a polychordal approach to serial harmony - part 1 - online version

7/26/2019 A Polychordal Approach to Serial Harmony - Part 1 - Online Version

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Barnaby Hollington

A PolychordalApproach toSerial Harmony

Part I


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CONTENTS

Introduction 2

Chapter 1 Interval-Class Set Taxonomy 4

Chapter 2 Tonal Consonance and Dissonance

Chapter ! "our #ro$lematic Terms% &tonality' #ost-Tonality' 12

#antonality' #olytonality

Chapter 4 The #olychordal &pproach 1(

Chapter ) The Art of Thinking Clearly% & #olychordal and Serial 21

&nalysis

Chapter ( Serialism' #olychordally Conceived 44

*i$lio+raphy 4(


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INT,ODCTION

My research aims to test the effectiveness of a consciously polychordal approach in

clarifying so-called atonal/post-tonal1harmonic trajectories to the listener. My term for the

system I am developing is the Polychordal Approach. I anticipate that my research ill

demonstrate that this method allos a composer to minimise the cognitive opacity! that"

according to #erdahl $1%&&'" (ic)s $1%%1'" Meyer $1%*+'" ,arus)in $1%%" !&' and other

commentators" renders much atonally/post-tonally-conceived harmony particularly serial

harmony - inaccessi0le to most listeners.

,he Polychordal Approach is founded on the premise that" ithin the 0ounds of eual

temperament" all pitch-class sets have potential tonal implications as commentators such as

23ti $1%4&'" Parncutt $!%' and Ad5s $!1!' maintain. Most pitch-class sets have multiple

latent tonal centres i.e." they are inherently polychordal. A fe pitch-class sets possess only

one possi0le tonal centre i.e. they are inherently tonal. Anypitch-class set can therefore 0e

spaced" hori6ontally and/or vertically" in such a ay as to ma7imise the audi0ility of its tonal

connections i.e." in all 0ut the simplest cases" spaced as a polychord. In the process" one or

more tonal centres ithin the pitch-class set ill 0e perceived as the focal point$s' or tonal

anchor$s' of the pitch-class set. ,his is the )ey to meaningful cognition of any conceiva0le

succession of pitch-class sets 0y the listener8 the )ey to consistently avoiding the cognitive

opacity that ill otherise continue to prevent atonally/post-tonally-conceived music from

ever communicating meaningfully ith a ider audience.

1 Post-tonal and atonal are the generally accepted terms for the type of

harmonic territory that my msic tends to co!er" I hold that neither term

satisfactorily describes my msic" #oreo!er$ both terms pose certain more

general problems% for e&ample$ certain composers concei!e of their msic in

atonal or post-tonal terms$ bt 'hether listeners al'ays percei!e

atonally(post-tonally-concei!ed harmony in the same terms in practice is another

matter" )he *estion 'ill be e&plored frther in +hapter ,"

2 erdahl$ ."% +ogniti!e +onstraints on +ompositional Systems in GenerativeProcesses in Music$ ed" /ohn A" Sloboda 0&ford% +larendon Press$ 1334$ p"2,1"


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,he primary purpose of this paper is to demonstrate the or)ings of the Polychordal

Approach" through harmonic analysis of sections of The Art of Thinking Clearly" for solo

piano. ,his analysis" in 9hapter 4" forms the main 0ody of this paper.

:ut 0efore any cogent discussion of the Polychordal Approach can ensue" three sets of

terminological and conceptual hurdles must first 0e overcome;

1. deal ith each of the three uestions outlined a0ove" preparing the ground

for an e7planation of the rationale informing the Polychordal Approach in 9hapter . 9hapter

4 ill analyse the or)ings of this approach in practice" dissect some of the harmonic

phenomena created through employing the Polychordal Approach in conjunction ith

serialism" and in the process - refute #erdahls claim $1%&&' that serialism can only result in

harmonic opacity" from the listeners perspective.>9hapter * concludes 0y considering some

of the 0roader implications of this Polychordal Approach to serial harmony.

, I do not dispte erdahls assertion 01334 that the harmony in serial and other

atonally(post-tonally-concei!ed 'or5s sch as Bole6s Le Marteau Sans Matre

01784 is cogniti!ely opa*e" I do contend$ ho'e!er$ that serialism can berendered tonally clear" Schoenberg and Berg both attempted to achie!e this$ 'ith

mi&ed reslts" 9ebern 01,,4 also held that his o'n serial harmony cold be

interpreted in tonal terms" After 1:7$ the generation of composers 'ho too5

o!er serial techni*es chose to 'rite as thogh tonal implications can and shold

be eradicated from ne' msic altogether$ typically reslting in cogniti!e

opacity" Bt if pre!ios generations of serial composers ha!e failed to achie!e

harmonic clarity$ from the listeners perspecti!e$ it does not follo' that crrent or

ftre generations of serial composers cannot attain the desired coherence"

Indeed$ the analysis in +hapter 7 'ill demonstrate that certain serial techni*es

can also be sed to increaseharmonic intelligibility$ in themsel!es$ if s5ilfllyhandled"


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C.TE, 1

INTE,/&0-C0&SS SET T&ONO3

=or the purposes of my research" Interval-9lass '

ostensi0ly apitch-class set ta7onomy - hich is currently in idespread use. In see)ing to

demonstrate the effectiveness of the Polychordal Approach over all conceiva0le harmonic

territory ithin eual temperament" I intend to use not only all conceiva0le interval-class

sets" 0ut e7ploit each portion of this e7pressive territory roughly eually" over my

composition portfolio as a hole. Individual pieces ill concentrate on different portions of

this territory" to different e7pressive ends.

=ortes convoluted classification system precludes sufficiently concise and accurate

presentation of the full range of tonal and e7pressive implications possi0le ithin eual

temperament for my purposes.


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=urthermore" =ortes system treats inversionally-related sets e.g. the minor triad and

the major triad - as identical.@vidently" the minor and major triads do not possess the same

e7pressive ualities" and the same can 0e said of other numerous other inversionally-related

sets in 0oth estern and non-estern music. Bor can the tonal implications of inversionally-

related sets 0e considered identical. ,he Polychordal Approach to post-tonal/pantonal

harmony is concerned ith these tonal implications. ,herefore" any system that does not

distinguish 0eteen major and minor" nor 0eteen any other inversionally-related interval-

class or pitch-class sets" ould hinder my research.

As a conseuence of listing the elements of each set as a num0ers indicating neither the

pitch-classes" nor $directly' the interval-classes" nor $if the set is inverti0le' hich ay up the

interval-classes are" =orteian pitch-class set analysis is an unnecessarily la0yrinthine"

tortuous affair. ,oo often" the system effectively does little more than muddy the ater. Perle

$1%%' has persuasively demonstrated ho this has adversely affected =ortes on analytical

or).

I have therefore devised a simpler" more elegant and more consistent system" hich

avoids each of the pitfalls descri0ed a0ove; Interval-9lass


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listed first therefore C1"!">"*E. ?nder Interval-9lass *.

?nder this ne system" inversions of interval-class sets are listed as separate sets. All

that is necessary to discover the inversion of an interval-class set is to reverse its num0ers;

1!>* 0ecomes *>!1. :ut the 1 is then placed first; so *>!1 0ecomes 1*>! $listed as a distinct

set from 1!>*'. In other ords" to invert an interval-class set under my system" one )eeps the

first num0er in place" and flips the rest around; 1!>*" flipped" 0ecomes 1*>!" 1!!+ 0ecomes

1+!!" 1!4 0ecomes 14!" and so on. ,he major triad" >4" flipped" 0ecomes the minor triad"

>4. 1!+!" flipped" remains 1!+!" i.e. 1!+! is uninverti0le. ,hus" unli)e =ortes system"

Interval-9lass


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TON&0 CONSON&NCE &ND DISSON&NCE

,he ne7t terminological and conceptual minefield reuiring clearance" 0efore any

meaningful discussion of the Polychordal Approach can ensue" is the uestion of consonance

and dissonance. ,here are several conflicting definitions. ,hese fall under to 0road

categories tonal $Plomp and #evelt" 1%*4'4 or sensory $#erdahl" 1%&&'" versus

conte7tual $(ill" 1%&*' or musical $#erdahl" 1%&&'.*,hese descri0e to very distinct

phenomena" hich happen to coincide for most of the history of estern art music. 9ontrary

to assertions from 2osen $1%+4' and other authors" 0oth tonal and conte7tual dissonance can

remain meaningful" and can 0e coherently and consistently descri0ed ith reference to

atonally/post-tonally-conceived music" in hich the to do not coincide - here tonal

dissonances have 0een emancipated" i.e. no longer necessarily serve as contextual

dissonances.

Plomp and #evelt $1%*4' and other commentators employ the terms tonal dissonanceand tonal consonance to descri0e the sensory difference 0eteen intervals" and therefore

also 0eteen chords. $#erdahl and others prefer sensory consonance/dissonance'. According

to Plomp and #evelt" a tonally dissonant interval produces audi0le 0eats 0eteen the

partials of the to tones. ,hese 0eats occur hen the difference 0eteen the to freuencies

falls ithin a critical 0andidth. Plomp and #evelt thus identify the most tonally dissonant

interval as the minor !nd" folloed 0y the major !ndand the tritone. ,hrough the centuries"

there have 0een many divergent rationalisations of tonal consonance and dissonance. ,he

resultant hierarchies are generally uite similar. I consider Plomp and #evelts hierarchy and

definition the most persuasive" for the purposes of my research.

7 Plomp$ " and e!elt$ 9"/"#"% )onal +onsonance and +ritical Band'idth$ in

Journal of the Acoustical Society of America ,3 01874$ pp"7:3-78C"

8 )his is a broad generalisation" Hill 01384 identi;es three types of de;nition"

+a6den 013C4 identi;es fourteen separate de;nitions$ bt 'as na'are ofPlomp and e!elts 'or5 at the time of 'riting"


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Another group of theorists and composers" including :a00itt $1%*4'" 2osen $1%+4'"

9a6den $1%&'" Gamien $!1/1%+*'" and Anderson" have argued that consonance and

dissonance can only properly 0e defined ithin the conte7t of a specific piece of music" or" at

0est" a given set of harmonic rules as defined in a particular era. 2osen $1%+4'+rites;

CA dissonance isEH any musical sound that must 0e resolved" i.e.

folloed 0y a consonance; a consonance is a musical sound that needs

no resolution" can act as the final note" that rounds off a cadence.

,onal or sensory consonance/dissonance is crucial to the Polychordal Approach.

9onte7tual or musical consonance/dissonance is less central to my concerns" although not

insignificant.

ne may measre the relati!e le!els of tonal dissonance bet'een

inter!al-class sets simply by conting the instances of the more tonally

dissonant inter!als 'ithin each set$ according to Plomp and e!elts

hierarchy" n that accont$ borro'ing my terminology from


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9e can then add the second and si&th nmbers of .ortes inter!al-

class !ectors for each set 0i"e" the rest of the dissonance !ector4$ to

determine the nmber of mild dissonances i"e" tones 024 or tritones 084"

.or tetrachords$ 11 hierarchical strata ense%

Strong

Dissonances

Mild

Dissonance

s

Sets

, semitones 2

tones(tritone

s

111

2 2tones(tritone

s

1123$ 11:8$ 1177$ 118:$ 1132$ 1717

1

tone(tritone

11,$ 11,$ 1213$ 1:18

C

tones(tritone

s

1,1

1 2 122$ 12,8$ 12:7$ 128,$ 122$ 1,28$

1,82$ 1:27$ 172:$ 17:2$ 182,$ 18,2$

1221 127:$ 1,,7$ 1:72$ 17,,C 1,::$ 1,7,$ 1:,:$ 1::,

C : 2228$ 2:2:, 22::2 227,$ 2,27$ 2,,:$ 2:,,$ ,,,,1 2,:,

)he follo'ing chart sho's similar hierarchies for all Inter!al-+lass Sets

containing ;!e elements or fe'er" Inter!al-+lass Sets 0I+ Sets4 of the same

color contain the same nmber of strong dissonances" 9ithin each sch

category$ higher ro's contain more mild dissonances? lo'er ro's contain

fe'er of these" In!ersionally-related sets 'ill al'ays contain the same

le!els of tonal dissonance$ and are no' sho'n side by side" )his chart 'ill

sce for the analytical prposes of this research paper 0+hapter 74? later

in my research$ I 'ill e&pand this classi;cation to inclde all ,7C I+ Sets"


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f the ,7C possible I+ Sets 01,8 of 'hich are listed abo!e4$ only 3 are

tonal consonances% silence$ C 0nison4$ , 0minor ,rd(ma=or 8th4$ :3 0ma=or

,rd(minor 8th4$ 7 0perfect :th(7th4$ ,:7 0the minor triad4$ ,7: 0the ma=or


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triad4 and ::: 0the agmented triad4" f all the I+ Sets ser!ing as

contextual consonances throghot the history of 9estern art msic ntil

the late 1thcentry$ there are only types 0if one incldes silence4 of

the 3 possible tonally consonant I+ Sets listed abo!e" )he e&ception is :::0the agmented triad4" )his ob!ios shared inherent trait none of these

I+ Sets contains tonally dissonant inter!al-classes? 'hereas$ of the other

,:, possible I+ Sets$ all but one contain tonally dissonant inter!al-classes

casts dobt o!er +harles osens claim that a consonance or dissonance

shold only be de;ned contextually%

It is notH the human ear or nervous system that decides hat is

a dissonance" unless e are to assume a physiological change

0eteen the thirteenth Ccentury" hen only ths" 4thsand octaves

could serve as contextual consonancesE and fifteenth century

Chen >rdsand *thscould also serve as contextual consonances"

0ut thscould no longer do soE.&

)hat all contextual consonances in the history of con!entional 9estern

tonal grammar are also tonal consonances$ by Plomp and e!elts criteria

and that$ ::: aside$ no other tonal consonances are possible - 'old

sggest that the hman ear does$ in fact$ decide 'hat is a dissonance"

9hilst I+ Sets possess measrably !ariable inherent le!els of tonal

dissonance$ in practice these dierences do not correspond directly to

perceptible le!els of tonal dissonance" ather$ I+ Sets constitte only the

;rst and most straightfor'ardly analysable of se!eral factors in

determining perceptible tonal dissonance le!els" )he other rele!ant

factors are spacing$ register$ dynamics and timbre" #y research is not

3 osen$ +"% ibid"$ p",," #oreo!er$ osens implication that perfect : thsne!er

ser!e as contextual dissonances bet'een the enaissance and the 2Cthcentry

is not strictly accrate" )hroghot this period of 9estern msical history$

althogh a :thcold not ser!e as a cadential resting-point in itself$ : thsfre*ently

do appear at cadential resting-points$ in the middle or pper !oices of larger

chords" +onsidering that in con!entional 9estern tonal msic$ chordsJ are

KconsideredL dissonant if they inclde e!en a single dissonant inter!al KHill$ +"%

+onsonance and Missonance in The e! "arvard #ictionary of Music

0+ambridge$ #ass" and ondon$ 1384$ p"1L$ if : thshad trly been considered

dissonant bet'een the enaissance and the 2Cth

centry$ :ths

'old ne!er ha!eappeared at cadential resting-points o!er that period bt they do"


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primarily aimed at ;nding failsafe methods of measring tonal dissonance

le!els althogh later in my period of research$ I 'ill be measring the

eect of Polychordal spacing on percei!ed le!els of tonal dissonance$ and

considering the eect of register" Ho'e!er$ the classi;cation of I+ Sets byrelati!e tonal dissonance le!els enables me to de;ne dierent e&pressi!e

territory in broad terms from piece to piece$ or from passage to passage "

+onse*ently$ I can test the eecti!eness of the Polychordal Approach

across the entire a!ailable so-called post-tonal harmonic spectrm"

.or e&ample$ The Art of Thin$ing %learly$ analysed in +hapter 7$ ses !ery

tonally dissonant I+ Set material - yet in practice$ the polychordal spacing of

those sets reslts in a mch lo'er le!el of perceptible tonal dissonance than the

I+ content 'old sggest" I 'ill demonstrate this flly at a later point in myresearch"


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C.TE, !

"O, #,O*0E&TIC TE,S% &TON&0IT3' #OST-TON&0IT3' #&NTON&0IT3'

#O03TON&0IT3

>'1" Auhagen $1%%'" Ad5s $!1!' and"

most significantly for the purposes of my research" Parncutt $!%'.11Parncutt argues that it is

impossi0le for a composer to eliminate tonal implications from their music" unless they also

eliminate pitch altogether" as happens in 1'. ,he term atonal" therefore" ought only to apply to music hich e7cludes pitch;

rddegrees of a minor scale or the * th" +thand &thdegrees of a major

scale" suggesting that either the HC9E or the HC@0E may 0e heard as a point

of reference. ,he major-third $-semitone' intervalC-classE em0edded ithin

H CI9 &" e.g. 9" D0" @E suggests that its reference pitch is HC9E"

regardless of hether the pattern is heard as Beapolitan" Ara0ic or =lamencoH1>

...it is surprising that many pc-set Cpitch-class setE theorists tacitly consider

all pc-sets a priori to 0e euivalent or value-free" as if they had no tonal

implications or as if tonal implications did not e7ist. 9an the tonal

1C ne can certainly ta5e the !ie' that e!en 'ith s KSchoenberg$ Berg$

9ebernL there is still a tonic present I certainly thin5 so" 9ebern$ A"% The Pathto the e! Music$ 9"eich$ ed"$ "Blac5$ transl"$ 0Bryn #a'r$ Pennsyl!ania%

)heodore Presser$ in association 'ith Dni!ersal Ndition$ 18,01,,44$ p","

11 Parnctt$ "% )onal Implications of Harmonic and #elodic ) n-)ypes in

Mathematics and %omputing in Music$ )"


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implications that e learn from music simply disappear $hich is

psychologically implausi0le'" or are they ar0itrary $hich is psycho-

acoustically and ethnomusicologically implausi0le'1

...in real music heard 0y real human 0eings" pc-sets ill alays have

tonal implications"

17

In other ords" if a composer conceives of their harmony in so-called atonal or post-

tonal terms that is" denies or ignores the uestion of tonal implications in their harmony

altogether" or considers that tonal associations only appear ithin certain harmonic territory"

and that other territory remains free of such associations - this ill not prevent most or

perhaps any listeners from hearing" feeling and attempting to conte7tualise those freuent

unintended or unac)noledged fleeting tonal references that inevita0ly 0ecome audi0le. $Ffcourse" e7periences vary according to each listeners prior musical e7perience" and

psychological disposition.' It ould 0e unreasona0le and unor)a0le for a composer to

demand that listeners suppress this part of their perception and cognition" particularly

considering that identification of tonal implications tends to induce the impression of

understanding the harmony on some level.

If the term atonal" as commonly understood" is unsatisfactory" the same can 0e said of

post-tonal. ,he premise 0ehind post-tonal is that only certain elements of the )ind ofharmony in uestion have tonal implications and that these can only ever represent isolated

instances of deconte7tuali6ed" post-modern references to a 0ygone system ithin a ider

musical conte7t in hich tonal implications are otherise supposedly either a0sent or

irrelevant" as in atonality.

Parncutts article $!%' is restricted to the tonal implications of trichords" and does not

address the uestion of multiple roots or tonics ithin a single sonority i.e. polychords.

Fnce these are considered $the Polychordal Approach'" it is possi0le to identify all the

possi0le hypothetical tonal implications of any given set of pitch-classes. It might have 0een

tempting to conclude that all so-called atonal or post-tonal voca0ulary is therefore in fact

theoretically polytonal" al0eit to greatly varia0le levels of intelligi0ility in practice"

depending on ho clearly any given voca0ulary is presented" in polychordal terms. (oever"

the term polytonal" as generally understood" implies that each of the multiple tonalities

1: Ibid"$ p"128"

17 Ibid"$ p"128-"


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present in any one sonority is clearly sustained over a reasona0ly e7tended period" as in

Milhaud or


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Ff course" in


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C.TE, 4

The #olychordal &pproach

,he Polychordal Approach is founded on the folloing hypothesis;

1. ithin eual temperament" each of the >4 possi0le Interval-9lass . ,he same holds true for any conceiva0le succession of sets of pitch-classes. I.e.

provided that a composer consistently ma7imises the audi0ility of polychordal

connections" any conceiva0le succession of sets of pitch-classes can 0e rendered

harmonically intelligi0le" and therefore harmonically meaningful to most listeners.

,o illustrate point 1" let us e7amine a relatively tonally am0iguous or so-called atonal

set of pitch-classes - C@" =" =O" N" 9" D0E" 0elonging to I9 . ,his set of pitch-

classes contains numerous clearly diatonic su0sets" including;

N0 +th CN0" D0" =E

9 major C9" @" NE

=O +th C=O" 9O" @E

= %th C=" 9" $@'" NE

1 )here has been considerable debate among msicologists and composers

concerning 'hether one can$ in fact$ percei!e more than one simltaneos tonal

centre" Gan den )oorn 013,4$ Babbitt 01:4$ Hindemith 01:24$ Ba5er 013,4

and .orte 01774 ha!e arged that only one tonal centre can be percei!ed at any

one time" )ymoc65o 02CC24$ )ars5in 01C4$ #ilhad 012,4$ Stra!ins5y$ and

)hompson and #or 0124 all claim the contrary" Perhaps the most persasi!e

cases ha!e been made by 9olpert 02CCC4 and Hamamoto$ Botelho and #nger

02C1C4$ 'hose research sggests that some listeners can percei!e mltiple tonal

centres$ and others cannot" As a rle$ listeners familiar 'ith 9estern Art #sicare more li5ely to percei!e se!eral tonal centres simltaneosly"


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D0 major +th CD0" =" 9E

@ minor %th C@" N" =OE

It is easy enough to present theset of pitch-classes C@" =" =O" N" 9" D0E vertically as a

polychord i.e. as a chord in hich to or more of the possi0le diatonic su0sets are clearly

audi0le. ,here are many possi0le solutions. ,he first five of the si7 solutions 0elo involve

clear registral separation of the to or three diatonic units. ,he si7th is perhaps a little more

am0iguous" as the registers are interoven. Bevertheless" the si7th chord is also clearly

audi0le as a polychord. ,he first to chords occur in The Art of Thinking Clearly" to 0e

analysed in 9hapter 4.

here to or three diatonic su0sets are emphasised in this ay through spacing" the

tonal centres of each of these su0sets are then clearly percepti0le as rival" simultaneously-

sounding tonal centres. =urthermore" of these tonal centres" one often dominates. In the first

of the si7 chords a0ove" it is reasona0le to e7pect that most" if not all listeners ill perceive 9

major more strongly than N0+ and hile some listeners may onlyperceive 9 as the tonic"

others may also perceive N0 as a rival" if ea)er tonic.

!

In the second chord" due to 0oth thegreater registral pro7imity of the to su0sets" and the 9 major su0set shifting to the first

inversion" 9 major ill still normally 0e perceived as the stronger tonal centre" 0ut rather less

emphatically than in the first chord. In all cases a0ove" the loest tonal centre happens to 0e

the strongest =O+ for the third and fourth chords" =% for the fifth" 9 major for the si7th.

I maintain" furthermore" that here one pitch is clearly perceived as tonally stronger" all

of the other pitches in the sonority can also 0e heard" to some e7tent" modally in relation to it.

2C See Hamamoto$ Botelho and #nger 02C1C4"


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2C

=or e7ample" the = in the second chord a0ove can 0e heard 0y an astute listener not only as

the +thof N0 major or minor" 0ut also simultaneously as the thof 9 major. ,he top D0 can 0e

heard not only as the 4thof N0 major or minor" 0ut also as the flattened supertonic of 9" and

the N0 as a 0lue note in 9" as ell as a rival tonic in its on right.

Indeed" if any one of the constituent elements of any conceiva0le non-diatonic set of

pitch-classesere ar0itrarily assigned as a hypothetical tonic" each of the other pitch-classes

could 0e defined in relation to that tonic according to either a estern or non-estern mode"

or else a hy0rid of to e7isting modes. In the case of C@" =" =O" N" 9" D0E" the presence of

three consecutive semitones prescri0es hy0rids in all cases - 0ut hichever ar0itrary tonic one

assigns" there are numerous hypothetical solutions. (ere are solutions for each of the *

possi0le tonics ithin the set. @ach solution is a hy0rid of a estern scale and a (industani

,haat;


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,o give a more e7treme e7ample" the set of pitch-classes C9" 9O" D" DO" @" =" =O" NE

could" for e7ample" 0e construed as a 0lend of part of the 9 major scale C9" D" @" =" NE and 9

,odi C9" D0" @0" =O" NE. If a composer ere to desire to present that particular set of pitch-

classes in those specific terms" and ere sufficiently s)illed" all pitch-classes ithin the set

could then 0e rendered audi0le in terms of their relation to the tonic 9.


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functional can potentially can 0ecome applica0le to other e7isting or future types of

harmony" 0esides conventional estern tonal grammar provided that the ne types of

harmony also appear to function" i.e. possess the folloing;

1. 9learly percepti0le harmonic centricity throughout a piece of music

metaphorically" as a series of harmonic locations.

!. ,he presence of harmonic movement 0eteen various locations.

>. Discerni0le patterns governing this movement.

,he harmonic conventions e hear in operation in the estern tonal tradition create the

illusion of purpose" simply 0ecause the patterns and the inherent internal logic are so

transparent.


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C.TE, )

THE ART OF THINKING CLEARLY% & #O03C.O,D&0 &ND SE,I&0 &N&03SIS

Ff my recent compositions" The Art of Thinking Clearly $!1>' serves the most usefully

as an introduction to the Polychordal Approach. @ach of my recent or)s covers distinct

harmonic ground" and employs one or more devices not found elsehere in my or). :ut the

com0ination of territory and techniues covered in The Art of Thinking Clearly ena0les a

clearer e7position more of the )ey concepts than ould 0e possi0le via discussion of Partita

$!1>'"Prosthesis $!1'$ T%o Sketches $!1' or &elvet 'evolution $!1'.

Example 1% The tone-ro9 used in The Art of Thinking Clearly

All of the pitch-class material in The Art of Thinking Clearly derives from a single 1!-

tone ro. An analysis of any ros constituent I9


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2:

Ff course" one could dra up similar charts for all possi0le he7achords" heptachords"

etc. :ut from the smaller I9


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27

Baturally" the inversion of this ro freuently used throughout the piece - ould

produce a similar-loo)ing chart" 0ecause any to inversionally-related I9


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28

transparent manner" consistently throughout The Art of Thinking Clearly" this ould suggest

that the Polychordal Approach is also li)ely to succeed in its aims" hen applied to more

consonant material given that such material tends to 0e more tonally e7plicit. !!

e can no analyse the or)ings of the Polychordal Approach more closely.

Example 2&% #entachordal Cycle' $ars 1-'polychordally conceived

The Art of Thinking Clearly opens ith a >*-0ar harmonic arch. In uasi-*-0ar harmonic arch. In the e7ample 0elo" I have

divided each chord into to or more su0-chords" to illustrate the presence of multiple tonal

centres in each sonority. I have shon hat I perceive as the strongest tonal centres ithin

each chord in 0old" 0ut I am more than happy to ac)noledge the pro0a0ility that some22 )here are$ ho'e!er$ t'o signi;cant e&ceptions" f the more consonant I+ Sets$ :::0the agmented triad4 and ,,8 0the diminished triad4 are problematic in this regard%'hilst they are relati!ely tonally consonant 0according to Plomp and e!elts criteria4$they are also inherently tonally unstable 0in conte&tal terms4" As noted in +hapter 2$ ofthe 3 tonallyconsonant I+ Sets$ ::: is the only one ne!er to ha!e ser!ed as a contextualconsonance in the history of 9estern tonal grammar" )he inherent instability of ::: and,,8 'old appear to e&plain 'hy$ in the corse of composing Partita$ in 'hich thea!ailable 1 trichordal I+ Sets 'ere spread e!enly o!er t'o tone-ro's 'hich generatedmost of the pitch-class material$ ::: and ,,8 became notably nder-represented inpractice% sbconsciosly$ I 'as a!oiding this harmonic !ocablary to some e&tent" )henderlying harmonic aim of my ne&t 'or5 - Prosthesis ) 'as$ therefore$ to e&plore the

tonal implications of ::: and ,,8 in order to test the eecti!eness of the PolychordalApproach on this more tonally ambigos material" Nlse'here$ it is remar5able thatnone of Babbitts ostensibly all-trichordal ro's actally contain either of these trichords the other 1 trichords 0in .ortes classi;cation$ only 1C4 are al'ays present$ bt :::and ,,8 ne!er featre at all" )hese so-called all-trichordal ro's are employed in manyof the 'or5s from Babbitts middle period$ inclding Paraphrases 014$ #ual 013C4$Ars%ombinatoria 01314$ the Piano %oncerto 01374$ Lagniappe 01374$ The %ro!ded Air01334 and %onsortini 0134" #ead 01:4 claims that the omission 'as made becase$nli5e other trichordal I+ Sets$ ::: and ,,8 cannot be ordered to represent the forclassical transformations Koriginal$ in!ersion$ retrograde$ retrograde in!ersionLnambigosly$ and the latter K,,8L is the single trichord type that cannot generate anaggregate Kof all 12 pitch-classesL" 0p"177-84" Perhaps$ consciosly or other'ise$ theinherent tonal 0rather than serial4ambigity of ::: and ,,8 may ha!e played a greater

part than Babbitt 'old e!er ha!e admitted$ or that #ead 'old e!er admit"


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2

listeners ill in some instances perceive other su0-chords more strongly than I do" including

su0-chords not shon 0elo. I do not specify 1 stinversions" !ndinversions" etc" partly 0ecause

these ill 0e self-evident" and partly to avoid unnecessary clutter.

,he polychordal content of this &-chord e7tract may 0e summarised" chord 0y chord" as

follos;

1. ,he tonal centre of the first chord is emphatically :. ,a)e aay the top @0" and e

have a straightforard : minor % thchord. 2espell the top @0 as a DO" remove the D"

and this time e have a : major % th 0oth su0-chords are anchored 0y the same

pitch-class" :.

!. I perceive the strongest tonal centre in the second chord as an @ major +th" ith the th

$A' included" 0ut there is an additional" rival tonal centre; the loest > pitches form a

:0 major +th.


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23

>. ,he 9 major triad included in the third chord esta0lishes 9 as the prevailing tonal

centre" ith the = and =O percepti0le as the thand sharpened thithin a 9 major

tonality. A faintly percepti0le secondary tonal centre is an =O flattened % th" given that

=O is the loest note of the chord.

. In the thchord" : major is again the overriding tonal centre" e7cept this time in the

!ndinversion $the first chord as in the root position'. ,he 9O is percepti0le as a % th8

the 9 natural as a flattened %th. I am tempted to hear a secondary sonority" ith an @0"

N0 and 9 over a non-e7istent A0 possi0ly 0ecause I have internalised the harmonic

cycle enough to anticipate the A0 in the folloing sonority. hether any other

listeners ould hear this possi0le secondary focus is another matter. Parncutt $!%'

and other commentators certainly ac)noledge the cognitive possi0ility of a listener

hearing a tonal centre that does not actually sound in the chord.4. ,he fifth chord is the most tonally am0iguous" ith at least five tonal su0-chords"

and at least four plausi0le tonal centres. ,he highest four pitches could 0e heard in

terms of either a :0 major + th $R th' or an @0 flattened + th $Rth'. ,he lo A could

also 0e cognised as a sharpened thin @0" suggesting that @0 is perhaps the strongest

tonal centre 0ut this is far from clear-cut" to my ears. A0 and D are other plausi0le

tonal centres.

*. ,he si7th chord might 0e heard in terms of an overriding = minor %th 0ut to some

listeners" perhaps the same pitches might appear to 0e anchored around :0. Fne

could also sense the loest four pitches as the first four notes of the ,odi ,haat"

starting on @.

+. 9O major ould seem the prevailing tonal centre in the seventh chord" ith the : and

9 $enharmonically :O' percepti0le as the +thand sharpened +threspectively.

&. ,he eighth chord is the same as the first. It is audi0le as a uasi-cadential resting

point a return to a tonic esta0lished in the first chord.

Another nota0le feature of this chord seuence is that each chord contains one or to

pitch-classes in common ith its adjacent chords. ,hese common notes are in most cases

registrally fi7ed" thus acting as pivots 0eteen successive sonorities. ,his voice-leading

decision as intended to further clarify the harmonic movement. :ut 0esides presenting each

successive chord as e7plicitly as possi0le" in polychordal terms" and clarifying the voice-

leading as much as possi0le" there is another contri0uting factor to the harmonic cohesion of

this opening chord seuence one that helps to e7plain hy the return to the first chord in

0ars 4-+ feels li)e a perfect cadence.


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2

Example 2*% #entachordal Cycle' $ars 1-' erially conceived

,hat factor is the serial mechanism 0ehind the pitch-class content. ,his claim might

seem contentious and impro0a0le" certainly to advocates of #erdahls dictum that serialism

causes cognitively opaue harmony $and it freuently does e7actly that'. =urthermore" most

serial devices are not directly intelligi0le to listeners. :ut this does not render all serialism

esthesically meaningless" if the composer is sufficiently s)illed.

,he serial mechanism in operation here governs all harmonic movement from 0ars 1 to

>*. ,he folloing ta0le summarises the content of each successive chord until 0ar *" in serial

terms;

@lement 1 ! > 4 * + & % 1 11 1!

Pitch-9lass 9O : @0 D A :0 A0 @ N = =O 9

9hord 1 7 7 7 7 7

9hord ! 7 7 7 7 7

9hord > 7 7 7 7 7

9hord 7 7 7 7 7

9hord 4 7 7 7 7 7

9hord * 7 7 7 7 79hord + 7 7 7 7 7

9hord & 7 7 7 7 7

@ach successive pentachord shifts along the series" 0ut retains alternatively either one or

to pitch-classes from the previous chord. ith each ne group of pitch-classes comes a

corresponding shift in tonal centres8 given sufficiently clear polychordal spacing and good

voice-leading $including" in this case" pivot notes'" these tonal shifts ill appear audi0ly

coherent. :ut 0esides hearing ne harmonic ground ith each ne chord" one can sense"from 9hord onards" that the seuence of tonal centres heard in 9hords 1-> is no 0eing

developed approached again" 0ut ith some changes - as shon 0y the 0lue lines and red

crosses. ,he second 0lue line" through 9hords 4" * and +" traces through similar harmonic

territory" in a similar seuence to 9hords !" > and ; in pitch-class terms $and therefore tonal

terms'" 9hord 4 is uite similar to 9hord !" 9hord * is uite similar to 9hord >" and 9hord +

is uite similar to 9hord . 9onseuently" folloing 9hord +" e expect something li)e

9hord &. ,he satisfaction of hearing 9hord & is enhanced further" 0ecause;


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,C

1. 9hord & is identical to the 9hord 18 one senses that a cycle has 0een completed.

!. ,he tonal centre is unam0iguously :.

>. : seems to 0e recurring at fairly regular intervals; in 9hord 1 $root position'" then

9hord $!ndinversion'" and no returning decisively in 9hord &.

. ,he loest to pitches in 9hord &" : and 9O" are carried on from 9hord +"

0oosting their tonal eight.

Ff course" the success of this type of mechanism depends entirely on hether the tonal

centres of each chord can 0e rendered sufficiently audi0le in the first place. ,he folloing

e7ample shos the tonal and serial movement" side 0y side;


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,1

Example 2C% "ull #entachordal Cycle' $ars 1-!(

In theory" the serial cycle 0egun in 0ars 1-+ ould have lasted for ! chords" consisting

of three su0-cycles of the )ind demonstrated a0ove" in hich the first chord is identical to the

eighth. ,he entire seuence" in terms of elements 1-1! of the tone ro" ould have proceeded

as follos;

9hord 1 9hord ! 9hord > 9hord 9hord 4 9hord * 9hord + 9hord &

>-+ *-1 1-! 1-)

-+ +-11 1-! !-* )-:

:-1 1!- -& +-11 11-> !-* *-1 :-1

In practice" I chose to set up the chord consisting of elements :-1of the ro as a uasi-

dominant $= 0eing the unam0iguous tonal centre in this case'" ith the first chord elements

1-) serving as a uasi-tonic $anchored on :'. I then appended the uasi-dominantto the end

of su0-cycle !" as the ne7t chord in the seuence" repeated su0-cycles 1 and ! 0efore finally

presenting su0-cycle >" and completing the cycle. In uasi-


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,,


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,:


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,7


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,8

Example !&% Chord ultiplication inAppalachian !pring


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,

,he transpositions of the initial A0 major cell could scarcely have 0een rendered more

aurally transparent. If one ere to present these pitches as a 0loc) chord in the manner shon

a0ove $(armonic Aggregate'" the highest nine pitches ould certainly still articulate the

three relevant tonal centres in an aurally coherent and satisfying manner. If the lo A0 ere

then moved don an octave" the same could 0e said of the entire chord.

If e consider a hypothetical alternative" in hich 9opland had alloed the

transpositions of the original A0 major cell to overlap registrally" the results ould have

remained tonally intelligi0le to the listener" so long as the pitches ere presented melodically

and not as a chord. ,he folloing hypothetical e7ample demonstrates that" ith the registral

overlap and a 0loc) chord presentation" any sense of tonal coherence ould disappear"

despite the harmonic clarity of the original cell;


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,3

In order to render the 0loc) chord shon a0ove tonally intelligi0le to the listener" the

composer ould need to separate out the various transpositions of the A0 major cell

registrally.

Example !*% Chord ultiplications in The Art of Thinking Clearly' $ars !


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,


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:C

In the * chord multiplications shon a0ove" the tonal centres are easy to analyse" giventhat one original cell-type is then replicated elsehere and this is aurally evident. @.g. in the

first chord multiplication shon" the loest A0 % R cell $i.e. A0" :0" D0" @0' is replicated

as further % R cells higher up. #i)eise" the minor */4 cells in the second multiplication"

the major %thcells in the third multiplication" the % thsin the fourth multiplication" and so on.

I made three types of change to the hypothetical models. @ach of the three )inds of

modification originated from the same concern; to ma7imise the audi0ility of the multiple

tonal centres.

,he reasoning 0ehind the first type of change is straightforard; lo registers normally

reuire ider intervals 0eteen pitches" to ensure aural clarity. ,herefore" in each case shon

a0ove" one of the loest pitches is moved don an octave. In the fourth and fifth instances"

this the second-loest pitch. In all other cases" the loest pitch is moved don. I do not

alays ma)e this type of adjustment; e.g. in the seuence of chord multiplications in 0ars

1!%-1>4 this as never harmonically necessary.


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:1

,he second type of change resulted from a desire to avoid octaves" in this instance.

Again" this is not alays necessary; in those instances" I )eep the octaves. =or e7ample" over

0ars 1!%-1>4" there are eleven instances of octaves 0eteen right and left hand;

In 0ars >+&-&" hoever" the presence of octaves in the second and fourth chord

multiplications ould have interfered ith clear perception of the multiple tonal centers. In

general" here the root notes of these tonal centres are dou0led at the octave" as is the case in

0ars 1!%-1>4" the octaves prove aurally satisfying; they reinforce the tonal anchors. In the

second and fourth e7amples shon on pages >*-+" hoever" the octaves reinforce pitches

hich otherise ould not have 0een perceived as tonal centres conseuently" the

polychordal spacing ould have 0een rendered more opaue and less aurally convincing" had

I alloed the hypothetical models to stand in this particular te7ture. In serial terms" therefore"

I cheated" and moved certain pitches in the right hand up or don a semitone" as indicated

0y the diamond-headed notes. Ff course" this type of tin)ering further increases the tonal

comple7ity of the sonorities in uestion" 0ut such increases serve merely to add harmonic

spice to the chord" rather than upset the 0alance of tonal centres" as the octaves ould have

done.

,he third and final amendment to the hypothetical serial models occurs in the fifth chord

multiplication $page >+'. In this case" had I )ept the highest note of the original cell as the

loest note of its upper transposition" and vice versa" the highest five pitch-classes of the

resultant chord multiplication ould have matched the loest five pitch-classes

$(ypothetical $A'. ,his is due to the interval-class distance 0eteen the loest and highest

pitch-classes of the original chordal cell" N and D0 respectively evidently" to tritones


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:2

ma)e up an octave. Again" the resultant octaves are not necessarily a pro0lem; in the third

chord multiplication $p.>' the to = major %thssound perfectly ell either side of a : major

%th. In the fifth chord multiplication" the addition of a minor >rd 0eteen the various

transpositions of the original cell simply created a richer harmonic effect.

Fn one hand" these sonorities could only have 0een conceived through serial thin)ing; in

all cases" the similarities 0eteen the actual sonorities and their hypothetical serial models far

outeigh the small divergences. (oever" as soon as the results are not as harmonically

convincing as they might 0e" I opt to 0rea) serial e7actitude. ,he overriding concern is ith

setting up multiple tonal centres" and presenting these audi0ly and euphoniously. I

consistently steer clear of those serial techniues that actively hinder harmonic intelligi0ility

0y their very nature. :ut rather more serial devices can prove helpful in this respect than is

commonly thought. 9ertainly" 0asic retrogrades" inversions and transpositions can 0e

employed to aid tonal coherence. ,he same is true of certain types of serial cycle $as shon

in @7ample !' and chord multiplication $@7ample >'. #i)eise" Messaiens chords of

transposed inversion device is essentially serial" and can enhance harmonic clarity" if

intelligently used. I have used this techniue freuently elsehere" 0ut it only appears once"

fleetingly" in The Art of Thinking Clearly" so discussion of this techniue ill have to ait

until a later point in my research.

Example 4% 7=rene> ,otations6 in $ars 2))-2()' as perceived tonally

,he final serial techniue used in The Art of Thinking Clearly to 0e discussed here"

li)eise" facilitates tonal coherence 0y its very nature. ,he term Grene) rotation is my on;

it as first employed 0y Grene) in (amentatio )eremiae Prophetae $1%'. ,he techniue

has 0een idely discussed" particularly ith reference to


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:,

Gnussens use of Grene) rotations in*lourish %ith *ire%orks $1%&&' esta0lishes one pitch-

class A as a tonic;!>

Niven the importance of the pitch A inH the linear rotations CGrene)

rotationsE outlined a0ove" this pitch starts to assume the function of a focal

point to the harmony" an easily recogni6a0le modal tonic hich guides the

ear through the many simultaneous comple7ities of the musics te7tures"

,his e7planation argua0ly holds true in*lourish %ith *ire%orks" 0ut does not apply to

my use of Grene) rotations in The Art of Thinking Clearly and elsehere" nor ould it apply

to other instances of this device in the or) of certain other composers" including


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::


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:7

=ocusing" for the time 0eing" on the three Grene) rotations out of conte7t $


47/52

:8

In the first Grene) rotation 0o7 in the e7ample a0ove" 9 functions as a tonic throughout"

rather than the pivot note $=O'. ,his happens" despite =O repeating four times" and the 9

occurring only once. ,he repeated Ns and @s/@0s function audi0ly as the dominant and

major/minor mediant respectively8 all other pitches" including the =O pivot" are perceived in

terms of their relation to 9. ,his is partly due to the structural placing of 9 ithin a larger

conte7t" partly due to the sfor6ando" and partly 0ecause it lasts longer than the other notes.

In the second Grene) rotation 0o7" the percepti0le tonal centres shift uic)ly" ith the

pivot note" A" assuming a ne function each time. Initially" it is heard as the +th of :

major/minor. In 2otation :" it is heard as the mediant of an =O minor + th. In 2otations 9 and

D" it is heard as the mediant of an = major + th. ,hus" the major/minor +th sonority of the

original melodic cell proves more important than the repeated pivot note in esta0lishing tonal

centres. ,his is a 0y-product of an intentional spacing decision; ere the outline of the initial

+thnot so strongly felt" perhaps the repeated As might have 0egun to sound li)e tonics an

option I chose not to ta)e in this instance.

In the third Grene) rotation 0o7" the upper voices in the right hand radically alter the

harmonic perspective of the left hand line. I have only shon some of the more poerful

tonal centres" each of hich coincides ith an occurrence of the pivot-note" =. At the start of

2otation :" the = is heard as the dominant of a :0 major % th$ith a flattened supertonic'. At

the start of 2otation 9" the = functions 0riefly as a tonic" as mentioned a0ove. At the end of

the e7ample" immediately after 2otation D" the = is heard enharmonically as an @O" operating

as the mediant of 9O major.

,his e7ample illustrates that here Grene) rotations succeed in facilitating harmonic

clarity" they do so mainly 0ecause they audi0ly transform a distinctive initial harmonic cell

e.g." in the first Grene) rotation 0o7" a -note chromatic cluster" and in the second Grene)

rotation 0o7" a major/minor +th. Ff course" this can only or) if the initial harmonic cell is

presented sufficiently clearly. ,he fact that a pivot note also happens to repeat is normally of

secondary importance. ,o suggest that Grene) rotations esta0lish harmonic cohesion mainly

through the 0anal repetition of a single pitch!ould 0e to sell the techniue short.

2: Anderson does not *ite sggest this$ bt one might easily misinterpret thestatement"


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:

C.TE, (

SE,I&0IS' #O03C.O,D&003 CONCEI/ED


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:3

)eep the composer from inadvertently riting passages that sound tonal CI

partly disagreeEH they do not preclude the possi#ility of functionalism Cmy

italicsE.!+

H the fact is that the compositional constraints imposed 0y the rules of

serialism are much less comprehensive than those imposed 0y the traditional

constraints involved in the composition of tonal music.!&

,he CserialE system isH patently incomplete. ,hat is" it provides no rules for

H" H" for choosing pitch register Cmy italicsE" H" H" and so on. !%

I hold that the Polychordal Approach allos serially-conceived harmony to 0e heard

functionally. ,he Polychordal Approach imposes an additional set of rules onto e7isting

serial rules. ,his ne grammar consistently aims at one thing; to clarify hatever harmonic

voca0ulary might 0e thron up 0y the serial mechanisms. In doing so" the Polychordal

Approach allos far greater cognitive transparency than as hitherto possi0le serially. It

argua0ly completes the rules of serialism or" more accurately" possi0ly represents one of

several potential means of completion" some of hich have yet to 0e devised. In so doing" the

Polychordal Approach is intended to ena0le serialism to finally achieve '"Dectet $!&' and+agatelle $!+'" I generally prefer to

)eep using 0oth" side 0y side.

2 Ibid"$ p"2:2"

23 Ibid"$ p"2:C"

2 Ibid"$ p",C,"


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:

SE0ECTED *I*0IO?,.3

Ad5s" ,homas" and '" pp.1*-1.

Auhagen" olfgang; @7perimentelle ?ntersuchungen 6ur Auditiven ,onalitTts0estimmung

in Melodien" in/0lner +eitr1ge 2ur Musikforschung" vol.1& $Gassel; :osse" 1%%'.

:a00itt" Milton; ,he 4" no> $uly

1%%'" p.>&.

:a00itt" Milton; ,he '.

(amamoto" Mayumi" :otelho" Mauro" and Munger" Margaret P.; Bon-Musicians and

Musicians Perception of :itonality inPsychology of Music" vol.>&" no $Fcto0er !1'"

pp.!>-4.

(ic)s" Michael;


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7C

(ill" 9hristopher 9.; 9onsonance and Dissonance in The .e% 7arvard Dictionary of Music

$9am0ridge" Mass. And #ondon; (arvard ?niversity Press" 1%&*'" pp.1%+-1%%.

(indemith" Paul;A Composer4s 5orld$ 7ori2ons and (imitations$9am0ridge" Mass.;

(arvard ?niversity Press" 1%4!'" reprinted in Contemporary Composers on Contemporary

Music" @7panded @dition" ed. @lliott


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2osen" 9harles;Arnold Schoen#erg $Be Wor); Ji)ing Press" 1%+4'.

nly" vol.1" no> $Fcto0er 1%&+'.

,hompson" illiam =. and Mor" $1%>>''.

hittall" Arnold; Serialism $9am0ridge; 9am0ridge ?niversity Press" !&'.

olpert" 2ita

a polychordal approach to serial harmony - part 1 - online version

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