Bartók’s Polymodality

The Dasian and Other Affinity Spaces

José Oliveira Martins

Abstract The article proposes that a construct I call the Dasian space provides an effective framework to
interpret harmonic aspects of scale relations in twentieth-century polymodality, particularly in the music of
Bartók. Based on Bartók’s intuition that the pitch space modeled after his notion of polymodal chromaticism
retains integral “diatonic ingredients,” the Dasian space (named after the medieval homonymous scale) estab-
lishes a system of relations between all potential diatonic segments, without relying upon traditional con-
straints, such as complete diatonic collections, harmonic functions, or pitch centricity. The Dasian space is a
closed, nonoctave repeating scalar cycle, where each element is identified by a unique coordination of pitch class
and modal quality. The dual description of each element enables both the specification of location in a given cycle
and the emergence of a group structure, whose generators—named transpositio and transformatio—are also
characteristic musical motions and relations. The proposed analytical methodology is probed in a couple of
short pieces of Bartók’s Mikrokosmos and in the third movement of his Piano Sonata. The article argues that,
unlike other tonal and atonal classic approaches, the Dasian framework enables the analyst to reconcile the
constructional character of a Bartókian idiomatic feature (the combination of distinct and integral scale strata)
with the interpretation of harmonic space in terms of scale-segment interaction and formal processes. The
article then contextualizes the structure of the Dasian space within a larger class of constructs, which I call
affinity spaces, by generalizing some of its group-theoretical properties that model relations between nondia-
tonic scalar materials. The analytical pertinence of affinity spaces is probed in Bartók’s “Divided Arpeggios,”
an intriguing posttonal piece appearing late in the Mikrokosmos set.

Keywords Béla Bartók, Mikrokosmos, Piano Sonata, polymodality, polytonality, diatonicism, chromaticism,
scale theory, rotational form

the combination of distinct diatonic layers is a characteristic marker

for early twentieth-century composers such as Béla Bartók, Igor Stravinsky,
and Darius Milhaud. A variety of terms, such as polymodality, polytonality, poly-
chordality, and polyscalarity, have been used to describe the layered construc-
tion of multidiatonic passages, and yet the harmonic implications for such

I would like to express my gratitude to Richard Cohn, Ian Quinn, Dmitri Tymoczko, and two anonymous
readers of this journal for their thought-provoking commentary and insightful suggestions at various
stages of this project.

Journal of Music Theory 59:2, October 2015

DOI 10.1215/00222909-3136020 © 2015 by Yale University 273

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 274 JOURNAL of MUSIC THEORY

preferential deployment remain poorly understood.1 The prevailing analyti-

cal activity developed for this repertoire in the second part of the last century
often downplays the scalar integrity of combined diatonic materials, thus
questioning the syntactical pertinence of scale segments in the global orga-
nization of pitch space. In particular, tonal and atonal harmonic interpreta-
tions of Bartók’s scalar layering often compromise, or simply override, the
coherence of individual segments. Models sensitive to tonal associations pro-
pose extended tonal frameworks that fuse the combined diatonic strands
to project prolongational and centric readings, whereas models concerned
with progressive and atonal aspects of the style explore how layered diatonic
elements engage with the syntactic possibilities of the twelve-tone chromatic
space.2 In short, tonal approaches reduce the resultant chromaticism of the
combined strata into deeper-level diatonicism, whereas atonal approaches
scatter surface diatonicism into deeper-level chromaticism.
While the combination of diatonic layers in the twentieth century has
been understood as both providing a strong link to the past and signaling
progressive harmonic innovations,3 the rift between the compositional pro-
cedure and the established interpretative analytical activity invites a number
of fundamental questions. If Bartók and other composers are using an
expanded tonal framework, why are tonal routines often hard to detect or
define? And if composers are using an atonal framework, why are tonal mate-
rials privileged in the combined deployment? Specifically, which resources
and properties of the pitch system are explored by the interaction of layered
diatonic materials, and how are such interactions best interpreted in light of
local and global harmonic relations?
This article sketches an answer to these questions by proposing an ana-
lytical approach to some of the harmonic implications of Bartók’s polymodal
practice. A preliminary and useful interpretation of this practice is offered by
the composer’s own notion of “polymodal chromaticism.”4 Bartók coined the
term using a short musical illustration (see Example 1), which superimposes
two contrary-motion diatonic melodies (four flats over one sharp), wedging
toward pitch class C. Bartók ([1943] 1992, 367) interprets the passage as the

1 These terms variously refer to a compositional technique 3 Fosler-Lussier (2001) discusses “conservative” versus
and a stylistic label. A more general designation often used “progressive” aspects of the “two Bartóks” in the context
is diatonic posttonality. of the reception of his music during the Cold War in Europe.

2 For a selective sample of analytical approaches to Bar­ 4 The term polymodal chromaticism, coined by Bartók
tók’s music that integrate the diatonic strata within models ([1943] 1992, 365–71), has been adopted and extended by
of expanded tonality, see Travis 1970, Waldbauer 1990, some post-Schenkerian models of prolongational structure
Morrison 1991, and Lerdahl 2001, 333–43. For approaches (see especially Waldbauer 1990; Morrison 1991; Lerdahl
projecting the diatonic strata into chromatic (pitch and 2001). The concept is also explored in a nonprolongational
pitch-class) space, see Perle 1955, Forte 1960, Antokoletz manner by János Kárpáti (1994).
1984 and 2000, Cohn 1988 and 1991, Wilson 1992, and
Bernard 2003. Some of these approaches are discussed in
more detail below.

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 Martins

José Oliveira Martins Bartók’s Polymodality 275

Martins, Ex. 1 (correx 7/24/15):

� �� �� � ��
�� �� � �� �� �
�� � �

� � � ( �) � � �
� � ��
� �� �

Example 1. Bartók’s musical illustration of “polymodal

chromaticism” (Bartók [1943] 1992, 367, example 4)
Martins, Ex. 2 (correx 7/24/15):

superimposition of integral modal segments or scales, which combine to

� = 170
� space and share a �common final (and fifth): �
Allegro molto

3 �� � � 1 � � 3 �� � �
exhaust the chromatic
2 � �� 2 � ��
Ar

� �� �� � � �� �� 8 � � pentachord � �� 38
� � 8 �� � � As 4the
� result � � � � a� Lydian
� � �of superposing � � 4and � Phrygian � � 4with � � a� common
� �
� fundamental tone, we get a diatonic pentachord filled out with all the possible
� 3 � � ���degrees. These
flat2 and sharp
4 � different �
seemingly �� chromatic
41 � the���altered
38 �
degrees, 24 � however, �� �
� �are 38
� 8 totally ��in their function from chord degrees ��
of the chro-
matic styles ofAl the previous periods. A chromatically-altered note of a chord is

� �
� � chord.
� � In our
in strict relation to its Ar nonaltered form: it is a transition leading to the respec-

� of the� following � polymodal

� � � � � � � � � �
� � not� altered
� 2degrees� �
tive tone chromaticism, however, the
3 � 24 and
� ��sharp
� tones 3 � � �
� � � � � � � 1 � 38
7

� � 8 �� � �� ��� 8 4 4 ���
� � � modal
flat are at all; they are diatonic ingredients
of a diatonic scale.
� �
Bartók’s often-cited� formulation � � � � ��polymodal � �
� 3 2 ��� �
� 3 24 namely, � � �� attempts
� �� ��chromaticism ��
41 38
for
�
� reconcile � ��� (1) the global chromatic
� 8 to 4 two�� theoretical �� 8 intuitions, that ��
space combines diatonically coherent strata and that (2) an underlying har-
� � �
Al
monic principle (“a common fundamental tone”) coordinates the strata. 5
�� �While
� intended
Ar
�
38 � � � � 2 � �� �� �� ��� 3 � �� � 2 �� ��� �� � his music uses multiple con-
as a corrective to the view that
���“maze� ��of keys” � ��(Bartók
� � ��[1943] � 68
13

4 8 � � 4
� � �� �� fundamental”
�� �� � ��for ��combina-
� ��
current tonics, which result in a polytonal
1992, 366), the stipulation of a single “common
tions of strata often fails sustained analytical scrutiny for � numerous polymo- �
� � reveal that projected pitch centers are
� 3
�� � �
dal � �� �
passages:
often 2not common
�
analyses
� �� 83tones
of the
� ��� �involved,
repertoire
� � to� the2 strata ���� �� available ��
�� � ��common ���tones 68
� �� are
� 8 4
not deployed as� pitch �� 4 �
�� Al layers
centers, or combined �� � simply �� do��not
� project ��
�� salient
pitch centers.
In contrast, this article explores Bartók’s intuition that “all possible flat
or sharp tones” in a polymodal deployment are not “altered” or “chromatic”
but, rather, “diatonic ingredients” within their respective layers. I argue that
Bartok’s insight entails that the resulting harmonic space relies on the scalar

5 The insistence on a “single fundamental tone” for the erential” elements (Morrison 1991, 182–83) from distinct
combination of layers has caught the attention of the strands but also are frequently confronted with passages
Schenkerian tradition, which led to the development of that fail to provide a common tonic to the combined layers,
(abovementioned) pitch models that support prolongation which in turn undermine a stable referent for large-scale
in Bartók’s posttonal environment. Such centric readings, progressions.
however, not only rely on the fusion of “departure” or “ref-

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 276 JOURNAL of MUSIC THEORY

integrity and interaction of distinct diatonic segments.6 Specifically, the arti-

cle advances a framework for multidiatonic harmony, which interprets dis-
tinct diatonic strata in terms of harmonic distance within a single integral
system that embeds all diatonic segments. This framework bypasses attribu-
tions of pitch centers, diatonic scale-degree functions, and the default invo-
cation of complete diatonic collections.7 (These aspects are still available if
analytically useful but are not considered crucial for the understanding of a
global polymodal space.)
The central feature of this inquiry is a cyclic model of diatonic scalar
segments, as they interact within chromatic space. I refer to this construct as
the Dasian space, since it generalizes some aspects of the medieval Dasian
scale. Specifically, the space generalizes affinity relations at the perfect fifth
of the four modal qualities that a given note can assume in any diatonic col-
lection and explores the group-theoretical properties that result from the
coordination (order pairs) of pitch classes and modal qualities.
After introducing the conceptual framework for the Dasian space and
probing its analytical potential in a couple of brief pieces from the Mikrokos-
mos, this article examines how polymodal activity in the third movement of
Bartók’s Piano Sonata engages the exploration of pitch space. The patterns
that emerge suggest that Bartók’s preference for compositional deployments
that preserve distinct diatonic strata do not merely explore “color” or texture
but participate in local and large-scale tonal/pitch strategies, which in turn
shape aspects of form.8
The article extends the principle of intervallic affinities underlying the
Dasian space to a larger class of constructs, which I call affinity spaces, by
generalizing some of its group-theoretical properties that model relations
between nondiatonic scalar materials. The analytical pertinence of affinity
spaces is probed in the modeling of pitch space for Bartók’s “Divided Arpeg-
gios” (Mikrokosmos no. 143). The article concludes by briefly inventorying
some of the theoretical and analytical implications for the understanding of
polymodality set forth by the proposed framework.

6 This insight is reinforced by Alfred Casella’s (1924, 159– 8 I am not suggesting that Bartók’s music (or the theoreti-
60) claim that “‘polytonality’ signifies, to be sure, the inter- cal apparatus I am proposing) is stylistically “neomedieval.”
penetration of diverse scales; but, it likewise assumes—in However, I am proposing that a number of pitch space
the very nature of things, the survival of the original scales.” properties (such as interval affinities, modal qualities, and
relations) relevant to medieval conceptions of pitch space
7 Traditional approaches consider complete diatonic scales
are also appropriate and useful to understand the inter­
as hypostasized entities. Those entities are thought to con-
action of diatonic (and other scalar) layers that character-
struct a tonal or modal space in which relationships obtain
ize twentieth-century polymodality. Several authors have
between entire scales. By invoking a part of the scalar
developed approaches that study the interaction or combi-
entity, one inherently also invokes its entirety, and thus
nation of scalar spaces for the music of Bartók (Kárpáti
when distances between scale segments are measured,
1994; Gollin 2007), Debussy (Hook 2008; Tymoczko 2004),
they are routed through distances between the entire
Ives (Lambert 1990), Milhaud (Harrison 1997), Reich (Quinn
scales with which the segments are associated. The tradi-
2002), Ravel (Kaminsky 2004), and Stravinsky (Tymoczko
tional approach acknowledges that elements between
2002) in ways that are variously resonant with the analyti-
those scales might be shared, but it insists that those ele-
cal approach proposed in this article.
ments remain conceptually distinct.

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 José Oliveira Martins Bartók’s Polymodality 277

Figure 1. The medieval Dasian scale

The Dasian space

The structure of the proposed Dasian space is built upon the pattern of inter-
vallic affinities embedded in the medieval Dasian scale.9 The scale, illustrated
in Figure 1, is a stacking of four TST (tone–semitone–tone) tetrachords,
consistently separated by a tone of disjunction, plus two “residual” notes.
Each tetrachord embeds the four modal qualities—protus, deuterus, tritus, and
tetrardus—which are defined by the relative position of a note with respect to
the interval pattern of the tetrachord. The scalar arrangement of the Dasian
scale privileges affinity relations at the perfect fifth, given that notes four
steps apart share the same modal quality (i.e., the same pattern of neighbor-
ing tones and semitones). Affinity relations at the fifth, however, conflict with
octave relations in the scale, since notes seven steps apart in the scale have
different modal qualities. In other words, the consistency of modal replica-
tion at the fifth “modulates” or “curbs” a strict diatonic pattern, such that
the Dasian and the diatonic scales intersect only in a span of a major tenth.
Norman Carey and David Clampitt (1996) refer to this contiguous span as
an octave-fifth region (see Figure 1, solid lines beneath the scale). As they
explain, the relation between octaves and fifths in diatonic space is reversed
in the Dasian scale: “Just as the periodicity at the octave in the usual diatonic
pitch space allows two voices to move perpetually in parallel octaves, while
voices moving in parallel fifths will encounter a diminished fifth, in the
dasian scale, the reverse is the case: parallel fifths may be maintained per-
petually, while motion in parallel octaves will eventually result in an aug-
mented octave” (124).
The interval structure and associated affinity relations of the Dasian
scale can be generalized into a Dasian space construct (Figure 2). The space

9 The Dasian notation and its associated scale are dis-

cussed in the medieval Musica enchiriadis and Scolica
enchiriadis treatises (Erickson 1995). Interval affinities are
discussed in Pesce 1987, Cohen 2002, and Atkinson 2008
in the context of medieval theory in the early middle ages.

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 278 JOURNAL of MUSIC THEORY

Figure 2. The Dasian space. (2.1) The four modal

qualities—protus, deuterus, tritus, and tetrardus,
reinterpreted as mq(2), mq(3), mq(0), and mq(1)—are
structured by the four interlocked circles of fifths; the
twelve pitch classes are structured by the twelve scalar
cycles of modal qualities. (2.2) Extended region of six
partially overlapping octave-fifth regions (reckoned from
a C modal center) that contains all pitch classes C and G
in Dasian space. (2.3) The operations transpositio (p) and
transformatio (f ).

is a closed structure in two ways. First, it stands as an ordered cycle of forty-

eight space elements (outer circle in Figure 2.1), where each pitch class (pc)
appears four times at different modal quality (mq) positions, and conversely,
each modal quality appears twelve times at different pitch class equivalences.
For generalization purposes, the four medieval modal qualities are relabeled
as cyclic order positions, where protus corresponds to mq(2), deuterus to mq(3),
tritus to mq(0), and tetrardus to mq(1). As a result, the ordering 〈mq(0), mq(1),
mq(2), mq(3)〉 graphically privileges the continuity of whole-tone scalar adja-
cencies, where radii in the figure mark off the semitone between neighboring
positions mq(3) and mq(0). Each space element is thus uniquely defined as an
ordered pair coordinating a pitch class x and a modal quality y: [pc(x), mq(y)],

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 José Oliveira Martins Bartók’s Polymodality 279

where 0 ≤ x ≤ 11 and 0 ≤ y ≤ 3.10 Second, the space exhausts several increasingly

larger diatonic structures, which are related by perfect fifth (twelve times):
semitones (which indicate unique scale-step locations in the space), “Dorian”
tetrachords (TST), guidonian hexachords (TTSTT), and locally diatonic
octave-fifth regions.11 Thus conceived, the Dasian stands both as a scalar space
(where the order of modal qualities interacts with register) and a pitch-class
space (where each element assumes octave and enharmonic equivalence).
Figure 2.2 reframes within a Dasian framework Bartók’s illustration of
polymodal chromaticism discussed in reference to Example 1. The region of
the circle that contains all pitch classes C and G is segmented into six overlap-
ping diatonic regions. The modal labels, clockwise from Phrygian to Lydian,
indicate the diatonic mode of each region’s pitch-class context, reckoned from
a C modal center (or tonic). As proposed in Bartók’s formulation of polymo-
dality, the Dasian modeling preserves the scalar integrity of modes but mea-
sures distance between them in terms of their chromatic mismatches rather
than in terms of the distance between their pitch centers.12 In addition, the
partial overlap of scale segments in neighboring modes suggests a continuity
and harmonic gradation in polymodal space rather than a discontinuity of
distinct modal or diatonic affiliations, whereby relations between different
segments are routed directly through a measurable distance between space
elements.
The system of modal quality affinities and pitch-class replications in the
Dasian space induces relations between nonadjacent (pc, mq) elements that
can be conceived as privileged navigation modes or intraspace motions. Each
modal quality affinity relation is structured by one of the four interlocked
circles of fifths (internal circles in Figure 2.1). In a dynamic view of this prop-
erty, the motion of (multiples of) four “steps” (or clockwise stations) in the
space retains a given modal quality and induces a change of pitch class by (mul-
tiples of) T7 (mod 12). Let us label this characteristic motion as transpositio (p),
thereby adopting to the context of the Dasian space the medieval term that

10 While conceived in different terms and designed for a consistency, completion, and locality. The article general-
distinct repertoire, Steve Rings’s (2011) formulation of a izes these properties in the context of affinity spaces, of
theoretical framework that combines scale-degree qualia which the Dasian space is a particular example.
and pitch-class chroma underlying a tonal generalized inter-
12 Although not a primary focus of this article, the notion
val system (GIS) structure captures aspects akin to the
of pitch centricity, or referential element as privileged com-
relation between modal quality and pitch class proposed
mon tone to different modal segments, can be modeled by
here. The theoretical framework is developed in his chap-
the operation of transformatio introduced below by “fus-
ter 2.
ing” two or more occurrences of the same pitch class in
11 Martins 2009 shows that seven-note diatonic segments the Dasian space. The notions of pitch centricity and set-
embedded in Dasian space (bounded by the transformatio referentiality are invoked in the analysis addressing the
relation as described below) constitute a unique symmetri- various contexts for pentatonic themes throughout the third
cal set-class (“host set”), which establishes a privileged movement of Bartók’s Piano Sonata, as well as in the move-
correspondence with the space based on the properties of ment’s closure.

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 280 JOURNAL of MUSIC THEORY

refers to a transfer of an interval pattern to a different position (or location)

related by affinity in the diatonic scale.13 In Figure 2.3, solid transpositio
arrows exemplify the recurrence of p, (F, 0) → (C, 0) → (G, 0) → (D, 0), or
pitch-class transposition by T7 that retains mq(0), or tritus.14
Conversely, pitch-class replication in the space is structured by a cycle
of modal-quality order positions. The corresponding motion of (multiples of)
seven steps retains the same pitch class and induces a change of modal qual-
ity by (multiples of) 1 descending order position (mod 4). Let us label this
characteristic motion as transformatio (f ), thereby also adopting the charac-
teristic medieval term that refers to a change of a note’s surrounding interval
pattern, or formation (i.e., a change in its modal quality).15 In Figure 2.3, solid
transformatio arrows track the motion f, (C, 3) → (C, 2) → (C, 1) → (C, 0),
which exemplify the retention of pitch class C over an incremental descent in
modal quality order positions. However, the continuation of a seven-“step”
motion, exemplified by the dotted f arrow from (C, 0) to (C ♯ , 3), enacts a
change of pitch class by T1 (mod 12) while continuing the descent in modal-
quality order position from mq(0) to mq(3). Let us also label this motion
transformatio, since it extends the change of modal quality to the (chromatic)
alteration itself (which in turn effects the formation, or reconfiguration, of
the surrounding interval pattern).16 In short, there are two types of outcomes
for a transformatio move: either a pitch-class retention when the move is
applied to mq(3), mq(2), or mq(1), or a pitch-class change (by T1, mod 12)
when applied to mq(0).17 A complete forty-eight-transformatio cycle results
from the succession of twelve complete four-modal-quality cycles of descend-
ing order position (where each four-modal-quality cycle goes through the
four pitch-class occurrences).
The structure of the complete forty-eight-transformatio cycle is isomor-
phic with the structure of the complete (clockwise) forty-eight-step cycle,
which results from the succession of the twelve complete four-modal-quality
cycles of ascending order position (the forty-eight-step cycle is the default
cyclic representation of the Dasian space used in Figure 2). Let us define the
two possible outcomes for a step move s: either s effects an “ascending whole-
tone” (T2, mod 12) when applied to mq(0), mq(1), and mq(2), or s effects an
“ascending semitone” (T1, mod 12) when applied to mq(3).

13 A historical and theoretical examination of the terms 16 Using as framework the medieval Dasian scale, the
transpositio and transformatio in medieval writings appears writer of the Enchiriadis treatises labels changes in modal
in Pesce 1986 and 1987. quality due to “chromatic” inflections of the given chant as
vitia. See Atkinson 2008, 128–29.
14 Rather than using an integer notation for the first ele-
ment pc(x) of the ordered pair [pc(x), mq(y)], this article 17 The formula regulating the unique value of pitch-class
instead uses the traditional letter name designation for change occurring in transformatio is generalized further
pitch classes: C for 0, C ♯ /D ♭ for 1, and so on. below for all affinity spaces.

15 The mnemonics (later conceived as space generators)

p and f stand for the initial letter in the words positio and
formatio, respectively.

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 José Oliveira Martins Bartók’s Polymodality 281

In short, the structure of the Dasian space forms a cyclic group of order
48 and can be expressed in the context of a generalized interval system (GIS).
In the GIS triple (S, IVLS, int), S is the family of forty-eight [pc(x), mq(y)]
pairs; IVLS forms a cyclic group of forty-eight intervals, of which p, f, and
s are group generators; and int is a function mapping S × S → IVLS, such
that int([pc(x), mq(y)], [pc(w), mq(z)]) = int([pc(w – x)], [mq(z – y)]), where
pitch class and modal quality intervals are computed in mod 12 and mod 4,
respectively.18
The Dasian framework and associated analytical methodology are now
probed in two short Mikrokosmos pieces, which illustrate the space’s versatility
for modeling polymodal relations.19 Bartók’s “Diminished Fifth” (Mikrokos-
mos no. 101) is frequently discussed as an instance of octatonic usage, but the
piece also engages consistent polymodal diatonic relations. The piece can be
organized in six phrases, which are paired up to form three larger sections,
according to motivic and harmonic considerations: mm. 1–11, A A′ | mm.
12–25, B A′ | mm. 26–44, B′ A.20 The pitch reduction in Figure 3.1 shows the
piece systematically superimposes scale segments (“Dorian” (0235) tetra-
chords) related by T6 (hence the piece’s title), which combine into resultant
octatonic collections. In the first section, the tetrachordal superimpositions
of both phrases produce OCT23; in the second section, the first phrase starts
with OCT01 and the second phrase returns to OCT23; and finally, the last
section runs through the three different octatonic collections (starting with
the remaining OCT12, followed by OCT01, and returning to OCT23 in the
closing phrase).
While a traditional reading contextualizes each tetrachordal strand
with respect to a global octatonic macroharmony for each phrase, 21 a linear
modal component cuts across phrase boundaries and addresses the sequence
of octatonic collections and tetrachordal partitions throughout the piece. The

18 The concept of GIS is developed in Lewin 1987, chaps. different pieces of Bartók and not to explain matters of
2–3. In the group structure of the Dasian space, the gen- scale interaction or resultant harmony within a single
erators s and f yield the entire space, whereas p alone piece. Martins (2006b) proposes a construct related to the
does not (iterations of p generate only twelve elements). Dasian space, called Guidonian space, which is used as
Multiple (x) iterations of a given generator are expressed in a background scalar framework to model diatonic relations
the form px , fx , and sx . While focusing on a different set of in Stravinsky’s “Hymne” (Serenade in A). The Guidonian
pitch properties, Edward Gollin’s (2007) notion of multi­ space is based on a pitch-class cycle that results from the
aggregate cycles, structuring compound interval cycles, stacking of a 2–2–1 modular unit, which efficiently embeds
is amply consonant with the approach developed here; I three diatonic modal qualities.
address several aspects of Gollin’s construct later in the
20 The formal arrangement proposed observes primarily
article, when considering the larger framework of affinity
thematic (motivic) and imitative relations. Other formal
spaces.
arrangements are naturally possible, such as a rondo-like
19 Bachmann and Bachmann (1983, 85–87) identify a pat- scheme.
tern (referred to as the Lydian octachord scale cycle) that
21 I use the term macroharmony in the sense proposed by
corresponds to a registrally disposed Dasian scale pattern.
Dmitri Tymoczko (2011, 4) as “the total collection of notes
While relevant for mapping Bartók’s musical materials,
heard over moderate spans of musical time.”
however, the pattern is used merely as a grid or back-
ground compendium for themes and motives drawn from

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 Journal of Music Theory 59.2: Music Examples, p. 45

282
Martins, JOURNAL
Fig. 3.1: of MUSIC THEORY

(3.1)
A A' B A

OCT23 OCT23 OCT01 OCT23

 
    
 
E  -Dorian
       


T6 12–19 19–25

          
 
A-Dorian
  


B' A

OCT12 OCT01 OCT23

  
 B  -Dorian
         
A-Dorian


T6 26–29 30–34 35–44

    E  -Dorian
E-Dorian         


Martins, Fig. 4.1 (correx 7/24/15):


Ar
    Br
    
 f

1–9; 10–16 f -1 f 17–24


f -1

       


Al Bl

Figure 3. Bartók’s “Diminished Fifth”

(Mikrokosmos no. 101). (3.1) Pitch reduction of the
piece based on the superimposition of “Dorian”
(0235) tetrachords. (3.2) Octatonic and diatonic
relations of Dorian tetrachords as projected in
Dasian space.

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 José Oliveira Martins Bartók’s Polymodality 283

crossed succession of tetrachords in the second section (from B to A′) yields

the diatonic modal pair (E ♭ -Dorian, A-Dorian), unifying the entire section.22
In the third section, the climatic phrase B′ moves by T7 to the pair (B ♭ -Dorian,
E-Dorian) before returning to the crossed succession of the previous pair of
modal collections to finish the piece.23
The projection of Dorian tetrachords in Dasian space in Figure 3.2
shows that the piece explores two contiguous Dasian segments in opposite
parts of the space. Adjacent Dorian tetrachords making up the modal collec-
tions are related by p, and octatonic tetrachordal pairs are related by p 6 occu-
pying polar positions in the space. This reading privileges the diatonic conti-
nuity of the piece as it corresponds to contiguity in the space at the expense
of octatonicism, which results from the superimposition of harmonically dis-
tant tetrachords.24 The Dasian/modal reading also accounts for the two pen-
ultimate notes of the piece (G and D ♭, circled in Figure 3), which are foreign
to the octatonic reading but extend the three contiguous Dorian-tetrachordal
segments.
Figure 4.1 is a pitch reduction for “Melody against Double Notes”
(Mikrokosmos no. 70), which superimposes modally distinct scale segments in
the left and right hands. Up to m. 16, both hands present (0257) tetrachords,
suggesting two “gapped” scale segments Al (left hand), D–E–G–A, and Ar
(right hand), F♯ –G ♯ –B–C ♯ . In the first phrase (mm. 1–9), the right hand plays
a melody that emphasizes the note F♯ and the left hand plays a chordal accom-
paniment, whereas the second phrase (mm. 10–16) reverses the tetrachordal
roles, with a left-hand inverted melody emphasizing the note A and a right-
hand chordal accompaniment. After m. 17, gapped tetrachordal segments
are filled by the notes F and A ♯ , respectively, resulting in inversionally related
(02357) pentachords D–E–F–G–A and F♯ –G ♯ –A ♯ –B–C ♯ .
The ambiguity of diatonic affiliation of right- and left-hand tetrachords
has interesting implications for polymodality and harmonic progression. Given
the lack of a defining semitone, each gapped tetrachord can be projected at
two alternative locations in Dasian space, marked A and B in Figure 4.2. As

22 The scalar arrangement of E ♭ -Dorian can be particularly “chromatic” semitones A ♭ –A and D–E ♭ that occur across
heard across phrase boundaries in mm. 19–20. Notice the strata. This distinction underlies the argument that this is a
motivic imitation between hands and {E ♭ , C, B ♭ } as sharing polymodal piece. The suggestion is that diatonic semitones
aspects of both the B phrase (pitch content) and the ensu- occur “within” a given diatonic collection or, in Dasian
ing A phrase (motive). terms, between contiguous modal qualities (deuterus-t­
ritus), that is, [pc(x), mq(3)] → [pc(x+1), mq(0)], whereas
23 George Perle (1990) discusses the interaction of octa-
chromatic semitones occur “between” distinct diatonic
tonic and diatonic modes in Bartók’s “Song of the Har-
collections or, in Dasian terms, [pc(x), mq(y)] → [pc(x+1),
vest” (44 Violin Duos no. 33). The compositional design
mq(w)], where y and w are modal qualities other than
of the duet is similar to the piece analyzed here (except for
mq(3) and mq(0), respectively. In the case of the opening
the ending of the duet, which “resolves” the octatonic ten-
octatonic collection, these chromatic relations are A ♭ –A,
sion by turning to a single diatonic mode).
(A ♭ , 1) ↔ (A, 2); and D–E ♭ , (D, 1) ↔ (E ♭ , 2). Both “chromatic”
24 This reading argues for a perceptual as well as a theo- semitones distance p 5 + s (i.e., 5-times-transpositio +
retical distinction between what we could refer to as “dia- 1-step) in the space.
tonic” semitones C–B and F–G ♭ that occur within strata and

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 284Martins, Fig.J O4.1
U R (correx
N A L o f 7/24/15):
MUSIC THEORY

      
(4.1)

  
Ar Br

 f

1–9; 10–16 f -1 f 17–24


f -1

        

Al Bl

Figure 4. Bartók’s “Melody against Double Notes” (Mikrokosmos no. 70).

(4.1) Superimposition of “gapped” tetrachordal and “filled-in” pentachordal segments.
(4.2) Location and progression of superimposed segments in Dasian space.

suggested in Figure 4.1, the prominent F♯ in the right-hand melody (mm. 1–9)
can be heard to “fill the gap” of the left-hand tetrachord, suggesting a “major”
pentachord D–E–F♯ –G–A; conversely, the left-hand A of the inverted melody
(mm. 10–16) fills the gap of the right-hand tetrachord, suggesting the “minor”
segment F♯ –G ♯ –A–B–C ♯ . Both F♯ and A “major and minor third imprints” cor-
respond to symmetrical transformatio moves ( f and f −1) in Dasian space. Fig-
ure 4.2 shows that a counterclockwise transformatio (dotted arrow) signals
the change of modal quality or function from Ar to Al for pitch class F♯ , that
is, f −1, (F♯ , 2) → (F♯ , 3); conversely, a clockwise transformatio (dotted arrow)
signals the reinterpretation from Al to Ar for pitch class A, that is, f, (A, 1) →
(A, 0). These mutual “imprints” suggest the necessary semitonal relations to
define segmental locations in Dasian space.25
The filling in of tetrachordal segments also impacts the assessment of
harmonic distance and tension. The segments at A are more closely located

25 Notice that these pentachordal scale segments are not

uniquely affiliated to a single diatonic collection but are
uniquely located in Dasian space.

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 José Oliveira Martins Bartók’s Polymodality 285

in Dasian space, given their minimal step mismatch created by G/G ♯ in each
hand, that is, between (G, 0) and (G ♯ , 3). These segments are also symmetri-
cally located in the space around C ♯/D and G/G ♯ , reflecting the symmetry
created by the combination of tetrachords in pitch space. The arrival of notes
F and A ♯ (m. 17), which are integral to each segment (located at [F, 0] and
[A ♯, 3], respectively), result in a symmetrical shift to locations at B, via f and f −1
(solid arrows in Figure 4). The increased mismatch between segments (F/F♯ ,
G/G ♯ , A/A ♯) is reflected on the increased harmonic distance of segments in
the space, in which the inversional relation between pentachords is reflected
in both the Dasian and pitch space.26

Polymodality in the third movement of Bartók’s Piano Sonata

After drafting the ideas framing polymodal chromaticism, Bartók ([1943]

1992, 368) lists his Piano Sonata (1926) among works where “this principle is
put into action—at least partially.” While he did not specify which movements
or passages in the Sonata might better “activate” the principles of polymodal
chromaticism, the convocation of various combinations of diatonic materials
in the third movement invites scrutiny. However, rather than relying on the
composer’s restrictive formulation of modal superposition projecting a single
pitch center, I examine the gradual unfolding and cumulative effect of com-
binations (simultaneous and successive) of scale segments modeled by the
Dasian system of relations. My general analytical goal is to understand how
imprints of scalar layerings onto Dasian space might shape tonal trajectories
(polymodal syntax), which participate in coherent formal processes. In par-
ticular, the analysis explores how both local and global trajectories of poly-
modal patterns, which create a kind of “inner” form, relate to and qualify rota-
tional aspects of the movement’s “outer” form, traditionally viewed as a rondo.27
A brief sketch of the movement’s formal layout helps to frame the ana-
lytical argument. The rotational scheme summarized on Table 1 organizes
the piece into four rotations, the last of which is condensed and functions as
a coda. Each rotation results from the reiteration of a cyclic sequence of what
we can refer to as blocks, which include time units of various lengths, from
whole sections to brief passages, whose characteristic thematic materials, or
prominent textural changes, result in significant formal markers.28

26 The superimposition of key signatures (zero and five 27 Analytical accounts of the movement as a rondo appear
sharps) along with the inversional relations between hands in Somfai 1990, Wilson 1992, Konoval 1996, and Susanni
could be traditionally interpreted as the diatonic modes 2001. The historical scope and character of rotational form
D-Dorian (lower pentachord) and C ♯ -Dorian (upper penta- is proposed and discussed in Hepokoski and Darcy 2006
chord). While this (symmetrical and modal) reading is also (see esp. app. 3, 611–14). For the aspects of rotational
captured in Dasian space, I suggest focusing not on whether form in Bartók, see Keller 2011.
we hear (poly)tonics for minor and major pentachords but,
28 The notion of formal “block” is traditionally used in the
rather, on the interaction and contrapuntal deployment of
context of Stravinsky’s abrupt discontinuities of material
mismatched and harmonically distant strata.
and texture. The idea of block stratification was first pro-

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 286 JOURNAL of MUSIC THEORY

Table 1. Rotational form and block sequence for the third movement of Bartók’s Piano Sonata

Figure 5 sketches formal blocks in each rotation and aligns them

according to thematic and textural materials. Blocks A initiate each rotation
with a thematic refrain, characterized by a pentatonic pattern that descends
from an anchoring pitch class F♯ 〈−2, −3, −2, −2〉 (which I refer to as the fixed
refrain theme; see Example 2). Traditional attributions of the form as a
rondo are based on the movement’s cyclic alternation between fixed refrains
in blocks A and episodic thematic variants (or movable episodic themes) in
blocks E and in the additional block G. Given that episodic themes promi-
nently feature transposed pentatonic variants of the refrain, there is consid-
erable disagreement on what constitutes the contrast between refrains and
episodes and which events mark the boundary between them.29 Rather than
weighing on the different analytical criteria for hearing contrast required in
a rondo scheme, however, I propose to focus on the cyclic or rotational
sequence of blocks A–F, which remains mostly invariant with respect to order
and overall thematic content. In the rotational scheme, the main thematic
statements (blocks A, E, and G) provide higher formal stability, while the
remaining passages (blocks B, C, D, and F) project transitional, dynamic, and
closing formal functions.
My argument unfolds three related analytical claims. First, the analysis
shows that the various polymodal combinations and textures in the move-
ment are efficiently modeled by the system of diatonic relations of the Dasian
space and result in several tonal patterns that reinforce and establish internal
relationships between blocks within the proposed rotational scheme. Focus-
ing the analytical lenses on rotation 1, I propose that the pitch exploration
in the first three blocks creates a larger continuous unit. In particular, the

posed in Cone 1962 and further developed in van den sodes at block D. For Paul Wilson (1992, 78), the contrast
Toorn 1983. Because some of the blocks or “components” required for a sense of alternation between the rondo sec-
in the third movement of Bartók’s Sonata (such as blocks tions results from “changes in harmonic setting, rather
C and D) are temporally brief, they are better understood than a true change of theme,” such that the alternating
as important changes of texture and harmonic material binary scheme groups A and B, on the one hand, and C, D,
rather than constituting sections per se. E, and F, on the other. Yet other analytic studies (Konoval
1996; Susanni 2001) consider block B as providing the
29 László Somfai (1990, 546, 547) sees the movement in
most prominent contrast to block A, and practically all sec-
a binary (ritornello-episode) scheme but describes it as “a
tions other than A are considered as episodic material.
monothematic rondo, for the episodes are, from the the-
Paolo Susanni (2001) considers block C as transition,
matic standpoint, variants of the ritornello theme”—the
whereas Michael Konoval (1996) does not consider block
basic contrast lies between the “agitated mixed meters
A (at m. 157) to be a true ritornello but, rather, a variant
of the ritornello” and the “smooth ostinato background” of
(episode) with transitional function.
episodes. As such, Somfai sees blocks C as the closing
of an extended ritornello and locates the beginning of epi-

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Blocks: A B C D E F G

Rotation 1
                 
       
mm: 1 20 38 40 53 84
    
Martins, Fig. 5 (correx 7/24/15):

Rotation 2
 
           
            
mm: 92 111 137 139 143 156
     

Rotation 3
            
  
      
    
mm: 157 175 192 194 205 222 227
  
José Oliveira Martins

Rotation 4
            
  
      
Journal of Music Theory 59.2: Music Examples, p. 46

mm: 247 262 264 265 281

  

Figure 5. Block sequence, alignment, and rotational scheme of prominent thematic material in the third movement of Bartók’s
Piano Sonata
Bartók’s Polymodality

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287
 � �� � � �

OURNAL of MUSIC
288
Martins, Ex. 2 J(correx 7/24/15): THEORY

� = 170
Allegro molto
� � �
3 �� � � 24 � � � � � � � � � � � � �� � � 24 � � � � �
Ar

� � 8 �� � � � � � � �� � � � � � � 41 � ��� 38 � � � � � � � � ��
38
�
� 3 � 24 � � ��� � 41 �
��� 3 � 24 � � � �� � 38
� 8 �� �� 8 ��
Al

� �
�� � �
Ar

� � � � � � �� � � �� �
3 24 3 �� � � 24 � � � �� � � � � � � 14 � �� 38
� � �� � � 8
7

� � 8 �� � �� � � � �� ��
� �
�� � � � �� �
� 3 � 24 � ���� � ��� 83 24 �� � ��� �� � � 41
��� 38
� 8 � � �� ��

� � �
Al

�� � �
Ar
� �� � � � � � � � � �
� � � � 2 � � � � �� 3 �
83 � 24 � �� � ��� �� �� �� �� �� �� �� �� 68
13

4 8 �
� � �� � �� � � � � � �
� �
� �
� �� � � � �� �
�
�
� �� 3 � � � 2 � �� � ���� ��
3 24 �� � ��� �� ��� � �� 68
� 8 � �� 8 4 ��� Al �� � �� �� �� ��

Example 2. Opening refrain (mm. 1–19), which is framed by a pentatonic pattern that
descends from an anchoring pitch class F♯ 〈−2, −3, −2, −2〉

imitative counterpoint in block B sets up a dynamic Dasian trajectory, which

dramatizes the static polymodal tension between the melody and accompani-
ment of the refrain in block A and progresses to the marked polychord of
block C. The superimposition of diatonic segments in block C, in turn, neatly
demarcates an extended Dasian region (explored in previous blocks) and
culminates a process of symmetrical expansion for the first three blocks in
the rotation. The last three blocks in the rotation also present a distinct pat-
tern of continuity. The transitional block D and the concluding polychord in
block F exhaust the Dasian regions not previously enacted, whereas the suc-
cession of phrases of the episodic theme in block E reenact the polymodal
tension of superimposed layers in the opening refrain. The analysis thus
shows that the structured exploration of the entire Dasian space is exactly
completed within a full rotational cycle.
Second, while pentatonic patterns frame both fixed and movable themes,
the specific Dasian regions explored depend on the distribution of semitones
that “fill in” each pentatonic framework. As such, pentatonic frames function
as potentialities for a variety of polymodal inflections in the piece. The anal-

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 José Oliveira Martins Bartók’s Polymodality 289

ysis shows that the polymodal scope of fixed refrains is significantly expanded
in later rotations, and the relation between superimposed layers of scale seg-
ments changes gradually at every rotation.
Third, in contrast to attributions of a “weak closure” for the piece (Wil-
son 1992, 84), the Dasian approach clarifies and reinforces the claim for a
strong closure. I propose that closure in the piece entails two related large-
scale gestures: first, a tentative closure is offered with the insertion of a new
episodic variant or theme (in block G) at the end of rotation 3; a second,
more emphatic and definitive closure is offered through the dissolution of
the refrain’s harmonic and thematic features in the contracted block C/D in
rotation 4. I argue that the alteration of formal patterns at the end of the
movement also provides a solution for the compositional problem posed by a
cyclic form: how to articulate a closure without leading into a new rotation.

Polymodality in rotation 1

Perhaps the most suggestive passages of polymodal gradation in the move-

ment are blocks B, where short scale segments overlap and modulate through
different diatonic regions. These blocks function as transitional sections,
providing some thematic and textural relief from the previous refrain of
blocks A leading into the dense polychords of blocks C.30 Example 3 illus-
trates the corresponding passage in rotation 1 (mm. 20–38), which builds up
a forward-driving counterpoint of partially overlapping diatonic scale seg-
ments, culminating in the strident polychords at m. 38. Figure 6.1 presents a
scalar reduction of the passage, segregating between right (r) and left (l) hands
and temporally successive segments (B1 through B4).31 There’s an overall
“modulation” from a “D-Mixolydian” region (B1l ) toward an “E-Mixolydian”
region (B4), accomplished through upward transposition by perfect fifth of
certain scale segments on both hands.32 In Figure 6.2, these two regions (B1l
and B4) form a symmetrical layout in the space by overlapping the space ele-
ments (A, 1) and (B, 2). (These two elements assume a central role in the
connection of block B to the polychord of block C, as discussed below.) In the
overall harmonic progression of block B, the succession of segments shifts
clockwise through contiguous space rather than signaling specific modal

30 Benjamin Suchoff, the editor of Béla Bartók Essays, and r and l refer to right- and left-hand components of tem-
chooses a passage from the third movement of Bartók’s porally simultaneous strata (although at times this is not
Piano Sonata (mm. 127–37) as an illustration of polymodal strictly the case, as in the vertical chords of the refrain,
chromaticism from a list of possible pieces indicated in which are played by both the left and the right hands but
Bartók’s notes for the lecture (Bartók [1943] 1992, 368). are labeled as Al).
The passage chosen displays a counterpoint of distinct
32 Although we might well hear pitch class D as a pitch
scalar segments culminating on a superposition of the
center in B1l (mm. 20–25), modal finals are used here for
two-flat diatonic collection and the white-note collection.
labeling purposes only.
31 The analysis sets up the following labeling system: the
first letter refers to the active block, integers distinguish
temporally successive segments in order of appearance,

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 Journal of Music Theory 59.2: Music Examples, p. 41

Martins, Ex. 3 (correx 9/25/15):

290 JOURNAL of MUSIC THEORY

� �� �
6 2 � ���� � ��� 5 1 ��� 6 2
4� �
20 B1r

� � �
�� 8 4 � 8 8 � 4

� �
6 meno 24 85 � � � � � 41 �� 68 � � � � � � � 24
� 8 � � � � �� � �� �� � � � � �� �
B1l “D-mixolydian”
B2r

�� � � � � �
2 � ��� � ��� 5 14 � ��� 85 � � � � � � � �� � 83 �� � � 85
25

� � 4 �� �� 8 � �� �� � � � � � �� � � �
� ff �
� 2� � � � � � �� � � � �� � �
� 4 �� �� 85 � � � � � 41 � 85 � � � 83 � � 85
�
B4

�� �� � � ��
B3r poco a poco stringendo

�� � � �
5 � � �� � 38 � � �� �� �
85 � � � � � � 38 � � � � 85 �� �� � � �� � �� � � ��
31

�� 8
� �
� � � � �� � 3 �� �
� 5 � � �� � �
� 8 8 � 85 � � �� �� � � 38 � � 85 � � ��
�� � �� � �� �� �� � �
� � � B4
� � ��
B2l B3l “E-mixolydian”

� �
al

�� �� � � � ��� � ��� �� �
� �� �� �� � �� �� �� ��
Cr

� � � �� �� �� � 2 � � ����
� ���� ����
D

� �� � �� � � ���� �� �� �� �� �� ��
�� 4 � � � 85
37

��
� �� �� � �
�
� � � ��� �� � �� �� 24 � � ���� �
�� � ����
� ��
���� ��� � ���� � � ��
85
� � � �� �� �
�� �� �� �
Cl

Example 3. Block B (mm. 20–37), block C (mm. 38–39), and beginning of block D (m. 40) in
rotation 1

affiliations. The motion through melodic peaks in the right-hand F♯ –C ♯ –G ♯

and the motion of bass notes G–D–A in the left hand is modeled by transpo-
sitio (p), (F♯ , 3) → (C ♯ , 3) → (G ♯ , 3), and (p), (G, 1) → (D, 1) → (A, 1).33 The
succession of scale segments also creates occasional chromatic mismatches,
as indicated by the dotted arrows in Figure 6. These mismatches (F/F♯ , C/C ♯ ,
G/G ♯) are modeled by transformatio (f ), (F, 0) → (F♯ , 3), (C, 0) → (C ♯ , 3), and
(G, 0) → (G ♯ , 3).

33 Wilson (1992, 81) also notes the T 7 upward motion in

both hands as part of the transformation of set-class 7–35
(i.e., from D-Mixolydian to E-Mixolydian).

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 Journal of Music Theory 59.2: Music Examples, p. 47

José Oliveira Martins

Martins, Fig. 6.1 (correx 7/24/15): Bartók’s Polymodality 291

(6.1)
1–19 20–25 28–30 31–33 34–37 38–42
Ar B1r B2r B3r B4 Cr
p

�� �� � �
p

�� � � � � �� � � ���
�� � �� �� �� � �� � � �� �� � � ��
�� � �� ��
�� f f f2

� �� � � �� � � �� �� � �
f
�
f -2

� � �� � � �� � � � �� �� � �� � ���� ( ��)
� �� � �� � � p p
“D-mixolydian” “E-mixolydian”
Al B1l B2l B3l B4 Cl

Martins, Fig. 7.1 (correx 7/24/15):

Rot. 2

D1 D2 E1 E2 F1 F2 F3 F4 F5 Ar

�� � � �� �� � � � ��
�� �� �� �� �� �� �� �� �� � �� �� � � �
�� �� � �� ��� �� ��� �� �� ��
��
�� (D2)
f -1

�
43–46 47–52 53–76 77–83 84–91 92

�� �� �� �� �� ��
�
(D1)

Figure 6. Activity of scale segments in blocks

A, B, and C of rotation 1. (6.1) Pitch reduction
and contrapuntal deployment of scale
segments. (6.2) Progression of segments
within block B in Dasian space. (6.3) Dasian
location for block A and symmetrical split for
the arrival of block C.

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 292 JOURNAL of MUSIC THEORY

The Dasian trajectory unfolded by the transitional block B dramatizes

and amplifies the polymodal relation and harmonic distance of combined
layers in block A, that is, between the melodic theme (Ar, fixed theme) and
chordal accompaniment (Al). The theme is articulated in two phrases, mm.
1–8 and 9–16, the second of which repeats and reinforces the first phrase (see
Example 2).34 The scalar reduction of block A layers in Figure 6.1 super­
imposes the diatonic segment A–B–C ♯ –D ♯ –E (built on the fixed pentatonic
frame, bracketed off in the figure) and a chordal accompaniment, which uses
the collection D–E–F♯ –G–A–B. (While the registral deployment of the
accompaniment privileges stacks of “fourths” and “fifths,” the bass descend-
ing motion A–G–F♯ –E emphasizes the scalar aspects of the collection.) A sim-
ple way to conceptualize the harmonic distance between accompaniment and
melody is to extend their respective pitch-class materials upon the line of fifths
G–D–A–E–B–F♯–C ♯ –(G ♯)–D ♯ . The fixed pentatonic frame takes up the con-
tinuous segment A–E–B–F♯ –C ♯ (in bold), the accompaniment takes up the
segment between G and F♯ , and the melodic theme takes up the segment
between A and D ♯ , overpassing G ♯ (more on this below). The two layers over-
lap the segment A–E–B–F♯ and are mismatched by D/D ♯ . Figure 6.3 maps
scale segments for the melody (Ar) and accompaniment (Al) in Dasian space,
assigning them precise locations due to their respective embedded semitones.
Therefore, while the layers (melody and accompaniment) share some com-
mon pitch classes, the Dasian projection clarifies their distinct, albeit har-
monically close, affiliations. We can thus interpret the Dasian trajectory in
block B to first expand to the counterclockwise region of Al (notice the semi-
tone B–C in B1l and the whole tone F–G of B1r)35 and then to gradually
unfold and bridge the distance between the segments that correspond to Al
and Ar. Furthermore, the mentioned transpositio relations between right-hand
melodic peaks (F♯ , 3) → (C ♯ , 3) → (G ♯ , 3) explicitly bridge that distance.36
The strategy of region expansion from block A to block B culminates in
block C, thus suggesting a formal continuity across these three sections. The
rhythmic culmination of block B is also a sectional overlap with the begin-
ning of block C (see Example 3, mm. 38, 39, and 42), which articulates the
collection G ♯ –A–A ♯ –B–C–D–E–F♯ , the most dissonant simultaneity thus far
in the piece, the set-class (0123468t).37 The collection is registrally split into a

34 In later rotations, the second phrase of refrains and epi- 37 Both Somfai (1990) and Wilson (1992) consider that
sodes is varied and expanded. block C articulates the boundary between ritornello and
episodes (chords in block C are included in the ritornello
35 While there are three potential locations in Dasian space
for Somfai and excluded for Wilson). As pivotal overlap
to accommodate B1r, F–G–C–D, its chosen location pro-
between large-scale sections, block C signals both the
vides both the lower boundary F and the gap C for block A.
arrival of the previous block B and sets up the (major sec-
This analytic choice can be understood as the fulfillment of
onds) material that characterizes the rest of the rotation.
an analytic opportunity.

36 G ♯ , the melodic peak of the passage, or (G ♯ , 3) in Dasian

space, also fills in the gap between C ♯ and D ♯ in the line of
fifths connecting the scalar materials of Al and Ar.

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 José Oliveira Martins Bartók’s Polymodality 293

quasi-symmetrical polychord, superimposing the segments Cl, A–(B ♭)–C–D–E,

and Cr, E–F♯ –G ♯ –A ♯ –B.38 The rhythmic continuity between blocks B and C is
emphasized by the scalar goals of the outer voices, arriving on pitch classes A
and B on the downbeat of m. 38 (marked by open note heads on Figure 6.1).
The analysis suggests a reinterpretation of these notes from the scalar con-
text of B4 to melodic prominence in the extreme registers of the split poly-
chord. This reinterpretation is modeled in Dasian space via a symmetrical
split of transformatio relations, which take pitch classes A and B from B4 to
Cl and Cr, respectively (see connected open note heads in Figure 6.1). Solid
arrows in Figure 6.3 capture the transformatio reinterpretation f −2,(A, 1) →
(A, 3), and f 2,(B, 2) → (B, 0). The split polychord of block C borders the
entire space covered by the previous blocks A and B, and in turn Cr and Cl
are themselves related by transformatio through the common pitch class E
(f |3|: (E, 0) ↔ (E, 3), shown by the dotted line in Figure 6.3).39 This transfor-
matio relation (and axis of symmetry) through the common pitch class E
between superimposed segments in opposite parts of the space also plays a
crucial role for the movement’s closure.
From block D up to the closing block F of rotation 1, whole-tone dyads
combine in various ways to produce different textures and assume several
harmonic roles. Therefore, the proposed formal and Dasian (extended dia-
tonic) continuity of blocks A, B, and C finds, to some extent, a counterpart in
the continuity across episodic blocks D, E, and F, although the Dasian model-
ing of the latter set reveals distinct polymodal patterns. Figure 7.1 registers
pitch reductions of all (simultaneous and successive) scalar activity in these
three blocks, and Figure 7.2 interprets such activity in terms of Dasian loca-
tions and resultant segment relations.
The episodic theme in block E is framed by a pentatonic pattern, which
is transposed by T5 or p −1 from the refrain to B–A–F♯ –E–D (bracketed off
in Figure 7).40 But unlike the refrain, the second phrase (mm. 74–83) expands
the polymodal scope of the theme. The first phrase (mm. 53–72) and the

38 The registrally reinforced subset D–E–F ♯ is also drawn thermore, this subito forte supplies E3, which was absent
symmetrically from both segments. The cluster at m. 38, from the Cl downbeat. While we can consider the poly-
although (abstractly) symmetrical about the axis E/A ♯ , is chord at m. 38 both as partially segmented and partially
not literally split symmetrically by the left and right hands. fused, it nonetheless results from the superimposition of
Instead, B ♭ is absent in the left hand (hence the paren­ the Dasian segments Cr and Cl.

39 The set D–E–F ♯ articulated in fortisimo in mm. 38, 39,

thesis in Cl ). In the absence of a defining semitone for
the left-hand pentachord—which would be either A–B ♭ or
and 42 reinforces the symmetrical layout of the split in a
B–C—my analytical choice falls on A–B ♭ , not only because
number of ways: it highlights the pitch-class symmetry by
fifths A–D in the bottom and B–F ♯ on the top registers and
it allows for the symmetrical transformatio but also because
it is reinforced by the left hand at m. 43. This is in line with
reinforces pitch class E as the pitch-class axis of the collec-
the compositional logic of “gap filling” demonstrated in
tion and note of transformatio.
the previous sections. Also, in the reduction of block C, I
am assigning F ♯ to the right hand, thus considering the F ♯ 3 40 Reflecting the actual deployment of pentatonic mottos
in the left hand to double the F ♯ 6 of the right hand. I justify in the musical surface, the analytical Dasian projections
this somewhat arbitrarily analytical move because F ♯ 3 dis- privilege minimally gapped scale segments.
appears from the left hand at the subito forte (m. 38). Fur-

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 Journal of Music Theory 59.2: Music Examples, p. 42

Martins,
294 Ex. 4J O
(correx
U R N A L7/24/15):
of MUSIC THEORY

�
�
�� �
E1 a tempo E2

� � �� � �� �� �� � �� � �� � �� � � �� � �
� � � �� � �� � �� � �� � ��� � �� � ��� � � �� � �� � �� � �� �� ��� �� ��� �
74

�� �
�
� � �� � � � �� � �� � � � �� � �� � � � �� � � �� � � �� � � �� � �� �
� � � �� � � �� � �� � � � �� � �� � � � ��
� �� �� � �� �� � �� �� � �� �� � �� �� � �� �� � �� �� � ��

Example 4. The second phrase of the episodic theme, mm. 74–83

beginning of the second use the segment E1, D–E–F♯ –G ♯ –A–B–C ♯ , while in
the course of the second phrase, there’s an inflection to E2, C–D–E–F♯ –G–
Martins, Ex. 5:
A–B (see Example 4, mm. 74–81). As the two segments present different sca-
�� �contexts upon the same
Più vivo � = 184
lar pentatonic framework, they also refer to different
� � � � ��
� � � � � �� � � � in
�� � �� �����
� � � � � �
58 � ���� m. �
harmonic regions and Dasian locations. The syncopated vocal “whoop”
24 �at� the boundary �� ��between �� E1
�� and � E2, � � �
42

�� 77, standing �� �� ��
seems to �� �
dramatize ��the
� shift �� between segments: � D1 the note A is associated (in different octaves) to G
� ♯�, suggesting the connection of � different occurrences of pitch class A in
C
� 5 � ��� �
� � ��� 24 � � � � � � � � � � ��
�
�
and G
� 8 � the��space � via transformatio f −1: (A,� 0) → � � (A, 1).41 Compared with the poly-
�� � � �� � � � � �� � �
modal state of the refrain (of block A), the expansion at the end of the second
phrase �
�
of the episodic theme retains the region that corresponds to the
� � �� refrain’s� ��accompaniment but� shifts
� � ��� � � � �� � �� � � � �� � �� � � � �� � ��
47

� � �� � � �� � � �� �� �
(by p −1) the region of the first phrase. In
�other �
words, the polymodal expansion now occurs in successive rather than
phrases,�as is the case between melody and accompaniment in
�
simultaneous
D2

� � the� �opening � � � ��refrain.42� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��

� �� � As a �formal
� �� � �� ��block
process, �� ��D rarefies�� �� �the � ��rhythmic�� �� �outburst
� �� �� of��block �� ��C
� E1and serves as a transitional antechamber for the arrival of the episodic theme
� � � � in�block � Example 5). In this process, whole-tone dyads first combine �
� E� (see
� � � � � � � �� ��� � ���
to
� � � � � � � � � � � � � � � �
53

�� � � � � � � � � � � � � � � � � �
create the two-flat diatonic collection C–D–E –F–G–A–B (D1, mm. 40–46), � � � ♭ � � � � ♭� � � �
� while later the dyads D ♭ –E ♭ and A ♭ –B ♭ modulate to a corresponding four-flat
�
� ��� ��� ���scalar �� �� �� �in � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �43� ��
diatonic collection (D2, mm. 47–48). As shown in Figure 7.2, the modeling of
� � � �� activity �� �� �� block �� ��D �mostly � �� ��maps �♯� into �� ��uncharted
�� �� ��Dasian �� �� territory,
�� �� �� bor- ��
� dered by the elements (E, 3) and (F , 0).

41 Somfai (1990, 547) refers to the sforzando syncopation the note B5 is the highest note. Depending on the ana-
in m. 77 as imitating a vocal “whoop.” The registral disso- lyst’s temporal focus, one could say that m. 38 anticipates
ciation between G–A and G ♯ –A helps project both associa- m. 53 or that the m. 38 re-elaborates m. 53. At m. 53,
tions for pitch class A. however, the pitch class B stands as the crux of super­
imposition of two Dasian spaces, corresponding to the
42 There is further reason for associating blocks E and A:
chordal and linear contexts of the note B at that point.
the episodic theme is bookended by material that maps
When pitch class B is understood as part of the segment
B–B ♭ –A ♭ , it reenacts the same location as in m. 38, since
into similar Dasian locations as those covered by Cr and Cl.
The vertical sonority at m. 53 G–A ♭ –A–B ♭ –B is transposed
the determining dyad is B–B ♭ , while as the initiator of the
thematic melodic segment B–A–G ♯ –F ♯ –E, pitch class B is
by T–1 (mod 12) from the chromatic partitioning of the clus-
ter at m. 38 G ♯ –A–A ♯ –B–C (the vertical sonority at m. 53
“channeled” (by transformatio) one counterclockwise sta-
tion to E1, and the determining dyad becomes A–G ♯ .
obtains from combining the whole-tone dyads from block
D with the pitch class B from block E1). Furthermore, these
moments are registrally associated because in both cases 43 The note B ♭ (in D1) fills the gap (B ♭ , 0) left open at Cl.

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 Martins, Fig. 7.1 (correx 7/24/15):
José Oliveira Martins Bartók’s Polymodality 295

(7.1)
Rot. 2

D1 D2 E1 E2 F1 F2 F3 F4 F5 Ar

�� � � �� �� � � � ��
�� �� �� �� �� �� �� �� �� � �� �� � � �
�� �� � �� ��� �� ��� �� �� �� ��
� �
(D2)
f -1

�
43–46 47–52 53–76 77–83 84–91 92

�� �� �� �� �� ��
�
(D1)

Figure 7. Activity of scale segments and collections in blocks D, E, and F of rotation 1.

(7.1) Scalar and chordal reductions. (7.2) Dasian locations and relations between scale
segments.

Taken together, blocks A–D exhaust almost the entire Dasian space
except for the segment (C ♯ , 1)–(D ♯ , 2)–(E ♯ , 3)–(F♯ , 0), which assumes an
important role in the closing of block F leading into rotation 2. Block F (mm.
84–91) alternates two polychords that superimpose each of the whole-tone
dyads retained from block D (through block E) to the opposing whole-tone
collection (or its subset), resulting in F1, G–A/B ♭ –C–D–E (mm. 84 and 86),44
and F2–F5, A ♭ –B ♭/F–G–A–B–C ♯ –D ♯ –E ♯ (mm. 85 and 87–91). The buildup
of a complete whole-tone collection in block F neutralizes or disrupts the

44 B ♭ –C–D–E (which is a transformatio echo of m. 80,

with B ♭ instead of B) is superposed to the dyad G–A (cir-
cled in Figure 7.2), thus reinforcing the position of Cl
(where B ♭ was absent).

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 Martins, Ex. 5:J O U R N A L o f MUSIC THEORY
296

�� �
Più vivo � = 184

� � ���� �� � �� � ��� �� �
�� � �� �� �� �
5
42
� � �� ����� 24 �� �� �� �� �� � � �� �� �� ��
�� 8 �� �� ��
� �� �
5 � ��� �� �
C
�
D1

�� � ��� 24 � � � � � � � � ��
� 8 �� ��
�� �� � � �
�� �� � � �� � �
�� � �

� �
� � �� �� �� � �� �� �� �� � � � ��� � � � �� � � � � �� � � � � �� �
47

� � �� � � � �� �� ��
�
�
D2

� � � � � � � �� � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��
� �� � �� �� � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��
�
� �� � � �
E1

� � � �� �� � � � � �� � � � � �� � � � � ��� � �� ��� � ��
� � � �� �� � �� � � � �� � ��
53

� � � � � � � � � � � �
�
�
� ��� ��� ��� ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��
� � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��
�

Example 5. End of block C, block D, and beginning of block E

diatonic/Dasian pattern beyond a 2–2–2 pattern, which is uncharacteristic of

the scalar fabric suggested in all previous sections of rotation 1; it is thus fit-
ting for assuming a liquidating function. The modeling of whole-tone collec-
tions onto the Dasian space implies the partial intersection of every other
whole-tone tetrachord (2–2–2). Therefore, “opposite” whole-tone tetrachords
are reached in the cycle as a result of building the scale up to the octave
completion (2–2–2–2–2–2) (see F2–F5 in Figure 7.2).
As F5 reaches the last tetrachord of the whole-tone collection B–C ♯ –D♯ –E♯,
it occupies the only remaining segment in the Dasian cycle not yet covered in
rotation 1, with the exception of (F♯ , 0) associated by semitone to (E ♯ , 3).
However, the semitone E ♯ –F♯ can be heard via the voice-leading connection
between E ♯ , the top note of the whole-tone collection, and F♯ , beginning rota-
tion 2 (mm. 91–92). The exhaustion of Dasian space in rotation 1, achieved
only by the arrival of F♯ initiating rotation 2, is suggestive of a rotational
reading in which the beginning of a new rotation also concludes the previous
rotation.45

45 The scalar context provided by the refrain in rotation

2 (block A) reinterprets the note F ♯ (ending rotation 1)
via transformatio f −1: (F ♯ , 0) → (F ♯ , 1) to its original space
location.

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 José Oliveira Martins Bartók’s Polymodality 297

Polymodal expansion in refrains of rotations 2, 3, and 4

The analysis of the episodic theme in block E showed that pentatonic frames
are pliable to different scalar contexts and thus adaptable to distinct (Dasian
or diatonic) harmonic regions. Figure 8 registers the various scalar contexts
for blocks A over the four rotations, accommodated by the fixed pentatonic
framework F♯ –E–C ♯ –B–A (bracketed in the figure). Compared with the seg-
ment of the opening right-hand melody (Ar, rotation 1), the theme’s poly-
modal scope is expanded considerably in rotations 2–4 through the use of
various registers and scalar patterns.
The modeling of polymodal relations in Figure 8 exposes the double
nature of the Dasian space as both a scalar space (where the order of modal
qualities interacts with register) and a pitch-class space (where each element

Figure 8. Mapping of blocks A (Ar and Al) throughout the four rotations

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 Journal of Music Theory 59.2: Music Examples, p. 43

298 JOURNAL of MUSIC THEORY

Martins, Ex. 6:

�
A2r

� ��
�� � � �� � � �� � � �� �� � � �� �
A3r

� � ��� �� � � � ��
5 �� � � � 2 � �� � 58 � � � � � � � 83 � � � 5 �� � � � 24
100

�� 8 4 ��� 8
ff
� � ��
� �� � � �� � � �� �� � � � �� 3 � �� �� � � � � �
5 24 �� � ���� 85 ���
85 � � � � 24
� 8 8 ��
Al

�� �
�
� � �� � ��� � � �� � � �
2 �� � 58 � � � � � � 24 � � 68
105

� � 4 � � � � ���� � ���� � � ���� � ���� � � ���� ����

�
� � �
� �� � � ff
24 � � � � �� 85 � �� �� �� � � 24 �
� � � � � � � � �� �� �� � �� 68
� � �� � � � � �� � � �� ��
�

Example 6. Second phrase of the refrain (block A) in rotation 2, mm. 100–110.

assumes octave and enharmonic equivalence). At times, the polymodal

expansion results from the assignment of different chromatic inflections to
distinct registers in the same melodic statement, thus differentiating by reg-
ister two Dasian regions. For instance, the beginning of the theme’s second
phrase expansion in rotation 2 (see Example 6) projects the extended space
A2r, A–B–C ♯ –D ♯ –E–F♯ –(G ♯)–A ♯ in mm. 100–103, thus reaching A ♯ in the
upward expansion (m. 100) and retaining A of the pentatonic frame in the
melodic descent (m. 102). (This relation is modeled by f −1: [A ♯ , 3] → [A, 0].)
In contrast, the polymodal expansion later in the second phrase results from
the chromatic inflection of D ♯ (m. 102) to D (m. 106). While the inflection
D ♯/D occurs in the same register, their respective integral segments corre-
spond to distinct Dasian regions (A2r to A3r). The modeling of the passage
as involving distinct Dasian locations thus rests on considering elements as
pitch classes, where the mismatch D ♯/D is modeled by f −1: (D ♯ , 3) → (D, 0).46
In this analytical case, hearing the passage in terms of pitch-class equivalence
is actually facilitated by the three- or four-octave registral reinforcement of
the theme.
Figure 8 also shows that, in refrains of rotations 2 and 4, the expanded
polymodal region of the melodic theme (Ar) extends to the Dasian location

46 The analysis of Figure 7, which refers to changes in

the scalar context occurring in the same musical register
(block E), also rests on the notion of space elements as
pitch classes.

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 José Oliveira Martins Bartók’s Polymodality 299

(Al) of the accompaniment, which is retained in its original location. In con-

trast, the Dasian location Al for the chordal accompaniment in rotation 3
projects the location G ♯ –A–B–C ♯ –D ♯ –E, which is shifted significantly from its
initial location in rotation 1 and now overlaps with the theme’s A1r. One of
the consequences of this drastic shift in space location in rotation 3 is that Al
“vacates” a Dasian region, which is later occupied by the episodic themes in
block E (and the additional block G) as part of the strategy for the move-
ment’s closure.

Pentatonic frames and the movement’s closure

To ultimately assert harmonic and thematic stability over the episodes, the
formal scheme for a classical rondo or baroque ritornello would typically
grant a final statement of the refrain’s theme in the home key to close the
movement. The ending of rotation 4, however, features a rhythmically driven
passage built on the textures typical of blocks C/D (and also F), lacking most
of the thematic and harmonic features that characterize the refrain. This
strategy is at odds with a typical global scheme for a rondo and has been
interpreted as providing a weak and unprepared closure to the movement
(Wilson 1992, 84).47 Here, however, I argue that this strategy is effective,
given that formal patterns at the end of the movement provide a two-phase
articulation of the piece’s closure, substituting for the refrain’s role of cul­
minating a rotation and launching a new one. First, a tentative closure is
sketched by the interpolation of the episodic theme (block G, mm. 227–47),
which features a pentatonic frame anchored on pitch class E, “impersonat-
ing” the formal location of the fixed refrain that would initiate rotation 4;
a second, more emphatic closure is offered by the final E-Phrygian collec-
tion, which dissolves the refrain’s harmonic and thematic features in block D
(mm. 268–81) at the end of rotation 4. The following transformational and
polymodal approaches show that the ending section not only completes a
process for the entire rotation but also reinforces the completion of a pattern
that structures pentatonic frames in both refrains and episodes throughout
the movement.
In Figure 9, each of the five pentatonic patterns used in the piece (the
fixed refrain, F♯ –E–C ♯ –B–A, and the four movable episodes, B–A–F♯ –E–D,
C ♯ –B–G ♯ –F♯ –E, A–G–E–D–C, and E–D–B–A–G) is represented as a continu-
ous five-note segment embedded in the extended line of fifths from C to G ♯ ,
bordered by F and D ♯ (more on this below). Two theoretical observations are

47 Wilson (1992, 84) argues that the “the concluding class set techniques), the final E ♮ in the upper voice cannot
events [of the movement] contain some element of the be connected with the large-scale structural beam that links
arbitrary or the unprepared” (referred here to blocks C/D of other structural events “because of its weakness in con-
rotation 4) and thus provide only a “weak closure.” In Wil- text and obvious difference in function.”
son’s analysis (combining extended Schenkerian and pitch-

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 300 JOURNAL of MUSIC THEORY

Figure 9. Line of fifths (from C to G ♯) embedding all pentatonic frames used in the movement
(refrain and episodes)

relevant for the movement’s large-scale organization: these are the only five
pentatonic patterns that contain pitch class E, and in addition to the refrain’s
anchoring F♯ , the anchoring notes for each of the four movable pentatonic
frames (namely, B, C ♯ , A, and E, circled in Figure 9) also constitute the notes
of the refrain’s complete pentatonic pattern (boxed in Figure 9). As such, the
interpolation of the “impersonator” episode anchored on pitch class E (block
G) at the end of rotation 3 not only breaks the large-scale alternation between
refrains and episodic themes but also completes a large-scale pentatonic pat-
tern formed by the five anchoring notes.48 However, despite the episode’s role
completing the large-scale pentatonic pattern, and its formal position “imper-
sonating” a refrain, it provides only a partial closure for the movement as
the theme’s consonant stability is undermined by a “dominant” pedal and a
chordal collection that suggests B-aeolian (the bass note B is retained from
mm. 222–26 in the previous block F and kept throughout the section).49
The movement’s tentative closure in rotation 3 sets up a transpositional
scheme that is reinforced and re-elaborated in the rotation 4. Figure 10.1
presents a transformational scheme between pentatonic frames structuring
refrains and episodic themes throughout the four rotations.50 The additional
episode at the end of rotation 3 sets up a large-scale T10 transpositional ges-
ture between frames anchored on pitch class F♯ (refrain, block A) and pitch
class E (episode, block G). This T10 closing gesture is retaken in rotation 4,

48 Wilson (1992, 83) notices that the initiating notes make connection from rotation 3 into 4 is done by a whole-step
up the pentatonic collection of the initiating F ♯ fixed theme, E–F ♯ (and after a breath mark), thus somewhat weakening
although not venturing into the implications (for form and the urge for a rotational recycling. This observation is further
closure) of such relation. supported by the mapping of rotation 3 in Dasian space,
where only (F, 3) fails to be covered in the Dasian space.
49 The voice-leading connection between the closing block
Given the unique locations of semitones in the space, this
G in rotation 3 and the beginning of rotation 4 also informs
note’s absence undermines the possibility for the semi-
tonal voice leading into F ♯ .
the E-anchored theme’s attempt to close the movement.
While the voice-leading connection at corresponding for-
mal places from rotation 1 into 2 and rotation 2 into 3 was 50 Transposition (T) is here used conventionally as operat-
done through the semitone E ♯ –F♯ , linking the closing whole-­ ing in (chromatic) modulo 12 pitch-class space.
tone collection with the opening F♯ -theme, the voice-leading

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 Journal of Music Theory 59.2: Music Examples, p. 44

Martins, Ex. 7 (correx 7/24/15):José Oliveira Martins Bartók’s Polymodality 301

allargando

�� � � �
B

� � � � �
14 � 85 41 � 38 � � � � 58 �� � � � 24
259

�� ��� � �� � �� � � �� �
� � �� � � �� �
� � � � � �
� 1 � � �� � ���
� 4 � �� 85 � 41 � � ��� 38 85 24

D
� = 184
C

���
accel. al Vivacissimo

2 � � � � �� � �� � � �� � �� � � �� � �� �� � � � �� � ��
264

� � 4 � � ��� � ��� ���

� �� �� � �� �� � �� �� � � �� ��
�� �� � �
� �� � � �
2
� 4 � � �� �� � �� ��� ��� ��� ��� � �� ��� ��� ��� ��� � �� ��� ��� ��� ��� �� �� �� �� � � �� �� �� ��
�� � � �
�� �

��� ���
D

� � �� � �� �� � � � �� � �� �� � � �� � �� � �� � ��
270

��� ���
�� � �� �� � � �� �� � �� �� �� ��
� �
� �� � � � ��� �� �� �� � � � � � � � � � �� �� �� �� �� �� �� ��
� � � �� �� �� �� � � � � � � �� �� �� �� � � �� �� �� �� � � � � � � � �
�

�� ���
� �� � �� � �� � �� � �� � �� � �� � �� � �� � �� ��� �
276
�
�� �� �� � �� �� � �� �� � �� �� ��
ff
� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � �
cresc.

�� � � � �
� � � � � � � � � � � � � � � �� �� � �� �� �� ����
�
� ��
�

Example 7. Ending of rotation 4: blocks B, C, and D

from the pentatonic refrain anchored on F♯ to the closing pentatonic frame

E–D–B–A–G, embedded in the E-Phrygian sonority (see Example 7, right
hand and thumb of left hand in mm. 273–81).
The T10 closing gestures of rotations 3 and 4, however, are differently
composed. In rotation 3, T10 is composed by 〈T3, T7〉 via the episode anchored
on pitch class A (m. 205, block E), whereas in rotation 4, T10 is composed by
〈T5, T5〉 via a “rotated” pentatonic frame F♯ –E–D–B–A in the contracted block
B (mm. 262–63), which retains F♯ in the top register. These two transpo­
sitional pairs have a number of manifestations at the musical surface and
middle-ground levels throughout the movement. In rotation 4, the marked

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 T3

A
157
artins, Fig. 10.1

artins, Fig. 10.2:

E
143

on 15 February 2018
Rotation 2
T7

by UNIV CA IRVINE user

A
92



T2
(10.1)


Rotation 1 Rotation 2 Rotation 3 Rotation 4


53

E

302 (correx 7/24/15):

Blocks: A E A E A E G A / B C / D
JOURNAL

T5

Rotation 1
T2 T10 T10

A
                 
T7 T7 T5 T5

T10 
T5
T5 T3


mm. 1

T7


T7

T5




Blocks:


T2
T10 T3 T7 T2
                                 

T3
     
 T7    



  
mm. 1 53 92 143 157 205 227 247 262 264 273–281
    
Martins, Fig. 10.2:

of MUSIC THEORY

(10.2)
T5 T3


 T10 
T2

T5
 
T7
Journal of Music Theory 59.2: Music Examples, p. 48

Figure 10. Deployment of pentatonic thematic frames in the movement. (10.1) Transpositional scheme between fixed pentatonic
frames of refrains and movable episodes throughout the four rotations. (10.2) Structural melodic intervals used in the refrain
throughout the four rotations.

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 José Oliveira Martins Bartók’s Polymodality 303

polychord (block C, mm. 264–67) that initiates the final gesture is registrally
disposed into 〈T3, T7〉, where the upper register dyad A–E (superimposed to
the left-hand sharp notes F♯ –C ♯ –G ♯) partially anticipates the T10 arrival of
the final E-Phrygian collection.51 The T10 relation between pentatonic frames
anchored on F♯ and E at the beginning and ending of the rotation goes
through a pliable process: the T5 from block A to B retains F♯ as the “head
note” (but a different pentatonic “body”); then the arrival of block C poly-
chord shifts the F♯ to the bass, while the top register moves to the “head” note E
(without a pentatonic “body”); and finally, block D brings the “head” note E
and the pentatonic “body” together, concluding the remaining T5. The final
E-Phrygian collection is articulated by three pentatonic frames 〈−2, −3, −2,
−2〉 (anchored by D, A, and E), which are superimposed as 〈T7, T7〉, thus offer-
ing a counterpart and creating a convergence point to the 〈T5, T5〉 spanning
the rotation. Similarly, the spanning of T10 in rotation 3 is counteracted by T2
between pentatonic episodes from rotation 1 to 2. Figure 10.2 diagrams the
main intervals underlying the melodic gestures in the refrain, which are pre-
cisely those that are unfolded by the transpositional scheme between penta-
tonic frames throughout the movement just discussed.52
Given that pitch class E is included on all pentatonic frames discussed
for the piece and is the ultimate goal (as pentatonic anchor, pitch center, and
transpositional arrival) for the movement, we now consider how the succes-
sion of episodes interacts with the four modal qualities for pitch class E. Fig-
ure 11.1 presents the various scalar contexts for pentatonic frames (brack-
eted) and the common tone pitch class E (circled) in all movable episodes
(blocks E and G) in rotations 1–3. Figure 11.2 situates these segments with
respect to their Dasian locations and the four modal qualities that pitch class
E assumes in the space (circled). In rotation 1, the movable theme anchored
on pitch class B at first defines (E, 1) and later expands to (E, 2), given the
change in scalar context (as discussed above; see Figure 7). In rotation 2, the
movable C ♯ -anchored theme significantly expands the polymodal scope of its
constituting segments, by using both (E, 1) and (E, 0).53 In rotation 3, the
movable A-anchored theme is split into two segments, each of which includes
one of the two remaining modal qualities for pitch class E, (E, 2) and (E, 3),
although the latter mq(3) is only weakly asserted at the theme’s conclusion
(m. 212), given the lack of an explicit semitone E–F in the segment.54 The
tentative closure provided by the E-anchored episode (block G) actually fails
to reinforce the position (E, 3) and is instead situated in the context of (E, 1),
a position already amply covered in previous episodes.

51 Both Wilson (1992, 79) and Somfai (1990, 548) consider 53 (E, 0) is also characteristic of the refrain given the
m. 264 as initiating the movement’s coda. Konoval (1996) embedded semitone D ♯ –E.
notices that the F ♯ below E at m. 264 delays the large-scale
descending bass line A–G–F ♯ –E.
54 (E, 3) is also touched upon very briefly in the context of
E–F (block F, rotation 3, m. 219).
52 The transpositional scheme 〈T 3, T 7〉 is particularly active
in the refrain’s melodic profile F ♯ –A–E in rotations 2–4.

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 Journal of Music Theory 59.2: Music Examples, p. 49

304 JOURNAL of MUSIC THEORY

Martins, Fig. 11.1 (correx 7/24/15):

(11.1)
E1 E2

�� � � �� �� � � � �
� �� �� � �� � �
Rotation 1
� �
mm: 53–77 77–83

�� �� � �� �� �� �� � � �� �� �� �� �
�� �� �� �
Rotation 2
�
mm: 143–148 149–153

� � �� � � � � ( ) � �
(�) � �� �
Rotation 3
�
mm: 205–210 211–216

� � �� � �
Rotation 4
� (�)
mm: 227–247

Figure 11. Modal qualities for pitch class E in episodes. (11.1) Scalar contexts for pentatonic
frames (bracketed) of movable episodes (blocks E and G) and changing scalar context for
pitch class E (circled). (11.2) Dasian locations of thematic episodes (blocks E and G) and
changing modal qualities for pitch class E (circled).

In contrast with the formal process where rotations are launched by the
initiating power of refrains, the movement’s closure avoids a new rotational
beginning by asserting a clear pitch space differentiation from the refrain.
Figure 12 shows that scalar materials in the contracted rotation 4 explore an
extended Dasian continuity encompassing the four modal qualities for pitch
class E.55 The ending brought about by the “E-Phrygian” collection (block D)
not only camouflages the thematic pentatonic frame within the open-fifths
ostinato but also provides a differentiated Dasian location from the opening
refrain at the farthest end of the extended region.
The formal role for the marked polychord (block C, m. 264) as a bound-
ary event is interpreted a superposition of layers that bridges the opening and
closing regions. The polychord’s wide registral ambitus (F♯ –C ♯ –G ♯/A–E),
which entangles F♯ in the bass and E on the upper register as discussed above,
corresponds to a distant relation in Dasian space, connecting the most coun-
terclockwise pitch class F♯ (F♯ , 3) with the most clockwise pitch class E (E, 0). 56

55 The thematic material corresponding to block B starts 56 Alternatively, the analysis could choose to view the
around m. 254. The “dramatic” moment at m. 264 can “polychord” as a scalar segment E–F ♯ –G ♯ –A–C ♯ , but this
easily be related to the beginning of the pivotal block C interpretation would miss the superposition (“poly”) of lay-
because of its change of texture and point of overlap ers and its wide registral distribution.
between sections. What follows m. 264 recalls the rhyth-
mic ostinato of block D (left hand) and the wide register
(right hand) of block C.

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 José Oliveira Martins Bartók’s Polymodality 305

(11.2)

Figure 11 (continued). Modal qualities for pitch class E in episodes. (11.1) Scalar contexts for
pentatonic frames (bracketed) of movable episodes (blocks E and G) and changing scalar
context for pitch class E (circled). (11.2) Dasian locations of thematic episodes (blocks E and
G) and changing modal qualities for pitch class E (circled).

After m. 264, the pitch material gradually abandons the polychord and cen-
ters upon the diatonic collection E–F–G–A–B–C–D–E, which is precisely
mapped into the contiguous counterclockwise space to (F♯ , 3), the bass of
block C. The fifth A–E of the E-Phrygian collection is retained from and
related to the polychord by transformatio f −2 (Figure 12, solid arrows).
The closing function of the white-key diatonic collection is further
emphasized through its large-scale symmetrical opposition to the refrain’s
melodic theme. The axis of symmetry inverts the four-sharp diatonic collection
that characterizes the refrains (A–B–C ♯ –D♯ –E–F♯ –G ♯) to the final E-Phrygian
collection (C–D–E–F–G–A–B) via transformatio (f |3|) between (E, 0) and
(E, 3). This large-scale polymodal juxtaposition reverses the semitonal asso-

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 306 JOURNAL of MUSIC THEORY

Figure 12. Dasian modeling of symmetry and closure in rotation 4

ciation for pitch class E (E–D ♯ vs. F–E), helping to reinforce its projection as
a pitch center.57 The common tones in the polymodal juxtaposition include
pitch classes E, A, and B, which are projected as vertical chords precisely at
the end of the opening refrain (rotation 1, mm. 17–19) and the beginning of
the closing E-Phrygian collection (rotation 4, mm. 268, 271, and 273). 58 Fur-
thermore, the final chord for the piece (E–F–B–D–E) reaches pitch class E,
in both top and lower registers, reinforcing octave-bounding centricity on
pitch class E and condensing marked elements of the three previous rota-
tions: the notes E, F, and D correspond to axes (related by transformatio) that
connect distinct scale segments within polychords and mark the boundaries
between the extended areas between refrains and episodes in rotations 1, 2,
and 3, respectively. In short, the closure for the movement, far from “arbi-
trary” or “weak,” is, rather, effective and powerful.

Affinity spaces and Bartók’s “Divided Arpeggios”

Having considered a model of polymodality that correlates the combination

of distinct diatonic strata with the exploration of the Dasian space, we now

57 The diatonic juxtaposition of the four-sharp diatonic to notions of “encirclement” (Gilles 1989) and “disposition
white-note diatonic reaches exactly one station beyond pairs” (Morrison 1991). Lerdahl 2001, 333–41, examines
both boundaries in the line of fifths activated for all penta- the prolongational potential of double leading-tone forma-
tonic frames, as discussed above (see Figure 9). The tions in Bartók.
explicit diatonic ending is thus an extension of the penta-
58 As discussed above (see Figure 6), the notes E, A, and
tonic procedures of the piece and also completes the
B fill a crucial role at the arrival of the polychord at m. 38 in
double leading-tone figure to the common note E. Bartók’s
rotation 1 (as axis of symmetry and as bass and top notes,
compositional procedure of suggesting a pitch center via
respectively).
double leading-tone attraction has been addressed by the

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 José Oliveira Martins Bartók’s Polymodality 307

turn to a more flexible notion of polymodality, which models combinations

of intervallically consistent, nondiatonic scale strata. I develop the analytical
framework of affinity spaces for this enlarged notion of polymodality by gen-
eralizing the central properties of the Dasian space, namely, nonoctave affin-
ity relations and space closure. These properties are structured by a general
modular-unit formula that gives rise to diverse configurations of pc-mq
cycles. The analytical potential of such an enlarged framework is probed by
considering pitch relations in Bartók’s “Divided Arpeggios” (Mikrokosmos no.
143). In particular, the analysis shows that motivic and contrapuntal relations
of the piece’s arpeggios are structured by two distinct affinity spaces, where
the respective operations of transpositio and transformatio model salient
relations between different types of arpeggiated tetrachords. 59
Sections A (mm. 6–25) and A′ (mm. 50–80) of “Divided Arpeggios”
focus on combinations of “gamma-chord” (3–5–3) segments, characteristic
of Bartók’s music; the contrasting B section (mm. 30–46) turns to minor-
seventh chord (3–4–3) arrangements and other stacks of thirds.60 Tetra-
chordal relations throughout the piece suggest the construction of two closed
pitch-class spaces illustrated in Figure 13. Space A assumes two concurrent
cyclic forms (or cocycles 0 and 1) correlated to sections A and A′; space B
assumes a single cyclic form correlated to the contrasting B section. Each of
the spaces stacks its respective “modular” tetrachord (bracketed in the cycles)
separated by pitch-class interval 3. This arrangement results in modular pat-
terns of 3–3–3–5 (or its rotations) for space A and 3–3–3–4 (or its rotations)
for space B. The analysis shows that cocycle 0 of space A is completely
exhausted at the opening of the exposition in mm. 6–11 by arpeggios of adja-
cent gamma chords, whereas cocycle 1 is exhausted at the corresponding
opening of the re-exposition in mm. 50–54. The B section does not exhaust
its corresponding space: the opening pair of juxtaposed minor-seventh chords
(bracketed in space B) gradually expands by various stacks of (over and
under) “thirds,” resulting into an overall symmetrical exploration of space B.

59 Although the main ideas concerning the theoretical analytical focus. Seminal work on the relevance of single
framework and analytical applicability of “affinity spaces” interval cycles for structural relations in the music of Bar­
have been conceived and developed independently within tók includes Perle 1977 and Antokoletz 1984. The notion of
the scope of my Ph.D. dissertation (Martins 2006a), sev- compound interval cycles is first explored by Philip Lam-
eral aspects of its conceptual underpinning and analytical bert (1990) regarding the relevance of cyclic alternation of
relevance intersect with the work of Gollin (2007, 2008) on two intervals (what he calls “combination cycles”) for the
multiaggregate cycles. In particular, Gollin (2007, 157–63) compositional language of Charles Ives.
frames the analysis of Bartók’s “Divided Arpeggios” in
60 Ernö Lendvai (1971, 44–47) constructs “gamma chords”
reference to multiaggregate cycles whose graphic repre-
as the partial superimposition of two (tonic and dominant)
sentations coincide with those proposed here. That our
of his “axes systems” (1–16). In a post-Schenkerian read-
approaches were developed independently only attests, I
ing of the piece projecting a large-scale tonal framework,
think, to the strength of the ideas involved. However, our
Ivan Waldbauer (1982) sees the small-scale juxtaposition
approaches differ significantly in a number of important
of gamma chords as a somewhat “mechanical” procedure,
aspects concerning the conception of the cycles/spaces,
which is balanced and overcome by the developmental
focus and relevance of certain properties, and analytical
and contrasting B section.
modeling of musical relations. Below, I point to some of
these distinctions in reference to both the theoretical and

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 308 JOURNAL of MUSIC THEORY

Figure 13. Two affinity spaces, 3–3–3–5 (A) and 3–3–3–4 (B), suggested in Bartók’s “Divided
Arpeggios,” displaying interval affinities at pitch-class intervals 2 and 1, respectively

The modular construction of both spaces resembles the Dasian’s stack

of “Dorian” tetrachords separated by a whole tone of disjunction (2–2–2–1
modular unit, or its rotations). The unique pitch-class interval in a given
modular unit (1 for the Dasian, 5 for space A, and 4 for space B) induces
twelve unique locations in each of these cyclic spaces, whereas a recurrent
interval normalizes the remaining “steps” within the modular units. Also, just
as the Dasian space creates interval affinities every four steps at pitch-class

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 José Oliveira Martins Bartók’s Polymodality 309

interval 7, so spaces A and B create affinities every four steps at pitch-class

interval 2 and pitch-class interval 1.
Let us now generalize the structure of affinity spaces by lifting value
constraints of pitch-class intervals (“step” sizes) within a modular unit.61 The
formula p = nr + q (congruence mod 12) generalizes the modular unit of
affinity spaces, where p, r, and q are pitch-class intervals (mod 12) and n is a
positive integer: p corresponds to the interval of transpositio, measuring the
affinity in the space, n is the number of recurrent normalized steps r, and q
is a unique step interval in the modular unit. Given this formula, the Dasian
space and the two affinity spaces A and B can be expressed, respectively, as
7 = 3×2 + 1, 2 = 3×3 + 5, and 1 = 3×3 + 4, (congruence mod 12).62
For a given affinity-space formula, the number of distinct, transposi-
tionally related cycles produced (cocycles) is determined by the interval value
of transpositio p (the interval of affinities in each cycle). If p is 1, 5, 7, or 11
(i.e., if p is coprime with 12), the result is a single closed cycle, as is the case
of the Dasian and space B. If, however, p is 2, 3, 4, 6, 8, 9, 10 (i.e., if p is a divi-
sor of or has a common multiple with 12), the result is concurrent cocycles,
whose number equals the value of p (or 12 − p, when p is 8, 9, or 10). In the
case of space A (where p = 2), it comprises two cocycles (0 and 1). In any affin-
ity space, there are always n + 1 interlocked p-cycles.63
As explored for the Dasian space, the pattern of (nonoctave) affinities
leads to a unique correspondence between pitch class and modal quality. In
a given affinity space, each of the various occurrences of a pitch class is
uniquely associated to a distinct position in the modular unit (and conse-
quently to a different interval pattern surrounding that position). The range
of modal qualities in a space corresponds to the cardinality of the modular
unit (n + 1) and is ordered from 0 to n, such that the unique interval q always
spans between positions n and 0 in adjacent modular units. The four modal

61 Step (or s), the directed (clockwise) interval (congru- the Dasian space would take the form of a (2, 1, 2, 2)-cycle.
ence mod 12) between adjacent (pc, mq) positions in a Special cases of compound interval cycles are what he
cycle, is also defined as a group generator below. calls “multiaggregate cycles,” which are compound inter-
val cycles that “run through the tones of more than one
62 The formula for the modular unit positions the unique
aggregate” (143). Gollin’s approach is particularly inter-
interval q at the “end” of the unit. We can, of course, state
ested in investigating the conditions and analytical uses of
a less restrictive formula for a modular unit by allowing
the property of maximally even distribution of occurrences
other intervals to sum up to the interval of transpositio.
of the same pitch class in multiaggregate cycles (mea-
However, this design ensures the homogeneity of recur-
sured by the number of steps in a distribution vector).
rent steps, on the one hand, and the presence of a unique
What I refer to as affinity spaces A and B used in the analy-
interval that results in (twelve) unique locations in any
sis of Bartók’s “Divided Arpeggios” thus take the form of
affinity space, on the other. As is the case of semitones in
(3, 5, 3, 3)-cycles and (3, 4, 3, 3)-cycles in Gollin’s formula-
the Dasian space, unique intervals are markers of locality
tion (the analysis of the piece appears at 157–63).
and harmonic differentiation in the affinity space. Gollin
(2007, 146) uses a less restrictive formula for the genera- 63 Gollin (2007, 147) points out the “sufficient, but not
tion of what he calls “compound interval cycles,” which necessary condition”: if the “sum of the component gen-
are “generated by a repeated pattern of two or more dis- erating intervals is co-prime with 12,” the result is a multi-
tinct intervals” and takes the form (x, y, z, . . .)-cycle, aggregate cycle.
where x, y, z, . . . , are ordered pitch-class intervals. As such,

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 310 JOURNAL of MUSIC THEORY

qualities of spaces A and B are marked from 0 to 3 inside the cycles of Figure
13. Generalizing the procedure developed for the Dasian space, we can thus
conceive of affinity space elements as ordered pairs (pc, mq), where each
pitch class is assigned to every available modal quality and each modal quality
is assigned to every pitch class.
The ordered pair (pc, mq) assignment of affinity spaces induces a
closed group structure, in which the operations transpositio (p), transforma-
tio ( f ), and step (s) act as generators of the space, specifying three kinds
of motion between ordered pairs.64 Group structures for spaces A and B are
captured in Figure 14 by two graph representations for each of the spaces:
one of the graphs combines the generators p/f, and the other combines the
generators p/s. All graphs have four closed paths corresponding to four
(interlocked) p-cycles, resulting from the recurrence of p, and thus p 6 = I
(identity) in space A (in both cocycles 0 and 1), while p 12 = I in space B.65 In
space A (Figure 14.1) generator increments of either f or s run through twenty-
four group elements only (i.e., half of the total forty-eight elements), while in
space B (Figure 14.2) generator increments produce all forty-eight group
elements. Any ordered pair (pc, mq) can serve as a reference point for the
origin of motion on the corresponding graph, and a given overall motion
might follow alternative (but group-theoretically equivalent) paths using dif-
ferent generators. For instance, consider three alternative paths for the over-
all motion from (C, 3) to (A ♯ , 3) in space A (both ordered pairs are hosted
in cocycle 1). One path moves by increments of the transpositio generator
in either the p/f or the p/s graph: p–p 2–p3 –p 4 –p 5 (or p−1) yields (C, 3)–(D, 3)–
(E, 3)–(F♯, 3)–(G ♯ , 3)–(A ♯, 3). Another path follows increments of the trans-
formatio generator in the p/f graph: f–f 2–f 3 –f 4 yields an alternation between
cocycles that runs through (C, 3)–(C, 2)–(C, 1)–(C, 0)–(A ♯, 3). Finally, a third

64 As proposed for the Dasian space, we can describe the depending on whether it is applied to mq(0) (modal quality
structure of any affinity space as a GIS (Lewin 1987). In the in position zero) or to some other modal quality mq(n ≥ 1).
triple (S, IVLS, int), S corresponds to the set of all ordered In other words, applying transformatio to a group element
pairs (pc, mq) in a given affinity space, IVLS is the group either retains the pitch class descending its modal quality
of intervals composed of the generators transpositio (p), by one, that is, f: [pc(x), mq(y)] → [pc(x), mq(y – 1)], for 0 ≤
transformatio (f ), and step (s), and int is a function mapping x ≤ 11 and 1 ≤ y ≤ n, or (if the element is in modal quality 0)
S × S into IVLS, such that int([pc(x), mq(y)], [pc(w), mq(z)]) substitutes for another pitch class in modal quality n. The
= int[pc(w) – pc(x), mq(z) – mq(y)], where pitch class and interval producing a pitch-class substitution under f is
modal quality intervals are computed in mod 12 and mod always equal to r – q (mod 12), that is, f: [pc(x), mq(0)] →
n + 1. While the set of generators p, f, and s might be ana- [pc(x + r – s), mq(n)], for 0 ≤ x ≤ 11, and r, s, and n defined
lytically useful, reiterations of a single generator are not all by the modular unit formula. It is interesting to note that
required to produce the entire space. As such, the exhaus- while the value of p (transpositio) determines the existence
tion of a complete affinity space might require the compo- of single cycles or cocycles in any given affinity space, it
sition of generators. The generators p, f, and s can be does not affect the value for the interval of pitch-class
described as p = int([pc(x), mq(y)], [pc(x + p), mq(y)]), that substitution under f. For a more detailed discussion of
is, the interval spanning from a given pc(x) to pc(x + p) (or affinity space group-theoretical properties, see Martins
pc[x + (n × r + q)]) of the same modal quality in the (clock- 2009, 505–7.
wise) adjacent modular unit. (Notice the double usage of p
65 The p -cycles correspond to four 2-cycles in space A
as a generator in IVLS and pitch-class interval in int.) The
and to four 1-cycles in space B.
generator transformatio (f ) has two possible outcomes

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 José Oliveira Martins Bartók’s Polymodality 311

Figure 14. Group structure (graph representation) for spaces A and B using generators p, f,
and s. (14.1) Space A. (14.2) Space B.

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 312 JOURNAL of MUSIC THEORY

path follows increments of the step generator in the p/s graph: s −1–s −2–s −3 –s −4
(or s 20) yields (C, 3)–(A, 2)–(F♯ , 1)–(E ♭, 0)–(A ♯ , 3). In short, while the opera-
tions p 5, f 4, s −4 suggest different paths, they yield the same overall motion in
space A and are group-theoretically equivalent.66
Some linear and contrapuntal combinations of tetrachords in Bartók’s
piece are modeled by strong recurrences of affinity space generators. In addi-
tion, the somewhat surprising polymodal superimpositions of tetrachords in
the closing passages of sections A and A′ give rise to consistent generator
relations. Figure 15.1 sketches gamma-chord activity in section A (mm. 6–22)
and traces some relations modeled by transpositio and transformatio. The
overall gesture of the passage is shaped by an arched bass movement C–E–
G ♯ –F♯ –E–D–B ♭, which encompasses three phases: in phase 1 (mm. 6–11), pairs
of stacked gamma chords ascend by a major third (bass: C–E–G ♯); in phase 2
(mm. 11–13), a similar pairing of tetrachords descends by a whole tone (bass:
G ♯ –F♯ –E–D), which reverses and fills in the previous ascending motion; and
in phase 3 (mm. 14–22), a closing gesture prolongs the superimposition of
two nonadjacent gamma chords.
Figures 15.2–15.4 capture tetrachordal activity via affinity-space rela-
tions. Figure 15.2 shows that pairs of stacked gamma chords (adjacently
located and related by p in cocycle 0) move by p 2 (major third bass ascent) to
exhaust the entire cocycle in phase 1. Figure 15.3 models the whole-tone
descent of tetrachordal pairs in phase 2 by three consecutive moves 〈p −1, p −1,
p −1〉, which fall short of undoing the 〈p 2, p 2〉 of phase 1. This descent is also
characterized by a two-pitch-class overlap between top and lower notes of the
arpeggiated figures. This overlap connects different instances of the same
pair of pitch classes in different parts of cocycle 0 and is modeled by f −2.
For instance, the two top notes (F♯ , 0) and (A, 1) that finish phase 1 in m. 11
(see Figure 15.1 and 15.3) are related via f −2 to the two lower notes (F♯ , 2) and
(A, 3) that initiate phase 2 in the same measure. Throughout phase 2, pitch-
class overlap between arpeggios continues to be modeled by f 2, but the pat-
tern of overlap is broken at the end of m. 13, where the top notes (C, 0) and
(E ♭, 1) do not find corresponding lower counterparts at (C, 2) and (E ♭, 3)
(signaled by X over an arrow in Figures 15.1 and 15.3).67 This moment cor-
responds to a break in both transformatio and transpositio patterns and
avoids the pre­mature closure that the opening gamma chord C–E ♭ –A ♭ –B
would bring to the passage. Instead, phase 3 restates in the left hand the
gamma chord B ♭ –C ♯ –F♯ –A that finished phase 1, and later superimposes the
new gamma chord C ♯ –E–A–C in the right hand. This suggests that phase 3
provides a sense of relative closure by filling in a harmonic distance opened

66 All affinity-space groups are commutative, allow for 67 Gollin (2007, 159) makes a similar claim, remarking that
paths formed by the combination (words) of generators, “Bartók exploits the ordered, pairwise distribution of pitch
and yield the following relations: if f a = s b and p a = s c , then classes of the (3, 5, 3, 3)-cycle in the stretti that follow the
f c = p b , where a, b, and c are integers modulo m for f m = cyclic unfoldings in the exposition and reprise, using the
s m = I (identity), and modulo n for pn = I. cycle’s invariant dyads to align new cycle segments.”

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 ns, Fig. 15.1:

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Phase 2

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(15.1)
Phase 1 Phase 3

p b. c. d. 14–22
p
      a.      
p
 
p
 
f -2 f -2 f -2
              
f
 
p        
        
p2 p -1
   
mm. 6 8 10 11 p -1 12 13 14–22
p -1
  
p2
 
f -2

Figure 15. Gamma-chord activity in section A (mm. 6–22). (15.1) Gamma-chord relations in phases 1, 2, and 3 modeled by transpositio and
transformatio. (15.2) Phase 1: transpositio relations traced in affinity space A. (15.3) Phase 2: transpositio and transformatio relations.
(15.4) Phase 3: transformatio relations between gamma-chords in different cocycles 0 and 1. (15.5) Phase 3: relation of materials displayed
on the s/f pitch lattice alternating cocycles 0 and 1.
Journal of Music Theory 59.2: Music Examples, p. 50

José Oliveira Martins

Bartók’s Polymodality

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313
 314 JOURNAL of MUSIC THEORY

Figure 15 (continued). Gamma-chord activity in section A (mm. 6–22). (15.1) Gamma-chord

relations in phases 1, 2, and 3 modeled by transpositio and transformatio. (15.2) Phase 1:
transpositio relations traced in affinity space A. (15.3) Phase 2: transpositio and transformatio
relations. (15.4) Phase 3: transformatio relations between gamma chords in different cocycles
0 and 1. (15.5) Phase 3: relation of materials displayed on the s/f pitch lattice alternating
cocycles 0 and 1.

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 José Oliveira Martins Bartók’s Polymodality 315

Figure 16. Symmetrical exploration of materials and transformatio relations in section B

at the passage from phase 2 to phase 3. Figure 15.4 shows that the pair (B ♭, 2)
and (C ♯ , 3) of B ♭ –C ♯ –F♯ –A (phase 3) distances f −2 from (B ♭, 0) and (C ♯ , 1) of
the arpeggio at the end of phase 2, thus retaining the transformatio relation
that characterizes phase 2. This f −2 distance is filled in by the superimposed
gamma chord C ♯ –E–A–C of phase 3, which finds its space location in cocycle
1. This harmonic relation is better captured in Figure 15.5 by an s/f lattice
coordinating step relations on the rows with transformatio relations on the
columns.68 As the vertical dimension in the lattice alternates between cocy-
cles, the right-hand C ♯ –E–A–C in cocycle 1 mediates left-hand gamma chords
of cocycle 0 by filling in the harmonic gap between those events. Common
notes C ♯ and A between left and right hands in phase 3 are related by f and
become the melodic goals of the section in m. 25.69 The contrasting section
B turns to an overall symmetrical exploration of affinity space B, as shown in
Figure 16, where the transformatio arrows signal the change of modal quality
for some pitch classes.
Phase 1 in the re-exposition (mm. 50–55) completely exhausts cocycle
1, but the direction of the exploration now runs counterclockwise 〈p −2, p −2〉,
matching the reversal of the melodic major-third descent in the arpeggios.
The notion of harmonic mediation is also useful to approach gamma-chord

68 Oblique lines traversing rows and columns of space A 69 Superimposed gamma chords B ♭ –C ♯ –F ♯ –A and C ♯ –E–
mark off the unique perfect-fourth intervals on the rows A–C stand at 〈f, s〉.
and the note change produced by transformatio on the
columns.

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 316 JOURNAL of MUSIC THEORY

Figure 17. s/f lattice structuring gamma-chord relations in re-expository phases 2 and 3
(mm. 55–63, section A′)

relations in corresponding phases 2 and 3 (mm. 55–63) of section A′ and the

coda (mm. 68–80). However, I am arguing that, unlike events in the exposi-
tion, re-expository events spanning from phase 2 to phase 3 fail to be har-
monically mediated. This mediation, however, is ultimately achieved by the
final gamma chord in the coda. Figure 17 presents the s/f lattice structuring
gamma-chord relations in re-expository phases 2 and 3. Phase 2 is abridged
on a single arpeggio (F–A ♭ –D ♭ –E–G–B ♭ –E ♭ –F♯ , mm. 55–56), whose top notes
(E ♭, 0)–(F♯ , 1) stand at f −2 from (D ♯ , 2)–(F♯ , 3), which both end phase 1 (in
E ♭ –F♯ –B–D, m. 55) and initiate phase 3 (in D ♯ –F♯ –B–C ♯♯, m. 57).70 Having
explored exclusively cocycle 1 in phases 1 and 2 of the re-exposition, phase
3 brings a new gamma chord on the left hand (B ♯ –D ♯ –G ♯ –B, mm. 57–63)
located in cocycle 0, thereby also reversing the expository exploration of
cocycles. However, while the concluding phase 3 reenacts the superimposi-
tion of gamma chords,71 the left-hand tetrachord (B ♯ –D ♯ –G ♯ –B) does not
mediate the harmonic space spanned from phase 2 to phase 3 but, rather,
stands harmonically further. This harmonic distance is fleetingly mediated
by the slight pattern deformation at the end of m. 57. The fleeting note C at
the top of the arpeggio (the last note of m. 56) forms a perfect fourth with
the previous note G and thus alters the location of the unique perfect-fourth
interval that characterizes gamma chords. Such deformation induces a (C, 0)
location in cocycle 0 (Figure 17, asterisk).72
The coda presents isolated gamma chords interspaced with “appoggia-
tura” figures, thereby creating a gestural fragmentation appropriate for the

70 In the exposition, the left-hand gamma chord B ♭ –C ♯ – 71 As in the exposition, superimposed gamma chords
F ♯ –A (which initiates phase 3) returns to the same tetra- (D ♯ –F ♯ –B–C ♯♯ and B ♯ –D ♯ –G ♯ –B) also stand at 〈f, s〉 in the
chord that concludes phase 1, thereby bookending the re-exposition (mm. 57–63), and common tones E ♭ and B
entire phase 2. In m. 11, however, {F ♯ , A} initiates the trans- also become melodic goals of the passage in m. 66.
formatio overlap, while in m. 14, {B ♭ , C ♯ } (the remaining
72 The (C, 0) location for the last note in m. 56 is confirmed
by the ensuing D ♯ on m. 57, which juxtaposed with C is
dyad of the gamma chord) is engaged in the relationship.
heard as (D ♯ , 1).

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 José Oliveira Martins Bartók’s Polymodality 317

Figure 18. s/f lattice for gamma-chord activity in the coda (mm. 68–80): The last gamma-
chord E–G–C–E ♭ finally fills in the harmonic gap.

piece’s conclusion. The material revisits and condenses procedures developed

on section A of the piece and ultimately provides the harmonic gap media-
tion created in mm. 55–57 (fleetingly covered by the note C in m. 56). Figure
18 maps gamma-chord activity of the coda into the s/f lattice, showing that
the four gamma chords articulated in mm. 68–78 (C–E ♭ –A ♭ –B, D–F–B ♭ –D ♭,
E ♭ –F♯ –B–D, and F–A ♭ –D ♭ –E), while gesturally fragmented, stand at adjacent
rows and columns in the lattice (i.e., they create scalar and harmonic adjacen-
cies in the affinity space A). The last gamma chord of the piece (E–G–C–E ♭,
mm. 79–80) fulfills two roles: it continues the melodic adjacency of gamma
chords set forth in the coda (bounded by a solid line in the figure), and it
touches upon (C, 0)–(E ♭, 1), finally arriving upon the harmonic mediation
thus far unfulfilled in section A′ (mm. 55–56; dotted line).
In conclusion, the analytical approach developed in the article to exam-
ine the combination of scalar layers in early twentieth-century music, partic-
ularly in Bartók, suggests a fertile interaction between diatonicism and chro-
maticism. Based on the intuition (proposed by Bartók and others) that scalar
layers are not fused but retain their diatonic integrity when combined in
certain (polymodal) contrapuntal textures, the article proposes conceptual
constructs (the Dasian and other affinity spaces) that share properties of
both scalar and pitch-class spaces in order to better understand harmonic
relations in multidiatonic passages. This contrasts with the prevailing tonal
and atonal analytical models developed in the second half of the century for
this music, which espouse either a deeper level diatonicism or chromaticism.
The analytical argument developed in the article suggests that composers
such as Bartók have explored the combination of layers not only to create new
textures or characteristic harmonic states but also to shape and contribute to
large-scale tonal strategies. In addition, the morphologies of affinity spaces
sustaining the analytical framework have deep historical roots as models for
the understanding of the relation between melodies and scales. The central

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 318 JOURNAL of MUSIC THEORY

properties of those spaces (in particular interval affinities, and the interac-
tion of pitch class and modal quality) are awakened from dormancy in our
familiar materials and put to new analytical uses, while bypassing some (at
times) limiting features usually associated with those materials (such as the
specification of pitch centers, enharmonic spellings, and labels for and affil-
iations to complete scales). The generalization of those properties to non­
diatonic contexts proposed in the article also allows us to better understand
how compositional strategies in the twentieth-century scalar tradition cre-
ated new harmonic relations that contributed to the overall exploration of
chromatic space.

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José Oliveira Martins received his Ph.D. in music history and theory from the University of Chicago
and currently is principal investigator in music and the humanities at the Research Center for Science
and Technology of the Arts (CITAR), Universidade Católica Portuguesa–Porto. His work centers on
aspects of multilayered harmony in twentieth-century music. He has held faculty appointments in
music theory at the Eastman School of Music and the University of Iowa.

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