Page 1
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 1/12
MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Computer Music Journal.
http://www.jstor.org
The Hierarchical Structure of Time and MeterAuthor(s): Masato YakoSource: Computer Music Journal, Vol. 21, No. 1 (Spring, 1997), pp. 47-57Published by: MIT PressStable URL: http://www.jstor.org/stable/3681218Accessed: 03-10-2015 22:42 UTC
F R N S
Linked references are available on JSTOR for this article:http://www.jstor.org/stable/3681218?seq=1&cid=pdf-reference#references_tab_contents
You may need to log in to JSTOR to access the linked references.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of contentin a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship.For more information about JSTOR, please contact [email protected] .
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 2
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 2/12
Masato
ako
Department
of Acoustic
Design
Kyushu
Institute of
Design
Minami-ku 9-1 Shiobaru
4-chome
Fukuoka-city,
Fukuoka
815,
Japan
[email protected]
h e
ierarchical
tructure
o
i m e
a n d
e t e r
This article considers the
interrelationships
of
hier-
archy, time,
and meter
in
music. Music is an
art
form that
organizes
time and reflects different attri-
butes of time.
A
unique point
in
the flow of time is
the
present.
In
physical
terms,
the
present
is a
point in time; in practical terms, however,the pres-
ent
is understood
to have
a
certain
span,
the extent
of which varies
depending
on the context
in
which
the
present
is
placed
(Mach 1906;
Husserl
1928;
Fraisse
1963,
1982). Similarly,
n
music,
the
present
has
a
time-span.
For
example,
the
present
could
mean the note
A,
the first
subject,
or the
develop-
ment,
etc.,
depending
on
the
context
in
which the
music
belongs.
If
a
random
point
in
time
happens
to
be
in the
development
section,
the
section
began
before that
point
and is
likely
to
continue
after it.
Basically,
the selection of
a
musical
present
is also
attributable to the characterof a piece of music (a
musical
style). However,
the
selection is not the
sole
choice for a
piece.
The
interpretation
of
a
piece
of
music
varies,
depending
on
the
time-span
that
is
adopted
as the
present.
There
are a
number
of
possible
time-spans
for a
musical
present.
This
fact
is
inseparable
from
the
hierarchy
that
is
latent
as
a
structural
element
in
music.
Specifically,
that
different
time-spans
can be
adopted
as
the
present
indicates the existence of
hi-
erarchical
evels
in
music,
from
higher
strata
(longer units)
to lower
strata
(shorter
units).
In
other
words,
the
ability
to select a musical
present
in
a
number of
time-spans
is a
positive
indication
of
the hierarchical
structure
of
music.
In
music,
meanwhile,
a unit of time
adopted
as
the
present
is maintained under
musically
intrinsic
dynamics
to
form
a
piece.
One
element of
such
dy-
namics
is
meter. When the
time-span
selected as
the
present
is
repeated cyclically
to
a
beat,
the
char-
acter
of the musical
present
is
maintained,
and
the
time structure that is
intrinsic in music is formed.
First,
a discussion on
hierarchy
in music
is
pre-
sented. This
is followed
by
the introduction of
a
model
to
consider the
relationships
between multi-
ple time-span choices, hierarchy,and meter. A the-
ory
of musical
time,
based on
hierarchy,
s then
de-
veloped.
Hierarchy
nd
Perception
A
hierarchical structure
has
a
structure
similar
to
a
tree
model,
where
each
element
converges
to a sin-
gle
source.
The
advantage
of
introducing hierarchy
to
understand
a
phenomenon
is due
to one's
ability
to
take
up
any
random
part
in a
system,
and
clarify
its position within the system. In other words,by
adopting
a
hierarchical
perspective,
the
position
of
a
part
within
the whole
becomes clear. Music
is
perceived
and
recognized
whatever
its structure.
Perception
cannot be
separated
rom
hierarchy.
For
example,
the
question
of
whether an
element in
music should be
perceived
in
relation
to
the
piece
as a
whole or
independently
as a
part
becomes a
question
of
whether to
place
emphasis
on
the
per-
ception
of
a
higher
hierarchical
stratum
(the
whole)
or a
lower stratum
(a
part),
and is
thus reduced to a
problem
of
hierarchy.
A
hierarchical
cognizance
without the
immediacy
of
perception
is also
pos-
sible,
as we can
have
a
bird's-eye
view
of
a
hierarchi-
cal
analysis
model
in
our
mind,
and
recognize
both
the whole and
the
parts simultaneously.
For
ex-
ample,
H.
Schenker
(1956)
describes an
Ur struc-
ture
into which a
random
movement of
tonal
mu-
sic
can
be condensed. But
this is
cognizance
of
music
through
a
spatial
representation
of
a
hierar-
chical
model
image
that
is unrelated to
actual
per-
ception.
Problems
specific
to
perception
and time
cannot
be resolved
by
considering
the
hierarchy
of
Computer
Music
Journal,21:1,
pp.
47-57,
Spring
1997
?
1997 Massachusetts Institute
of
Technology
Yako
47
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 3
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 3/12
Figure
1.
A
tree
model,
in
which
the rules
for
one
hi-
erarchical
stratum
are
linked to the
rules
in
the
strata above
and
below.
he
linking
rules
from
the
individual
perceptions
to
the
core
are
transparent.
music
solely
from
cognizance
through
spatial repre-
sentation.
Hierarchical
trata nd
Representation
As
a
prerequisite
to
viewing
hierarchy dynamically
in relation to
time,
the
general
characteristics
of hi-
erarchy
and
perception
will first be
considered.
There
are
basically
two
types
of
hierarchy
n
percep-
tion:
(1)
where a
representational
level and
a
physi-
cal
level
are found
in
combination at
each
stratum
of the
hierarchical
process;
and
(2)
where
only
phys-
ical levels
exist,
with no
representational
evel.
First,
we
will
consider
type
(1).
If
a random
physi-
cal
group
has
attributes
that cannot be
prescribed
by the physical level within that groupalone, but
are
stipulated by
other
levels,
a hierarchical
repre-
sentation is used
in
the
terminology
for
categoriza-
tion or
analysis.
In
type
(1),
a stratum
becomes
a
method for
representing perception
and
cognizance.
A
representational
level exists for each
upward
shift of
perception
to a
higher
stratum.
For
ex-
ample,
selective attention
in
the
cognitive
pro-
cess is formed
by
a
set of
physical
level and
repre-
sentational levels
in
perception
(Broadbent
1958);
attention here is
established
representatively (sym-
bolically). However,
although
the
separation
of the
representational evel and the physical level is a suf-
Figure
2.
A
tree
model
whose
linking
rules
from
the individual
perceptions
to
the core are not
trans-
parent.
ficient condition for the existence
of
a hierarchical
structure,
it
is
not a
necessary
condition. In the
hi-
erarchical structure
of
type
(2),
the
possibility
must
be considered
of a
case
where the
physical explana-
tion itself
fulfills
its function
in
the
structure of a
phenomenon,
so
that there is no
need
to
resort
to
representation.
In
type (2),
the
hierarchy
of
phenom-
ena is understood as
a
combination of materials.
Therefore,
in
the case of
type
(2),
hierarchy
is
latent
in
phenomena;
when
it
is
represented,
the
represen-
tation is
made
as a
discovery.
Linking
ierarchicalevels
In a
hierarchical
structure,
what
are first
discerned
are the rules prescribingeach stratum. Formed in
each
stratum,
these rules
are attributes that are
spe-
cific to the
stratum.
They
serve as
keys
to
the
dis-
covery
of a
representational
evel that is
specific
to
a
hierarchical
stratum.
Rules,
therefore,
exist in hi-
erarchical
strata,
regardless
of
whether
or
not
repre-
sentational levels can be
discovered.These rules be-
come
keys
to
progressive
abstractions until the
framework
of the
structure is
extracted. Even
if
the
structural framework s not
directly
perceived,
it
is
understood as a
simplified
model
(see Figures
1
and
2).
The rules for
one hierarchical stratum
are
linked
to the rules in the strata above and below it. In Fig-
48
Computer
Music
Journal
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 4
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 4/12
Figure
3. The nested
hierar-
chical
structure
of
rule
and
medium.
The level
of
sound
that
composes
a
note is
S1;
the
level
of
a
single
note that
forms
the
note
unit is
S2;
the level
of
a
unit
or
phrase
compris-
ing
a
number
of single
notes is
S3;
and a musical
unit
comprising
a number
of
S3 stratum units
is S4.
A rule
implemented
in
a
particular
stratum
func-
tions
in the stratum above
as a
stratum-forming
medium.
Si
S2
53
S4
Medium
C
Rule
Medium C Rule
Medium C Rule
ure
1,
the
system
of
rules
from the individual
per-
ceptions
to the core is
apparent.
For
example,
H.
Schenker
(1956)
reduces
a
melody
to an
apparent
Ur
structure.
Fred
Lerdahl
and
Ray Jackendoff
(1983)
also
present
a tree model for
a
bird's-eye
de-
scription
of hierarchical
rules,
but rather than
as-
sume
an
Ur structure
into which
melody
can
be
condensed,
they
concentrate on
describing
the
rules
that
provide
keys
to condensation.
In
contrast,
in
Figure
2,
the
representational
ev-
els for
reaching
the
Ur
structure are not
apparent.
For
example,
E. Narmour
(Narmour 1977,
1983;
Meyer 1973)presents
an
analysis
model based
on
implication
realization,
and
proposes
the
direction
from
the
low-level data
up.
However,
his
analysis
model is not reduced to a tree model. M. Yeston
(1976)
also
attempts
a
description
of the hierarchi-
cal structure of
melody
based on the metric struc-
ture,
but
does
not
aim
at a
reduction to a
tree
model. The hierarchical model of
N.
Ruwet
(1972)
is characterized
by
descriptions
of
equivalence,
repe-
tition,
and
transformation,
but whether the
pres-
ence
of a
work
can
ultimately
be retained
in
a
spe-
cific hierarchical level remains a
question.
In
this
case, the hierarchicalanalysis method does not aim
at a reduction
to a
tree
model,
but instead concen-
trates
on
linking
rules between
higher
and lower
strata.
Rules
nd
Media
I have
described how
individual hierarchical strata
have
rules for
representation
levels,
even
if
they
are
not
apparent.
Let
us now consider the
underlying
characteristics of rules at each stratum. In a hierar-
chical
structure,
a rule
is linked to
upper-
and
lower-strata rules. When rules are stacked
step-
wise
in this
way,
the stratum
below
the rule-
implementing
stratum must
become
the
formal
and
physical
material
from
which
the stratum
above
it is
formed.
In
other
words,
a lower
stratum
forms the medium
for
rules
in
the
higher
stratum.
Medium
here
can
be
described as the soil in which
rules are sown. For
example,
the
format
for
sound
that
forms
representation
rules
(such
as
a
note's
pitch
or
length)
is a medium.
Similarly,
a rule
implemented
in a
particular
stratum
functions
in
the
stratum above
as a
stratum-forming
medium
(Yako 1992b).
Considered
thus,
a
set of rule
and me-
dium
has
a
nested
structure,
as
shown
in
Figure
3.
Let us assume that
S1
is the level
of sound that
composes
a
note,
S2 is
the level of a
single
note
that
forms the note
unit,
S3
is
the level of a
unit
or
phrase comprising
a number of
single
notes,
and S4
is
a
musical unit
comprising
a number of
S3 stra-
tum units.
In
this
system,
an
aggregate
of S2
rules
forms
the
media
for S3
rules; likewise,
an
aggregate
of
S3
rules
forms
the
media for S4.
Inter-Penetrationf Hierarchical
trata
Generally speaking,
one
condition of a
hierarchical
structure
is
that each stratum is
independent,
i.e.,
strata are
discretely
structured.
A
representational
rule is an
attribute
that
is
specific
to an
individual
stratum,
and
its
scope
is
limited
to a
specific
stra-
tum.
Therefore,
rules,
when
represented,
exist dis-
cretely
between
higher
and lower
strata.
In
con-
trast,
media can be
interlinked
indiscretely,
as
long
as the restriction of
discreteness
is maintained
by
representationallevels (see Figure4). Forexample,
Yako 49
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 5
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 5/12
Figure
4.
Inter-penetration
between media in
nested
hierarchical
structure.
The
sound
format
in
S1
that
forms
the medium
for
S2
rules can
also
form
the
me-
dium
for
S3
rules.
Media
can
be interlinked
indis-
cretely.
S1 S2
S3 S4
Medium
C Rule
(physical)
Medium
C
Rule
(physical)
e d i u m u l e
the
S1
medium
for
S2 rules is
in no
way
different
(other
than
quantitatively)
from
the
S1
medium for
S3 rules. Thus
it
can be said
that,
in
contrast to
rules,
media have a
strata-linking
unction. For
example,
the sound format
in
S1
that
forms the
me-
dium for
S2 rules
can also form the medium for S3
rules
without
any change. Similarly,
the
series
of
note units
in
S2,
which is
the
medium
for
the
phrase
in
S3,
can
form,
if
extended,
the medium for
S4 rules.
In
this
way,
a medium is
open-ended
to-
ward
higher
strata,
and has the function of
linking
strata.
A
particular
situation that needs to be
considered
is
that when the media
attributes
of
strata
Sn-1
and
Sn-2 invade stratum
Sn,
the
rules
specific
to
Sn
may
not be realized. For
example,
when an
ex-
tended sustained note is
accompanied
by
melisma
with
pitch
and
duration,
the
sound attributes of
S1
flow
into
S3 with the hierarchical
structure of
strata
S1
and
S2
remaining
indistinct.
In
such a sit-
uation, the medium attribute of Si, i.e., sound
level,
may
extend
the
perception
of
sound and
make it
difficult for that
perception
to
lose
signifi-
cance
by
the
higher
stratum rule. This
type
of inva-
sion of medium occurs
primarily
from a
lower stra-
tum
to a
higher
stratum.
In
the
case as
described,
t
must be
kept
in
mind that an
invasion of a
medium
from a lower
stratum
may
impede
the uncondi-
tional
recognition
of
higher
strata rules. This
pro-
vides a basis
for
viewing
hierarchical
strata not stat-
ically,
but as
having
competitive
interrelationships.
The
nesting
model for
rules and media that
fol-
lows is a new presentation and discussion.
Toa
Theory
f Time
In the
previous
section,
a
general
discussion on
hi-
erarchy
and
perception
was
presented.
In
reality,
however,
music
is
not
necessarily
recognized
trans-
parently
in
terms of
hierarchy.
Even
if
music is rec-
ognized
as
having
a hierarchical structure
on the
one
hand,
it
cannot avoid
being perceived
on the
other within the frameworkof time. The percep-
tion
of
music,
therefore,
can be
described as the
act
of
following
its
hierarchical
process
accompanied
by
the
passage
of time.
However,
the
character
of
time does not allow a
merely
mechanical
pursuit
of
music's hierarchical
structure;
a
particular
charac-
teristic of
time
may
obstruct a hierarchical
cogni-
zance.
The
PresentMaintained
y
Media
A
musical
present
has a
time-span
whose
length
varies,
according
to the context.
Because a time
pe-
riod
for
the
present
can be
note
A
or the
first sub-
ject,
the
span
of the
present may
be
discontinuous
with
intervals,
and the
present
can be
so
broad as
to
encompass
the entire
discontinuous
zone
(Mach
1906;
Fraisse
1963).
E.
Husserl's
Retention
(Husserl
1928)
assumes that
the
past
is
saved in
the
present.
He
argued
that
consciousness of
the
present
is ac-
companied by
retention: the
present
is like a
body
of
a comet
(Kdrper
es
Komet),
and the
past
is the
comet's tail (Schweif des Komet). In other words,
50
Computer
Music
Journal
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 6
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 6/12
depending
on
how the
present
is
viewed,
its
span
could be
long
or short.
Meanwhile,
when a musical
present
is consid-
ered
from
a
hierarchical
perspective,
it is the strata-
linking
function
of media that maintains
the
pres-
ent
in
music.
A
medium,
regardless
of
whether
or
not
a
corresponding
rule
exists,
links
a
lower stra-
tum
with
a
higher
stratum
and
brings
into the
higher
stratum
the
characteristics
of
the
present
that
are
selected
in the lower stratum.
At the same
time,
a
rule,
when
represented,
has
a basic characteristicof focusing on one stratum.
Therefore,
a
rule that is
represented
s
independent
of
a
musical
present.
Whereas the
hierarchically
in-
discrete character of
media
basically
extends and
maintains the
present,
the
hierarchically
discrete
character
of rules extends the characteristics
spe-
cific to each stratum fora static construction of
music. The
generation
of
a
pitch
class
(the
S3
rule)
through
differentiation
of
pitch
is an
example
of
the latter.
In
any
case,
a rule comes
in
contact
with
the
present
only
through
a medium.
In
this
way,
hi-
erarchy
can be understood also as a
temporal
exten-
sion of media.
L.
B.
Meyer
also seems
to
have
recognized
the
conflicting relationship
between medium and rule
while the medium forms
continuation. Mr.
Meyer
argues
that
a
redundant
note
movement
delays
a
de-
cisive note movement to induce emotional
move-
ment
(Meyer
1956).
If
decisive note movement is
read as rule and redundantnote movement
as
me-
dium,
it
can
be
said
that a medium is
maintained
while
in
conflict with rules to form emotional ac-
tivity.
However,
Mr.
Meyer
does
not
recognize
that
rules and media
nest
hierarchically.
Sound
s Presence
The
Present
n
S1)
Music is
actually
experienced
through
sound. Con-
tact
with
sound
in
music
is
restricted to the
pres-
ent-the
present
that
appears
as sound indicates a
point
in music. A
note
unit
(such
as
S2)
is indi-
cated
by sound,
which is the
presence
in
S1;
it
is re-
alized
through
the
medium of sound as
presence.
Sound as
presence
also forms the medium for note
groups at strata S3 and above through its strata-
linking
function,
and acts as
a
presence
at the
higher
strata.
Thus,
the
present
at
S1
that is
pre-
sented as sound is transferred
o
higher
strata with
the
passage
of
time,
affirming
the
span
of an
in-
stant as
a
present
in
music,
and
giving
direction
to
the actualization
of music. When the
indication
of
a
point
through
sound
is
implemented
throughout
a work
of
music,
the
piece
is
completed
and
comes
to an
end.
TheRepetitionf Notes
A
musical
work memorizes
past
notes.
This
means
that
a work
is
recognized by retaining
notes in the
mind
through
perception.
Generally
speaking,
the
difficulty
of
forming
note
memory
is
in
proportion
to
the
temporal
distance between the note that is
being
sounded now and the notes
that
have
already
been sounded. The
time
mechanism
promoting
the
formation
of this
memory
is
the
repetition
of
notes. Notes
presented
in
repetition
are
retained
in
memory
more
easily
than
are
single
notes.
Repeti-
tion of notes is also related to the formation of the
present.
The unit of
repeated
notes
forms
the
span
of
the
musical
present.
In
this
way,
repetition
of
notes
can also
be
seen
as a mechanism for
prescrib-
ing
the
present
in
music.
I
will next consider
how
the
repetition
of
notes
interacts with
individual
strata of music.
S1
and
Repetition
Whereas S2
and
above,i.e., combined notes, arepre-
sented as
facts,
S1
or
sound format is
described sta-
tistically.
This
is because
describing
a
form
of
sound
in
a
piece requires
a
process
for
obtaining
the
average
rom
a
continuous acoustical
presenta-
tion
of notes. For this
presentation
of
statistical
form
to be received as a
fact,
it
must be
repeated
(Desain
and
Honing 1992).
Through
repeated pre-
sentation,
the
sound of a note
presented
statisti-
cally
is
realized, memorized,
and
prescribed
as a
fact.
Therefore the
S1
note,
as a
medium
for S2
and
S3,
makes clearer the
representation
attributes of
S2 and S3 by repetition. It anticipates the clarifica-
Yako
51
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 7
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 7/12
Figure
5.
The S3 rules
in
a
nested
hierarchical struc-
ture.
Although
the unit
of
range for
S3 consists
of
only
several
S2
notes,
the
number
of
rules increases
substantially
S1 S2
S3
Sound
C
Representation
Rule
Note
unit
C
Primary
Rule
C
Secondary
Rule
C
Tertiary
Rule
tion of time that is specific to S2 and S3, which is
created
by
differentiation
of
pitch.
As
the medium
for
repetition,
sound works as
a
brake
against
mak-
ing
the
time
specific
to S2 and
S3
an
unconditional
primary
factor.
Repetition
f
S2
The smallest unit
allowing
music to be
interpreted
is the
S2
note
unit,
which is
basically
seen
on
a
score
(e.g.,
note
A in
a
score).
If
S2
is
adopted
as the
span
for the
present,
the
variability
of musical
inter-
pretation
is limited to the
scope
of
interpretations
at
S2 and above. The
single
note
in
S2 is itself ho-
mogeneous.
Let us assume that the note is re-
peated.
The
present
established
by
the
single
note
in
level
S2
is maintained
by repetition.
In
other
words,
repetition
creates a
present
that can also be
interpreted
as the
past.
For
example,
when lis-
tening
to
music,
consecutive notes sound continu-
ous;
however the
first
note
is
no
longer
actually
heard when the second note is heard. The notes
that
have
stopped sounding
also
remain
in the
con-
tinuation
of
melody-making
notes,
and should be
linked to
the last note
of
the
continuation.
E.
Hus-
serl calls this the
orientation effect
(Richtungs-
wirkung)
of retention.
In
this
way,
retention of
the
past
is
established
at
any
random
point
in a
melody.
Rule n S2
If
each
repetition
is
the
present,
repetition
in itself
does not become a momentum for differentiation.
The
present
created
by
repetition
is a
present
that
is an extension of time past. On the other hand, dif-
ferentiation
is created
by
a
present
that
is
an
escape
from
the
past.
At this
point,
it
should not be over-
looked that
the
S2
note
unit
possesses
a
representa-
tion
rule. A note unit is
accompanied
by
the
repre-
sentation of
pitch
name
and duration. This
representation
becomes
the momentum toward dif-
ferentiation,
and this
differentiation between the
symbols
represented
n
S2 is clarified in the
rule
in
S3.
Thus,
the
way
time is structured
in
S2
is
deter-
mined
relatively
through repetition
and
differentia-
tion
of
pitch.
Ruleand
Time n
S3
The
parameters
of
pitch
and duration that are
speci-
fied
in
S2 as described above
are
open-ended
toward
higher
strata
(S3
...
).
The
S3
unit is formed
by
sev-
eral notes.
Although
the unit of
range
for S3 con-
sists
of
only
several S2
notes,
the
number
of
rules
increases
substantially (see
Figure 5) (Narmour
1983).
In
general,
it can
be said
that the
higher
the
stratum,
the more
multiplex
the
characteristics
of
rules. The
present
in
S3
is time that is
multiplexed
using
as
rules
the
momentum toward
repetition
and differentiation in S2.
Each
combination of
pitch
and duration
allows for further
progression
and variation of
rules,
and the
rules
in
S3,
which
capture
the momentum
of the
music
toward
differ-
entiation,
form
the medium for S4 and
above.
Time
n
S4 and
Above
Rules in S4 are formed using as medium the rules
that
have
multiplied
and
become varied
at S2. To
52
Computer
Music
Journal
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 8
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 8/12
Figure
6. Meter
is
a
repre-
sentation
rule
in
S3,
which
as a
cycle
of
pulse
forms
it-
self
a medium
for higher
strata
and establishes
dif-
ferent
rules in S4 and
above.
S1
S2
S3
S4
9-
Sound
C
Representation
Note
Unit
C
Meter
(Rule)
Cycle
of Pulse
C
Primary
Rule
C
Secondary
Rule
C
Tertiary
Rule
the extent
that rules have
increased,
time at
higher
strata is
more
dependent
on
memory
or
knowledge.
Compared
to S3
and
below, therefore,
time at S4 be-
comes
more artificial and
arbitrary.
The
increasing
rules aremore
dependent
on the individual
musical
style,
and
are therefore more
difficult to
generalize.
As
a
result,
the
higher
the level of
time,
the less re-
stricted
it is
by
the
perception
of the
present;
in
other
words,
time at
higher
strata loses its
intrinsic
characteristics
and is more
easily
represented
spa-
tially.
This
non-temporal aspect
of rules at
higher
strata comes
into collision
with the
presence
gener-
ated
by
the lower stratamedia
through
the strata-
linking
function,
thereby creating dynamics unique
to
the music.
Meter
We
have considered the
relationship
between hierar-
chy
and
time;
now we will
position
meter within
the schematic thus farpresented.
In
S1, S2,
and
S3,
the
momentum-creating
time was
repetition.
Repe-
tition
is a
movement that builds on and extends
it-
self.
Specifically, repetition
in
S2 is
made in the
form
of a
continued
pulse.
In
this
continuation,
a
space
is formed that allows the
implementation
of
various rules. When
a
cyclical
imprint
is made on
the continued
pulse,
meter is established
(Meyer
and
Cooper
1960).
There are other
attempts
to
give
some account of the
process
of metric selection
(Steedman 1977;
Longuet-Higgins
and Lee
1982;
Lerdahland
Jackendoff1983;
Povel and Essens
1985; Palmer and Krumhansl 1990).Here, meter is
formed with the
physical
level of the
cycle
of main-
tained
pulse.
The
cycle
of maintained
pulse
is
equal
to the unit
range
of S3. This
range
of S3 cre-
ates time
points,
and events
occurring
at these
time
points
can
form an
equivalence
class.
Here,
meter as a rule
in
S3
gives
these time
points
an
identity independent
of
tonal, motivic,
harmonic
accents
(Benjamin 1984). Riding
on this
cycle
of
the
range,
meter establishes
different
rules
in
S4
and above
(Lester
1986).
Defined
in
the
context
of
hierarchy,
herefore,
meter would be a rule
in
S3,
which as a
cycle
of
pulse
forms the medium not
only
for S4 rules but also for rules
in
higher
strata.
Therefore,
if
accompanying
rules are removed
from
meter, only
the
characteristic
of
the
medium, i.e.,
maintaining
the
present
which is
cyclically
re-
peated,
would remain.
In
Figure
6,
the nowness of
perception
is main-
tained
through
a
cyclical
repetition
of
meter accom-
panied
by
a number of notes.
However,
the effect of
meter becomes more
pronounced
in
the extensive
time
flow
constituting
a
piece
of music
in
a
higher
stratum rather than
in
a lower stratum.
Meter as a
cycle
of
pulse
forms a medium and
permeates
into
the
higher
strata
by
repetition
and
actualization,
progressively creating
rules
in
this
permeation
process.
And as with
sound as a
presence,
meter
provides
a brake
against
rules
being
alienated from character-
istics intrinsic in time.
Specifically,
the
nowness of
note units as a medium is distributedin a
piece
of
music
through
the strata
linking
the function
of
meter as a
cycle
of
pulse,
so
that at
any
random
point
in
the
music,
a
uniform,
tightly
knit
time
structure is maintained. As a result, the present is
maintained
throughout
the
piece
of music.
Yako
53
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 9
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 9/12
Figure
7.
The
relationship
between
time
passing
and
musical
presence.
As
the
time-span
from
p
in-
creases,
the distance
from
perception
increases and
the
passing
speed
of
a
phe-
nomenon
slows down.
Alienation
.
from
Perception
,
Past
Future
Passing
of Phenomenon
TheHierarchical
unction
f Meter
Here we will
clarify
the
role of meter in music. At
the
beginning
of
a
piece,
when
only
one note has
been
perceived,
the
presence
of that
single
note be-
comes
the
totality
of
the
piece.
Actually,
however,
music
is
composed
of
many
notes.
How should
the
present
be
interpreted
once the music has
begun
and a
little time has
passed?
Let us assume a broad
range,
and
say
that the
present encompasses
the
time when the
music started until the time the cur-
rent note
is sounded
(perceived).
As
a note is
added,
the
entire time
becomes the time of
perception
from
the
starting point,
po,
until the
added note be-
comes
the
present.
The
present
is a time
extension
of the
starting
point.
In
this
case,
the
present
at a
random
point
p,
can be taken
in
two
ways (point
in
time
pn
or the
span
between
po
and
pn).
However,
these two types of present could have been taken at
any
point
in the continuation
of
notes to
pn.
At
point p,,
therefore,
the memories
at
each
point
of
time
in
the section
Pm-Pn
are accumulated
and in-
tegrated.
The
multiplicity
of
the
present
allows the
selection
of the smallest unit
corresponding
o
each
stratum
in
music to be formed.
Perception
comes
in
contact
with
the rules formed at each stra-
tum
through
the
memory
of
the
many presents
ac-
cumulated at
p,.
Hierarchical strata here are
formed
along logarith-
mic time.
As
the
piece proceeds,
therefore,
there is
a decreasein the number of times that section po0-
p,,
the
span
of
the
present,
coincides with the
breaks
in
the
integral
strata of the
higher
strata
(for
example,
the last note
in
measure
2, 4,
or
8).
Thus,
it becomes
more
difficult
to attain a
present
as
a
time
extension
of the
original
point
Po.
As a
result,
the
span
of the
musical
present
is shrunk
to
p,
and its
neighborhood.
In other
words,
as
the
piece proceeds,
the characteristic of the
present
that
existed at the
start of the
piece
becomes
ineffective.
Meter is
a musical
mechanism
for
avoiding
the
situation
of time at
point
p.
in
a
piece becoming
in-
effective.
When a
cyclical
repetition
of meter acts
as
a
higher
strata
medium,
its
strata-linking
func-
tion
begins
to
operate.
Through
this
function,
the
role
of
time
at
po,
the
starting
point
of the
piece,
and at
p,
a random
point
in the
piece,
is
consid-
ered to be the same.
In
other
words,
regardless
of
how
far
away
in time
pn
is to the
starting point
po,
the section po-p, can be adoptedas the present in
the same
way
that
it
is
immediately
after the
start
of the
piece.
As a
result,
the
hierarchical
time
struc-
ture
in the
neighborhood
of
p,
is
formed
in
the
same
way
that
it
is
in
the
neighborhood
of
po.
By
this introduction of
meter
as
medium,
the
tightly
knit
time
structure of
music based
on
the
starting
point
is
established
throughout
the
piece.
A
TimeModelf
Hierarchy
nd
Meter
Let us consider a model of time and hierarchythat
includes meter
(see
Figure
7).
The
horizontal axis
54
Computer
Music
Journal
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 10
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 10/12
in
Figure
7
shows
the
time-span
from the
past
to
the
future, sandwiching
the
present;
it contains
the
musical
phenomena
that occur
before
and after
the
present
point
p.
The
vertical
axis shows
the dis-
tance
from
the
unique
point
called the
present
(i.e.,
perception).
Perception
is thrust
outward as
phe-
nomena
through
p
from
the future
toward
the
past.
In terms of
the
passing
speed
of a
phenomenon
rela-
tive to
time,
passage
is
quickest
at the constricted
time
point
p.
On
the other
hand,
as the
time-span
from
p
increases,
the distance
from
perception
in-
creases and the passing speed of a phenomenon
slows
down.
In
this
figure,
the
greater
the
distance,
the
higher
the
hierarchical
stratum.
Figure
7
is
a
model
where
the
past
and
the future
are
transpar-
ent, i.e.,
where
the
perception
at a
lower stratum
is
directly
linked to a
higher
stratum.
In
Figure
7,
the
gradient
of
time to the distance
from
perception
is
constant,
and
can be shown
by
a
straight
line.
If
the
characteristics
of
time and
perception
are
taken
into
account,
however,
Figure
7
must
be mod-
ified.
First,
the
present
is not
a
point
but
a
span.
This
means that
the
perception
of the
present
em-
braces
specific points
in the
past
and the
future,
and that
distancing
from
perception
does not
occur
up
to
the
points
embraced.
Second,
the distance
from
perception
is
rapidly
accelerated
with
the sud-
den increase
of
rules at S3 and
above,
and
the bi-
ases
of
perception,
oblivion,
and the future's
unpre-
dictability
all serve to
reinforce this
tendency.
Therefore,
the distance
from
perception
is not
in
proportion
to the
time
axis, rather,
t increases
in a
broadening
curve.
The
shape
of the
model of time
and
hierarchy
would be
as shown in
Figure
8 rather
than
as
in
Figure7. Figure
8 shows
how the
present
p,
instead of
being
a
point,
has a
span
along
the
time axis and
is
spread
out
before and after.
At the
same
time,
the
greater
the distance
from the
pres-
ent,
the
greater
the distance
from
perception.
In
Figure
8,
as
in
Figure
7,
the
passing speed
of a
phenomenon
is maximized
in
the
neighborhood
of
the
present
p.
However,
since
the constricted sec-
tion is not a
point
but a
span, Figure
8 shows
the in-
exorable
passage
of time and how the
perception
of
the
present
comes
in
contact
with the
phenome-
non not
only
at
point
p
but also
in
the
neighbor-
hood of point p. Again as in Figure 7, the passing
speed
of a
phenomenon
is
quick
near
the con-
Figure
8. The
alienation
from
perception
is
rapidly
accelerated
with
the sud-
den
increase
of
rules
at S3
and
above,
and
becomes
infinite
at the
point
where
the
curve
is
perpendicular
to the time
axis. Meter
is
conveyed
by
the
passing
speed
near
point p
to
travel
straight
through
the
model
in
a
waveform,
re-
gardless
of
the
flexion
of
the curve.
Past
Future
IHigher
Straum
Meter
flienation
P
from
W
So
Lower Stratum
Perception
Higher
Stratum
Passing
of
Phenomenon
stricted
point
p
and becomes
slower as the time-
span
from
point
p
increases.
In
Figure
8,
the dis-
tance from
perception
becomes
infinite at the
point
where the curve is
perpendicular
o the time
axis,
and
the
speed
of
a
phenomenon
becomes
zero. At
this
point,
the
phenomenal
structure
in a
higher
stratum is released
from the
compulsory
restriction
of the
equation
of
time
=
perception,
and becomes
free.
Essentially,
the
irreversibility
of
time is dis-
solved,
and a
phenomenon
is
represented
and
recog-
nized
spatially.
As
previously
described,
the
higher
the stratum
and therefore
the more
knowledge-
dependent
the rules
are,
the more
arbitrary hey
be-
come because
they
are freed from
perception
and
can be structured
artificially.
The
phenomenon
of meter and
melody
crossing
at
right
angles (becoming
unrelated)
also
explains
that it becomes difficult
for the
hierarchical
model,
which
depends
on metric
accents,
to
converge
ulti-
mately
into a tree model.
As described
above,
that there is a
point
in
time
close
to the
present
p
where the
passing speed
of a
phenomenon
becomes zero is
important
in
the con-
sideration
of the time structure
throughout
a
piece
of music.
Using
the
present
when a note is
sound-
ing
as the reference
point,
let us
compare
section
Op, he passageof time from the beginning of the
work
to the
present,
and section
pe,
the
expected
Yako
55
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 11
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 11/12
passage
of
time from
the
present
to the end of
the
work.
It
will be seen that
the
increase
or decrease
in
Op/pe
or
pe/Op
s
not
in
proportion
to the
simple
increase
or decrease
in time.
Here,
the music's
start-
ing
and
ending points
need to be
determined
as
unique
points.
The
organization
of
the
ending
points
is
a
cadence.
However,
in a work of
music
having
a
finite
length,
the
shape
of
the model
in
Fig-
ure 8
is maintained
regardless
of whether
the
model
is taken at
the
beginning
of the
piece,
in
the
middle,
or at
the end.
Thus,
the
time structure
close to the presentp can avoidthe intervention of
progressive
time which
sees the
start and
the end
of a
piece
of music
as
unique points
in time.
Let us now
indicate
meter
in
Figure
8.
The time-
span
taken for
the
present
at the
constricted area
in
Figure
8 determines the
cycle
of the beat. That
is
to
say,
the
unit of
time-span
having
a constant
pass-
ing speed
forms the musical
present
of the
beat,
i.e.,
the
cycle. By
means
of
this
cycle,
meter
pres-
ents the
neighborhood
of
the
present
in the music.
Meter as
a
cycle
of
pulse
is
then
conveyed by
the
passing speed
near
point
p
to
form the medium
for
higher
strata,
and travels
straight through
the
model in a waveform
regardless
of the flexion
of
the
curve.
The
continuation
of the
permeation
of
meter
into the
higher
strata is thus established.
The
phenomenon
of meter
and
melody
crossing
at
right
angles
(or becoming
unrelated)
explains
the
fact that
it becomes difficult
for
L.
B.
Meyer's
and
M.
Yeston's
hierarchical
model,
which
depends
largely
on metric
accents,
to
converge
ultimately
into a tree model.
Meter
propagates
to
the
higher
strata the time
structure
near
point p
in
the
lower strata.
In
this
way,
meter works
as an
inhibiting
factor,
preventing
the
higher
strata rules
from
being
alienated from
the intrinsic characteristics
of
time,
thereby
estab-
lishing
nowness
in music. Such characteristics
of
meter can be seen from
Figure
8.
Summary
By considering hierarchy
and time and
describing
a
model,
I have
presented
the characteristics
unique
to the musical present and their relationship to a
higher-strata, spatially represented
musical
percep-
tion,
as
well as
the function
of
meter
within this re-
lationship.
The discussion
can be summarized
as
follows:
*
The
link between a
higher
stratum
and a lower
stratum
can be shown
in the
nesting
sche-
matic
of medium
and rule.
*
Rules
are structured
discretely
between
strata,
but media areindiscrete
and have a strata-
linking
function.
*
Specific
contact with
music is made
only
through
the
presence
of
sound
in
level
S1.*
Pitch and duration
are addedto sound at S2 to
form a note unit.
The note unit
forms a me-
dium
for
rules
in S3 and above.
*
The
present
is maintained
by
the
repetition
of
a note unit
(S3),
and the formative
power
of
music is
generated by
the maintenance
of
pulse.
This forms
a medium
for rules in S4 and
above.
Meter is a
cyclical
maintenance of
pulse.
*
Meter as
a
cycle
of
pulse permeates
the
higher
strata
by
forming
a medium and
being repeated
and actualized. Thus the present is maintained
throughout
a
piece
of music.
*
At the
highest
strata
(which
are more knowl-
edge dependent),
rules
are freed
from
the re-
striction of
a
present perception,
and are
repre-
sented
spatially.
*
When meter as a
cycle
of
pulse
brings
into
play
the
strata-linking
function of a
medium,
the nowness of a note
unit
is distributed
within a
piece
of
music,
and a
tightly
knit
time structure similar
to the
beginning
of
the
music is maintained at
any
random
point
within the piece.
*
Meter forms a
neighborhood
of the
present
at
lower
strata
and
propagates
n a
waveform into
spatially
represented
time at a
higher
stratum.
It
provides
a
brake to
prevent
the rules at
higher
strata from
being
alienated from time-
intrinsic characteristics.
Acknowledgments
This
article
is a modification
of the author's
previ-
ous paper in Journal of the Japanese Society for Aes-
thetics
(Yako
1992a).
56
Computer
Music
Journal
This content downloaded from 128.250.144.144 on Sat, 03 Oct 2015 22:42:54 UTCAll use subject to JSTOR Terms and Conditions
Page 12
7/17/2019 Hierarchical Structure of Time and Meter- Masato Yako
http://slidepdf.com/reader/full/hierarchical-structure-of-time-and-meter-masato-yako 12/12
References
Benjamin,
W.
1984.
' A
heory
of Musical
Meter. Music
Perception 1(4):355-413.
Broadbent,
D. E.
1958.
Perception
and
Communication.
Oxford,
UK:
Pergamon.
Desain, P.,
and
H.
Honing.
1992.
Music,
Mind and Ma-
chine.
Amsterdam,
The
Netherlands: Thesis.
Fraisse,
P.
1963.
Psychology of
Time.
New
York:
Harper.
Fraisse,
P.
1982.
Rhythm
and
Tempo.
n D.
Deutsch,
ed.
The
Psychology of
Music. New York:
Academic.
Husserl,
E.
1928.
Vorlesungen
ur
Phdinomenologie
des
Inneren
Zeitbewufltseins.
zweiter Abschnitt in Husser-
liana
X
[Neudruck
1969].
Hague,
The
Netherlands:
Martinus.
Lerdahl,
E,
and
R.
Jackendoff.
1983.
A
Generative
Theory
of
TonalMusic.
Cambridge,
Massachusetts:
MIT
Press.
Lester, J.
1986. The
Rhythms of
Tonal Music.
Chicago,
Il-
linois:
Southern Illinois
University
Press.
Longuet-Higgins,
H.
C.,
and
C. S. Lee.
1982. The
Percep-
tion
of
Musical
Rhythms. Perception
11:115-128.
Mach,
E.
1906.
Erkenntnis und
Irrtum-Skizzen zur
Psy-
chologie
der
Forschung.
Leipzig, Germany.
Meyer,
L.
B. 1956. Emotion
and
Meaning
in
Music.
Chi-
cago,
Illinois:
University
of
Chicago
Press.
Meyer,
L. B. 1973.
Explaining
Music.
Berkeley,
California:
University
of
California Press.
Meyer,
L.
B.,
and G.
W.
Cooper.
1960. The
Rhythmic
Structure
of
Music.
Chicago,
Illinois:
University
of
Chi-
cago
Press.
Narmour,
E.
1977.
Beyond
Schenkerism.
Chicago,
Illi-
nois:
University
of
Chicago
Press.
Narmour,
E.
1983. Some
Major
Theoretical
Problems
Concerning
the
Concept
of
Hierarchy
n the
Analysis
of TonalMusic. Music
Perception
1(2):129-199.
Palmer, C.,
and C.
Krumhansl.1990 Mental
Representa-
tions for
Musical Meter.
Journal
of
Experimental Psy-
chology 16(4):728-741.
Povel, D.,
and P. Essens.
1985.
Perception
of
Temporal
Patterns. Music Perception 2(4):411-440.
Ruwet,
N. 1972.
Langage,Musique,
Poesie.
Paris,
France:
Seuil.
Schenker,
H.
1956. Der Freie Satz.
Vienna,
Austria: Uni-
versal
Edition.
Steedman,
M.
J.
1977.
The
Perception
of
Musical
Rhythm
and
Meter.
Perception
6:555-569.
Yako,
M.
1992a. Hierarchical
Structureand
Meter.
Jour-
nal
of
the
Japanese
Society
for
Aesthetics
43(3):12-23.
Yako,
M.
1992b.
Perception
of
Decaying
Sound
and the
Sound of
the Shamisen.
Journal
of
the
Musicological
Society of
Japan38(1):19-35.
Yeston,
M.
1976.
The
Stratification
of
Musical
Rhythm.
New Haven, Connecticut: YaleUniversity Press.
Yako
57