serialism as practice the argenteuil hall

Ha

nd

ou

t for

“Ea

rly Se

rialis

ms in

the

Un

ited

Sta

tes”

Ric

ha

rd H

erm

an

n, U

niv. o

f Ne

w M

exic

o,

ha

rha

r@u

nm

.ed

u

in th

e p

an

el d

iscu

ssio

n

Th

eo

ries a

nd

Ae

sth

etic

s: A

n H

isto

rica

l Re

co

nsid

era

tion

of S

eria

lism

as P

ractic

e

SM

T M

ee

ting

, Mo

ntré

al, 8

-11

pm

, 31

Oct. 2

00

9

the

Arg

en

teu

il Ha

ll

NB

: co

pyrig

hte

d m

ate

rials

exce

rpte

d w

ithin

are

pro

vide

d u

nd

er th

e

“fair u

se

” pro

visio

ns o

f the

Un

ited

Sta

tes C

op

yrigh

t Co

de

.

Table 1

Five Technical M

yths on Am

erican Serialism Inspired by Straus (

JSA

M 2008 vol 2, no. 3)

Examined in Som

e Prominent T

heoretic/Com

positional Sources for Early Am

erican Serialism

1Richard Hermann, Univ. of New M

exico, harhar@unm

.edu, “Early Serialisms in the United States,” in the panel discussion

Theories and Aesthetics: An Historical Reconsideration of Serialism as Practice, SM

T Meeting, M

ontréal 2009

Co

well,

Yasser,

Sch

oen

berg

,H

ind

em

ithL

eib

ow

itz,

Ru

fer,

New

Mu

sic

al

Evolv

ing

Sty

le a

nd I

dea

Craft M

usic

Schoen

berg

Com

p. w

/1

2 T

on

es

* *

* *

* *

Resou

rces (

‘30

)T

on

ality

(‘3

2)

(‘5

0, le

ctu

res ‘3

4-‘4

6)

Com

p. v

. I (

‘42

)School (

‘49

)(‘5

2/

’54

)

Seria

lism

As

Itʼs one techniqueno m

entionHe argues for historical

See 4th Tech.

“the real music

Evolutionary/historical rhetoric

Co

mp

ositio

nal N

orm

among several

of this termnecessity but recognizes

Myth below;

of our time” and

(14-23)

and not mentioned

other practice. (216-7, 244)perhaps itʼs

“the international

* *

* *

* *

by name.

”...no tone is repeatednot possible.

language of music”

Ro

w a

nd

PC

“..in which each see below within the series and that

*** (v) Evolutionary rhetoric

No

n-R

ep

etitio

ntone is independent...”

it uses all 12 tones of theno com

ment

“...a subtle device”no repetition within the series

no

8ves, P

C e

qu

ality

(41) comm

ent on 8vechrom

. scale...” (218)of repeating som

e...im

mediate repetitions of

* *

* *

* *

(39)“avoidance of 8ve dbl” (219)

of the tones...” (122) notes are ... possible” (87)

Seria

l as R

igo

r“...retrograde, inverse

”,,,a ʻfreeʼ use***

“group II [aggreg.]”consist of

“avoids doubling 8ve” (90)

or P

urtiy

melodic line, etc.” (41)

of the twelve-tone“their order in the set has

...must get along

successive or***

“consistency... straight chord, as exem

plified always been strictly ob-without the tones

simultaneous

thematic, expository,

forward logic” (41)by Schönberg and

-served. ...slight digressione�, f, a�, and b�.”

presentations ofsource, throughout, but

his group or to poly- ..when the set had already(219)

the complete row

no esp. special claims

* *

* *

* *

phonic constructions become fam

iliar to the ear.”a fragm

ent of it,of rigor beyond that

Seria

l as A

nti-T

on

al

emphasizes interval

based upon it” (351) (226)“tonal analysis

or several forms of

***

combinations, ratios,

***step already taken by

given here.”it.” (102) ***

”In [tonalityʼs] place, as

harm. series. No

“[Schoenberg] theDebussy, W

agnerof Schoenbergʼs

”The Suspensionthe bearer of a new tonality,

comm

ent on tonalitym

ost extreme of the (216)

Op. 33 a. (219)

of the Tonalthe twelve-note series.” (23)

* *

* *

* *

atonalists” (250*)System

” (74)***

12

x1

2 M

atr

ix

“...an ingenious ***

no comm

ent“M

agic Square,... same

as th

e S

eria

l Sp

ace

method of geo-

no mention

no mention

48 row forms (102)

letters form sam

e words ...

metric diagram

...” (41)(O

, I, R, RI)sam

e meaning vert & hrz”(48)

Table 1

Five Technical M

yths on Am

erican Serialism Inspired by Straus (

JSA

M 2008 vol 2, no. 3)

Examined in Som

e Prominent T

heoretic/Com

positional Sources for Early Am

erican Serialism

2Richard Hermann, Univ. of New M

exico, harhar@unm



T Meeting, M

ontréal 2009

Dallin

,K

ren

ek

,P

ersic

hetti,

Bab

bitt,

Martin

o,

Perle

,

20

-Cen

t.Com

p.

Exte

nts

& L

imits

20

th-Cen

t Harm

.(‘5

5, ‘6

0, ‘6

1)

Sou

rce S

et

Seria

l Com

p. &

Ato

nal

* *

* *

* *

(‘5

7, ‘6

4, ‘7

4)

Seria

l Tech. (

‘60

)(‘6

0)

(‘6

1)

(‘6

2)

Seria

lism

As

one optionissue not addressed

one optionother options

issue not addressedissue not addressed

Co

mp

ositio

nal N

orm

among others

among others

dominate (ʻ55, 38)

* *

* *

* *

No

n-R

ep

etitio

n,

“Octaves and

not addressedno m

ention”weighted aggregates”

”No note appears more than

no

8ves,

unisons areoperator yields pc

issue not addressedonce within the set.” (3)

Pc e

qu

ality

generally duplication (ʻ61, 83)

Here set = row-class mem

ber.

* *

* *

* *

avoided,” (ʻ74, 196)“...hexachord as

Seria

l as R

igo

r“...the strict

strict, multi-dim

ensional”A basic order

an independentm

ath used, but deals“An unam

biguous ordering is

or P

urtiy

adherence to the(aka Total or Integral

of tones, all twelve unit led to [Schoen-with orderings both

assumed; but the degree to

(“M

ath

” to

o)

order of the seriesSerialism

), “premeditated”

or fewer, may be

-bergʼs] using itlinear and harm

onicwhich this ordering actually

in all parts.” (ʻ74, 196)used as a unifying

without regard toof unordered partitions

determines the general m

us-* *

* *

* *

basis...” (262)fixed ordering...”

-ical procedures varies greatly

Seria

l as A

nti-T

on

al

“...methods of

not specificallyChordal form

ations(ʻ55, 42)

from one work to another,” (2)

tonal organizationaddressed but

that arise from”Negatively, there

issue not addressed***

ceased to function.strongly im

pliedserial...have little

was the motivation

“Atonality ... liberate[s]... the

To replace them,

or no function in... against the trans-

chrom. scale from

... functionSchoenberg devised...”

a scalar tonal-ference to 12-tone

-al associations.”

(ʻ74, 190)sense.” (262)

comp. of criteria

“[Schoenbergʼs 12-tone] * *

* *

* *

belonging to triadicm

ethod [had] relevance to

12

x1

2 M

atr

ix

displays one forUses various rotational

no mention

music.” (ʻ55, 40)

noneprobs. of atonal com

p.”(1-2)

as th

e S

eria

l Sp

ace

Schoenbergʼsoperators that vitiate

“set matrix, containing

***4

th Qt (ʻ74, 195)

this sort of matrix

all row forms” (ʻ61, 82 )

P, I, R, RI = 48 in “row-class”

Table 2

Issues of Three Early Sophisticated C

ritics of Serialism W

riting in the United States Identified and Located

Reti, !

Cone, "

Meyer,a

Tonality in M

odern Music

Analysis T

odayM

usic, the Arts, and Ideas

(‘62 earlier eds. ‘58, ‘60)

(‘60)(‘67 som

e essays 50s & early 60s)

****************************

(1) Nature: acoustical m

aterials (65)

not addressed, #(245-48)

of music, contrary to

(2) Against isom

orphism of pitch

not addressednot addressed, #

(248-53) and tim

e

(3) Analogies to m

ath and physics(150)

(176)(253-62)

models specious

(4) False argument of

(56)not addressed

(263-65) “historical necessity”

(5) Cognitively ill form

ed(“57”)

not addressed(266-74)

(6) Quite D

ifficult to Learn(274-79)

(7) Serial Theory &

Analysis

(58)(174)

im

plied in (2) and (5) aboveD

escriptive and Prescriptivebut not explanatory

(8) Too far from

the traditional(150-51, im

plied)not addressed, #

# His “Beyond Analysis” PNM

6, 1 (A-W ʻ67): 33-51

! 12 counts Schoenberg,

" Com

ment on several

a He presents no analysis of serial m

usic.recom

poses passages from Schoenbergʼs O

p. 33a4

th Qt opening, opening

bars each of the openingand W

ebernʼs Op. 27 by reflecting the originals in

of Berg, Vln. Cto.& Boulez,to Schoenbergʼs O

p. 33apitch space (m

irror symm

etry, aka inversion).1

st Pno. Sonata opening(registrally dispersed, stepwise m

elody)Also takes on total serialism

in this way.and locating a structural downbeat in bar 12 O

p. 27, III.

Richard Hermann, Univ. of New M

exico, harhar@unm



T Meeting, M

ontréal 2009

Richard Hermann, Univ. of New Mexico, [email protected], “Early Serialisms in the United States,” in the panel discussion Theories and Aesthetics: An Historical Reconsideration of Serialism as Practice, SMT Meeting, Montréal 2009

Figure 1 Annotated Exs. 5 and 6 from Schoenberg’s “Composition with Twelve-Tones” (1950)

Semicombinatoriality at Index 5 and not at Schoenberg’s “inversion of Basic Set down a perfect 5th, I5” NB: I8 here means the 8ve away from the Basic Set.

Row-forms use comment: The 1st Theme is in the top staff with its antecedent presented as the 1st H and the consequent presented as the 2nd H of the Basic Set (henceforth BS). It is accompanied in thebottom two staffs by the same two hexachordspermuted via a simple rotation. Note thatthe internal orderings of the Hs are maintained.

With only the exception of the pc dyad {3,7}in the middle staff’s 3rd bar, both the BS row-form and its accompanying Hexachordally rotated version of the BS have their pitches presented in asimple linear fashion.

Texture notes: Superimposition (here a variant of the traditional theme and accompaniment homophony)

fig. 1, page 1

Notes on row-classmembers: The BS (basic set) and the BStransposed down a P5thhold 2 pcs invariant.

The BS and the inversionof the BS down and 8vehold four pcs invariantper Hexachord. (Hence-forth hexachord is abbreviated as H).

etc.


fig. 1, page 2


Notes on row-class members: cross-related 8ves occur as the result of the near simultaneous presentatins of BS and its inversion down an 8ve, “I8.” There is no effort to project the pitchesof the BS and its “I8” in pitch-space as mutual reflections. such as in Webern’s practice.

Texture notes: Superimposition (here in a traditional three voice free counterpoint)

Notes on row-class members: Here the retrograde of the BS trans-posed down a major 2nd is presented twice in succession.

Note that the top staff is a fundamental bass-like analysis ofthe temporal ordering of the row-forms in the passage.

Oval enclosed harmonic dyads show that the odering is a partial ordering of the unfolding of the row-form. One would need to know the ordering of the BS beforehand to assign order numbers to this passage Theorder numbers of these dyads appearsto form the pattern of two dyads withthe smaller ordinals above (in p-space)the higher ordinals followed by one dyadwith the higher ordinal above the lowerordinal. This pattern does not partition out an associated ic design.

Note how in the 2nd row-form thatorder position 7 of the penultimate baris out of order.

Texture notes: Juxtaposition successively of single disjunct row-forms in an antiphonal design.


fig. 1, page 3


Notes on row-class members: A single BS row-form is partitioned into two voices viacombinations of adjacent and non-adjacemt order positions. The BS is repeated and repartioned into two via another assignment of adjacent and non-adjeent order positions.

Again, the bottom staff is a fundamental bass-like analysis of the order positions.

The 1st segment of the horn’s pcs forms a member of set-class 4-23 {0257}, and both voice’s 1st segments freature ic palindromes: horn <2,3,2>; bsn <2221222>.

Texture notes: A duo in free counterpoint featuring alternating oblique motion.

Notes on row-class members: R hexachordal combinatoriality is featuredwith the I and RI forms.

Note the unavoidable cross-related 8ves between the end of the 3rdbar and the start of the 4th.

Texture notes: Another duo in free counterpoint featuring alternating oblique motion.


fig. 1, page 4


Notes on row-class members: Schoenberg presents two pairs of row-forms each transposed by a diminished 5th.The first involves the BS, and the second does theinversion of the BS starting on thesame pc, B-flat.

Due to the ic 6 between some adjacent pcs in the BS, this results in swapped appearance in the transposed BS by ic 6. This is found across the bar line between the first two measures of the excerpt. Hence no com-binatoriality here.

Schoenberg marks the tetrachord partitions with numbered brackets. Note how in the transposed BS the2nd and 3rd tetrachordal partitions are presented simultaneously and they preserve their internal orderings.

Texture notes: A two voice point of imitation followed by a three voice free homophonic passage.

Notes on row-class members: Here the three tetrachordalpartitions are presented such that the 1st two tetrachordalcells of the partitioned BS are swapped in order. Each cell retains its internal order.

Texture Notes: Overlap is featured between the 2nd and 1st tetrachordal cells of the partition and also between the1st and 3rd cells. This is a free contrapuntal texture in 5parts.

NB: Schoenberg calls these tetrachordal partitions thatmain tain their internal ordering “small sets.”

Notes on row-class members: the right hand staff presents the BS ininversion and transposed a diminished5th. The left hand presents the BS.They are arranged in time such thatthe 2nd H of the BS occurs simul-taneously with the 1st H of the I form.No combinatoriality results as thesesuperimposed Hs’ set intersection ispcs {9, e}. Texture notes: This passage presents a two voice canon in inversion.



fig. 1, page 5

Notes on row-class members: Schoenberg writes about Ex. 16:“I used ..a device...derived fromdouble counterpoint of the tenthand twelfth, which allows the addition of parallel thirds to every part involved. By transposing BSa third up (BS3) and INV a thirddown (INV3), ...” The passage below in Ex. 17 shows its use inthe 5th variation.

Of course as can be quickly confirmed, such doubling precludes combinatorialitywhen the segment includes theinterval of doubling as this BSdoes.

While the cello chordal accompan-iment does not maintain ordering of INV starting with G’s as myhandwritten order numbers show, the pc diagram beneath shows thatreading the three chords downwardsreveals a permutation that suggestsa conscious design.

Texture notes: A traditional homophonic theme and accompaniment texture

Texture notes: Antiphonal.

Notes on row-class members: See Ex. 16 above.

pcs expected linear order: 7 1 e 2 0 8 3 4 t 9 6 5

pcs in top to bottom order: 2 e 7 1 0 3 8 4 t 5 9 6

1st chord 2nd chord 3rd

Table 3a

Some Elem

entary Row

-Class R

ealization-Classes derived and som

ewhat extended from

Schoenberg’s“C

omposition w

ith Tw

elve-Tones” (1950) as a influence for W

uorinen’s Row

-Form R

ealization Strategies in his Piano Variations (1963)

Richard H

ermann, U

niv. of New

Mexico, harhar@

unm.edu, “Early Serialism

s in the United States,” in the panel discussion

Theories and Aesthetics: An Historical R

econsideration of Serialism as Practice, SM

T Meeting, M

ontréal 200

Music segm

ent assignment to a realization-class is an ordered quadruple consisting of the follow

ing:1

. Pc S

ub

dim

en

sio

n 1

mem

ber a

ssig

nm

en

t;

2. P

c S

ub

dim

en

sio

n 2

mem

ber a

ssig

nm

en

t (n

ull o

r s

uccessiv

e ju

xta

po

sitio

n fo

r s

ing

le r

ow

-form

presen

tatio

ns);

3 S

eq

uen

tial L

ayo

ut fo

r S

ing

ly P

resen

ted

Ro

w-F

orm

mem

ber a

ssig

nm

en

ts; a

nd

4 S

eq

uen

tial L

ayo

ut fo

r S

imu

ltan

eo

usly

Presen

ted

Ro

w-F

orm

s m

em

ber a

ssig

nm

en

ts.

PC Subdim

ension 1, kinds of mem

bership for Individual Row

-Form Presentations

(includes minim

ally incomplete presentations)

Ord

ered

Ord

ered

Partia

lly O

rd

ered

Ord

ered

Partitio

n C

ells

Perm

ute

d P

artitio

n C

ells

Perm

ute

d P

artitio

n C

ells

linearly

vertic

ally

(mix of linear &

vertical)b

ut u

no

rd

ered

with

in c

ells

bu

t cell o

rd

er m

ain

tain

ed

*

& p

erm

ute

d w

ithin

cells

PC Subdim

ension 2, kinds of mem

bership for Aggregate Results for Sim

ultaneously Presented Row

-Forms

Even

Co

mb

inato

ria

lityW

eig

hte

d A

ggregate

s w

ith s

ign

s

Weig

hte

d A

gg

reg

ate

s w

itho

ut

Ro

w-fo

rm

s d

o n

ot a

pp

ear a

s p

art

of lo

cal d

esig

n p

ro

cesses e

ng

ag

ing

sig

ns o

f local d

esig

n p

ro

cesses

of a

n a

gg

. co

mp

letio

n d

ue to

min

imal p

c d

up

licatio

ns**

en

gag

ing

min

imal p

c d

up

licatio

ns

mo

re th

an

min

imal p

c d

up

s.

Sequential Layout, kinds of mem

bership for successive Singly Presented Row

-Forms

(E

ach

pair

, trip

let, q

uarte

t, etc

. of r

ow

-form

s r

eceiv

es a

n a

ssig

nm

en

t)

Jux

tap

osed

Overla

pp

ed

***

(temporal gap betw

een row-form

s)

Sequential Layout, kinds of mem

bership for Simultaneously Presented R

ow-Form

s (E

ach

pair

, trip

let, q

uarte

t, etc

. of r

ow

-form

s r

eceiv

es a

n a

ssig

nm

en

t)

Su

perim

po

sitio

nD

oveta

iled

(precisely or roughly starting and stopping together)(not precisely or roughly starting together)

NB:

(1) T

he r

ealiz

atio

n-c

lass is

a d

escrip

tion

of th

e m

usic

al s

eg

men

t’s d

istin

ctiv

e fe

atu

res a

nd

the p

ossib

le c

om

bin

atio

ns r

ep

resen

t the e

ntitie

s o

f a

no

min

al s

pace.

(2) R

ota

tion

is n

ot d

iscu

ssed

in S

ch

oen

berg

’s a

rtic

le, a

nd

su

rp

ris

ing

ly, I

have n

ot fo

un

d it o

perativ

e in

Wu

orin

en

’s v

aria

tion

s; th

us, it d

oes n

ot

ap

pear in

this

cla

ssific

atio

n s

ch

em

e. T

he r

ow

-cla

sses a

re th

us d

efin

ed

by th

e c

lassic

al T

TO

s.

(3) G

estu

res s

uch

as th

em

atic

, ath

em

atic

, imita

tion

, can

on

, an

tiph

on

al a

nd

so

forth

are c

on

sid

ered

asp

ects

of s

tyle

an

d n

ot s

eria

l pro

pertie

s.

* T

his

is S

ch

oen

berg

’s “

sm

all s

ets

.” *

* E

.g. W

eb

ern

’s in

varia

nce r

ealiz

atio

ns w

ith e

ven

ind

ex

nu

mb

ers *

**M

ultip

le O

rd

er P

ositio

n N

um

bers

Table 3bSegm

ent Assignm

ents to Classical Serial R

ealization-Classes for som

e of Schoenberg’s examples from

his “C

omposition w

ith Twelve-Tones” (1950) and excerpts from

Wuorinen’s Piano Variations (1963)

1R

ichard

Herm

ann, U

niv. o

f New

Mexic

o, h

arh

ar@

un

m.e

du, “E

arly

Seria

lism

s in

the U

nite

d S

tate

s,” in

the p

anel d

iscussio

nT

heorie

s a

nd A

esth

etic

s: A

n H

isto

rical R

eco

nsid

era

tion o

f Seria

lism

as P

ractic

e, S

MT

Meetin

g, M

ontré

al 2

009

PC Subdim

ension 1PC

Subdimension 2

Sequential LayoutSequential Layout,

Indiv. Row

-Forms

Simult R

-Fs, Aggregate Status

Single Row

-Forms

Simultaneous R

ow-Form

s

Com

poser, Location

Schoenberg, Ex. 6 (WW

Qnt)

top stave:O

rd. linearly Even C

ombinatoriality

(not applicable,Superim

posed

bottom staves:

Partially Ord.

excerpt too short)


Qnt)

top stave:O

rd. linearlyN

ot Aggregate C

ompleting

JuxtaposedSuperim

posed

bottom stave:

Ord. linearly

(considered R-F polyphonically)

Juxtaposed


Qnt)

bottom staves:

Partially Ord.

(null)Juxtaposed

(null)


Qnt):

Ord. linearly

(null)Juxtaposed

(null)


Qnt):

top stave:O

rd. linearly“Even C

ombinatoriality”

(na)D

ovetailed

bottom stave:

Ord. linearly

(na)

Schoenberg, Ex. 13a (Suite)

top stave:O

rd. linearlyW

eighted Aggregate

(na)Superim

posed

bottom stave:

Permuted Part. C

ellsw

/o serial design processes(na)

Ordered w

ithinengaging m

inimal pc dups.

Schoenberg, Ex. 15 (Suite)O

rd. linearly(null)

(na)(na)

(but tetra cells rotated & overlapped)

Schoenberg, Ex. 15a (Suite)

top stave:O

rd. linearlyW

eighted Aggregate

(na)D

ovetailed

bottom stave:

Ord. linearly

w/o serial design processes

Table 3bSegm

ent Assignm

ents to Classical Serial R

ealization-Classes for som

e of Schoenberg’s examples from

his “C

omposition w

ith Twelve-Tones” (1950) and excerpts from

Wuorinen’s Piano Variations (1963)

2R

ichard

Herm

ann, U

niv. o

f New

Mexic

o, h

arh

ar@

un

m.e

du, “E

arly

Seria

lism

s in

the U

nite

d S

tate

s,” in

the p

anel d

iscussio

nT

heorie

s a

nd A

esth

etic

s: A

n H

isto

rical R

eco

nsid

era

tion o

f Seria

lism

as P

ractic

e, S

MT

Meetin

g, M

ontré

al 2

009

PC Subdim

ension 1PC

Subdimension 2

Sequential LayoutSequential Layout,

Indiv. Row

-Forms

Simult R

-Fs, Aggregate Status

Single Row

-Forms

Simultaneous R

ow-Form

s

Com

poser, Location

Schoenberg, Ex. 16a (Orch Var)

top stave:O

rd. linearlyEven C

ombinatoriality

(na)Superim

posed

bottom stave:

Ord. vertically

Schoenberg, Ex. 17 (Orch Var)

top stave:O

rd. linearlyW

eighted Aggregate

(na)Superim

posed

bottom stave:

Ord. linearly


(only first Hs presented &

repeated)

***********************

Wuorinen, m

ms. 1-6, P2:

Partially Ord.

(null)O

verlapped(null)

Wuorinen, m

m. 5-8 (m

. 9 silent)

RI2:

Partially Ord.

Weighted A

ggregateO

verlappedD

ovetailed

P2:Partially O

rd.w

/o serial design processesO

verlapped

Wuorinen, m

. 10, P2:O

rd. vertically(null)

Juxtaposed(null)

Wuorinen, m

m. 36-39

R6:

Partially Ord.

(null, chordal presentationsJuxtaposed

Superimposed

(H1s m

ult ord pos

RI7:

Partially Ord.

H1s, share all pcs, then H

2s interleaved)Juxtaposed

then H2s interleaved)

Wuorinen, m

m. 49-51

I7, 1st trichord:

Ord. vertically

Weighted A

ggregateO

verlappedSuperim

posed

RI0:

Partially Ord.


Overlapped

P5, ord pos 2-5:O

rd. linearlyJuxtaposed

(a juxtaposed insert)

Wuorinen, m

m. 51-54:

[4 juxtaposed segments of card. 3 that could have m

ultiple row-form

s mem

bership. They do not create an aggregate, 7 pcs]

Wuorinen, m

. 56, P6:O

rd. vertically(null)

Juxtaposed(null)

Wuorinen, m

m. 186-90:

[row-form

segments easily assigned superim

posed, juxtaposed, and overlapped with segm

ents with m

ultiple possible assignments superim

posed, 5 segments in total,

this passage starts one bar before the start of the 12th var. ]


Table 4 p. 1 of 2


Table 4 p. 2 of 2

Richard Hermann, Prof. of Music, Univ. of New [email protected], not for distribution without written permission

Discrete (non-rotational)

Set-Class: no. of instances Prime Form Locations given in adjacent P form order position numbers.

10-1: 2 [0-9] 0-9, 2-e10-6: 1 [0-4, 6-t] 1-t,

2/6 ≅ 33% (max. possible, 50%)9-1: 2 [0-8] 0-8, 3-e9-5: 2 [0-4,6-9] 1-9, 2-t

2/12 ≅ 17% (max. possible, ≅ 33%)8-1: 2 [0-7] 0-7, 4-e8-5: 2 [0-4, 6-8] 1-8, 2-9,8-9: 1 [0-3, 6-9] 3-t

3/28 ≅ 11% (max. possible, ≅ 18%)7-1: 2 [0-6] 0-6, 5-e7-5: 2 [0-3, 5-7] 1-7, 4-t7-7: 2 [0-3, 6-8] 2-8, 3-9

3/38 ≅ 8% (max. possible, ≅ 16%)6-1: 2 [0-5] 0-5, 6-e6-z3: 2 [0-3, 5,6] 1-6, 5-t6-z6: 2 [0-2, 5-7] 2-7, 4-96-z38: 1 [0-3,7,8] 3-8,

4/50 = 8% (max. possible, 14%)5-1: 2 [0-4] 0-4, 7-e5-3: 2 [0-2,4,5] 1-5, 6-t5-6: 2 [0-2, 5,6] 2-6, 5-95-7: 2 [0-2, 6,7] 3-7, 4-8

4/38 ≅ 11% (max. possible, ≅ 21%)4-1: 2 [0-3] 0-3, 8-e4-2: 2 [0124] 1-4, 7-t4-5: 1 [0126] 3-6,4-7: 2 [0145] 2-5, 6-94-8: 1 [0156] 5-84-9: 1 [0167] 4-7

6/28 ≅ 21% (max. possible, ≅ 32%)3-1: 2 [012] 0-2, 9-e3-2: 2 [013] 1-3, 8-t3-3: 2 [014] 2-4, 7-93-4: 2 [015] 3-5,6-83-5: 2 [016] 4-6, 5-7

5/12 ≅ 42% (max. possible, ≅ 83%)

Table 5 Wuorinen, Piano Variations (1963): Study of Set-Class Membership formed from Adjacent Order Positions

Var No. Measure No./pg nos Nos. of Beats/Tempo Duration Elapsed Time, Approx.

1 1-25/pp. 1-3 67.5/eighth = 112 45.8”27/eighth = 168 48” est1 fermata, very short1 rit. not long 0’ 48”

2 26-59/pp. 3-6 116/eighth = 112 62.14”2 fermatas both breve 64” est 1’ 52”

3 60-69/p. 6 37.5/to the end all 20.09”are eighth = 112 2’ 29.5”

4 70-82/pp. 7-8 42.75 22.96” 2’ 52.46”

5 83-94/pp. 8-9 35.5 19.01”1 fermata breve 19”50” est. 3’ 11.96”

6 95-104/p. 9 25.5 13.66”1 fermata non troppo 16.04”1 fermata long! almost until silence 3’ 28”

7 105-137/pp. 9-11 181.75 97.36”1 fermata breve 98” (1’ 38”)rit. not long 5’ 06”

8 138-153/pp. 11-12 39 20.89”1 fermata breve 22” est 5’ 28”

9 154-164/pp. 12-13 17.75 9.51” 5’ 37.51”

10 165-176/pp. 13-14 25.75 13.8” 5’ 51.31”

11 177-187/pp. 14-15 33.5 17.96” 6’ 09.27”

12 188-203/p. 15 26.5 14.19” 6’ 23.46”

13 204-218/pp. 15-16 13.5 7.24” 6’ 30.7”

14 219-237/pp. 16-17 70 37.5” 7’ 08.2”

Easley Blackwood’s recording lasts 9’ 20”.Charles Wuorinen Music of Two Decades Volume 1Music & Arts CD-800 (1994)P.O. Box 771, Berkeley, CA. 94701

Table 6 Wuorinen, Charles. Piano Variations, 1963Information on lengths of each variation

Richard Hermann, Prof. of Music, Univ. of New Mexico

Var. No. Duration Contour No. Absolute Contour Adjusted*

1 0’ 48” 11 072 1’ 04” 12 083 0’ 20.09” 07 05**4 0’ 22.96” 09 05**5 0’ 19.50” 06 046 0’ 16.04” 04 037 1’ 38” 13 098 0’ 22” 08 059 0’ 09.51” 01 0110 0’ 13.8” 02 0211 0’ 17.96” 05 0412 0’ 14.19” 03 0213 0’ 07.24” 00 0014 0’ 37.5” 10 06

Var. No. Duration Contour No. Absolute Contour Adjusted*13 0’ 07.24” 00 009 0’ 09.51” 01 0110 0’ 13.8” 02 0212 0’ 14.19” 03 026 0’ 16.04” 04 0311 0’ 17.96” 05 045 0’ 19.50” 06 043 0’ 20.09” 07 05**8 0’ 22” 08 054 0’ 22.96” 09 05**14 0’ 37.5” 10 061 0’ 48” 11 072 1’ 04” 12 087 1’ 38” 13 09

* Creelman, temporal discrimination at ≥ 10% difference. Hence, new adjusted contour durations based on Creelman induced equivalence-classes. This is Psychologically untested at this level of duration and for this type of situation. See Cogan and Escot, Sonic Design, pp. 240-242. This collapses the 14 valued contour space for durations to a 10 valued space.** This pair of values in the “equivalence-class” is slightly greater than 10%, hence transitivity fails (22.96 – 2.296 = 20.664; 20.664 > 20.09) in this one instance.

428”/14 variations = 31” (30.57142857143”) per variation average.Shortest variation is almost 23% of the average length, 7/31 = 0.22580645161 ≅ 23%.Longest variation is approx. 316% of the average length, 98/31 = 3.16129032258.Longest var is almost 14 times that of the shortest var.

Table 7 Wuorinen, Charles. Piano Variations, 1963More information on lengths of each variation

Richard Hermann, Prof. of Music, Univ. of New Mexico

Variation N

umber

1 2

3 4

5 6

7 8

9 10

11 12

13 14

Absolute

Durations

Duration

Contour

Values(after M

arvin)

Friedmann

Contour

Differences

Adjusted

Contour

7 8

5a 5c

4 3

9 5b

1 2

4 2

0 6

Values

Adjusted

Contour

Differences

0’ 48 1’ 04

0’ 20.09 0’ 22.96

0’ 19.50” 0’ 16.04”

1’ 38” 0’ 22”

0’ 09.51” 0’ 13.8”

0’ 17.96” 0’ 14.19” 0’ 07.24”

0’ 37.5”

11 12 7 9 6 4 13 8 1 2 5 3 0 10

+ + – – + – – + + – – +

–

+ – + – – + – – + + – – +

??

0

12

3

45b

6

7 8

9

Transitivity Graph of

Creelm

an Adjusted

Contour Values

(duration differences >= 10%)

5a5c

Ric

hard

Herm

ann, U

niv

. of N

ew

Mexic

o, h

arh

ar@

unm

.edu,

“Early

Seria

lism

s in

the U

nite

d S

tate

s,” in

the p

anel d

iscussio

n

Theorie

s a

nd A

esth

etic

s: A

n H

isto

rical R

econsid

era

tion o

f

Seria

lism

as P

ractic

e, S

MT

Meetin

g, M

ontré

al 2

009

Table 8 A

nalysis of Duration C

ontours for Wuorinen, Piano Variations (1963)

References“Early Serialisms in the United States”

1

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References“Early Serialisms in the United States”

2

Myhill, John. “Musical theory and music practice.” Journal of aesthetics and art criticism 14 Dec. 1955: 191-94.

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serialism as practice the argenteuil hall

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