articulorogers chopin (1)

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1/8

Chopin, Prelude in A Minor, Op. 28, No. 2Author(s): Michael R. Rogers

Source: 19th-Century Music, Vol. 4, No. 3 (Spring, 1981), pp. 244-250Published by: University of California PressStable URL: http://www.jstor.org/stable/746697.

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2/8

ehearings

Chopin,

Prelude

in A

Minor,

Op.

28,

No.

2

MICHAEL

R.

ROGERS

The nineteenth

century

abounds in

examples

of tonal

ambiguity.'

Chopin's

Prelude

in

A

Minor

provides

one

particularly

striking

exam-

ple

of

a

romantic

work whose

beginning

ob-

scurities

and

subsequent

meanderings

have

mystified

theorists

and

performers

for decades.

In

fact,

in a

lecture

in

Warsaw

in

1883,

the

pianist

and

pedagogue

Jan

Kleczyriski

recom-

mended that this

prelude

should never be

per-

formed

because it was too "bizarre."2

More

re-

cently,

Leonard

Meyer

has

viewed

this

work

as

the

psychological

establishment

and

distur-

bance

of

various

melodic

and

harmonic

pro-

cesses.3

I would like to

suggest

that

some of

the

harmonic and melodic

ambiguities

in

this

Chopin prelude

are

interlocked

with

and

un-

derpinnedby

durational codes. These

temporal

configurations

can often

help

to

clarify

the

meaning

of

crucial

but

puzzling

pitch

events

by

organizing

their

pacing

and

location

and

thereby

creating

a

unity

of

larger

temporal

patterns.

It

is

possible

that one of these

macrorhythmic

organizational

principles

is the

golden

section,

which has been

recognized

since

Greek

antiquity

in the visual arts

but

which

has an

equally fascinating

existence in

the

temporal

realm.4

1For

just

three

of

many

discussions of this

subject

see

Leonard

Bernstein,

"The

Delights

and

Dangers

of

Am-

biguity,"

The Unanswered

Question:

Six Talks at Harvard

(Cambridge, 1976), chap. 4;

Daniel

Coren,

"Ambiguity

in

Schubert's

Recapitulations,"

Musical

Quarterly

60

(1974),

568-82;

and

David

Epstein,

"Ambiguity

as

Premise,"

Be-

yond Orpheus:

Studies in Musical

Structure

(Cambridge,

Mass.,

1979),

chap.

8.

2Jean

Kleczyriski,

Chopin's

Greater

Works,

trans. Natalie

Janotha (London,

1922), p.

47.

3Leonard Meyer, Emotion and Meaning in Music (Chicago,

1956), pp.

93-97.

4A

gradually

emerging

body

of

literature has been

exploring

this connection between the

golden

section

and musical

structure. One

of the

earliest articles

to

comment on

this

relationship

is

J.

H.

Douglas

Webster's,

"Golden-Mean

Form

in

Music,"

Music &

Letters

30

(1950),

238-48. The

best-known studies are by E. Lendvai and are found in nu-

merous

journal

articles

and

summarized

in

Bdla

Bartdk:

An

Analysis of

His

Music

(London, 1971).

Four

recent dis-

sertations

include

Jane

Perry Camp, Temporal Proportion:

A

Study

of

the Sonata

Forms in the

Piano Sonatas

of

Mozart

(Ph.D.

diss.,

Florida

State

University,

1968);

Clive

Pascoe,

Golden

Proportion

in Musical

Design (D.M.E.

diss.,

University

of

Cincinnati,

1973);

Michael

R.

Rogers,

The Golden Section

in

Musical

Time:

Speculations

on

Temporal Proportion

(Ph.D.

diss.,

University

of

Iowa,

1977);

and

James

A.

Rothwell,

The

Phi Factor: Mathemati-

cal

Proporitions

in

Musical Forms

(Ph.D. diss.,

University

of Missouri at Kansas City, 1977). For information on the

conscious

use

of

the

golden

section as a

compositional

de-

vice

see

Jonathan

D.

Kramer,

"The Fibonacci Series

in

20th-Century

Music,"

Journal

of

Music

Theory

17

(1973),

100-50.

0148-2076/81/010244+07$00.50

@

1981

by

The

Regents

of

the

University

of

California.

245


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Plate 1:Chopin, Prelude n A Minor,op. 28, no. 2. Facsimile of the composer'sautograph n the Polish National Librar

Reproduced

by permission

from

the

facsimile

of

the

24

Preludes,

op.

28,

edited

by

W.

Hordynski

(Krakow, 19


4/8

19TH

CENTURY

MUSIC

The term

golden

section,

although

a rela-

tively

recent

one,

refers

to a

solution

of

the old

geometricalproblem of dividing a given length

into extreme and mean

ratio.

In

this

division,

the ratio of the short

to the

long segment

equals

the

ratio

of the

long

to

the whole

seg-

ment

(fig. 1).

This

can

be stated

in

even

more

general

terms

for

any

two identifiable

and

sepa-

rate

quantities (of

length,

time,

number,

a-

mount,

etc.):

the lesser

is to the

greater

as

the

greater

is to the sum.5

0.0 .618 1.000

A P

B

PB AP

=.618

P

-

AB

Figure

1:

The Golden-Section Division

The solution to the problem of dividing a

line

in

this

manner was first recorded

by

Euclid

(c.

300

B.C.)

in

Book

VI, proposition

30

(Ele-

ments).

The

golden

ratio

can

be

deduced

through geometric

proofs

and can

be

expressed

mathematically

as

V2(V5-1),

or

more

simply

as

.618034.6

Any quantity,

then,

that is

approxi-

mately

62%

of

another

quantity

creates and

exists

within a

golden-section relationship.

In

this paper, golden sections have been viewed

and

calculated

as

general

areas rather

than

as

precise spots.

A

self-imposed

margin

of

error

of

2%

either

way

(60-64%)

has been

allowed.

Using

this

process

means that

in a

hypothetical

composition of 100 measures (of steady tempo)

the main

golden

section could

occur

anywhere

between

measures 60-64.

Golden

sections

of

this

unit,

in

turn,

could

fall

within the same

percentage

boundaries.7

The cumulative sense

of

tonal focus

in

Chopin's

Prelude

in A Minor

is directed

by

a

deep-level

undercurrent that

exhibits

golden

sections at critical

points

during

the

unfolding

of the work.Figure2 illustrates the form of this

composition

as

it relates to

the

basic

melodic

and

harmonic/tonal

structure. The three-note

motives labeled "a"

and "b"

are hewn from

the

same intervallic

set;

"b"

is but

a

reordered

n-

version

of

"a"

(fig. 3).

This

three-note set

per-

meates the

motivic

material of

the

prelude

and

appears

in

a variant

form

only

at mm.

14-16,

where

"a"

becomes

A-E-F

instead

of A-E-G.

The pattern of common tones between ad-

jacent

"a" and "b" motives is broken

in

only

two

places:

at the

beginning

of the third

phrase

(m. 14)

and at the final

cadence.

These

two

"breaks" n the

descending pattern

constitute

formal

cleavages marking

off the minor sev-

enth as a

structural

melodic

interval.

The

pitch

A

entering

on

beat

2

of

m.

14

thus

articulates

the

growth

of this

composition

in

several important ways. In addition to signal-

ing

the

only

appearance

of

a

motivic

variant,

breaking

the "a" and "b"

pattern

of

the

opening

measures,

it

also

initiates a

new descent

of

a

minor seventh.

Its

position

on beat

2

of

m. 14

occurs at the

golden

mean

of the

overall

melodic

descent

of both

minor

sevenths-that

is,

the descent from the first

sounding

of

E in

m. 3

to the first

sounding

of

B

in m.

21

(fig. 4).

5A

technical

distinction can

be

made

among

the

following

comparable

terms:

golden

ratio,

golden

section,

and

golden

mean.

A

ratio

is the

quantitative comparison

of two items

(XIY)

while a

proportion

is the

equality

of

the two ratios

(XIY=S/T).

In

figure

1,

the ratio of PB/AP would

be an instance of a golden ratio and could be expressed nu-

merically.

It

is,

however,

the

larger

and more

inclusive

proportional relationship

involving

all

three

units

of

length

(PB:AP:AB)

that is

properly

called the

golden

section

(or

golden proportion).

The

golden

mean would be that

specific

point

P

on the

line AB that

permits

this

larger

relationship

to exist.

6The full

expression

of this number is

impossible

since it is

irrational;

its value has been

computed

to

4598

decimal

places

by

the Elliot

803

computer

of Bath

University.

A full

mathematical

proof

is

provided

in

Newman

W.

Powell,

"Fibonacci

and the Golden Mean:

Rabbits, Rumbas,

and

Rondeaus," Journal of Music Theory 23 (1979), 234-35;

and

a

method

of

construction in

H. E.

Huntley,

The Divine

Proportion (New

York,

1970), p.

27.

7Some

studies

of

the limitations

of human

auditory

dis-

crimination

involving

short durations

(up

to two

seconds)

have

suggested

that

time

segments

within

a

range

of 10%

difference cannot be

distinguished.

Whether these same

limits hold

true for the

much

longer

durations

of

complete

compositions

is

not

yet

clear. At

any

rate

the

4%

margin

span

that

I

have set

for

my analyses

seems

reasonable.

See

C. D.

Creelman,

"Human

Discrimination

of

Auditory

Duration," Journal of the Acoustical Society of America 34

(1962), 582-93,

and Leonard

Doob, Patterning

of

Time

(New

Haven, 1971).

246


5/8

REHEARINGS

a

b

a

b

b'

b

D

vi

V

G :vi V

I

Ala: IV/IV

IV

V

NOTE:

Ties areemployedto call attention to common-tone connections between "a" and "b"motives;

the

melodic

structure

of mm.

3-7 is notated an octave

higher

than in the score

on the

assumption

that

a

registral

shift need not obscure the

general

downward thrust

of the

first twelve

measures;

the

half

notes,

stemless

black

notes,

beams, etc.,

are

employed

to show

structural

units at

varying

hierarchical

levels.

The

harmony

of mm. 11-13

is

representedby

a

question

mark

since

any

selection

of

roots

would

have to

be made somewhat

arbitrarily.

Figure

2: Structure in

Chopin's

Prelude in A

Minor

a

a

a inverted

and

inverted

transposed=b

Figure

3:

Motivic Units

45 beats

27

beats

m. 3

m. 14

m.

21

45+27=72 total

beats;

45172=63%

Figure

4:

Golden

Section

within

the

Overall

Melodic Descent

What

is

even

more

remarkable,

though,

is that

it also representsthe first appearanceof A as a

tonic

(though

its

"tonicness"

does not become

apparent

until

m.

15 when

the

bass

slips

down

from

an

F-a French sixth

sonority

sounds

above

and

beautifully

sets

up

the

resolution-

to

an

E,

exactly

dividing

the

time

span

of the

entire

composition

into

a

golden

section).

This

appearance

of a

dominant-functioning

E

thus

dispels

the

harmonic and

tonal

fog

that

blan-

keted this prelude from its beginning and that

had

become

especially

thick

during

mm.

11-13.

M.

15,

then,

provides

a

convincing

a.

5 mm. 3 mm.

m.

3

m.

8

m.

11

5+3=8 total

measures;

/%=63%

b.

17 beats 10 beats

(bass)

m. 14

m. 18

m.

21

17+10=27

total

beats;

17/27

=63%

Figure

5:

Golden

Sections

within

the

Two Minor-Seventh

Descents

set-up

for the eventual

tonic

goal

which

is

re-

vealed here for the first time to be A minor.

Additional

golden

sections are created on a

melodic

level within each of the

first

and

sec-

ond

appearances

of

the minor-seventh

descent.

The

golden

mean of

the first

descent

falls

at the

beginning

of

m.

8,

which is not

only

the

begin-

ning

of the second

phrase

but

also

the

point

at

which the B-minor

chord,

construed

perhaps

as

a

submediant of

D,

is

first

heard

(fig. 5a).

The

second descent may be dividedin a similar way

on beat 3

of

m.

18

just

as

the E

returns

in

the

bass

reconfirming

its

dominant

role

(fig. 5b).

247


6/8

19TH

CENTURY

MUSIC

Returning

to

the

larger

overall

aspects

of

tonal

and

temporal

shaping,

further

examples

of golden sectioning resonate throughout this

composition

on

lower

hierarchic

levels

and

tend

cumulatively

to

bolster

and

mimic

the

main

division

that

has

already

been mentioned

at m.

15

(see fig.

6).

The

golden

mean

of the

longer

"half"

occurs

in m.

9

at the first

intro-

duction of

A

(both soprano

and

bass)

while the

golden

mean of that

segment

is

cut

off on the

downbeat

of

m.

6-the

first

emphatic

cadence

of the piece. A reverse (shortersegment before

the

longer)

or

passive

golden

section module

(working

backwards

from the end

within the

shorter"half" of the entire prelude)is situated

in

the middle of

m.

18-once

again

emphasiz-

ing

this return of an

influential bass

pitch.

These areas at mm.

6, 9, 15,

and

18

produce

a

kind of

gradually

emerging

and

increasingly

fo-

cused

view of tonal

centricity-m.

6

represent-

ing

a

decoy,

m.

9

an

ambivalence,

m.

15 a

pre-

diction

about the

ending,

m. 18 a

fortification,

and the

ending

itself,

of

course,

the

final

affirmationof our expectations.

15/24

=

63%

[reverse

20/32

=

63%

9/15 =

60% 22/36

=

61

%

module]

12

14

20 beats

beats

beats

22 beats

m.

6

m.

9

m. 15

m. 18 m. 24

NOTE:

The

silent

downbeat of

m. 24

represents

the end of the

piece;

the arrival n

m.

18

occurs

in the

middle of

a

measure.

The

measure numbers are

approximations

since

the arrivals are

considered

as

areas extended

into

a

given

measure

rather

than

simply

as

the

point

of the downbeat.

For

the

shorter

divisions,

golden

sections

have

been calculated

in

beats

rather han

measures

for

greater

precision.

Figure 6: Main and Sub-Level Temporal Golden Sections

(Chopin's

A-Minor

Prelude)

This

process

of

embedding

one

golden

sec-

tion within

another within

yet

another

finally

makes

this

prelude

work

as a series of

signals,

strategically placed

and

deliberately

paced,

which

regulate

the harmonic

ambiguities

and

help to foreshadowthe ultimate establishment

of tonal

stability-a

stability

that

arrives,

in

this

case, just

in the nick of

time-at the

very

end.

Keeping

the

Chopin

A-Minor

Prelude

in

mind

as a

kind

of

central

image,

we can com-

pare

it with

one

other

example.

The

first

movement

of

Beethoven's

Piano Sonata No.

14

in

C#

Minor,

op.

27,

no.

2 serves as

an ar-

chetype of golden-section form in general and

represents

the kind

of

temporal

model with

which

Chopin

was familiar.

The

Beethoven

example

also

represents

a

model of tonal

clar-

ity

quite

unlike

the

Chopin prelude.

In

the first

movement of

the

Beethoven,

we

are

given regularly

expanding

tonal

blocks.

Each new

point

of

arrival

develops

from

the

preceding tonal area and simultaneously pre-

pares

for the next. The

feeling

of

gathering

strength

is

inescapable

as each new

goal

ca-

dence relates

temporally

both

to

what has

gone

before

and

to what

is

going

to

follow.

The har-

monic

control is

metered

out,

though,

so that

while

the arrivals are

spaced

further and fur-

ther

apart,

their durational ratios to one

another remain constant.

The first unit is simply mood setting and

key

establishment,

as

C#

minor is affirmed

by

the cadence

in m.

5

M.

9 in E

(relative

major),

248


7/8

m.

15

in B

major/minor (dominant

of relative

major)

and

mm. 23-25

arriving

n

and reinforc-

ing F#

minor

(subdominant)

build

up

succes-

sive

golden

sections.

We

can understand

why

the

following

dominant

prolongation

is

exactly

fourteen

measures

long

as each stretch of

time

hangs

in the

memory

and forecasts future

landing

points.

Beethoven's intuitive sense

of

timing

holds off the release of the dominant

prolongation (and

its

accumulating

tension)

by

delaying

the

tonic return

until m. 42-the

boundary

set

up

by

the earlier

progression

of

golden-section

arrivals

and

itself

the

progenitor

of the

concluding boundary

at

the

movement's

end

in m.

69

(fig. 7).

We

have

in

operation

here

both the

wait of time and the

weight

of time.

REHEARINGS

42/69

=

61%

25/42 = 60%

15/25

=

60%

9/15

=

60%

mm.

9 15

23-25

42

69

c$

c#

E

B/b

ft

c# c#

i

-----

i

III

V/III

iv

V

---------

i

i

(prolongation)

Figure

7:

Proportional

Harmonic

Areas

in

Beethoven's

Piano

Sonata

No.

14,

op.

27,

no.

2

Not

only

is

each

of

the main

key

areaswell

prepared

by

a

functionally

sturdy pre-dominant

/dominant/tonic

cadence,

but the

melodic

phrase structure grows in exactly the same

proportional

ncrements and

in

the

correspond-

ing

places

as the

harmony.

The

whole move-

ment takes

shape through

a

process

of

both

multiplication

and

addition,

and

its

forward

momentum is

continually

renewed

through

geometric expansion.

Tonic

reinforcements

(mm.

51,

60,

66)

in

the final

section are

in

re-

verse

proportion

(short/long

as

in

Chopin)

and

decelerate the motion to a stop. Even the last

four

measures-all tonic

arpeggiation

and sim-

ple

block

chords-are essential

in

the

overall

scheme so that the end

does

not

come

too

soon.

It

is,

once

again,

the

timing

of the

goals

and not

simply

their content

that

gives

satisfaction.

It

is not

enough

in

such

precisely equilibrated

examples

to

say

that the

form is

regulated by

the

golden section;

the

form,

in

fact,

is

a

golden

section.

The

Chopin example

fails to

employ

the

familiar

tonal landmarks

of

Beethoven and

proceeds

from

its

murky

beginning

by only

gradually

exposing

the terminal

goal.

Nev-

ertheless,

the

control

of

its

shaping process

is as finely tuned as in Beethoven and the tradi-

tional

classical

temporal pattern

lingers

on.8

It

is

important

to remind

ourselves

in

these

temporal

analyses

that serious

study

remains

in

the

beginning stages,

even

though

these

stages may

be

very

revealing.

European

and

Ameri-

can

music

theory

are

only slowly

recovering

from

the

centuries-long

fixation

on

pitch

that

was dic-

tated

by

Rameau's

tonal

harmony.

Even

in

the

scien-

tific sphere, ime remains themoremysteriousfacetof

the

space-time

continuum.9

In

this short

paper,

I

do

not wish to

speculate

on the

aesthetic

appeal

of

temporal

golden

sec-

8The

prevalence

(and

almost

ubiquity)

of

golden

sections

throughout

the

so-called

common-practice period

is well

documented in many of the studies listed in fn. 4.

9Robert

Cogan

and Pozzi

Escot,

Sonic

Design

(Englewood

Cliffs, 1976), p.

304.

249


8/8

19TH

CENTURY

MUSIC

tions or

on

the

quality

of

their

organizational

power.10

But

I

should like to

suggest

that

obscure

and

equivocal

pitch

details

in this

Chopin prelude

may

be

illuminated

by

discov-

ering

their tactical

timing

as well as

their

sub-

stance.

We

can

say

that the

appearance

or

ex-

plication

of

tonality

and its

myriad

blurrings

s

not

merely

a

harmonic

phenomenon

but

also a

temporal

one-perhaps..-

ven

mainly

temporal.

1oFora discussion

of

these matters

see

chap.

V of

my

disser-

tation,

cited

in fn.

4.

Liszt,

Fantasy

and

Fugue

for

Organ

on

"Ad

nos,

ad

salutarem

undam"

R.

LARRY TODD

During

his

first few

years

at

Weimar,

Franz

Liszt

produced

a

series

of instrumental

works

in which he

employed

a new

type

of modified

sonata

form.

He

was

probably

inspired

to in-

vestigate

novel

formal

plans

at this

time

by

his

more or

less

abrupt

turn to orchestral

music,

and to orchestral

music

of a

new, revolutionary

kind.

Concurrently

with the first few

sym-

phonic

poems-Tasso,

Ce

qu'on

entend sur la

montaigne,

Prometheus,

and Les

Prdludes--

Liszt

composed

several

keyboard

compositions

that are

no less

revolutionary

in

form.

This

group

of

works,

which

culminated

in the

great

B-Minor

Piano

Sonata

of

1853,

includes the

organ

Fantasy

and

Fugue

of

1850

on

the

pseudo-chorale

"Ad

nos,

ad salutarem

undam"

from

Meyerbeer's

Le

Proph~te.

The

Fantasy

and

Fugue

was

Liszt's

first

organcomposition,

and

it

may

be

that the

novelty

of

the

medium,

like the

novelty

of the orchestra

at this

time,

encouraged

an

innovative treatment of

form

and

design.

Liszt

had been

predisposed

for

some

time

toward

a structural

renovation

of

the sonata

principle.

As

early

as 1837

he

argued

for

a

freer

For Fenner

Douglass

treatment

of

form

in

a review of

some

piano

works

by

Schumann. That

composer's

F-Minor

Piano

Sonata,

op.

14,

had

appeared

n

1836

as

the Concert

sans

orchestre,

and

this

peculiar,

seemingly

contradictory

titlel

prompted

the

reviewer

to

consider its formal

characteristics.

Liszt was

quick

to

point out,

in

fact,

that the

work was more a sonata than a

concerto,

yet

he

vigorously qualified

that observation:

"Though

we have made

this

distinction

[i.e.,

between

the sonata

and

concerto

forms],

we

by

no

means

intend

to

assign

each of these

composi-

tional

types

to a

specific

and

unyielding

formal

division."2

To

buttress his

argument,

Liszt

pointed

to several

structural innovations

in re-

cent

concerti:

Field had introduced

an

Adagio

in

the first movement

of his last concerto

(no.

7

0148-2076/81/010250+12$00.50

@

1981

by

The

Regents

of

the

University

of

California.

'Apparently suggested

to Schumann

by

the

publisher

To-

bias

Haslinger,

as related

by

J.

W. von Wasielewski in

Robert

Schumann

(Dresden,

1858),

p.

153. See also Linda

Correll

Roesner,

"The

Autograph

of

Schumann's

Piano

Sonata in

F

Minor,

Opus

14,"

Musical

Quarterly

61

(1975),

128-29.

21"Robert

Schumann's

Klavierkompositionen

Opus

5,

11,

14," in Gesammelte Schriften von Franz Liszt, ed. and

trans.

L.

Ramann,

vol. II

(Leipzig, 1881),

pp.

105-06: "In-

dem

wir

diesem Unterschied

feststellen,

wollen

wir

durchaus nicht

gesagt haben,

dass

wir

jeder

dieser

Kom-

positionsgattungen

einen

bestimmten

und unveriinder-

lichen Zuschnitt

beilegen

wollen."

250

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