rahn jay african rhythm european mt

8/11/2019 Rahn Jay African Rhythm European MT

1/20

Turning the Analysis around: Africa-Derived Rhythms and Europe-Derived Music TheoryAuthor(s): Jay RahnReviewed work(s):Source: Black Music Research Journal, Vol. 16, No. 1 (Spring, 1996), pp. 71-89Published by: Center for Black Music Research - Columbia College Chicagoand University of Illinois Press

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2/20

TURNING

THE

ANALYSIS AROUND:

AFRICA-DERIVED RHYTHMS AND

EUROPE-DERIVED MUSIC THEORY

JAY

RAHN

As

Samuel

A.

Floyd

Jr.

(1993,

1)

has

emphasized,

analysis

is an

activity

that

was

developed

as

a

way

of

examining chiefly Europe-derived

works

of music.

Susan

McClary

and Robert Walser

(1994,

77)

have stressed fur-

ther that the

Europe-derived discipline

of music

theory

has

notoriously

neglected rhythm

in

favor of

abstract

patterns

of

pitch

and

form.

I

feel

that

Europe-derived theory

and

analysis

have not handled well the

fol-

lowing

topics, long

and

widely

acknowledged

to

be

important

for

Africa-

derived

traditions: constant

syncopation,

off-beat

phrasing, turning

the

beat

around,

backbeat,

cross-rhythm,

anticipation

(as

contrasted with

re-

tardation),

constant

repetition

of

rhythmic

and

melodic

figures,

the relat-

ed

phenomena

of

call-and-response

phrasing,

riffs,

vamps,

time-lines,

particular

additive

rhythms,

metronomic

pulse

(or

approximately,

what

ethnomusicologists

have

referred to

as

the metronome

sense)-all

this

grounded

in

the

unique

forward momentum

of

unflagging

rhythms

highlighted by Floyd (1991, 268, 279;

cf.

Floyd 1993, 1, 4).

Attempted

below is a

technical account

of Africa-derived

syncopation

that

might

serve as an

alternative to

orthodox,

Europe-derived

accounts.

Rather than

depicting

syncopated rhythms merely

as deviations from a

four-square

metrical

hierarchy,

I

try

to

show

how

they

can be

portrayed

as

highly integrated

wholes

in

their

own

right.

Such wholes favor a

dif-

ferent

aesthetic

imagery

than

has been usual

in

Europe-derived analyses

of

rhythm: specifically,

(1)

an

imagery

of

complementation, long

estab-

JAY

RAHN

is associate

professor

and music

coordinator,

Fine

Arts

Department,

Atkinson

College, YorkUniversity. His writings include A TheoryforAll Music: Problemsand Solutions

in the

Analysis

of

Non-Western

Forms

(University

of

Toronto

Press,

1983)

and,

with Edith

Fowke,

A

Family

Heritage:

The

Story

and

Songs of

LaRenaClark

(University

of

Calgary

Press,

1993).

71


3/20

BMR

Journal

lished in

Europe-derived

discourse on

pitch

structure

in

post-tonal

music

(e.g., John

Rahn

1980)

but little

developed

in orthodox

rhythmic analysis;

(2)

metaphors

of

braiding

and

circularity,

only

recently

developed by

the

technical

apparatus

of

post-tonal theory;

and

(3)

pendularity

and

cyclism,

long recognized

as central

values in the

aesthetic of

Africa-derived

music

that

Floyd

has identified

by

the

expression

"Call-Response"

but

largely

neglected

in

Europe-derived analyses.

Such

images

contrast

with

con-

ceptions

that

presume

as

central

such

metaphors

and

concepts

as

the

fol-

lowing:

relentless

linearity;

the

arrow

of

time;

immediate

expectancy;

constantly

frustrated,

teleological, goal-directed

processes

(e.g.,

of

desire

in Schenkerian

analysis); segmentation;

and

asymmetric hierarchy.

These

two

conceptual

groupings

greatly

differ in

their

possible

analytic

conse-

quences

and

can

thrive

independently.

Nonetheless,

I

try

to

show

that

one can

shift from

one

to

the

other,

in

whole

or

part,

or

even

hold to

both

at

once,

without

contradiction.

There

are at

least

two

reasons for

seeking

such

multiplicity

in

analysis.

First,

as

Floyd

has

outlined,

interactions

between

audience

and

perform-

ers,

and

among

performers

themselves,

are

important

components

of

the

Call-Response

aesthetic.

Second,

reception

and

appropriation

of

Africa-

derived

idioms and

genres

have

been

persistently multiple

in

their

sensi-

bilities for

at least a

century.

Thus,

Gary

Tomlinson

(1991,

240),

for

exam-

ple,

advocates

decentered

analysis

of

jazz

to admit

a

variety

of

vantage

points

rather than

legislating

a

singular,

authoritative

perception.

By

contrast,

Europe-derived

music

theory

has

tended

to

eschew inter-

pretative

diversity

in

favor

of

readings

that

are

convergent, singular,

un-

ambiguous,

exclusive-indeed,

it

could

be

said,

canonic,

authoritative,

correct,

or to

use a

long

discredited

term from

the

early

decades

of

eth-

nomusicology,

authentic. In

intercultural

settings,

intracultural

ensemble,

self-delectative solo

performance,

even

reading

or

listening

to

perfor-

mance that

is

mediated

(e.g.,

by

recording

or

notation),

one can

find

com-

munity

where

one

might

have

been led to

expect,

as

a

musico-social

value,

conformity

(cf.

McClary

and

Walser

1994,

79).

Decentered

analysis

would

reserve a

spot

for

the

interlocative

self

that can

change

places

with

another

(Holquist

and

Liapunov

1990,

xxvi;

cf.

Bakhtin

1990,

22-23),

even

if

the

acts of

others,

which

can

comprise

invention,

sounding,

moving,

and,

of

course,

listening,

are

merely

imagined

or

recalled

on

the basis

of

sounds or

other

signifiers.

Quite

early

in

the

history

of

African-American music

analysis,

Winthrop Sargeant

(1938,

58-64)

identified the

rhythm

of

Figure

la as

es-

pecially typical

of

jazz

and

other

Africa-derived

styles.

Jazz

analysts

long

have

recognized

that this

basic 3+3+2

rhythm

also

can take

such

forms as

those

notated in

Figure

lb.

Ernst

Bornemann

(1946),

Gunther

Schuller

72


4/20

Rahn

*

Turning

the

Analysis

Around

(1968,

24),

and Marshall Steams

(1956, 142)

traced

such 3+3+2

patterns

to

African

musical traditions. Don

Knowlton

(1926,

581), quoting

an

unidentified

African-American

guitarist,

had used the

expression

"sec-

ondary rag"

for

rhythms

of this sort.

Shortly

thereafter,

Aaron

Copland

(1927, 10-12)

noted that the

temporal organization

of these

"foxtrot"

pat-

terns contrasted with

Europe-derived

practice,

suggesting

unsyncopated

measures of

I,

],

and

A

as

notationally

preferable

to

syncopated

measures

of t.

Referring

to

the basic

pattern

of three

syncopated

tones within four

quarter-note

beats,

Sargeant

designated

these

rhythms

"three-over-four,"

and like later

analysts

traced

them to

Africa,

crediting

this

important hy-

pothesis

to

the much

earlier,

prodigious

investigations

of

Nicholas

J.

G.

Ballanta-Taylor

(1922).1

Figure

1.

3+3+2 and

close

variants

a.

Basic

3+3+2

rhythm

J.

J. J

b. Close variants

J.

J

J J

bJ.

JJ

m;

J

As in

such

parallel

cases as

the

so-called

Scotch

snap

of

strathspeys

and

flings

and the canzona

rhythm

of

Renaissance

song

and

closely

related in-

strumental

music,

one can ask

whether

the

3+3+2

figure

is

structurally

special

or

privileged.

Or does

this

figure

stand out

stylistically

only

be-

cause of its

disproportionately

frequent

occurrence

in

traditions of

Africa

and

the

African

diaspora?

Is

this

rhythm,

and

others

closely

related

to

it,

to be

understood in

its

own

terms,

or

only

through

the

filter or

lens

of Eu-

rope-derived

common-practice

rhythmic

theory?

I

present

two

groups

of

interpretations

of

this

and

other,

cognate

patterns:

one

group

set in

a Eu-

rope-derived

framework

of

common-practice

concepts,

the

other in

an

emerging

paradigm

of

recent

music

theory.

In a

Europe-derived,

common-practice

conception

of

meter,

the

basic

3+3+2

rhythm

can

be

considered

both

dotted and

syncopated.

In

nota-

tional

terms,

the

beginning

of

the

second dotted

quarter

is

unprepared

because no

note

appears

on

the

second

quarter

and

unresolved

since

there is

no

note on

the

third

quarter.

Extending

this

conception

slightly,

1.

Ballanta-Taylor,

who had

just

arrived in

the

United

States from

Sierra

Leone via

a Mus.

Bac.

degree

at

Durham

University

and

studies of

harmony

with Sir

John

Stainer

(on

which

see

Cuney-Hare

1936,

347-348),

published

his

pathbreaking

technical

account of

African

and

closely

related

jazz

rhythms

partly

in

response

to

critical

opinions

that

had

been

voiced

in

both

the Boston

Transcript

and the

Negro

Musician.

73


5/20

BMR

Journal

the

attack of the second dotted

quarter

can be

considered to be both

pre-

pared

and resolved

texturally

if

the

part containing

the

3+3+2

rhythm

is

accompanied

by

another

portion

of the texture

that has notes

starting

on

the second and third

quarters-as

in a

walking

bass line in

swing

and

later

styles,

a stride bass

in

piano rags

and other

march-related

forms,

or

steady strumming

in

quarters

by banjo

or

guitar

in New

Orleans,

Chica-

go,

and other

early jazz

idioms

(see

Fig.

2).

Figure

2.

Preparation

and

resolution

of syncopation

of

the

basic 3+3+2

rhythm

Basic 3+3+2

rhythm

J.

J. J

Quarter-note

pulsation

J

J

J J

Combined

rhythm

J J

In

a

monophonic

break or

unaccompanied

introduction,

the

second

dotted

quarter

is not

prepared

or

resolved

explicitly-neither

within

the

3+3+2 line

itself,

nor

in

another,

simultaneous

portion

of the

texture. If

an

immediately

earlier or later

passage

has

established a

quarter-note

pulsa-

tion,

the

second

dotted

quarter

could

be

considered

prepared

and

re-

solved in

an

extraordinary

sense;

that

is,

implicitly,

or

to use a

logical

term,

modally,

i.e.,

by

virtue

of

notes that

could

have

begun

on

the

sec-

ond and

third beats

and

thus could

have

formed

time-interval

matching

relations with earlier

or later

pairs

of

notes. Even

without a

preceding

or

following

passage

of

pulsating quarters,

one can

understand

the

second

and

third

quarters

as

being modally

implied by

the end

of

the 3+3+2

rhythm,

providing

an

instance of

what

Sargeant

appears

to

have

meant

by

"internal"

syncopation;

that

is,

syncopation

relative

to

a

metrical

hier-

archy

which can

be

inferred on

the

basis of

the

rhythm's

own

notes,

irre-

spective

of

other

passages

or

parts

of

the

texture.2

Sargeant's

observation

that

anticipations

are

more

frequent

in

the

early

jazz

he

surveyed

than

are

retardations

can be

understood

in

at

least

two

ways.

Disregarding

such

pitch

structures as chord

progressions,

one can

take

his

generalization

to mean

that

an

anticipatory

transformation

of a

measure of four even

quarters

into an

eighth,

two

quarters,

and a dotted

2.

Such

inferrable

pulses,

which

can

be

"generated"

in

theory

via a

modal

axiom,

e.g.,

along

the

lines of

Jay

Rahn

(1978),

could be

considered to

correspond

to the

metronome

sense in

ethnomusicological

accounts.

74


6/20

Rahn

*

Turning

the

Analysis

Around

quarter,

for

example,

is more usual than its retardative

counterpart

(actu-

ally,

its

temporal

inversion or

retrograde)

consisting

of a dotted

quarter,

two

quarters,

and an

eighth

(see

Fig.

3).

Figure

3. Variants

of

basic

quarter-notepulsation

Basic

quarter-note pulsation

J

J

J

Anticipatory

variant

b

J

J

Retardative

variant

J

J J

b

Widely

affirmed since

1938,

Sargeant's

hypothesis

also can

be under-

stood

in terms of both

pitch

and

rhythm.

If

so,

it

would

predict

that har-

monic

or

contrapuntal

anticipations

are

more

frequent

than

suspensions.

The second dotted

quarter

of a 3+3+2

rhythm might

anticipate

a

change

of chord on the fifth

eighth

from a chord on the first

eighth

or

might

re-

solve,

in

retardative

fashion,

a

suspended

dissonance heard

on the third

eighth

that had been

prepared

on

the first.

In

anticipations,

melody

notes

can be heard

as

leading

or

directing

chord

changes

or

progressions;

metaphorically,

d incites or

provokes

Y

(see

Fig.

4a).

In

suspensions,

chord

progressions

can be

heard as

leading

or

directing

changes

of

melody

notes or

melodic

progression;

Y

incites or

provokes

c

(see

Fig.

4b).

In this

way,

Sargeant's

hypothesis

would

predict

that

melody

leads

chords,

not vice versa.

The

preceding

analyses

of 3+3+2 and

other,

closely

related

syncopated

rhythms

presume

four-square

meter as a

framework and

understand

3+3+2

not

in,

or

on,

its own

terms but

rather in

terms of a

putatively

more

privileged,

hierarchical

metric

structure

consisting

of

stronger

and weak-

er beats and

subdivisions.

Recent work in

theory

and

analysis

yields,

to

extend

Gunther

Schuller's

suggestive

idea

(1968,

6-10),

a

more

democra-

tic, or,

one

might

say,

non-metrocentric,

construal.

This

understanding

is

clearly

connected

to

concepts

of

metronome

sense,

smallest

rhythmic

units,

basic

pulses,

nominal

values,

and the

density

referent in

ethnomu-

sicology

(cf.,

e.g.,

Nketia 1974, 125-138). Also related to Leonard B.

Meyer's

notion of a

flat

hierarchy

(1973,

90),

the

technical

elaboration of

these

essentially

non-hierarchical

conceptions

has

not been

realized with-

in

any

of

the

analytic

traditions

just

cited.

Instead,

technical

resources for

75


7/20

BMR

Journal

Figure

4.

The 3+3+2

rhythm

with

anticipation

and

suspension

a.

Anticipation

b.

Suspension

Consonance/dissonance

-.------

c-

c------..

c

d--c--

----.

.----

Melodic

rhythm

J.

J

.

J

.

J.

J

Chords

x

Y

x

Y

z

Chord

rhythm

J

J

J

J

J J

comprehending

non-metrocentrism

have resided at

the

edges

of

what

might

seem the least

relevant of

formulations,

namely,

the

large body

of

post-tonal

theory

developed

mostly

to deal

with

pitch

structures in

twen-

tieth-century Europe-derived avant-garde music.

A

key

component

of

this shift

of

paradigm

for

rhythmic

analysis

is

Mil-

ton

Babbitt's

theory

of

time-points

(1962).

Published more than

three

decades

ago,

Babbitt's

formulation of

time-points

was

developed

not

from

existing

theories

of

rhythm

and

meter-nor even

from

traditions de-

veloped

in

common-practice

theory

to

describe

intervals,

scales,

and

chords

(which

remain

influential

throughout jazz

theory

and

analysis).

Instead,

Babbitt's

time-points

emerged

as an

extension of

classic,

twelve-

tone

serialism.

Babbitt's

insight

was that

time-intervals between

the

at-

tacks or

beginnings of notes can produce temporal structures parallel to

those

formed

by

pitch-intervals.

A

second,

heterodox

stage

of

paradigm

development

involved

the em-

pirical

finding

that

there are

structural

parallels

between

syncopated per-

cussion

ostinatos or

time-lines,

in

several

African

traditions,

and the

dia-

tonic

scale

(Pressing

1983;

Jay

Rahn

1983;

Jay

Rahn

1987).

The

diatonic

scale or

collection is

labeled

7-35 in

Allen

Forte's standard

listing

of

sub-

sets

of the

12-semitone

aggregate

(Forte

1973,

Appendix

I;

John

Rahn

1980,

140-143).

The

collection 7-35

comprises

not

only major

or

Ionian-

024579E

(cf. CDEFGAB,

or

TTSTTTS,

or in

semitones, 2212221,

or with

eighths

corresponding

to

1

and

quarters

to

2,

the

rhythm

J J

h

J J J

)-but

also the

remaining

modes

(Dorian,

Phrygian,

etc.)

and

their

rhythmic

counterparts (e.g.,

2122212

or

J JJ J J

'J for

Dorian).

Such

diatonic

time-

lines have

long

been

well

documented for

music in

such African

cultures

76


8/20

Rahn *

Turning

the

Analysis

Around

as

Ewe,

Ghana

Jones

1959, 93, 112,

121, 138, 170;

Koetting

1970, 129;

Ladzekpo

and

Pantaleoni

1970,

21;

Ladzekpo

1971,

14;

Pantaleoni

1972,

21, 59a;

Chemoff

1979, 85, 86, 119,

120);

Ashanti,

Ghana

(Koetting

1970,

136);

Yoruba,

Nigeria (King

1960, 53;

King

1961,

i-xxxviii;

Koetting

1970,

135);

and

Venda,

South Africa

(Blacking

1967,

151-153; 1970,

33,

43, 46, 47,

50).

Time-lines of this kind

appear

also in

Mwenda

Jean

Bosco's

guitar

pieces,

which stand at the

beginning

of the

important

modem

guitar

tra-

dition of Zaire

(Rycroft

1962,

100)

and have

been transcribed for

music

of

otherwise

unidentified

cultures

in

Ghana

(Jones

1954,

59;

Nketia

1963a,

89),

Benin

(Kolinski

1967,

16,

20),

and,

more

generally,

West

Africa

(Ekwueme

1975-1976,

30).

Understood

in

terms

of

pitch

and

time,

number-theory

proofs

have

shown how far

certain

features of

this diatonic

structure

extend into

cog-

nate

rhythms

and

micro-tonal,

non-12-semitone scales.

The most

impor-

tant of

these

theorems

have

appeared

in the

already

epochal

study

of

maximally

even

sets

by

John

Clough

and

Jack

Douthett

(1991)

and

in

Clough's

recent

treatment

of diatonic

interval

cycles

(1994).

Increasingly,

this

body

of

theoretical lore

has

shown close

structural

connections with-

in,

between,

and

among

the

7-tone/12-pulse rhythms

just

discussed,

as

well

as

diatonically

structured

rhythms comprising

5

tones

among

12

pulses,

5

or 3

among

8,

and 9 or

7

among

16

(see

Fig.

5).

These

rhythms

underlie

and

even

form

explicit

components

of

African-

American

and

African-Hispanic

traditions of

the Western

hemisphere.

As

time-lines,

they

have

permeated

traditional

music of

West,

Central,

Southern,

and

East

Africa

as well

as

pan-African

idioms of

recent

decades. In

addition

to the

traditions

and

7-note/12-pulse

time-lines

just

cited,

music

using

time-lines of the

more

general

sort

illustrated

in

Figure

5

has been

transcribed in

studies of

the

following

cultures:

Ga,

Ghana

(Nketia

1958,

22);

Akan,

Ghana

(Nketia

1963b,

102,

106, 110, 114,

118, 122,

124, 126,

128, 130,

132, 136,

138);

Luba,

Zaire

(Rycroft

1962,

100);

Ngbaka-

Maibo,

Central

African

Republic

(Arom

1976,

509);

Hausa,

Nigeria

(Raab

1970, 100;

Besmer

1974,

58);

Tonga,

Zimbabwe

(Jones

1964,

11-12);

and

Shananga-Tsonga,

Mozambique

(Johnston

1971,

69).

Theorems

concern-

ing

this

more

general

sort of

diatonic structure

draw

attention

to several

affinities

among

these

traditional

time-lines and

show

how

they

are both

distinct

from and

yet

highly

compatible

with

other,

four-square,

hierar-

chical

patterns.

Some

mathematical

results

bearing

on

these

rhythms

are

quite

prob-

lematic,

for

they

resist concrete

interpretation.

For

instance,

arresting

for-

mulations

by

John

Clough

and

Gerald

Myerson

(1985)

prove

that in dia-

tonic

structures of

the sort

just

illustrated,

for

each

kind of two-note

set

(i.e.,

each

kind of

interval or

dyad,

namely,

second,

third,

fourth,

fifth,

77


9/20

BMR

Journal

Figure

5.

Diatonic

rhythmsfor

which

theoremsand

proofs

appear

n

Clough

and

Douthett

(1991)

and

Clough

(1994)

7

tones/12

pulses

JJDJJJJ

JbJJJJJ

DJJJdJJ

JJJDJJJ

JJDJJDJ

5

tones/12

pulses

JJJ.JJ. JJ.JJ.J .JJ.JJ JJ.JJ. .JJJ.J

5

tones/8

pulses

.Jf.J.t.J

W.

JJ

J

JJ.J

JJ.J.

3

tones/8

pulses

J..J

J.JJ.

JJ.J.

9

tones/16

pulses

JJJJ.JJJJ

JJJ.JJJJJ

JJ~JJJJJJ

J.JJJJJJJ

.JJJ

JJJJ

JJJbJJJJ^

JJDJJJJfJ

JJJJJJJ

JJJJJJJ

7

tones/16

pulses

JJJJ.JJJ. J.JJJ.J JJ.JJJ.JJ J.JJJ.J JJJ.JJJJ.

JJ.JJJJ.J

J.IJJJ

J

etc.)

there are

two

types.

For

example,

the

types

are

minor

and

major

for

seconds,

thirds, sixths,

and

sevenths;

perfect

and

augmented

for

fourths;

perfect

and

diminished for fifths.

Similarly,

for

each kind

of

trichord

or

three-note set in a

diatonic

structure

(whether

of

pitch

or

time),

the

num-

ber of

types

is

three;

for

example,

the

types

of

triads

in a

diatonic

scale

are

diminished,

minor,

and

major.

The

types

of

trichords

that

comprise

con-

secutive

steps

or scale

degrees

are

similarly

threefold: from bottom to

top,

major-second-plus-major-second

(e.g.,

C-D-E,

F-G-A,

and

G-A-B),

major-

second-plus-minor-second

(e.g.,

D-E-F

and

A-B-C),

and

minor-second-

plus-major-second

(e.g.,

E-F-G

and

B-C-D);

and

so

forth,

for all

the

possi-

78


10/20

Rahn

*

Turning

the

Analysis

Around

bilities.

Translated

into

time,

this

highly

abstract

set of features

holds

for

all

the

8-, 12-,

and

16-pulse rhythms

listed above. What

such

precise

nu-

merical relations

among

types

or kinds of

things

actually might

sound

like has not been

as clear as the

proofs

themselves. Other

aspects

of

these

rhythms,

formulated

just

as

mathematically,

seem more

directly

connect-

ed

to both

perception

and

performance.

These features can

be

translated

into concrete

terms

that

involve

neither

abstract kinds or

types

of

things,

nor

precise

numbers

(see

Jay

Rahn 1995

on the

general

translatability

of

abstractions into

concrete

things

for

music).

Concepts

of

evenness,

coherence,

and

proportionacy

developed

in

these

formulations

arguably

correspond

to the

smoothness shared

by

the

time-lines listed

above.

These

concepts

also

define

analogies

between the

time-lines and

much

plainer

successions

of,

for

example,

even

quarters

or

dotted

quarters,

or

quarter-eighth

pairs.

The

concept

of

maximum

match-

ing

among

time-intervals

helps

define a

way

in

which

these

rhythms

can

be

heard

as

integral,

unified

wholes or

temporal

Gestalten and

distin-

guishes

them from

maximally

redundant

(and

maximally

even)

succes-

sions

of

indefinitely repeated

eighths,

or

quarters,

or

dotted

quarters,

etc.,

and from such

minimally

redundant

(but

nonetheless

maximally

even)

successions as

eighth-quarter

and

quarter-half

pairs.

Among

maximally

even

rhythms,

another

concept,

namely

individua-

tion

(closely

analogous

to

the

notion of

affinities

in

medieval

modal the-

ory), expresses

a

further

contrast

between

diatonic

rhythms

and

those

in

which

the

most or

fewest

time-intervals

match

each other in

size.

Diaton-

ic

rhythms

are

maximally

individuated in the

sense that

each

of their

notes

bears

a

unique

constellation

of

relations to

every

other note.

For in-

stance,

in

the

5-tone/8-eighth-pulse

diatonic

rhythm

J

hJ

J the

third

note

appears

1

eighth

and

3

eighths

after but

2

and 3

eighths

before

other

notes

in

the

rhythm,

whereas

the fifth

note

appears

1

and 3

eighths

after

and 2

and 4

eighths

before other notes.

Irrespective

of

how

often the

rhythm

might

be

repeated,

this

individuation is

sustained

for

all

five

notes

in

the

pattern,

which

thus

constantly

renews

itself

through

note-to-note

diver-

sity. By

contrast,

every

note

in

a

(nonetheless

maximally

even)

succession

of

straight

eighths

is like

every

other

note,

as is

every

other

note

in an

in-

definitely

repeated

(and

again,

maximally

even)

rhythm

of

quarter-

eighth pairs.

A

rhythm

need

not

be

even,

coherent,

proportionate,

etc. in

order to

be

maximally

individuated.

Any rhythm

that

cannot

be

fully

subdivided or

partitioned into

repeated

segments

will, on

subsequent

repetition,

be

maximally

individuated in

this

technical

sense;

for

example,

J

J

D.J

s

con-

trasted

with,

for

instance,

J .bJ

J,

which

subdivides into

two

statements

of J .b.

Whereas

particular

notes

within

a

repeated

four-square rhythm

79


11/20

BMR

Journal

stand out

as the clearest candidates

for

being

heard as

strongest,

second

strongest,

etc.

beats or

parts

of

beats,3

the

diatonic

rhythms

considered

here elude such linear

ordering.

Instead,

they

can be

understood as

com-

prising

cycles

of

(approximate)

half-measures

analogous

to

half-octave

cycles

that have

been discerned for

scales

(see

Jay

Rahn

1977).

Within

such

cycling processes,

these

rhythms

can be

understood

as

twisting

or

tunneling

through

time,

forming

a

special,

braided

structure.

As

recently

formulated

by Clough

(1994),

the

braided

structures char-

acteristic of

diatonic

rhythms

can

be

contrasted

clearly

with

four-square

rhythms.

For

example,

in

the 7-tone

rhythms

discussed

above,

every

tone

is

immediately

inside a

pair

of

tones, i.e.,

a

single step away

from

each.

Each tone of

this

pair

is the same

number

(i.e., two)

of

steps away

from

the other and

yet

another. In

turn,

each

of these is four

steps

from

the

other and another

still;

and so on.

The entire

pattern

comes full

circle. Be-

cause of the

clumsiness of

Europe-derived

rhythmic

notation,

this

point

Figure

6.

Braided,

circular

structures in

7-tone/12-pulse

rhythm

C

D

E

F G

A

B

C

D

E F

G

A

B C

D

E F

G A

B...

J

J

D J J

J

D

J

J

J

J

J

J

J J

JJ

4

,...

a.

Each

C is one

step

earlier than

a

D;

each

D

is

one

step

earlier

than an

E

Each

C is two

steps

earlier than

an

E;

each

E

is

two

steps

earlier

than a

G

Each

C

is

four

steps

earlier

than a

G;

each

G is four

steps

earlier than

a

D

Each C is eight

steps

earlier than a D; each D is

eight

steps

earlier than

an E

(cf.

first line in

(a):

"Each C

is one

step

earlier

...")

b.

C's are one

step

later than

B's;

B's are

one

step

later

than

A's

C's are

two

steps

later than

A's;

A's are

two

steps

later

than

F's

C's

are four

steps

later

than

F's;

F's are

four

steps

later than

B's

C's are

eight

steps

later

(i.e.,

one

step

later)

than

B's;

B's are

eight

steps

later than A's; i.e., C's are one step later than B's, and B's are one step

later

than

A's

(cf.

first

line

in

(b):

"C's are

one

step

later

...")

3. For

example,

in

Jay

Rahn

(1978).

80


12/20

Rahn

*

Turning

the

Analysis

Around

is more

easily

illustrated

by

pitch

letters,

as

in

Figure

6a. In a

fully

circu-

lar

approach

to

time,

one

step

earlier has the same

sense as six

steps

later;

two

steps

earlier,

the same

sense as five

later;

. . .

and

eight steps

earlier

has the

same

sense as one

step

earlier or six

steps

later.

Reversing

direc-

tions

yields Figure

6b. Each tone

(e.g.,

C)

is related to

each other tone

(e.g.

D,

E,

G,

and

B,

A,

F)

in

a

distinct

way.

Each

tone shares with all other

tones all the distinct

ways.

Just

as the

center of

a

circle is not on

the circle

itself,

each tone

stands outside the

tones with

which it is related in

this

manner. In

Figure

6,

C is

distinct from

D,

E, G,

and

B, A,

F;

D

is distinct

from

E, F,

A,

and

C, B, G; E,

from

F, G, B,

and

D,

C, A;

and

so

forth,

for all

tones

in

the

rhythm.

Proofs

in

number

theory

show that such

completely

braided,

truly

cir-

cular

structures can arise

only

in

3-, 5-, 7-, 9-,

etc.

tone

successions like

those listed

in

Figure

5.

By

contrast,

in

the 4-tone

pattern

of

Figure

7a,

no

circularity

arises.

Each such

4-step process

converges

on a

single

beat.

This is

arguably

a model of

four-square

time,

which

seems not to

renew

itself but instead

quickly

reaches a

clear limit where it

remains.

Indeed,

if

one

step

later

corresponds

to three

steps

earlier,

this model

comprises

a

technical basis for the

Europe-derived

norm

of 4-measure

phrasing.

In

re-

verse,

this

patterning corresponds

to

Figure

7b.4

The last technical

points

to be considered here involve

variability

and

complementarity.

The

eighths

that

are not

sounded

in

a

diatonic

7-

note/12-eighth rhythm

(stems

upward)

form a

5-note/12-eighth rhythm

(stems

downward)

which is

also

diatonic

(in

the

special

sense

employed

here):5

Each

diatonic

rhythm

listed

above

has

a

unique complementary

partner

which

is itself also a

diatonic

rhythm.

Not

only

is

this

relationship

be-

tween paired diatonic rhythms mutual or reciprocal, but such pairs over-

lap

maximally. By way

of a

single

illustration,

if

the

stems-downward,

5-

note/12-eighth rhythm just

presented

begins

an

eighth

earlier,

each

of its

notes

coincides

with

its

stems-upward,

7-note/12-eighth

partner

(see

Fig.

8).

Another

possibility

for

tacit motor mediation

of

sounding

notes is for

the

unheard

portion

of

the

gesture

to

peak

at the

temporal

mid-point

be-

tween

sounds,

so that the

major-mode

time-line

4.

If 0 is

the

first,

3 the

fourth,

6

the

seventh,

and 12 the

thirteenth

measure,

this

process

could be

considered to model standard

12-bar

blues, highlighting

its

cyclical, pendular,

tonic-permeated,

harmonic

structures,

in

contrast with the

teleological

delay

of tonic

reso-

lution

(via

counterpoint)

in

Europe-derived

Schenkerian

analysis.

5. The

orderly

notations

of

7-note time-lines

by

Kolinski

in

Blesh

(1958,

Ex.

42;

cf.

34,

384)

suggest

his

awareness of their

connection with diatonic modes and

complementary

rela-

tionship

with anhemitonic

pentatonic

rhythms.

81


13/20

BMR

Journal

JJ)fJJJ

would produce the following as its shadow.6

rP' '

r

P' '

Although complementarity

and

overlap might

seem

highly

abstract as-

pects

of relations

between

rhythms,

read

into them after the

fact,

as it

were,

I believe

they

can be understood

quite concretely

in

connection

with Erich von Hornbostel's reflections on African

drumming:

Figure

7.

Convergent

4-tone

cycle

a.

J J

J J

J

J J J J...

w x

Y

Z

W

X

Y

Z W...

W's

are one

step

earlier than

X's;

X's are

one

step

earlier than

Y's

W's are two steps earlier than Y's; Y's are two steps earlier than W's

W's are four

steps

earlier than

W's,

and

(trivially)

W's are four

steps

earlier

than

W's

W's are

eight steps

earlier than

W's,

and

(trivially, again)

W's are

eight

steps

earlier than

W's

b.

Reading

underlined letters from

right

to

left, and,

for the

parenthetical

statements, italicized letters from left to right

J J J J

J

J J J J

J

J J J

w

X Y Z

W

X

Y

Z W X

Y Z

W

W

is one

step

later

than Z

(cf.

W

is three

steps

earlier than

Z)

and Z

is

one

step

later than

Y

(cf.

Z

is

three

steps

earlier than

Y)

W

is

two

steps

later than Y

(cf. W is six steps earlier than Y) and Y is

two

steps

later than W

(cf.

Y

is

six

steps

earlier than

W)

W

is four

steps

later than

W

(cf.

W

is twelve

steps

earlier than

W)

6.

Compare

the

structure of

rast

in

North

African

maqamat.

82


14/20

Rahn

*

Turning

the

Analysis

Around

Figure

8.

Overlap

of complementary

diatonic

rhythms

7-tone/12-pulse

J

J

J J J

5-tone/12-pulse

r

r r

r

r

African

rhythm

s

ultimately

founded

on

drumming.Drumming

can be

re-

placed

by

hand-clapping

or

by

the

xylophone;

what

really

matters

s

the

act

of

beating;

and

only

from this

point

can African

rhythms

be

understood.

Each

singlebeating

movements...

two-fold:

he

muscles re

trained

nd

released,

the hand

s

lifted

and

dropped.Only

the second

phase

is stressed

acoustically;

but

the

first,

inaudible one

has the motor

accent,

as it

were,

which

consists

in

the

straining

of

the muscles. This

implies

an essential contrast between

our

rhythmic

conception

and the

Africans';

we

proceed

from

hearing, they

from

motion;

we

separate

the two

phases by

a

bar-line

and commence the

metrical

unity,

the

bar,

with the

acoustically

tressed

time-unit;

o

them,

the

beginning

of

the

movement,

the

arsis,

is

at the same time the

beginning

of

the

rhythmical

figure;

up-beats

are

unknown

to them.

(Hornbostel

1928,

52-53;

emphasis

added)

Despite

the

presumptuous,

arguably

essentialist,

us/them

contrast

of

Hombostel's

account,

a

number of

his

points

remain of

value. For African

traditions,

John

Blacking

(1955a;

1955b;

1961)

stressed-as has

John

Baily

(1985;

1992)

for

other,

non-Africa-derived

music-that not

only

drum-

ming,

hand-clapping,

and

mallet

performance,

but

performing

in

gener-

al

can

be

understood as

highly

patterned

motor

activity.

To

extend Horn-

bostel's

approach,

the

complement

of the

3-note/8-eighth rhythm

heard

as J. J. J

need

not

be

understood as

mere

sounds

nor

as a mere

abstraction

from

sounds.

Rather,

such a

rhythm

can

be

considered to

comprise

a

fully

concrete

motor

rhythm

that

results not

just

from

motions

corresponding

to

the

sounds

actually

heard

but

also from

motions,

unheard

but felt

by

performers,

which

occur

between

these

sounds and which

complete

or

complement

the

rhythm

actually

heard

(see

Fig.

9).

In

this

manner,

motor,

and

plausibly

social,

aspects

of

such

rhythms

can be

drawn into

other-

wise

purely

sonic

analysis.7

Incorporating

something

like

the

Effort values used in

Labanotation for

dance

(Bartenieff

and

Lewis

1980),

one

might gain

a

more

comprehensive

understanding

of

such

a

rhythmic

practice.

In this

regard,

Hombostel's

dichotomy

between strained-stressed-lifted and

released-inaudible-

dropped portions

of

a

movement

cycle

might

not

be

fully

appropriate.

7.

Conversely,

Baily's

motor

grammar

of

instrumental

performances

(1985;

1992)

omits

a

detailed account

of

time-interval

relations.

83


15/20

BMR

Journal

Figure

9.

Complementary

diatonicmotor

rhythms

Totalrhythm

b

f f

Sounding portion

J

J

Silent

portion

r

r

r

English

was not his first

language,

and

for all we

know he

never

under-

took traditional African

drumming.8

And the

maximum felt

contrast

of

the

unheard,

medial

portions

of

movement with

heard

parts

that

result

in

sounds

might

not

always

correspond precisely

to the

attacks of the

stems-

down

eighths

in the

above

notations.

Nonetheless,

a

technical

account of

the remarkable

time-interval

structures involved

merely

in

the

sounding

portions

of these

rhythms

provides

a

framework within

which

one

might

enhance

sensitivity

to

the

larger

motor

processes

within

which

they

occur. In

fact,

though

often

responded

to as

if

they

were

legislative

and

unquestionable,

mathematical

forms of

analysis,

including

the

framing

of

proofs,

can be

understood as

discovery

procedures-ways

of

imagina-

tively exploring regions

where

common

sense,

which in

music

theory

often seems

to have consisted of

insufficiently analyzed

Europe-derived

concepts,

has failed to secure

satisfactory

models or

representations

of ex-

perience.

Despite

the enhanced

understanding

that

results from

taking

motor

ac-

tivity

into

account,

analysts

are

unlikely

to

adopt

a

view of

music

exactly

parallel

to the

following

iconoclastic

perspective

on

speech

advocated

by

pioneer

motor

phonologist

R.

H.

Stetson:

"Speech

is rather a

set of

move-

ments

made

audible

than a

set of

sounds

produced

by

movements"

(quoted

in

Kelso and Munhall

1988,

58;

see

also Kubik

1979,

228 for a

ten-

tative

musical

rendering

of

this

doctrine).

An

important

difference is

that

all

spoken

languages,

as

such,

are

founded on

structures that

are

signifi-

cantly

articulative

in

ways

not

shared

by

such

arguably

musical

activities

as one

encounters in the

production

of

chance and

electronic

forms.

Nonetheless,

there is

sufficient

basis to

understand

the

notes of

music as

symbols,

not

merely

of

sounds heard in

certain

ways

but

also of

sounds

8. In

seeming

contrast to

Hornbostel's

account of

drumming,

Blacking

(1955b,

51-52)

as-

sociates a

feeling

of release with the

portion

of a hand

movement in which a

flute

player's

hand

is

away

from,

rather than

on,

the instrument.

84


16/20

Rahn

*

Turning

the

Analysis

Around

produced

in certain

ways;

and to hear such sounds with a motor

imagery

that

can,

in

principle,

be shared

with

others

(see,

e.g.,

Kubik

1972,

29).

A

plausible

case

in

point

is

off-beat

clapping,

which has

accompanied

the

spread

of

African

musical

practices

throughout

the world.

Often

in

Africa-derived

music,

constantly

syncopated playing

and/or

singing

is

accompanied

solely by

off-beat

clapping.

Problematic for

a

merely

sonic

analysis

of

music,9

this effect is

thoroughly

straightforward

if

one

accepts

the axiom

that

every

act of

performance implies

a

counter-act;

every

clap

of

hands

that

is

heard

implies

a

pair

of hands

unclapped

and

unheard;

every key pressed,

a

key

released;

etc.

Such

alternations

are

cyclic

and

highly compatible

with diatonic

braiding.

Diatonic

braiding

involves

processes

comprising

powers

of

two,

which

sequence

as 2n

=

1, 2, 4,

8,

16,

...,

and moduli

that

comprise multiples

of 4

units,

e.g.,

4, 8, 12,

16

...

(see

discussion

above).

Off-beat

clapping

is

pendular,

well

modeled

by

the

sequence

-ln

=

-1,

+1, -1, +1,

...

Each

of

its

cycles

has two

parts,

each

the

opposite

of

the

other,

neither

being necessarily

first nor second.

Capable

of

forming

epi-

cycles

within

larger

cycles

(like

dancers

turning

alternately

to left

and

right,

within

a

larger,

counter-clockwise

circling),

off-

(or,

for

that

matter,

on-)

beat

clapping

provides

orientation

enough

to

specify

that the

beat

has been turned

around,

helically

or

toroidally,

as

it

were,

within each

cycle

of

such

an

odd-numbered

figure

(i.e.,

of

3,

5,

7,

etc.

notes),

as

in

Fig-

ure

10,

which

constantly

renews

itself

by

assimilating

cycles

that

cross-

cut,

intensify,

subdivide,

or

multiply

each other.

Figure

10.

Cross-cutting

of

diatonic

7-note time-line

by

pendular

clapping

7-note

ime-line

r

r r

rJ

Pendularclapping

r

J

r

r

r

r

Although

much

hearing-as

can

be

shared,

much

can be

divergent.

The

diatonic

rhythms

discussed here can

be varied

considerably

without los-

ing

their

relevance. As

Figure

11

illustrates,

diatonic

rhythms

can sustain

extra

or

omitted notes

without

losing

the

characteristics of

evenness,

etc.

outlined

above,

if

the variant

forms

are

heard

as versions of

a

diatonic

rhythm

rather than

independently

as

rhythms

in their own

right

(see

Jay

Rahn

1991,

44-49).

Such

variants

of

diatonic

rhythms

can also

be heard in

a

four-square

manner and in

many

instances reward

substantially

such

9.

For

example,

Arom's elaborate

theory

founders on this

point

(1991, 202-205).

85


17/20

BMR

Journal

an

approach.

For

example, adding

a note at the fourth

eighth

of the

basic

5-pulse/8-eighth rhythm

of

Figure

3 results in

J

J

.

f

J which

can

be

heard not

only

as

a

"chromatic" version of

the diatonic

rhythm,

but

also

as an

unproblematic

realization

of

four-square

t

meter.

Figure

11. Diatonic

5-note/8-pulse

rhythm

and variants

involving

extra

and

omitted notes

Diatonic

5-note/8-pulse

rhythm

J

J

6-note variant JJ b . b

4-note variant

J

b

J

J

In

fact,

each

group

of diatonic

rhythms

includes at

least

two

(mutually

retrograde) patterns

which,

even

without

being

varied,

involve

little

or

no

syncopation

relative to a

particular

four-square

Europe-derived

framework.

8-eighths:

cf.

J:

b

J

J

and

J

J.J

12-eighths:

cf.

4 +4

(or

?):

J J

and

JJ

JJJ

16-eighths:

cf.

+

:

J J

J

JJJ

and

JJJ

JJJJ

Europe-derived analysis,

even in

ethnomusicology,

has

tended

to

insist

that

only

one

interpretation

is

possible

on

any

particular

occasion.

For in-

stance,

Mieczyslaw

Kolinski

(1973,

502)

asserted

that

"one

is

absolutely

unable to

perceive

[a

particular

African

piece]

at the same time in

g

or i."

Nevertheless,

within

the

doctrine

of

Gestalt

psychology,

which

analysts

like

Kolinski

have taken

as a

basis

for such

skepticism,

one

finds

Wolf-

gang

Kohler

saying,

in

connection with

an

ambiguous,

"Rubin's"

draw-

ing

(i.e.,

of

the

duck-rabbit

variety,

which

provides

a

visual

parallel

to

the

question

of

hearing

or

feeling

a

particular

passage

in

I

and/or

in

i-or

for

that

matter,

according

to

a

four-square

and/or

diatonic

framework):

"Under

certain

unusual

conditions both

objects

may

be

seen

at the

same

time"

(Kohler

1947,

183).10

Just as

there is

no

contradiction in

seeing,

on

10.

Blesh

(1958,

39-40)

interprets

psychologically

what

he calls

"time-shifts"

that

result

from

"off-beat

ostinatos"

(cf.

off-beat

clapping,

above)

as

follows:

"the

placing

of

the mea-

sure

division

tends to

shift,

the

counting

is

advanced

and

the

off-beat

becomes

the

princi-

pal

beat,"

noting

further

the

similarity

of

this

phenomenon

to

"various

types

of

optical

il-

lusion

and

the

seeing

of

complementary

hues

during

color

fatigue."

86


18/20

Rahn

*

Turning

the

Analysis

Around

any

particular

occasion,

a

given drawing

both

as

a

drawing

of

a

duck and

as

a

drawing

of a

rabbit,

there is no contradiction in

hearing

or

feeling

a

passage

as

simultaneously

both

in I

and

1,

nor

as,

at

the

same

time,

both

four-square

and diatonic in

rhythm.

It

would

be

contradictory

to claim

to

have

perceived

a

passage,

at a

particular

time,

both as

four-square

and

not

as

four-square.

Generally,

however,

such

baldly contradictory

claims

do not arise

in

analysis, although

other,

non-contradictory

claims

(e.g.,

of

the

duck-and-rabbit

sort)

are

cited to

justify singular, convergent

under-

standing.

If

diatonic

rhythms

can be

experienced divergently,

even within

a

sin-

gle experience,

how

can

one enhance skill

in

their

production

and

per-

ception?

Although

the

topic

is

vast,

a few

(admittedly

anecdotal)

obser-

vations seem called for.

Much of

the

material

developed

for

curricula in

rhythm reading

and dictation

appears

to

presume

that

unsyncopated

rhythms

are

substantially

easier

than

syncopated rhythms.

To be

sure,

syncopated rhythms appear

more

complicated

on

the

page, especially

if

conveyed

with a multitude of ties.

However,

in

my

own

experience,

stu-

dents who read and take down both diatonic and

unsyncopated rhythms

from the

outset encounter less

difficulty

later on.

Whereas a retardative

approach

to

syncopation

encourages

a

sudden

grasping

at a note

just

after the beat it follows and an

anticipative

strate-

gy

favors

staying

a

beat ahead of the

music

(e.g.,

in

one's

imagination),

diatonic

rhythms

reward

sustaining relatively

large

wholes

(e.g.,

of

8,

12,

or 16

units)

that

could

continue,

with or

without

variation,

indefinitely

far in the

future. As in so

many

other

regions

of

pedagogy,

here it seems

best

to

begin

not at the

beginning

(i.e.,

atomistically,

in

bottom-up

fash-

ion,

from

beat

and

subdivision

to

measure,

phrase,

and

piece)

nor at

the

end

(i.e.,

from

piece

to

part,

in

the

top-down

manner of

obscurantist

cog-

nitive

theories)

but

rather

in

the

middle

(i.e.,

inside-out,

from

substantial,

significant

units toward both the

parts

that

constitute

them

and

the

larg-

er

wholes

they

form

with one

another).

REFERENCES

Arom,

Simha. 1976.

The use of

play-back

techniques

in

the

study

of oral

polyphonies.

Eth-

nomusicology

20,

no.

3:483-519.

.

1991.

African polyphony

and

polyrhythm:

Musical

structure and

methodology.

Cam-

bridgeshire,

U.K.:

Cambridge University

Press.

Babbitt,

Milton.

1962.

Twelve-tone

rhythmic

structure and

the electronic

medium.

Perspec-

tives

of

New

Music 1:49-79.

Baily,

John.

1985. Music

structure and

human

movement.

In

Musical structure and

human

cog-

87


19/20

BMR

Journal

nition,

edited

by

Peter

Howell,

Ian

Cross,

and Robert

West,

237-258.London:

Academic

Press.

.1992. Musicperformance,motorstructure, nd cognitivemodels.In Europeantud-

ies in

ethnomusicology:

istorical

evelopments

ndrecent

rends,

dited

by

MaxPeter

Bau-

mann,

Artur

Simon,

and Ulrich

Wegener,

142-158.

Wilhelmshaven,

Germany:

Florian

Noetzel

Verlag.

Bakhtin,

Mikhail.1990.

Author and hero in

aesthetic

activity.

n Art and

answerability:arly

philosophical

ssays

by

M. M.

Bakhtin,

dited

by

Michael

Holquist

and Vadim

Liapunov,

4-256. Austin:

University

of

TexasPress.

Ballanta-Taylor,

icholas

J.

G.

1922.

Jazz

music and its

relation

to

African

music. Musical

Courier

4,

no.

22

(June

1):7.

Bartenieff,

rmgard,

nd

Dori

Lewis. 1980.

Body

movement:

oping

withthe

environment. ew

York:

Gordon

and

Breach

SciencePub.

Besmer,Fremont.1974. KidanDaranSalla.Bloomington:AfricanStudiesProgram, ndiana

University.

Blacking,

ohn.

1955a.Some notes

on

a

theory

of

rhythm

advanced

by

Erichvon

Hornbostel.

African

Music

1,

no. 2:12-20.

.

1955b.

Eight

flute tunes

from

Butembo,

East

Belgian

Congo. African

Music

1,

no.

2:24-52.

1961.

Patternsof

Nsenga

kalimba

music.

African

Music

2,

no.

4:26-43.

1967.

Venda

hildren's

ongs.

Johannesburg:

Witwatersrand

University

Press.

. 1970.

Tonal

organization

n

the

music

of

two

Venda nitiation

chools.

Ethnomusicol-

ogy

14,

no. 1:1-56.

Blesh,

Rudi.

1958.

Shining rumpets:

history

fjazz.

2nd

ed. New York:

AlfredA.

Knopf.

Bornemann,

Ernst.1946.A

critic ooks t

jazz.

London:

Jazz

Music Books.

Chernoff,JohnMiller. 1979.African

hythm

ndAfrican

ensibility.

Chicago:

University

of

Chicago

Press.

Clough,

John.

1994.

Diatonic interval

cycles

and

hierarchical

tructure.

Perspectives

f

New

Music

32,

no.

1:228-253.

Clough,

John,

and

Jack

Douthett. 1991.

Maximally

even sets.

Journal

of

Music

Theory

35:93-173.

Clough,

John,

and Gerald

Myerson.

1985.

Variety

and

multiplicity

n

diatonic

systems.

Jour-

nal

of

Music

Theory

9:249-270.

Copland,

Aaron. 1927.

Jazz

structureand

influence.

ModernMusic

4,

no. 2:9-14.

Cuney-Hare,

Maud.

1936.

Negro

musicians nd

their

music.

Washington:

Associated

Publish-

ers.

Ekwueme,LazarusE. N. 1975-1976.Structuralevels of rhythmsandform nAfricanmusic.

African

Music

5,

no.

4:27-35.

Floyd,

Samuel

A.,

Jr.

1991.

Ring

shout

Literary

tudies,

historical

tudies,

and

black

music

inquiry.

BlackMusic

Research

ournal

1,

no.

2:265-287.

.1993. On

integrative

nquiry:

Towarda

common

scholarship.

CBMR

Digest

6,

no.

1:1-6.

Forte,

Allen.

1973.The tructure

of

atonal

music.

New

Haven,

Conn.:

Yale

University

Press.

Holquist,

Michael,

and Vadim

Liapunov.

1990.

Introduction.

Art

and

answerability: arly

philosophicalssaysby

M. M.

Bakhtin.

ustin:

University

of Texas

Press.

Hornbostel,

ErichMaria

von.

1928.

African

Negro

music.

Africa

:30-61.

Johnston,

Thomas

F.

1971.

Shananga-Tsonga

rum and bow

rhythms.African

Music

5,

no.

1:59-72.

Jones.

A. M. 1954.Eastand

west,

northandsouth.

African

Music

1,

no. 1:59-62.

.1959.

Studies n

African

music.

2

vols.

London:

Oxford

University

Press.

.

1964.

African

metrical

lyrics.African

Music

3,

no.

3:6-14.

Kelso,

J.A.S.,

and

K.

G.

Munhall.

1988.

R.

H.

Stetson's

"Motor

honetics":

retrospective

di-

tion.

Boston:

Little,

Brown.

88


20/20

Rahn

*

Turning

the

Analysis

Around

King,

Anthony.

1960.

Employments

of

the

"standard

pattern"

n Yoruba

music.

African

Music

2,

no.

3:51-54.

1961. YorubaacredmusicromEkiti. badan,Nigeria:IbadanUniversityPress.

Knowlton,

Don.

1926.The

anatomy

of

jazz. Harper'sMonthly

Magazine

April):578-585.

Koetting,

James.

1970.

Analysis

and

notationof WestAfricandrum

ensemble music.

Select-

ed

Reports

,

no.

3:115-146.

Kohler,

Wolfgang.

1947. Gestalt

psychology.

nd ed. New

York:

Liveright.

Kolinski,

Mieczyslaw.

1967.

Recent trends in

ethnomusicology.

Ethnomusicology

1,

no.

1:1-24.

1973.A

cross-cultural

approach

o

metro-rhythmic

atterns.

Ethnomusicology

7,

no.

3:494-506.

Kubik,

Gerhard.

1972.

Transcription

f African

music from silent

film.

African

Music

5,

no.

2:28-39.

1979.Patternperceptionand recognition n Africanmusic. In Theperformingrts:

Musicand

dance,

dited

by

John

Blacking

and

Joann

W.

Kealiinohomoku,

21-249.

The

Hague:

Mouton.

Ladzekpo,

Kobla. 1971.

The

social

mechanicsof

good

music:

A

description

of

dance

clubs

among

the

Anlo-speaking

people

of

Ghana.

African

Music

5,

no.

1:6-22.

Ladzekpo,

Kobla,

and

Hewitt

Pantaleoni.

1970. Takada

drumming.

African

Music

4,

no.

4:6-31.

McClary,

usan,

and

Robert

Walser. 994.

Theorizing

he

body

in

African-American

music.

Black

MusicResearch

ournal

4,

no. 1:75-84.

Meyer,

Leonard

B.

1973.

Explaining

music.

Berkeley:

University

of

California

Press.

Nketia,

J.

H.

Kwabena.

1958.

Traditional

music of the Ga

people.

African

Music

2,

no.

1:21-30.

. 1963a.

African

music n Ghana.

Evanston,

ll.:Northwestern

University

Press.

.

1963b.Folk

songsof

Ghana.

Legon:University

of

Ghana.

.

1974.

Themusic

of Africa.

New York:W.

W.

Norton.

Pantaleoni,

Hewitt. 1972.

Toward

understanding

he

play

of

sogo

in

atsia.

Ethnomusicology

16,

no. 1:1-37.

Pressing,

Jeff.

1983.

Cognitive

isomorphisms

n

pitch

and

rhythm

in world

musics:

West

Africa,

he Balkans

and

Western

tonality.

Studies

n

Music 17:38-61.

Raab,

Claus. 1970.

Trommelmusik

erHausa n

Nord-West-Nigeria.

unich:

Kommissionsver-

lag

Klaus Renner.

Rahn,

Jay.

1977.

Some

recurrent eaturesof

scales. In

TheoryOnly

2,

no.

11/12:43-52.

.1978.

Evaluating

metrical

interpretations.

erspectives

f

New Music

16,

no. 2:35-49.

.1983. A theoryorallmusic:Problemsndsolutions n theanalysis fnon-Westernforms.

Toronto:

University

of Toronto

Press.

1987.

Asymmetrical

ostinatos n

sub-Saharan

music:

Time,

pitch,

and

cycles

recon-

sidered. In

Theory

Only

9,

no. 7:23-38.

.1991.

Coordination

f

intervalsizes in

seven-tone

collections.

Journal

f

Music

Theo-

ry

35:33-60.

Rahn,

Jay.

1995.

A

non-numerical

predicate

of wide

applicability

or

perceived

ntervallic

e-

lations.

Muzica

6,

no.

2:23-33.

Rahn,

John.

1980.

Basicatonal

heory.

New

York:

Longman.

Rycroft,

David.

1962. The

guitar

improvisation

of

Mwenda

Jean

Bosco

(Part

II).

African

Music

3,

no.

1:86-101.

Sargeant,Winthrop.

1938.

Jazz:

Hotand

hybrid.

New York:Arrow

Editions.

Schuller,

Gunther.1968.

Early azz:

Its

rootsandmusical

evelopment.

ew York:

OxfordUni-

versity

Press.

Stearns,

Marshall.1956.The

tory

of

jazz.

New York:

Oxford

University

Press.

Tomlinson,

Gary.

1991.

Cultural

dialogics

and

jazz:

A

white historian

signifies.

Black

Music

Research

ournal

1,

no.

2:229-264.

89

rahn jay african rhythm european mt

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