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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
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BEAT-CLASS MODULATION
IN
STEVE
REICH
S MUSIC
63
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mf
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Ronic-
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(6-10Ox)
(6-lOx) (6-lOx)
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11:
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(24x)
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n
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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
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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
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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
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MUSIC THEORY SPECTRUM
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
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MUSIC THEORY SPECTRUM
25 (2003)
I30
31
f
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1
,^
^ Y^r
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
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L<+
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7
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yt
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~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
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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.
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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.
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MUSIC THEORY SPECTRUM
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,
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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
8/18/2019 Beat-Class Modulation in Steve Reich's Music
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MUSIC THEORY
SPECTRUM
25
(2003)
D
C
s
B
10
+
1
^
B
A
I
,
-
i21
tonic
I
D
S S
B
^
r_
'
N
9
10 0
1 2
not
quite
a
J
stream
J
streams
N
456 89 10 0 1 2 4
L1
------------I
I
I I
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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bc: 2
A %,9b I-1
:11
V
*i-
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4 2 4 5
C
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o. stream
I
6 6%, 7 1
I I
- 7 -
-
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i
. 1I IJ
290
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8/18/2019 Beat-Class Modulation in Steve Reich's Music
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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
8/18/2019 Beat-Class Modulation in Steve Reich's Music
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BEAT-CLASS MODULATION IN STEVE REICH
S MUSIC
E19
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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
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MUSIC THEORY
SPECTRUM
25 (2003)
bc:
0
4 5 6
J
=
ca. 92
D
C
10
14
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
B
C
D
20
22 23 0
D
C
I
p
treams
I(meass4 b s io t e eal
(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
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BEAT-CLASS
MODULATION IN
STEVE
REICH S MUSIC
Pitch-class collection
1
Pitch-class
collection
Pc
content
ofcanonic oicepairs
t
R70
[sounding
Bb
=
0]
at R71
live and
4:
[8,B,1,3} pivot.
[8,B,1,3}
collections
2
and
5:
{B,1,3,5,7,8}
hold
[8,B,1,3,4}
3 and 6:
[B,1,3,5,7,8}-
[8,B,1,3}
f8,9,B,1,4,6}
9
and
10:
{3,9}
and
[9}
[6,9,1}
in common
-------------------........
J=
ca. 184
D
C
C
S
S
10
(rx~-n
a)
,K ^ L
-.-
iJ
l -i , k r.l0
J
(
x3
I
k
live
2
3
4
5
6
9
10
2bl
^f,b
i
l
.
f
'
K'
e 1 L h>
'
XI
l
'
L-''F :III:Y'
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'ri
rr
X
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AH
i'
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9-
f I I It
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Y
i^
^^^y
^^y^jp
fVlyvy^.^
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1 1
J stream
I
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o. stream
i
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-
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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
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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.
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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
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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
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