beat-class modulation in steve reich's music

8/18/2019 Beat-Class Modulation in Steve Reich's Music

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Society for Music Theory

Beat-Class Modulation in Steve Reich's MusicAuthor(s): John RoederSource: Music Theory Spectrum, Vol. 25, No. 2 (Autumn, 2003), pp. 275-304Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/3595433

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PBeat-Class odulation

n

Steve

ReichsMusic

JOHN

ROEDER

A beat-classmodelof rhythm,employedby Cohn andothersto analyze extural orm in Steve

Reich's

arlyphase-shifting ompositions,

s

here

enlarged

o embrace he

concepts

of

beat-class

"tonic"

nd

"mode,"

efined

formally

by

analogy

o

pitch-class

onality.Using

these

concepts,

analyses

f

Reich'smore

recent

music-Six

Pianos,

New York

Counterpoint,

nd

The

FourSections-

demonstrate

ow

form-creating

rocess

of

pitch

and

rhythm

result

rom the

specific

manner

n

which

repeated atterns

rebuilt

up,

varied,

nd

combined

polyphonically.

WIDELY

PERFORMED,

IMITATED,

AND

anthologized,

Steve Reich's"minimal"music of the 1960s and

early

1970s

proved

surprisingly

usceptible

to

a

model

of

rhythm

developed

for

very

different

music. It

was

in

the

context

of

twelve-tone

composition

that Milton

Babbitt'

first

proposed

conceiving

rhythm

analogously

to

pitch

by

using

the

integer

residuesmodulo 12 to

represent

the metric

location

of

event

attacks

(rather

han the events'

durations,

as

did the Darmstadt

composers).

Later

scholars

applied

the

concept

of

set

to the

rhythms

of non-serial

music;PressingandAnku,for instance, reatedworld musics

that were

inspirations

for

Reich's

compositions.2

But

the

most

detailed

analytical

application

of

this

rhythmic

model

was

Richard

Cohn's

study

of content and

large-scale

form

in Reich's

Phase

Patternsand Violin

Phase.3

Each of these

"phase-shifting"

pieces,

like a

canon,

combines a

repeated

pattern

with a

delayed

statement

of

the same

pattern

n an-

other voice. As the

piece

progresses,

he

temporal

interval

of imitation

between

original

and

imitated voices varies

systematically,

rom

one beat

up

to the whole

length

of the

pattern.

Noting

the "formal

esemblances

between the struc-

tures of metric cycles and the twelve-pitch-classuniverse,"

Cohn

pursued

the

consequences

of the idea that "much of

the

technology

developed

for

atonal

pitch-class analysis

is

transferable

o

the

rhythmic

domain."

Adopting

terminology

suggested

by

Dan

Warburton,4

e

represented

ach

repeated

pattern

as a beat-classset-a

rhythmic

analog

of

a

pitch-class

set-that

denotes which beats

are attacked

in

the

pattern.

This model facilitated

analysis

of

the

varying

attackdensities

that result

from the

systematic phasing

of

beat-class

sets;

specifically,Cohn analyzedhow density in these pieces de-

velops

toward

and

away

from

saturation,

or

the

"beat-class

aggregate,"

n

which

every

beat is attacked.

Formally,

ener-

ating

the beat-class

aggregateby

phasing

a

particular

beat-

class set

against

itself is

analogous

to

generating

the

pitch-

class

aggregate by taking

the union

of

transpositions

of

a

particular

pitch-class

set. Cohn's

paper

demonstrated

how

the

large-scale

textural

design

of

these

pieces

could

be un-

derstood,

by

considering

processes

analogous

o

the

transpo-

sitional combination

of

pitch-classsets,

to manifest

proper-

ties of the small-scale

beat-class sets

themselves.

Babbitt 1962.

Pressing

1983,

Anku 1988.

Cohn

1992.

4

Warburton

1988.

275

I

2

3



BEAT-CLASS MODULATION

IN

STEVE

REICH

S MUSIC

63

16

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1681

1691

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11:

:111:1 :111:

1 .

mf

E

Ronic-

"dorian"

mode "build-up"

(6-10Ox)

(6-lOx) (6-lOx)

l

11:

-

111

-

:11:

-

: 111:

-

~

,

11

,

:111:

-

(6-lOx)

(6-lOx)

(6-lOx)

l ##1l

-

111

-

111

-

111 111- S\} 11

}

111

Q2

(24x)

(2-4x)

(2-4x)

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:I

1

:I1

n

mf

pc

content:still

{

1,2,4,6,7,9,B

}

I

ncipits

on beat-class

41

EXAMPLE I.

[continued]

beat class sets that are not

transpositionally

elated.

Patterns

change

content

during

some

pieces,

and some

pieces

super-

impose patterns

of

differing

content

and

periodicities.

Tex-

ture is

also

freer.

Ensembles

are

larger

and more

diverse,

and

individual

parts

fade

in

and

out.

Pulsing large

chords,

often

partitionedinto overlappingand shifting components, ap-

pear

simultaneously

with

phased patterns,

or

alternating

with them.

The form

of

these

more recent

compositions

s not

simply

a matter of

beat-class-aggregate

formation. Reich

himself

describes orm

in terms of

changes

of

mode and

key,

devel-

opments

of

timbre

and

register,

chord

progression,

tempo

modulation,

and

metric fluctuation.7

His abandonment

of

phasing

for other

formative

processes,

while still

maintaining

the

repeatedpatterns

of his earlier

music,

raises

some

inter-

esting questions

about his current

technique.

What

function

do these

patterns

play

in the more

variegated

textural

and

harmonicdesigns?What motivatesthe particular hoices of

pitch-transposition

and beat-class

transposition,

or,

more

generally,

how

are tonal and

metric

processes

coordinated?

This

paper proposes

some

ways

of

answering

hese

ques-

tions

by developing

a

model

that shows how

both

tonality

7

Reich

1977,1986,

and 1991.

1

2

3

4

5

6

1741

11=:

:l

tl(Q2)

:~

I

:I

-

L :

:L

-

----

]

_)

:

11

^ - A - ' 7

- t

277



MUSIC THEORY SPECTRUM

25

(2003)

and meter

depend

on

pitch,

harmonic,

and other accentual

features

of the

patterns

as

they

are

combined

polyphonically.

First,

an informal

examination

of Reich's

transitional

music

of the

early

1970's

motivates

the focus on accent. Formalism

is then

developed

to

represent

how accentscombine,defin-

ing

the

percepts

of

beat-class

"tonic"

and "mode."

Excerpts

from

two

of

Reich's

mature

works

from

the 1980s will be

an-

alyzed

to

show

how their

pattern

combinationsare

designed

to

produce

large-scale

modulations

of

pitch-class

and beat-

class

tonics,

and

thus to create

musicalform.

The

role of

accent in

large-scale

process

is evident from

even a cursorylistening to Reich'stransitionalpieces. Ex-

ample

1

shows

a

representative

excerpt

from Six

Pianos

(1973).

As

it

begins,

at

R55,

all

instrumentsare

playing,

and

the

pitch

relations

among

their

materials

are clear.Pianos

1,

2,

and 3

repeat

distinct

eight-beat

patterns,

abeled

Q1,

Q2,

and

Q3

respectively.

Q1

is

an exact

pitch transpositionup

a

perfect

fifth of

Q2. Q3

doubles the

highest

three

pitches

of

Q1

an octave

lower,

but

substitutes

D3

and

A3

for

Ql's

F#4

and

B4.

Imitation is

evident

in

two

other

parts.

Piano 4

plays

the same patternas Piano 3 (Q3) but one eighth-note beat

later.

n terms of

beat-class

theory,

his

canon

can

be

symbol-

ized as

tl(Q3),

where

tn

signifies

"time

transposition delay)

by

n beats."

This

paper

uses

lower-case to minimize confu-

sion with

pitch-class

transposition,

upper-case

T.)

Similarly

the

pattern

played

by

Piano 5 can

be

expressed

as

t6(Ql),

that

is,

as

the

pattern

of

Piano

1

delayed

by

6

eighths.

As the

music

continues,

some

clear

pitch processes

emerge

from these

specific

time-

and

pitch-transpositional

relations. Although all parts draw their pitches from the

same diatonic

scale,8

he dense

imitation

might

seem

to fore-

stall the

emergence

of

any

one of the

pitch

classes as a

tonic;

indeed,

on

any given

beat most

members

of the

collection are

8 Since

Q2

is a

5-23[02357]

diatonic

pentachord,

its combination

with

its

T7

transpose,

Ql,

yields

the diatonic

heptachord

[1,2,4,6,7,9,B).

attacked.

Nevertheless,

the

registration

and

rhythm

of the

pitch

classes

up

until R60 establish

D

as

a tonic

or,

at

least,

as

a

persistent

chord root.9

Specifically,

he lowest

pitch,

D3,

and the

highest,

F#5,

suggest

the

constant

presence

of a

D

major

riad;both

pitches

are

always

approachedby leap, giv-

ing

them stress

and

therebysuggesting

that

they

function

as

stable chord

tones.The

priority

of

these

pitch

classes

is also

enhanced

by

their metrical

regularity:

ne of them is

attacked

every

quarter

note due to

the

particular

ntervalof imitation

between

pianos

3 and

4,

and between

pianos

1

and 5.

Starting

at

R60 the

same diatonic

collection

is main-

tained,

but

a

new

tonalitybegins

to

be established

by

changes

that

shift

emphasis

to different

pitch

classes in the

collec-

tion. The changesare indicatedby annotationson Example

1.

First,

at

R60,

the

low-register patterns

that accented

D3

fade out.

Then

at

R64 Pianos 1

and

3

begin patterns

that,

although

similar

n

contour to

Ql

and

Q3

and use

the same

collection,

place

Es

at the

registral

xtremesof the ensemble.

Accordingly,

here is a modulation

to

E

dorian,

mediated

by

the unvaried

Q2.

Some

metrical

ambiguity

is evident

especially

during

R55-60,

as

the

pianos engage

in the imitation described

above.'lTwo differentmetricalinterpretations f the passage

are

analyzed

n

Example

2,

which

shows

the combination

of

all voices at

R59 and

labels the

eight

eighth-note

beats

with

integers

from

0 to

7,

following

the

conventions of beat-class

theory.

Attending

to

the lowest notes

in the

texture,

one can

hear

pairs

of

D3s

repeated

n

a

rhythm

of 5+3

eighths.

(The

fast

tempo,

quarter

=

192,

makes the

second of each

pair

dif-

g

Reich

names

the tonalities

analyzed

here

in his foreword to the

score of

Six Pianos

(1977).

Io Other

analysts

have

noted similar

metric

fluctuations

in other music

by

Reich. Cohn

(1992)

remarks

that the

downbeat

"floats" n

some

of the

phase-shifting pieces,

and Gretchen Horlacher

(1994)

has documented

several

intriguing

instances of

metrical

ambiguity

and

process

in Reich's

later

works.

The transitional

Musicfor

Pieces

of

Wood

provides

another

clear

example

of how Reich's interest

changed

from

phasing

to the

build-up

of

canons

involving ambiguities

of

downbeats.

278



BEAT-CLASS MODULATION IN STEVE REICH S MUSIC

Interonsetdurations downbeat?

in

the

F#5

stream:

- --

J-

--._

.

_

-

---J.

--

beat class:

0 1 2 3

4 5

6

7 0 1 2 3

45

6

7

leaps

to

registral-boundary cs

'_ 44

.

*

.

444:

v t

3

#v

v

y

v v v

Cp

v

7

7

v

P P

registral-boundary

'

pc

is

attacked

very

I

]

quarter

note

Interonset

durations

J,

____-_,

-J.

X

_.--. ------

_

-

_--

-

in the

D3

stream:-

t

downbeat?

t

EXAMPLE

2.

Pitch-classmphasis,ulse,andcompeting

ownbeats

in SixPianos,

R59.

ficult to hear as

a

distinct

event,

and the first of each

pair

is

introduced

by

leap,

making

the onset of the first more

marked.)

The

greater

regular

accent,

and so the sense of

downbeat,

accrues o

the onset of the

longer

of these

two in-

teronset durations,5, which alwaysoccurson beat-class 0.

The

second

interpretation

attends to the

highest pitches,

where one

could hear beat-class

4

as the

downbeat

since the

longer

member of

the

repeated

interonset-duration

series

2+6

regularlybegins

then.

The

downbeat

ambiguity

resolves

abruptly

at

R61,

when

pianos

3 and 4

drop

out. But the sense of

beat-class

4 as

an

alternative

downbeat

returns

soon after the

modulation,

as

shown

in

the

latter half of

Example

1.

The

build-up

in

pi-

anos4 and5 starting n R67 regularlyaccentsbeat-class4 as

the

beginning

of a

group

of

eighth

notes,

even

though

the

pattern

when

completed

(in

R74)

turns

out

to

be

a

beat-

class-transposition

f

piano

2

by

one

beat,

not four.

This

analysis

suggests

that the

questions

of

rhythm

and

pitch surrounding

Reich's

recent music

may

be addressed

by

considering

the

function of accent

in

the

repeated

patterns.

To focus

the

inquiry

further,

and to

establish

a

basis

for a

more formal

and

precise

model

of

accent,

let us examine a

more recent

composition.

The

passage

shown

in

Example

3 occurs

during

the first

movementof New YorkCounterpoint1985). It beginswith a

single

clarinet

presenting,

without

build-up,

a

repeatedpat-

tern

lasting

12

eighth

notes.

(Reich's

score

is

written

in

B

b,

but

for convenience

I

will referto the

pitches

as

they

are no-

tated,

not as

they

sound.)

As

above,

beat

classes are labeled

conventionallyby

integers,

with beat-class

(bc)

0 as the first

beat

in each measure.

Since

the zeros indicate

notated mea-

sure

beginnings,

bar lines

may

be omitted

for

clarity

in this

and

subsequent

examples.)

Thus the

repeated

pattern places

attacks on the set of beat classes[0,4,5,7,9,11), which I will

call

Q1.

In

R8-R33

a

six-voice texture

develops

that is

imitative

but

not

exactly

pitch-canonic.

It

proceeds

in

two

stages.

During

R9-R19 two more

patterns,

abeled

Q2

and

Q3,

are built

up loudly,

then faded and

transferred

o

other

voices. Their

build-ups

are

irregular

and

rapid,

not

gradual

and

attack-by-attack

ike

those

in

Six Pianos.

Although

these

279




25

(2003)

voices have

the same

pitch

content,

their

pitches vary

in

order and

duration;

or

example,

n

Q1

the

EL6

s

long

and

followed

by

a

short

G5,

while in

Q2

it is short

and followed

by

a

long

B65.

Nevertheless their beat-class sets are

transpo-

sitionally

related: Q2, {0,2,4,5,9,10}, is t5(Ql), and Q3,

[0,1,3,5,7,8},

is

t8(Ql),

that

is,

t3(Q2).

The

combination

of

these

transpositions,by

the

way,

does not create the beat-

class

aggregate,

or

beat-class 6 is

never attacked.

In the second

stage

of this

excerpt,

R20-R33,

three more

patterns

enter,

abeled

Q4,

Q5,

and

Q6

on

Example

3. Each

pattern

rapidly

and

irregularly

uilds

up

a beat-classset that

is identical to

a

pattern

n

the first

stage-Q4

builds

up

the

same

beat-class

set as

Q1,

Q5

builds

up

Q2's

set,

and

Q6

Q3's. So, the same beat-class sets are built up in the same

order,and,

moreover,

he

beat-class

aggregate

s

not attained

at the end of the second

stage

either.

However,

the

pitch

content of these later

patterns

is different and

generally

lower than that of the

originals.

These

differencesarise from

a

specific

relation

among

the

patterns:

each

pattern-pitch

n

the

second

stage

is a

tenth below the

pitch

at

the same beat

class

in

the

corresponding irst-stage pattern.

(The

few ex-

ceptions

to this rule

are necessitated

by

the limited

range

of

the clarinets,and yet also contributesignificantlyto large-

scale

process,

as will be

shown.)

Confronted with this

evident

compositional

scheme,

we

can

focus the

questions

raised

earlier.

Since the

ending

com-

bination is not the

aggregate,

what

are

appropriate

ways

to

characterizethe

rhythmic

form,

if not in

terms of

aggre-

gates?

And since the

imitative

processes

are

not

strictly

canonic,

what

design

regulates

or results from

the

specific

ways

that the

patterns

build

up

and

vary

their

content and

their time-

and

pitch-transpositional

elations?

As was the case

with

Example

1,

it seems to

me

that

all

these

questions

can be

answered

by attending,

in

detail,

to

the

accentual

properties

of

the

patterns

and of

their

combi-

nations,

and

by

modeling

them

appropriately.

Rather than

treating

all

attacks

in

a

pattern

as

equally weighted,

as

in

previous

beat-class-set

theory,

the model should

incorporate

the

accentual

distinctions

that

pitch

and

rhythm

create

among

them.

Although no previous research has attempted such a

model

specifically

or

Reich's

music,

recent

rhythmic

theory

provides

a sound basis

for

such an

investigation,

by clarifying

the nature

and

typology

of

accent.1l

It defines accent

as

a

perceivedemphasis,

at

a

point

in

time,

that

may

arise in at

least threedistinct

ways:

from

perceivedchanges

n

pitch,

du-

ration,

loudness,

and

in

more

complex

musical

processes

of

harmony,

imbre,

and

texture;

rom

expectations

of

regularity

such

as

meter;

and from

the

perceivedfunction

of the events

at that timepoint in the structure f melodic and harmonic

segments.

This

general

conception

suits Reich's music

fairly

well,

but it will be

necessary

to define

the various

types

of

accent

much more

specifically,

n order

to

understand

heir

interactions

and contributions

o

rhythmicprocess.

To

begin

this

task,

Example

4 defines "intrastream"c-

cents,

meaning

accents that

arise

within each individual

voice

in

a

texture

(more

complex types

of

accent,

such

as

changes

n

registral

density,

which

result rom the interaction

of all concurrentvoices, are also important,and will be dis-

cussed

below).

The definitions are

expressed formally

for

precision,

and

in

order

to

distinguish

accents that are

specific

to

Reich's

monophonic

patterns

from more

general types.12

Each

is

instanced in

Example

5(a),

which

analyzes

the

ac-

centual structure

of

Q1.

*

An accent

of

climax

appears

at the onset of an event

whose

pitch

exceeds those

of

the

preceding

and subse-

quent

events.

In

Example

5(a),

the first

E&6

oes

not take

such an accent,since no event precedes t, but all subse-

quent

El6s

do.

So

do

all

B%gs,

ince

each is

preceded

and

followed

by

lower

pitches.

11

Berry

1976,

Lerdahl

&Jackendoff

1983,

Kramer

1988.

12 Some of

these

definitions formalize

verbal

descriptions

such

as

those

in

Lerdahl

&Jackendoff

1983,

17.

280




25 (2003)

I30

31

f

6V

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' 1 7 ~ ~

1

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r-_-= ,7S

A

r

W

bb Y

r

Y

7

7

4

I

A_

-

__d_ __ .-m

mf

M*

-^

-

bc

tonic

321

reverts

o bc

0

.

I

E331

Q6

=

Q3

Live

-|

-lb

_,

?

.

L<+

Cl.

' ' 1

7

-

Cf2,

5

-d^^f

yt

fXAML

Y3.f

cni

Cl.CI.~~~

~EXAMPLE

3.

[continue]

EXAMPLE

.

[continued]

*

An accent

of

nadir

appears

at each onset of each event

whose

pitch

is

equal

to

or

lower than the lowest

pitch

so

far,

and that is lower than the

immediately

preceding

and

following

events.

Thus,

in

Example

5(a)

an accent of

nadir

appears

at

each

onset

of

F4,

since it is the

lowest

pitch

in the

passage.

At

higher troughs

in

the

contour,

such

as at the

onsets of

Al5,

there

is no such

accent.

*

An

accent of

(interonset)

duration

appears

at the onset

of an

event

that is

much

longer

than

the

preceding

event,

or

when

the

time

to

the

next

onset

is

much

greater

than

the

time

sincethe last onset.13

13

The tenuto marks on the

score

are

interpreted

here

simply

as

directing

the

performer

to

hold the

note

for

its

entire notated

value.

Any dynamic

Q5

=

Q2

29h

A

I

Live

C1.

CI.

1,4

Cl.

2,

5

(bc 8)

build-up

of

Q6

9)

A.-

A

^

FI4

1

Cl.

(0

blib

rv

P

pV

r

1

^

p

>

V

r Vr

C CV r

,

V

284

&

-'

^

'*_

^

I



BEAT-CLASS

MODULATION IN STEVE REICH'S MUSIC

Given a

monophonic

stream

S

presenting

a series

of n

non-overlapping

vents

of

the

form

(pitch,

duration,

imepoint

of

attack):

S

=

((pl,dl,tl),

(p2,d2,t2),

(p3,d3t3)..

.,

(Pn,dn,t))

such

that,

for all i

(1-i<n),

t+1

t.+d..

Quantifythe pitchesPiacording o the integermodel of pitch (Rahn 1980), and model pitch differences intervals)as integers.

Find a

duration

of

which

every

timepoint

t. and durationd.

can be

expressed

as

an

integer

multiple. Quantify

this

duration

as

1,

and

quantify

he

ti

and

di

accordingly

as

integers.

At t. there is

an

accent of

symbolized

by

iff

Climax

C

Pi

>

Pi,_

and Pi

>

pi+l

Nadir

N

Pi

<

Pi-1

and

Pi

<

Pi+l

and

pi

<

pj

for

1

<

j

-

i

(Interonset)

Duration D d. >> d or ti - ti > t. - ti

Subcollection

shift

S

There is an

integer

k

<

i suchthat 0

<

Pi

-

Pi k

I

<

2

(semitones)

and

there is

noj:

i-k

<

j

<

i such

that

0

<

I

Pik

-

Pj

I<

2

(semitones)

Beginning

of

B

(local)

ti

-

ti_

>

1,

and there exists

m

>

i such

that

for

all

j:i

j <

m,

d.

=

1

connected series

andt.

=t.

+

1

J+1 J

Pulse

There is an

accent

of

one

of the

types

defined above at t.

-T and at

t.

-2T;

or

there is a

pulse

accent

at t.

-

T

and an

accent

of one

of the

typesdefined above at t - 2T andat t. - 3T

Attack

Pi

exists

[an

event

(not silence)

is

attacked

at

ti]

EXAMPLE

4.

Typesof

intrastream accent

in

Reichs

music.

*

Accents

of

subcollection

hift

originate

in

the

special

pitch

context

of

Reich's music:

diatonic scales

organized

into

rooted triads

that are

extended,

as in

jazz, by

tertian

"tension

tones."

In

the

patterns

Reich

composes

from

such

collections,

he

change

from

a

givenpitch

to

an

adja-

emphasis

added

by

the

performer

would,

of

course,

increase

the

accent

on the note's

onset.

cent

pitch

in the diatonic

scale marksa

change

of

harmony,

more than

do

leaps,

which often

simply

extend the

pre-

vailing

tertian

sonority

without

changing

the

root.14Ex-

ample

5(b)

illustrates

such a

change

within

Ql:

once

the

14

The rooted subcollections

I am

positing

to

underlie

Reich's music

may

thus be

understood,

in William

Benjamin's

(1984)

terms,

to

constitute

"images"

whose

"shift"create

accent.

285



BEAT-CLASS

MODULATION

IN

STEVE

REICH S MUSIC

*

Regularly

epeating

durationsmarked

by

accent nduce

a

pulse

stream,

which itself accents

timepoints metrically.17

For

instance,

a series of

equal

durations

n

Q1

quickly

es-

tablishes

a

half-note

pulse,

as follows:

First,

the

accents

on beat-classes0 and4 projecta half-note duration, tart-

ing

from beat-class

4,

that

is

expected

to

be realized at

beat-class

8.18

Although

no event marks

beat-class

8,

the

recurrences

f accent a half-note

later,

on

the

next beat-

class

0,

then

again

on the

following

beat-class

4,

confirm

the half

note

as

a

repeated

duration,

and

so

createsa

pulse

stream.

The

stream

is

symbolized

in

Example

5(a)

as

a

horizontalline

linking

vertical strokes that

denote

when

pulse

accents

occur,

according

to the formal definition

given in Table 1. Isolatedpulse accentsmay also be pro-

duced,

under the

given

definition,

without

linking

into

continuous

streams;

n

Ql,

pulse

accent

appears

on

beat-

classes

9,

11,

1

(since

9 and

11

are

accented),

and

3,

but

the accents needed to

establish a

continuous

quarter-note

streamare

crucially acking

at

beat-classes5

and 7.19

Although many

of

these definitions are

consistentwith

other theorists'

reatment

of

accent,

I do

not

intend

their for-

mality

to

suggest

that all

these accents are

aurally

salient

in

all music. Nadir accent,for example, s arguablynegligible n

the

more usual

styles

of music that

presents

a

given melody

only

once or

twice.

These

accents can be heard

in

Reich's

(1983,

51-2).

But in

passages

dominated

by

the

build-up

of

patterns,

this makes

a

very

minor contribution.

17

More on the nature of

pulse

streams

can

be found in Roeder

1994. The

concept

of

pulse "layers,"

reated

most

thoroughly

in

Krebs

1999,

is sim-

ilar,

although

it

is

not

usually

construed

as

a source of metrical

accent.

I8 The

conception

of durational

"projection"

s taken from

Hasty

1999,

although

it

is not

part

of his

agenda

to

explain

its

connection

to tradi-

tional notions of

metrical

accent.

19

Under this definition

an

event

does not

take accent

simply

because

it

is notated

on

a

strong

beat. This

seems

consistent

with

practice:

per-

formances

of Reich's

music

supervised by

the

composer

do

not stress

notated

downbeats.

music,

however.

ndeed,

it is

precisely

he unusual

eatures

of

his music-its

repetitiveness

and

redundancy-that

permits

the listener

to

focus on

such

accentual ubtleties

as

nadir,

and

then

to

consider

their

participation

n

distinctive,

arge-scale

rhythmicprocesses.The formal definitions provide a basis

for

a

precise

description

of

rhythmic

form,

as

we shall

see,

andalso for

the evaluationof such

descriptions.

The

analysis

n

Example

5

shows

how the distribution

of

accent

among

the beat classes

in

Ql

varies in both

quality

and

quantity.

Some beat classes

take

more

types

of accent

than

others,

as demonstrated

by

the

tally

in

Example

5(c).

Beat-class

accentuation

also varies

over time:

some beat

classes in later

repetitions

of

Ql

have

different accents

than

the correspondingbeat classes in its first statement,because

some

accents,

like climax and

pulse,

take time to

establish.

Moreover,

when

a

pattern

s

building

up,

the accent

one at-

tributes to its

attack varies

considerably

with the

degree

of

completeness

of the

pattern.

When

one attends to

accent,

one

hears

hardly

any

exact

repetition

in

this

nominally

"repetitive"

usic.

To

express

this

diversity

t is

not sufficient to

represent

rhythm simply

as the

collection

of all attackedbeat

classes,

as

has

been

done

for Reich's

phase

music.

A

tally

of accent

types

on each

beat

class,

as

suggested

n

Example

5(c),

is somewhat

better. It does

not

account

well

for differences

n accentual

quantity,

because

it

does

not

weight

the various

types

of

ac-

cent,

and because

some

accents

of a

given type

are

stronger

if,

for

instance,

they

involve

greaterchange.

But even with-

out such

weighting

the

tally

facilitatesa

description

of

the

rhythm

of

Example

5(a):

during

that time

span

a distinctive

series of

accent

types

consistently repeats, promoting

the

perception

of beat

classes;

at beat-classes

0, 4,

and

11,

the

most

types

of accent

appear,

while consistent but fewer

types

of accent

appear

at

other

beat classes.

When we evaluate

au-

rally

the

strength

of these

accents,

beat-class 0

clearly

stands

alone as most

accented,

since it

is the

highest

and

longest

event,

while beat-class

11

sounds weaker than beat-class

4

(and

0),

but

stronger

hanothers.

287




25 (2003)

This

description

suggests

a formal

analogy

between

the

accentual

organization

of

rhythm

and modal

organization

of

pitch,

one

that extends

and enrichesthe

analogy

Cohn

made

between beat-class

sets

and

"atonal"

pitch-class

sets.

Music

may be understoodas "modal"o the extent that its pitches

are

heard as instances of

pitch

classes

organized

in a func-

tional

hierarchy.

he

structurally

most

importantpitch

class,

called

the

tonic,

acts as a

reference

or the

collection,

in

that

the

other

pitches

are

named

as "scale

degrees"according

to

the intervals

they

form with

the tonic.

The ensembleof these

intervals,

ogether

with information

about

the relativestruc-

tural

importance

of

the

non-tonic

pitch

classes,

constitutes

the mode.20 or

instance,

the

D-major

section

in

Example

1

is distinguishedfrom the E-dorian section not by its pitch-

class

content,

which is the

same,

but

because a different

pitch

class

is

presented

as the tonic. Since

the other

pitch

classes

form

different ntervals

with

E than

they

do with

D,

and

since

they,

too,

are

accented

differently-for example

B

is more

prominent

at R64 than at R55-the

mode of these

two sections is

different.

The

concepts

of

tonic and

mode also seem

appropriate

or

expressing

he consistent structuraldistinctions

that Reich's

rhythms

make

among

beat

classes. I define the "beat-class

tonic"

of a

time

span

as the

beat

class

that,

in a

given

context,

acts as

a

reference or

the

other

accented

beat

classes,

n the

sense

that

one

perceives

their

temporal position

in

terms

of the interonset durations

from it

to them.

Although

the

meaning

of

"beat-class tonic"

thus

overlaps

with that

of

"downbeat,"

find

the

term "tonic"

more

apt.

It avoids con-

fusion

with

notated

downbeats,

which often have no audible

status

in

Reich's

performances;

t facilitates the

description

of

competing,

even

conflicting,

tonics,

and of

changes

and

20

This

prescriptive,ompositionally

riented efinition f

mode

res-

onateswith

recent esearch

n

music

sychology.

or

nstance,

utler&

Brown

994

demonstrate

ow

onality

that s,

tonicand

mode)

may

be

cognized y ocating

rare"

ntervals ithin

a

given

diatonic

et,

nter-

vals hat

are

understood

o

span

and hereforeo mark

pecific

cale

degrees

membersn

a

major

rminor

ey.

ambiguities

that the

term "downbeat"

may

exclude;

and

it

emphasizes

similarities

n

the

way

that

Reich

changes

beat-

class

tonics and

pitch-class

tonics

through

the

use of

pivot

collections,

which

will be

discussed

below.

The distributionof differentlyweighted accentsprovides

a basis

for

characterizing

what

I call the "beat-class

mode"

of

the

passage.

It can

be

determined

by

an

analysis

ike

that of

Example

5,

which

locates the

most accented

beat

classes-

taking

into account

both

the numberof

different

types

of

ac-

cent

on each beat class

and

the

weight

of each

of those

accents

-and

labels each

of them

by

the number

of beats

from the

tonic to it.

Just

as

pitch-class

mode

is identified

with

refer-

ence

to triadic or

otherwise distinctive

interval

structures,

the beat-classmode is identifiedby matchingthe most ac-

cented

beat classes

with distinctive

series of durations.

If

these

modally

significant

beat

classes

create

a

pulse

stream,

then

the "mode" f

a

pattern

s

tantamount

to

its

meter,

but

in

many

cases

they

do

not,

such

as in the

passage

from

The

FourSections iscussed

below.

Usually,

however,

he tonic

be-

longs

to the beat-class

set

that

characterizes

he

mode,

just

as

the

tonic

pitch

class

belongs

to

the tonic triad.

Let

us consider

this

analogy

of

rhythm

and

pitch

more

specifically

in

the

context of

New York

Counterpoint,

Ex-

ample

3. Rehearsals

8-9

project

F

as

pitch-class

tonic

by

pitch-specific

features

of

the

pattern

evident

in

Example

5(a).

F recurs

regularly

as

the

lowest

pitch,

acting

as a

pedal

point.

The

other

most

accented

pitch

classes

sound

like

chord

factors

of an

F-rooted tertian

harmony--AN

s a minor

third

over the

root,

El

a

minor

7th. Root

movement,

such as

it is

(Example

5[b]),

leads

toward

F. The intervals hat all the

pitch

classes

form

with the

tonic

are consistent

with

the dis-

tinctivestructures

f the

minor

and

dorian

modes.

Analogously,

0 is

projected

as beat-class tonic

by

intrinsi-

cally

rhythmic

features

of

the

pattern.

It is the

first

accented

beat

class,

and at

its first

two

attacks t takes

more

types

of

accent

than

does

any

preceding timepoint.

Although

by

R9

beat-classes

4

and

11

present

as

many

accent

types,

beat-

class

0 still takes the

greatest

accent

of

climax and

duration,

288



BEAT-CLASS MODULATION

IN

STEVE

REICH S MUSIC

and it

contributes

o

two

pulse

streams.

The other accented

beat

classes

relate

to

the

tonic in

a

distinctive

way.

Beat-

classes

4

and 8

belong

to

a

tonic-including

pulse

stream

hat

measures the

time

span

of

Q1

into

three

equal

durations.

The beat classjust precedingthe tonic is stronglyaccented

and

belongs

to a

set

of beat-classes

{11,1,3}

that

suggests

but

does not

quite

sustain another

pulse

stream.This distinctive

ensemble of

accents,

and

their

temporal

relation to

the

beat-

class

tonic,

constitutes he beat-class

mode.21

As

a

further illustration

of

beat-class

modality,

consider

Example

6,

which

analyzes

accent

n the

build-up

of

Q2,

be-

ginning

at R9.

Recall

that

the

complete

Q2,

as a

beat-class

set,

is

t5(Ql).

If

Q2

presented

exactly

Ql's

series of

pitches

and durations-as it would in Reich'sphase-shiftingpieces

-then

the

beat-class tonic would shift

to

beat-class

5,

con-

forming

to

the

time

transposition.

ts mode

(expressing

how

its

time

span

is divided

by

pulse

and

other

accents)

would

re-

main

the

same.

(Generally,

exact

time

transposition,

like

pitch transposition,

changes

tonic but

not

mode.) However,

even

though

Q2

contains the same

pitches

as

Ql,

the order

and duration

of

Ab5,Bb5,

and

El6

in

it

are

different,

and

so

the distribution

of

accent

in

Q2

is different.

This

affects

the

beat-class

mode:

in

Q2,

accent

supports

wo

half-note

pulse

streams,

one

containing

beat-classes

{8,0,4},

and the

other

{1,5,9}.22

Beat-class

0 in

Q2

has more accent than

does

the

21

Beat-class

mode resembles theoretical constructs of tala in North

Indian

classical

music,

which are

distinguished

by length

and

by

the

beats that receive

the

most accent. See

Clayton

2001.

Tala,

however,

are

not

usually

built

up

or

phased.

22

The

coexistence of

these two

pulses

can be characterized as

the

"dis-

placement

dissonance" D4+1 in terms of Krebs 1999. Such a

descrip-

tion is

certainly

conceivable for minimal music; indeed Krebs's

analysis

of form in Schumann's

music,

which narrates a succession of states of

metrical consonance

and

dissonance,

resembles

my

accounts of

form

in Reich's

music. What

especially distinguishes

our

approaches,

how-

ever,

is

my

focus on

shifting

beat-class tonics

(which

are

not contem-

plated

in

Krebs's

theory)

and their correlation with

changes

of

pitch-

class-modality.

transpositionally

orresponding

eat-class

7

in

Q1,

and beat-

class 9

in

Q2

has

less accent

than does

the

transpositionally

corresponding

beat-class

4

in

Q1,

so

stream

[8,0,4}

is

stronger

and stream

{1,5,9}

is weaker

thanwould be the case

underexacttransposition.

The

changes

also affect

the beat-class tonic.

In

the com-

plete

Q2,

at

R12,

beat-class

4

takes

as

many types

of

accent

as does

beat-class

5,

so

at first

glance

it

might

seem that

ei-

ther

of

them

could act

referentially.

But the

specificway

in

which

Reich builds

up

Q2-another

crucial difference

be-

tween

it and

Q--is

decisive

in

establishing

which

of these

two

beat classes

s the tonic.

Beat-class

4

is

the first accented

beat

class,

and at its

first three attacks there

is

more accent

than at any preceding timepoint. Although by R12 beat-

class

5

presents

as

many

accent

types,

beat-class

4

still

takes

the

greatest

accent of

climax,

and it contributes

o more

pulse

streams.

Contrary

o

what

one

might

have

expected

from the

t5

relation

of the beat-class

sets,

then,

the

pitch reordering

and

the

build-up

of

Q2

make beat-class

4

referential.

Comparing

the

analyses

of

Ql

and

Q2

in

Examples

4

and

5,

it

is evident that

both

patterns place

their climaxon

their

respective

tonic,

and

both articulatea

complete

pulse

stream,including

the

tonic,

that measures

their time

spans

into

three

equal

durations.

In

terms

of

pulses

and accent

of

climax,

then,

Q2 (at

R12)

and

Q1

have

the same mode.

This

is

analogous

to the

similarity

we intuit between two

F-minor-seventh chords

in which the chord

factors

are

dif-

ferently

voiced and doubled.

Moreover,

these two

examples

of beat-class

modality

il-

lustrate

a

process

that

is essential

to

the

form of

Reich's

music.

Changes

in

tonic

or mode-which

I

will

call beat-

class

"modulation"-create

large-scale

contrast,

progression,

and

return,

analogous

to

processes

of

pitch-class

tonality.

These

modulationsarise

from

changes

in

the

membership

of

the beat-class

collection

itself,

or from

changes

in

the

types,

strength,

and

placement

of

accent

within a

continuing

col-

lection.

Sameness of

mode,

which

is essential to formal

processes

of

closure,

arise n

patterns

with differentbeat-class

289



MUSIC THEORY

SPECTRUM

25

(2003)

D

C

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B

10

+

1

^

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A

I

,

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tonic

I

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not

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a

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456 89 10 0 1 2 4

L1

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EXAMPLE

6. Accent in the

build-up

ofQ2.

sets and

tonics,

as

long

as

the most accentedbeat classes re-

late to their

respective

onics

in

the same

modally

character-

istic

way.

The

variations

n

Reich's

patterns exemplify

these

theoretical

situations,

as

we shall see.

With

this

model, however,

I am

not

suggesting anything

more than a

formal

correspondence

between

rhythm

and

pitch.

Modality

is

perceived differently

in these

two do-

mains,

so I do not claim that the "distinctive"tructures hat

characterize

pitch-class

modes

(triads,

which

are

asymmetri-

cal subsets

of

the

total

chromatic)

are

perceptually quivalent

to those that

characterize

beat-class modes

(usually

pulse

streams,

which

are

symmetrical

subsets of the

beat-class

aggregate).

Yet

the

correspondence

runs much

deeper

that

has been

previously

discussed,

and

I

will show

that such a

"modal"

conception

of

rhythm

is essential to

understanding

metrical and

other

large-scale

processes

in

Reich's

post-

phase

music.

* * * * *

When

patterns

combine

polyphonically,

heir

accents

in-

teract

richly

to affect

beat-class

tonic

and mode. To a

certain

extent the

modality

of a

particular

polyphonic passage

de-

pends upon

both the

relative

prominence

of the voices and

the context

that

precedes

t. For

example, during

the build-

up

of

Q2,

when it is

loud,

the accentual

structure

analyzed

in

Example

6 dominates the

texture,

stressing

beat-class 4.

But since

the

pulse

stream

characterizing

he

mode of

Q2,

[8,0,4),

is

beat-class-identical

with the

modal

pulse

stream

of

Ql,

and

since

beat-class

0 is accented

in

Q2

nearly

as

much as

beat-class

4,

the

combination

of

Q2

with

Q_

does

not

change

the tonic or mode established

by

Q_.

Q2

has a

different

tonic,

as

analyzed

n

Example

6,

only

if it is

played

in

isolation

from its true

context.

At

R13,

as

Q2 fades,

its

prominence

diminishes,

so one

becomes

more

aware of its

interactions

with

Q1.

Intrastream

ccents

still

may

be

heard,

but interference

among

the streams

affects

their

salience.At

R14,

when

Q1

and

Q2

are

equally

oud,

their

combination,

analyzed

in

Example

7(a),

denies accent

of

contour and

duration to

some beat

classes

that are accented

when either

is

played

alone.

For

example,

in

Q_

the

Bb5

at

beat-class 9

took a

pitch-contour

accent

because

it was

preceded

and

followed

by

lower

pitches,

F4 and

A15.

However,

the

Bb5

at

beat-class5 in

Q2

has a such a

long

durationthat it covers

the

F4

in

Q1

when the

patterns

are

combined;

consequently,

the

Bb5

at beat-class 9 no

longer

has

pitch-contour

accent,

because t

no

longer

follows

a

lower note.

The

pitches

added

by

Q2

to

Q1

also

change

the

moments

where

we

senseshifts

of

subcollection:

for

example,

beat-classes 2

and

5,

which

D

C

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6 6%, 7 1

I I

- 7 -

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V -1 V

i

. 1I IJ

290



292

MUSIC THEOR

beginning

of

the

excerpt,

we

see that beat-class

0

still has the

greatest variety

of

accent,

and that beat-class

4

has also

gained variety.

Moreover,

beat-class 0 still

predominates

n

the

strength

of its

accents,

and beat-classes0

and

4

together

reinforcethe beat-class mode characterizedby the {8,0,4}

pulse

stream. But

the mode

is now colored

by

another and

weaker

pulse

stream

that

arises

from

multiple

accents on

beat-classes

5

and 9.

In

the

following

music,

as

Q3

is

built

up

and

combined

with

Q1

and

Q2,

the

accentual

profile

adjusts again

in

an

apparently

calculated

manner.

Like

Q2, Q3

as

a

beat-class

set is

a

transposition

of

Q1,

and it

contains the

same

pitches

as

Q1

but in a

slightly

differentorder.

ust

as

the

build-up

of

Q2 emphasizedbeat-class4, the build-upof Q3 emphasizes,

by

means of

durationaland

metrical

accents,

beat-class

8

of

the

pulse

stream

[8,0,4}

established

by

Q1.

Accents of sub-

collection

shift

within

Q3

strengthen

he beat

classes of this

mode. At

R19,

as

Q3

fades to

the

loudness

of

Q1

and

Q2,

the accentstructure

again adjusts,

as

analyzed

n

Example

8.

Beat-class

0

is accented

strongly

andin

nearly

every

possible

way,

and

although

other

pulse

streamscan

be

discerned,

he

one that includes

beat-classes

[8,0,4}

is

supported

best

by

the

most number of

accent-types.

Across

the other

beat

classes,

accent

s

spread airlyevenly,rendering

he tonic

and

mode

susceptible

o

furtheralteration.Beat-class

6

stands as

the notable

exception:

it

is

not even accented

by

pulse.

Interpreted

n

context of the model of beat-class

modality,

this

lack

of

emphasis

s

designed

to

negate utterly

the

possi-

bility

of

duple

meter-that

is,

it

clarifies the

triple-meter

mode

by

denying

the

simplest

alternative.

To summarize:

during

he

first

stage

of

New

York

Counter-

point,

beat-class 0

has

been

established as

tonic.

Then,

as

beat classes and accents

multiply

in the

build-up

of new

voices,

first

beat-class 4 then

beat-class 8 become

more

prominent.

By

R19

a

texture

is

achieved in

which

nearly

every

beat

class is

similarly

accented,

except

those that define

the

mode and tonic.This

analysis

revealsa

rhythmicprocess

essential to

this

movement,

and

to

many

of Reich'srecent

Y SPECTRUM

25 (2003)

pieces.

As will be demonstrated

below,

the

accentual

focus

caused

by

the

build-ups

and

by

the interaction of

repeated

patterns

shifts from beat class

to beat

class,

analogous

to

changes

of

pitch-class

tonic in a

tonal

composition.

The

modulationof beat-classtonics has its own immanentlogic

quite

distinct from

that of the

pitch-class-modulatory

processes

t

resembles

formally.

To

understand his

logic,

let

us returnto

Example

3 and

examine

its second

stage.

In

this

passage,

as

during

the first

stage,

the

loud

build-up

of

each

pattern

adjusts

he

types

and

weights

of accent on each beat class.

As each

pattern

matures

and

then fades into the

accompanimental

exture,

t interacts

with the

established

patterns.

Thus the

resulting

ensemble

does not remainconstant,but is subject o changesof mode

and

tonic.

The

pitch-class

collection

also

undergoes

ormally

similarbut

not

exactly

coordinatedmodulation.

Specifically,

although pattern

Q4

builds

up

the same

beat-class

set as the

original

pattern

Q1,

its

particular

pitch

series and

build-up

have a

very

different

rhythmic

impact,

even

shifting

the accentual focus of the

entire

ensemble.

It

begins

in R20

by loudly stressing

beat-classes 9

and

11,

distracting

attention

from

the

still

referentialbeat-class0.

At

R21

it

marks

beat-class

4

with an accent

of

beginning,

while

still

omitting

beat-class

0. As

three

voices now accent

beat-

class 4 the same

way,

that beat class

suddenly

and

decisively

assumesthe role of tonic.

Meanwhile,

the

accents

still sus-

tain the

pulse

stream

{8,0,4},

continuing

to

measure he

pat-

tern's

time

span

in the

previously

established manner.

Changing

the

tonic this

way

while

maintaining

he mode

is

analogous

to

changing

the

key

from

F

minor

to,

say,

Ab

minor,

in

which

the

new

tonic is

a member of

the

mode-

defining

tonic triad

of

the

original

key.

Coincidentally,

he same new

pitches

that causethe beat-

class modulation also restructurethe

ongoing

pitch-class

collection.

Since

each

pitch

in

Q4

is

a

tenth below the corre-

sponding pitch

in

Q1,

Q4's

pitches

at beat-classes

4

and

5

are lower than

any preceding

pitch.

The

lowest, Ab,

insinu-

ates itself

as the

new referential

pitch

class,

a

change

that



BEAT-CLASS MODULATION IN STEVE REICH

S MUSIC

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EXAMPLE8.

Accent in

the

equal-loudness

combination

of

Q3,

Q2,

and

Q1.

is solidified

as

a modulation at R22

by

the introductionof

a new

pitch

class,

Dk.23Thus the

beginning

of the second

stage

establishes both a

new

beat-class tonic

and a

new

pitch-class

tonic

via

structurally

imilarmodulations.

Reich's

specific

choices of

pattern

and

build-up

in the

fol-

lowing

music

can

be

similarly explained,

with

reference

to

beat-class

modality.

The

build-up

of the next

pattern,

Q5,

introduces he

same

beat classes

n

the same order as did

Q2

in R9-R12.

The

resulting

stress

on

beat-class

4

functions

now to confirm

its role

as

tonic.

(See

the

annotations

to

R24-R27

in

Example

3.)

The

build-up

of

Q5

(still

mimick-

ing

that

of

02)

is

designed

to hold

off its

lowest

pitch,

F3,

until

the

very

end,

at R28. As the

new lowest

pitch,

the

F

will

change

the

pitch-class

tonic

and

reemphasize

beat-class

0,

so

delaying

its

entrance

prolongs

the

previous

pitch-class

23

Similarhanges f tonicoccurustafter hebuild-uphownn Ex-

ample

is

complete.

twice-repeated

eries f

pulsing

hords,

rawn

from

he

opening

f the

movement,

nd

each

asting

everalterations

of the

repeated patterns,

successivelypresents

bbm7,

DbM7,

and

Fm(add

)

chords.

The

series nimates

he

unchanging

itch

classes-

notably

b

and

A--in

the

patterns y varying

heir

ntervallic

ela-

tions o the

changing

oots.

and

beat-class tonics as

long

as

possible.

Once

F3

enters,

R28-R31

project

rhythmic

ambiguity,

as two different

beat

classes

sound

equally

accented and referential.

One

might

characterize

his as a "doublebeat-classtonic

complex".)

The

final

build-up

in

this section

(of Q6)

begins

by

stressing

beat-class

8,

as did its beat-class-set

homonym

Q3.

Because

the

F4

attacked

then

is

not

strongly

accented,

however,

the

beat-class

tonic

stays

on

4.

However,

at R32

a

beginning

ac-

cent on beat-class

0,

reinforced

by

a

grouping parallelism

with

R21

and

by

the

multiplicity

of coincident accents n

the

other

voices,

changes

the beat-class tonic.

At

the

end

of

the

passage,

then,

formal

closure

is

achieved

as both the

pitch-

class

and the beat-classmodes return o

their

original

states.

The

theory

of

beat-class mode

thus enables

one

to

de-

scribe

rhythmic

directionand

goals.Accordingly,

t

provides

a

means of

answering

the

questions

about Reich's

post-phase

music,

raised

above,

which

cannot

be

addressed

by

an "atonal"

theory

of beat-class sets.

Through

it we understand hat the

purpose

of

combining

beat-class

sets

is

not to

achieve

the

beat-class

aggregate,

but to create

a

progression

of

beat-class

tonics across

large

spans

of

time,

taking advantage

of

the

modes shared

by

the

pattern

combinations.

The

notion

of

rhythmic

closuretakes

on

the

precise

sense of

a

return o

the

293



MUSIC THEORY

SPECTRUM

25 (2003)

bc:

0

4 5 6

J

=

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D

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10

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15

16 20

S

D

B

C

D

22

23

0

I

I

4 5 6

10

14

15

16

(measures

24

beats

into four

equal

durations)

S

S

S

S S

D

C

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C

D

20

22 23 0

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p

treams

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(measures

24 beats into three

equal

durations)

Iton

tonicl

EXAMPLE

9.

Accent

andpulse

streams at R44

ofNew

York

Counterpoint.

original

beat-class

tonic and

mode,

as at a tonal cadence.

Variations

n

patterns

themselves

are understood as

part

of

the

modulatory

process,

when combinations

of exact beat-

class

transpositions

do

not

provide

the

clarity

of

mode

and

tonic

required

or these

large

formal

processes.

So are the

ir-

regularbuild-ups;

for

example,

Q5

and

Q6

are built

up

in

the

same

way

as

Q2

and

Q3

because

they play

similar

roles

in

shifting emphasis

from beat-class

0

to beat-classes

4 and

8,

respectively.Finally,

he choice of

pitch-transposition

of a

tenth

from

earlier to later

patterns

can

be

explained

as

the

best

one to

minimize

interferencewith the establishment

of

subcollection-shift

accents,

while

introducing

a

lower

regis-

ter in which accents

can

act to

change

both

pitch-class

and

beat-class tonics.

A

remarkable

eature

of the

densely

imitative

web

that

Reich weaves in this movement is the

persistent

clarity

of

the

{8,0,4}

pulse

streamand

of the

tripartite

mode in

which

it

measures he

patterns'

ime

spans.

However,

the

composer

does

not

alwaysprefer

o maintain

a constant meter.

Indeed,

the

opening

of

the second movement

of the same

work,

New

York

Counterpoint,

onfronts

the

listener

immediately

with

a

very

dynamic

modality.

Example

9

analyzes

accent and

pulse

streams

in the

passage,

which

repeats

a

pattern

lasting

24

sixteenth

notes.

The brackets above and below

the

score

show

that two

pulse

streams

with

different

durations

are

ar-

ticulated

concurrentlyby

regular

accent.

The

dotted-quarter

pulse

stream

arises

principally

rom

accents of subcollection

shift,

while

accents

of

durationand

beginning (supported

by

slurs)

coordinate

to

produce

the half-note

pulse

stream.

Neither

of these streams ncludes the

tonic

(0,

accented

in-

tensely

by

duration,

contour

and

pulse),

but

they

are

syn-

chronized

so that

they

measurethe

pattern's

ime

span

into

equal

durations

both

triply

and

quadruply.

The metrical

ambiguity

created

by

the

pattern's

artful accentual

design

deepens

as the movement

develops.

Its

largest-scale

consequences

are

not

manifested,

how-

ever,

until

the

last movement of

New

York

Counterpoint,

when

both

pitch-class

and

beat-class

modes

and tonics

un-

dergo gradual asynchronouschanges.

The

modulations

are

most

striking

in the

excerpt

shown

in

Example

10. At

R70

l

I

294



BEAT-CLASS

MODULATION IN

STEVE

REICH S MUSIC

Pitch-class collection

1

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3 and 6:

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9

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10:

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Beat-classmode1

T

T

EXAMPLE

IO.

Pitch-class

and

beat-class modulations

in

the

third

movement

ofNew

York

Counterpoint.

a dux trio

of

clarinets

(notated

on

the

top

staff)

is

chased

in

canon

by

a comes

trio

(notated

on

the

second

staff)

at the

quarter-note

unison.24

On the lower

staves,

two bass clar-

24

A few

added

notes end

some

flair

o

the live clarinet

part,

but this

aug-

mentation

f

the

beat-class ollection oes

not affect he

modeor tonic.

inets

synchronize

their

changes

of

pitch

class but are never-

theless also

in

rhythmic

canon,

as

will

be shown below.

The

low

El

in clarinet

10 acts as the

pitch-class

tonic,

casting

the

segment

in the mode of

an

Eb7

chord with a raised fifth

and eleventh.

The table

above the

score

lists the

pitch-class

content of

each

canonically

related

pair

of instruments.

At

9.

T T

HT

T

T T

N

I

I

I

T

T T

T

iOf. . Y. H. I. D. 1* /

295

combined:

/, Y,

,,

I,

,

,j

'n

T' r-



BEAT-CLASS

MODULATION IN STEVE

REICH S MUSIC

x

t2(x)

0

I I

I

I I

I

I

I

I

0

2 45

7

9 11 12 16 17 19 21 22

0

2

4

5

6 7 9 1112 16 17 18 19 21 2223

0

2

4 67 9 11

13

14 18 19 21

23

0

2 45

8 1112

14

16 17 20

23

0

3

4 6 8 9 12 1516 18 20 21

i

i

I

I i i

I I

I

Y t12(Y)

0 2 45

7 9 1 12

16 17 19 21

2

0 2 45 7

9

1112 1617

19

21 22

0

2

4 67

9 11

13

14 18 19

21

23

0

2

5

6

8

1011

14

1718

20

22

23

0 34

6 89 12 1516 18 2021

0

2

45

7

9

11

12 16 17

19 21 22

0 1

2 3 45

7 8 9

101112 1617

19

2021

22

0 1

3 5

7 8 10 12 14 15 19

20

22

0

3 6

7 9 1112 15 119 21 23

0

3

I

I

I I

I

I

I

I

I

toA

toA

U

t6,18,23}

t2A

t8B

toB

toB

n

tgB

=

{0,4,8,12,16,203

(4-cyclic)

toA

toA

U

(6,18,23}

t2A

t2B

U

to}

toB

toB

n

(t2B

U

o})

=

{0,6,8,18,201

toA

toA

U

{1,3,8,10,203

t3A

t3B

toB

toB

n

t3B

=

10,3,6,9,12,15,18,21}

(3-cyclic)

EXAMPLE

II.

How

transposition

of

subsetscreates

he

beat-class

modulation

in

R71-73.

10

continue those of

R71,

and

only

the

temporal

imitation

between

the

bass clarinets

shifts,

from

t8

to

t2,

thus

matching

the time

delay

in the

upper-voice

canon. But this

slight

change

affects the

beat-class

mode

by

breaking up

the

pre-

ceding

half-note

pulse

stream.

This

is

symbolized by

the

dashed brackets under

R72

in

Example

10,

and is

also

evi-

dent in

Example

11,

which

shows that the intersection of the

beat-class sets of bass clarinets 9 and 10-the

low-register

accents

of textural

density-can

no

longer

be

generated

cyclically

by

beat-class interval

4.

(Clarinet

9 adds an extra

attack to its

pattern,

at beat-class

0,

to

keep

the tonic

clear.)

This

modal

uncertainty proves

transitory.

At

R73,

when

the

pitch-class

content reverts to

that

of

R70,

the

beat-class

accentuation

changes directly

to another

mode,

again

by

simply

changing

the interval of

imitation. The outer

voices,

clarinets 2/3 and

10,

continue to

present

the same

rhythms

71

2/3

live

4/5/6

9

10

2/3

live

4/5/6

9

10

73

2/3

live

4/5/6

9

10

triple

meter

Y.

transition

quadruple

meter

297



298

MUSIC THEOR

as

they

have

done since R70.

However,

the

beat-class set

of

clarinets

4, 5,

and 6

changes

from

t2

to

t3

of

clarinets2

and

3,

and the

beat-class set of clarinet

9 also

changes

from t2

to

t3

of the

clarinet

10-that is the comes

voices

increase their

delay by one beat. Now the beat-classsets of the bass clar-

inets,

whose

intersection was a

4-cycle

at R71

and a

sym-

metricalbut

noncyclic

beat-class

set

at

R72,

intersect n a

3-

cycle

at R73.

The audibleresult s a new

dotted-quarter-note

pulse

stream,

symbolized by

the bracket under R73 in

Example

10,

that creates a 12/8 meter.

Thus

the

beat-class

modulation from R71 to

R73 is achieved with the

utmost

minimum of means.

It

is

mediated

by

the

set

of beat classes

at

R72

that

the two modes have in

common,

exactly

analo-

gous

to the

common-tone modulation between the

pitch-

classcollections n

the

passage.

Other recent

compositionsby

Reich contain

many

similar

passages,

n

which

slight

but

structurally

elling changes

to

patterns

and their

imitative

relations

create

formally signifi-

cant

modulations

of

pitch-

and

beat-class.

They

are

most

impressive

n his

works for

large

ensemble that

juggle

several

different

patterns

at

once.

Consider,

as

a

final

example,

the

opening

of the last movement of TheFourSections

for

or-

chestra, 1987).

At different

paces

and

times

during

this in-

troduction our

different

patterns

are

built

up,

each of which

is

distinguished

by

instrumentation,

egister,

durational

con-

tent,

and

attack

density. Example

12

displays

their

com-

pleted

forms and

analyzes

their

beat-class-combinational

structure.

Starting

at

Rill,

middle

register strings

and mallet

in-

strumentsbuild

up

a

predominantly ighth-note

rhythm

nto

a two-line beat-class canon,

fully

completed at R122, in

which

one

voice

lags

three

eighth-notes

behind the

other.

From

R113-R120

trumpets

1

and 3 build

up

an

apparently

unrelated

pattern,

which features a

variety

of

durations,

yet

also

suggests

an

exact

pitch

and

beat-class

canon,

without

ex-

Y

SPECTRUM

25

(2003)

plicitly

stating

it.25In the

percussion,

brass,

and

low instru-

ments at R115 a

build-up begins

of a

different,

noncanonic

pattern,

completed

at R124.

All three of these

patterns

are 20

eighth

notes

long. Lastly,

at R118 the

high strings

and

winds

buildup a pattern wice as long-40 eighth notes-featuring

very long

durations;

his resolves nto a

t10

canon at R125.

Within this

complex,

asynchronous

aggregation

of

dis-

crete

patterns,

beat-class

mode

and

tonic

fluctuate n a

con-

trolled and

progressive

manner. The

build-up

starting

at

Rill,

analyzed

n

Example

13(a),

has two

principal

formal

functions.

First,

it

clearly

establishes the beat-class

tonic:

beat-class0 takesthe most

accent,

and 0 is the first

beat class

to mark a

regularly

ecurring

duration

(20

eighths,

the dura-

tion of most of the patterns).Second,this passagealsoestab-

lishes a distinctive beat-class

mode,

but

only

after

raising

several

mutually

incompatible possibilities.

Initially,

accents

on

beat-classes

16,

0, 4,

and

8

project

a

series

of half-note

durations.

However,

this

potential

half-note

pulse

streamis

vitiated at R112

by

the

shifting

of accent to beat-classes

{0,3,6,9},

which

suggest

a

dotted-quarterpulse

stream in-

commensuratewith both

the

half

notes

and

the

20-eighth

durationof

the

patterns.

At

R113

the first

trumpet's

attacks

measure

he 20

eighths

into

two

equaldurations,suggesting

a

regular five-quarter pulse

stream,

likewise

incompatible

with the

previouslysuggested possibilities.

Finally,

at

R114

the next

stage

in

the

string-vibraphone uild-up

establishes

consistent accent on

beat-classes

{0,6,10,16}-not

a

regular

pulse,

but still distinctive

and

persistent

enough

to

serve

as

the beat-class mode.

As

in New York

Counterpoint,

eat-class

modulation be-

gins

as soon as mode and

tonic

are secured.At R115

(Ex-

ample

13[b])

clusters

in

the

pianos

and

trombones

strongly

25

To see

the

canon,

compare

the

two

trumpet parts starting

at

the

re-

peated,

accented

eighth-note

Es. In each

part,

there

follows

a

quarter

rest,

then a half-note

D#,

then

eighth-notes

C# and

F#,

separated

by

an

eighth

rest.



BEAT-CLASS

MODULATION IN STEVE REICH'S MUSIC

Vib

'

,

Vn.

.

Vib.

2,

,

,

120a

Tpt.

1

7

1no. ,

24

Tpt.i|,*

-

HXv r

f

Brass,

m.

#

-

v

Timp.

~___

-if f

Tpt.

1

Tpt.

3

{0,1,2,3,5,6,7,8,10,11,12,13,15,16,17,18}

=X

{0,1,3,4,5,6,8,9,10,11,13,14,15,16,18,19}

=

tl3(X)

{2,4}

u

{6,7,10,14,16}

1

t6

t8

1

{8,10}

u

{14,15,18,2,4}

{0,6,7,10,19}

{0,10,20,28,30}

=

Y

{0,10,20,30,38}

=

to0(Y)

EXAMPLE 12.

Patterns n

the

opening

f

thefourth

movement

of

TheFourSections.

accent beat-class

10. As

this

beat

class

belongs

to

the estab-

lished

mode,

and since the mode is

transpositionally

nvari-

ant

at

tlo,

the

tonicity

of

beat-class 0

begins

to falter.

By

R117 the further

build-ups

of the

patterns

cooperate

to ac-

cent beat-class 10 far more than beat-class 0,

making

the

modulation definite.

Thus,

the

entrance of the

high strings

in

R118 sounds

metricallystrong,

even

though

it is notated

on

a

different beat than the

beginning

of

the

pattern

in

Rl11.

After this new

beat-class tonic is

established,

however,

the

completion

of the

build-ups

in

R120-R124

and the

pitch

variations

in the

highest parts provide

new

accents.

The

completed

canon

in

the middle

strings

and

mallet

in-

struments

emphasizes

both beat-classes0 and 10. The low

instrumentsalso accent

both of these beat classes.

Starting

at

R120,

the

high

instruments

place

contour accents on

two

different

points

of

the

40-eighth-note spans,

but

up

until

R125

(see

the

upper system

in

Example

13[c]),

these

always

299



302

~~~~~~~~MUSICHEORY SPECTRUM

25 (2003)

Bc 1)

persists

as

C

T

D

D D

D

N

N

Accent on bcs 0 and 10

equalizes

t-I-

4

L7

I

(c)

Modulation back to beat class

0.

EXAMPLE

13.

[continued]

D

124

. 19

C

T

Vn.,

w.w.

Vn.

2, Va.,

Vib.

1,

2

Tpt.

Pno.,

Tbn.,

Hn.

Vn.,

w.w.

Tpt.

Pno.,

Tbn.,

Hn.

302

beat-class modulation in steve reich's music

Documents