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1/31
Yale University Department of Music
Harmonic Resources in Bartk's "Fourths"Author(s): Richard S. Parks and Bela BartkReviewed work(s):Source: Journal of Music Theory, Vol. 25, No. 2 (Autumn, 1981), pp. 245-274Published by: Duke University Presson behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843651.
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HARMONIC RESOURCES
IN
BARTOK'S
"FOURTHS"
Richard
S.
Parks
Bela
Bart6k
may
be
counted
among
those remarkable
omposers
of
the
early
twentieth
century
whose vision
profoundly
altered the
way
we hear and
experience
music. His works
are
intuitively
accessible,
but
have
proven
difficult
to
penetrate
analytically,
or his
handling
of
pitch
materials seems to resist conventionalanalytic approaches.His music
has
been
characterized s
tonal,
for
instance,
yet surely
this
label
means
something
different
for
Bart6k
than
for
composers
a
generation
or
so
older,
since the harmonic and
contrapuntal nterrelationships
hat were
essential
n
their music
are
only
vestigial
n
his.
This
paper
contends
that atonal
theory
can
provide
a
better
approach
to Bart6k's
secrets,
ncluding
he nature of
tonality
in his
music
and the
extent to
which it
serves as
a
source of control. The issue of
tonality
will be
explored
in
the
light
of
remarks
by
the
composer
himself,
and
invarianceand linearitywill be examinedas possiblesources of tonal
function. This
essay
focuses
upon
a
single
composition,
but it
reveals
aspects
of
process
and
structure
important
in other works
by
Bart6k
as well.
"Fourths"
provides
an
excellent
object
for
analysis,
because
it is an
uncomplicated
piece, exhibiting
the
extraordinary
conomy
of
means
characteristic
of
all
Bart6k'sworks.
The
piece
is
reproduced
n
Example
1.
245
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Allegro
non
troppo,
a
1 4
'K
f
P
A
Si
a-
A
f
I
W--
a a
II
f"
11< J
21
=
from
MIKROKOSMOS,
Vol.
V,
1940
by
Hawkes and
Son
(London)
Ltd.;
renewed
1967.
Reprinted by
permission
of
Boosey
and
Hawkes,
Inc.
Example
1.
Bart6k,
"Fourths,"
Mikrokosmos
no.
131
246
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26
p
316
41
46
f
31
--ll?
Example
1
(continued)
247
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As the
title
"Fourths"
implies,
Number 131
of Bart6k's didactic
series,
Mikrokosmos,
treats
that
interval
as
a
performance
problem.1
Although
piano
studies
which
address
his
particular
ask are not
com-
mon,
one earlier
example,
Debussy's
"pour
les
Quartes"
from
the
Douze tudes, invites comparison.2Harmonicfourths are pervasive
in both
hands
hroughoutDebussy's
study,
as
in
Bart6k's,
but
Debussy's
study
admits diminished
and
augmented
fourths
as
well as
perfect
fourths. In
addition,
Debussy
frequently
injects
extraneous ones
(that
is,
non-fourth
related),
and
while some
passages
are
entirely chordal,
the two
hands
exhibit some
rhythmic
independence
throughout
the
work.
Bart6k
also
employs
harmonic ourths
in
both hands
throughout,
but,
unlike
Debussy,
he
uses
only
perfect
fourths,
avoids
extraneous
tones,
and does
not
emphasize ndependence
of
line.
The
only change
in texture which
occurs
(in
mm.
35-42)
is
still
based
upon perfect
fourths.
Indeed,
every
note
in
the
piece appears
as a memberof
a
perfect
fourth.
Of
course,
this concentration
upon perfect
fourths in each hand
does not
exclude
other intervals
arising
from the vertical coincidence
of fourths
in
right
and
left
hands and
their
coalescence
nto
chords.
The formal
plan,
shown in
Figure
1,
divides the
piece
into
nine
sec-
tions which
are
defined
by
changes
n
thematicmaterial
or
by
repetition.
As
the
diagram
shows,
symmetry
in the
form of statement-contrast-
return is sparse;sections D1 and D2 (separatedby E) constitute an
exception.
In
general,
the
piece
evolves
through
a succession of
inter-
related
but
contrasting
blocks of
material.Certain
unifying
featuresare
obvious
and include the
pervasive
use
of
perfect
fourths in each hand
already
mentioned. Most
thematic
shapes
also
prominently
feature a
stepwise
contour
resembling
a
neighbor-note onfiguration,
as illustrated
in
Example
2
which shows
the
upper
line from the
beginnings
of
sec-
tions
A
through
D.
In
section E of
the
formal
plan (mm.
35-42),
the
stepwise
shape
is less
obvious; nonetheless,
it
exists
in
the bass of
mm. 35-36, 37-38 and39-40.
A
glance
at
the details
of
the
piece
from section to section
points
first
to the
examination of four-note vertical sonorities as an obvious
basis
for
segmentation;
Bart6k's
articulative
markings
n
the
form of
slurs
suggest
another
way
of
grouping
and hearing)
ones.
Only
Section
E
(mm.
35-42)
does
not
lend
itself so
conveniently
to
these
segmenta-
tion
procedures;
he
remaining
ections
consistently yield
tetrachords.
Example
3
shows vertical
segmentation
or mm. 1-3
of
Section Al
and
for the first measureof each succeedingsection except for section E.3
Example
4
shows
a
representative egmentation
nto
four-note
collec-
tions
based
upon
slur
markings
n
each
hand for the initial bars of sec-
tions
Al,
B and
C1.
Section
E
(mm.
35-42)
does
not
lend
itself
to the
same
segmentation
process.
It seems reasonable
n
mm.
35-36 to hear the first
six
eighth-
248
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Al
A2
B
C1 C2 D1
1 -- 4 5 -- 8 9 -- 16 17 -- 20 21 -- 30 31 --
Figure
1. FormalPlan
for "Fourt
4
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Al
2 3 4
B
9
10
C1
7
19
D1
Example2
250
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4-26
4-8
4-26
4-26
4-9
Al
/,(10,
1,3,6)
q
1,
,4,5)2
/(10'1'
316)
1
3
(10,1,
31,6
(2,3,18,9)
4-8 4-20
B
(56,10
1t
4-23
(7,8,3)
4-26
4-8
4-26
(C1
"o1.
3,-,
)
D
J
3,
6,
8,1)
4-8
4-26
4-23
D2
,4,
5)
,3,
6)
F
(10o,o,,1
4-23
IFF
Example
3
251
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Al
4-8
4-26
4-23
S(0,
1,5,6)
2(10,
1, 3,6)
3
3 68)
10,1
3
4)
10,
6)
(9,10,2,3)
4-8 4-26
4-8
B
4-23 C1
4-8
L
-
(35,8,10)
(0,
17
(6,7,11,0)
4-84-8
Example
4
4-20
E 6-324-20
(10,0,2,3,5,7)
Example
5
252
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notes
as
a unit
comprised
of
six
pitches arranged
n
superimposed
perfect-fourths,
ollowed
by
a tetrachord
or the last two
eighths
(Ex. 5).
Measures
37-38
may
be
similarlypartitioned
and
yield
a
transposition
of the first six-note set followed
by
a
different
four-note
set.
Measures
39-40 repeat the materialof mm. 35-36 an octave lower, while mm.
41-42
yield
a three-note
set-the
only
trichord
which contains
two
perfect
fourths.
Figure
2
displays
the
locations of
all sets derived romthe
segmenta-
tion
processes
just
described.4
The
economy
of
Bartok's
four-note set
vocabularly
s
remarkable;
egmentation
revealsthat
only
five
different
tetrachordsoccur
throughout
he
piece.
All
appear
by
m.
9,
and
four of
the five are
introduced
by
the
end of
m.
3.
A
variety
of
transpositions
occur
for each
tetrachord,
and
these
are listed in
Table
1, along
with
the
approximate
number of times each
transposition
occurs.
(This gives
some indication of the
relative
emphasis
given
these forms
both individ-
ually
and
collectively.)
The
five
sets cited
may
be
expressed
by
the
following interval-arrays:
5-4-5
(4-26),
5-6-5
(4-8),
5-1-5
(4-9),
5-5-5
(4-23)
and 5-3-5
(4-20).
In
Example
6,
the
sets
are
transposed
o
C
in
orderto facilitate
compar-
isons.5
The
six-note
set which
appears
n
two
transpositions
n
mm.
35-
40
may
be
expressed
as
5-5-5-5-5;
the
three-noteset of
mm.
41-42
as
5-5. The affinity of all sets to the perfect fourth (interval-class ) is
obvious.
In order to
understand
the
nature
and
degree
of
selectivity
which
Bart6k
has
imposed upon
this
piece,
one must consider
certain
special
characteristicsof the
tetrachords
used relative to
all
possible
sets
of
four
pitch-classes.
Forte
lists a total
of
29 distinct
four-note
pitch
class
sets.
Of
these,
only
one
contains
a
total of
three
of
interval-class
(hereafter
abbrevi-
ated
i.c.)
in
its interval
vector;
no
set
contains more than
three.6
Eight
of the other 28 sets contain no i.c. 5 at all, while twelve sets each
contain
only
one. The
remaining
eight
sets all include
two
of
i.c. 5
in
their
interval
vectors.
Overall,
of 29
distinct
four-note
sets,
only
nine
contain two or more
perfect fourths,
and
only
one
contains
as
many
as
three
(in
other
words,
a
true
"quartal"
hord:
5-5-5).
As
might
be
expected,
all
of
Bart6k's
five
sets contain at
least
two
perfect
fourths
(Table
2
displays
the
interval
vectors for
the
five
sets),
and the set
with three
perfect
fourths
is
an
important
member
of
the
list.
(It appearsapproximately
44
times n the piece andby this standard
is
emphasized
less
than
two
other sets: 4-26
and
4-9.)
However,
Bart6k's
selectivity
extends
beyond
the
high
concentration
of
perfect
fourths,
since his five
sets
share an
additional
property;
they
all
repli-
cate
themselves n
inversion
and
thus haveno
distinct
nversional
orms.7
Of
the nine sets
containing
at
least two
perfect-fourths,
ix
possess
this
253
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4)
Figure
2. Chart
of
"Fourths,"
showing
tetrachordal
egmentatio
groupings
f
pitches,
as
well
as
largeraggregates
erived
rom forma
Key:
Sets
designated
A
=
4-26
(5-4-5)
SECTION:
B
=
4-8 (5-6-5)
C=
4-9 (5-1-5) BAR
OS.:
( G
D
=
4-23
(5-5-5)
E
=
4-20
(5-3-5)
PITCHES:
6
5 6
3
[B
]
[A
]
1
0
1 10
3
4 3 6
[B
] [A
]
10 11 10
1
A
B
A
A
(Secondary segmentation: 8-6 (10,11,0,1,3,4,5,6)
SECTION:
BAR
NOS.:?
PITCHES:
6
3 6
8 6 10
8
10
[A]
[D]
[
D
]
[
D
1
10
1 3 1 5
3
5
3 6 3 2 3 11 0 11 0
[A
]
[B
]
[B ] [B ]
10
1
10 9
10 6
7
6 7
A
A
A
C
A
B D
E B D
0,1,3,4,5,6)
7-20
(10,9,8,6,3,2,1)
8-14
(0,11,10,8,7,6,5,3)
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[(9's'E'Z'T'O'OT'
'Z'T)
iZ-8
(E'S'9'L'8'ot'tt'0)
tT-8 [(9'S'E'Z'
'9'L'S'0I'TI'0)
1T-8 (?'S'9'L'8'0I'II'0)
1'-8
('6'8'L'9'
''T','0)
6-8
1
W-
\'
Sl l Sl
Sl H
H
L 9
9
L
9
9
L
6
[ 8]
[
]
[
a]
0
TT
TT
0
TT
T
o
z
?
S
?
S
S
9
S S
?
3
]
[
0
]
[
o]
[
]]
[ ]
[[
]
[
]
[]
01
8 01
01 11 01 01
1
9
8
OT 8
OT
OT TT
OT OT 8
9
O
O
O:
[(t'Z'E'9'8'6'ot)
0Z-L]
(9'S't,'E'T'O'TT'OT)
9-8
(6'8'L'9'E'Z'T'O) 6-9
WS'
V
V V
3
3
oT TT
oT
ot 6
L
L
[ 5]
[5
]
[5]
[5
]
E E E
0
0
T T
0
I T
E
T E
] [a] []
[ ][
a
]
[
9 9
9 9
8
9 8
@
0
@&:SONO
G
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ON
(Figure
2,
continued)
SECTION:
(C2)
BAROS.:
PITCHES: 6 4 4 3 5 6 4 3 5 6
]
[
D
]
[B]
[D
]
[
D
]
[B]
[D0]
1
11
11
10 0
1
11
10
0
1
2
5
6
4
3
5
6 4
3
[
B ]
[
D]
[B] [D]
[
B
[
D] [B]
[D]
9
0
1
11
10
0
1
11
10
E
B A
B
A B
A B
A
8-14
(9,11,0,1,2,4,5,6)
8-6
(10,11,0,1,3,4,5,6)
SECTION:
BAR
NOS.:
PITCHES:
11 10
11 8
11
10
11
8
11
10
[B
]
[B
i
[A
]
[
A
]
[B
]
[B
i
[A
J
[
A
]
[B
]
[B
I
6
5
6
3
6
S
6
3
6
5
8 9 8 11 8
9
8
11
8
9
[
B
]
[B]
[A]
[A]
[
B
]
[
B
]
[A]
[A]
[
B
]
[
B
]
3
4
3
6
3
4
3 6
3 4
A
B
A A
A
B
A
A
A
B
8-6
(3,4,5,6,8,9,10,11)
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14/31
SECTION:
BAR
NOS.:
PITCHES: 10
10
10
5
5
5
6
1
6
2,7,0,5,10,3
1
9,2,7,0,5,10
8
2,7,0,5,10,3
1
10,3,8
6-32
E 6-32
A
6-32
E 3-9
8-14
(7,6,5,3,2,1,0,10)
8-14
(2,1,0,10,9,8,7,5)
8-14
(7,6,5,3,2,1,0,10)
SECTION:
(
BARNOS.:
?
(
PITCHES:
6 5 6 3 6 5
6
3
6
5
6 3
6
5
[
B
]
[
B
]
[ A
] [A
[
B
]
[
B
]
[
A
[A
]
[
B
i
[
B
i
[
A i
[A]
[
B
]
[ B
1 0 1 10
1 0 1 10
1 0
1
10
1
0
3 4 3 6 3 4 3 6 3 4 3 6 3 4
[B
] [8] [A
i
f
A
[
B]
[B]
[A]
[A
]
[B
]
[B3]
[A
]
[A] [B4]
[B
10
11
10
1 10
11 10
1 10 11
10
1 10
11
A
B
A
A
A B
A
A
A
B
A
A A
B
8-6
(10,11,0,1,3,4,5,6)
tO
;1
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Table
1.
List
of
tetrachordal
sets
used
in
"Fourths,"
their
transposi-
tions,
and the number
of
times each
appears.
Set
Number
of
Total
for
Label
Transposition
Occurrences EachSet
A
10,1,3,6
60
4-26
3,6,8,11
29
5,8,10,1
1
90
B
11,0,4,5
15
4-8
0,1,5,6
15
10,11,3,4 16
9,10,2,3
4
5,6,10,11
17
6,7,11,0
7
1,2,6,7
2
4,5,9,10
4
3,4,8,9
8
88
C
2,3,8,9
3
4-9 0,1,6,7 1
4
D
1,3,6,8
11
4-23
5,7,10,0
4
3,5,8,10
8
7,9,0,2
2
10,0,3,5
10
11,1,4,6
5
44
E
7,8,0,3
7
4-20
1,2,6,9
4
5,6,10,1
2
13
Table 2. Interval
vectors
for
tetrachords
isted
in
order
of
appearance.
Set
Label
I
Interval
Array
IntervalVector
A
=
4-26
5-4-5
[012120]
B
=
4-8
5-6-5
[200121]
C
=
4-9
5-1-5
[200022]
D= 4-23
5-5-5
[021030]
E
=
4-20
5-3-5
[101220]
258
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self-replicating
feature: the five
above,
plus
set 4-6
(which
can
be
representedby
the
interval
array
5-5-1).8
The
reason
for
the
latter's
absence
n
this
piece
will
be
seen
shortly.
Besides these common
properties
(an
abundance
of i.c.
5 and
self-
replication),
the five sets also have a potential for contrast in their
interval
contents-a
potential
which
Bart6k
exploits.
Sets
4-26 and
4-23 contain neither minor seconds
(i.c. 1)
nor tritones
(i.c.
6),
but
both contain
major
econds
(i.c.
2)
and minor thirds
(i.c.
3).
In
contrast,
sets
4-8
and 4-9 exclude
major
seconds
and minor
thirds,
but
do
con-
tain
minor
seconds
and
tritones.
A
dichotomy
thus exists
among
these
four
sets;
they may
be
grouped
into
two
categories
based on their
inclusionof
i.c.'s
1,
2,
3,
and
6.
The fifth
tetrachord
4-20)
fits
neither
category, for it lacks one of the intervalscharacteristic f each (i.c. 2
and
i.c.
6)
and
contains
ntervals
excluded from both
(i.c.
1 and
i.c.
3).
Bart6k's
choice
of
these five
particular
sets can
be
understood,
in
part,
by returning
to the
performance
problem
he
poses: namely,
playing perfect
fourths
in
both
hands
at
all
times.
Example
6 shows the
sets with
pitches
arranged
o
that each
is
partitioned
into two
perfect
fourths connected
by
another
interval.
The
connecting
interval
varies,
ranging
rom
i.c.
1,
to
i.c.'s
3,
4,
5
(the
"quartal"
hord)
and
6.
Interval-
class
2
cannot serve
as
the
connecting
nterval,
since it
would
produce
a
redundantpitch-classamongthe two perfect-fourths, husreducing hat
set
to
a
trichord
Ex.
6,
shown
in
parentheses).
The
reason
for
avoiding
he
remaining
elf-replicating
et 4-6
(5-5-1)
becomes
clear;
ts
pitches
cannot be
arranged
n a
way
which
permits
a
partitioning
nto
two
disjunct
perfect
fourths,
that
is,
a
perfect
fourth
cannot
be
assigned
to
each
hand
or to each half
of
a
melodic-harmonic
unit
under
a
slur
(compare
Exs.
8 and
9).
The
remaining
hree
tetra-
chords
of
the nine
that contain
two
or
more
perfect
fourths,
4-14,
4-22,
and
4-16
(whose
fourths could be
distributed n
interval
arraysas
5-5-3,
5-5-4 and 5-5-6
respectively)
are
eliminated
for the
same
reason.
The
elucidationof
tetrachordal
onstruction
n
"Fourths"
proceeded
from
a
direct
approach
to
segmentation
which
relied
upon
obvious
associationsof
adjacent
pitch-classes
on the surfaceof the
piece
(in
the
form of
chords,
or
adjacent
fourths
linked
by
slurs).
This
may
be
designated
a
primary
method
of
segmentation.
A
secondary
method,
less
obvious
but
nonetheless
important,
examines
aggregates
of tetra-
chords as they accumulatefrom section to section. A judicious parti-
tioning
of
sections into
smaller
units
(phrase
members,
pairs
of
adjacent
chords,
and
so
forth)
yields
collections which
are
indeed
of
interest.9
Figure
2
shows this
secondary
segmentation
with
the
resultant
arge
sets indicated
by
brackets and labeled
beneath the
integer
chart.
The
initial
two
measures f
the
piece (the
first of two
phrase
members)
display
259
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A
(4-26)
B
(4-8)
C
(4-9)
5
-
4
I
5
.
6
-5-
J
-
5
D
(4-23)
E
(4-20)
not
possible
Example6
set A
(4-26)
inversion
on
D
5 - 5
5
-4-5
Example
7
4-6
4-14
4-22
4-16
5 - 5 - 1 5 -5 -3 5-5-4 - 5 -6
Example
8
set 4-26 or
Example
9
260
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8/9/2019 Harmonic Resources in Bartk's Fourths - Richard S. Parks
19/31
Table
3. Intervalvectors for
3,
6,
7 and 8 note sets used
in
"Fourths."
Set
Name
Vector Interval
Array
3-9 [010020] 5-5
6-32
[143250]
5-5-5-5-5
7-20
[433452]
5-3-5-5-5-5
7-22
[424542]
5-6-5-5-6-5
7-35
[254361]
5-5-5-5-5-5
8-6
[654463]
5-5-5-6-5-5-5
8-9 [644464] 5-5-5-3-5-5-5
8-14
[555562]
5-5-5-5-5-8-5
8-23
[465472]
5-5-5-5-5-5-5
262
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(7-22)
(7-20).,.
.00
(7-35)
(4-8)
(8,-
4 - 9 )
8 - 9 )
l i
(4-20)
(4-23)
..
-14)
(4-26)
(8-23)
(6-32)
(3-9)
Figure
3.
Matrixof
Subset and
Superset
Relations
n
"Fourths"
263
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chords,
in
spite
of
the
fact
that it is
a
"quartal
hord." Set
6-32 is most
often omitted as
their
supersetalthough
t, too,
is
a
quartal
chord.
According
to
Benjamin
Suchoff,
Bart6k described
this
piece
as
bi-
tonal,
alluding
to
Eb
minor and
Gb
major
as
providing
concurrent
onal
centers.10Conservative iews of tonal
organization for
example,
those
expressed
by
Schenker
and even
Hindemith)
hold such
a
conception
of
tonality
as
inherently contradictory.
On the
other
hand,
advocates
of a
more liberal attitude
admit the
possibility
of
multiple
keys,
but
caution
that elements
must be
separated
n
such
a
way
that
the ear
may
unam-
biguously
distinguish
the two
centers.11
Clearly
such
centers are not
without
ambiguity
in
"Fourths,"
since
neither
register
nor timbre
separate
he
Eb
and
Gb
triads.
Perhaps
a
re-examinationof
pitch
class
content will shed light on Bart6k's allusion to this mixture of major
and
minor tonalities. Set
4-26 in its initial
transposition
10,1,3,6)
is
the
predominant
sonority
for the
piece,
shown
by
the tabulations
n
Table
1. Its four notes
includeboth the
Gb
major
6,10,1)
and
Eb
minor
(3,6,10)
triads,
but
they
are
completely
fused,
so that
the role
of
tonic
triad
cannot
be
assigned
o
either
with
certainty.
12
On
the other
hand,
these
four notes
together
do
provide
a referential
ntervallicand
pitch-
class
collection
for the
piece. They
occur
in cadences which
mark
im-
portant
points
of
subdivision
n the formal
plan
(for example
mm.
4,
8,
16, 30, and 46). Also, the pitch-classcontent of the set providesmost
of
the
stressed
pitches
of
the outer lines
in termsof
accentuation,
and
registral
and
durational
emphasis.
Example
10 isolates
these
pertinent
tones
(notated
in
open
noteheads)
and shows
how
they
are connected
through
linear
motions.
(Note
that
pitch-class
6
is often notated
as
F?
in the
lower
voice,
rather than as
Gb
)
The
unfolding
skips
between
tones
of the
referential
sonority throughout
the
piece
are almost
invariably
filled out
by stepwise
motion.
These
linear motions
are
charted
in
Example
11.
(Again,
the
referential
sonority
is
always
represented
using
open
noteheads
while
connecting
tones,
passing
or
neighboring,
are
filled
in.)
In
Example
11,
graph
II
is a reduction
which
better
displays
the
long-range
inear motions
of
graph
I.
This
example
demonstrates
he
extent to which
the
pitch-class
content
of
the referen-
tial
sonority
is
prevalent
hroughout
he
piece
and
also
how its
presence
increasesand
attentuates
from section
to section.
13
In
Example
11
four
simultaneous
lines
are
displayed
in which
both treble
and bass
parts
focus
upon
members
of the
referential
sonority;
where
they
do
not,
only the appropriate uterline is shown.Thehands(andthus the treble
and
bass
registers)
exchange
fourths
frequently.
This
may
be seen
in
m.
2,
beat
2.
The
linear
graphs
how
how these
exchanges
areconnected
by stepwise
motions
reflecting,
on the surface
evel,
similar
connections
which
occur
over
longer spans.
An
examination
of
all set-forms
used
in
the
piece
reveals
that the
pitch-class
content
of
the
referential
onority
264
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A1,A2
B
C1
17-20
-
22
1i
4
5-8 9 11
13
15
-
16
1
,--
.
,
/L..=I.M-I,-
so.
.
..12-
_
C2
23 24 25
26 27
28 29
30
Dl
E
D2
F
31
-
34
,
35-3637-3839-4041-42
43-46
47
-
50
I
?
Example
10
265
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I
Ioldtrexg
is1
?M
WINI=,
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h)
17
13
14
15
16
E.18
19
2 0
contin
_
.
,
M-
Ex.
11
(continued)
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0)
00
I
I
.1
....
.,
r......
..
......
....h
e,
23
24
25
26
283
I,"~~~ ."
."b
'
-
--
v
,,-
__,_,,-__-_____..... ...._____.
28
29
30
Ex.
11
(continued)
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32
33
34
;35
b
10Ex
c o n t i n u e d )
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04
;
VI slt
#lfC~
0
iI
Ccd
1w,
t i l
op
lr
'o,
fW~~c
O
I
-
d
I
II
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(4-26
as
10,1,3,6)
is well
represented;
ery
few set-forms
appear
which
do
not contain
at least one
pitch-class
romthis
transposition,
and
most
include
two
or
three.
It
is
apparent
that
the
process
of
selecting
pitch-class
resources,
which accounts for the remarkablyhomogeneoussound-colorof this
composition,
is
inextricably
wedded to the didactic
problems
which
Bart6k
designed
or
the
pianist:
the
playing
of
perfect
fourths
in
various
contexts-one
in each
hand,
two in each
chord,
and two
in each
pair
of
adjacent
ntervals
n each
hand.
4
It
might
be
tempting
to
conclude
that the
pianistic
problem
is
responsible
or the set
vocabulary.
But the
temptation
must
be
resisted,
for
even a
piece
preoccupied
with
perfect
fourthsin each
hand
could
employ a broadercatalog
of
resources-for
example by including
more
or
fewer
pitches
for each hand or
by duplicating
pitches
between
hands.
In this
regard
two notational details
are
of
interest.
First,
the "ossia"
ending,
mm.
47-50
(which
Suchoff
says
"should
be
played")
adds
a
fourth
pitch
to
eachmotivic
unit
of each
hand.is
Perhaps
n
afterthought
arose
from
a
sense
of
inconsistency
in sound
color
here
in
the
original
version,
for the
pianistic
consistency
is
greater
with
dyads
(since
each
hand
is
assigned
dyads
throughout
the rest
of the
piece).
Also,
the
G
of m. 35
(left
hand)
is not sustained
nto
m.
36,
although
he
F
on
the
treble staff is, with the resultthat the first threeeighthsof m. 36 each
contain a
four-note
sonority. Retaining
the G
would
have resulted
in
five-note
sonorities,
and
dropping
the
F would
have left
three
note
sonorities-all
inconsistent
with
the
norm
of four note sonorities
established
n the first
34
bars.16
While the tetrachordal
et
vocabulary
s limited
to
five,
their
varying
dispositions
from
section
to section
throughout
the
work
complement
the formal
plan.
Certain
ets
(in
certain
ranspositions)
hus
predominate
for each
section.
Sections
Al,
A2,
D1
and
D2 are
dominated
by
set
4-26,
with set 4-8
appearing
ntermittently
as a foil
(since
its
tritone,
lacking
in
4-26,
supplies
a
sharp
contrast n
sound).
Sections
C1
and C2
are
dominated
by
set 4-8
(in
several
transpositions),
and
here set
4-26
serves
as a foil.
Section
B
provides
new sonorous
color
through
its
quantitative
emphasisupon
sets
4-23 and 4-20. Set
4-26
appears
only
at the
end of
Section
B
as
a
cadential
sonority.
Section
E
departs
rom
the
predominantly
vertical texture
of
the rest
of the
piece,
displaying
few
tetrachords
n favor of the hexachord
comprised
of
superimposed
fourths,6-32. SectionF is dominatedby the quartal etrachord,4-23.
But
although
the
use
of tetrachords s
closely
tied
to
the
formal
plan,
the
large
sets revealed
by
the
secondarysegmentation
are
not.
For
example,
set 8-6
appears
n
Sections
A1,
C1,
D1
and
D2,
but not in
C2.
Contrastbetween
sections
may
therefore be
associated
with
changes
n
the
vocabulary
f
tetrachords,
but
not
to
changes
n the
largeraggregates
271
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formed
by
their
conjunctions
into
octachords.
Instead,
the
larger
aggregates
luctuate
independently
of
the formal
plan.
In
conclusion,
a kind
of
tonal
process
unfolds from section
to
sec-
tion
and is discernable
n
the fluctuation
of
pitch-class
ontent
vis-i-vis
the referentialsonority. Sections Al, A2, C2 and D2 are repletewith
articulations
of
that
sonority,
but
in
Sections
B
and
E
it
appears
nfre-
quently.
The referential
sonority
in its
referential
ransposition
domi-
nates
the
beginning
and
ending
of the
piece.
It
provides
he
pitches
for
the
framing
structure,
nterlaced
with
linear
connections,
which
shapes
the outer
parts.
Also,
its
constituent
pitch-classesappear
frequently
as
members
of different sets and
transpositions
n
the
outer
sections,
so
that
we are never
far
from
hearing
at least
a
portion
of this
referential
collection.
It
is
interesting
to
observe that
the
descending hird,
Gb
to
Eb,
so
prominent
at the
beginning
of
the
piece,
closes
it
as well.
Initially,
Gb
predominates
(by
accent,
duration
and
iteration);
at the
end,
Eb
predominates
by
the
same means.
One
can
hear this as
simply
another
aspect
of the
dichotomy represented
by
the bitonal idea
to which
Bart6k
alluded.
It seems
likely
that
Bart6k's
rigorous
control and
selectivity
of
pitch-
class
resources,
though
tied to
pianistic
and
pedagogical
onsiderations,
was
not
merely
a
by-product
of
them,
but
rather
was
purposeful-perhaps
unconscious,but not accidental. Theevidencesuggests hat Bart6kwas
intrigued
with
the
compositionalpossibilities
of
subtle
interconnections
and
differences
residing
within
a
very
limited
number
of
pitch
combina-
tions.
Obviously
Bart6k
understood the
nature
of his
pitch
materials.
Analyses
of No. 9 of
the
Fourteen
Bagatelles
or
Piano,
opus
6
(1908),
the
third movement
of
the
First
String
Quartet,
opus
7
(1908),
the
second
movement of the
Piano
Sonata
(1926)
and
the
middle move-
ment of
Contrasts
(1938)
reveal
a
similar
harmonic
consistency
and
orderliness
of construction.
A
systematic
examinationof
many
works
spanning
Bart6k'sentire
career
would
surely
provide
new
insights
into
the
scope
and
nature
of
his
compositional techniques.
Moreover,
like
most
of
his
pioneering
contemporaries,
Bart6k
remained unattached
to
any
school
or
group,
yet
his
music
shares
many
characteristics
with
theirs. Set-theoretic
analysis
offers
a more concrete
basis
for
comparison
than has
been
available
in
the
past
and enables us
better
to
appreciate
the
kinship
they
shared
both
in historical era
and
in
the effort
to
create
a
new
musical anguage.
272
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8/9/2019 Harmonic Resources in Bartk's Fourths - Richard S. Parks
31/31
one
of
its forms
(5-35).
Furthermore,
the
six-note
set
of
mm. 35-40
(6-32)
is
its
most
characteristic hexachordal
superset.
11.
Leon
Dallin, Techniques of
Twentieth-Century
Composition,
3rd
ed.
(Du-
buque,
Iowa:
Wm.
C.
Brown, 1974),
p.
133,
is
typical
in his
assertion
that
"For polytonality to be consciously perceived, the two keys must be relatively
pure
and
adequately
separated
in
register
or
timbre."
12.
One
might argue
that
polytonality
in this
piece
resides
in
the
opposing
interval
roots
of
the
perfect
fourths
disposed
in treble and bass
registers:
Gb
for the
right
hand
and
Eb
in the
left. Without
questioning
the Hindemithian assertion
of
these tones as
roots,
it is
open
to
question
whether Bartok was aware
of
the
concept,
and
it
should be
emphasized
that his inclusion
of
the
qualifiers
"major"
and "minor"
suggests
a
more conventional view
of
what constituted
a
tonal
center.
13.
One is reminded
of
Stravinsky's
use
of
the
terms
"pole
of
sonority,"
"poles
of attraction," and "polar centers" in his attempt to describe his freer con-
ception
of
tonality
(or
"antitonality"-again,
his
term),
by
which
he
seems
to
mean
that a
pitch
class or
collection
of
pitch
classes could
serve as a stabi-
lizing
force in a
piece-however
unconventional this combination
might
be.
Igor
Stravinsky,
"The Phenomenon
of
Music,"
in
Poetics
of
Music,
trans.
Arthur
Knodel and
Ingolf
Dahl
(New
York:
Vintage
Books,
1947),
pp.
23-
46,
and
especially
pp.
37-40,
44.
14.
Bart6k's
preoccupation
with the
number
four
in
this
piece
is demonstrated
not
only
by
his
emphasis
on
fourths,
but
also
in
his
persistent
use
of
four-
note sonorities, in his choice of a duple meter featuring subdivision into four
eighth-notes
per
bar,
and
in
a
formal
scheme which divides
the
piece
into
four-measure
phrases.
(The
two
exceptions,
Section
C2
[mm. 21-30]
and
Section
E
[mm. 35-42],
still
incorporate
the
number
four into
their struc-
tures-in
the
first instance as
four-plus-two-plus-four
measures,
and in
the
last as
four-plus-four
measures.)
15.
Suchoff,
Bart6k's
Mikrokosmos,
p.
114.
16.
The three notes
of
mm.
41-42
represent just
such an
inconsistency
of
course,
but
this
does
not diminish
the
consistency
of
the
four-note
sonorities
so care-
fully
maintained
in
the
above-mentioned bars.
Indeed,
the
anomaly
of
mm.
41-42 is even more
striking
precisely
because it constitutes a
singular
departure
from the established
texture.
274