connection in the following order of
preference:
(1)
ek attaches to ei as a strong right prolongation
(2)
ek attaches to ej as a left progression
(3)
ek attaches to ei or ej as a weak prolongation
(4)
ek attaches to ei as a right progression
(5)
ek attaches to ej as a strong left prolongation

b. Time-span segmentation: If there is a time-span that contains ei and ek but

not ej, choose the connection in which ek is an elaboration of ei; and
similarly with the roles of ei and ejreversed.

Using these conditions and rules as a guide, we may proceed to the first
analysis, found in diagrams 1a-e. In diagram 1a-I, we observe that the
passage under question (the coda) is right-branching, signifying a departure
from the starting point (m. 126). At first glance, this might seem
counterintuitive to one's conception of the ending of a movement, where,
especially in a tonal context, emphasis is often placed on a return to opening
material, which would be indicated in tree notation with a left branch. It is
important to keep in mind, however, that as a coda, this passage should be
viewed as an extension of the piece (and hence as a departure). The branch
that stems from m. 126 would connect back to the rest of the prolongational
tree through a left branch stemming from m. 1, as shown in diagram la-2.

Indiana Theory Review Vol. 18/2

6
Diagram la-I. Deep structure of Coda

m. 126

145

152

161

Diagram 1a-2. Deep structure of Bart6k's Fourth String Quartet, I

m. 1

126~--Coda---161

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

Using only the salience conditions of figure 1 to select the head of each
time-span, the set of graphs given in diagrams 1b-d yield many problematic
results due to the relative importance they give to supposedly more salient
events at the expense of clearly important tonal events, which achieve less
prominence than they deserve. Two of these events in particular deserve
special attention here. The first is in the opening passage of the coda, mm.
126-34, shown in diagram 1b (see foldout).7 The motion to the dominant that
frames this opening section should, it seems, play an important role in the
overall structure of the coda as an initial departure away from the opening
tonic. This departure would be represented graphically as a right
progression, and the node would be at a fairly high level. Yet in terms of the
salience conditions, this opening section is dominated by the repeated dyad
D-A (CelloNiola and Violin I, respectively), which occurs consistently
throughout mm. 130-34. Hence, as the graph indicates, application of the
salience conditions alone requires that the first occurrence of D-A in m. 130
become the head of the prolongational region spanning mm. 130-34, then
connect to the main (left) branch at level c. Subsequent occurrences of the
D-A dyad are subsumed at level c, but serve as the heads of their respective
regions at level d. These connect to the first D-A dyad as strong right
prolongations at level d.
The second event worthy of mention is a quasi-subdominant F-E dyad in
m. 145 which, according to diagram la-I, connects to the tree as a left
progression to the quasi-dominant ofm. 152 (G#/D). This latter event in tum
connects as a left progression to the main (right) branch. This series of
nested left progressions thus represents a gradual relaxation into the final
octave Cs of the piece. Let us examine the F-E branch more closely,
however. In diagram lc (see foldout), we observe that the highest level
branch coming off of this F-E branch connects at level c; tracing that branch
back to its origin leads to the B~-B~ of m. 135. Following the level d
reduction of the passage established by these two boundaries (mm. 135-45;
see reduction beneath the score), we observe that the B~-B~ dyad initiates two
stepwise lines in contrary motion (bottom voice: B~-C#-D-C#-D-(D)-F; top
voice: B~-A-G-A~-G-(G)-E). This line is displayed quite clearly on the tree
through a series of progressions and prolongations at levels c and d, but it
misses the larger point of the passage, which is the repeated departure from
See my grouping analysis (brackets beneath the music, level e), which begins with a
statement of octave Cs ("tonic") and closes on a G-D dyad ("dominant").
7

Indiana Theory Review Vol. 18/2

and return to (that is, prolongation of) the F-E dyad heard throughout. The
two stepwise lines (salient because of their registral prominence) undermine
this F-E prolongation, reducing the appearance of the dyads to mere surface
level (level e) right progressions. Rules of tree construction do not permit
branch crossing; therefore, these F-E dyads may not connect in any way,
given this configuration.
Other aspects of an analysis constructed solely with salience conditions
work well, however. For example, the G#-D dyad of m. 152, shown in
diagram 1d (see foldout), which serves as the penultimate quasi-dominant
before the return to the octave Cs at the end of the piece, is salient due to its
extreme register and dynamic. It is thus selected as the head of the
prolongational region spanning mm. 152-56, connecting to the main (right)
branch at level a as a part of the series of nested left progressions mentioned
earlier. Also, most of the surface level (level e) decisions made based on
salience conditions alone seem reasonable. In short, it is at the highest levels
of structure that problems arise when stability conditions are ignored in
favor of salience.
But how do we integrate these conditions? How do we move from this
set of graphs to one that more accurately incorporates stability and salience
in this excerpt, in which clearly both tonal and atonal elements are at work?
Before we can proceed to the graphs found in diagram 2, we must discuss
the issues that are engaged when salience and stability elements both playa
role in a musical passage. Clarification of these issues will provide
justification for the analytical decisions made in diagram 2.
A useful starting point for this discussion might be Lerdahl and
Jackendoff s GTTM, in which the authors address the issue of surface
salience versus stability in tonal music by warning against the confusion of
an event's structural importance with its surface salience. To illustrate their
point, they use an excerpt from a Bach chorale, shown in figure 3, which
contains a IV chord at the downbeat of m. 1. This chord is salient not only
because of its strong metrical position, but also because of the relative height
of its soprano and bass lines (both local maxima). In terms of structural
importance, however, this IV chord is subservient to the opening tonic and
the immediately following 16. As such, the authors conclude that this chord
is better interpreted as an "appoggiatura" chord that is reduced out at a fairly
low level. They conclude from this example an already intuitively obvious
point: that the most striking event is not always the most structurally
significant event:

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

Figure 3: Excerpt from Bach chorale

We do not deprecate the aural or analytic importance of salient events; it is

just that reductions are designed to capture other, grammatically more
basic aspects of musical intuition. A salient event mayor may not be
reductionally important. It is within the context of the reductional
hierarchy that salient events are integrated into one's hearing of a piece. 8

Recall that it was only the complete absence of stability conditions in

atonal music that led Lerdahl to replace them with salience conditions. If
tonal implications are present in a piece of music and play a role in the
structural importance of the piece, however, a reductive analysis must return
to the same relationship between salience and stability conditions that
Lerdahl and lackendoff establish in GTTM. Simply put, what is the most
striking is not necessarily the most structurally important, and when these
two conditions conflict, stability conditions assume greater importance.
Charles Morrison, in his analysis of the final movement of Bartok's Fourth
String Quartet, emphasizes this point: "On occasion the externally defined
stability conditions will be shown to support interpretations which actually
contradict those founded on salience factors alone. In such instances,
stability conditions will override salience conditions.,,9 But he goes on to say
that this does not mean that salience conditions play no role in the
prolongational design of the piece. While Morrison is referring specifically
to the movement he analyzes, I would like to extend this claim to include
much of the pitch-centric literature. Salience continues to play a role at
middle and lower levels of structure, and certainly continues to affect a
listener's perception at these levels.
Lerdahl and Jackendoff, 109.
9 Morrison, 180.
8

10

Indiana Theory Review Vol. 18/2

We now tum our attention to diagrams 2a-d, which incorporate the tonal
elements that were lacking in the first set of analyses of this coda passage.
Notice, first of all, that in diagram 2a (facing page) the basic shape of the
prolongational tree remains intact: the coda begins with a motion away (right
progression) from the opening octave Cs, while the second and third large
sections feature harmonies that participate in a general relaxation (series of
nested left progressions) into the final closing cadence. The events that
participate in these large-scale structural motions have been chosen not on
the basis of salience, however, but on the basis of stability conditions.
Therefore, the head of the second large section is at m. 135, not m. 145. In
diagram 2b (see foldout), a graph of the opening section of the coda (mm.
126-34), it is the G-D dominant dyad of m. 134 that connects to the main
(left) branch at level c instead of the repeated D-A dyad heard throughout
mm. 130-34 (compare diagram 2b to diagram 1b). This presents a perfect
example of a situation in which the more salient event (by virtue of its
repetition) is not the more important event (by virtue of its harmonic
function). Indeed, the G-D dyad, by comparison, is heard for just one eighthnote duration, yet because of its structural importance it is chosen as the
head of this particular prolongational region. The D-A dyad of mm. 130-34
assumes a less important role in the overall structure, connecting at level d
as a left progression to the branch originating from the G-D dyad.
In diagram 2c (see foldout), a graph of the second section (mm. 135-45),
the F-E dyad ofm. 135 is chosen as the head of the entire region, connecting
at level a (not shown) to the rest of the tree. All subsequent F-E occurrences
attach to this initial branch, either directly as strong right prolongations, or
indirectly as nested strong right prolongations (depending on the level of the
graph-c or d-at which one looks). Finally, in diagram 2d (see foldout), the
pseudo-dominant G~-D dyad from the closing section (m. 152) connects to
the main (right) branch at level b as a left progression, but in this case it is
selected as the head of its prolongational region on the basis of both
structural importance and salience. In short, diagram 2d appears the same as
diagram 1d. Here, then, is an example of an event that is simultaneously the
most salient and the most structurally important, proof that the two are not
necessarily always in opposition. Furthermore, in each of these diagrams
salience conditions, though replaced at the highest level by stability
conditions, do continue to play an important role in the branching decisions
made at the middle and lower levels.

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

11

Diagram 2a. Deep structure of Coda (revised)

m.126

135

152

161

Yet for all of the improvements these graphs make over the diagram 1
graphs, there remains one troublesome aspect-namely, the graphic
representation of the series of strong prolongations of the F-E dyad in the
second large section of the coda (diagram 2c) fails to simultaneously
represent the salience of the two stepwise lines discussed earlier. The graph
is unable to reveal these relationships due to certain well-formedness rules
built into the theory that forbid branch crossings of any kind at any level. As
a result, musical passages that feature compound textures with many
different ideas being developed simultaneously (such as this one) are
shortchanged upon analysis. These events are clearly salient at this middle
level of structure, and should be represented in some way on the graph.
There are two ways to approach this dilemma. First, a simplistic solution to
represent these stepwise lines on a prolongational tree would be to allow an
additional preference rule that would permit limited branch crossing at lower
levels in order to include graphic representation of compound textures such
as this one. A revision of this section of the tree (mm. 135-52) to include the
stepwise lines following this proposed solution appears in diagram 2c,
version 2 (see foldout). One does not need to be a zealous layerist to feel

12

Indiana Theory Review Vol. 18/2

sudden pangs of anxiety over this graph. It is graphically messy, and it

significantly muddles the picture of what transpires in this passage.
Another, perhaps more reasonable, approach would be to explain the
compound texture in a different way, thus eliminating the need for crossed
branching. In this case, we might view the F-E dyad as a pedal point,
initiated not at m. 135 but one measure earlier, m. 134, and "held" through
reiteration-made necessary by the instrumentation-until m. 145. Thus the
initial F -E dyad would still connect as a left branch to the rest of the tree at
level a, but the subsequent F -E attacks would no longer need to be
represented graphically, leaving room for graphic representation of the
stepwise lines at levels d and e. A sketch of this alternative is shown in
diagram 2c, version 3 (see foldout).
One final issue needs to be addressed. How is a prolongational tree
structure that integrates stability and salience conditions an improvement on
other prolongational attempts to describe pitch-centric music? For example,
theorists such as Felix Salzer and Roy Travis have applied Schenkerian
theory to music of this type in an attempt to reveal prolongational as well as
voice-leading structures. Travis's analysis of the first movement of Bart6k's
Fourth String Quartet is excerpted in figure 4 to include just the coda; this
excerpt will serve as a foil to my tree structures in diagram 2.
For conservative Schenkerians, adaptations of the theory for the purpose
of analyzing any music outside of the "Bach to Brahms" historical
boundaries are simply unacceptable. For those followers who are more
liberal, many of these same analyses are quite simply unconvincing. As
Lerdahl asserts, "on both sides the dissatisfaction provides small comfort,
because the basic intuitions that Schenkerian theory addresses-the sense
that musical material is elaborated, the recognition of local and global linear
connections-also need to find a place in any theory of atonal music."l0
Central to the dissatisfaction on both sides is the very issue this paper
addresses: prolongation. As a further problem, Schenkerian theory deals not
with reductions from foreground events, but rather with elaborations of
background events. This reduced emphasis on the musical surface and
relationships inferred from them is especially troublesome when dealing
with twentieth-century music, a musical environment in which listeners,
often unable to use their knowledge of the traditional tonal system to make
harmonic sense of contiguous events, struggle desperately to find coherence.
10

Lerdahl, "Atonal Prolongational Structure," 67.

Buehrer, Prolongational Structure in Bartok's Pitch-Centric Music

13

Figure 4. Travis' middleground analysis of Bartok's coda

l
) ~v

; in cont rary motion

-_...... ""

-...:p

,;

'~~~~-;::~:---- ~~
.... ..

"",,"--,;

---.!:~Vi

IN

l: . . . . . . . .---.----..-.. . -.. . . . . --..-.--I

..J

@
~

Nevertheless, Schenkerian analyses of twentieth-century music, such as

Travis's analysis of Bartok's Fourth String Quartet, do exist. His
11
middleground reading of the coda is shown as figure 4. Notice that all of
the primary harmonic regions shown in the trees of my diagram 2 are present
here as well: the motion to V in m. 134, the prolongation of the F-E dyad
across mm. 134-45, the subsequent motion to the dominant (m. 152), and the
final motion to tonic (m. 161). Viewed in strict Schenkerian terms, this
analysis is problematic because the fundamental line does not receive proper
harmonic support. The 3 is supported by subdominant harmony, while the 2,
which first occurs in m. 152, is supported by A~ (G~) and only loosely
receives dominant support through an alignment symbol connecting it to the
G of m. 160. But even if we allow for these departures and take a more
Travis gives a middleground graph of the entire movement in his article. See his
example 3, pp. 302-308.
11

Indiana Theory Review Vol. 18/2

14

liberal view, this analysis remains problematic because it fails to incorporate

salient events along with structural events into this middleground reading.
Salience in Schenker's theory is a surface phenomenon, and as such is only a
factor at the most local levels of structure. Hence, all of the stepwise lines
that receive higher prominence in my version 3 of my diagram 2c are
reduced out of Travis's analysis as he moves from the foreground to the
middleground. In addition, in m. 130 Travis favors a motion to F (IV) in his
middleground reading, which, though arguably a structural arrival point on
the way to the G-D dominant dyad in m. 134, completely ignores the clearly
salient D-A dyad reiterated throughout mm. 130-32 (see my diagram 2b
graph, level d).
The foregoing discussion reveals that in order to best describe and
uncover prolongational structure in the pitch-centric music of Bartok, one
must be willing to embrace both salience and stability conditions in a new
way. The potential pitfall of this duality has not gone unnoticed. Morrison,
for example, anticipated criticism of his analysis with the following remarks:
The fact that criteria for prolongation ... at the highest level of structure
are said to be syntactical and conceptual in nature, while those for
prolongation at the middle level are said to be psychological and
perceptual in nature, may well be problematic from a purely "structuralist"
. 0 fVIew.
'
12
pomt

Yet it is precisely the fact that we are dealing with different types of
structures and diverse organizational systems that demands a reexamination
of the roles that salience and stability conditions should play in this music.
We have seen what these roles cannot be: salience cannot completely
dominate stability (as the graphs in diagram 1 showed), nor can stability
completely dominate salience (as Travis's analysis showed). It is only
through the responsible integration of the two in a way that prefers stability
at the highest levels of structure, but salience at the middle and lower levels,
that yields true analytical insight into the prolongational structure of
Bartok's pitch-centric music.

12

Morrison, 195.