harmonic resources in bartók's fourths - richard s. parks

8/9/2019 Harmonic Resources in Bartk's Fourths - Richard S. Parks

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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2/31

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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3/31

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



4/31

26

p

316

41

46

f

31

--ll?

Example

1

(continued)

247



5/31

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



6/31

Al

A2

B

C1 C2 D1

1 -- 4 5 -- 8 9 -- 16 17 -- 20 21 -- 30 31 --

Figure

1. FormalPlan

for "Fourt

4



7/31

Al

2 3 4

B

9

10

C1

7

19

D1

Example2

250



8/31

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



9/31

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



10/31

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



11/31

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)



12/31

[(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



13/31

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)



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



15/31

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



16/31

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



17/31

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



18/31


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



20/31

(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



21/31

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



22/31

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



23/31

I

Ioldtrexg

is1

?M

WINI=,



24/31

h)

17

13

14

15

16

E.18

19

2 0

contin

_

.

,

M-

Ex.

11

(continued)



25/31

0)

00

I

I

.1

....

.,

r......

..

......

....h

e,

23

24

25

26

283

I,"~~~ ."

."b

'

-

--

v

,,-

__,_,,-__-_____..... ...._____.

28

29

30

Ex.

11

(continued)



26/31

32

33

34

;35

b

10Ex

c o n t i n u e d )



27/31

04

;

VI slt

#lfC~

0

iI

Ccd

1w,

t i l

op

lr

'o,

fW~~c

O

I

-

d

I

II



28/31

(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



29/31

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



30/31


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

harmonic resources in bartók's fourths - richard s. parks

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