8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
1/62
Yale University Department of Music
Toward a Formal Theory of Tonal MusicAuthor(s): Fred Lerdahl and Ray JackendoffSource: Journal of Music Theory, Vol. 21, No. 1 (Spring, 1977), pp. 111-171Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843480
Accessed: 14/04/2010 10:38
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=duke.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].
Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize,
preserve and extend access to Journal of Music Theory.
http://www.jstor.org
http://www.jstor.org/stable/843480?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=dukehttp://www.jstor.org/action/showPublisher?publisherCode=dukehttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/843480?origin=JSTOR-pdf
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
2/62
TOWARD
A FORMAL THEORY
OF TONAL MUSIC
Fred Lerdahl and
Ray
Jackendoff
INTRODUCTORY
REMARKS
We take
the
goal
of
a
theory
of
music to
be a
formal
de-
scription
of
the musical
intuitions
of
an
educated
listener.
By
"musical intuitions" we mean
the
largely
unconscious knowl-
edge
which a
listener
brings
to
music
and
which allows him to
organize
musical sounds
into coherent
patterns.
By
"educated
listener"
we
mean
not
necessarily
a trained
musician
but
a
listener who is
aurally
familiar with the musical idiom in
question.
Such a listener is able
to
identify
a
previously
unknown
piece
as
an
example
of
the
idiom,
to
recognize
ele-
ments of
a
piece
as
anomalous
within the
idiom,
and
gener-
ally,
to
comprehend
a
piece
within the
idiom.
The
"educated
listener" is an
idealization.
Rarely
do two
people
hear the same
piece
in
precisely
the same
way
or
with
the same
degree
of
richness.
Nonetheless,
while it is
conceiv-
able to hear a piece any way one wants to, there is normally
substantial
agreement
on
what
are
the
most
natural
ways
to
hear a
piece.
Our
theory
is
concerned
not
with
particular
in-
stances of
hearing,
which
are
always
subject
to a
degree
of
111
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
3/62
variability,
but with
the
idealized
underlying
competence
which
the
educated
listener
brings
to
bear
in
understanding
tonal
pieces.
A
theory
about
a
particular type
of music
is,
ideally,
a
sub-
set of a
theory
of all music. We are
constructing
our
theory
of
tonal music with this
larger
perspective
in
mind.
While
many
of our
specific
rules are
applicable
only
to tonal
music,
the
basic
components
of
the
theory
are
designed
to
accommodate
music
of
different traditions
and
historical
periods.
A
reader
who
is at all
acquainted
with
contemporary
lin-
guistics
will observe
that the
goals
we
have
set for
ourselves
are
in
some
ways parallel
to the
goals
of transformational
generative grammar, which seeks to describe the linguistic
intuitions
of
a native
speaker
of
a
language
and
to
discover
those
aspects
of
particular
grammars
which are
common to
all
languages.
Indeed,
our
way
of
thinking
about
music is
pat-
terned
after the
methodology
of
linguistics
in
that we
demand
strong
motivation,
formal
rigor,
and
predictive
power
for
every part
of the
theory.
On
the
other
hand,
we do
not
ap-
proach
music
with
any preconceptions
that the
substance
of
our theory will look at all like linguistic theory, since language
and
music are
on the face
of it
different
manifestations
of
human
cognitive
capacity.
Previous theories
of tonal
music have
not met such
de-
mands
of
rigor
and
prediction.
Even
Schenker's
theory,
which
can
be
construed
as
having
much in common
with
the
genera-
tive
approach
to
linguistics,
is
at
bottom
inexplicit.
One
of
the
virtues
of a formal
theory
is not that it is
necessarily
more
"true," but that, even where incorrect or inadequate, it clari-
fies issues
precisely.
To elucidate
in what
sense our
theory
is
modeled
after lin-
guistic
theory,
we
must
mention a
common
misconception
about
transformational
grammar.
It
is often
thought
that a
Chomskian
generative
grammar
is an
algorithm
that
grinds
out
grammatical
sentences;
this view
suggests
that
a
genera-
tive music
theory
should be
a
device
which
composes pieces.
There are two errors in this view. First, the sense of "gener-
ate"
in the term
"generative
grammar"
s not that of an elec-
trical
generator
which
produces
electricity,
but
rather the
mathematical
sense,
in
which
it
means
to
describe
a
(usually
infinite)
set
by
formal
means.
Second, Chomsky
distinguishes
112
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
4/62
the
"weak
generative
capacity"
of
a
theory
from
its
"strong
generative
capacity."'
A
grammar
weakly
generates
a set
of
sentences
and
strongly
generates
a set
of structural
descrip-
tions,
where
each structural
description uniquely specifies
a
sentence,
but not
necessarily
conversely.
It is the notion of
strong
generation
which
is
overwhelmingly
of
interest
in
linguistic
theory.
The
same
holds
for music
theory,
since
a
theory
which
weakly
generates "grammatical"
tonal
pieces
would
tell
us
nothing
interesting
about their
structures.
A
strongly
generative theory
of
tonal music
would
not
merely
give
a
description
of what
pieces
are
grammatical.
Rather,
it
would
have to
specify
each
tonal
piece
together
with its structural description, i.e., a specification of all the
structure
which the
educated
listener
infers in
his
perception
of
the
piece.
If
a
given
piece
cannot
be
heard as
tonal,
the
theory
should be
unable
to
give
it a
structural
description;
if
a
piece
can
convincingly
be
heard
in
several
ways,
the
theory
should
give
it a
different
structural
description
for
each
way
of
hearing
it.
We
have
found that a
generative
music
theory
must
not
only assign structural descriptions to pieces, but must differ-
entiate
the
structural
descriptions
along
a
scale of
coherence,
weighting
them as
more or
less
"preferred"
ways
of
hearing
a
piece.
Thus,
the
theory
is
divided
into
two distinct
parts:
well-
formedness
conditions,
which
specify
possible
structural
de-
scriptions;
and
preference
rules,
which
designate,
out
of
the
possible
structural
descriptions,
those
that
correspond
to
the
educated
listener's
hearing
of
any
particular
piece.
There are various criteria for determiningwhich structural
descriptions
of
a
piece
are
"preferred."
Among
these,
of
course,
are our
own
intuitions.
Furthermore,
beyond
all
seemingly
self-evident
intuitions,
a
"preferred"
structural
description
will
tend
to
relate
otherwise
disparate
elements in
a
satisfying
way
and
to
reveal
surprising analytic
insights.
In
addition,
we are
aware
of
relevant
research
in
experimental
psychology
and are
concerned
that its
findings
be in
agree-
ment with our theoretical constructions. Criteriawithin the
theory
itself
include the
internal
consistency
of
the
rules
and
their
generalization
from
particular
instances
to
the
entire
body
of
classical
tonal
music. We
utilize
within
the
frame-
work
of
the
theory
such
psychologically
primary
notions
as
113
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
5/62
parallelism,
articulation,
and
stability. Finally,
the nature
of
our structures must be
psychologically plausible
in
that
their
complexity
must result from the interaction
of a
fairly
small
number
of
processes,
each
of which
taken in
isolation
is
rela-
tively simple.
Although
it
would
be
possible
within
the
theory
to
gener-
ate new tonal
pieces,
we
have chosen
only
to
generate
struc-
tural
descriptions
for
existing pieces.
It
would
seem
to be
inherently
less
rewarding
to
specify
normative but dull
pieces
than
to
develop
structural
descriptions
for works
of
lasting
interest.
Moreover,
in the
former
case,
there
would
always
be
the
danger,
through
excessive
limitation of the
possibilities
in
the interest of conceptual manageability, of oversimplifying
and
thereby establishing
shallow
or incorrect
principles
with
respect
to music in
general.
For
the common
conception
of
a
transformational
genera-
tive
grammar
as
merely
a
sentence-generating
device
is
mis-
taken
in
a further
respect.
Linguistic
theory
is
not
simply
concerned
with the
analysis
of
a limited set
of
sentences;
rather it considers
itself a branch
of
psychology,
concerned
with making empirically verifiable claims about one complex
aspect
of
human
mental
life,
namely
language.
Similarly,
by
putting
our
emphasis
on the
musical intuitions
of the
edu-
cated
listener,
and
by
taking
as our
sample
a
highly
complex
body
of
music,
we are
asserting
that the
analysis
of
pieces
of
music,
though
not
without
a
great
deal
of intrinsic
interest,
is
not
an end
in
itself.
Rather
the
goal
is
an
understanding
of
the
mental
process
of musical
perception,
a
psychological
phenomenon. From this viewpoint, our theory of music is not
just
an
analytic system,
but makes
strong
claims
about
the
delimitation
of
possible
theories
of musical
cognition.
By
regarding
music
only
as
apprehended
structure,
we
are
deliberately
avoiding
the difficult
issue of musical
meaning.
Whatever
music
may
"mean,"
it
is in no sense
comparable
to
the semantic
component
in
language;
there
are no
substantive
parallels
to
sense and
reference
in
language,
or to
such seman-
tic
judgments
as
synonymy, analyticity,
and entailment.
It is
in
the domain
of
syntax
that the
linguistic approach
has rele-
vance
to music
theory.
Yet even
here
there are
no
substantive
parallels
between
musical
structure
and
such
grammatical
categories
in
language
as
noun, verb,
adjective,
noun
phrase,
114
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
6/62
verb
phrase,
and
so
forth. The
concepts
of
musical structure
must
be
developed
in terms
of
music
itself.
We
likewise
avoid
the
issue
of
aesthetic
value.
Nevertheless,
that we
are
dealing
with
works of art rather than
"mental
objects"
from the everyday world, such as sentences, is to a
degree
problematic.
Whereas
a sentence
normally
has a
defi-
nite
meaning,
structural
ambiguity
in a
work
of
art
is
com-
mon.
And
whereas a
sentence
normally
has a definite
function,
it is in
the
nature of a
work of art that it is
appreciated
and
contemplated
from various
points
of
view and with various
purposes
in
mind,
not
only by
different
people
but
by
the
same
person
on
different
occasions.
These
differences,
how-
ever,do not mean that the understandingof a work of art can
take
any arbitrary
form
whatsoever; rather,
they
mean
that,
to an
extent,
multiple
understandings
are
possible,
desirable,
and
even
inevitable.
In
constructing
our
music
theory,
we
have
accounted for this
state
of affairs
by
building
a
system
of
interactive
components
and
by
emphasizing
the
"pre-
ferred"
nature
of the
resulting
structural
descriptions.
Under
our
conception
of
music
theory,
then,
the under-
standing of a piece of music by the idealized listener consists
in
his
finding
the
maximally
coherent
structural
description
or
descriptions
which
can
be
associated
with
the
piece's
se-
quence
of
pitch-time
events.
Maximizing
coherence involves the
interaction of
a
number
of
different
domains
of
analysis,
each
of
which
must
be
repre-
sented
in
the
structural
description.
There
are four
with which
we will be
concerned
here,
to
be
termed
grouping
analysis,
metrical analysis, time-span reduction, and prolongational
reduction. As an initial
overview,
we
may
say
that the
group-
ing
analysis
assigns
group
boundaries
to
the
music in a hier-
archic fashion
at
every
level of
a
piece.
The
metrical
analysis
assigns
a
hierarchy
of
strong
and weak
beats. The
time-span
reduction
designates
"structural
beginnings"
and
"structural
endings"
of
groups,
and
assigns
to the
pitches
a
hierarchy
which
relates
them
to
the
grouping
and
metrical
structures.
The
prolongational
reduction
assigns to the pitches a hier-
archy
which
expresses
harmonic
and melodic
continuity
and
progression;
it is
the
closest
equivalent
in
our
theory
to
Schenkerian
analysis.
There
are
some
abstract
properties
common
to
these
four
115
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
7/62
domains
of
analysis.
First,
each
domain
partitions
a
piece
into
discrete
regions,
organized
hierarchically
in
such
a
way
that
one
region
may
contain
other
regions,
but
may
not
partially
overlap
with
other
regions.
For
example, Figure
1
a
represents
a possible kind of organization. Figure lb represents an im-
possible organization:
at
j,
two
regions
overlap
within
level
2;
and
at
k,
a
boundary
in level
3
overlaps
a
region
in
level
2.
Another
property
common to these
domains
is that the
processes
of
organization
are
essentially
the same at all hier-
archic levels.
A related
point
is
that the
processes
of
organiza-
tion of these domains
are
recursive, i.e.,
capable
of indefinite
elaboration
by
the same
rules.
Other
aspects
of musical struc-
ture, however, are not hierarchic in nature. For the present
we shall
ignore
these
dimensions.2
As mentioned
above,
the rules which
assign
structural de-
scriptions
are
categorized
as
well-formedness
rules,
which
assign
possible
structures,
and
preference
rules,
which
select
coherent structures from
possible
structures.
In
addition,
transformational
rules,
which convert structures into other
structures,
are
needed
for
special
cases
(such
as
elisions)
not
generatedby the well-formednessrules. Although transforma-
tional
rules have
been
central to
linguistic theory, they
play
a
peripheral
role in
our
music
theory,
at
least
at its
current
stage
of
development.
The
following
discussion
of
the
organization
of
the
theory
will be
informal.
Emphasis
will
be
placed
on how the
compo-
nents
work
in
principle.
To
give
a
complete
account
would
exceed
our
present
purpose,
which
is to
convey
in
general
the
goals, operations, and implications of the theory as a whole.
We
begin
with
a discussion
of
rhythmic
structure in terms of
grouping
analysis
and
metrical
analysis.
Then we
develop
the
two
modes
of
pitch
hierarchization,
time-span
reduction and
prolongational
reduction.
Finally,
we
apply
the
completed
system
to
a
piece
of some
complexity.
RHYTHMICSTRUCTURE
When
hearing
a tonal
piece,
the listener
naturally
organizes
the sound
signals
into
units such
as
motives, themes,
phrases,
periods,
theme-groups,
sections,
and the
piece
itself.
Our
generic
term for these
units
is
"group."
The
grouping
analysis
116
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
8/62
Figure
la
lb
3)
..
.
J
4)
k
a
Figure 2a
2b
: *
/
0,~?
117
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
9/62
picks
out the
groups
and indicates them
in
the
structural
de-
scription
by
slurs
beneath
the musical
notation. The
grouping
well-formedness
rules restrict the
possible
grouping
structures
to the
kind
of hierarchic
organization
discussed
above,
where
Figure
1a was
possible
and
Figure
lb was not
possible.
Under
these
conditions,
both
Example
1
a
and
Example
Ib are
possi-
ble
groupings.
However,
while
Example
la would
appear
to
show
the
cor-
rect
grouping,
Example
lb
is
absurd.
In
order to
select
the
actually
heard
grouping
or
groupings
(such
as
Example
la),
as
against
all the
merely
possible
groupings
(such
as
Example
lb),
we
develop
the
grouping preference
rules. These
provide
the criteria for pickingout groups, and are classified according
to
principles
of
(a)
articulation
of
boundaries, (b)
parallelism
in
structure,
and
(c)
symmetry.
Group
boundaries
are articulated
by
such factors as dis-
tance
between
attack
points,
rests,
slurs written into the
music,
change
in
register, change
in
texture,
change
in
dy-
namics,
and
change
in timbre.
A
further
articulatory
device
is
the
harmonic
cadence,
which from
the
phrase
level
upward
normally signifies the ending of groups;this will be discussed
later.
Parallelism
in structure involves
some
kind
of
repetition
or
similarity
in the
music,
such
as
a
motive,
a
sequence,
a
section,
and so
forth. The
similarity
is
particularly
crucial at
the
begin-
ning
of
groups;
for even
if
they
diverge
later
on,
they
are
still
perceived
as
parallel.
In tonal
music,
parallelism
is the
major
factor
in all
large-scale
grouping.
Related to parallelismis the principle of symmetry, which
states that
the
ideal
subdivision
of
any
group
is
into
equal
parts.
In
Example
la,
all the
groupings
are
assigned
by
the
parallelism
rule
(reinforced
by
various
articulatory
criteria),
except
for the
groupings
marked
s,
which
are
due
to
the
sym-
metry
rule;
thus,
each 4-measure
group
subdivides
not
only
into
1
+
1
+
2,
but,
at the
next
level,
into 2
+
2.
A
grouping
transformational
rule is
required
to account
for grouping overlaps, which as such do not meet the well-
formedness
conditions
of hierarchic
organization.
The trans-
formational
rule
relates
well-formed
underlying
groupings
to
the
musical
surface,
thereby preserving
the
sense that
overlaps
are variations
on
normal hierarchic
grouping.
Thus,
in
Exam-
118
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
10/62
Example
la.
Beethoven: Sonata
Op.
2,
No.
2,
Scherzo
^^f
$
r f
1
ff
r>TI
p
lb
P
A-
$
T
^-
z
C
$%
4^
v.?
t^
m J j J
^
v$^ .^^t^ v^rf^^' ^
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
11/62
pie
2,
the
events at
u
and
v
function as
both
endings
and
beginnings
of
groups;
these
overlaps
are
disentangled
at
under-
lying
levels.
We turn
now
to metrical
structure,
which
is
independent
of
grouping
structure but interactive with it. Given
appropriate
musical
cues,
the listener
will
instinctively
infer a
regular,
hierarchic
pattern
of
beats
to
which
he
relates
the
actual
musical sounds.
The
metrical
analysis
assigns
to a
piece
such a
pattern
of beats and
indicates
them
in
the structural
descrip-
tion
by
dots beneath
the
musical notation and above
the
grouping
slurs,
as in
Example
3.
Each
level
of
dots
represents
a
marking-off
of the
music
into equal time-spans. A dot at a particularlevel representsa
judgment
that
that
moment
in the
music is a
beat
at that
level. If a
beat
at
a
particular
level
is felt
to
be
"strong,"
or
"down,"
it
is a
beat
at
the next
larger
level
and
receives an
additional
dot.
(Thus,
metrical
structure
does
not exist
with-
out
at least
two
levels of
dots.)3
The
process
is the same
whether
at the
level of the
smallest
note value
or
at
a
hyper-
measure
level. The
notated
meter
is
usually
an intermediate
metrical level.
Theoretically,
the
dots
could
be built
up
to
the level of a
whole
piece.
However,
the
perception
of relative metrical
stress fades
over
long
timespans.
In
addition,
at
large
levels,
metrical
structure
is heard
within
the context
of
grouping
structure,
which
is
rarely regular
at
such
levels;
and
without
regularity
there can be
no
metrical structure.
Therefore,
metrical
structure
is a
comparatively
local
phenomenon.
The metrical well-formedness rules assure the hierarchic
condition
that a beat
at a
particular
level
must also
be
a
beat
at all
smaller
levels. Characteristics
of
metrical
well-formed-
ness in
classical
tonal
music
include
the
equal
spacing
of beats
and
the
provision
that at
each
successive
level the
distance
between
beats
must be
either
two
or
three times that
of the
immediately
lower
level. Musical
styles
of other cultures
and
historical
periods
often
require
more
complicated
rules
of
metrical well-formedness; the rhythmic complexities of tonal
music
arise
from
the
interaction
of metrical
structure
with
grouping
structure
and
pitch
structure.
It
hardly
needs
emphasizing
that bar
lines and
beams
be-
tween
notes
are notational
devices
and
not
part
of
the
physi-
120
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
12/62
Example
2.
Mozart:
Sonata
K.
279,
I
Example
3.
Bach:
"O
Haupt
voll
blut und
Wunden"
r.,N
I i I
I,
"
/t4
J
LJ
i
.
LL
t rbr
r r
r
C r
CJ
I
n I
j j
J
u Jj
*0
It*
0.
.
'
I.
'
*
r
?
?
e
*
0
0)..
.
.
I.
- -
I,
..
, T14 _
- I
...
dr
I
- I
I
II
--
I
I
I I I
_I
I
I I
I
?
? ?
? ?
*
?
?
e
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
13/62
cal
signal,
as
pitches,
durations,
dynamics,
and timbre
are.
From
the
physical .signal
the
metrical
preference
rules
assign
to
a
piece
the
actually
heard
metrical
structure
(or structures)
instead
of the
myriad
well-formed
but
inappropriate
metrical
structuresapplicable to it. For example, they select the metri-
cal
structure
in
Example
3 rather
than one in a
triple
meter,
or
one
which
places
the
strongest
downbeat
of the
passage
on
the
opening
event.
The metrical
preference
rules
can be classified
according
to
principles
of
(a)
cues
for
strong
beats, (b)
parallelism
with
grouping
structure,
and
(c)
regularity
of
pattern.
The
cues in
the music
for
relatively strong
beats
include
such
factors
as
attack, accent, change of dynamic, registerof pitch, harmonic
change,
and
suspensions.
Added
to
these is the listener's
tendency
to ascribe
parallel
metrical
structures to
parallel
grouping
structures
(especially
rhythmic patterns
which
form
groups).
If there
is
any
regularity
to
these
various
cues,
the
listener
extrapolates
an
entire
metrical
hierarchy,
which
he
will renounce
only
in
the
face of
strongly
contradictory
cues.
Syncopation
takes
place
when
there are
strong
contradictory
cues which yet are not strong enough to overridethe inferred
pattern.
A metrical
transformational
rule is needed
for metrical
overlaps,
in
which
a shift
in the metrical
structure occurs
in
such a
way
that
the
same
moment
in a
piece
serves
a
double
metrical
function.
The
transformational
rule
deletes
one
set
of
dots
in favor of
the other
at
the
musical
surface.
In
Figure
2a,
the weaker
metrical
function
is
deleted;
in
Figure
2b,
the
stronger is deleted. (It may be helpful to think of the largest
level of
dots
as
representing
the measure
or half-measure
level.)
Although
it
is conceivable
for metrical
overlaps
to
arise
independently,
they
generally
happen
as
a
consequence
of
grouping
overlaps
which
take
place
on
relatively
weak
metri-
cal stresses.
On the
other
hand,
if a
grouping
overlap
takes
place
on a
relatively
strong
metrical
stress
(as
in
Example
2),
no metrical
overlap
results.
Using
familiar
terminology,
we
call a
combined
grouping
and
metrical
overlap
an elision.
We
further
distinguish
between
elisions
in which
the weaker
metrical
function
has been
deleted
(as
in
Figure
2a),
and
elisions
in
which
the
stronger
metrical
function
has
been
122
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
14/62
deleted
(as
in
Figure
2b).
Example
4 is
typical
of
the
former
kind;
the
effect at
s is one of
jarring
reorientation.
In
the
latter
kind,
the effect is rather
one of
retrospective
awareness
that
a shift
has
taken
place;
there is an
example
of this
in
the
Schumann
song
analyzed
later in this
paper
(pp.
149ff).
Some
general
reflections
are in order
concerning
the
organ-
ization of the
grouping
and
metrical
components.
It seems
to
us essential that the two not
be confused
inadvertently.
It
is
tempting,
for
example, perhaps
on the
basis of a
misleading
spatial analogy,
to
attribute
duration to the
concept
of
metri-
cal
stress,
so that first
a
beat,
then a
measure or
motivic
grouping,
and
finally
a
phrase
receives
"strong"
or
"weak"
markingsas the analysis proceeds to higher levels. In ourview,
however,
the notion of "beat"
or
"metrical stress"
can
only
be understood
as a
point
in
time,
without
duration;4
hence
our use
of dots to
signify
metrical structure.
Beats are
correct-
ly
analogous
to
equidistant
geometric points
rather
than
to
the lines
drawn between
them;
while
rhythms
and
groups
have
duration, then,
beats do not. The
metrical
component
assigns
metrical stress
not to
groups
but
to beats
within
groups.
Of course the
listener senses
that a
group
has a
certain
weight
if
within it
there is a
strong
beat. But this
does
not
mean
that the
group
as
a
whole
receives
metrical
weight;
for
the weak beats
are all
equally
weak.
In
Example
5a,
for in-
stance,
the
E-flats are
metrically
equal,
a
fact
obscured
in
Example
5b,
in
which
metrical
properties
appear
to be in-
cluded within a
grouping
notation.5
Just as groups as such do not receive metrical stress, metri-
cal
structure as
such
does not
possess any
inherent
grouping.
Whether a
weak beat is
heard
as an
afterbeat or
as an
upbeat
is
entirely
a
matter of the
grouping
associated with
it.
Similar-
ly,
at
a
larger
level of
grouping, say
the
paradigmatic
ante-
cedent-consequent
pattern
in
Figure
3,
there is
nothing
in
the
metrical
structure
which
prevents
the half
cadence
in m. 4
from
resolving
as a full
cadence to
the
tonic in
m.
5.
It
does
not so resolve only because of the intervening group bound-
ary.6
Figure
3
poses
a
question
of a
different
sort.
A
phrase
can
be
roughly
characterized as the
lowest
level of
grouping
which
has
a
structural
beginning,
a
middle,
and a
structural
ending
123
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
15/62
Example
4.
Haydn: Symphony
no.
104,
I,
mm.
17ff.
i$f
r
f
r
0
1
8
i
j^
.
a
r
4^
r
Y)r
r J
8
-
.
.
.
?
Example
6.
Beethoven:
"Hammerklavier"
Sonata
op.
106,
I
j
f
a
-
t
'
t
*
*
;
'r
'f
r
r
rI
"""=~~''(t
i
cedent)
~
_ ~
ntec
de nt
124
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
16/62
: **
:
:
r'
'
I
?
':
, '
:
.
.
.
I
j[
fr'f
TS
'
rvm
f=
u=ro
.
-
-
L'
@
*
\
'F
rr
rJ
JJ
j
'r
r
ir
J
J
(extended
consequenT)
125
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
17/62
Example 5a. Mozart: Symphony no. 40, I
ibb
t
n ,
n
n
?
.
?
.
?
.
. .
?
? ? ?
?
? ?
?
?
Sb
;9
kr
f
,
r'
i
v
i
1,
/
,Example
7. Bach
Example
7.
Bach:
"0
Haupt"
126
*
.I
I
._
....
,
-,
-
I
'..,
?
,
,,
.
? ,
?
? ,
?
1
1 t
1
1 ?
J
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
18/62
Figure
3
I
w%%V I-
measure:
1
2
3
4
5
6
7
8
Figure 4a
(a'I
pitch
events:
W z
4b
(b)
(C)
w
y
z
4c
w
x
y
z
127
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
19/62
(a cadence).
Thus,
a
phrase
normally
has a kind of structural
gravity
at or
near its
group
boundaries.
If
groups
as such have
no metrical
stress,
and
if metrical
structure
is
an
extrapolation
of
multi-leveled, regularly spaced beats,
how are
these
points
of
gravity
accounted
for in the
structural
description?
In
Figure
3
there
is no conflict
between
metrical stress and
structural
weight
in
mm.
1
and
5;
but
mm.
4
and
8,
where
the cadences
occur,
are
metrically
relatively
weak
(depending
on
the
example,
the
cadences
might
even
happen
in
the second
halves
of these
measures).
One
solution
might
be
to move the
second
strong
metrical
stress of
each
phrase
from
the down-
beats
of the third
and seventh measures
to the
points
at which
the cadences actually occur.7 However, there are two strong
reasons
against
such a
revision
of the
metrical
component.
First,
it
would
entail
surrendering
one
of the formal
proper-
ties
common
to
all the
domains
under
discussion,
namely
that
the
processes
of
organization
are
essentially
the same at
all
hierarchic
levels.
Secondly,
it
would
mean
giving
up
the tradi-
tional
distinction
between
cadences
which
take
place
at
weak
metrical
points
and cadences
which
take
place
at
strong
metri-
cal points. The latter are particularlyimportant for achieving
large-scale
arrival
and-in
conjunction
with
grouping
overlaps
-continuation.
Thus,
in
Example
68 the
consequent
phrase
is
extended
so that
the cadence
at
q
arrives
on a
strong
beat
and
overlaps
with
the
succeeding phrase.
According
to the
pro-
posed
revision,
the
cadence
of the antecedent
phrase
at
p
also
would
have
to
be
metrically strong,
with
the result that
the
metrical
distinction
between
p
and
q
would be
lost.
This is
plainly not acceptable.
The
points
of
gravity
in a
phrase,
then,
are not to
be
inter-
preted
as
metrical
phenomena.
Rather, they
are
hierarchically
important
elements
produced
by
the
interaction
of
pitch
structure
and
grouping
structure;
and
they
stand with
metrical
structure
in a
contrapuntal
relation,
so to
speak,
which
under-
lies
much
of the
rhythmic
richness
of
tonal
music. The
metri-
cal
component
therefore
remains
as
originally
set forth.
As
will be seen below, the proper distribution of structural
weight
in a
phrase
emerges
in the
analysis
as
part
of
the
time-
span
reduction,
which
treats
a
phrase
as an elaboration
of its
beginning
and
ending.
128
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
20/62
PITCH
REDUCTION
Both the
time-span
reduction and
the
prolongational
re-
duction
assign
hierarchic structures
to all
the
pitch
events in
a
piece.9
To
represent
these two kinds of
reduction,
we have
invented two somewhat different
"tree"
notations;
these
visu-
ally
resemble,
but are
substantially
different
from,
the
tree
notation
utilized
in
linguistics. Linguistic
trees
represent
"is-
a"
relations: a noun
phrase
followed
by
a
verb
phrase
is
a
sentence;
a
verb followed
by
a noun
phrase
is
a verb
phrase;
and
so
forth. Our musical
trees,
however,
do not involve
gram-
matical
categories.
The
fundamental
relationship
which
they
express is that of a sequence of pitch events as being an elabo-
ration
of
a
single pitch
event.
The
dominating
event,
that
of
which a
sequence
of
events is an
elaboration,
is
always
one
of
the events in
the
sequence;
the
remaining,
subordinate events
in the
sequence
are
heard
as
relatively
ornamental.
"Reduc-
tion"-the
process
of
recursively
substituting single
events for
sequences
of
events-can be
thought
of
as the
inverse of
elab-
oration.
In the following exposition we begin with the time-span
component
and
then turn to the
prolongational
component.
In
both
cases,
after
outlining
the notations
and the
basic
principles
of
reduction,
we
apply
the
components
first
to
an
abstract
antecedent-consequent
pattern,
then
to the
opening
eight
measures of
Mozart's
Sonata,
K.
331.
In
the
tree
notation
for the
time-span
reduction,
a
"right
branch"
(
X),
in
which
a
line
to the
right
attaches
to a line
to
the left, denotes the subordination of an event to the preced-
ing
event
within
that
region
at that
level;
a "left
branch"
(
X),
in
which
a
line
to
the
left
attaches
to a
line to the
right,
denotes the
subordination of
an
event to the
following
event
within
that
region
at that level.
The
well-formedness
condi-
tions
for these
trees
prohibit
both the
crossing
of
branches,
as
in
Figure
4a,
and
the
assignment
of
more
than one
line of
the
tree
to the
same
event,
as in
Figure
4b.
(The
letters in
paren-
theses signify reductional levels.)
The
relevant
notion of
elaboration in
the
time-span
reduc-
tion is
elaboration
at
successive levels
within
more or
less
equally
spaced,
discrete
time-spans.
Within
each
time-span,
or
region,
a
dominating
event must
be
found;
that
is,
all
other
129
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
21/62
events in
the
region
are elaborations
of
that event.
Thus,
at
level
(c)
in the well-formed tree
in
Figure
4c,
x is in
the
region
of
w
and
is
an elaboration
of
w;
y
is
in
the
region
of
z
and
is
an
elaboration
of
z.
At level
(b),
all
four
events are in the
same
region,
and z
(together
with x and
y)
is an elaboration
of
w.
Before
the
rules
which
establish
the tree
structure
can
be
applied,
it is
necessary
to select the
regions
of
application
at
every
hierarchic
level
within
a
piece.
These
regions
for all but
the most local
levels of
analysis
consist of the
groupings
assigned
by
the
grouping
preference
rules. Within the
lowest
grouping
level,
smaller
regions
are
chosen in terms of levels of
metrical structure. In these smallerregions, a given weak beat
is bracketed with
the
preceding
strong
beat,
unless the
pre-
ceding strong
beat
is
separated
from
the weak
beat
by
a
group
boundary;
in
this
latter
case,
the
weak
beat
is
bracketed
with
the
following strong
beat.
In musical
terminology,
this means
that
a weak
beat
is an afterbeat
unless it is situated
at the
beginning
of a
group,
in
which
case it
is
an
upbeat.
See
Exam-
ple
7;
the
brackets
indicate
the
sub-group regions
of
applica-
tion. In a given tree, each level of branchingcorresponds to a
region
of
application
for
the
preference
rules.
Given these
regions
of
application,
the
preference
rules for
the
time-span
reduction
choose
the
syntactically
most co-
herent reduction
(or
reductions)
from
all the
possible
but
mostly implausible
reductions
of a set of
pitch
events.
Syn-
tactic coherence
in
this domain
can be
thought
of
in terms
of
stability.
These
preference
rules are
classified
as
(a)
those
which ascertain the most stable structure (the tonic), and (b)
those which
establish
the
hierarchy
of relative
stability
in rela-
tion
to the most
stable
structure.
The tonic-at
any
level,
local
or
global-is
selected with
reference to
the available
pitch
collection and cadential
struc-
ture at the
appropriate
level.
The
rules of
relative
stability
or
instability
are,
in
broad
musical
terms,
the
principles
of
relative
consonance
or disso-
nance. For example, a local consonance is more stable than a
local
dissonance;
a triad
in
root
position
is
more stable
than
its
inversions;
a chord
is more
stable
if its melodic
note
is the
same
pitch-class
as
its
root;
the
relative
stability
of
two
chords
can be determined
by
the relative
closeness
to the local
tonic
130
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
22/62
of
their
roots on the circle
of
fifths;
conjunct
linear
connec-
tions
are more
stable
than
disjunct ones;
a
pitch
event
is
more
stable
if it is in a
metrically
stronger position;
and
so
forth.
The
preference
rules
use all of these criteria
in
picking
out
which event in a
given region
is
dominating.
Thus
(and
let
us assume here
the
"preferred"
grouping
and
metrical
structures)
both
Example
8a and
Example
8b
are
well-formed
time-span
reductions.
The
preference
rules
select
the
reduction in
Example
8a as
opposed
to
any
other
possible
reduction,
such as
Example
8b.
In
these
examples,
and
in
all
succeeding time-span
reduc-
tions,
a
notation
formally
equivalent
to the
tree
appears
beneath the music proper: the relation of pitch structure to
metrical structure
is
expressed by
the
placement
of
the
syn-
tactically
most
significant
event
within a
bracketing
on
the
strongest
beat within
that
bracketing.'? Although
the
two
notations
express
the
same
information,
we
retain
both
in
the
time-span
reduction
because,
as
will
become clear
in
longer
examples,
they
serve
somewhat
different
purposes.
The one
below the
music
is useful in
hearing any
particular
hierarchic
level. The tree-which, though unfamiliar, is easily compre-
hended
with
a
little
effort-gives
a
picture
of all
the
levels in
relation to
one
another;
moreover,
it
is
illuminating,
in
con-
nection
with a
particular
piece,
when
compared
to the
pro-
longational
reduction.
At
relatively
local
levels,
the tree
for the
time-span
reduc-
tion
correlates
with
the
metrical
analysis
to
produce
the
paradigmatic
situations in
Figure
5. Both
(h)
and
(i)
pertain
to afterbeats: in (h) the event on the afterbeat is the less
stable
event,
such
as
a
passing-
or
neighboring-tone
or
-chord;
in
(i)
the
event on
the
downbeat is
the
less
stable
event,
such
as
a
suspension
or an
appoggiatura-tone
or
-chord. Both
(j)
and
(k)
pertain
to
upbeats:
in
(j)
the
event
on
the
upbeat
is
less
stable than
the event
on its
associated
downbeat;
in
(k)
the
event
on
the
downbeat
is
less
stable than
the
event
on
its
associated
upbeat.
In
these
ways,
the
relation
of
syntactically
significant events to metrical structure is made clear. At large
levels,
where
metrical
structure
is
no
longer
hierarchically
operative,
the
right
and left
branchings
simply
denote
subor-
dination
within
grouping
structure.
Instances
of
all
four
paradigmatic
situations
occur
in
Exam-
131
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
23/62
Example
8a.
Mozart:
Sonata
K.
331,
I
8b
(a)
tli
I
:
I
I
N
-
I
K
r
Pr
p
r PF
P
I)
r
pr
r prF
(e)
I
I-
-
W
L###9
L1
P
r
rr
^
(b)
#
i
1
(a)
#
s--
132
(e)
(
:';rr r
;
Lfr
il(d)
$
#
<
I
I
IjI.
?(c)-i
i
i
(b)
1
.
I
11
(a)
0#
11
J Ad
J.
J J -
{8tPtI,
_i m m
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
24/62
Figure
5
(h)
(i)
(
(
A
. X
X
*
Figure6
(9)
level
(h)
=
[b4
[c
[
[?c
level(g)
=
[b8]
[Cs]
measure: 1
2
3
4
5
6
7
8
*
0
* *
.
.
4
4
8
133
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
25/62
pie
9;
h, i,
/,
and
k in the music stand
for
Figure
5
(h), (i),
(j),
and
(k),
respectively.
(The
sub-phrase bracketing
is
the
same
as
in
Example
7.)
To
carry
the
reductional
process
any
further
than we
have
in Example 9, it is necessary first to develop conceptions of
the "structural
beginning"
and
the "structural
ending"
of a
group.
By
structural
ending,
we
mean
either
a
V-I
progression
when
it
appears
at the end
of a
group,
i.e.,
the
full
cadence,
or
those
variations
on the
full
cadence
known
as the
half
cadence
and
the
deceptive
cadence.
The cadence
is
designated
as a
syntactic
unit,
both
elements
of which
are retained at
the
appropriate
levels
in the
time-span
reduction,
with
the
first
element subordinate to the last.l1
By
structural
beginning
we
mean the
most
stable
event
early
in
a
group
in
which
there
is a structural
ending.
There
must
normally
be
at least
one
intervening
event
(the
"mid-
dle")
between
this stable
event and the cadence
if the former
is
to be
designated
as
a
structural
beginning.
Thus the
smallest
grouping
levels,
such as those
specifying
motives,
usually
do
not have structural
beginnings
and
endings;
all
groups
from
the phraselevel on up do have them.
The
structural
beginning
and the cadence
of a
group
are
specially
labeled
in the
reduction,
with
b
standing
for "struc-
tural
beginning"
and
c for "cadence."
They
dominate
hier-
archically
all other
events
in a
group.
As
a
visual
aid,
we
place
a
number,
signifying
the
number
of measures
spanned,
within
each
grouping
slur beneath
the
music;
the
same number
ap-
pears
as
a
subscript
to
the
b
and
the c
for each
group.
Thus
in
Figure 6 each b and each c receives the subscript "4" at level
(h)
in the
reduction,
since
they
begin
and
end
four-measure
groups.
However,
the
first b and
the last
c are also
the struc-
tural
beginning
and
the cadence
for the
entire
eight-measure
group;
therefore,
at
this
next
region
of
application,
they
re-
ceive the
subscript
"8"
and
are
retained
at level
(g).
In effect
this
labeling
creates
a
double-layered pitch
hier-
archy
between
those
events
which
are b's
and c's and
those
which are not. At local levels, all other
events
in a
group
are
subordinate
to
the
group's
structural
beginning
and
ending.
At
more
global
levels,
structural
beginnings
and
endings
are
subordinate
or
dominating
by
virtue
of
the hierarchic
struc-
ture
of
the
groups
for which
they
function.
At
the end of
the
134
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
26/62
Example
9.
Bach:
"O
Haupt"
/
\
/
\
/
o)(\
Ab
b
(a)(a)
(b)
(b)
C;
c1
I
I
I
I
F
r
r
r
r
'
'
r
, , ,,,
r
J
J
J
J
J
J
((#
i
i
X
J-r
j
j
I
w
f,
r Pfrf
I ij jI
I
(b)i
(a)
135
?
?
?
?
*
? ?
?
?
? ?
?
?-
?
*
?
?
*
:
*J
?
.
,P
44
j j j j
-
j
.
( WI----wj 6 - p - p tp FF -1-
I
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
27/62
reductional
process,
there
remains
only
the structural
begin-
ning
and
the
structural
ending
for the
piece
as a whole.
We
are now
in a
position
to
investigate
a
complete
time-
span
reduction.
Example
10
gives
the
music12
and
Example
11 represents its analysis. By this point, Example 11 should
on the whole
be
self-explanatory.
The
following
remarks are
in the
nature
of
annotations.
1.
The best
way
to "read"
Example
11
is first to examine
the
grouping
and
metrical
analyses,
then
to hear the
levels in
rhythm,
as notated
beneath
the
music,
in
their reductional
order.
If the
analysis
is
correct,
each level should
sound
like a
natural simplification of the previous level. Any difficulties
which
the
reader
initially
has
in
deciphering
the
tree
can be
cleared
up
by
a
step-by-step
comparison
with
the notation
underneath.
2. Level
(f)
in
Example
11 has
already
been reduced
from
the actual
music
(Example
10)
to what
is felt
to
be
its
small-
est beat
level.
This
is a
convention
which
we
always
follow
in
constructing
the
time-span
and
prolongational
reductions
for
a piece, partly because syntactically unimportant detail is
thereby
eliminated,
and
also because
the
two
reductions
do
not
significantly
diverge
beneath
this level.
3. It would
be
possible
to
assign
further
low-level
grouping
slurs within
these
eight
measures.
However,
since
the
indica-
tions
for
these
groupings
are somewhat
conflicting,
and
since
such
groupings
would
not
affect
the
analysis
as
a
whole,
we
choose
not to
assign
them.
Therefore,
the weak
beats
within
the lowest groupingslurs are simply bracketed as afterbeats.
4. The
structural
beginnings
and
cadences
are labeled
at
each
level,
but
they
do
not receive
subscripts
until the
reduc-
tional
process
has reached
the
grouping
levels
for which
they
function.
5.
The
selection
of which
events
are
dominating
is
straight-
forward
except
in mm.
3 and
7 at
level
(d).
In m. 3 there
is a
conflict
in
the
preference
rules between
the
metrically strong-
er position of the F-sharp-E-A
chord and
the
more
stable
structure
of
the
V6
chord.
The
F-sharp-E-A
chord
is
chosen
for
reasons
having
to do with
the
phrase
as a whole:
the
regu-
larity
of harmonic
rhythm
is
preserved,
and
a
descending
line
in
the bass
from
the
tonic
to
the dominant
is created.
The
136
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
28/62
Example
10. Mozart: Sonata
K.
331,
I
Andante
grazioso.
A I
(ŝa rr%p-r
f*-*
a
(-)
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
29/62
K,
,
I
k I K .
.
I
K I
.
f,
P
rr
rr
r
r
G
rPt
d
r
Gr
r
'r
rr,
r
P
^r2i
3
,r
rr
r
r
P
r
'
r
pr
p
pC
r
cJ
4
8
(e) on>
I
I
i
ii i
,
I
i
.
i
I
i1
[)
g
~
J
2
_
2
'
tb 2 2 2 2t
c
J
(C)
*'Io
|I,
1,
;
B
114)
(C
1^^
C4
i
4
4
4
ca
,
aYI
8c
c
-
~~~~~~~(a)
,a>ll D
I b 3
r4
C
138
Example
11
n.,
A I
k I
K
I
I
I
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
30/62
same
chord is
chosen
in m.
7
because
of
the
parallelism
with
m.
3.
6. Note that if the
cadences were
not labeled and
retained,
in
m.
4 the
I
chord instead of the
V
chord would have
been
selected at level (d). The labeling also causes the retention at
levels
(d),
(c),
and
(b)
of
both elements
of
the
full
cadence
in
m. 8.
7. Levels
(c)
and
(b)
can
be construed
as
Schenkerian
"background"
evels.
Schenker's own
analysis
of
this
passage13
gives
the
melodic structural
weight
to
the
E
in
the
first
mea-
sure rather
than to
the
opening
C-sharp.
In
our
theory,
it
would
be
possible
to achieve his
result
by
applying
the
"arpeg-
giation rule," i.e., by regardingthe first measure as a broken
chord
and therefore
compressing
it
into one
event
placed
on
the
downbeat
of
the measure.
However,
such
a
decision
pro-
duces difficulties
at
intermediate
structural levels
(especially
level
(d)).
8.
In
the
large
levels
(b)
and
(a),
the sense of
metrical
struc-
ture
has
faded to the
point
that it is
largely inoperative.
The
events
at these
levels in the
notation
beneath the
music
do
not receive rhythmic values because there are no more dots in
the metrical
analysis
to which
such
values
could be
related.
It
is at
this
point
in
the
reductional
process
that
pitch
events
can
be
thought
of as
rhythmically
free-floating.
9.
It
would
perhaps
be
sufficient
to
stop
the
reductional
process
at level
(b).
Level
(a)
simply
selects
the
most
stable
event
from
those
available at level
(b).
As is
often
the
case,
the
most
stable event here
is the
last
chord; thus,
the
rest of
the piece is a left branch to it. This situation can be inter-
preted
to
mean
that,
in a
sense,
all
the
events of
a
piece
except
the
last
constitute a
delay
of
the
moment of
complete
repose,
which
is
the
ending.
If
the
point
of
maximal
stability
happened
in
the
middle
of
a
piece,
there
might
be
no
reason
for
the
piece
to
continue.
10. We
have
considered these
eight
measures
as
if
they
were
a
complete
piece.
If
the
entire
theme,
which is
18
measures
long,
were
analyzed,
the
opening event would eventually be
labeled
b18
and thus
would
dominate all
other
events in
the
first
eight
measures.
11.
Observe
how the
geometry
of
the
tree,
while
accurately
conveying
the
hierarchy
of
pitch
events,
also
mirrors
visually
139
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
31/62
the
partitioning
of the
piece
into
time-spans
(groupings
and
bracketings).
Moreover,
the tree
notation makes
visually
clear
the interaction of the
pitch
structure
with
the
metrical struc-
ture at all
pertinent
levels.
There
are
important
hierarchic
aspects
to
the
pitch
struc-
ture
in
Example
10 which do not
emerge
in the
time-span
reduction
in
Example
11. To
capture
these
aspects,
we must
develop
the
domain
of
analysis
called
the
prolongational
re-
duction.
In the
parameter
of
rhythm
in this
domain,
events
come before or after other
events,
but
they
are not
measured
according
to some metrical
conception.
The
relevant
notion
of elaboration is elaboration by harmonic and melodic con-
nection.
There are two
kinds of elaboration
in the
prolongational
reduction:
prolongation,
in
which
a
pitch
event
is
elaborated
into
two
or
more
copies
of
itself;'4
and
contrast,
in
which
a
different,
relatively
ornamental,
pitch
event is
introduced.
A
prolongation
is
represented
by
two branches
extending
from
a
small circular
node,
as in
Figure
7a. Neither event takes hier-
archic priority; rather,both events are thought of as an exten-
sion over
time of what
at a
deeper
level is the same event.
In
contrast,
hierarchic subordination
is
designated by
right
and
left
branching,
as in the
time-span
reduction.
However,
where-
as
in the
time-span
reduction
these
branchings
only
indicate
the
subordination
of
one event to
another,
in
the
prolonga-
tional reduction
they
receive
a further
interpretation.
A
right
branch
signifies progression
in a
piece,
whether
at
a
local
level
as in Figure 7b or at a large structural level. A left branch is
utilized
only
at local
levels and
signifies
delay
in
relation
to
the
bass,
as
at level
(c)
in
Figure
7c.
(Since
metrical
structure
does
not
play
a
role in
the
prolongational
reduction,
Figure
7c
represents
a
suspension
considered
only
with
respect
to its
pitch
structure.)
The well-formedness
conditions
and
the
preference
rules for
the
prolongational
reduction
are
similar
to
those for
the
time-
span reduction.
The well-formedness
conditions
preclude
the
crossing
of branches
and
the
assignment
of more
than
one
branch to
the
same event.
Given a
sequence
of
pitch
events,
the
preference
rules
select
a
hierarchy
according
to
principles
of
140
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
32/62
Figure
7a
(a)
(IM
--
_
-
U
*--
(a)
(b)
(b)
J>I^vwv-Y
I
-
I
||
[ba]
[C4]
[b,]
[cs]
measure: 1 2 3 4
5
6 7 8
time-span
reduction
time-span
reduction
(a)
>
(a)
(b)
)
(b)
(b)
>
(c)
>
(c
(C)
)
(C
I- s- ^-y
I-
^^
--1[
1 2 3 4 5 6 7 8
prolongational
reduction
7b
7c
(a)
(b
O
1
(a)
(b)
I
*?
Figure
8
141
i .
1i=
-I1dn
.
f
Ii
11
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
33/62
stability largely
resembling
those
for the
time-span
reduction.
Meter is
disregarded;
prolongations
are
maximized.
At
any given
level down
to the
phrase
level,
the
sequence
of
events
available
to
the
preference
rules
for the
prolonga-
tional
reduction
is
precisely
the same
sequence
as at the
equivalent
level in the
time-span
reduction.
In other
words,
the
hierarchy
of
b's
and
c's as
designated
in the
time-span
reduction
heavily
determines
the
prolongational
reduction.
Again,
we illustrate
with
a
typical
antecedent-consequent
pattern.
In the
time-span
reduction
in
Figure
8,
levels
(a),
(b),
and
(c)
refer
to events
labeled
as
structural
beginnings
and
endings;
these events
at
equivalent
levels become
the material
for the prolongational reduction in Figure 8.
Even when
precisely
the
same events are
available
at a
given
level
for
the two
kinds
of
reduction,
the reductions
draw
radically
different
connections
among
these events.
For in-
stance,
in
Figure
8
the
V chord
in the
full cadence
in m.
8 is
a
left branch
to the
ensuing
I
chord
in the
time-span
reduction,
but
is a
right
branch
from
a
prolonged
I
chord
in the
prolon-
gational
reduction.
In the
former
case,
it
is
a
left branch
be-
cause it is within the time span of the final tonic; in the latter
case,
it is a
right
branch
because
it
progresses
to
the
final
tonic.
(In
both
cases,
it
is subordinate
to
the
tonic
according
to
principles
of
stability.)
Large-scale
right
branching
in
the
prolongational
reduction
always
indicates
significant
syntactic
"progress"
n a
piece.
Note, too,
that
at
level
(c)
in
Figure
8
the
prolongational
reduction
brings
out
connections
of harmonic
identity
not
captured in the time-span reduction. The tree for the time-
span
reduction
would
look
the
same even
if the
b
for
the
consequent
phrase
were
an
entirely
different
chord;
in the
prolongational
reduction,
however,
such
a
change
would
pro-
duce
a
right
branch
at
level
(c)
instead
of the
prolongation
represented
there.
On
the
other
hand,
the
tree
for
the time-
span
reduction
expresses
grouping
structure, something
not
conveyed
in the
prolongational
reduction
itself.
Thus,
for
example, the tree for the prolongational reduction cannot
differentiate
between
the
full cadence
in m.
8,
and
the
phrase-
ending
half cadence
followed
by
the
phrase-beginning
tonic
chord in
mm. 4-5.
142
8/17/2019 Jackendoff Toward a Formal Theory Tonal Music
34/62
From the
phrase
level
down
to the smallest beat
level-that
is,
for all events
which
are not
b's and
c's
except
for the
most
local details-the
prolongational
reduction
is
constructed
without
reference to
the
time-span
reduction.
As
a
result,
within this
region
the two reductions
usually
differ not
only
in the connections
made but
in the actual
sequences
of
events
at
corresponding
levels.
To facilitate
reading
the
prolongational
reduction,
we
give,
beneath the music in a
non-rhythmic
notation,
the
actual
sequence
of events for each level from
the
phrase
on
down.
(See
Example
12.)
Unlike the
tree, however,
this
secondary
notation
does
not
convey
the
types
of
elaboration within a
sequence of events.15For each level from the phraseon up, it
is
sufficient to mark the events
which,
as
b's
and c's
in
the
corresponding time-span
reduction,
have
caused the
equiva-
lent
sequence
of events in the
prolongational
reduction;
this
marking
is
accomplished
by
placing
the
corresponding
letters
in
parentheses
at the
appropriate places
just
below the
music.
Example
12
represents
the
prolongational
reduction
for
the first
eight
measures of
Mozart's
K.
331.
Remarks:
1. The most local level in the
prolongational
reduction
(level
(h))
corresponds
with
the
"lowest
beat level"
as
deter-
mined
in
the
time-span
reduction
(Example
11,
level
(f)).
In
placing
this
lower
boundary
on
the
prolongational reduction,
we are
in
effect
claiming
that
beyond
this
level
local detail
is
not
of
prolongational
significance.
2. Level
(a)
represents
the level
of
abstraction
at
which the
A
major
root
position
triad is
totally
unelaborated. The
high-
est level of the prolongational reduction for any classical tonal
piece always
results in
an
undifferentia