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Some Ideas about Voice-Leading between PCSetsAuthor(s): David LewinSource: Journal of Music Theory, Vol. 42, No. 1 (Spring, 1998), pp. 15-72Published by: on behalf of theDuke University Press Yale University Department of Music

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SOME

IDEAS

ABOUT

VOICE-LEADING

BETWEEN

PCSETS

David

Lewin

1.

Preliminaries

Definition 1.1.

Given

pitch-class

sets X

and

Y,

a voice

leading

from

X

into

Y will

(until

further

notice)

mean

a function V

which

maps

each

member

x

of

X

into

some

member

y

=

V(x)

of

Y.

For

instance,

if

X={D,F,A}

and

Y={E,G#,B),

then

a certain voice

leading

from X

into

Y is

defined

by

the

function

V(D)=E, V(F)=E,

V(A)=G#.

For this

occasion

I

have

mentioned

the

members

of

X in the

order of

"root,"

"third,"

nd

"fifth,"

but the

function is

consideredto be

defined on X

as

an

unordered et. The

constituentmembersof X

have

to

be

mentioned

in

some

order,

on the

page

or

as

I

speak

them-but the

order

s

not

relevant

o the

definitionof

the

function.

We

could

represent

he

above voice

leading

function

by

a table

with

arrows

(Figure

1).

As we

read down the

left column

of

the

figure,

the

members of

X

are

listed

in

alphabetical

order.

Once

again,

the

order is

arbitrary,o far as the functionitself is concerned.

Not

every

member

of Y

need be a

value of

a

voice-leading

function

as

defined.

In

Figure 1,

for

instance,

we

see that

the

member

pc

B

of

pcset

Y

does

not

appear

as

a

value

of the

function.

Furthermore,

ome mem-

15






A

-

G#

D-E

F-E

Figure

1

bers of

Y

may

appear

more

than

once as function

values.

In

Figure

1,

the

member

pc

E of

pcset

Y

appears

wice as a function value.

Definition 1.2.

We shall

say

that

the various

quasi-musical

passages

of

Example

1

instance

or

project

the

voice-leading

of

Figure

1.

Here we are

dealing

with

pitches

in

register,

rather

han

pitch

classes.

In

each

of the

passages,

some

pertinent

musical

featureassociates

each

pitchof the firstchord with some pitchof the secondchord,in a manner

instancing

or

projecting

he functionat

issue.

In

the

homophonic

extures

of

Examples

la

and

ib,

the

pitches

are associated

by

a familiarcriterion:

the

highest pitch

of the

firstchord associates

with the

highest pitch

of

the

second

chord,

and

so forth.

In

Example

Ic,

a

motivic

rhythm

associates

the first

pitch

of

the

melody

with the

fourth,

he second

with

the

fifth,

and

the third

with

the sixth.

In

Example

Id,

the first

pitch

of the violin asso-

ciates

with

the

second

pitch

of the

violin,

and so forth.

The curious

reader

will

recall

or discover

other

musical

criteria,

by

which

the

abstract

voice-

leading functionbetween the pcsets may be instancedin pitched pas-

sages.

Note 1.3. The

retrograde

of

Example

la

points

up

a

deficiency

in our

mathematical

model so far.We cannot

constructa

mathematical

unction,

from

pcset

{E,G#,B

into

pcset

(D,F,A},

that

is

being

instanced

or

pro-

jected by

the

retrogradepassage

of

Example

la.

If V were such

a func-

tion,

what

would

be

V(E)?

Given

what

we shall

understand

y

a mathe-

matical

"function,"

V cannot have

two

different

values

(viz.

both

F

and

D) atone andthesameargument viz. E).Anotherproblem s alsoexhib-

ited

by

the

retrograde

f

Example

la.

If

there

were

a

pertinent

unction

V

mapping

{E,G#,B}

into

{D,F,A},

what would

be

V(B)?

The

example

contains

no

pitch

named

B,

so

we cannot

assign

any

valueto

V(B).

Later

on,

we shall

expand

and

generalize

the

notion

of formal

"voice

leading"

between

pcsets,

so as to obviate

these

problems

with the current

model.

(We

shall

do

so without

having

to constrain

pcsets

X

and

Y,

in

general,

to be

of the

same

cardinality.)

For the

present,

however,

we

shall

stick

with the

current

model,

as

given

by

Definition

1.1.

There

will

be

plenty

to discuss.

Definition

1.4.

Example

la

highlights

a

significant

property

of

the

voice-leading

V

that

is

being

instanced.

The

pitch

D5 in

Example

la is

16






(a) (b)

(c)

(d)

Vln.

Via.

(A)

Vc.

-

Example

1

led to

a closest

possible

pitch

of

the

E

harmony,namely

E5;

the

pitch

A4

in the

example

is led to

a closest

possible

pitch

of the E

harmony,namely

G#4;

the

pitch

F4

is

led to

a closest

possible

pitch

of

the

E

harmony,

namely

E4.

We

shall call the

voice-leading

V

a

maximally

close voice

leading

between the

two

pcsets,

because it

admits the

projection

of

Example

la.

Example

lb,

in

its

arrangement

f

pitches,

projects

the

same "maxi-

mal closeness" of the voice leading.Even though pitch classes are not

fixed

in

one

register

or

another,

we

can

speak

of

"closeness"

n

this con-

nection,

meaning

something

like the

following.

For

any pitch-in-some-

register

whose name is

D,

some

pitch

whose

name is E lies

as-close-as-

or-closer-than

ny

pitch

whose name is

B,

or

any pitch

whose

name is

G#.

For

any

pitch-in-some-register

whose

name is

F,

some

pitch

whose

name

is E

lies

as-close-as-or-closer-than

ny

pitch

whose

name

is

B,

or

any

pitch

whose

name is

G#.

And for

any


whose

name

is

A,

some

pitch

whose

name is

G#

lies


ny

pitchwhose name is B, or any pitchwhose name is E.

We

call the

pc

voice

leading

"maximally

close"

rather

han

"closest,"

because

in

some

situations

here

may

be more

than one

maximally

close

voice

leading.

For

instance,

consider the

pcsets

X=

D,G}

and Y=

{

C,E,G

}.

In

a

maximally

close

voice

leading,

the

pc

G of

set X

must be

led to the

pc

G of

set

Y;

but

the

pc

D

of set X

may

be

led

either to the

pc

C of set

Y,

or to

the

pc

E of

set

Y.

(We

are

assuming

an

equally

tempered

12-note

articulationof

the

octave

in

this

context.)

So in

this

particular

case

there

are two

maximally

close

voice

leadings

of X

into Y.

The

voice

leading

V(D)=C, V(G)=G is

maximally

close: for

any pitch-in-some-

register

whose name is

G,

some

pitch

whose

name is

G

lies

as-close-as-

or-closer-than

ny pitch

whose

name is

C,

or

any pitch

whose

name is

E,

and

also for

any


whose

name is

D,

some

pitch

17






whose name is C lies


ny pitch

whose name is

E,

or

any pitch

whose

name

is G. The

different

voice-leading

V(D)=E,

V(G)=G

is

also

maximally

close: for

any pitch-in-some-register

whose

name is

G,

some

pitch

whose name is G lies


any pitch whose nameis C, or any pitchwhose name is E, and also for

any pitch-in-some-register

whose name is

D,

some

pitch

whose name is

E

lies as-close-as-or-closer-than

ny pitch

whose name

is

C,

or

any pitch

whose name is G.

Definition

1.5. We shall

say

thatthe

quasi-musical

passages

of

Example

la

and

Example

lb

manifest

the

maximal

closeness of the

pc

voice-

leading.

In

the

terminology

of Definitions

1.2

and

1.5,

Examples

Ic

and

Id

instanceorprojectthemaximallyclose voice-leading(i.e. thevoice lead-

ing,

which is

maximally

close),

but

they

do not

manifest

the

maximal

closeness

in

their

pitch

relations.

Examples

2abc instance

other

voice-leadings

of

X=

{

D,F,A}

into Y=

{E,G#,B

.

Example

2a instances the

voice-leading

V(A)=G#,

V(D)=B,

V(F)=E.

The

example

manifests a

significantproperty

of the

new voice

leading.

Definition

1.6. We shall call

the

voice

leading

at issue the

downshift

voice

leading

of

pcset

X into

pcset

Y.Each member

pc

of X is led to the

next

pc

of Y

encountered

n

a "downward"

irection.

Since

pitch-classes,

being independent

of

register,

do not

strictly

move

"up"

or

"down,"

we

shall

do well to formulate

he definition

more

rigorously:

for

any pitch

whose name is

A,

the next

pitch equal

to or

lower than

that

A,

whose

name is

a

memberof

pcset

Y,

is some

G#;

for

any pitch

whose name

is

D,

the next

pitch

equal

to or lower thanthat

D,

whose name

s a

member

of

pcset

Y,

is some

B;

for

any pitch

whose name

is

F,

the

next

pitchequal

to

or lower than that

F,

whose name

is a

mem-

ber of

pcset

Y,

is some

E.

In

that

sense,

the

voice-leading

function

V(A)=G#,

V(D)=B,

V(F)=E

is "the downshift

voice

leading"

of X=

{D,F,A}

into

Y={E,G#,B}.

Example

2b instances

the

voice-leading

V(D)=E,

V(F)=G#,

V(A)=B.

The

example

manifests

a

significantproperty

of the

new

voice

leading.

Definition

1.7. We shall

call the voice

leading

at

issue the

upshift

voice

leading

of

pcset

X into

pcset

Y.

Each

member

pc

of X is led to the

next

pc of Y encountered

n

an

"upward"

irection.

More

rigorously:

or

any

pitch

whose

name

is

D,

the

next

pitch

equal

to

or

higher

than that

D,

whose name

is

a

memberof

pcset

Y,

is some

E;

for

any pitch

whose name

is

F,

the

next

pitch

equal

to

or

higher

than

that

F,

whose

name

is a member

of

pcset

Y,

is some

G#;

for

any pitch

whose

18






(a) (b)

(c)

(d)

Tenor Trombones

. . . .

IIII

,I

Cb. Tuba

8ba-

- - - - - - -

(FB)

Example

2

name

is

A,

the next

pitch equal

to

or

higher

thanthat

A,

whose name is

a

memberof

pcset

Y,

is some

B. In

that

sense,

the

voice-leading

function

V(D)=E, V(F)=G#, V(A)=B

gives

the

upshift

voice

leading

of X=

{D,F,A}

into

Y={E,G#,B .

Of

course,

from

a traditional

point

of view the

passage

of

Example

2b

is "bad": t leads

F

to

G#

through

an

augmented

second,

and it sounds

a

parallel

fifth as well.

Still,

the

voice-leading

of the

pitch

classes is

not

musically

irrelevant.

Example

2c shows how

Wagner

uses

exactly

this

voice-leading

when the tenor

trombonesand contrabass

uba sound the

last version of

the Fate

Motive,

ten

measures before the

end

of

Die

Walkiire.

D

is led

to

(the

pitchclass)

E in

the contrabass

uba;

F is

led to

G#

in

the second

trombone;

A

is led

(essentially)

to

B in

the first trom-

bone.

In

the

music,

the

upshift

voice

leading gives

a marveloussense of

"rising"

o the

final

cadentialmotive

(despite

the

seventh-leap

down

in

the

tuba,

of which

more

later).

While

we have been

hearing

he

rising

A-

to-B

in

the

melody

of the

Fate Motive for

a

long

time

before,

and

the

ris-

ing

F-to-G#

as well

in

an

inner

voice,

the

bass

of

the motive

has so far

descendedfrom

D

to

C#,

rather

han

rising

(in

the

pitch-class

sense)

from

D to E.

The

upshift

voice

leading

mimics the

rising

of the

flames,

and of

Wotan'shopes for the future.

Wagner

does not

eschew the

augmented

second

in

the second

trom-

bone. He

does,

however,

mitigate

he

effect

somewhat

by

the

quarter-note

6chapp6e

on

G#

n

the first

trombone,

anticipating

he

second trombone's

G#.

We are

alreadyvery

familiarwith

this

aspect

of the

Fate Motive.

The

third

rombone,

not

shown,

leads

F

to

E-but that is not

a salient

feature

in

the

presentation

f the

Motive here.

The

6chapp6e

G#

distracts

somewhatfrom

the

effect

of

the

"parallel

fifth"

between first

tromboneand

tuba. More

efficacious

yet

is

Wagner's

wonderfulconceit, leadingthe tubafrom D to E down a seventhrather

than

up

a second.

In

our

terminology,

Wagner

nstancesthe

upshift

voice-

leading

here,

but does

not

manifest

that

voice-leading

in

the

tuba

part.

The

seventh-leap

down

destroys

he

"parallel"

ffect

of

the

fifths

between

first trombone

and

tuba.

It is

a

cheap

trick,

but it is one of

those

cheap

19






tricksthatturn o

magic

in

Wagner's

hands.

Specifically,Wagner, eaping

down to the lowest note

in

the

tuba,

suggests

that the

D

above

it

repre-

sents an

overtoneof the

fundamental

ow

E,

rather hana seventh

degree

of

its

Phrygian

scale.

Thus his

progression

mimics or

parodies

the

fun-

damental-bassstructureof a traditionalV-I cadence, transforming he

archaic

organalgesture

nto a

specimen

of functional onal

harmony,

nd

thereby bringing

his archaic

myth

into the thick of

nineteenth-century

German

culture.

There

is a

special

reason that the

organal

sense of the

upshift

voice

leading

sounds so archaic here. When a

fundamentalbass

is active

and

moving up (pitch

class

wise)

by step,

there s

a

particular

ormative

pro-

cedure

for

writing

functional

harmony

n

the

usual

four-part

exture:

tu-

dents

are instructed o lead the three

upper

voices of thattexture

down-

in

contrary

motion to the fundamental

bass-manifesting

the downshift

voice

leading. Example

2d

illustratesnormative

procedure,

n

the

con-

ventional

4-part

exture,

or

voicing

the

progression

of a

D

minor

triad o

an

E

major

riad

n the

upper

voices. The fundamentalbass

part,

marked

"(FB),"

is

assigned

solid noteheads without

stems. The

three

upper

voices manifest

downshift

voice-leading

of X=

{

D,F,A}

to Y=

{E,G#,B

.

The

innervoices

of

Example

2d are

actuallypresentedby

the

third

and

fourth

trombones

n the

Wagnerpassage.

This

too-beyond

the seventh

leap

down

in the

tuba-encourages

us to be aware hatfunctional

unda-

mental

bass

harmony

s latent

in

the

passage.

But the

hypothetical

unc-

tional

model

of

Example

2d,

with its downshiftvoice

leading

n the

upper

parts,

destroys

an

essential feature

of the

Fate

Motive,

the

characteristic

upward

urn at the

end of

its

melody.

The characteristic

upwards

rise

in

the

Motive

melody,

from

its

beginning

on

A to its

ending

on

B,

is

exactly

what

generates

he

upshift

voice

leading

of

Example

2c.1

2. More

analyses;

intervallic

considerations

2.1.

Example

3a

represents

he

central

progression

rom the

opening

of

George

Crumb's

Makrokosmos

or solo

piano.

The

progression

s clouded

in

performance

by

various

aspects

of Crumb's

"Darklymysterious"

ex-

ture:

the

piano's

register

is

murky,

the

dynamic

is

"pppp

sempre",

the

damper

and

una corda

pedals

are

to

be held down

throughout

he

passage,

and each

of

the chords

on the

example

is

inflected

by

a

grace-note

chord

a

tritone

ower.

Still,

we

get

"ca. 3

sec."

to listen to each

chord

depicted.

Then too

the

passage

appears

ix

different

imes

during

he

course

of the

work-either atthe indicated evel or atritonehigher-so we haveagood

deal

of

opportunity

o sort

out the acoustical

signal.

The

open

noteheads

of

Example

3b

transpose

he

progression

of Exam-

ple

3a into a

more

normal

keyboard

register.

The filled-in

noteheads

of

Example

3b

provide

a fundamental

bass

for

the

succession

of minor

tri-

20






(a)

(b)

8ba

- -

J

Example

3

ads above.The fundamental ass

proceeds

unidirectionally

own

through

half of a

chromatic

scale.

Despite

this

chromaticism,

he

example

never-

theless

manifests

n

its voice

leading

the traditional dvice

given

to

begin-

ning

studentsof diatonic

harmony

who have to

connect,

in

a

traditional

texture,fundamental-positionriadswhose basses are relatedby step:

move

the

upper

three

voices,

in

contrary

motion to the

bass,

to the

next

available

notes of the new

harmony.

We

already

discussed this

normative

advice in connection with the

Wagner

passage

studied above.

Using

our

presentterminology,

we

can

say

that

each

open-notehead

riadof

Exam-

ple

3b moves to the

next,

not

only

instancing

but also

manifesting

upshift

voice

leading.

Even

thoughadjacent

riadsof the

progression

re

n different

chordal)

inversions,

even

though

he

upper

ine

displays

a

variety

of

melodic inter-

vals, the

progression

of chords still has a

homogeneous

characterabout

it,

because

the total amountof motion

upwards

rom each triadX to the

next

triad

Y

is

always

9

semitones,

in

the

following

sense.

The

fifth of

each

triad

moves

up

to the

open-notehead

oot of the next

through

4 semi-

tones;

the

open-notehead

root of each triadmoves

up

to the

third of the

next

triad

through

2

semitones;

4

plus

2

plus

3

is 9. The series of

inter-

vals

just

mentioned s

allegorized

n the

melodic

triple-interval ycle

man-

ifested

by

the

upper

voice

of the harmonic

texture: the

rising

melody

C-E-F#-A-C#-Eb-F#

oves

through

half

of

an

interval

cycle generatedby

intervals

of

4, 2,

and 3

semitones

n

cyclic

succession.

(And

when the

pas-

sage

is

transposed

a

tritone,

the

transposed

melody

completes

the

entire

cycle.)

To

be

sure,

the

particular

ntervallic

numbershere can be

analyzed

as

if

they

were

a

secondary

feature,

resulting

from

the

consistent manifes-

tations

of

upshift

voice-leading

as

one

chord moves

to

the

next,

together

with the

fact that each

open-notehead

hord

pc-transposes

he

preceding

one

by

a

pc-semitone

down.2

Still,

the

constancy

of those

numeric

values,

fromeach stage of the progression o the next, is a featureworthnoting

in

its own

right.

We

shall return o the

point

in

later work.

2.2.

Example

4a

represents

one

complete

cycle through

an

analogous

progressionusing

major

rather

hanminor

triads,

and

a

fundamental

ass

21






(a)

(b)

I

-,•

TL

,M

Example4

(a) (b)

(c)

Bassoon

e

e3IL3

3

uI

(ins)ments3 3 1

Bass Clarinet

Example

5

moving

down

in

whole

rather han

half

steps. Segments

of the

progres-sion are

very

audibly foregrounded

at

several

points

in Liszt's

Fantasia

quasi

Sonata,

"Apres

une lecture

du Dante."3The

segment

from D

through

F#

harmonies

appears,

one

harmony

per

measure,

mmediately

preceding

he

last

eight

measuresof the

piece

(mm.

364-68).

The entire

cycle

from

D

harmony

onwards s

projected

n mm.

228-32;

the music

then

continues,

during

mm.

232-34,

with

another

cyclic segment

as

far

as the

Ab

harmony,

which receives a

substantial

prolongation

over

mm.

234-37.

Analogous

formal features

obtain,

in

Example

4a,

to all those

which were

noted

in connection

with

Example

3b.

In

particular,

one

observes thatthe total numberof semitones

shifted

upward

emainscon-

stant

from each chord

X

to the next chordY of the

open-noteheadpro-

gression.

In

the Sonata

itself,

Liszt

develops Example

4a

out

of his

second

theme,

whose

beginning

(mm. 103-107)

is sketched

n

Example

4b.

2.3.

Example

5a

copies

certain nstrumental

arts

rom

the

beginning

of

the

Magic Sleep

motive toward he end of Die

Walkiire.

he bass clarinet

carries he acoustic bass line for the progression. t is not a fundamental

bass line

throughout

but,

like the solid-notehead

undamental

bass lines

of

earlier

examples,

it moves

throughout

n

contrary

motion to the sense

of

the

uppermost

hree voices

in

the

passage.

Those

three

voices,

in the

present

example,

are

projected

by

the

two flutes

and the first bassoon.

22






They

move

downwardsor remain

stationary

hroughout, xcept

that

the

bassoon moves

up

from

the

F

of

the third chord to the

F#

of the

fourth

chord.

(We

shall

devote some attention o that

later.)

At the

parenthesized

E

major

harmony

he instrumentation

hanges;

so does the instrumentalvoice-leading over the next four chords.Al-

though

the four

beats

starting

at the

E

major

harmony

are

basically

se-

quential

to the four

beats

starting

at the

Ab

major

harmony

of

Example

5a,

the

sequence

s

inexact-inexact not

only

in

instrumentation nd

reg-

ister,

but also

in

harmony.

The

chordthatfollows the

E

majorharmony

s

not a G

major

5/3

(which

would

be

analogous

to

the

B

major

5/3 of

Example

5a);

rather,

t is an

Eb

6-chord.4

The

following

chord

also

varies

from

its

model.

For these

reasons we shall confine

our attentionhere

to

Example

5a

here.

Example5a omits the two clarinetparts,which fill in the harmonies

above the bass clarinet and

under the

upper

three voices. While

their

voice

leadings

are

of

interest,

I am

treating

hem as

subordinate o

Exam-

ple

5a for

present

purposes.

Also

omitted from

Example

5a is the

little

harp

arpeggio

that accents the

beginning

of the

motive,

at the

Ab

chord.

The most

seriousomission from

Example

5a is the

timpani

roll

on

high( )

E,

that

begins

at

the

beginning

of

the

example

and

continues

throughout

all

of

the

Magic

Sleep

music.

Whatthatroll is

doing

makes sense

only

in

a

larger

context.5

Example5b focuses on thethreeuppervoices. The numeral3 appears

above the note

heads between

the

first chord and

the second. This

indi-

cates that the total

shift of the

instrumental oices

as written s 3

semi-

tones- the first

flute

moving

down

by

1

semitone and

the bassoon

by

2,

while the

second flute remains

stationary.

The

numeral 3 also

appears

above the note

heads

between the

second chord

and the

third.This

indi-

cates that

the

total shift of the

instrumental

voices

between

those two

chords is also 3

semitones-each

of the

three

instruments

moving

down

here

by

1

semitone. Between the

third chord and

the

fourth,

the

instru-

ments once

again

move

by

a totalof 3 semitones-the second

flute mov-

ing

down

by

2

and

the bassoon

up

(NB )

by

1.

Between

the fourth

chord

and

the

fifth,

the

instruments

move

by

a total of

1

semitone-only

the

first flute

moves,

and

it moves

by

a

semitone.

The

progression

rom the

first

to the second

chord

of

Example

5b

in-

stances

downshiftvoice

leading

between the

pcsets,

and also

maximally

close voice

leading.

Furthermore,

t

manifests

both

those voice

leadings.

The

same

is

true of

the

progression

rom

the

second chord to

the

third,

and

the

progression

from the

fourth

chord to the

fifth.

The

progression

from

the third

chord to

the

fourth,

in

the

example,

both instances

and

manifests

maximally

close voice

leading

between

the

pcsets.

That is

the

rationale,

presumably,

or

leading

the bassoon

up

here. To

manifest

downshiftvoice

leading,

the

bassoon

would have

to

drop

rom

F down

to

23






middle C-not

only

would that

make a

glitch

in the vertical

sonority,

t

would also shift

pitches by

a total

of

seven,

rather han

three,

semitones.

Thatwould

put

a

big rip

in

the

systematic

succession

3-3-3-1 of the

num-

bers above the

example.

The successionis suggestivein connectionwiththe bass clarinetpart,

copied

over

in

Example

5c.

The bass clarinet

projects

the

rising

succes-

sion of intervals

3-3-1. This

analogy

to the

basically-descending

3-3-3-1

structure

of

Example

5b

projects

an

interesting

sort of mirror

relation

between

bass

clarinet

below,

and three-note chords

above. The

rising

solo bass

clarinet

n some sense "balances" he

more-slowly-descending

upper

three

voices as

a

group.

One notes that the

role of the

Bb

suspen-

sion

in

the

firstflute

is crucialhere. Without hat

syncopation,

he

instru-

mental

part-leading

and close

shifting

in

Example

5b

would

give

a

numericalpatternof 3-3-4, not 3-3-3-1.

2.4.

Example

6a transcribes

aspects

of the

music

when the

voice enters

the last

movement

of

Schoenberg's

econd

string

quartet.

The voice

sings

"Ich

fiihle

luft...,"

doubled

by

the 'cello.

The

top

staff of the

example

indicatesthe

homophonic

accompaniment

n

the

upperstrings.

The

pas-

sage depicts

a

"magic"

moment-the

"luft"

s

"von anderem

planeten."

In

that

respect

Example

6a

invites

comparison

with the

beginning

of

Wagner's

Magic

Sleep

music,

just

studied

in

Example

5. The musical

textures have some

interesting

similaritiesas well. In

particular,

one

notes

in

Example

6a,

as

in

Example

5a,

the

rising

bass

line in

contrary

motion

against

a

generally

descending

homophonic

group

of three

upper

voices.6

In

the

Schoenberg

quartet,

he

rising

bass

line sets a

pitch-motive

that

is

already

quite

familiar

rom

earlieron

in

the

piece.

Below

the

upper

staff

of the

example,

numbers

are written

between

consecutive

chords

in the

upper

voices.

These numbers

mean

just

what

the

analogous

numbers

meant

n

Example

5b.

They give

the total

number

of semitones

traversed

by

the three

voices,

as

they

move from

their

notes

in one chord

to

their

notes

in

the

next chord.

Thus

in

Example

6a,

to

get

to the

chord

after

the

eighth-note

A,

violin

1 moves4

semitones,

violin

2

moves

3

semitones,

and the

viola moves 3 semitones.

The number

10,

that

appears

on

the

example

at

this

point,

is the

sum of

4, 3,

and

3.

The numbers

on

Example

6a

are

striking.

Relatively

few

numerical

values

appear,

andthe

ones that

do

appear

ecur

quite

a bit

during

he

pas-

sage.

(We

should

recall that

they

are not

numbers

mod

12,

but

rather

pos-

itive

real

numbers

rom

among

a

potentially

unlimited

amily

of

positive

integralvalues.)Specifically,onlythe numbers2, 3, 7, and10appeardur-

ing

the

passage.

Example

6b

shows

how

these

numbers

develop

intervallic

aspects

of

the

"Ich

ftihle

luft"

motive. The

numbers

n

Example

6b

are also

positive

24






8va

-

---

(a) 3 2 10 2 3 7 3 7 3 (b)

7 3

tL

I

Ich flih- le luft

10

Example

6

integers,

counting pitch-semitones

from lower to

higher pitches

in

the

motive,

which are

respectively

earlier

and

aternotes of the motiveas

well.

The reader s invited ocompare herelationbetweenExamples6a and

6b

in

this

regard,

o the relation

between

Examples

5b and 5c. The

bass-

clarinet

part

n

Example

5c is not a

prominent

motive in

Die

Walkiire,

ut

Wagner

uses the intervallicseries 3-3-1 a fair

amount

n

his music

gen-

erally,

and the series becomes a

very prominent

motive

in

Parsifal.

3.

An

anecdotal excursion

3.1.

Roger

Sessions

wrote

in

1951 an

essay

entitled "TechnicalProb-

lems of

Today"

o conclude his

harmony

extbook.'The end of the

essay

discusses

atonality

and

twelve-tone

composition.

Sessions refers to

the

first

movement

of

Schoenberg's

Fourth

Quartet,

op.

37,

citing

"sensa-

tions

which..,

.suggest...

the

key

of

D,

even

specifically

D

minor."8

Pointing

out the

difficulty

of

exploring

such

sensations

using any

re-

ceived

theoretical

terminology,

Sessions continues:

"Such

quasi-tonal

sensations are

simply

evidence that

[the]

ear has

grasped

the relation-

ships

between

the

tones,

and has

absorbedand

ordered hem. It is a

mis-

take to

regard

such

sensations as

connected

exclusively

with

the tonal

system

as such. The

intervals,

and their

effects,

remain

precisely

the

same;

two

tones a fifth

apart

still

produce

the effect of the

fifth, and,

in

whatever

degree

the

context

permits,

will

convey

a

sensation similar to

that of a

root and its

fifth,

or of a

tonic and its

dominant."9

3.2. In

the

late

1950s,

when

I

was

studying composition

with

Sessions,

he

told me an

interesting

tory

that

follows

up

on the

passage ust

quoted.

This is

the

story,

as I

recall it.

Shortly

after

having

finishedthe

harmony

book, Sessions (who was living in Berkeleyat the time) visited Schoen-

berg

in

Los

Angeles.

Curiousas

to

Schoenberg's

views on

these

matters,

Sessions

asked

Schoenberg

what he

thought

of

Sessions's

remarkson

the fifth.

Schoenbergexpressed

firm

disagreement.

Sessions

then

asked

25






Schoenberg

how he

(Schoenberg)

n

fact did conceive of the fifth

as an

interval.

Schoenbergexpressed

interest

n

that

question

and,

after some

thought,

said

something

ike: "It's

slightly

more than

half an octave."

That

s the end of Sessions's

story,

as

I

recall it. When he told it

to

me,

my firstreaction was thatSchoenbergwas being ironic or sarcastic,as

Beethoven

was often wont to be when irritated

by

comments

or

ques-

tions.

But

on

later

reflection,

I

no

longer

think that.

I

think, rather,

hat

Schoenberg

was

trying

to describe

a serious musical

conception.

3.3. Let

us consider

for instance he two

pitches

C4 and

G4,

which

span

a fifth.

In

moving

from C4 to

G4,

or

vice-versa,

we

are not

manifesting

maximally

close

voice

leading

(between

the

singleton pcsets).

To mani-

fest

that

leading,

we would

have

to

move C4 down a fourth o

G3,

or

G4

up a fourthto C5. The situationcomes aboutpreciselybecausethe fifth

is

larger

han

half an octave.

But it

is

only slightly

more than half

an octave. That

is,

if we move a

pitch

by

any

smaller

pitch

interval

n

12-tone

equal temperament,

nclud-

ing

a

tritone,

we are

manifestingmaximally

close voice

leading.

The fifth

is thus the

unique

smallest

pitch

interval

in

12-tone

equal

temperament)

which

does not

manifest

maximally

close

voice

leading

in

the

pertinent

situation.

0

3.4. Example7a gives the pitchesfor the theme at the beginningof the

Fourth

Quartet.

With two

exceptions,

all

the melodic intervals

between

successive

pitches

are

smallerthan half

an octave in absolute

size.

The

exceptions,

bracketed

on the

example,

are the

leap up

from

Bb

to

F,

and

the

leap

up

from

E

to

C. The

F

and

the C are the

only

two notes

of

the

theme

which take

tonic

accents,

and

their

accentuation

s

emphasized

all

the more

by

the

unique

(bracketed)

departures

rom melodic

close-shift-

ing

that herald heir

arrivals.

In

respect

to the

consideration

ust

discussed,

F and C sound

analo-

gous.

And that

encourages

us to hear them in a

transpositional

elation.

When

we do

so,

we

shall hearthat

F defines the

top

of the

ambitus

or the

first

hexachord

of the

theme,

while

C defines the

top

of the

ambitus

for

the

second

hexachord."

Example

7b

lays

out

the two

ambitus,

as

indi-

cated

by

the slurs.

Hearing

a

transpositional

elationbetween

F and

C,

we

can

to a

certain

extent

hear

a

transpositional

elation between

the

two

ambitus.

And

when

we do

so,

we can

easily experience

one

of the

quasi-

tonal

sensations

o

which Sessions

refers.

We can

specifically-given

the

first

five notes

of the

theme-hear

the A-to-F

ambitus

of the

first hexa-

chord

as

spanning

degrees

5-to-3

of a

D minor

mode;

then

we can

hear

the

transposed

E-to-C

ambitus

of the

second hexachord

panning

degrees

5-to-3

of

a dominant

A minor

mode.

That

is,

we

will be able

to

hearthe

T7-relationbetween

the

two

ambitus

as a tonal

fifth-relation.12

Examples

26






(a)

?--

--

--

,

-

(b)

(c)

Largo

=78

f,

a4

3 3

incipit

chant

I

H

I

I

cadence

(d)

I-

I

.

?

I

(e)

incipit

chant

cadence

incipit

chant

cadence

Example

7






7ab,

and the

commentary

o

far,

thus

appear

o endorse both

Sessions's

remarkson the

fifth,

and

Schoenberg'sresponse

to Sessions.

3.5.

Example

7c

gives pitches

and

rhythms,

with a few other

ndications,

for themelodyof the themein thequartet's low movement.Bracketson

the

example

articulate he

theme into three

segments,

which

I

have

called

the

incipit,

the

chant,

and the

cadence. The

chant

segment

is character-

ized

by

repeated

tones or

figures.

Bb

within the chant

segment

is not

repeated,

but

chanting

resumes

immediately

after the

Bb,

and continues

for some

time thereafter.

Though

the

E

is

technically

a

repeated

note,

it

sounds

so

strongly

as

part

of a conventional adence formula hat

I

do

not

count

it as

continuing

the chant

section;

rather

I

consider

the

thirty-

second

note

E

to be

anticipating

he

quarter-note

of the

cadence.

Example7d is an analog to Example7a. It gives the pitches for the

largo

theme,

and

brackets

those intervals that

depart

from

immediate

melodic

close-shifting, eaping

a fifth

or

morebetween

consecutive

ones.

The

leap

from

Bb

to

Gb

here,

like the minor

sixth

leap

in

Example

7a,

reinforces

the tonic

accent on the theme's

climax.

The

situation

as

regards

ambitus

is

different,

though.

The entire

melody

of the

largo

theme

lies within

an

ambitus

of

eleven

semitones,

and there

is

no such melodic

articulation

of the

hexachords,

each

span-

ning

an ambitus

of a minor

sixth,

as we observed

in

the earlier

allegro

theme(Example7b). Rather, he threebracketedeapsof Example7d go

with

the

pentachordal

egmentation

proposed

n

Example

7c: the

leap up

from

Ab

to

Eb

goes

with and

inflectsthe

incipit

segment;

he

leap

up

from

Bb

to

Gb

goes

with and

inflects the chant

segment;

and the

leap

down

from

E

to

A

goes

with and

inflects

(indeed

constitutes)

the cadence

segment.

Example

7e

depicts

the

leaps

in

that

context,

using

slurs to

group

the

incipit,

chant,

and cadence

segments.

Listening

to

this,

we

will hear

that

the

minor

sixth

leap

Bb-Gb

pans

the

total ambitus

of

the chant

segment.

When

we writeout theambitus or the three

segments,

n

Example

7f,

we

will

then

hear

the

incipit segment

also

spanning

a minor

sixth

in ambitus.

We

will further

hear the cadential

fifth

leap,

which is its own

ambitus,

lying

in

the

same

tessitura

as the other

two ambitus.

Following

the

analogy

of

Example

7b,

we

might

entertain

hearing

a

transpositional

elation

(T3)

between

the

incipit

and chant

ambitus.

The

idea

is

supported

when

we hearthatboth

the

leap

to

Eb

and

the

leap

to

Gb

are

leaps

up.

But-especially

absent

the

quasi-tonal

"fifth-relation"

f

Example7b-the transpositional

dea is not

so

strong

n

Example

7f.

An

inversional

elation,

between

incipit

ambitus

and chant

ambitus,

s corre-

spondingly

more

manifest.

The inversional

relation

of ambitus

is

sup-

ported

by

an

inversional

ayout

of entire

pitch-sets

in

register,

between

28






the two

segments:

G3-Ab3-B3-C4-Ek4,

eading

pitches

upwards hrough

the

incipit

segment,

inverts

into

Gb4-F4-D4-D64-B63,

eading pitches

downwards

hrough

he chant

segment.

The low

G of the

incipit

and

its

inversional

partner,

he

high

G6

of the

chant,

mark he

total ambitus

of

the

largotheme as a whole. The E64at the top of the incipitambitus nverts

to the

B63

at

the bottom of the chant ambitus.

Eb4

s

amply

marked

by

a

bracketed

eap.

B63,

in

the

chant,

is

not

approached

by

a bracketed

eap

(of

7 or

more

semitones).

But it is

approached

y

leap

(of

3

semitones

or

more),

and it

is

the

only

tone of

the chant

segment,

other than the

high

G6,

that is so

approached.

The chant

segment

also

singles

out B63 as

its

unique

tone that is not

repeated-in

particular

he

firsttone

in

the

theme

we have heardsince E6

that is not

repeated.

Inversional

hearing

of

the

layout

n

Example

7f is

further

trengthened

when we hearthat the two notes of the cadenceare' elatedby the same

pitch-inversion

hat

relates low G

to

high

Gb,

the

same

pitch-inversion

that

relates the

incipit

pitch-set

as a whole

to the

chant

pitch-set

as a

whole.

Indeed,

within the total

ambitusof the

theme,

E4

and A3 are

the

only

inversional

pitch-partners

hat

span

a fifth

(seven semitones).

Here

we

come

back

again

to the ideas

put

forth

by

Sessions and

by

Schoenberg

in

connection with the

seven-semitone

pitch

interval.

The

conventional

formulaic effect

of the

fifth

as a

quasi-tonal

gesture

for the

theme's

cadence is

startlingly

blatant-all the

more

so

when

the solo

'cello

softly

echoes the

falling-fifth

pitch-interval

mmediately

following

the

ending

of

the

largo

theme.

Example

7e

suggests,

in

addition,

that

we

might

want

to

explore

a

transpositional

or

inversionalrelation

between

the two

bracketed ifth-

leaps,

Ab-Eb

n

the

incipit

and

E-A

in

the

cadence.

I

believe this

would

indeed

be

fruitful,

but

I

do not

want to

take the

space

in

this

essay

to

pur-

sue

the matter

more

extensively.13

4.

Another

application

of

close

voice-leading

theory

to

melodic

analysis

4.1.

Example

8a

gives

the

first four

pitches

of the

theme from

Schoen-

berg's

first

movement.

For

purposes

of

melodic

theory,

the

pitches

are

articulated nto an

opening

segment

{D4,

C#4,

A3

and a

continuation-

pitch

Bb3.

Arrows

are drawn

from each

pitch

of the

opening segment

to

the

continuation-pitch,

nd each

arrow s

labeled

by

the

absolute

number

of

semitones

(either

down

or

up)

between

the

pair

of

pitches

involved.

The total amountof shifting,4+3+1

=

8 semitones, is given above the

example.

In

the

context of

section

3

above,

one

observes

thateach

arrow-

number s

less than

7.

Consequently,

he

example

manifests

(in

our

tech-

nical

terminology)

maximally

close

voice-leading

between the

pcset

29






total

=

8

total

=

16

total

=

22

total

=

26

(a)

(b)

(c)

4

(d)

8

4 8 4

total

=

20

total

=

30

3

0 4

to

or6I,5I2tot2l

4

total = 42 total = 54

(g)9

(h)

6r

3

f

f8

?

W s

Example

8

{D,C#,A

and the

pcset

{Bb}.

Evidently

the

progression

rom

{D4

to

{C#4

manifests the same feature, and so does the progressionfrom

{D4,C#4}

to

{

A3).

Example

8b

manifests

maximally

close

voice-leading

between the

pcsets

{D,C#,A,Bb}

and

{F). (All

arrowson the

example

are labeled

by

numbers ess than

7.)

Given

the

registers

or

the first our

pitches

here,

an

octave

F is

required

o

manifest the

close

voice-leading.

Given the con-

ventions under

which

Schoenberg

was

working

in

this

composition

(or

simply

the conventionof

melody

as

monophony),

Schoenberg

has to

pick

only

one

pitch

to

represent

he

pcset

{

F

}-either

the

pitch

F3 or

the

pitch

F4.

Examples

8c and 8d show how F4

(in

Example

8c)

represents

he

idea of

manifesting-maximally-close-voice-leading

etter,

n

the

context,

than does F3

(in

Example

8d).

While some arrows of

Example

8c

are

labeled

by

numbers7 or

greater,

he total of those numbers s

only

22.

The

corresponding

otal of

number

n

Example

8d is 26.

So,

despite

the

immediate

fifth-leap up

to F4

in

the theme

(as

discussed

in

section 3

above),

where

there

is

no such

large leap

to F3

in

Example

8d,

Schoen-

berg's

choice of

F4,

to

represent

he

pitch

class

F

here,

involves a certain

amount

of melodic

husbandry.

That

is

very

much not the

case as

regards

the

high

C

of the theme.

Example

8e

gives

the

pitches

of the

theme

up through

E4,

and then

continues

with C4 rather han

Schoenberg'sC5.

The

example

demon-

strates that

this

layout

of

pitches

in the

melody

would manifest

(in

30






our technical

sense)

maximally

close voice

leading

between

the

pcsets

{D,C#,A,Bb,FE6,EE)

nd

{

C).

That

s,

all the numerical abels for

arrows

on the

example

are less than 7.

Schoenberg's

choice of

C5

here,

rather

than

C4,

is a

very

violent

melodic

gesture

n our

present

context.

In

fact,

one can see thatif arrowsweredrawn rom the firstsevenpitchesof the

example,

to

Schoenberg's

actual continuation

pitch

C5,

the

numerical

label for

every

arrowwould

be 7 or

greater.

4.2.

Example

8f addresses the theme from the slow movement.

The

example

manifests

maximally

close

voice

leading

between the

pcsets

{

C,B,G,Ab,Eb,Db,D,Bb

and

{

Gb

.

The

total of the arrow-numbers

s

30.

Example

8g

shows

(with

its arrow-total f

42)

how the

continuation-pitch

Gb3

represents

he

close

voice-leading

better,

han

does

the

continuation-

pitch Gb4in Example8h (with its arrow-total f 54). Example8g thus

better

represents

melodic

husbandry

n

the

context;

Schoenberg's

actual

pitch

in

Example

8h enacts a certain

disruption

of

such

husbandry.

5.

Maximally

uniform

voice-leading;

applications

to

melodic and harmonic

analysis

Our theoretical

tudy

has so

far

singled

out

three

particular

pecies

of

voice

leading

as

regards

pcsets,

namely

maximally

close,

downshift,

and

upshift

voice

leading.

Weshallnow

study

another

species.

Definition 5.1.

Given

pcsets

X

and

Y,

a

voice-leading

V

from X

into Y

will

be

called

maximally

uniform

f

it

differs as little as

possible

from

a

straight ransposition.

The

definition s not

expressed

rigorously.

Before

shoring

t

up

in

that

respect,

we will

find it

helpful

to

inspect

an

example,

an

example

which

shows the

usefulness of the

concept

for

melodic

analysis.

5.2.

Example

9a articulates everal

trichords

rom

the

right

hand at the

beginning

of

Schoenberg'spiano

piece

op.

11,

no. 1. We

shall

ignore

the

solid

notehead

at

B3,

below

the

open

noteheadsof

trichord

i),

until

fur-

ther

notice.

The

texture

suggests

that we

hear trichord

ii)

as

somehow

analogous

to

trichord

i).

Specifically,

trichord

ii)

is

heardas

"almost

a

transposition"

f

trichord

i).

Joseph

Straus,

n

a

recent

article,

referred

to the

phenomenon

as

"near

ransposition."14

Figure

2

fleshes out

the

idea

with

some

numbers.

The

analysis

hears trichord

ii)

as "almost

T(-3)"

of

trichord i), the

pitches

differing

rom

exact

transposition

y

only

one

semitone.

Accord-

ingly,

using

the

idea of

Definition

5.1,

we can thinkof

Figure

2

as

mani-

festing

(in

our

technical

sense)

maximally

uniform

voice

leading

from

the

pcset

{B,G#,G}

into the

pcset

{A,F,E}.

The

relevant

voice-leading

31






(a)

0

(i)

(ii) (iii)

(iv)

(v)

(b)

ps

T3

ps

T_5

T_2 T_5

T7

T_5

(i)

(ii)

(iii)

(c)

•

T2

IT2

I'v

To

ps

T_1

(iv)

(v)

ps

T_5

(d)

Example

9

function

for

the

pcsets

is

V(B)=A,

V(G#)=F,

V(G)=E.

The voice

leading

of

pitch

classes

is

"maximally

uniform"

because the

pitch

classes

involved can

be

projectedby

pitches-in-register

o

as to

manifestnumer-

ically

Straus's

"near

ransposition"

s

"nearly"

s

possible, given

the

spe-

cific

pcsets

at issue.

5.3.

The

voice-leading

functionabove is not

the

only

maximally

uniform

voice

leading

from

(the

pcset

of)

trichord

i)

into

(the

pcset

of)

trichord

(ii).

Figure

3 manifests another

maximally

uniformvoice

leading.

This

analysis

hears

trichord

ii)

as "almost

T(-2)"

of trichord

i).

The

difference from exact

transposition

s

again

only

one semitone. So max-

imal

uniformity

is a

property

of the

voice-leading

function

V(B)=A,

V(G#)=F,

V(G)=F,

as well as the

voice-leading

function of

Figure

2

above.'5

The

foreground

texture of

Schoenberg's

piece

does

support

more

overtly

the

hearing

of

Figure

2,

but-as we shall

later see-the

hearing

of

Figure

3,

with its

idea

of

"almost

T(-2),"

is

quite pertinent

o

the

composition

as well.

We

might

note

parenthetically

hat the

voice-leading

for

Figure

3

is

the downshift

voice-leading

for

the

pcsets

involved.'6

Figure

4,

hearing

richord

iii)

as

"almost

T(-5)"

of trichord

ii),

man-

32






ifests a

maximally

uniform

voice

leading

from

(the

pcset

of)

trichord

ii)

to

(the

pcset

of)

trichord

iii).

The

maximally

uniformvoice

leading

dif-

fers from

strict

ransposition,

nce

again, by

only

one semitone.That

dif-

ference of

"only

one semitone"

helps

us

hear a

certain

constancy

in

the

progressionof trichordsso far:(i) progressesto (ii) by near-transposi-

tion,

with an

offset of

"only

one

semitone";

hen

(ii)

progresses

o

(iii)

by

near

transposition,

gain

with

an

offset

of

"only

one

semitone."17

The

analytic

commentary

o far

has

suggested

two

useful

definitions.

Definition

5.4.1. Given a

maximally

uniform

voice

leading

V from

pcset

X

to

pcset

Y,

the

pseudo

transposition

numberN of V

is the

positive

or

negative

number

of

pitch

semitones

by

which Y

is

"almost

TN of

X,"

in

a

pertinent

pitch

manifestation

of

V.

Thus the pseudotranspositionnumbers or the voice leadingsmani-

fested in

Figures

2,

3,

and 4

are the

negative

numbers

-3, -2,

and

-5

respectively.

Definition

5.4.2. Given

a

maximally

uniform

voice

leading

V from

pcset

X

to

pcset

Y,

the

offset

number

of V is

the

positive

(absolute)

real num-

ber of

semitones

(either

up

or

down or

both)

by

which

Y

differs from

TN(X),

in

a

pertinent

pitch

manifestationof V

Thus 1

(absolute)

semitone is the

offset

number

or each of

the

voice

leadings

manifested n

Figures

2, 3,

and

4.18

B4

-

A4

via

2

semitones down

G#4

-

F4

via

3 semitones

down

G4

-

E4

via 3

semitones

down

Figure

2

B4

-

A4

via

2

semitones

down

G4

-

F4

via

2

semitones

down

G#4

-

F4

via

3 semitones

down

Figure

3

A4

-

E4

via 5

semitones

down

F4 -- C4 via 5 semitones down

E4

--

B4

via

6

semitones

down

Figure

4

33






5.5. We

continue

now

with

further

analysis

of

Example

9a. On that

example,

trichord

iv)

has

a

parenthesized

olid note

head

at the

pitch

B4.

The

pitch

does

not

actually

sound

in

the

music,

but a

good

case can

be

made for a notion

that

B3,

sounding

an octave

below,

represents

the

unsoundedB4. Example 9a suggests the rationaleby including in its

depiction

of trichord

i)

the

pitch

B3

with

a solid note head.

During

the

music for

trichord

i),

the

right

handsounds

B3

under

G4;

B3 there

fairly

clearly

recalls to the

ear

the

opening

B4 of

the

trichord,

and

of the

piece.

Thus,

when

B3-under-G4

ounds

again

at the

beginning

of trichord

iv),

one can

imagine

the

parenthesized

B4

as

implied

to some

extent,

putting

trichord

iv)

into the same

pitch-contour

s the otherfour trichords

f

the

example,

and

suggesting

trichord

(iv),

specifically,

as a variantof tri-

chord

(i).

More exactly, trichord(iv), with the high B4, manifestsmaximally

uniform voice

leading

from trichord

(i).

It

is

specifically

a

"pseudo

T(- 1)"

of

trichord

1.

with

offset number

1 (Figure

5a).

It

is

also a

"pseudo

TO" f trichord

i),

again

with offset number

1

(Figure

5b).

Trichord

iv)

with

the low

B3

instead

of the

high

B4 does instance

the

voice

leading

of

Figure

5a

in

its serial

ordering,

but

it

does

not

manifest

that voice

leading

in

the

registral

contourof its

pitches.

Using

the

high

B4

for trichord

iv),

we

can

verify

that

every

one

of tri-

chords

(i)

through

iv)

in

Example

8a is related

o the next trichord

via

(at

least

one)

maximally

uniformvoice

leading

withanoffset of 1 semitone.

That continues

the sort

of

"constancy"

n

the

progression

of

trichords,

that we observed

earlier or

trichords

i)

through

iii).

Example

9b,

using

the

high

B4 for trichord

iv),

gives

a

useful

"pseudo

network"

or

inspect-

ing

some of the salient

pseudo-transpositional

elationships

among

the

five trichords.

On the

example, "psT(-3)"

stands for

"pseudo-transposi-

tion down

3

semitones,"

and so

forth.

"T2"

stands

for

"(ordinary)

rans-

position up

2

semitones";

n

the

present

context we

can

regard

his as

a

species

of

maximally

uniform

voice

leading, namely"pseudo-transposi-

tion

by psT2,

with an

offset of

0

semitones."

One could make

another

(well-formed)

pseudo-network

ike that

of

Example

9b,

displaying

richords

ii)

and

(iv)

as

pseudo-transpositions

f

trichord

i)

by

-2

semitones and

0 semitones

respectively.

However,

since

these

relations

nvolve

major-third

or ic4)

dyads

within the

trichords,

t

might

be more relevant

to

make a

regulartranspositional

network

like

that

of

Example

9c,

charting

the

transpositional

progressions

of

major

third

(ic4)

dyads

among

the various

trichords.

The

example

shows

how

the major hirdsof trichords ii) through v) move steadily through

a cir-

cle-of-fifths

progression,

T(-5),

T7,

T(-5).

The T2 relations

between

dyad

(ii)

and

dyad

(iv)

can thus

be read

as two

steps through

that

circle-of-

fifths;

so

can the

T2

relation

between

dyad

(iii)

and

dyad

(v).

Those

T2

relations

correspond

with the T2 relations

among

trichords

ii)

through

34






(a)

B4

-

B4

via

0 semitones

G#4

--G4

via

1

semitone

down

G4- F#4

via 1 semitone

down

(b)

B4

-

B4 via 0 semitones

G#4

--G4

via

1 semitone

down

G4-G4

via

0

semitones

Figure

5

(v)

on

Example

9b. On

Example

9c,

the TOrelationbetween

dyad

(i)

and

dyad

(iv),

involving

the

{B,G}

dyads

of

the

corresponding

richords,

addresses

quite cogently

the

way

that

B3-under-G4

recurs

n the

music.

As we read

along

Example

9c

from

dyad

(i),

through

dyads

(ii)

and

(iii),

to

dyad(iv),

we see how the TOrelation

participates

n the

algebraic

rela-

tion TO

=

T2(T(-2))

=

(T7

T(-5))(T(-2)).

Example9c, using a solid notehead, ncludes the pitchA3 underthe

dyad

{F4,A4

}.

The music sounds A3

simultaneously

with

F4,

in

a

tex-

ture

analogous

to that of the

preceding

B3-under-G4.This

strengthens

the

T(-2)

relationbetween the

dyads

I{G4,B4}

and

I{F4,A4},

and thence

the

psT(-2)

relationbetween trichords

i)

and

(ii).

To be

sure,

the

foregoing

considerations

oncerning

he

melody

at the

opening

of

op.

11

are far

from

adequate

o constitute"an

analysis"

of

the

passage.

Among

other

hings, they

do not

carry

he

analysis

of

the

melody

through

he entire

phrase,

nor do

they

considerevents

in

the music below

theregisterof Example9a, nordo theytakeintoaccountvarious nverted

forms of trichords

(i),

(ii),

and

(iii)

that

appear

within the

tessitura

of

Example

9a

during

he

passage,

suturing

he

melodic

continuity.

Still,

the

discussion does indicate

how

maximally

uniformvoice

leading

can be a

useful

idea

to

bring

into

an

analytic

context.

To

suggest

how the

present

ine

of

analysis

might

continue,

Example

9d

sketches other

prominent major-third

dyads

in

the

music,

beyond

those of

Example

9c.

Especially striking

s the return f

{

G,B

I

and

{

F,A

at

the

right

of the

example,

now

superimposed

n

a

lower

register

within

a cadentialharmony.The music embeds the major hirdsportrayedhere

within various

ransposed

or

inverted

orms

of trichords

i), (ii),

and

(iii).

6.

Review, cautions,

and

formalities

6.1. We

have now

studied several

special

species

of

voice

leading

from

one

pcset

to another.One

species

is

maximally

close

voice

leading

(Def-

initions 1.4

and

1.5,

sections

3

and

4,

Examples

1, 5b, 6, 7,

and

8).

Another is

downshift

voice

leading (Definition1.6, Examples2ad, 5b).

Another s

upshift

voice

leading

(Definition

1.7;

Examples

2bc, 3ab,

4a).

And

another s

maximallyuniform

voice

leading

(section

5,

Example

9,

Figures

2,3,4,5).

When we

bring

our intuitions o bear

upon

these

conceptions,

we

must

35






exercise caution: he

formalisms

do

not

always

behave

n

ways

we

might

casually suppose they

do.

6.2.

Caution:

We have

already

observed

(in

Note

1.3)

thata musical

pas-

sage (e.g. Examplela) may instance-even manifest-a certainformal

voice

leading,

while the

retrograde

f that

passage

does not instance

any

"voice

leading"

at

all,

in

the

formal

sense

of Definition 1.1.

6.3.

Caution:

In

particular,

he

retrograde

of the

upshift

voice

leading

from

X to Y

need

not

be,

in

general,

the downshiftvoice

leading

from

Y

to X. For

instance,

Figure

6a

displays

the

upshift

voice

leading

from

pcset

{C,E}

into

pcset

{F,G},

while

Figure

6b

displays

the

downshift

voice

leading

from

{F,G}

into

{C,E}.

The formershifts notes

by

a total

of 6 semitones(up);the latterby a total of 4 semitones(down).The ret-

rogrades

of

Figures

6a and 6b are not voice

leadings

in

our

formal sense.

If

we are

tempted

o intuit

(erroneously)

hatthe

retrograde

f

upshift

voice

leading

from

X into Y must be downshift

voice

leading

from Y

to

X,

that

is

partly

because

of a certain

symmetry

n

the

words and letters

that

express

the

thought,

and

partly

because

we are

intuiting

he situation

symbolized

by Figure

7.

The

figure

displayspcsets

X

={xl,

x2,

....

xN}

andY

=

{yl, y2

....

yN}

that have

the

same

cardinality.

Furthermore

he

pitches

instancing

the various

pcs

of X andY can be laid out in

registral

orderhere so that

xl

is

below

(or

the same

as)

y

1,

which is

(strictly)

below

x2,

which is

below

(or

the same

as)

y2,

which is

(strictly)

below

x3,

and

so forthuntil

we reach

y(N-1),

which is

(strictly)

below

xN,

which is

below

(or

the

same

as)

yN.

In

short,

we

could

say

that normal

orderings

of

the two

pcsets

"dovetail"

though

hat

s

not

quite

precise enough

to allow

for the

possibility

that

any

xn

might

be the same

as

yn).

In

these

special

circum-

stances

the

retrograde

f

Figure

7

will indeed be

a formal

voice

leading,

and

t

will

indeed

nstance he

downshiftvoice

leading

from

Y into X. But

the

special

circumstances

are

very

restrictive.

6.4.

Caution:

In

general,

it is more

likely

that the

retrograde

f a

maxi-

mally

uniform

voice

leading

will be a

maximally

uniform

voice

leading,

when

the

two

pcsets

involved

have

the same

cardinality.

But

this

phe-

nomenon,

while

more

likely,

is

by

no means

necessarily

the case.

Figure

8,

which

reproduces

Figure

5,

illustrates

he

point.

Figure

8a

illustrates

maximally

uniform

voice

leading

(via

pseudo-

T(-1)) from thepcsetX = {B,G#,G into thepcsetY = {B,G,F# . Theret-

rograde

of

Figure

8a

does indeed

illustrate

maximally

uniform

voice

leading

(via

pseudo-TI)

fromY

into X.

However,

Figure

8b

is a different

story.

It illustrates

another

maximally

uniform

voice

leading

from

X to

Y

36






(a)

C

--

F via 5 semitones

up

E-

F via 1 semitone

up

(b)

F

-

E via 1 semitone

down

G-

E

via 3 semitones

down

Figure

6

yN

xN

y(N-1)

x(N-1)

y2

x2

yl

xl

Figure

7

(a)

B4

--

B4 via 0

semitones

G#4 -G4 via 1 semitonedown

G4

--

F#4

via

1

semitone

down

(b)

B4

--B4

via 0 semitones

G#4-G4 via 1 semitonedown

G4

--G4

via 0 semitones

Figure

8

(now

via

pseudo-TO)-and

the

retrograde

f

Figure

8b,

which does not

engage

all of

Y,

is not a formalvoice

leading

at all.

6.5. Caution:It is of course

true

that

the total numberof semitones tra-

versed,

in one

maximally

close voice

leading

from X into

Y,

must be the

same as

the

total number

of

semitones

traversed,

n

any

other

maximally

close voice

leading

from

X into

Y.

That is

inherent n the definition of

"maximally

close."

But the total number of semitones

traversed,

n

a

maximally

close voice

leading

from

X

into

Y,

need not

necessarily

be

the

same

as the

total

numberof semitones

traversed,

n

a

maximally

close

voice

leading

from

Y into X.

Figure

9 illustrates

he

point. Figure

9a

gives

two

maximally

close

voice

leadings

from

{F,A,C

}

into

I{F#,G#,A#},

ach

via a total of

4

semitones.Figure9b gives

a

maximally

close voice

lead-

ing

from

F#,G#,A#

into

I{F,A,C

,

via a

total of

3

semitones.

The

pcsets

{

F,A,C}

and

{F#,G#,A#

here

have no

common

tones,

and

they

are

of

the

same

cardinality.They

do

not, however,

"dovetail" n the sense

discussed

37






(a)

F-F#

via

1

semitone

A-

G#

or

A#

via

1

semitone

C

-

A#

via

2

semitones

a total shift of 4 semitones

(b)

F#-

F

via 1

semitone

G#

-

A

via

1

semitone

A# --

A via

1

semitone

a total shift of 3 semitones

Figure

9

above

-both

F#

and

G#

come between

F

and

A

in

a close

registral

order-

ing

of

representative

itches.

7.

Maximally

close voice

leading

of set

classes;

a scordatura

fantasy

7.1.1.

Suppose you

want

to

retune

your

violin so

that

all its

open

strings

soundnotes of

the

F

majorharmony.Maximally

close

voice

leading

ndi-

cates

ways

of

doing

so with

(in

some

sense)

as little overall

strainas

pos-

sible.

Figure

10 indicates he

two

possible maximally

close voice

leadings.

Each

maximally

close

voice

leading

involves

a

total shift

of

5

semi-

tones.

G

-

F

or

A

via

2 semitones

D- C via 2 semitones

A

-

A

via 0 semitones

E--

F

via

1 semitone

Figure

10

7.1.2.

Now

suppose

you

wish to retune

your

viola so that

all

its

open

strings

sound

notes

of the

pcset

{F,A,C

}.

Figure

11

gives

the

maximally

close voice

leadings.

Each

maximally

close

voice

leading

involves a total shift

of

4

semi-

tones. As

one

sees,

we

do

"better"

etuning

he viola

here,

than

we

did

retuning

he

violin,

which necessitated

a

total

shift

of 5

semitones.

C

-

C via

0 semitones

G

--

F or

A

via

2 semitones

D -- C via

2

semitones

A --

A

via 0 semitones

Figure

11

7.1.3.

Now

suppose

you

wish

to

retune

your

viola-in-Bb

so that all

its

open

strings

sound

notes

of the

pcset

{

F,A,C

,

and

similarly

for

your

38






viola-in-A,

and

similarly

for

your

viola-in-Eb.

Figures

12(a), (b),

and

(c)

give

the

maximally

close

voice

leadings.

Each

maximally

close

voice

leading

involves

a

total shift of 3

semi-

tones. As one

sees,

we

do

even "better"

etuning

he

viola-in-Bb,

or

the

viola-in-A, or the viola-in-Eb,than we did retuningthe regularviola,

which

necessitated

a

total shift

of

4

semitones.

(a)

B--A

via

1

st

F-F

via 0 sts

C

--C

via

0

sts

G

- F

or

A

via

2

sts

(b)

A-A

via 0 sts

E -F

via 1 st

B

-C

via 1 st

F#

-F

via

1

st

(c)

Eb--F

via 2 sts

B

--A

via 1 st

F -F via

0

sts

C -C via

0

sts

Figure

12

7.1.4. Can

we

do

yet

better,

using

some other instrument n

our

fantasy

family

of

transposing stringed

instruments

n

all

possible keys?

The

answer

is no:

retuningany

other such instrument

as

stipulated

will

re-

quire

a total

shift

of more

than 3

semitones.The

following paragraphs

f

this

section

prove

the

fact;

the

uninterested

eader

may

take

my

assur-

ance,

and

go

on to

section 7.2.

(A)

Suppose

our

fantasy

nstrument as no

open

strings

hat

belong

to

the

pcset

{

F,A,C

}.

Then each

string,

n

retuning,

must shift

by

at

least

I

semitone,and the total shift will be at least4 semitones.

(B)

Suppose

our

fantasy

instrumenthas

exactly

one

open

string

that

belongs

to the

pcset

{

F,A,C

}.

Inspecting

he

pertinent

egment

of the

cir-

cle-of-fifths,

that s

Ab-Eb-Bb-F-C-G-D-A-E-B-F#,

e

see thatour instru-

ment

must have

open

strings

eitheron

Ab-Eb-Bb-F,

r on

G-D-A-E,

or

on

D-A-E-B,

or

on

A-E-B-F#.

The

open strings

of

any

other

nstrumentwill

have either no common

tones

with

{

F,A,C

},

or

two common

tones with

F,A,C

.)

We have

already

nvestigated

he

cases G-D-A-E

(violin)

and

A-E-B-F#

(viola-in-A);

the violin

needed

more than3

semitones'shift

to

retune,

andthe viola-in-Awas one of the three"best"

nstruments,

eed-

ing only

3

semitones' shift. We

see that the

viola-in-Ab,

uned to

Ab-Eb-

Bb-F,

requires

4

semitones for a

maximally

close shift into

I{F,A,C ,

as it

retunes o A-F-A-F.

And

the

viola-in-D,

tunedto

D-A-E-B,

also

requires

4

semitones for

a

maximally

close shift

into

{

F,A,C

},

as it

retunes o

C-

A-F-C. In

sum,

every

instrument n

our

family explored

so

far,

except

for

viola-in-Bb

or

viola-in-A,or

viola-in-Eb,

requires

4

or more

semitones'

shift to

retuneas

required.

(C) Suppose

our

fantasy

nstrumenthas

two

open strings

hat

belongs

to the

pcset

{

F,A,C},

andthat

those

open

strings

sound

A

and C.

Then the

instrument s

a viola

(or

cello,

etc.),

and we

know

that

t

requires

4

semi-

tones' shift to

do its

job.

39






The instrument annot

have

open strings

on both F

and

A. So we

have

now

only

to

investigate

nstruments

with

open strings

hat soundF and C.

One of those is

the

viola-in-Bb,

with

open strings sounding

Bb-F-C-G,

already

seen to be one of our three

"best" nstruments.

Another is the

viola-in-Eb,with open stringssoundingEb-Bb-F-C, lreadyseen to be

another

of

our threebest.

The

only

remaining

nstrument o

investigate

s

the

violin-in-F,

with

open strings sounding

F-C-G-D. We

see

that this

instrument

equires

4

semitones to retuneto

F-C-F-C

or

F-C-A-C,

so it

is not one

of the "best" nstruments.We

have now

completed

our verifi-

cation that

the violas

in

A,

Bb,

and

Eb

are the "best"

nstruments,

nd

the

only

"best"

nstruments.

7.2. Each of our

stringed

instrumentshas

its

open strings

on some

fourth-chordof pitch classes (pcset of Forte set-class 4-23), and each

possible

such fourth-chord

pcset

in

that

set class

4-23)

corresponds

o

one of our

fantasy

nstruments.

Accordingly

we

can

say

that

I

Bb,F,C,G

,

{A,E,B,F#

,

and

{Eb,Bb,F,C}

re the best members

of

set class

4-23,

as

regards

maximally

close voice

leading

into the

pcset

[FA,

C).

And

we can

single

out the

pertinent

maximally

close

voice

leadings

as the best max-

imally

close voice

leadings

from

set-class 4-23 into the

pcset

{FA,

C).

And we can

speak

of

3

semitones as the

tightest

total

shift

possible, from

set-class

4-23 into the

pcset

{FA,

C).

7.3. For

the

remainder

f section 7 we shall write

"vaBb"

or

{Bb,FC,G},

the

particular

cset representedby

the

open

strings

of the "viola in

Bb,"

and

so forth

for our

other

"violas";

we

shall

write "FM"for

the

pcset

{F,A,C}.

Let Y be some

major-triad

et.

Suppose

we want to

know which are

the best members

of

set-class

4-23,

as

regards

maximally

close voice

leading

into the

pcset

Y. We can write

Y=T(FM)

for some

transposition-

operation

T;

then

the desired

sets

are

T(vaBb),T(vaEb),

and

T(vaA).

In

like

fashion,

ifY is some minor-triad

et,

write

Y=I(FM)

for some inver-

sion-operation

;

then

I(vaBb),I(vaEb),

and

I(vaA)

are

the

best members

of

set-class

4-23,

as

regards

maximally

close

voice

leading

into

this new

Y.19

The total

shift,

for

any

one

of the voice

leadings

involved

above,

will

be

by

3

semitones.

In

that

sense,

we can

say

that

3

semitones

s the

tight-

est

total

shift

possible, from

set class 4-23 into set-class

3-10

(the

set

class of

{

F,A,C

}).

As

regards

our scordatura

antasy,

we can

put

the

formal

idea

above

into more

intuitive

anguage.

The

best

we

can do as

regards

minimal otal

shift

of

voices,

when we

try

to

lead the voices

of

any

fourth-chord

f

pcs

into some

harmonic triad

of

pcs,

is to lead

via some

transposed

or

inverted orm

of

Figures

12(a), (b),

or

(c)-and

that best

will involve a

total

shift

of 3 semitones

in the voice

leading.

40






(a)

C

--

C via 0

sts

F

-

F

via

0

sts

A

-

Bb

via 1 st

(b)

C

-

B via

1

st

F

-

E

or

F#

via

1 st

A

-

A via

0 sts

Figure

13

7.4.

The ideas

of sections

7.2 and 7.3

can be

generalized,

o discuss

the

best members

of

set class

/X/,

as

regards

maximally

close voice

leading

into the

pcset

Y,

the best

maximally

close

voice

leadingsfrom

set-class

/X/

into the

pcset

Y,

the

tightest

total

shiftpossible,

from

set-class

/X/

into

the

pcset

Y,

and

the

tightest

total

shift

possible,from

set class

iX/

into set-

class

/Y.

7.5. We must

take to

heart the cautions

of section

6.5 in

all

this,

not

expecting

things

to

behave

more

symmetrically

han

hey actually

do. For

instance,

while

vaBb,vaEb,

and vaA are

the best

membersof

set

class

4-

23 for

maximally

close

voice

leading

into

{F,A,C

,

with a total voice-

leading

shift of 3

semitones

n

each

case,

it is not true

thatthe threeviola-

sets are also

the best members of set

class

4-23

as

regardsmaximally

close voice

leading

rom

{F,A,C

},

nor is

it true that the

maximally

close

voice

leadings

in that

regard

are

by

3

semitones'

shift.

Figure

13ashows that

{

F,A,C}, when led maximallyclosely into the

open strings

of

vaBb,

or

of

vaEb,

nvolves a total shift

of

only

1

semitone,

not

3.

vaBb

and

vaEb-involving

only

1

semitone

shift in each case-are

therefore

among

the best membersof

set

class

4-23

for this

purpose.

Fig-

ure

3b

shows that

maximally

close voice

leading

of

{F,A,C}into

vaA

requires

2

semitones' total

shift;

accordingly

vaA

is

not one of the best

membersof set class

4-23,

into which to lead

{F,A,C

}

maximallyclosely.

7.6. The idea

of

looking

for the

"tightest"maximally

close

voice lead-

ings,

from

[membersof]

a set class into a

pcset

or into anotherset

class,

is

built into the idea

of "maximalcloseness"

in

a natural

way.

Formally,

we

can look in a similarfashion

for

the

"tightest"upshift

voice

leadings,

from

[membersof]

a set class into a

pcset

or into anotherset class. But

the

idea

is

not

so

obviously justified by

the nature

of

"upshifting."

The

same

goes

for

downshifting.

As for

maximally

uniform voice

leading,

there

is

not much formal content to the idea

in

that

context.

If

pcset

X

maps

into

pcset

Y via

pseudo-transposition sTn

with an

offset of

s

semi-

tones, thenTi(X) will mapintoY via pseudo-transposition sT(n+i)with

an offset of s semitones. The

amount

of offset

being

the

same in

both

cases,

we

cannot

say

that

one offset is

"tighter"

han the

other,

or that

Ti(X)

is "better" hanX

as

regards

maximally

uniformvoice

leading

into

Y.

All

transposed

orms of X

are

equally

"good."

41






8.

Mapping

scales into other

scales;

intonation

and

temperament

8.1.1.

We

can

regard

any pcset

as a formal "scale."

In

that

context,

we

can regardour formal voice leadingsas ways of adjusting he tuningof

certain

scales to fit into or

adjust

to

the

intonationor

temperament

of

other

scales.20

The

idea

of

"retuning"

as indeed

already

been introduced

in

the context of our

scordatura

antasy.

8.1.2.

In

this

connection,

we

can

reconceptualize

our

pitch

classes,

as

taking

values

along

a

(potential)

continuum

f

values

modulo the octave.

There is no abstract

or

formalreason

to

confine the

pitch

classes to

only

twelve

possible

values within an

equally-subdivided

ctave.

8.1.3.

We

shall, however,

continue to consider all

"pcsets"

o have

only

a

finite

numberof members.It would be

possible

to

generalize

the math-

ematics

so that

many

of our

results

could

apply

as well to

certain

nfinite

sets

of

pitch

classes.21We shall have

plenty

to

talk about

without

explor-

ing

that

generalization, hough,

and

there s no

reason

to

intimidate

ead-

ers

without

pertinent

mathematical

ackground.

8.2.1.

Let

EQDOD

be a

pcset

comprising

a

12-note

equal

subdivision

of

the

octave;

let

EQHEP

be a

pcset

comprising

a 7-note

equal

subdivision

of the

octave,

one of whose

notes coincides with some note of

EQDOD.

Clough

and Douthett

pointed

out that downshift

voice

leading maps

EQHEP

onto a 7-35

subset of

EQDOD.22

Upshift

voice

leading

would

work

in

the same

way.

We

can

explore

this

phenomenon,

and other

re-

lated

phenomena,

rom

a

somewhat

different

vantage

point.

8.2.2.

To

do

so,

let us subdivide

he octave

into 84

(

=

7

times

12)

equal

intervals, abeling

the

points

of

subdivision

with numbers

0

through

83

inclusive,

point

0

being

the common tone of

the two sets.

The

tones

of

EQDOD

will then be labeled as

0, 7, 14, 21,

28, 35, 42, 49, 56, 63, 70,

and

77,

while

the

tones

of

EQHEP

will

be labeled

as

0,

12, 24, 36, 48, 60,

and 72.

The

top

row of

Figure

14a,

labeled

"HEP,"

ives

the tones of

EQHEP

as

just

labeled.

The

second row

of the

figure,

abeled

"UP,"

ligns

beneath

those

numbers

he

membersof

EQDOD

into which the

heptatonic

ones

are

upshifted.

Thus

the

tone 0 of

EQHEP

shifts

up

to the

tone 0

of

EQDOD,the tone 12 of EQHEPshiftsupto the next available one 14 of

EQDOD,

the tone

24 of

EQHEP

shifts

up

to the next

available one 28

of

EQDOD,

and so

forth.

The

process

translates

nto our

setting

the

alge-

braic observations

of

Clough

and

Douthett.Via the

upshift,

the notes

of

42






(a) (b) (c)

HEP 0 12

24

36 48 60 72

1

13

25

37 49 6173

2

14 26 38 50 62 74

UP 0

14

28

42

49

63

77

7

142842496377

7

142842566377

DIST

0

2

4

6

1

3 5 6

1

3

5

0 2 4 5 0 2 4 6 1

3

TOTD 21/84 = 1/4octave

Figure

14

EQHEP

are

mapped

onto the notes of the 7-35 set

{0,14,28,42,49,63,77

}

within

EQDOD.

The third

row

of

Figure

14a,

labeled

"DIST,"

ives

the distancesof the

heptatonic

notes from their

upshifted

mages

in the

7-35 set. 0

is

0 minus

0;

2

is

14

minus

12;

4

is

28

minus

24,

and so forth. Note 0 of

EQHEP

need not be changed at all, to get to note 0 of EQDOD; but note 12

of

EQHEP

must be "bent"

by

2/84 of an

octave,

to

get

to

note 14

of

EQDOD;

and

note 24

of

EQHEP

must be "bent"

by

4/84 of

an

octave,

to

get

to

note 28 of

EQDOD;

and so

forth. The total amount

of

"retuning"

that takes

place,

to

map

EQHEP

into

EQDOD

via the

upshift,

is

thus

(0+2+4+6+1+3+5)

eighty-fourths

of an

octave,

which

is

21/84 or

one-

quarter

of an octave.

That

is

quite

a substantial

amountof

retuning,

and we

might

say

that

the

7-35 does not

represent

EQHEP

very closely.

This is

irrelevant o

the purposesof Clough and Douthett,but the observationhas a certain

cogency

on its own.

Indeed,

f

our

purposes

were those of

practical

eth-

nologists

trying

o

make faithful

ranscriptions

f Martian

EQHEP

music

into a

Tellurian

EQDOD

system,

the observation

would be

important

nd

telling.

We

might

try

to

do

"better,"

n

the sense of

section

7

above,

by

transposing

EQHEP

about so as to

get

a closer

upshift

fitting

into

EQDOD.

But we

would not succeed.

Figure

14b

shows what

happens,

for

example,

when we

transpose

EQHEP

by

1

eighty-fourth

of

an

octave.

The

top

row of

the

figure gives

the

notes of the

transposed

EQHEP,

{1,13,25,37,49,61,73} instead of {0,12,24,36,48,60,72}. The second

row of

Figure

14b

shows

how

the

transposed

EQHEP

now

shifts

up

into

EQDOD:

note

1

of

"EQHEP+

"

shifts

up

to

note 7

of

EQDOD;

the

other

notes of

EQHEP+1

shift

up

to

notes

14

etc. of

EQDOD

as

before.Where

EQHEP

itself,

in

Figure

14a,

shifted

up

to

the

"G

major

7-35"

of

EQDOD,

EQHEP+1,

n

Figure

14b,

shifts

up

to the "D

major

7-35"

of

EQDOD.

Most

significantly

for our

purposes

here,

the

total

amount

of

retuning

nvolved

in

the

upshift

remainsthe

same for

Figure

14b,

as

for

Figure

14a,

namely

6+1+3+5+0+2+4

eighty-fourths,

or

one-quarter,

f

an octave.

Figure

14c

shows what

happens

when we

transpose

EQHEP

by

yet

another

ighty-fourth

f an octave:

he

7-35

image

is

now an

"A

major

43






7-35";

the

total amount of

retuning

involved

in

the

upshift

remains

the

same for

Figure

14c,

as for

Figures

14b

and

14a,

still and

again

5+0+2+4+6+1+3

eighty-fourths,

or

one-quarter,

f

an octave. Should

we

continue

ransposing

EQHEP

up by eighty-fourths

f an

octave,

the same

patternwould continue;we would never"improve"he total amountof

distortion nvolved.23

8.2.3.

We

can,

however,

do

considerably

better

n

this

respect by using

maximally

close voice

leading

of

EQHEP

nto

EQDOD,

rather

han

up-

shift voice

leading.

Figure

15

demonstrates.

In

Figure

15a,

instead

of

tuning

all

notes

of

EQHEP

"sharp,"

we tune

some

sharp

and some

flat,

retuning

n

each case to

get

to

the nearestavail-

able note of

EQDOD.24

The row

of

Figure

15a

labeled

"MCL"

abulates

the "maximallyclose" notes of EQDODto the notes of EQHEPthatlie

visually

above them on

the

figure.

As

before,

the

row

labeled

"DIST" ab-

ulates

the

distances,

between

notes

of

EQHEP

and the

closely-retuned

notes of

EQDOD.

The total amount

of

retuning

s now

only

1/7 of an

octave,

ratherthen 1/4 of an

octave,

so the

"distortion"

f

EQHEP

is

much less.

(The

total

amount

of

retuning

here,

1/7

of an

octave,

is

only

one

scale-step

of

EQHEP.)

Figures

15b and 15c show what

happens

when

we

begin transposing

EQHEP

by

incrementsof

1/84 octave.

As

with

the

upshift

voice

leadings

of

Figure

14,

the total amount

of

retuning

here,for themaximallyclose voice leadingsof Figure15, is not affected

by

such

transpositions.

We

cannot

single

out

any transposition

f

EQHEP

as "best"

or

"maximally

good")

in

this

respect.

Still,

Figure

15a

has a

decided

conceptual

advantage

over

Figure

14a,

in

a

certain

respect.

Where

Figure

14a shifted

the notes of

EQHEP

nto

the

notes

of

"the

G

major

7-35" within

EQDOD,

Figure

15a-with

much

less

retuning-shifts

the notes

of

EQHEP

nto the

notes

of

"theC

major

7-35."

So,

if we attribute ome kind

of

conceptualpriority

n this context

to the

unique

common tone of

EQHEP

and

EQDOD,

and

if we

attribute

some sort of functional tonal

meaning

to the 7-35

sets,

Figure

15a

re-

spects

these

priorities

and

meanings

where

Figure

14a

does not.25

8.3.1.

In the

work

just completed,

we saw that there were no

particular

"best"

ranspositions

of

EQHEP,

or the

mappings

underconsideration.

Thatwas due to a

numberof

factors,

ncluding

he

symmetries

of

EQHEP

and

EQDOD.

When we come to examine less

symmetrical

tructures,

we

shall see

that the land lies

quite differently.

8.3.2.

Let

us,

for

instance,

explore

formal

voice-leadings

of a

just

major

scale into

a 12-note

equal-tempered

cset.

For

present

purposes,

we shall

represent

all

notes at issue

by

numbers rom 0

through

1199,

giving

or

approximating

hem

by

whole numbersof cents

modulo

the 1200-cent

44






(a) (b)

(c)

HEP 0 122436486072

1

132537496173

2

142638506274

MCL 0

14

2135 49 63 70 0

14

28 35 49 63 70

0

14 28 35 49

63 77

DIST 0

2

3

1 1

3

2

1 1 32 0 2 3 2 0 2 3 1 1 3

TOTD 12/84 = 1/7octave

Figure

15

octave.

A

just

scale is then

given (approximated)

y

the

pcset

JUST

=

{0,

203, 386, 498, 702,

884,

1088},

where the note labeled 0 is the tonic

of

the

just majorsystem.

We shall

explore

ways

of

retuning shifting)

these

notes,

and the notes of

variously transposed ust

scales,

into notes of

EQDOD,

as

given

by

the numerical et

{0,

100, 200,

300,

400, 500, 600,

700, 800 900, 1000, 1100}.

The

top

four rows

of

Figure

16a are

analogous

to the

top

four rows of

Figure

14a earlier.The first row of

Figure

16a shows the notes

of JUST.

The second row

shows the notes of

EQDOD

into which

the JUST notes

upshift.

The

third

row

tallies the

distance,

n

cents,

from

each JUST note

to its

upshift

target

within

EQDOD.

The fourthrow

gives

the total

dis-

tance n

cents,

by

which the JUST

scale

gets

retunedwhen

upshifting

nto

EQDOD.

The fifth row of

Figure

16a-a new

featureof this

figure--describes

the "image set" of the upshift, that is the particularpcset of EQDOD

whose notes

appear

n

the second row of the

figure.

One

observes-per-

haps

with some

surprise-that

the

image

set is

not a 7-35 set. The

amount

of

retuning

here is so

substantial,

hat the

image

set is not

in

the Forte-

form that

conventionally represents

a

diatonic scale within

EQDOD.

(One

sees,

for

instance,

that the

image

set

here contains

three

adjacent

semitones of

EQDOD.)

That is

perhaps

not

so

surprising

when

one ob-

serves how

great

s the

total

distortion

of

the

upshift

here,

namely

239?.

In

Figure

16a

the

UT

note of the

just

scale

coincides

exactly

with a

note of

EQDOD.

Figure

16a+shows us what

happens

when we

transpose

the

just

scale

just

one

cent

up,

holding

the notes

of

EQDOD

fixed as

0,

100,

200,

etc.

(The

plus

sign

in

"a+"

represents

he

idea of

"just

a little

bit

higher.")

While the

upshift

targets

of the

transposed-just

RE,

MI,

FA,

SOL,

LA,

and

SI

remain

he same

(as

300,

400, 500, 800,

800,

and 1100

respectively),

the

upshift

target

of the

newly transposed

UT has

jumped

to

100,

where it

was 0

in

Figure

16a. The DIST

numbers or

the trans-

posed

RE

through

SI

all decrement

by

one

cent,

since

these notes

each

come

a cent

closer to

their

targets

n

EQDOD.

The DIST

number

or the

transposed

UT,

however,

umps

abruptly

rom 0

to 99

cents,

since the tar-

get

for the

transposed

UT

leaps

aheada

semitone

n

EQDOD.

The

TOTD

numberfor

Figure

16a+ therefore

rises a total of

930,

from

the TOTD

number

of

Figure

16a.

(93

=

-1-1-1-1-1-1+99.)

45






(a) (a+)

JUST

0 203 386 498 702 884 1088

1

204 387 499 703

885 1089

UP 0 300 400

500 800 900

1100

100 300 400 500 800 900 1100

DIST

0 97

14

2

98

16

12

99

96 13

1

97 15

11

TOTD 2390 3320

IMAGE SET:

a 7-22 a 7-30

(b) (b+)

JUST

2

205 388 500 704 886 1090 3 206 389 501705

887 1091

UP 100 300 400 500 800

900

1100

100 300 400 600 800 900

1100

DIST

98 95

12

0 96

14

10 97 94

11

99 95

13 9

TOTD

3250 4180

IMAGE

SET:

the 7-30

of

(a+)

a

7-35

("E

major")

(c)

(c+)

JUST

12 215 398 510714

8961100 13

216

399

511715 899

1101

UP 100 300

400 600 800 900 1100

100 300 400

600 800 900

0

DIST

88 85

2 90 86

4

0 87 84

1

89

85 3 99

TOTD

3550

4480

IMAGE SET:

the

"E

major"

7-35

of

(b+)

a 7-32

(d)

(d+)

JUST 14217400512716 898 1102 15 218401 513717 899 1103

UP 100 300

400 600

800

900

0

100

300 500

600 800 900

0

DIST

86 83 0

88 84

2

98

85 82 99

87 83

1

97

TOTD

441?

534?

IMAGE SET:

the 7-32

of

(c+)

a

7-32

(different

from

(d))

(e)

(e+)

JUST

16 219

402 514

718 900

1104 17 220403 515 719

901

1105

UP

100 300

500

600 800

900

0 100 300 500 600 8001000

0

DIST

84

81 98

86

82

0 96 83 80 97 85 81 99 95

TOTD

5270

6200

IMAGE SET:

the 7-32

of

(d+)

a 7-35

("D6

major")

(f)

(f+

=

g)

JUST

97300483

595799

981

1185 98 301484596800

982

1186

UP

100

300 500

600 800 1000

0 100

400 500 600

800 1000

0

DIST

3

0 17

5

1

19

15

2

99

16

4 0

18

14

TOTD

600 1530

IMAGE

SET:

the

D6

major

7-35

of

(e+)

a 7-22

(continued)

Figure

16






(g+)

T100(a)

JUST

99 302

485 597 801

983 1187

100

303 486 598

802 984

1188

UP

100

400 500 600 900 1000

0 100 400

500 600 900 1000

0

DIST 1

98 15 3

99 17 13

0 97 14 2 98 16 12

TOTD 2460 2390

IMAGESET:

another -22

(T100

of the

(a)

7-22)

the sameas

(g+)

T(-100)(f)

JUST -3

200 383 495

699 881 1085

UP

0 200 400

500 700

900 1100

DIST 3

0 17 5

1

19 15

TOTD

600

IMAGESET: a

C

major

7-35

Figure

16

(continued)

In

Figure

16a+ we see

that the

image

set of

the

upshift

has

changed

not

only

its content

butalso its

set class.

It is no

longer

a 7-22

set,

but

now

a 7-30

set. It is

still

and

again

not

some 7-35.

Figure

16b

transposes

the

just

scale

another

cent. At

this

point

the

transposed

FA

of the

just

scale

coincides

exactly

with a

member

of

EQDOD.

The

target

membersof

EQDOD

have not

changed,

from

Fig-

ure 16a+to Figure 16b,so each noteof the transposedust scale gains a

cent on

its

EQDOD

target,

n

passing

from

the DIST

numbersof

Figure

16a+ to

the DIST

numbers

of

Figure

16b.

The total

distance

thereby

decreases

1

for

each of the

seven

notes,

or a

total of

70,

as

we

pass

from

the

TOTD

of

Figure

16a+,

to the

TOTD

of

Figure

16b.

Figure

16b+

transposes

he

just

scale of

Figure

16b

"just

a

little

more."

Notes

UT, RE,

MI, SOL,

LA,

and SI each

come

one cent

closer

to their

upshifttargets

n

the

process;

he

target

or

note FA

now

leaps

ahead

99?.

So

TOTD

increases

abruptly

by

930

(93

=

-1-1-1-1-1-1+99),

as we

pass

from

Figure

16bto

Figure

16b+.

As

the

target

for

just-FA

leaps

a

semitone

ahead

in

Figure

16b+,

the

image

set

changes.

Finally,

it is

some form

of

7-35. But

it is

a

7-35 "in

the

wrong

key "

The

transposed

JUST

set here

is

only

30

sharp

from a

just

C

major

scale,

but the

target

set

within

EQDOD

gives

a

7-35

that

rep-

resentsan E

major

scale-set

within

EQDOD

(where

the

number0

always

represents

"C").

The

FA

of the

30-transposed

JUST

C-scale

upshifts

to

the RE

of

the E

major

7-35;

the

UT of

the

30-transposed

JUST

C-scale

upshifts

to the LA

of

the

E

major7-35,

and so

forth.

This

feature

llustrates

very

effectively

a

certain

tension between

the

principles

of

upshifting,

and

those of

modal

tonicity,

in

the

matters

at

hand. The

bizarre

mage

sets of

Figures

16a

through

16b

have

already

done

that in

a

different

way.

47






As we continue

transposing

he JUST scale

through

9 more

cents,

no

note of the

transposed

cales ever

attains

a

note

of

the

fixed

EQDOD.

The

target

notes

in

EQDOD,

for UT

through

SI

of the

transposing ust

scales,

remain the same. And with each

cent

of

transposition,

ach

just

scale-

degree comes one cent closer to its upshiftEQDOD target.Witheach

cent of

transposition,

hen,

TOTD

decreases

by

7?.

So

by

the

time we

have moved

from

Figure

16b+ to

Figure

16c,

9

transpositions

ater,

TOTD has decreased

9-times-70,

or

63?,

from the

4180

of

Figure

16b+

to the

3550

of

Figure

16c. The

image

set

of

target

notes remains

he

same

throughout

his

span

of

activity,

hat is

the "E

major"

7-35.

The rest of

Figure

16,

up

to the

stage

marked

"T(100)(a),"

proceeds

n

similar

fashion.

At the

transpositional

evels of

Figures

16d, e, f,

and

g,

some

note of

the

transposed ust

scale

coincides

exactly

with some note

of EQDOD.At thetranspositionalevels of Figures16d+, e+, f+, andg+,

which

happen

mmediately

after

d, e, f,

and

g

respectively,

he

pertinent

target

note

of the

upshift

leaps

ahead

to

the

the

next

note of

EQDOD,

making

TOTD

ump up abruptlyby

93?.

During

he

transpositional

pans

between

c+ and

d,

between

d+

and

e,

and between e+

and

f,

each note of

the

just

scale moves

a cent closer to its

upshift target

in

EQDOD

with

each

cent of

transposition,

o thatTOTD

drops

7?

in value for each cent

in

transposition.

Most

notably, during

the

80-cent

transpositional pan

between

Figure

16e+

and

Figure

16f,

TOTD

drops

80-times-70,

or

5600.

TOTD

drops

by

70 in the same

way,

between

Figure

16g+

and the table

marked

"T(100)(a)."

Our

transpositionalquantum,

roundingeverything

off

to the nearest

whole

cent,

is not

fine

enough

to reflect what

goes

on

during

a

continu-

ous

span

of

transposition

"between"

tage

f+

and

stage

g.

The two

stages

coincide

up

to the

nearestcent of

transposition.

f

we refined

our

quan-

tum of

transposition,

ay

to a thousandth

f a

cent,

we would observe

the

phenomenon

just

discussed,

which would

appear

at that

micro-level.

Stage f+

would

then have

an UT number

of 97.001

cents,

while

stage g

would remain

as

shown on

Figure

16,

with an

UT number of 98.000

cents.

TOTD

would

leap up

between

stage

f and the

new

stage

f+,

then

decline

by

7

milli-?

for each

milli-?

of

transposition,

during

the

span

between

the

new

stage

f+

and the new

stage g.

The

image

set

changes

between

stage

d and

stage

d+,

between

stage

e

and

stage

e+,

and so on. At all othertimes

the

image

set remains

he

same

from one

transpositional

evel

to the

next. At

stage

e+

the

transposed

ust

scale

(with

UT

=

17)

finally

focuses

upon

the "desired"

-35

upshift

tar-

get, theD6major7-35. Thistargetremains n placeduring he long span

between

stage

e+

(where

UT

=

17)

and

stage

f

(where

UT

=

97).

The

Db

major

7-35

is the

"desired"

arget

set

because

by

the time

the

just

UT

reaches

97,

we can

regard

ts

just

scale

as a

slightly-flat

D6

major

ust

scale,

rather

hana

sharp

C

major

ust

scale. It is remarkable

hat his

phe-

48






nomenon sets

in

at such an

early transpositional tage.

That

is,

in

some

sense we

are to

regard

he

just

scale of

stage

e+,

where the

just

UT

is

only

17,

as a

"very

very

flat

just

D6

scale" rather han

a

"somewhat

harp ust

C scale." We

observe

again

in

this

respect

the tension

upon

which

we

commentedearlier,between the "atonal"upshiftprinciple,and theprin-

ciples

of

classical

tonality

as

they impinge upon

the

structure f

diatonic

scales.

Figure

16(f),

with a TOTD

distortionof

only

600,

is the

unique

"best"

transposition

of the

just

scale,

in

our

earlier sense of that

parlance,

as

regardsupshifting

nto

EQDOD.

It is "best"

because TOTD is

here at a

unique

minimum.The

just

scale of

Figure

16(f)

need be distorted

only

600

as a

whole,

in

orderto

get

to its

upshift

image

set within

EQDOD;

all the other

99

transposed

ust

scales at issue

mustbe distorted

more.

The

last item of Figure 16, headed"T(-100)(f)," ransposes hejust scale of

stage

f

down

by

100

cents,

so

that ts UT has a

numericalvalue of

-3. The

just

scale

here

is

a

semitone

(1000)

lower

than he

just

scale of

Figure

16f;

its

image

set is

accordingly

a

semitone lower

than the

image

set of

Fig-

ure

16f-that

is,

its

image

set is

now the C

major

7-35 within

EQDOD.

"T(-100)(f)"

thus

displays

numerically

he moral of

the whole affair:

o

get

the "best"

results,

when

upshifting

a

just

scale into an

equally

tem-

pered

major

scale,

flatten the

just

scale-including

its tonic

note-by

(approximately)

3

cents.

Again

we see

the

tension-here

only

a

slight

twinge-between

upshifting

and

tonicity.

8.3.3.

Maximally

close

shifting

of

just

scales into

EQDOD

entails less

distortion han

does

upshifting.

It

also

avoids

abrupt

discontinuous

eaps

in

TOTD,

of the

sort we

observed n

Figure

16 between

stages

a

and

a+,

b and

b+,

etc. The

interestedreader

can

explore

such

matters n

more

specificity.

For

present

purposes,

t will

suffice

to

display

here

Figure

17a

and

Figure

17b.

(a)

(b)

JUST

0 203

386 498

702 884

1088 2

205

388

500704

886

1090

MCL

0

200 400

500 700

900 1100

0

200 400

500 700

900

1100

DIST

0 3 14

2 2

16 12 2

5

12

0

4

14

10

TOTD

49?

470

IMAGE

SET: he C

major

7-35

the C

major

7-35

Figure

17

The

top

row of

Figure

17a

displays

our

ust

scale

at the

transpositional

level UT

=

0,

where

its UT

coincides with

a

pitch

class of

EQDOD.

The

row

marked "MCL"

displays

the

maximally

close

target

notes within

EQDOD,

for

the

notes of

the

just

scale at

this

transpositional

evel.

49






TOTD,

the

total distance

(distortion)

nvolved

in

mapping ust

notes

to

their

EQDOD

targets,

s here

quite

modest.

49?

is

less

than

600,

the

best

we could

do

using upshift

voice

leading.

Furthermore,

he

image

set for

Figure

17a,

where the

just

UT

is

0,

is the

C

major

7-35.

This is a notable

improvement n Figure16a.Withmaximallyclose voice-leading, n Fig-

ure

17a,

we can

get

a

reasonablygood

fit of the

just

scale

into the

C

major

7-35

while

exactly preserving

he

tonicity

of

UT

=

0.

However,

Figure

17b

injects

a bit of caution into that observation.

t

shows

that

we

can

get

a

still

better

it if we

transpose

he

just

scale

2

cents

up.

In this

transposition

he

just

UT is

slightly

sharp

rom

the UT

of

the

target

7-35;

the FA of the

just

scale,

not the

UT,

now coincides

exactly

with its

pertinent

arget

note

in

the

equally-tempered

cale. Once

again,

we see some

tension,

albeit

slight

in

this

connection,

between

our"atonal"

voice-leadings and the principlesof tonicity.We can even observe a

slightly

"Lydian" uality

here,

in

the fact

that

Figure

17b

gives

the "best"

maximally

close

voice-leading

into

EQDOD,

among

all

possible

trans-

posed

forms of the

just

scale.

In

passing

from

Figure

17a to

Figure

17b,

some notes

of the

just

scale

get

more distant rom

their

targets

n

EQDOD.

Those are

the notes whose

numerical

values lie between

nOO nd

n49 inclusive

(where

n is

0, 1, 2,

..,

or

11).

For these

notes,

the

DIST

value increments

by

1?,

for each

cent

by

which the

just

scale is

transposed.

Thus,

in the UT

column and

theDISTrowof Figure17a,the 0 incrementsby 2 to the2 in theUT col-

umn

and

the DIST row

of

Figure

17b,

which

transposes

he

just

scale

2

cents

from

Figure

17a.

Likewise,

in the

RE column

and

the

DIST

row

of

Figure

17a,

the 3

increments

by

2 to the

5 in the RE column

and the DIST

row

of

Figure

17b,

which

transposes

he

just

scale 2 cents

from

Figure

17a.

Also,

in

the

SOL

column and the

DIST

row

of

Figure

17a,

the

2

increments

by

2

to

the

4

in the

SOL

column

and the

DIST row

of

Figure

17b.

In

passing

from

Figure

17a

to

Figure

17b,

othernotes

of

the

just

scale

get

closer to their

targets

n

EQDOD.

Those are the noteswhose numer-

ical

values

lie

between

n50

and

n99

inclusive

(where

n is

0, 1,

2,

....

or

11).

For these

notes,

the

DIST

value

decrements

by

10,

for

each cent

by

which

the

just

scale

is

transposed.

Thus,

in the

MI

column

and

the DIST

row of

Figure

17a,

the 14 decrements

by

2 to the

12 in the

MI column

and

the

DIST

row

of

Figure

17b,

which

transposes

he

just

scale

2

cents

from

Figure

17a. And

so forth

for the other 3 notes

whose

DIST

values decre-

ment

here.

In

passing

from

Figure

17a

to

Figure

17b,

3 of

the

seven

Stufen

are

incrementing

heir

DIST

values,

while the

other

4

Stufen

are decrement-

ing

their

DIST values.

Accordingly,

TOTD

diminishes

by

1e

for each

cent

of

transposition.

-1

=

3

times

1,

plus

4 times

-1.)

The curious

reader

can

take these observations

as a

point

of

departure

or more

thorough

50






exploration

of

maximally-close

scale-shifting

through transpositions,

here

in

particular

nd elsewhere

more

generally.

8.4. The materialof section

8.4-in

particular

ts arithmetic-will

get

fussy, and readerswho lose interestatany point may freely skipahead o

section 8.5. The section

will

study maximally-close

voice-leading map-

pings

from one

"isoquintal

cale" into

another:

By

an

isoquintal

scale I

shall mean

any

diatonic

scale

of

pitches,

or

the

corresponding

et of

pitch

classes modulo the

octave,

whose fifths are

all of

the same size. Familiar

examples

of

such

scales

are the

Pythagorean

cale

(all

fifths

ust),

the

12-

tone

equally-tempered

-35

(all

fifths

seven-twelfths

of an

octave),

other

equally-tempered

ystems

(e.g.

all

fifthseleven-nineteenths

f

an

octave),

the

quarter-comma

mean-tonescale

(four

fifths

=

two octaves

plus

a

just

majorthird),and the third-commamean-tone scale (three fifths = an

octave

plus

a

just

major

sixth).26

8.4.1. From a

more

formal

point

of

view,

we can consider

any

pcset

of

cardinality

7

that is

generated

by

some

interval

Q (for

"Quint"),

produc-

ing

distinct

pitch

classes we shall call

FA,

UT,

SOL,

RE,

LA,

MI,

and SI

so

as

to

satisfy

a

special

criterion.

The

special

criterion s thatwithin

any

given

octave

starting

on

some

UT,

the

pitch-representatives

f

our seven

pitch

classes come

in

registral

order

UT, RE, MI, FA, SOL,

LA,

SI.

Let

us writeV for the octave distance betweenpitches. (Duringthis section

we shall

refrain rom

using

the

symbol

V

for

anythingelse-e.g.

a voice-

leading

function.)

For the interval

Q

to

satisfy

the criterion

ust

formu-

lated,

necessary

and sufficient

algebraic

conditions are that

V<2Q

while

5Q<3V.

That

s,

two fifths

mustbe

greater

han

an

octave,

while five

fifths

must be

less than three

octaves. The reader

will

soon

see

why.

8.4.2.

Figure

18a

lays

out,

over a

broad

registral pan,

seven

pitches

that

represent

he

distinct

pitch

classes of

an

isoquintal

cale. The

figure

abels

RE, which is a centerof

symmetry,

with the number

zero.27

The SOL a

fifth lower then receives the

pitch

number

Q,

while the LA

a fifth

higher

receives the

pitch

number

Q,

and so

forth.

Figure

18b,

adding

or

subtracting

ctaves

(Vs)

as

necessary,brings

he

pitches

withinthe

span

of one

octave,

adjoiningpitch

UT' an

octave

above

UT. Fromthis

layout,

we see that

the whole

tone

up,

as

RE

minus

UT,

MI

minus

RE,

SOL minus

FA,

LA

minus

SOL,

or SI

minus

LA,

has size

2Q-

V,

while

the semitone

up,

as FA

minus

MI

or

UT' minus

SI,

has

size 3V-

5Q.

We

verify

algebraically

hat 5

formal

"whole-tones-up"

lus

2

for-

mal "semitones

up"give

anoctave:

5(2Q-V)

+

2(3V-5Q)

=

10OQ

5V +6V

-10OQ

V. The

algebraic

conditions at the

end of 8.4.1 above

amountto

saying

thatthe

formal

"wholetone

up"

and "semitone

up"

should

be

rep-

resented

by

positive

numbers.

51






(a)

FA

UT SOL

RE LA

MI SI

-3Q

-2Q -Q

0

Q

2Q 3Q

(b)

UT RE MI FA SOL LA SI UT'

V-2Q

0

2Q-V 2V-3Q

V-Q Q

3Q-V

2V-2Q

Figure

18

We

write "T" or

the whole tone

and

"S"

for

the

semitone;

thus

T

=

2Q-V;

S

=

3V-5Q.

8.4.3. Let us now considertwo isoquintalscales, with fifths of different

sizes. Let

us

call

the

scale

we

have

just

studied above

"the

upper-case

scale";

we have used

upper

case

symbols

for

its

scale

degrees

and inter-

vals. The other

soquintal

scale we shall call "the ower-case

scale,"

writ-

ing

its

degrees

as

ut, re,

mi,

fa,

sol, la, si,

measuring

ts

fifth,

its

whole-

tone,

and its semitone

as

q,

t,

and s.

We

shall

suppose

that

the lower-case

scale has

the smaller

fifth,

i.e.

that

q<Q,

and

we

shall

denote

by

the letter

d the differencebetween

the fifths:

d

=

Q

-

q.

Figure

19

lays

out the two sets as

segments

of their

respective

fifth-

cycles,

in the mannerof

Figure

18a. The sets are laid out

so

that

lower

case re

coincides

with

upper-case

RE.

Then,

as

the

figure

ndicates,

sol

=

re

-

q

will be a

distance

d above SOL

=

RE

-

Q,

ut

=

re

-

2q

will be a dis-

tance 2d above UT

=

RE

-

2Q

,and

so forth:

a

=

FA

+

3d,

la

=

LA

-d,

mi

=

MI

-

2d,

and si

=

SI

-

3d.

8.4.4.

Let

us

consider

the

voice-leading

of fa

into

FA,

ut into

UT,

sol

into

SOL,re intoRE, la intoLA , mi intoMI, andsi into SI. We shall call the

two scales

well

matched

when

this voice

leading

is a

unique

maximally

close

voice

leading

of the lower-casescale

into the

upper-case

cale,

and

we shall

restrict

our attention rom

now

on

to

well-matched

scales.

That

is,

we shall

from now on demand

hat a

be

closer to

FA,

than

t is to either

MI

or

SOL;

we shall demand

that ut be

closer

to

UT,

than

it is

to

SI or

RE,

and

so forth.

Our

demand

amounts to

demanding

that

3d

be less

than half the

whole-tone

of the

upper-case

scale-i.e.

that 3d

<

(1/2)T.

When that is

the

case,

fa = FA+3dwill be less thanhalf of the

way up

to the next avail-

able

SOL,

from

FA;

it will lie closer to

FA than it does to SOL.

And of

course,

being greater

han

FA,

fa

will

lie

closer to

FA

than

it does to the

MI below

FA. When 3d

is less than

(1/2)T,

2d

will

afortiori

be less

than

52






(1/2)T,

and

ut

will

be

less

than

halfway

fromUT to the next available

RE.

And

so

forth;

he

reader,

by inspecting

Figure

19,

can

verify

the

situation

as

regards

the other

scale

degrees.

Our

demand,

that the difference

d

between the

fifths

Q

and

q

should

be less

than a sixth of the

upper-case

whole-tone, is reasonableenough, seeing thatwe want both Q andq to

produce

sounds

we would

intuitively

be

willing

to

identify

in

character

(as

both

representing"perfect

ifths").

8.4.5. Now

imagine

the lower-case scale

in

Figure

19

transposed

by

an

interval

j

(positive, negative,

or

zero).

We shall call the

transposition

Stufen-preserving

when

there

s

a

unique maximally

close

voice-leading

from the

transposed

ower-case

scale

into

the

upper-case

cale,

and when

that

voice-leading maps fa+j

into

FA,

ut+j

into

UT,

and so

forth. So

the

demandwe made at theendof section 8.4.3 above can be

expressed

as a

demand that the

zero-transposition

f the lower-case scale

be Stufen-

preserving.

The

Stufen-preserving

ondition

will

obtain whenever the

following

considerationsobtain.

fa+j

should

be closer

to

FA,

than

t

is to eitherMI

or

SOL,

ut+j

should be closer to

UT,

than t

is

to either SI or

RE,

and so

forthfor the other five

scale

degrees.

8.4.6. Two

questions

now

suggest

themselves:

QUESTION

1: What are

the

Stufen-preserving

ranspositions

of

the

lower-case

scale? How

big

can the

interval of

transposition

et,

in

either

a

positive

or a

negative

sense,

before

maximally-close

voice-leadingmaps

some

degree

of

the

transposed

ower-casescale into a

different

degree

of

the

upper-case

cale?

QUESTION

2:

Among

all

the

Stufen-preservingranspositions

f

the

lower-case

scale,

is

therea "best"

one,

in

the sense of

minimizing

TOTD?

If

so,

what is

it?

The remainderof section 8.4 is devoted to answeringthe questions,

and

proving

hat

he

answersare

correct.

shall first

give

the

answers,

and

only

then

supply proofs.

That

will

enable a

determined

but

fatigued

reader,

having

noted the

answers,

to

skip

the

proofs

and

jump directly

to

section

8.5.

8.4.7. ANSWER

2:

Yes,

there is a

unique

"best"

transposition

of the

lower-case

scale,

as

regards

minimizing

TOTD,

the

total

distance tra-

versed,

when

each

degree

of

the

transposed

ower-case scale is

retuned

via maximally-close voice-leading to the correspondingdegree of the

upper-case

cale.

This

unique

best

transposition

s the one

where

=O,

the

one

given

in

Figure

19.

The best

transposition,

hat

s,

matches

re

exactly

with RE

(and

not

ut with

UT).28

ANSWER 1:

Set

BIGJ1

=

(1/2)T

-3d;

set

BIGJ2

=

(1/2)S

+

2d.

That

53






FA

UT SOL RE LA MI SI

fa

ut

sol re la mi

si

3d

2d

d 0 d 2d 3d

Figure19

is,

BIGJ1

s the differencebetween half the

upper-case

one and 3

inter-

vals of

d;

BIGJ2 is the

sum

of

half

the

upper-case

semitone

and

2 inter-

vals

of

d. BIGJ2 is

the sum of two

positive

numbers,

hence

positive.

Underthe

conditionswe demanded n section 8.4.4-that

the two scales

be

"well-matched"-BIGJ

1

will

also be

a

positive

number.

For the

j-transposition

f the lower-casescale to be

Stufen-preserving,

it is both

necessary

and sufficient

or

the absolute

value of the

transposi-

tion-interval to be less thanbothBIGJ1and BIGJ2.

8.4.8.

We shall now

show that

the

answers

above

are correct.

(This

is

where a

fatigued

reader

can

skip

ahead

to section

8.5.)

Figure

20 will be

a

helpful

visual aid.

RE

MI

FA SOL

LA

SI

UT'

RE'

re

mi

fa

sol

la

si

ut' re'

0 2d

3d d

d 3d

2d

0

Figure

20

The

figure

lays

out the

pitch

classes

of the

two

scales

as Dorian

modes,

emphasizing

he

symmetry

around

re

=

RE: mi lies a distance

of

2d below

MI,

while ut'

lies a distanceof

2d

above

UT';

fa

lies a distance

of 3d

above

FA,

while si lies a distance

of 3d above

SI;

sol

lies

a

distance

of d

above

SOL,

while la lies a distance

of d

below

LA.

We know that

the

0-transposition

of the lower-case

scale

is Stufen-

preserving-that

was the demand we made in section 8.4.4, that the

upper

and

lower case scales

be

well-matched.

Let us now-until

further

notice-restrict

our attention

o values

of

j

that

are

positive.

We

can visualize the

effect

of

j-transposition

on

the

lower-case

scale

of

Figure

20

by

imagining

the

lower-case

notes

shifted

a distance

of

j

to the

right

on the

figure;

some readers

may

find that

visu-

alization

helpful

in

connection

with

the

algebraicmanipulations

hat

fol-

low.

Suppose

thatour

positivej

is less thanboth

BIGJ and BIGJ2.

Since

j

is less than

BIGJ1,

3d+j

will be less

than

3d+BIGJ1,

which is

(1/2)T.

(BIGJ1

was

definedin section 8.4.7 above as

(112)T

3d.)

Since

3d+j

<

(1/2)T,

it

will

a

fortiori

be

the case

that

2d+j

<

(1/2)T,

and

that

d+j

<

(1/2)T,

and

that

<

(1/2)T.

Accordingly,

we will

have

54






FA

< fa +

j

=

(FA+3d)

+

j

< FA

+

(1/2)T

UT <

ut

+

j

=

(UT+2d)

+

j

< UT

+

(1/2)T

SOL

< sol +

j

=

(SOL+d)

+

j

< SOL +

(1/2)T

RE < re

+

j

=

RE+j

< RE +

(1/2)T

So

fa+j

is closer to

FA,

than

(to

MI

or)

to

SOL;

ut+j

is closer to

UT,

than

(to

SI

or)

to

RE;

sol+j

is

closer to

SOL,

than

(to

FA

or)

to

LA;

and

re+j

is closer to

RE,

than

(to

UT

or)

to

MI.

Also

la+j

is closer to

LA,

than

to

either SOL

or

SI,

because

LA

-

(1/2)T

< LA

-

d

=

la

<

la

+j

<

LA

+

(1/2)T

What about

mi+j

and

si+j?

Well,

we know

that

MI -

(1/2)T

< MI -

2d

= mi

<

mi +

j,

and

that

SI - (1/2)T < SI - 3d = si < si +j.

Thus

mi+j

is closer to

MI

than

to

RE,

and

si+j

is

closer

to

SI than to

LA. It

remains

only

to

ensure that

mi+j

<

MI +

(1/2)S (so

that

mi+j

is

closer to

MI

than

to

FA),

and that

si+j

<

SI

+

(1/2)S

(so

that

si+j

is closer

to

SI

than to

UT').

And the

assurance we

need

here

is

given by

our

assumption

hat

<

BIGJ2,

which is 2d +

(1/2)S.

Thus

mi +

j

< mi +

BIGJ2

=

mi

+

2d

+(1/2)S

=

MI +

(1/2)S

si

+

j

<

si

+

BIGJ2

=

si + 2d

+(1/2)S

<

si

+

3d

+

(1/2)S

=

SI

+

(1/2)S.

We have now

proved

the

following:

for

positive j,

if

j

is smaller than

both BIGJ

1

and

BIGJ2,

then the

j-transposition

f the

lower-case

scale

is

Stufen-preserving.

The

same sorts of

scrutiny

and

arithmetic an

be used

to

verify

the converse:for

positive

j,

if

the

j-transposition

of

the lower-

case

scale

is

Stufen-preserving

hen

j

must be

smaller than

both BIGJ

1

and

BIGJ2.We have now

verifiedANSWER

1

for

positive

values

of

j.

Focusing

upon positive

values of

j

that are

in

the

desired

range

(smaller

than both

BIGJ1

and

BIGJ2),

let us now

compute

TOTD,

the

total distancethroughwhich the degreesof thej-transposed ower-case

scale

must be

retuned o obtain

the

correspondingdegrees

of

the

upper-

case scale. The

visual field of

Figure

20 will

again

be

helpful

for

a num-

ber of

readers,

in

following

the

reasoning

here;

we

are

imagining

the

lower

case scale

shifted

a distance

of

j

to the

right

on

that

figure.

Since

fa+j

>

FA,

the

distance between those

two

degrees

is

(fa+j)

-

FA,

or

j+(FA-fa),

or

j+3d.

Likewise,

the

distance

between

ut'+j

and UT' is

j+2d,

the

distance between

sol+j

and

SOL is

j+d,

and

the

distance

between

re+j

and RE

is

j.

We

now have TOTD

=

(j+3d)

+

(j+2d)

+

(j+d)

+

j

+

THEREST,where THEREST s the sum of the distancesbetween la+j

and

LA,

between

mi+j

and

MI,

and

between

si+j

and SI.

We

simplify

the

equation:

55






TOTD

=

4j

+

6d

+

THEREST

Writing

"Inumberl"

or the absolutevalue of a

number,

we

can

express

the

value

of THEREST

as

I(la+j)-LAI

I(mi+j)-MII I(si+j)-SII,

which is

Ij-(LA-la)I Ij-(MI-mi)l Ij-(SI-si)l,

which is

Ij-dl

+

Ij-2dl

+

Ij-3dl.

Case

1: If

j

is less than or

equal

to

d,

then THEREST s

(d-j)

+

(2d-j)

+

(3d-j),

which is

6d

-

3j.

Then

TOTD

=

(4j

+

6d

+

THEREST)

=

(12d+j),

which is

bigger

than

12 d.

Case 2:

If

j

is

greater

than

d,

but less than

or

equal

to

2d,

then

THEREST s

(j-d)

+

(2d-j)

+

(3d-j),

which is

4d

-

j.

Then

TOTD

=

(4j

+

6d

+

THEREST)

=

(10d+3j),

which is

bigger

than

(10d

+

3d),

which is

bigger

than 12d.

Case

3: If

j

is

greater

than

2d,

but less than

or

equal

to

3d,

then

THEREST s (j-d)+ (j-2d)+ (3d-j),which is j. ThenTOTD

=

(4j + 6d +

THEREST)

=

(6d+5j),

which

is

bigger

than

(6d

+

10d),

which is

bigger

than 12d.

Case

4: If

j

is

greater

han

3d,

thenTHEREST s

(j-d)

+

(j-2d)

+

(j-3d),

which is

3j

-

6d. Then

TOTD

=

(4j

+

6d

+

THEREST)

=

7j,

which is

big-

ger

than

21d,

which is

bigger

than

12d.

So,

in all

possible

cases

for

j

positive,

TOTD

is

greater

han

12d.

But

12d is

exactly

the

TOTD

value for the

zero-transposition

f

the

lower-case

scale.

Looking

at

Figure

20,

we

can

see that

TOTD

=

(re-RE)

+ (MI-mi) +(fa-FA)+ (sol-SOL)+ (LA-la)+ (SI-si) + (ut'-UT'), which

is

0+2d+3d+d+d+3d+2d,

or

12d.

Thus,

of all

possible non-negative

val-

ues

for

j

here,

j=0

minimizes

TOTD;

he

zero-transposition

s the

unique

"best"of

all

the

non-negative ranspositions

or

the lower-case scale

that

are

Stufen-preserving.

Now

what about

the

possible negative

values for

j?

The

symmetry

of

Figure

20

suggests

what

is in

fact the

case,

that

shifting

the lower-case

scale a certain

amountto

the left is

simply

a

mirrorof

shifting

the same

amount

to the

right,

so that all

considerationsof "distances"

are

pre-

servedunder hemirror.Then fornegative , thej-transposeof the lower-

case scale

is

Stufen-preserving

f

and

only

if

-j

is

less

than

both

BIGJ1

and

BIGJ2.

And

of

all

non-positive

values for

j,

j=0

minimizes

TOTD;

the

zero-transposition

s the

unique

"best"of all

the

non-positive

trans-

positions

for

the lower-case

scale that are

Stufen-preserving.

So

among

all

transpositions

of

the

lower-case

scale

(both

non-nega-

tive and

non-positive)

that

are

Stufen-preserving,

he

zero-transposition

is the

unique

"best."We

ave now

verified he correctnessof

ANSWER

2

and

ANSWER

1.

8.4.9.

In

the above

work,

we demanded that

Ijl

be

smaller than both

BIGJ1 and BIGJ2.

A

natural

question

arises:

of the two

BIGJs,

which is

the smaller?

The

question

is

not central

o our

immediateconcerns

here,

56






but

perhaps

t should not

be left

hanging.

Uninterested

readers

are

en-

couragedagain

to move on

to

section

8.5.

BIGJ1

was defined as

(1/2)T

-

3d;

BIGJ2

was defined as

(1/2)S

+

2d.

BIGJ1

s then

greater

or smaller

han

BIGJ2,

accordingly

as

(1/2)(T-S)

is

greateror smaller han5d.That s the case accordinglyas (T-S)is greater

or

smallerthan

10d.

We

can call

(T-S)

the diesis

of the

upper-case

cale;

it is the

distance from

SIb

(a

fifth

below

FA)

to

SI,

or fromFA to

FA#

(a

fifth

above

SI).

So

BIGJ1

is

bigger

or smaller than

BIGJ2,

as

the

diesis

of the

upper-case

cale is

bigger

or

smallerthan

10

times the

interval

d.

8.5. At

several

points

now,

we

have

had occasion to

remarkon

tensions

between our "natural"

oice-leadingparadigms

on

the one

hand,

and

var-

ious

principles

of

diatonic

tonality

on

the

other.

The

matterdeserves

a lit-

tle extrahighlightinghere by way of close, for the ideas of upshifting,

downshifting,

and

maximally

close

shifting

all

engage

familiar

para-

digms

for voice

leading

that

continue to be

taught

n

a context of

com-

mon-practice

harmony.Examples

1

through

6

may

be reviewed in

this

connection.

9.

Voice

leadings

surjective

and

injective;

an

analytic

example

We shall now revert o using the symbol"V"as the name of a formal

voice-leading

function.

Definitions

9.1.1. A

function

f

from

an abstractset X

into an

abstract

set Y

is

surjective

when

every

element of

Y

is the

functional

image

of

some

x

in X.

(For

every y

in

Y,

there

exists

at least

one

x in X

such

that

y

=

f(x).)

f is

injective

when

no

two

distinctmembers

of

X

have the

same

image

in

Y.

(If

xl

is

not

the

same as

x2,

then

f(x

l)

is

not the

same as

f(x2).)

f is bijectivewhen it is bothsurjectiveandinjective.

The

italicized

terms,

introduced

by

a committee

of French

mathe-

maticians

shortly

after

WorldWar

II,

are now

in

fairly

widespread

use. I

myself

prefer

he

earlier

English

terms

"onto

Y"

for

"surjective

nd "one-

to-one" for

"injective."

Or

the

earlier

French terms

"surY"

and "uni-

voque";

have a

general

preference

or

vernacular

words,

even if

of Latin

origin,

over

highly

latinized

ones.)

The

readerwill

still

encounter he

ear-

lier terms in

the

literature.But

they

can be

awkward

yntactically;

"one-

to-one-and-onto,"

n

particular,

s

a

problematic

mouthful

as an

adjective,

where"bijective"rips smoothlyoff the tongue.

9.1.2. If X

and

Y

are finite sets

of

the

same

cardinality,

hen

a function f

from

X

into

Y

is

injective

if

and

only

if

it is

surjective.

The

readerwill

57






probably

ind this

easy

to intuit: f we

have a roomfulof

N

desks,

and

ask

N

students o seat

themselves,

then eitherthere will be

only

one

student

at each

desk,

with

no

desk

empty,

or else there

will be

at least one

empty

desk

and at least

one desk where2 or more

students

are

sitting.

9.1.3. Our

formalvoice

leadings,

which are

functions,

can

now be called

surjective,

njective,

or

bijective,

as

any particular

ase

may

warrant.

Fig-

ure

1,

for

example,portrayed

formal

voice

leading

of X

=

{

D,F,A

}

into

Y

=

{E,G#,B}

which was

neither

surjective

nor

injective:

V(D)=E;

V(F)=E; V(A)=G#.

The voice

leading

was not

surjective:

he member

B

of

Y

did

not

occur

as

a value

of the function

V. The voice

leading

was

not

injective:

the

members

D

and

F

of

X,

while

distinct,

had

the same func-

tion value.

The

voice-leading

function for

the Fate Motive at

the

end

of

Die Walkiire, n the otherhand,was both surjectiveandinjective-that

is,

bijective.

There,

we had

V(D)=E,

V(F)=G#,

and

V(A)=B;

every

mem-

ber

of

the

E

triad

appeared

as

a function

value,

and none

appeared

more

than

once.

9.2.1.

Figure

21

brings

the

foregoing

abstract onsiderations

nto contact

with some

specifics

that

will

shortly

be

applied

to a

Webern

analysis.

At

the

top

of the

figure

appear

two

pcsets

X

=

{BbL,B,C,Eb,F#}

nd

Y

=

{A,C,Db,Eb,E

.

The twelve rows of

the

figure

below

thatdetail

the max-

imally close voice leadings,fromthe twelve transpositionsof X intoY.

(NB

into not

onto;

the voice

leadings

are

not

in

general surjective.)

Let

us

consider,

for

instance,

the row of

the

figure

that contains

the

symbols

"T10"

and

"V10."

At the left

of the

row,

the members

of X

are

listed as

Bb,B,C,Eb,

and

F#.

The middle

list

in the row

gives

the

corre-

sponding

members

of

T10(X),

namely

G#,A,Bb,Db,

nd E. The

voice-

leading

function

V10

is the

unique

maximally

close voice

leading

of

T1O(X)

nto

(into)

the set

Y:

among

the members

of

Y,

A is closest

to

G#,

A is closest

to

A,

A is closest

to

Bb,Db

is closest to

Db,

and

E

is closest

to E. The

only displacements

nvolved hereareG#-to-Avia 1 semitone,

and

Bb-to-A

via

1

semitone. So

the total

distance

TOTD,

by

which

the

pcs

of

T10(X)

must be

adjusted

to

fit into

Y

via

V10,

is 2

semitones.

Among

the five

pcs

of

Y,

the

image

of

T10(X)

under

V10

contains

only

three;

V10

is

not

a

surjective

map.

The distinct

pcs

G#,

A,

and

Bb

of

T10(X)

are all

taken

by

V10

to

the

same

image

pc

in

Y;

V10

is not

an

injective

map.

For

each

N between

0 and

11

inclusive,

we can consider

a

map

WN

of

X into

Y:

WN

maps

each

pc

of

X, through

he

pertinentpc

of

TN(X),

to

the

pertinent

pc

of

Y,

as

given

by

VN-of-TN.

Using

the

terminology

ntro-

duced

earlier

in sections

5.1

through

5.4,

we

can

say

that

WIO

s

the

unique

maximallyuniform

voice-leading of

X into Y:It differs

from

the

straight

T10-transposition

f

X

by

an offset

of

only

2

semitones;

every

58






X =

{Bb,B,C,Eb,F#

;

Y =

{A,C,Db,Eb,E}

Bb,B,C,Eb,F#

TO--

B,B,C,Eb,E

-VO---

A,C,C,Eb,E

(TOTD

or

VO

s

4)

Bb,B,C,Eb,F#T1---

B,C,Db,E,G V1---

C,C,Db,E,A

TOTD

or VI

is

3)

Bb,B,C,Eb,F#

T2---

C,Db,D,F,G#

V2-

C,Db,Db

r

Eb,E,A

TOTD

or V2

is

3)

Bb,B,C,Eb,F#T3--- Db,D,Eb,F#,AV3--- D,D orEb,Eb,E,ATOTD or V3 is 3)

Bb,B,C,Eb,F#

T4--- D,Eb,E,G,Bb

V4-

Db

or

Eb,Eb,E,A,A

TOTD

or V4

is

4)

Bb,B,C,Eb,F#

T5---

Eb,E,F,G#,B

V5-

Eb,E,E,A,C

TOTD

orV5 is

3)

Bb,B,C,Eb,F#

T6-

E,F,F#,A,C

V6-

E,E,E,A,C

(TOTD

for V6 is

3)

Bb,B,C,Eb,F#

T7-

F,F#,G,Bb,Db

V7-

E,E,A,A,Db

TOTD

or

V7

is

6)

Bb,B,C,Eb,F#

T8-

F#,G,G#,B,D

V8-

E,A,A,C,Db

or

Eb

(TOTD

or V8

is

7)

Bb,B,C,Eb,F#

T9---

G,G#,A,C,Eb

V9-

A,A,A,C,Eb

TOTD

or V9 is

3)

Bb,B,C,Eb,F#

TIO--

G#,A,Bb,Db,E

V

IO-

A,A,A,Db,E

(TOTD

or

V

IO

s

2)

Bb,B,C,Eb,F#

TI

I-

A,Bb,B,D,F

V

I

1- A,A,C,Eb,E

TOTD

or

V

I

I

is

4)

Figure

21

other

WN

besides

W10

differsfrom

the

straightTN-transposition

f

X

by

3 or

more semitones'

displacement.

If

we are

analyzing

a

passage

of

music

with

a

texture hat

consistently

presents

chords

having

five different

pitch-classes,

then

the

image

set

of

W10,

despite

its formal

property

ust

discussed,

will

not sound

very

characteristic

n that

particular

texture,

as a

"pseudo-transposition

f

X."

W1O(X),on the

right

of

Figure

21, is

enharmonically

an A

major

triad,

and

that

set is not

likely

to

sound

intuitively,

n such a musical

context,

like a


f

X."

If

one listens

through

the

twelve

image-sets

for the

WN,

at the

right

of

Figure

21,

one hears

clearly

that

the

image

of

W2,

using

Eb

rather han

Db

where we have the

choice,

will

intuitively

seem much more like a


f

X"

in such a

context

than

will

the

image

of

any

other

WN,

W10

in

particular.

That is

largely

because the

image

of

W2

(using

Eb)

has 5

distinct

pcs,

just

like

X.

All other

WN-images

have

4

pcs

or

less,

and that

makes

t dif-

ficultto hear them

intuitively

as

"pseudo-transpositions"

f a 5-note set

in such a

context.29

Beyond

that,

we

do

not have to

displace

T2(X)

very

much,

to

get

to

its

V2-image.

The total

displacement

s

only by

3

semi-

tones,

and

that is

as small

as

or

smallerthan all

other

VN

displacements,

V10

excepted.

Abstractly,

t

will

be useful

(as

we shall see in an

upcominganalysis)

to

have a

term

for

the features hat make us

attach

a

special

meaning

to

T2,

V2,

and

W2

here.

We shall

say

that

W2, here,

is a

(unique) maximally

uniform

surjec-

tive

voice

leading,

of

X

into

Y.

It is also a

(unique)

maximally

uniform

injective

voice

leading

of X

into

Y,

and t is a

(unique)maximally

uniform

bijective

voice

leading,

of

X

into

Y.

The

terms

can

be

given

general

abstract

definitions,

as

follows.

59






Definition

9.2.2. Given abstract

pcsets

X

and

Y

(of

any

cardinalities,

pos-

sibly

different),

hen a

voice-leading

from

X

into

Y is a

maximally

uni-

form

surjective

voice

leading

when

it differs

from

straight

ransposition

of

X

into

TN(X)

by

a number

of offset

semitones that is

minimal

among

all surjectivevoice-leadingsof X intoY

Definitions

for

"maximally

uniform

injective,"

and for

"maximally

uniform

bijective"

voice

leadings,

are

analogous.

Thus

W2,

as

just

discussed

above

in

connection

with

Figure

21,

is a

maximally

uniform

surjective

voice

leading

for the

X and

Y

of

that

fig-

ure.

W2 is

not

a

maximally

uniformvoice

leading;

he

unique

maximally

uniform

voice

leading

is

W10.

W10,

however,

s

not

surjective.

If there

are no

surjective

voice-leadings

of

an

abstract

X

into

an

ab-

stract

Y

(which

will be the case when

Y

has a

greater

cardinality

han

X),

then there will be no maximally uniform surjective voice leadings,

though

there

will

be

at least

one

maximally

uniformvoice

leading.

There

will nevertheless be

plenty

of

injective

voice

leadings

when

Y

has a

greater

cardinality

han

X,

and

from

among

those we can

always

select

at

least

one

maximally

uniform

njective

one.

Likewise,

if

there are

no in-

jective

voice-leadings

of

X

into

Y

(which

will

be the case

when

Y

has a

smaller

cardinality

than

X),

then there

will be

no

maximally

uniform

injective

voice

leadings,

though

there

will

be

at

least one

maximally

uni-

form

voice

leading.

There

will nevertheless

be

plenty

of

surjective

voice

leadings

when Y hasa smaller

cardinality

han

X,

and from

among

those

we can

always

select at

least

one

maximally

uniform

surjective

one.

When

Y has

the

same

cardinality

as

X,

there

will

be

plenty

of voice

lead-

ings

that

are at once

injective,

surjective,

and

bijective;

among

those we

can

always

find at

least

one

with the desired

property,

hough

it

may

not

be

a

maximally

uniform

voice

leading

among

all

possible

voice

leadings

(includingnon-surjective

nes).

9.3. Example 10a

transcribes

aspects

of

mm. 3-5

(with pickup)

from

Webern's

piece

for

string quartet,

op.5,

no.5.

Of

the

five chords

in

the

upper

strings,

chord

2 is an exact

pitch-transposition

f

chord

1,

while

chords

4

and 5

are

exact

pitch-transpositions

f

chord

3. In this

context,

we can

easily

hear

chord

3 to a considerable

extent

as a

substitute or

some

pitch-transposition

f chord

2. The

progression

from

chord

2

to

chord

3 is bracketed

on

the

example.

The

pitches

of

chord

2 in

fact

represent

he

"pcset

X"

of

Figure

21,

and

the

pitches

of chord

3

represent"pcset

Y."

Example

10b

puts

the

X

chord

on the left and theY chord on the right,using solid noteheads.Between

them,

using

open

noteheads,

he

example

interposes

an

exact

pitch-trans-

position

of chord

2

two

semitones

higher.

Matching

his

example

with

the

T2/V2

row

of

Figure

21,

we hear how

the

pitch-structure

rojects

the

maximally

uniform

surjective

character

f

the

map

W2. Not

only

does

it

60






F31

(a)

ca. 48

etc.

PM

muted

T

2 T-2

(b)

,(c)

maximally

uniform

subjective maximally

uniform

Example

10

project

the

voice

leading

of

W2,

it

actually

manifests

hat

character

n its

registralarrangement

f

pitches.

(The

reader

may

want to review our use

of

the terms

"project"

nd

"manifest,"

s

those

were

discussed

n

sections

1.2 and

1.5.)

Example 10bindicatesthe displacementof threesemitones,from the

pcs

of

T2(X)

here

to

their

images

withinY.

That total

displacement

was

what madethe

W2

voice

leading maximally-uniform-surjective;

he three

semitones characterize he

"good

fit"

of

the

voice

leading-not

so much

as that

compares

o other voice

leadings

in

general,

but

rather

n

specific

connection with the

surjective

character

of

the

map.

W2

maps

X,

repre-

sented

by

chord

2,

onto all of

Y,

represented

y

chord

3. As

observedear-

lier,

"the

image

of

W2

... has

5 distinct

pcs,

just

like X.

All

other

WN-

images

have

4

pcs

or

less,

and that

makes

it

difficult to hear them

intuitively[inthisparticularmusicaltexture]as pseudo-transpositionsf

a 5-note set."

The

displacement

of three semitones relates

suggestively

to the

osti-

nato

pattern

n

the

'cello,

as that

instrument scillates

3

semitones down

and

up

between

E

and

C#.

This

sort of

relationship,

between bass

melody

and

homophony

n the

upper

voices,

is familiar

to us from earlier

study

of the

Magic Sleep

motive

(Example

5),

and

Schoenberg's

"ich

fiihle

luft"

setting (Example

6).30

The role

of

surjectivity

s

important

o savor

in

the Webern

example

because another eatureof themusic,a more

immediately

salientfeature,

prompts

us

to

hear

chord 3 as a

"pseudo-T2"

f chord

2.

That feature s

the

melodic interval

of

2

semitones between the

high

Bb

of chord

2

and

the

high

C of chord 3. The

readershould

appreciate,

nonetheless,

hatthe

61






melodic

major

second

is

hardly

sufficient

n

its own

right,

to

prompt

our

hearing

n

this

way.

Even

given

the

high

C

in

chord

3,

if

chord

3 did not

contain

five distinct

pitch

classes

(that

is,

if

the voice

leading

were

not

surjective),

we

would

be

considerably

ess

ready

to

hear

a

pseudo-trans-

positionalrelation n Webern's exture.

Example

10c

gets

at the

point by

instancing

(and

manifesting)

n its

pitch-structure

he

maximally

uniform

but

non-surjective)

oice

leading

of our

earlier

"W10"from

Figure

21. Here the

pitch-class transposition

T10 is

manifested

by

the

pitch-transposition

T(-2),"

which

maps

the

pitches

of

chord

2

down two

semitones each.

The

relation

of chord

2

to

the

A

major

harmony

n the

example

manifests

maximal

uniformity

n

the voice

leading.

Yet

if

the

A

major

chord

(of

the

example)

were to

appear

n

m.

4

of

Webern's

piece,

one

would

hardly

hear

it

intuitively

as

a "pseudo-transpositionown 2 semitones"of chord2. That is the case

even

though

the

A

major

chord has

5 distinct

voices,

and

even

though

it

does

instance

and manifest

a

formal pseudo-transposition

ccording

to

our

earlier

definition.

The chorddoes

not have

five distinct

pitch

classes,

and

this crucial

aspect

of

non-surjectivity

eavily

circumscribes

ur

cog-

nitive

perceptions

n the

particular

musical context.

For the

readerwho

wishes to

undertake

urther

analyses

of

maximally

uniform

voice

leadings, surjective

or

non-surjective,

njective

or

non-

injective,

I recommend

Schoenberg'spiano

piece

op.

11,

no.

1,

as a

source

forinterestingpassages-indeed I recommend n thatregardmuchof his

pre-twelve-tone

music without

key signature.

10.

Retrogradable

voice

leadings;

pcsets

with

doublings

10.1.As

we

have

so

fardefined ormal

"voice

eadings,"

nly bijective

ones

are

"retrogradable."

or

nstance,

as we

noted,

the voice

leading

of

Exam-

ple

1

a was not

retrogradable:

f we are

considering

he

pcsets

X=

{

D,F,A

andY=

{

E,G#,B

,

thenthevoice

leading

V(D)=E,V(F)=E,V(A)=GO an-

not

be

retrograded

o

some formal

voice

leading

V'

of

{

E,G#,B

into

{D,F,A

}.

We could

not have

both

V'(E)=D

and

V'(E)=F;

also

we

would

have no

idea how to

choose

some note

of

the D minor

riad

as

V'(B).

Mathematically,

ne

says

thatthe function

V

above

"does

not

have

an

inverse

[function]."

A

function

V

has

an inverse function

(retrograde

function

n our

sense)

if and

only

if

it is

bijective.

We

might

try

to

address

our

formal

problem

by

restricting

our

atten-

tion to

bijective

voice

leading

functions

only.

But

the restrictionwould

be

unbearable:

we do want to considerothersortsof functions.Forinstance

the

V

above,

that

maps

the

D minor

pcset

into

the E

major

set,

is the

unique

maximally

close voice

leading

for

the

pair

of

pcsets.

Another

example

is

provided

by

the

function

V

given

in

Figure

3

earlier:

V(B)=

62






A,

V(G#)=F,V(G)=F.

It is a

maximally

uniform

map

of the set we

called

"trichord

i),"

studyingSchoenberg's

op.

11,

no.

1,

into

"trichord

ii)."

As

we

saw,

this

non-bijective

map,

in

emphasizing

a relation of

pseudo-

transpositionby

T(-2),

turnedout to have

interestingconsequences

for

analysisof the passageat issue, particularly s concernedvariousmajor

thirds herein.

Beyond

these

examples,

t will

frequently

happen

hat

we wish to

con-

sider formal

"pcsets

with

doublings."

For

instance,

t

would

be

impossi-

ble

to

address

common-practiceharmony

(or

homophony

from the

Renaissance,

for

that

matter),

without

allowing

structures

hat address

progressions

ike that of

Figure

22.

(The

figure symbolizes

the refrain

"Jesumvon Nazareth" n

Schtitz's

St. John

Passion.)

Given the

symbols

employed,

we

cannot

use

mathematical

unctions

V

i, V2,

etc.

to

describe

what we surelywant to call "voice leading"here.We cannothave both

V1(G)=G (in

the

alto)

and

V1(G)=E (in

the

bass).

Nor can

we have

both

V2(E)=D (in

the

tenor)

and

V2(E)=F

(in

the

bass).

And

so forth.

We

could

try

to

addressthe

problemby

distinguishing

wo

different

pitch

classes,

in the first chord

of the

figure,

whose names contain the

symbol

"G."We

could

speak

of a mathematical

"set"

X={B,G1,D,G2}

for

the firstchord

of

the

figure,

and

of

a

"set"

Y={ C,G,E1,E2

for

the

sec-

ond

chord. Then we

could construct

a

bijective

function

V

mapping

X

onto

Y:

V(B)=C,

V(Gl)=G,

V(D)=E1,

V(G2)=E2.

This

function,

being

bijective,

can be

retrograded:

V'(C)=B,

V'(G)=G1,

V'(E1)=D,

V'(E2)=

G2. While such formalism s

suggestive,

and

definitely

worth

exploring,

it also

complicates

our

discourse n

ways

we

may

not

always

like.

Using

it,

for

instance,

we would not be

able

to thinkof the

firstchordan

instance

of the set

{G,B,D},

nor the

second chord

as an

instance of the

set

{

C,E,G

}.

In

general,

we would have to think of

{C,E,G

}, {

C1,C2,E,G),

{C1,C2,E1,E2,G},

{C,E,Gl,G2

},

and so forth

(including

sets of

indefi-

nitely large

cardinality)as

ormally

different

objects.

We would

have

to

develop

some

formalismthat

allowed this

possibility,

while

attaching

a

certain

priority

o the set

{

C,E,G

}

in the

context.While the

nicety

of

such

discrimination s

certainly

of

musical

interest,

and

certainly

deserves

exploration,

t

seems

needlessly

cumbersome

for

many purposes.

The

necessary

formalism would also

be somewhat

complicated

mathemati-

cally-and

hence

conceptually;

it

would

require

entities

beyond

the

boundaries

of

"set

theory"

as such.

Aside

from

common-practice

harmony

and

related

sorts of textures

(and

melodic

segments

with

recurring

notes,

et

alia),

one would

surely

also like to

consider ormal"chords"n

Stravinsky's

ate serial

music

that

consist

only

of

octaves

and

unisons,

and

aspects

of

"weighted

aggre-

gates"in

Babbitt's ater

music,

and other

structures

f

this

sort.

I

think that

the formalismof

"mathematical

elations"

s

optimal

for

our

purposes

here.

63






soprano:

B

C A

G G

alto: G G FED E

tenor:

D

E D

C C

B

C

bass:

G

E F

G C

Figure

22

Definition 10.2.1. Given abstractsets

X

and

Y,

a

relation

R

between X

and

Y

is

any

collection

(whatever)

of ordered

pairs

<x,y>,

where the

x

of

each

<x,y>

is

a

memberof

X

and the

y

of

<x,y>

is

a member

of

Y.

The readerwho

is

familiar

with

the notion

of "Cartesian

roducts"

an

see that a relation

R is

any

subset

(whatsoever)

of

the

Cartesian

product

XxY.

10.2.2. An

example:

et

X

=

{

cow,

goat, sheep,

owl, Pierrot,

moon

};

let

Y={moon,

cheese,

Columbine,

pussycat).

A

particular

ormal relation

between

X

and

Y

would be the set

R=

{<cow,

moon>, <cow, cheese>,

<sheep,

cheese>,

<owl,

pussycat>,

<Pierrot,moon>, <Pierrot,

Colum-

bine>, <moon,

moon>

}.

The reader

unfamiliar

with the

formalism of

"relations"

s invited

to notice

the

following

featuresof

this R: the mem-

ber

"goat"

of

X

never

appears

on the

left,

within

any

member-pair

f

R;

the members "cow" and "Pierrot" f

X

each

appear

more than once on

the left, within memberpairsof R.

10.2.3. The

reader'sattention

was drawn o those featuresof the

R

above,

because

they

would not

be

present

f R

were

afunction

(from

X

into

Y).

A

function from an abstract

X

into an abstract

Y,

strictly speaking,

is

a

particular

kind of formal relation

between X

and

Y. It

is,

specifically,

a

relation

R in which

every

member

of

X

appears

once

on

the

left,

within

some

member-pair

f

R,

and

only

once. We then write

"R(x)"

to mean

"the

unique y

that

appears

on the

right

of that

particularunique pair

<x,y> of the relation,where the given x appearson the left."

10.2.4.

As we

have

noted several

imes,

functions

hatare not

bijective

do

not have

inverse

(retrograde)

unctions. They

do, however,

have

inverse

relations.

(An

example

will

appearshortly.)

Indeed,

the

particular

irtue

of

relations

or our

present

work

s that

every

relationhas an inverse

ret-

rograde)

relation. For

example,

in 10.2.2

above,

we studiedthe

relation

R={<cow,

moon>, <cow, cheese>,

<sheep,

cheese>, <owl,

pussycat>,

<Pierrot,

moon>, <Pierrot,Columbine>,<moon,

moon>).

This relation

has the inverse(retrograde) elationR'={<moon, cow>, <cheese, cow>,

<cheese,

sheep>, <pussycat,

owl>, <moon, Pierrot>,

<Columbine,

Pier-

rot>,<moon,

moon>

}.

R' is a relationbetween the sets

Y and

X

of

10.2.2;

the inverse of

R'

is

R.

Let us now consider he

voice-leading

of X=

{

D,F,A

}

intoY=

{

E,G#,B

}

64






given by

the

function

V(D)=E, V(F)=E, V(A)=G#.

As

we have

observed

several

imes,

V

has no

inversefunction,

o

until

now

we havebeen unable

to formalizea

retrograde

oice

leading

[function]

or

Example

1a.

When

we

reconceptualize

V

as a

relation,

rather

han

a

function,

our troubles

clearup.Thinkingof V as therelation{<D,E>,<F,E>,<A,G#>}, we can

easily

enough

conceptualize

its inverse

or

retrograde

relation V'

=

{<E,D>,

<E,F>, <G#,A>

.

We

are no

longer

embarrassed,

n

our formal-

ism,

by

the

absenceof the

pitch

class

B

from

the left side

of

any pair

within

V';

nor

are we

embarrassed

by

the twofold

appearance

of

E

on

the

left

side

of

V'-pairs.

Furthermore,

his

V'

models

exactly

what we

want it

to,

when we intuit

a

"voice

leading"

from

the

second chord of

Example

la

to the firstchord

of that

example.

10.3.Example1 a representshe firstappearance f theTarnhelmMotive

in

Wagner'sRing

(Das

Rheingold,

II,

37-43).

In

each

G#

minor

harmony,

the

fourthhorn doubles

the

second on the

bass

G#.

We

shall

say

a

good

deal

more

later

on about

the

instrumentation

f the

passage.

Example

1

b

represents

the first

appearance

of the

Forgetfulness

Motive

(Giitterddimmerung,

,2,168-71).

Though

the

time

span

between

the two scenes is

huge,

we associate the

two

passages

for a

goodly

num-

ber of

reasons.

The

similar

instrumentaland

dynamic

textures are

of

course

striking.31

So are the

alternating

hords

at the

beginning

of each

passage.

So are various elements in the drama: he villain Alberichhas

supervised

Mime's manufacture f the the

magic

Tarnhelm;

he villain

Hagen-Alberich's

son

and heir-has

supervised

Gutrune's

preparation

of the

magic

potion.

Examples

11(c)

through

(h)

show how the

alternating

hords of

Ex-

ample

1

lb

can be derived from

the

alternating

hords

of

Example

1

la.

Example

1

c

gives

the

chords of

Example

1

a. Example

1ld

adds an F

below the

G#

minor

riad

of

Example

1

a,

turning

t

into a Tristan

Chord.

(Wagnerbegins

to

play

with that idea in Die

Walkiire;

t

becomes

a

very

characteristic armonic

ormula

n

all his

chromaticworks

hereafter-for

instance,

at

the end of

Tristan.)

Example

1

e

transposes

11

d down

a

major

third,

and

Example

1

If

retrogrades

)

lie.

Example

11

g

inverts he

voices

of

1

If

into

mostly

different

registers.

We

are now

very

close to the

chords

of

the

Forgetfulness

Motive. One

transformation

more

is

needed. It

is

shown

in

Example

1

h:

the first

horn

of the Motive

does not

alternate

C5

with

B4,

as

it

would

f

it

played

Example

11g.

Instead,

he

doubling

within

the C

minor

harmony

s

changed,

so

that

he

first

horn

doublesthe bass

G,

not themiddleC of the thirdhornas in 11g.Themelodyof theForgetful-

ness

Motive

alternates

G4 with

B4,

not C4 with B4.

The

melody

is

particularly

triking

because,

in

the

context of

the

C

minor

harmony

hat

opens

Example

1

h

(and

Example

11

b),

the

high

B4

projects

o

a definite

degree

the

character

f

a

leading

tone. In

that

capac-

65






ity

it makes

a

highly

pungent

effect,

since

there s

no C5 about

n Exam-

ple

1 h

(or

Example

1

b),

with

which the

high

B

can

alternate

melodi-

cally-like

the

high

B

in

Example

11g.

The effect

is

rendered

ven more

magical by

the

aspect

of the music omitted on

Example

1

b,

namely

a

continual rillof eight violins on G4-A4. This makesa "fuzzy"effect in

just

the

registralspan

between the first

horn's

G4

and

its

B4. The

effect

tone-paints

Siegfried's

intoxicated

befuddlement,

as he

falls

prey

to the

magic

potion.

It

also

calls

special

attention

o

the "be-fuzzed"

lternation

of

G4 with B4 in the first

horn.

One observes how

carefully

Wagner

ets

us

hearthe voice

leading

for

the

four

horns

in

Example

1

b,

staggering

the relevant

voice-leading

dyads

rhythmically

o

that

we shall

be sure to

appreciate

he

alternation

of

Eb

with

E in

the second

horn,

and the

alternation

f

C with

C#

in

the

thirdhorn,as well as the alternation f G withB in the firsthorn.

Example

1

i,

following

that

ead,

writes

in

open

score

the

instrumen-

tation

for

Example

1

la.

The

instruments,

we

see,

do

not follow

the

"SATB"

arrangement

f

11

a.

Figure

23 formalizes

the

voice-leading

of

Example

1

lh

by

a relation

called

"Vh"between

the first and

second

chords of

Example

1

h,

along

with the

inverserelation

Vh',

formalizing

he

voice-leading

back fromthe

second

chord

to the first.

The

dyads

of

Vh

and

Vh'

model

the individual

horn

parts

of

Example

1

b.

Similarly,

n

Figure

24

the

dyads

of relation

muted horns

(a)

muted

orns

(b)

.te

(c)

(d)

(e)

o

f)p

o

1+3

2+4

1+3

WL-?

RO

O;

P

Example

11

66






Vh

=

{<G,B>,

<C,C#>,

<Eb,E>,

<G,G>}

Vh'

=

{<B,G>,

<C#,C>,

<E,Eb>,

<G,G>}

Figure

23

Vi=

{<D#,E>,

<B,G>, <G#,B>'

<G#,E>}

Vi'=

{<E,D#>,

<G,B>,

<B,G#>,

<E,G#>}

Figure

24

Vi and its inverse

Vi'

model

the individual

horn

parts

of

Example

1

i

(up

to

the two final

chords).

ComparingFigure

24

with

Figure

23,

we can see the

origin

of

the

<G,B> and <B,G>

dyads

in Vh and Vh' (ForgetfulnessMotive): they

come

fromthe

<B,G>

and

<G,B>

dyads

of Vi andVi'

(Tarnhelm

Motive).

Likewise

we can see

the

origin

of the

<Eb,E>

and

<E,Eb>

dyads

in Vh

and

Vh'

(Forgetfulness

Motive):

they

come from the

<D#,E>

and

<E,D#>

dyads

of Vi

and Vi'

(Tarnhelm

Motive).

This is

quite

remarkable,

ince

(as

we saw

in

the

work of

Examples lid

and

1

e),

the

harmonicderiva-

tion

of

one

motive from the

other involves

a

proper

transposition.

Despite

that,

two of the most characteristic

oice-leading dyads

persist

untransposed.

Foremost

n this

respect

is the

<G,B>

of the

first horn

in

Example

1h

(and

Example

1

b),

a

dyad

to which we devotedconsider-

able discussion

earlier,

howing

how

Wagnerputs special emphasis

here

in

a number

of

ways.32

It would

be

impossible

to

express

any

of the

voice-leadings

Vh,

Vh',

Vi,

or

Vi'

as formal mathematical

unctions,

in

the mannerof sections

1

through

9 above. As formal

mathematical

elations,

they

do

pretty

much

what we want them

to for the

analysis:

they

are not bothered

by

doubled

notes

in

the

harmonies,

and

they

have formal

nverserelations hat

model

very aptly

the back-and-forth

alternations of chords

in

the musical

motives.

67






NOTES

1.

I

discuss

the motivic

melodic

gesture

A-B of the Fate Motive at

considerable

length

in

my

book,

Generalized

Musical Intervals

and

Transformations

(New

Haven:Yale

UniversityPress,1986),

184-88.

There

also

discuss

other,related,

aspects

of

act

2,

scene4.

2. If the

passage

were

merelyarpeggiating pwards

onsecutive lose

positions

of

the

f

minor

riad,

he

cycle

of

rising

melodic ntervals

wouldbe

5,

3,

and4 semi-

tones

as the

melody

moved

upwards

rom

C

to

F

to

Ab

andback o C.

The

melodic

interval-cycle

f

Example

b,

4-2-3-4-2-3 s

precisely

he above

cycle,

5-3-4-5-3-

4,

with

a

semitone

ubtractedromeach nterval

o reflect hetriads'

lipping

down

a

semitone

at

each

stage

of

the

progression.

3.

I

haveno idea to whatextent

he

progression

n Crumb's

osmologicalpiece

was

consciously

modeled after the

progression

n

Liszt's

cosmological piece,

or

unconsciouslymodeledafter t,or conceivedndependently.Froma certain"Jun-

gian"point

of

view the

question

s

irrelevant.)

4.

The

Eb

6-chord

maintains

certain

evel

of

dissonance,

which

he

G 5/3 does

not,

against

he

roll

on

E

in

the

timpani-about

which

more

will be said

ater.

5.

Virtually

ll citations f the

MagicSleep

motive

omitthis master troke f

orches-

tration.

Wotan

whose

"so kiisst

er die

Gottheit

on dir"

curiously

choes Hund-

ing's

"Heilig

st mein

Herd")

adences

n

C

minor;

he

Sleep

music

begins

on an

Ab

harmony,

nd he

timpani

ollson

E,

all at the samemoment.

The

Sleep

motive

(quasi-)

equences

o as

to

produce

hordson

Ab,

C,

or

E

at

every

ourthbeat.

The

Sleep

musicends on

an

E

chord,

ixteenhalf-beats fter he

beginning

f Exam-

ple 3a,thetimpani ollingon its E throughout. rom hatE chordon, the music

remains

n

E

major ight hrough

o

the end

of

the

drama,

with

only

a

brief

tonal

excursionat

"Loge,

hir'."

The

high

E,

where the

timpani

rolls,

is

the

bass

E

attained

y

the

rising

bass clarinet

at the end of

Example

3a;

it

is

also the

bass

E

attained

y

the

rising

cellos

at the end of

the entire

Magic Sleep passage.

t

thus

has a

strong

tructural

meaning

n its

specific

register

as

a

high

E.

In

addition,

t

interacts

coustically

with the bass

clarinet

and

later

cello)

notes below

it,

and

with the

clarinet

and

later

viola)

notes

in

its

registral icinity,

n

a

much

more

"magical"

ashion hanwould

a low

E

roll on

the

timpani.

Wotan's cho

of

Hunding

has

always

ntrigued

me.

Perhaps

Wotan

s

thinking,

at this

moment,

about he

fire-shortly-to-be

nhis

"hearth";

erhaps

e is recall-

ing

Fricka's

iradeon

behalfof

Family

Values;

perhaps

oth

references re

oper-

ating,

and

possibly

some others

as

well.

("Heilig

st

mein

Herd,"

fter

all,

can be

heardas

a

transformation

f

the

Ring

Motif.)

6. The

descending pper

trings,

n their

airly

highregister,

ollow

the

profile

of

the

four vowel

formants

n

"Ich

fihle

uft,"

which descend

against

he

rising

funda-

mental

requencies

f the

sung

pitches.

Such

resources or

"counterpoint"

ithin

a

single

line

of

sung

text

are second nature o

good

vocal

composers

and

play-

wrights),

but

nsufficiently

tudied,

t seems to

me,

from

an

acoustical/theoretical

pointof view.

7.

Roger

Sessions,

Harmonic

Practice

(New

York:

Harcourt,

race and

Company,

1951),

398-409.

8. Harmonic

Practice,

406.

9.

ibid.,

407.

68






10.

Benjamin

Boretz discusses related

aspects

of

the fifth

in

the

part

of his

doctoral

dissertation entitled "Musical

Syntax

(II),"

which

appeared

n

Perspectives

ofNew

Music

10,

no.1

(Fall-Winter

1971):

232-70.

Pertinent

discussion

appears

from

p.260

on.

Rather

than

focusing upon

maximally

close voice

leading

as

an

atonal

criterion,

though,

Boretz

is

concerned

with

a

synthetic

development

of a tonal

sys-

tem. The fifth with which he is concerned is

specifically

"the fifth" which divides

a

given

theoretical

tonic octave.

11. Of

course,

when we

know

the

piece

and its row

structure,

we will know that

the

hexachordal articulation of the theme is

highly

significant.

But even

listening

naively

to

the

opening

of the

music

in

itself,

we can hear

without much

effort how

the

twelve-note

theme can

be articulated into two

hexachords. The articulation is

supported locally by

the

accompaniment

texture,

which

projects

trichordal

pitch-

class

segments

from the theme

in

a

duple

metric

grid.

12.

The two

hexachords,

in

their totalities

(rather

than

simply

their

ambitus)

are

related

inversionally,

but

not

transpositionally.

Within the

theme, they

are laid out

in their

totalities so

as

to

support

the

inversional relation

in

the

registers

of

their

various

pitches.

All

that,

however,

does not render

ephemeral

our

hearing

a trans-

positional

fifth-relation between

their

two

ambitus

in

the manner

ust

discussed,

as

well as an inversional relation between

the

two

ambitus,

induced

by

the

hexa-

chordal totalities.

13. One

point might

be worth

making

within the context

of

Examples

7c

through

7f,

without

going

farther

nto

the

movement.

If we listen for

a

T3 relation

between the

incipit

and chant ambitus of

Example

7f,

and

also

listen for a T

(retrograde)

rela-

tion between

the

bracketed

fifth-leaps

of

Example

7e,

we shall attribute marked

structuringfunctions to the transpositional pitch-intervals of 3 and 1 in the con-

text. That seems

suggestive

in

connection

with

the

registral

layout

of the

incipit

and chant

pentachords.

The

incipit pentachord,

G3-Ab3-B3-C4-Eb4

reading

from

the

lowest

pitch up, gives

the

rising pitch-interval

series

1-3-1-3.

The chant

penta-

chord,

Gb4-F4-D4-Db4-Bb3

reading

from

the

highest

pitch

down,

gives

the

same

pitch-interval

series

in

a downward sense.

14.

"Voice

Leading

in

Atonal

Music"

in

Music

Theory

in

Concept

and

Practice,

ed. James

Baker,

David

Beach,

and Jonathan Bernard

(Rochester:

University

of

Rochester

Press,

1977),

237-74. Straus

does not

go

into

the

sorts of numerical

measurements that

the

present

article

will

develop.

Ian Quinn addressed related issues in a lecture, "Fuzzy Transpositionof Pitch-

Class

Sets,"

delivered at

the

meeting

of

the New

England

Conference

of

Music

Theorists

in

Amherst on March

30,

1996.

The

attitude

toward

listener

cognition

underlying

Quinn's

"fuzzy transposition"

differs from

Straus's

"near

transposi-

tion"

and

my "pseudo

transposition"

(as

I shall later call

it).

Quinn

formalizes,

addresses,

and

measures

an

attitude of "unsureness"or

"vagueness"

on

a

listener's

part.

Straus and

I

implicitly

conceive

a listener

who hears

(with

clarity)

a

certain

amount of exact

transposition

going

on,

and

a

certain

amount of

departure

from

that

exactness-i.e. one

or more

"wrong

notes,"

and

a

certain

preponderance

of

"right

notes."

15.

We

must

recall

that the definition

of

"voice-leading

function"

(Definition 1.1)

did

not

demand that

every

pc

of the set

"Y"

be a

function

value,

nor did it

prohibit

some

members of

"Y"

from

appearing

more

than

once

in

a

layout

like

that of

Fig-

ure 3.

Some readers

may

feel that the

voice-leading analysis

of

Figure

2

is some-

69






how

"better"han he

analysis

of

Figure

3,

because he function

f

Figure

2

maps

trichord

i)

onto

all

of trichord

ii).

I

believe

t

would

be

methodologically

ubi-

ous to make

such

an abstract alue

udgment.

As we shall

see,

the

major

hirds

{B,G)

and

{

A,F},

which

are

brought

nto

particular

rominence y

the

voice lead-

ingof Figure3, play

an

important

onstructiveole

through

he

passage.

t

would

be

awkward, think,

o

devalue his

hearing

priori.

n

section

9

we shall

develop

special erminology

or

describing

he

property

f

Figure

2

just

noted.Our ermi-

nology,

however,

will be value-free

n

the abstract.

In

the

music,

Figure

2 is a "better"

nalysis

o the extent

hatwe value he com-

plete

three-note ontourmotiveas

a

structuring

orce.

Figure

3 is "better"o the

extent hat

we value he

structuring

orce of the

major

hirds

{

G,B

}

and

{

F,A

.

I

do

not think our discourse

gains anythingby

invoking

he terms "better"

nd

"worse"

n

this

sortof

situation,

aluing

he

motive

more han he

major

hirds,

r

vice-versa,

nd

I

think he

terminology

makes t too

easy

for

us to dismiss

his

or

thataspectof ourhearingas inconsequential, evaluingt in advancensteadof

exploring

t

fully.

16.

Quinn's

"fuzzy

ransposition"

omes into

the

pictureagain

here,

o

the extent

we

are

unclear,

as

to

whether

we are

hearing"something

ike

T(-3)"

here,

or

"some-

thing

ike

T(-2)."

In the

present

ontext,

hough,

am

eschewing

he

interesting

issuesraised

by

that

approach.

ather,

am

mplicitly

upposing

hat

we hearboth

relationships

with

some

exactness),

nd hateach relation

ontributes

omething

specific

o our

istening xperience.

My implicit upposition

implifies

discussion

of the

passage,

butI

am

not

surehow

successfully

t can be maintained

s a mat-

ter of abstract

sychology.

Nor

am I sure

about

Quinn's

mplicit

psychological

stance, or thatmatter.)

17.The

sense of

"only

one

semitone

off"

is the

same,

whether ne

hears richord

i)

progressing

o

trichord

ii)

by

"almost

T(-3)"

or

"almost

T(-2)."

Hence

the sense

of

"only

one

semitone

off,"

o

be formalized

hortly,

rovides

omething

nfuzzy

(in

Quinn's

erminology)

bout

a listener's

erception

f alternative

ossible

rela-

tionships

n this situation.

18.

The offset

number

has an

interesting

elation o

a

measurement

alled

"diver-

gence,"

developed

by

Anton

Vishio

n a

forthcoming

issertation.

ishio's

"diver-

gence"

measures

departure

rom

perfect

mirrorwise

ontrary

motion

n voice-

leading;

our "offset"

measures

departure

rom

perfect

parallel

motion.

Vishio

(usually)

measures is

divergence

s a

pitch-class

ntervalmod12,wherewe are

(usually)

measuring

ffset

as

an

absolute

positive)pitch

nterval,

positive

ount-

ing-number.

hose

featuresof the

systematics

ould

be madesomewhat

lexible

in

both

directions.

19.

Notes

that

are

N

semitones

distant rom

each other

aretransformed

y

T,

or

by

I,

intonotes

that

are

N

semitones

distant

romeachother.

That eature

n

fact

can be

used to define

the

group

of

transposition

nd inversion

operations.

When

we

are

counting

emitone-distances,

e

cannot reatother

pc operations

n a similar

fashion.

20. Our heoreticalpparatus

ill

thereby

ome

ntocontact

with

a

good

dealof

recent

important

work

on

scales,

temperament,

nd intonation.To

be cited

in this con-

nection

are

Easley

Blackwood,

The

Structure

of Recognizable

Diatonic

Tunings

(Princeton:

rinceton

University

Press,

1985);

also John

Clough

and

JackDou-

thett,

"Maximally

ven

Sets,"

Journal

of

Music

Theory

35

(1991),

93-173;

also

70






John

Clough,

Jack

Douthett, N.Ramanathan,

and Lewis

Rowell,

"Early Hepta-

tonic Scales and Recent Diatonic

Theory,"

Music

Theory Spectrum

15

(Spring

1993),

36-58;

also Mark

Lindley

and Ronald

Turner-Smith,

Mathematical Mod-

els

of

Musical Scales: a New

Approach

(Bonn:

Verlag

ftir

systematische

Musik-

wissenschaft, 1993);

also

Eytan Agmon,

"Coherent

Tone-Systems:

A

Study

in

the

Theory

of

Diatonicism,"

Journal

of

Music

Theory

40.1

(Spring

1996),

39-58.

21.

Technically,

we could

consider

as a

"pcset" any

topologically

closed subset

of

the

circular

continuum

modulo

the octave. Readers

who

have a

background

in

point-

set

topology,

and

in

measure

theory,

can

explore

the

matter on

their

own.

22.

"Maximally

Even

Sets,"

cited in note

19.

23.

The state

of affairs would not be

different,

if

we tried

transposing

EQHEP

by

intervals other than increments

of 1/84.

24.

More

properly

speaking,

to a

maximally

close note of

EQDOD.

As

long

as a

trans-

position

of

EQHEP

has

a

common

tone with

EQDOD,

there will

be

only

one such

maximally close note. (Thatis because 7 and 12 have no common properdivisors.)

If

we were to

transpose

EQHEP

by exactly

1/24

of

an

octave,

the

transposed

set

would contain the

tone labeled

"3

1/2,"

equally

distant

from

0

below and

7

above.

25.

In some

sense,

we

might

say

that

Figure

14a is a

"Mixolydian"

structure:

while the

tone

common to

EQHEP

and

EQDOD

is a

"C,"

the

upshifted

image

of

EQHEP

gives

a

7-35 set whose

pc

content coincides

with that of a

functional "G

major"

scale.

26.

Blackwood

(Diatonic

Tunings)

points

out that the

heptad

generated

by

eleven-

nineteenths of an

octave,

in

nineteen-tone

equal temperament,

coincides

almost

exactly

with a

third-comma mean-tone

scale. Two

to the

(11/19)

power

is

1.49376

to five significant decimal places; the cube root of 10/3 is 1.49380. The two sys-

tems

are of course

conceptually

quite

different in

nature.

27. One

notes some

characteristic

tension between the

would-be

tonicity

of

UT,

and

the

formal

referentiality

of

RE,

involving

a

special

symmetry.

28.

Again

one

notes the

tension between the

would-be

tonicity

of

UT/ut,

and the

for-

mal

referentiality

of RE/re. The "best"

intonational match of the

two scales

has

a

certain

Dorian

character.The

formal

referentiality

of RE/re here

involves

not

only

the

abstract

symmetry

of

Figure

19,

but

also the

principle

of

maximally

close

voice

leading,

and the

principle

of

minimizing

the

amount

of

retuning

involved,

in

traversing

a

maximally

close

voice-leading map.

29.

I

keep

italicizing

"in such a context"" and the like because I do not intend

any

abstract

value

judgment

in

this matter. An

abstract

ormal

pseudo-transposition

need not

"sound

intuitively"

like a

straight

transposition,

in

order to have a struc-

tural

effect in a

passage.

In this

connection the

reader can review

note

15,

and the

Schoenberg

analysis

surrounding

it

(especially Examples

9c and

9d).

The

partic-

ular

Webern

passage

whose

analysis

will

soon

appear

does

emphasize

the norma-

tive

character

(for

Webern's

texture)

of chords that

contain five

distinct

pitch

classes.

30.

Besides the

structuring

nterval of

3

semitones

(up

and

down)

in

the

cello

melody,

another

interval stands

out

as

having particularstructuring

force there: it is

the

interval

0,

heard

between

successive

soundings

of the

note

E,

and between

suc-

cessive

soundings

of the note

C#.

The

interval

0 is

reflected

in

maximally

uniform

surjective

aspects

of

the

accompanimental

homophony

too: it

measures

the offset

from

straight transposition

between

chords 1

and

2;

it also

measures

the

offset

71






from

straight transposition

between chords

3 and

4,

and the offset from

straight

transposition

between chords

4

and 5.

Here,

the "offset of 0

semitones

"

occurs

because chord

2 is a

straight

transposition

of chord

1,

and

so forth.

The

parallel

registral

and instrumental voice

leading

between each

transpositionally

related

pair

of

successive chords manifests

(in

our

technical

sense)

both

maximally

uni-

form

and

maximally

uniform

surjective

voice

leading,

with

an

offset of 0

semi-

tones. In a more

colloquial

sort of

language,

we can

say

that the

recurring

pitches

in the cello are

related

in

this

way

to

exactly

parallel

voice

leading

in

the

upper

strings,

where that occurs

(between

chords

1

and

2,

between chords

3

and

4,

between

chords

4 and

5).

31.

Example

I lb is marked

"Sehr

langsam." Example

I1

a is

usually

performed

"Sehr

langsam"-I

have

always

heard it

at that

tempo-though

no edition

I have seen

indicates

any departure

from

the

"Hastig"

tempo

that

precedes

it. There is a

big

fermata

just

before

the

passage,

but that need not mean a

tempo change.

It is cer-

tainly there as a cue to the musicians, so they will allow time for the stage busi-

ness at

hand:

Mime

drops

the

Tarnhelm;

Alberich

picks

it

up ("hastig")

and exam-

ines it

carefully.

We

shall

later discuss

aspects

of the instrumentation

not manifest on

Examples

1 a and

1

b,

namely

the

disposition

of the four horn

parts

in

Example

I1

a,

and the

trill

by eight

violins on

G4-A4,

not

transcribed on

Example

I1

b.

32.

To be

sure,

one should

point

out that

the second and third horn

parts

of

Example

I li reflect a standard

technique

of brass

writing.

Brass

players--especially

if

playing

with crooks on

natural

instruments,

as

Wagner

notates

them-do

not like

to

play

dieses,

such as

G#-G4-G#-G4

nd so forth

in

the

Tarnhelmmusic.

Wagner's

voice-leading, in Example I li, enables the second and third horns to avoid the

diesis.

(Strauss

uses the

technique

to brilliant

effect

in

the now-notorious

opening

fanfare of

Zarathustra,

avoiding

the

major-minor

diesis in the

trumpet parts.)

The

origin

of

the

<G,B>

and

<B,G>

dyads

in the

voice-leading

of

Example

I

lii

may

well have

been that

straightforward

a

matter,

when

Wagner

was

orchestrating

Das

Rheingold.

By

the time he came to

compose

and orchestrate

Gotterdaim-

merung

though, many

years

later,

I believe

that he

was sensitive

to

the

potential

of

the

<G,B>

dyad

as structural

compositional

material,

isolating

and

highlighting

it

as

a

voice-leading

element

during

the

Forgetfulness

Motive.

As a

conductor,

Wag-

ner

might

well have

sung

over the

individual horn

parts

of

the TarnhelmMotive to

himself

while

orchestrating-or

even

while

composing-the

motive.

He

might

have

paid particular

attention

to

the

B-Gl-B-G0

(etc.)

in the third horn

there,

because

there

is a definite

danger

that

the

player

will

fluff the

Gis.

On a natural

horn

in

E

(which

Wagner stipulates

for

the music

here),

the concert

Gk

has to

be

lipped

down from

G#,

and

the

player

could

well be distracted

by

the

G~s

ust

sounded

by

the other

horns

immediately preceding

the

G4.

The diesis

I have

written

in

the second

horn

part

of

Example

I

lb is not

in

Wag-

ner's

score,

which

gives

(concert)

D#-E-D#-E.

(The

second horn

here is written

as

horn

in

E,

just

like

the

first horn

in

Example

I

lii,

with its

D#-E-D#-E

an

octave

higher.)

lewin - some ideas about voice-leading between pcsets

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