jackendoff toward a formal theory tonal music

8/17/2019 Jackendoff Toward a Formal Theory Tonal Music

1/62

Yale University Department of Music

Toward a Formal Theory of Tonal MusicAuthor(s): Fred Lerdahl and Ray JackendoffSource: Journal of Music Theory, Vol. 21, No. 1 (Spring, 1977), pp. 111-171Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843480

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2/62

TOWARD

A FORMAL THEORY

OF TONAL MUSIC

Fred Lerdahl and

Ray

Jackendoff

INTRODUCTORY

REMARKS

We take

the

goal

of

a

theory

of

music to

be a

formal

de-

scription

of

the musical

intuitions

of

an

educated

listener.

By

"musical intuitions" we mean

the

largely

unconscious knowl-

edge

which a

listener

brings

to

music

and

which allows him to

organize

musical sounds

into coherent

patterns.

By

"educated

listener"

we

mean

not

necessarily

a trained

musician

but

a

listener who is

aurally

familiar with the musical idiom in

question.

Such a listener is able

to

identify

a

previously

unknown

piece

as

an

example

of

the

idiom,

to

recognize

ele-

ments of

a

piece

as

anomalous

within the

idiom,

and

gener-

ally,

to

comprehend

a

piece

within the

idiom.

The

"educated

listener" is an

idealization.

Rarely

do two

people

hear the same

piece

in

precisely

the same

way

or

with

the same

degree

of

richness.

Nonetheless,

while it is

conceiv-

able to hear a piece any way one wants to, there is normally

substantial

agreement

on

what

are

the

most

natural

ways

to

hear a

piece.

Our

theory

is

concerned

not

with

particular

in-

stances of

hearing,

which

are

always

subject

to a

degree

of

111


3/62

variability,

but with

the

idealized

underlying

competence

which

the

educated

listener

brings

to

bear

in

understanding

tonal

pieces.

A

theory

about

a

particular type

of music

is,

ideally,

a

sub-

set of a

theory

of all music. We are

constructing

our

theory

of

tonal music with this

larger

perspective

in

mind.

While

many

of our

specific

rules are

applicable

only

to tonal

music,

the

basic

components

of

the

theory

are

designed

to

accommodate

music

of

different traditions

and

historical

periods.

A

reader

who

is at all

acquainted

with

contemporary

lin-

guistics

will observe

that the

goals

we

have

set for

ourselves

are

in

some

ways parallel

to the

goals

of transformational

generative grammar, which seeks to describe the linguistic

intuitions

of

a native

speaker

of

a

language

and

to

discover

those

aspects

of

particular

grammars

which are

common to

all

languages.

Indeed,

our

way

of

thinking

about

music is

pat-

terned

after the

methodology

of

linguistics

in

that we

demand

strong

motivation,

formal

rigor,

and

predictive

power

for

every part

of the

theory.

On

the

other

hand,

we do

not

ap-

proach

music

with

any preconceptions

that the

substance

of

our theory will look at all like linguistic theory, since language

and

music are

on the face

of it

different

manifestations

of

human

cognitive

capacity.

Previous theories

of tonal

music have

not met such

de-

mands

of

rigor

and

prediction.

Even

Schenker's

theory,

which

can

be

construed

as

having

much in common

with

the

genera-

tive

approach

to

linguistics,

is

at

bottom

inexplicit.

One

of

the

virtues

of a formal

theory

is not that it is

necessarily

more

"true," but that, even where incorrect or inadequate, it clari-

fies issues

precisely.

To elucidate

in what

sense our

theory

is

modeled

after lin-

guistic

theory,

we

must

mention a

common

misconception

about

transformational

grammar.

It

is often

thought

that a

Chomskian

generative

grammar

is an

algorithm

that

grinds

out

grammatical

sentences;

this view

suggests

that

a

genera-

tive music

theory

should be

a

device

which

composes pieces.

There are two errors in this view. First, the sense of "gener-

ate"

in the term

"generative

grammar"

s not that of an elec-

trical

generator

which

produces

electricity,

but

rather the

mathematical

sense,

in

which

it

means

to

describe

a

(usually

infinite)

set

by

formal

means.

Second, Chomsky

distinguishes

112


4/62

the

"weak

generative

capacity"

of

a

theory

from

its

"strong

generative

capacity."'

A

grammar

weakly

generates

a set

of

sentences

and

strongly

generates

a set

of structural

descrip-

tions,

where

each structural

description uniquely specifies

a

sentence,

but not

necessarily

conversely.

It is the notion of

strong

generation

which

is

overwhelmingly

of

interest

in

linguistic

theory.

The

same

holds

for music

theory,

since

a

theory

which

weakly

generates "grammatical"

tonal

pieces

would

tell

us

nothing

interesting

about their

structures.

A

strongly

generative theory

of

tonal music

would

not

merely

give

a

description

of what

pieces

are

grammatical.

Rather,

it

would

have to

specify

each

tonal

piece

together

with its structural description, i.e., a specification of all the

structure

which the

educated

listener

infers in

his

perception

of

the

piece.

If

a

given

piece

cannot

be

heard as

tonal,

the

theory

should be

unable

to

give

it a

structural

description;

if

a

piece

can

convincingly

be

heard

in

several

ways,

the

theory

should

give

it a

different

structural

description

for

each

way

of

hearing

it.

We

have

found that a

generative

music

theory

must

not

only assign structural descriptions to pieces, but must differ-

entiate

the

structural

descriptions

along

a

scale of

coherence,

weighting

them as

more or

less

"preferred"

ways

of

hearing

a

piece.

Thus,

the

theory

is

divided

into

two distinct

parts:

well-

formedness

conditions,

which

specify

possible

structural

de-

scriptions;

and

preference

rules,

which

designate,

out

of

the

possible

structural

descriptions,

those

that

correspond

to

the

educated

listener's

hearing

of

any

particular

piece.

There are various criteria for determiningwhich structural

descriptions

of

a

piece

are

"preferred."

Among

these,

of

course,

are our

own

intuitions.

Furthermore,

beyond

all

seemingly

self-evident

intuitions,

a

"preferred"

structural

description

will

tend

to

relate

otherwise

disparate

elements in

a

satisfying

way

and

to

reveal

surprising analytic

insights.

In

addition,

we are

aware

of

relevant

research

in

experimental

psychology

and are

concerned

that its

findings

be in

agree-

ment with our theoretical constructions. Criteriawithin the

theory

itself

include the

internal

consistency

of

the

rules

and

their

generalization

from

particular

instances

to

the

entire

body

of

classical

tonal

music. We

utilize

within

the

frame-

work

of

the

theory

such

psychologically

primary

notions

as

113


5/62

parallelism,

articulation,

and

stability. Finally,

the nature

of

our structures must be

psychologically plausible

in

that

their

complexity

must result from the interaction

of a

fairly

small

number

of

processes,

each

of which

taken in

isolation

is

rela-

tively simple.

Although

it

would

be

possible

within

the

theory

to

gener-

ate new tonal

pieces,

we

have chosen

only

to

generate

struc-

tural

descriptions

for

existing pieces.

It

would

seem

to be

inherently

less

rewarding

to

specify

normative but dull

pieces

than

to

develop

structural

descriptions

for works

of

lasting

interest.

Moreover,

in the

former

case,

there

would

always

be

the

danger,

through

excessive

limitation of the

possibilities

in

the interest of conceptual manageability, of oversimplifying

and

thereby establishing

shallow

or incorrect

principles

with

respect

to music in

general.

For

the common

conception

of

a

transformational

genera-

tive

grammar

as

merely

a

sentence-generating

device

is

mis-

taken

in

a further

respect.

Linguistic

theory

is

not

simply

concerned

with the

analysis

of

a limited set

of

sentences;

rather it considers

itself a branch

of

psychology,

concerned

with making empirically verifiable claims about one complex

aspect

of

human

mental

life,

namely

language.

Similarly,

by

putting

our

emphasis

on the

musical intuitions

of the

edu-

cated

listener,

and

by

taking

as our

sample

a

highly

complex

body

of

music,

we are

asserting

that the

analysis

of

pieces

of

music,

though

not

without

a

great

deal

of intrinsic

interest,

is

not

an end

in

itself.

Rather

the

goal

is

an

understanding

of

the

mental

process

of musical

perception,

a

psychological

phenomenon. From this viewpoint, our theory of music is not

just

an

analytic system,

but makes

strong

claims

about

the

delimitation

of

possible

theories

of musical

cognition.

By

regarding

music

only

as

apprehended

structure,

we

are

deliberately

avoiding

the difficult

issue of musical

meaning.

Whatever

music

may

"mean,"

it

is in no sense

comparable

to

the semantic

component

in

language;

there

are no

substantive

parallels

to

sense and

reference

in

language,

or to

such seman-

tic

judgments

as

synonymy, analyticity,

and entailment.

It is

in

the domain

of

syntax

that the

linguistic approach

has rele-

vance

to music

theory.

Yet even

here

there are

no

substantive

parallels

between

musical

structure

and

such

grammatical

categories

in

language

as

noun, verb,

adjective,

noun

phrase,

114


6/62

verb

phrase,

and

so

forth. The

concepts

of

musical structure

must

be

developed

in terms

of

music

itself.

We

likewise

avoid

the

issue

of

aesthetic

value.

Nevertheless,

that we

are

dealing

with

works of art rather than

"mental

objects"

from the everyday world, such as sentences, is to a

degree

problematic.

Whereas

a sentence

normally

has a

defi-

nite

meaning,

structural

ambiguity

in a

work

of

art

is

com-

mon.

And

whereas a

sentence

normally

has a definite

function,

it is in

the

nature of a

work of art that it is

appreciated

and

contemplated

from various

points

of

view and with various

purposes

in

mind,

not

only by

different

people

but

by

the

same

person

on

different

occasions.

These

differences,

how-

ever,do not mean that the understandingof a work of art can

take

any arbitrary

form

whatsoever; rather,

they

mean

that,

to an

extent,

multiple

understandings

are

possible,

desirable,

and

even

inevitable.

In

constructing

our

music

theory,

we

have

accounted for this

state

of affairs

by

building

a

system

of

interactive

components

and

by

emphasizing

the

"pre-

ferred"

nature

of the

resulting

structural

descriptions.

Under

our

conception

of

music

theory,

then,

the under-

standing of a piece of music by the idealized listener consists

in

his

finding

the

maximally

coherent

structural

description

or

descriptions

which

can

be

associated

with

the

piece's

se-

quence

of

pitch-time

events.

Maximizing

coherence involves the

interaction of

a

number

of

different

domains

of

analysis,

each

of

which

must

be

repre-

sented

in

the

structural

description.

There

are four

with which

we will be

concerned

here,

to

be

termed

grouping

analysis,

metrical analysis, time-span reduction, and prolongational

reduction. As an initial

overview,

we

may

say

that the

group-

ing

analysis

assigns

group

boundaries

to

the

music in a hier-

archic fashion

at

every

level of

a

piece.

The

metrical

analysis

assigns

a

hierarchy

of

strong

and weak

beats. The

time-span

reduction

designates

"structural

beginnings"

and

"structural

endings"

of

groups,

and

assigns

to the

pitches

a

hierarchy

which

relates

them

to

the

grouping

and

metrical

structures.

The

prolongational

reduction

assigns to the pitches a hier-

archy

which

expresses

harmonic

and melodic

continuity

and

progression;

it is

the

closest

equivalent

in

our

theory

to

Schenkerian

analysis.

There

are

some

abstract

properties

common

to

these

four

115


7/62

domains

of

analysis.

First,

each

domain

partitions

a

piece

into

discrete

regions,

organized

hierarchically

in

such

a

way

that

one

region

may

contain

other

regions,

but

may

not

partially

overlap

with

other

regions.

For

example, Figure

1

a

represents

a possible kind of organization. Figure lb represents an im-

possible organization:

at

j,

two

regions

overlap

within

level

2;

and

at

k,

a

boundary

in level

3

overlaps

a

region

in

level

2.

Another

property

common to these

domains

is that the

processes

of

organization

are

essentially

the same at all hier-

archic levels.

A related

point

is

that the

processes

of

organiza-

tion of these domains

are

recursive, i.e.,

capable

of indefinite

elaboration

by

the same

rules.

Other

aspects

of musical struc-

ture, however, are not hierarchic in nature. For the present

we shall

ignore

these

dimensions.2

As mentioned

above,

the rules which

assign

structural de-

scriptions

are

categorized

as

well-formedness

rules,

which

assign

possible

structures,

and

preference

rules,

which

select

coherent structures from

possible

structures.

In

addition,

transformational

rules,

which convert structures into other

structures,

are

needed

for

special

cases

(such

as

elisions)

not

generatedby the well-formednessrules. Although transforma-

tional

rules have

been

central to

linguistic theory, they

play

a

peripheral

role in

our

music

theory,

at

least

at its

current

stage

of

development.

The

following

discussion

of

the

organization

of

the

theory

will be

informal.

Emphasis

will

be

placed

on how the

compo-

nents

work

in

principle.

To

give

a

complete

account

would

exceed

our

present

purpose,

which

is to

convey

in

general

the

goals, operations, and implications of the theory as a whole.

We

begin

with

a discussion

of

rhythmic

structure in terms of

grouping

analysis

and

metrical

analysis.

Then we

develop

the

two

modes

of

pitch

hierarchization,

time-span

reduction and

prolongational

reduction.

Finally,

we

apply

the

completed

system

to

a

piece

of some

complexity.

RHYTHMICSTRUCTURE

When

hearing

a tonal

piece,

the listener

naturally

organizes

the sound

signals

into

units such

as

motives, themes,

phrases,

periods,

theme-groups,

sections,

and the

piece

itself.

Our

generic

term for these

units

is

"group."

The

grouping

analysis

116


8/62

Figure

la

lb

3)

..

.

J

4)

k

a

Figure 2a

2b

: *

/

0,~?

117


9/62

picks

out the

groups

and indicates them

in

the

structural

de-

scription

by

slurs

beneath

the musical

notation. The

grouping

well-formedness

rules restrict the

possible

grouping

structures

to the

kind

of hierarchic

organization

discussed

above,

where

Figure

1a was

possible

and

Figure

lb was not

possible.

Under

these

conditions,

both

Example

1

a

and

Example

Ib are

possi-

ble

groupings.

However,

while

Example

la would

appear

to

show

the

cor-

rect

grouping,

Example

lb

is

absurd.

In

order to

select

the

actually

heard

grouping

or

groupings

(such

as

Example

la),

as

against

all the

merely

possible

groupings

(such

as

Example

lb),

we

develop

the

grouping preference

rules. These

provide

the criteria for pickingout groups, and are classified according

to

principles

of

(a)

articulation

of

boundaries, (b)

parallelism

in

structure,

and

(c)

symmetry.

Group

boundaries

are articulated

by

such factors as dis-

tance

between

attack

points,

rests,

slurs written into the

music,

change

in

register, change

in

texture,

change

in

dy-

namics,

and

change

in timbre.

A

further

articulatory

device

is

the

harmonic

cadence,

which from

the

phrase

level

upward

normally signifies the ending of groups;this will be discussed

later.

Parallelism

in structure involves

some

kind

of

repetition

or

similarity

in the

music,

such

as

a

motive,

a

sequence,

a

section,

and so

forth. The

similarity

is

particularly

crucial at

the

begin-

ning

of

groups;

for even

if

they

diverge

later

on,

they

are

still

perceived

as

parallel.

In tonal

music,

parallelism

is the

major

factor

in all

large-scale

grouping.

Related to parallelismis the principle of symmetry, which

states that

the

ideal

subdivision

of

any

group

is

into

equal

parts.

In

Example

la,

all the

groupings

are

assigned

by

the

parallelism

rule

(reinforced

by

various

articulatory

criteria),

except

for the

groupings

marked

s,

which

are

due

to

the

sym-

metry

rule;

thus,

each 4-measure

group

subdivides

not

only

into

1

+

1

+

2,

but,

at the

next

level,

into 2

+

2.

A

grouping

transformational

rule is

required

to account

for grouping overlaps, which as such do not meet the well-

formedness

conditions

of hierarchic

organization.

The trans-

formational

rule

relates

well-formed

underlying

groupings

to

the

musical

surface,

thereby preserving

the

sense that

overlaps

are variations

on

normal hierarchic

grouping.

Thus,

in

Exam-

118


10/62

Example

la.

Beethoven: Sonata

Op.

2,

No.

2,

Scherzo

^^f

$

r f

1

ff

r>TI

p

lb

P

A-

$

T

^-

z

C

$%

4^

v.?

t^

m J j J

^

v$^ .^^t^ v^rf^^' ^


11/62

pie

2,

the

events at

u

and

v

function as

both

endings

and

beginnings

of

groups;

these

overlaps

are

disentangled

at

under-

lying

levels.

We turn

now

to metrical

structure,

which

is

independent

of

grouping

structure but interactive with it. Given

appropriate

musical

cues,

the listener

will

instinctively

infer a

regular,

hierarchic

pattern

of

beats

to

which

he

relates

the

actual

musical sounds.

The

metrical

analysis

assigns

to a

piece

such a

pattern

of beats and

indicates

them

in

the structural

descrip-

tion

by

dots beneath

the

musical notation and above

the

grouping

slurs,

as in

Example

3.

Each

level

of

dots

represents

a

marking-off

of the

music

into equal time-spans. A dot at a particularlevel representsa

judgment

that

that

moment

in the

music is a

beat

at that

level. If a

beat

at

a

particular

level

is felt

to

be

"strong,"

or

"down,"

it

is a

beat

at

the next

larger

level

and

receives an

additional

dot.

(Thus,

metrical

structure

does

not exist

with-

out

at least

two

levels of

dots.)3

The

process

is the same

whether

at the

level of the

smallest

note value

or

at

a

hyper-

measure

level. The

notated

meter

is

usually

an intermediate

metrical level.

Theoretically,

the

dots

could

be built

up

to

the level of a

whole

piece.

However,

the

perception

of relative metrical

stress fades

over

long

timespans.

In

addition,

at

large

levels,

metrical

structure

is heard

within

the context

of

grouping

structure,

which

is

rarely regular

at

such

levels;

and

without

regularity

there can be

no

metrical structure.

Therefore,

metrical

structure

is a

comparatively

local

phenomenon.

The metrical well-formedness rules assure the hierarchic

condition

that a beat

at a

particular

level

must also

be

a

beat

at all

smaller

levels. Characteristics

of

metrical

well-formed-

ness in

classical

tonal

music

include

the

equal

spacing

of beats

and

the

provision

that at

each

successive

level the

distance

between

beats

must be

either

two

or

three times that

of the

immediately

lower

level. Musical

styles

of other cultures

and

historical

periods

often

require

more

complicated

rules

of

metrical well-formedness; the rhythmic complexities of tonal

music

arise

from

the

interaction

of metrical

structure

with

grouping

structure

and

pitch

structure.

It

hardly

needs

emphasizing

that bar

lines and

beams

be-

tween

notes

are notational

devices

and

not

part

of

the

physi-

120


12/62

Example

2.

Mozart:

Sonata

K.

279,

I

Example

3.

Bach:

"O

Haupt

voll

blut und

Wunden"

r.,N

I i I

I,

"

/t4

J

LJ

i

.

LL

t rbr

r r

r

C r

CJ

I

n I

j j

J

u Jj

*0

It*

0.

.

'

I.

'

*

r

?

?

e

*

0

0)..

.

.

I.

- -

I,

..

, T14 _

- I

...

dr

I

- I

I

II

--

I

I

I I I

_I

I

I I

I

?

? ?

? ?

*

?

?

e


13/62

cal

signal,

as

pitches,

durations,

dynamics,

and timbre

are.

From

the

physical .signal

the

metrical

preference

rules

assign

to

a

piece

the

actually

heard

metrical

structure

(or structures)

instead

of the

myriad

well-formed

but

inappropriate

metrical

structuresapplicable to it. For example, they select the metri-

cal

structure

in

Example

3 rather

than one in a

triple

meter,

or

one

which

places

the

strongest

downbeat

of the

passage

on

the

opening

event.

The metrical

preference

rules

can be classified

according

to

principles

of

(a)

cues

for

strong

beats, (b)

parallelism

with

grouping

structure,

and

(c)

regularity

of

pattern.

The

cues in

the music

for

relatively strong

beats

include

such

factors

as

attack, accent, change of dynamic, registerof pitch, harmonic

change,

and

suspensions.

Added

to

these is the listener's

tendency

to ascribe

parallel

metrical

structures to

parallel

grouping

structures

(especially

rhythmic patterns

which

form

groups).

If there

is

any

regularity

to

these

various

cues,

the

listener

extrapolates

an

entire

metrical

hierarchy,

which

he

will renounce

only

in

the

face of

strongly

contradictory

cues.

Syncopation

takes

place

when

there are

strong

contradictory

cues which yet are not strong enough to overridethe inferred

pattern.

A metrical

transformational

rule is needed

for metrical

overlaps,

in

which

a shift

in the metrical

structure occurs

in

such a

way

that

the

same

moment

in a

piece

serves

a

double

metrical

function.

The

transformational

rule

deletes

one

set

of

dots

in favor of

the other

at

the

musical

surface.

In

Figure

2a,

the weaker

metrical

function

is

deleted;

in

Figure

2b,

the

stronger is deleted. (It may be helpful to think of the largest

level of

dots

as

representing

the measure

or half-measure

level.)

Although

it

is conceivable

for metrical

overlaps

to

arise

independently,

they

generally

happen

as

a

consequence

of

grouping

overlaps

which

take

place

on

relatively

weak

metri-

cal stresses.

On the

other

hand,

if a

grouping

overlap

takes

place

on a

relatively

strong

metrical

stress

(as

in

Example

2),

no metrical

overlap

results.

Using

familiar

terminology,

we

call a

combined

grouping

and

metrical

overlap

an elision.

We

further

distinguish

between

elisions

in which

the weaker

metrical

function

has been

deleted

(as

in

Figure

2a),

and

elisions

in

which

the

stronger

metrical

function

has

been

122


14/62

deleted

(as

in

Figure

2b).

Example

4 is

typical

of

the

former

kind;

the

effect at

s is one of

jarring

reorientation.

In

the

latter

kind,

the effect is rather

one of

retrospective

awareness

that

a shift

has

taken

place;

there is an

example

of this

in

the

Schumann

song

analyzed

later in this

paper

(pp.

149ff).

Some

general

reflections

are in order

concerning

the

organ-

ization of the

grouping

and

metrical

components.

It seems

to

us essential that the two not

be confused

inadvertently.

It

is

tempting,

for

example, perhaps

on the

basis of a

misleading

spatial analogy,

to

attribute

duration to the

concept

of

metri-

cal

stress,

so that first

a

beat,

then a

measure or

motivic

grouping,

and

finally

a

phrase

receives

"strong"

or

"weak"

markingsas the analysis proceeds to higher levels. In ourview,

however,

the notion of "beat"

or

"metrical stress"

can

only

be understood

as a

point

in

time,

without

duration;4

hence

our use

of dots to

signify

metrical structure.

Beats are

correct-

ly

analogous

to

equidistant

geometric points

rather

than

to

the lines

drawn between

them;

while

rhythms

and

groups

have

duration, then,

beats do not. The

metrical

component

assigns

metrical stress

not to

groups

but

to beats

within

groups.

Of course the

listener senses

that a

group

has a

certain

weight

if

within it

there is a

strong

beat. But this

does

not

mean

that the

group

as

a

whole

receives

metrical

weight;

for

the weak beats

are all

equally

weak.

In

Example

5a,

for in-

stance,

the

E-flats are

metrically

equal,

a

fact

obscured

in

Example

5b,

in

which

metrical

properties

appear

to be in-

cluded within a

grouping

notation.5

Just as groups as such do not receive metrical stress, metri-

cal

structure as

such

does not

possess any

inherent

grouping.

Whether a

weak beat is

heard

as an

afterbeat or

as an

upbeat

is

entirely

a

matter of the

grouping

associated with

it.

Similar-

ly,

at

a

larger

level of

grouping, say

the

paradigmatic

ante-

cedent-consequent

pattern

in

Figure

3,

there is

nothing

in

the

metrical

structure

which

prevents

the half

cadence

in m. 4

from

resolving

as a full

cadence to

the

tonic in

m.

5.

It

does

not so resolve only because of the intervening group bound-

ary.6

Figure

3

poses

a

question

of a

different

sort.

A

phrase

can

be

roughly

characterized as the

lowest

level of

grouping

which

has

a

structural

beginning,

a

middle,

and a

structural

ending

123


15/62

Example

4.

Haydn: Symphony

no.

104,

I,

mm.

17ff.

i$f

r

f

r

0

1

8

i

j^

.

a

r

4^

r

Y)r

r J

8

-

.

.

.

?

Example

6.

Beethoven:

"Hammerklavier"

Sonata

op.

106,

I

j

f

a

-

t

'

t

*

*

;

'r

'f

r

r

rI

"""=~~''(t

i

cedent)

~

_ ~

ntec

de nt

124


16/62

: **

:

:

r'

'

I

?

':

, '

:

.

.

.

I

j[

fr'f

TS

'

rvm

f=

u=ro

.

-

-

L'

@

*

\

'F

rr

rJ

JJ

j

'r

r

ir

J

J

(extended

consequenT)

125


17/62

Example 5a. Mozart: Symphony no. 40, I

ibb

t

n ,

n

n

?

.

?

.

?

.

. .

?

? ? ?

?

? ?

?

?

Sb

;9

kr

f

,

r'

i

v

i

1,

/

,Example

7. Bach

Example

7.

Bach:

"0

Haupt"

126

*

.I

I

._

....

,

-,

-

I

'..,

?

,

,,

.

? ,

?

? ,

?

1

1 t

1

1 ?

J


18/62

Figure

3

I

w%%V I-

measure:

1

2

3

4

5

6

7

8

Figure 4a

(a'I

pitch

events:

W z

4b

(b)

(C)

w

y

z

4c

w

x

y

z

127


19/62

(a cadence).

Thus,

a

phrase

normally

has a kind of structural

gravity

at or

near its

group

boundaries.

If

groups

as such have

no metrical

stress,

and

if metrical

structure

is

an

extrapolation

of

multi-leveled, regularly spaced beats,

how are

these

points

of

gravity

accounted

for in the

structural

description?

In

Figure

3

there

is no conflict

between

metrical stress and

structural

weight

in

mm.

1

and

5;

but

mm.

4

and

8,

where

the cadences

occur,

are

metrically

relatively

weak

(depending

on

the

example,

the

cadences

might

even

happen

in

the second

halves

of these

measures).

One

solution

might

be

to move the

second

strong

metrical

stress of

each

phrase

from

the down-

beats

of the third

and seventh measures

to the

points

at which

the cadences actually occur.7 However, there are two strong

reasons

against

such a

revision

of the

metrical

component.

First,

it

would

entail

surrendering

one

of the formal

proper-

ties

common

to

all the

domains

under

discussion,

namely

that

the

processes

of

organization

are

essentially

the same at

all

hierarchic

levels.

Secondly,

it

would

mean

giving

up

the tradi-

tional

distinction

between

cadences

which

take

place

at

weak

metrical

points

and cadences

which

take

place

at

strong

metri-

cal points. The latter are particularlyimportant for achieving

large-scale

arrival

and-in

conjunction

with

grouping

overlaps

-continuation.

Thus,

in

Example

68 the

consequent

phrase

is

extended

so that

the cadence

at

q

arrives

on a

strong

beat

and

overlaps

with

the

succeeding phrase.

According

to the

pro-

posed

revision,

the

cadence

of the antecedent

phrase

at

p

also

would

have

to

be

metrically strong,

with

the result that

the

metrical

distinction

between

p

and

q

would be

lost.

This is

plainly not acceptable.

The

points

of

gravity

in a

phrase,

then,

are not to

be

inter-

preted

as

metrical

phenomena.

Rather, they

are

hierarchically

important

elements

produced

by

the

interaction

of

pitch

structure

and

grouping

structure;

and

they

stand with

metrical

structure

in a

contrapuntal

relation,

so to

speak,

which

under-

lies

much

of the

rhythmic

richness

of

tonal

music. The

metri-

cal

component

therefore

remains

as

originally

set forth.

As

will be seen below, the proper distribution of structural

weight

in a

phrase

emerges

in the

analysis

as

part

of

the

time-

span

reduction,

which

treats

a

phrase

as an elaboration

of its

beginning

and

ending.

128


20/62

PITCH

REDUCTION

Both the

time-span

reduction and

the

prolongational

re-

duction

assign

hierarchic structures

to all

the

pitch

events in

a

piece.9

To

represent

these two kinds of

reduction,

we have

invented two somewhat different

"tree"

notations;

these

visu-

ally

resemble,

but are

substantially

different

from,

the

tree

notation

utilized

in

linguistics. Linguistic

trees

represent

"is-

a"

relations: a noun

phrase

followed

by

a

verb

phrase

is

a

sentence;

a

verb followed

by

a noun

phrase

is

a verb

phrase;

and

so

forth. Our musical

trees,

however,

do not involve

gram-

matical

categories.

The

fundamental

relationship

which

they

express is that of a sequence of pitch events as being an elabo-

ration

of

a

single pitch

event.

The

dominating

event,

that

of

which a

sequence

of

events is an

elaboration,

is

always

one

of

the events in

the

sequence;

the

remaining,

subordinate events

in the

sequence

are

heard

as

relatively

ornamental.

"Reduc-

tion"-the

process

of

recursively

substituting single

events for

sequences

of

events-can be

thought

of

as the

inverse of

elab-

oration.

In the following exposition we begin with the time-span

component

and

then turn to the

prolongational

component.

In

both

cases,

after

outlining

the notations

and the

basic

principles

of

reduction,

we

apply

the

components

first

to

an

abstract

antecedent-consequent

pattern,

then

to the

opening

eight

measures of

Mozart's

Sonata,

K.

331.

In

the

tree

notation

for the

time-span

reduction,

a

"right

branch"

(

X),

in

which

a

line

to the

right

attaches

to a line

to

the left, denotes the subordination of an event to the preced-

ing

event

within

that

region

at that

level;

a "left

branch"

(

X),

in

which

a

line

to

the

left

attaches

to a

line to the

right,

denotes the

subordination of

an

event to the

following

event

within

that

region

at that level.

The

well-formedness

condi-

tions

for these

trees

prohibit

both the

crossing

of

branches,

as

in

Figure

4a,

and

the

assignment

of

more

than one

line of

the

tree

to the

same

event,

as in

Figure

4b.

(The

letters in

paren-

theses signify reductional levels.)

The

relevant

notion of

elaboration in

the

time-span

reduc-

tion is

elaboration

at

successive levels

within

more or

less

equally

spaced,

discrete

time-spans.

Within

each

time-span,

or

region,

a

dominating

event must

be

found;

that

is,

all

other

129


21/62

events in

the

region

are elaborations

of

that event.

Thus,

at

level

(c)

in the well-formed tree

in

Figure

4c,

x is in

the

region

of

w

and

is

an elaboration

of

w;

y

is

in

the

region

of

z

and

is

an

elaboration

of

z.

At level

(b),

all

four

events are in the

same

region,

and z

(together

with x and

y)

is an elaboration

of

w.

Before

the

rules

which

establish

the tree

structure

can

be

applied,

it is

necessary

to select the

regions

of

application

at

every

hierarchic

level

within

a

piece.

These

regions

for all but

the most local

levels of

analysis

consist of the

groupings

assigned

by

the

grouping

preference

rules. Within the

lowest

grouping

level,

smaller

regions

are

chosen in terms of levels of

metrical structure. In these smallerregions, a given weak beat

is bracketed with

the

preceding

strong

beat,

unless the

pre-

ceding strong

beat

is

separated

from

the weak

beat

by

a

group

boundary;

in

this

latter

case,

the

weak

beat

is

bracketed

with

the

following strong

beat.

In musical

terminology,

this means

that

a weak

beat

is an afterbeat

unless it is situated

at the

beginning

of a

group,

in

which

case it

is

an

upbeat.

See

Exam-

ple

7;

the

brackets

indicate

the

sub-group regions

of

applica-

tion. In a given tree, each level of branchingcorresponds to a

region

of

application

for

the

preference

rules.

Given these

regions

of

application,

the

preference

rules for

the

time-span

reduction

choose

the

syntactically

most co-

herent reduction

(or

reductions)

from

all the

possible

but

mostly implausible

reductions

of a set of

pitch

events.

Syn-

tactic coherence

in

this domain

can be

thought

of

in terms

of

stability.

These

preference

rules are

classified

as

(a)

those

which ascertain the most stable structure (the tonic), and (b)

those which

establish

the

hierarchy

of relative

stability

in rela-

tion

to the most

stable

structure.

The tonic-at

any

level,

local

or

global-is

selected with

reference to

the available

pitch

collection and cadential

struc-

ture at the

appropriate

level.

The

rules of

relative

stability

or

instability

are,

in

broad

musical

terms,

the

principles

of

relative

consonance

or disso-

nance. For example, a local consonance is more stable than a

local

dissonance;

a triad

in

root

position

is

more stable

than

its

inversions;

a chord

is more

stable

if its melodic

note

is the

same

pitch-class

as

its

root;

the

relative

stability

of

two

chords

can be determined

by

the relative

closeness

to the local

tonic

130


22/62

of

their

roots on the circle

of

fifths;

conjunct

linear

connec-

tions

are more

stable

than

disjunct ones;

a

pitch

event

is

more

stable

if it is in a

metrically

stronger position;

and

so

forth.

The

preference

rules

use all of these criteria

in

picking

out

which event in a

given region

is

dominating.

Thus

(and

let

us assume here

the

"preferred"

grouping

and

metrical

structures)

both

Example

8a and

Example

8b

are

well-formed

time-span

reductions.

The

preference

rules

select

the

reduction in

Example

8a as

opposed

to

any

other

possible

reduction,

such as

Example

8b.

In

these

examples,

and

in

all

succeeding time-span

reduc-

tions,

a

notation

formally

equivalent

to the

tree

appears

beneath the music proper: the relation of pitch structure to

metrical structure

is

expressed by

the

placement

of

the

syn-

tactically

most

significant

event

within a

bracketing

on

the

strongest

beat within

that

bracketing.'? Although

the

two

notations

express

the

same

information,

we

retain

both

in

the

time-span

reduction

because,

as

will

become clear

in

longer

examples,

they

serve

somewhat

different

purposes.

The one

below the

music

is useful in

hearing any

particular

hierarchic

level. The tree-which, though unfamiliar, is easily compre-

hended

with

a

little

effort-gives

a

picture

of all

the

levels in

relation to

one

another;

moreover,

it

is

illuminating,

in

con-

nection

with a

particular

piece,

when

compared

to the

pro-

longational

reduction.

At

relatively

local

levels,

the tree

for the

time-span

reduc-

tion

correlates

with

the

metrical

analysis

to

produce

the

paradigmatic

situations in

Figure

5. Both

(h)

and

(i)

pertain

to afterbeats: in (h) the event on the afterbeat is the less

stable

event,

such

as

a

passing-

or

neighboring-tone

or

-chord;

in

(i)

the

event on

the

downbeat is

the

less

stable

event,

such

as

a

suspension

or an

appoggiatura-tone

or

-chord. Both

(j)

and

(k)

pertain

to

upbeats:

in

(j)

the

event

on

the

upbeat

is

less

stable than

the event

on its

associated

downbeat;

in

(k)

the

event

on

the

downbeat

is

less

stable than

the

event

on

its

associated

upbeat.

In

these

ways,

the

relation

of

syntactically

significant events to metrical structure is made clear. At large

levels,

where

metrical

structure

is

no

longer

hierarchically

operative,

the

right

and left

branchings

simply

denote

subor-

dination

within

grouping

structure.

Instances

of

all

four

paradigmatic

situations

occur

in

Exam-

131


23/62

Example

8a.

Mozart:

Sonata

K.

331,

I

8b

(a)

tli

I

:

I

I

N

-

I

K

r

Pr

p

r PF

P

I)

r

pr

r prF

(e)

I

I-

-

W

L###9

L1

P

r

rr

^

(b)

#

i

1

(a)

#

s--

132

(e)

(

:';rr r

;

Lfr

il(d)

$

#

<

I

I

IjI.

?(c)-i

i

i

(b)

1

.

I

11

(a)

0#

11

J Ad

J.

J J -

{8tPtI,

_i m m


24/62

Figure

5

(h)

(i)

(

(

A

. X

X

*

Figure6

(9)

level

(h)

=

[b4

[c

[

[?c

level(g)

=

[b8]

[Cs]

measure: 1

2

3

4

5

6

7

8

*

0

* *

.

.

4

4

8

133


25/62

pie

9;

h, i,

/,

and

k in the music stand

for

Figure

5

(h), (i),

(j),

and

(k),

respectively.

(The

sub-phrase bracketing

is

the

same

as

in

Example

7.)

To

carry

the

reductional

process

any

further

than we

have

in Example 9, it is necessary first to develop conceptions of

the "structural

beginning"

and

the "structural

ending"

of a

group.

By

structural

ending,

we

mean

either

a

V-I

progression

when

it

appears

at the end

of a

group,

i.e.,

the

full

cadence,

or

those

variations

on the

full

cadence

known

as the

half

cadence

and

the

deceptive

cadence.

The cadence

is

designated

as a

syntactic

unit,

both

elements

of which

are retained at

the

appropriate

levels

in the

time-span

reduction,

with

the

first

element subordinate to the last.l1

By

structural

beginning

we

mean the

most

stable

event

early

in

a

group

in

which

there

is a structural

ending.

There

must

normally

be

at least

one

intervening

event

(the

"mid-

dle")

between

this stable

event and the cadence

if the former

is

to be

designated

as

a

structural

beginning.

Thus the

smallest

grouping

levels,

such as those

specifying

motives,

usually

do

not have structural

beginnings

and

endings;

all

groups

from

the phraselevel on up do have them.

The

structural

beginning

and the cadence

of a

group

are

specially

labeled

in the

reduction,

with

b

standing

for "struc-

tural

beginning"

and

c for "cadence."

They

dominate

hier-

archically

all other

events

in a

group.

As

a

visual

aid,

we

place

a

number,

signifying

the

number

of measures

spanned,

within

each

grouping

slur beneath

the

music;

the

same number

ap-

pears

as

a

subscript

to

the

b

and

the c

for each

group.

Thus

in

Figure 6 each b and each c receives the subscript "4" at level

(h)

in the

reduction,

since

they

begin

and

end

four-measure

groups.

However,

the

first b and

the last

c are also

the struc-

tural

beginning

and

the cadence

for the

entire

eight-measure

group;

therefore,

at

this

next

region

of

application,

they

re-

ceive the

subscript

"8"

and

are

retained

at level

(g).

In effect

this

labeling

creates

a

double-layered pitch

hier-

archy

between

those

events

which

are b's

and c's and

those

which are not. At local levels, all other

events

in a

group

are

subordinate

to

the

group's

structural

beginning

and

ending.

At

more

global

levels,

structural

beginnings

and

endings

are

subordinate

or

dominating

by

virtue

of

the hierarchic

struc-

ture

of

the

groups

for which

they

function.

At

the end of

the

134


26/62

Example

9.

Bach:

"O

Haupt"

/

\

/

\

/

o)(\

Ab

b

(a)(a)

(b)

(b)

C;

c1

I

I

I

I

F

r

r

r

r

'

'

r

, , ,,,

r

J

J

J

J

J

J

((#

i

i

X

J-r

j

j

I

w

f,

r Pfrf

I ij jI

I

(b)i

(a)

135

?

?

?

?

*

? ?

?

?

? ?

?

?-

?

*

?

?

*

:

*J

?

.

,P

44

j j j j

-

j

.

( WI----wj 6 - p - p tp FF -1-

I


27/62

reductional

process,

there

remains

only

the structural

begin-

ning

and

the

structural

ending

for the

piece

as a whole.

We

are now

in a

position

to

investigate

a

complete

time-

span

reduction.

Example

10

gives

the

music12

and

Example

11 represents its analysis. By this point, Example 11 should

on the whole

be

self-explanatory.

The

following

remarks are

in the

nature

of

annotations.

1.

The best

way

to "read"

Example

11

is first to examine

the

grouping

and

metrical

analyses,

then

to hear the

levels in

rhythm,

as notated

beneath

the

music,

in

their reductional

order.

If the

analysis

is

correct,

each level should

sound

like a

natural simplification of the previous level. Any difficulties

which

the

reader

initially

has

in

deciphering

the

tree

can be

cleared

up

by

a

step-by-step

comparison

with

the notation

underneath.

2. Level

(f)

in

Example

11 has

already

been reduced

from

the actual

music

(Example

10)

to what

is felt

to

be

its

small-

est beat

level.

This

is a

convention

which

we

always

follow

in

constructing

the

time-span

and

prolongational

reductions

for

a piece, partly because syntactically unimportant detail is

thereby

eliminated,

and

also because

the

two

reductions

do

not

significantly

diverge

beneath

this level.

3. It would

be

possible

to

assign

further

low-level

grouping

slurs within

these

eight

measures.

However,

since

the

indica-

tions

for

these

groupings

are somewhat

conflicting,

and

since

such

groupings

would

not

affect

the

analysis

as

a

whole,

we

choose

not to

assign

them.

Therefore,

the weak

beats

within

the lowest groupingslurs are simply bracketed as afterbeats.

4. The

structural

beginnings

and

cadences

are labeled

at

each

level,

but

they

do

not receive

subscripts

until the

reduc-

tional

process

has reached

the

grouping

levels

for which

they

function.

5.

The

selection

of which

events

are

dominating

is

straight-

forward

except

in mm.

3 and

7 at

level

(d).

In m. 3 there

is a

conflict

in

the

preference

rules between

the

metrically strong-

er position of the F-sharp-E-A

chord and

the

more

stable

structure

of

the

V6

chord.

The

F-sharp-E-A

chord

is

chosen

for

reasons

having

to do with

the

phrase

as a whole:

the

regu-

larity

of harmonic

rhythm

is

preserved,

and

a

descending

line

in

the bass

from

the

tonic

to

the dominant

is created.

The

136


28/62

Example

10. Mozart: Sonata

K.

331,

I

Andante

grazioso.

A I

(ŝa rr%p-r

f*-*

a

(-)


29/62

K,

,

I

k I K .

.

I

K I

.

f,

P

rr

rr

r

r

G

rPt

d

r

Gr

r

'r

rr,

r

P

^r2i

3

,r

rr

r

r

P

r

'

r

pr

p

pC

r

cJ

4

8

(e) on>

I

I

i

ii i

,

I

i

.

i

I

i1

[)

g

~

J

2

_

2

'

tb 2 2 2 2t

c

J

(C)

*'Io

|I,

1,

;

B

114)

(C

1^^

C4

i

4

4

4

ca

,

aYI

8c

c

-

~~~~~~~(a)

,a>ll D

I b 3

r4

C

138

Example

11

n.,

A I

k I

K

I

I

I


30/62

same

chord is

chosen

in m.

7

because

of

the

parallelism

with

m.

3.

6. Note that if the

cadences were

not labeled and

retained,

in

m.

4 the

I

chord instead of the

V

chord would have

been

selected at level (d). The labeling also causes the retention at

levels

(d),

(c),

and

(b)

of

both elements

of

the

full

cadence

in

m. 8.

7. Levels

(c)

and

(b)

can

be construed

as

Schenkerian

"background"

evels.

Schenker's own

analysis

of

this

passage13

gives

the

melodic structural

weight

to

the

E

in

the

first

mea-

sure rather

than to

the

opening

C-sharp.

In

our

theory,

it

would

be

possible

to achieve his

result

by

applying

the

"arpeg-

giation rule," i.e., by regardingthe first measure as a broken

chord

and therefore

compressing

it

into one

event

placed

on

the

downbeat

of

the measure.

However,

such

a

decision

pro-

duces difficulties

at

intermediate

structural levels

(especially

level

(d)).

8.

In

the

large

levels

(b)

and

(a),

the sense of

metrical

struc-

ture

has

faded to the

point

that it is

largely inoperative.

The

events

at these

levels in the

notation

beneath the

music

do

not receive rhythmic values because there are no more dots in

the metrical

analysis

to which

such

values

could be

related.

It

is at

this

point

in

the

reductional

process

that

pitch

events

can

be

thought

of as

rhythmically

free-floating.

9.

It

would

perhaps

be

sufficient

to

stop

the

reductional

process

at level

(b).

Level

(a)

simply

selects

the

most

stable

event

from

those

available at level

(b).

As is

often

the

case,

the

most

stable event here

is the

last

chord; thus,

the

rest of

the piece is a left branch to it. This situation can be inter-

preted

to

mean

that,

in a

sense,

all

the

events of

a

piece

except

the

last

constitute a

delay

of

the

moment of

complete

repose,

which

is

the

ending.

If

the

point

of

maximal

stability

happened

in

the

middle

of

a

piece,

there

might

be

no

reason

for

the

piece

to

continue.

10. We

have

considered these

eight

measures

as

if

they

were

a

complete

piece.

If

the

entire

theme,

which is

18

measures

long,

were

analyzed,

the

opening event would eventually be

labeled

b18

and thus

would

dominate all

other

events in

the

first

eight

measures.

11.

Observe

how the

geometry

of

the

tree,

while

accurately

conveying

the

hierarchy

of

pitch

events,

also

mirrors

visually

139


31/62

the

partitioning

of the

piece

into

time-spans

(groupings

and

bracketings).

Moreover,

the tree

notation makes

visually

clear

the interaction of the

pitch

structure

with

the

metrical struc-

ture at all

pertinent

levels.

There

are

important

hierarchic

aspects

to

the

pitch

struc-

ture

in

Example

10 which do not

emerge

in the

time-span

reduction

in

Example

11. To

capture

these

aspects,

we must

develop

the

domain

of

analysis

called

the

prolongational

re-

duction.

In the

parameter

of

rhythm

in this

domain,

events

come before or after other

events,

but

they

are not

measured

according

to some metrical

conception.

The

relevant

notion

of elaboration is elaboration by harmonic and melodic con-

nection.

There are two

kinds of elaboration

in the

prolongational

reduction:

prolongation,

in

which

a

pitch

event

is

elaborated

into

two

or

more

copies

of

itself;'4

and

contrast,

in

which

a

different,

relatively

ornamental,

pitch

event is

introduced.

A

prolongation

is

represented

by

two branches

extending

from

a

small circular

node,

as in

Figure

7a. Neither event takes hier-

archic priority; rather,both events are thought of as an exten-

sion over

time of what

at a

deeper

level is the same event.

In

contrast,

hierarchic subordination

is

designated by

right

and

left

branching,

as in the

time-span

reduction.

However,

where-

as

in the

time-span

reduction

these

branchings

only

indicate

the

subordination

of

one event to

another,

in

the

prolonga-

tional reduction

they

receive

a further

interpretation.

A

right

branch

signifies progression

in a

piece,

whether

at

a

local

level

as in Figure 7b or at a large structural level. A left branch is

utilized

only

at local

levels and

signifies

delay

in

relation

to

the

bass,

as

at level

(c)

in

Figure

7c.

(Since

metrical

structure

does

not

play

a

role in

the

prolongational

reduction,

Figure

7c

represents

a

suspension

considered

only

with

respect

to its

pitch

structure.)

The well-formedness

conditions

and

the

preference

rules for

the

prolongational

reduction

are

similar

to

those for

the

time-

span reduction.

The well-formedness

conditions

preclude

the

crossing

of branches

and

the

assignment

of more

than

one

branch to

the

same event.

Given a

sequence

of

pitch

events,

the

preference

rules

select

a

hierarchy

according

to

principles

of

140


32/62

Figure

7a

(a)

(IM

--

_

-

U

*--

(a)

(b)

(b)

J>I^vwv-Y

I

-

I

||

[ba]

[C4]

[b,]

[cs]

measure: 1 2 3 4

5

6 7 8

time-span

reduction

time-span

reduction

(a)

>

(a)

(b)

)

(b)

(b)

>

(c)

>

(c

(C)

)

(C

I- s- ^-y

I-

^^

--1[

1 2 3 4 5 6 7 8

prolongational

reduction

7b

7c

(a)

(b

O

1

(a)

(b)

I

*?

Figure

8

141

i .

1i=

-I1dn

.

f

Ii

11


33/62

stability largely

resembling

those

for the

time-span

reduction.

Meter is

disregarded;

prolongations

are

maximized.

At

any given

level down

to the

phrase

level,

the

sequence

of

events

available

to

the

preference

rules

for the

prolonga-

tional

reduction

is

precisely

the same

sequence

as at the

equivalent

level in the

time-span

reduction.

In other

words,

the

hierarchy

of

b's

and

c's as

designated

in the

time-span

reduction

heavily

determines

the

prolongational

reduction.

Again,

we illustrate

with

a

typical

antecedent-consequent

pattern.

In the

time-span

reduction

in

Figure

8,

levels

(a),

(b),

and

(c)

refer

to events

labeled

as

structural

beginnings

and

endings;

these events

at

equivalent

levels become

the material

for the prolongational reduction in Figure 8.

Even when

precisely

the

same events are

available

at a

given

level

for

the two

kinds

of

reduction,

the reductions

draw

radically

different

connections

among

these events.

For in-

stance,

in

Figure

8

the

V chord

in the

full cadence

in m.

8 is

a

left branch

to the

ensuing

I

chord

in the

time-span

reduction,

but

is a

right

branch

from

a

prolonged

I

chord

in the

prolon-

gational

reduction.

In the

former

case,

it

is

a

left branch

be-

cause it is within the time span of the final tonic; in the latter

case,

it is a

right

branch

because

it

progresses

to

the

final

tonic.

(In

both

cases,

it

is subordinate

to

the

tonic

according

to

principles

of

stability.)

Large-scale

right

branching

in

the

prolongational

reduction

always

indicates

significant

syntactic

"progress"

n a

piece.

Note, too,

that

at

level

(c)

in

Figure

8

the

prolongational

reduction

brings

out

connections

of harmonic

identity

not

captured in the time-span reduction. The tree for the time-

span

reduction

would

look

the

same even

if the

b

for

the

consequent

phrase

were

an

entirely

different

chord;

in the

prolongational

reduction,

however,

such

a

change

would

pro-

duce

a

right

branch

at

level

(c)

instead

of the

prolongation

represented

there.

On

the

other

hand,

the

tree

for

the time-

span

reduction

expresses

grouping

structure, something

not

conveyed

in the

prolongational

reduction

itself.

Thus,

for

example, the tree for the prolongational reduction cannot

differentiate

between

the

full cadence

in m.

8,

and

the

phrase-

ending

half cadence

followed

by

the

phrase-beginning

tonic

chord in

mm. 4-5.

142


34/62

From the

phrase

level

down

to the smallest beat

level-that

is,

for all events

which

are not

b's and

c's

except

for the

most

local details-the

prolongational

reduction

is

constructed

without

reference to

the

time-span

reduction.

As

a

result,

within this

region

the two reductions

usually

differ not

only

in the connections

made but

in the actual

sequences

of

events

at

corresponding

levels.

To facilitate

reading

the

prolongational

reduction,

we

give,

beneath the music in a

non-rhythmic

notation,

the

actual

sequence

of events for each level from

the

phrase

on

down.

(See

Example

12.)

Unlike the

tree, however,

this

secondary

notation

does

not

convey

the

types

of

elaboration within a

sequence of events.15For each level from the phraseon up, it

is

sufficient to mark the events

which,

as

b's

and c's

in

the

corresponding time-span

reduction,

have

caused the

equiva-

lent

sequence

of events in the

prolongational

reduction;

this

marking

is

accomplished

by

placing

the

corresponding

letters

in

parentheses

at the

appropriate places

just

below the

music.

Example

12

represents

the

prolongational

reduction

for

the first

eight

measures of

Mozart's

K.

331.

Remarks:

1. The most local level in the

prolongational

reduction

(level

(h))

corresponds

with

the

"lowest

beat level"

as

deter-

mined

in

the

time-span

reduction

(Example

11,

level

(f)).

In

placing

this

lower

boundary

on

the

prolongational reduction,

we are

in

effect

claiming

that

beyond

this

level

local detail

is

not

of

prolongational

significance.

2. Level

(a)

represents

the level

of

abstraction

at

which the

A

major

root

position

triad is

totally

unelaborated. The

high-

est level of the prolongational reduction for any classical tonal

piece always

results in

an

undifferentia

jackendoff toward a formal theory tonal music

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