Hierarchical Structure of Time and Meter- Masato Yako

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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






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






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






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






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






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






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






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






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






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






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Documents

Hierarchical Structure of Time and Meter- Masato Yako