Extents and Limits of Serial Techniques (Ernst Krenek) [PDF]

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Extents and Limits of Serial Techniques Author(s): Ernst Krenek Source: The Musical Quarterly, Vol. 46, No. 2, Special Issue: Problems of Modern Music. The Princeton Seminar in Advanced Musical Studies (Apr., 1960), pp. 210-232 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/740372 . Accessed: 13/07/2013 16:12 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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EXTENTS

AND LIMITS OF

SERIAL TECHNIQUES By ERNST KRENEK

T

THE TITLE

HE propensityof presentmusical theoryforterminology originally to belonging mathematicsand physicsis characteristicof a styleof thinkingessentiallydifferentfrom earlier ways of viewing the subject matter.Although some of this language sounds merelypretentious,it has neverthelessadded usefultermsto musical discussion.One of these is the conceptof "parameter."It was introducedintorecentmusictheory by Dr. Meyer-Eppler,of the Instituteof CommunicationTheory at the Universityof Bonn, who was associated with the work of the electronic laboratoryof the West German Radio at Cologne. It is borrowedfrom mathematics,whereit means "a variable enteringinto the mathematical form of any distributionsuch that the possible values of the variable correspondto differentdistributions."' Serial organizationof a certainnumber of parametersof a musical process causes a certain number of other parametersto be left uncontrolled.A detailed studyof the relationshipsof these two areas was the purposeof the seminar.The titledid not, as was surmisedby some, hint of serial between accomplishmentsand shortcomings at a discrimination thinking. DEFINITION

Serial music was definedas a method of compositionthat has been developedas a sequel of the twelve-tonetechniqueinauguratedby Arnold Schoenbergaround 1923. While the serial conceptin that techniquewas embodied in the twelve-toneseries,i.e. an orderingof the pitchesto be adhered to throughoutthe course of the composition,the new idea of I American College Dictionary, New York, 1948, p. 879.

210

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Extentsand Limits of Serial Techniques

211

serialismencompassesall aspects (or "parameters") of the musical process,such as timbre,dynamics,articulation,and above all, time,i.e. durain tionof the individualsoundingelementsand theirmutual relationships In statements. serial time,subordinatingall theseaspectsto premeditated thisview the twelve-tonetechniqueappears to be a special, or limiting, of Newtonianmechanics case of serial music,similarto an interpretation of the of as a limitingexpression Special Theory Relativity,whichin turn has been explainedas a limitingexpressionof that General Theory. METHOD Anton Webern and Olivier Messiaen were mentionedas the bestknowngeneratorsof the new way of serial thinking,the formerbecause of the extraordinary impacthis workhas exercisedduringthe last twenty yearsor so, the latterabove all throughhis experimentswith "rhythmic rows" (or "modes") and his immediateinfluenceon such composersas Boulez and Stockhausen.The discussionthen turnedto the significance and consequenceof the gradual expansionof the musical area that was subjected to premeditatedorganization.It was recognized that serial orderingof the factorof time (i.e. premeditatedfixationof points of entranceand durationof the individual musical elements) caused funand meandamental changesin thestructure, appearance, perceptibility, ing of music. Thereforethe largerpart of the investigationwas devoted to the methodsof organizingseriallythe parameterof time.The discourse was mainlybased on myown workin the serialstylebecause myintimate knowledge of this work allowed succinct presentationof the relevant details, whereas the few available analyses of other composers' serial worksare frequentlyambiguousand far fromenlightening. THE

PRINCIPLE OF "ROTATION"

By rotationwe understanda procedurein which the elementsof a and progressively change theirrelativepositions givenseriessystematically conceivedin thatthe changes is in itself which a to serially plan according occur in regularphases.l I applied this principlefor the firsttime in a large choral work, 2 In his book, Die Komposition mit zw6ilfTiinen, Berlin, 1952, p. 113 ff. and passim, Josef Rufer points out that Arnold Schoenberg occasionally let neighboring tones of his rows exchange places, or groups of tones change their positions within the row. Rufer's discourse and the examples quoted show that this was done sporadically and mainly in order to create a musical context that would not have been served as well by adhering to the premeditated succession of pitches.

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The Musical Quarterly

212

LamentatioJeremiaeProphetae,3writtenin 1940 and 1941. The twelvetone seriesof thisworkreads thus: Ex. I --:--?---?------Each of its two constituentsix-tone groups is progressivelymodified by making the firsttone the last: Ex. 2 W-bL=-----I--

IBPLLf_~i=i~

'I

*.~~;~=

~;;r

~,

The patternsthus obtained may be called "diatonic" since they contain the same six tones.The rosterof patternsis doubled by transposing all those of the leftcolumn of Ex. 2 to begin on F, all thoseof the right column to begin on B. g~I).

i

1'

0

~

4e~-~-,i-.reL

'7

v

Kasscl. 3 B~ienreiter-VVerlag,

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Extents and Limits of Serial Techniques

213

These new patternsare "chromatic" because they eventuallyinclude all twelvetones.The rotationtakingplace was inspiredby the construction of the Greek modal scales and theirtranspositioninto one "characteristic"octave. The purposeof the operationwas not so much to make the serial designstricter, but ratherto relax it, insofaras the wide variety of available six-tone patternsmade it possible to remain within the frameof referenceof the twelve-toneserial techniquewithoutconstantly having to use complete twelve-tonerows. Thus it became possible to give various areas of the compositiondistinctiveharmonic flavors.At that time no attemptwas made to organize seriallythe selection and successionof the rotationalpatterns. A more consistentand systematicapplicationof the principleof rotation may be found in my orchestralwork, Circle, Chain and Mirror,4 The tone-row writtenin 1956 and 1957 forthe Basel Kammerorchester. of this work reads as follows: Ex.

3

2

I

4

4

3

5

5

6

6

7

7

8

9

12

0II

9

8

12

derivativeformsof this In the course of the compositiontwenty-four row are employed.The principleof derivationmay easilybe apprehended by comparingthe originalrow with its firstthree derivativeforms(the tones in theiroriginalsuccessionare numberedfrom1 to 12): Ex. 5 I

3

3

3

5

2

5

4

7

6

9

8

II

10 12

5

2

7

4

9

6

II

8

Z

7

Z

9

4

II

6

12

8

0o

10

The rotation taking place here consistsin forminga retrogradesuccession of each pair of two adjacent tones. Aftereleven such operations one arrivesat the complete retrogradeformof the original statement. The twelvefollowingderivatesrepresentthe retrogradeformsof the first transformation is identicalwiththe original. twelve,and the twenty-fifth The same procedure was applied to the invertedform of the original series (see Ex. 6). This arrangementsuggestedthe "circle" part of the titleof the work. SOriginal German title: Kette, Kreis und Spiegel. Biirenreiter-Verlag,Kassel.

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214

The Musical Quarterly

The sequence in which the forty-eight rowsthusobtained were used in the work was determinedby the decisionto have each originalform followedby the second of the two formsof the inversionwhich would have fortheirfirsttonesthe last tone of the precedingoriginal,while this inversionin turnwould be followedby an originalformbeginningwith the last tone of the precedinginversion.This interlocking arrangementis meant by the term "chain" in the title.The sequence of rows obtained through this operation may be partiallyseen in the followingtable (O=original, I-inversion, R=retrograde, RI-retrograde inversion): I

O 1

R

RI

8

6

12

_

10

4

4 ....._ __

10

12

6

8

8 8

........

1

6 10

4 ....1

2

. ..

12 4 10

...

2

6 2

5

8 ..

22 7

1i

9

3

3

etc.

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Extents and Limits of Serial Techniques

215

The symmetry resultingfromthisorganizationis obvious: The sequence 1, 6, 10, 4, 12 in lines 1 to 9 of the O column is identical with the sequence 1, 6, 10, 4, 12 in lines 13 to 21 of the I column. The same relationobtains as regardsthe sequences 12, 4, 10, 6, 2 in lines 4 to 12 of RI and 16 to 24 of R. The positionsof 18 between 01 and 6 and of 08 betweenI1 and 6 are equally symmetricaland correspondto the positions of the 8s in R and RI between 2 and 6 of RI and R respectively. Ex. 6 I OJ~

I

)C

~-

8

3

10

3

8

6

.

5

12

2

II

4

9

7

10

12

5

II

2

9

4

7

12

I0

II

5

9

2

7

8

11

5

7

6

"---

.

3

I

6

8

3

6

I

12

10

_b

9

..

2

4

4

The term"mirror"finallyrefersto the fact that the musical configuration that opens the work and is expressedin termsof the row 01 returnsin invertedformwhen the serial "conveyorbelt" produces the formI 1, in retrogradeinvertedformwhen the row RI 1 appears (not shown in the above table), and at the veryend of the work in termsof the form R 1. The remainingareas of the music are not any longer occupied by thematicstatement,development,recapitulation,and the like. Whatevermorphologicalkinshipmay be detectedbetweenadjacent sectionsis a resultof similaritiesof intervallicshapes that may occur in neighboringformsof the tone-row,the vicinityof which, however,is a consequenceof the premeditatedserial arrangementoutlined above and not dictatedby requirementsof a so-called musical nature. In thiscompositionno otherparameterbeside the successionof tones was seriallyordered.In this respectit belongs to the provinceof "classical" twelve-tonemusic. It transcendsthat provincein that it allows its structureto arisefromtheserialarrangementof the rotationalderivatives of its tone-row. The principleof rotation,which, as may be seen here, I discovered and utilizedforreasonsnot relevantto the evolutionof pan-parametrical significancewhen I became organization,turnedout to be of far-reaching

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216

The Musical Quarterly

interestedin that kind of organization.The point is that the notion of invariancyinherentby definitionto the concept of the series,if applied that eliminates of configurations to all parameters,leads to a uniformity or surprise.But unpredictability the last traces of unpredictability, ap"atonal" so-called of characteristic music, not to be only especially pears but desirable,or necessary,in any workof art. That the composerswho have made the most consistentattemptsat "total determinacy"are aware of this need transpiresfrom this utterance of Pierre Boulez: "L'inattendu, encore: il n'y a de creation que dans l'impr6visible devenant n~cessit6."5 that resultfromrotational Combinationof the various configurations elementsmeans that the serial constant with (non-rotating) procedure principleof order that governsone set is applied to another,unrelated set (as if one, for instance, would order the numbers from 1 to 5 alphabetically: fivefourone threetwo). Since thisis one of the definitionsof randomness,we meet here forthe firsttimethe factorof chance, which has attainedhigh significancein recentdevelopments. ROTATION AND TIME

Accordingto GyorgyLigeti's analysis8of Pierre Boulez's Structures for two pianos,' the composer has interpretedthe transpositionsof his twelve-tonerow to various pitch levels as a formof rotationand has transplantedthe resultsto the parameterof time in order to obtain an analogous sequence of derivativeformsof his timeseries. 5 "The unexpected, again: there is no creation except in the unforseeable becoming necessary" (Revue musicale, April 1952, p. 119, as quoted in Die Reihe, No. 4, Vienna, 1958, p. 71). It is interestingthat this statement almost verbatim sums up Carl Bricken's brilliant argument about "inevitability and the unexpected" in his analysis of Beethoven's Quartet Op. 18, No. 3 (Some Analytical Approaches to Musical Criticism, in Proceedings of the Music Teachers National Association for 1936, Oberlin; 1937, p. 262 ff.). In Bricken's discourse the "inevitable" is, of course, represented by those musical processes that appear to be most likely to occur within the frameworkof tonal harmony so that they constitute a predictable, "normal" set of events. The "unexpected," then, consists of the deviations from the norm introduced by the genius of the individual composer. In the case of serial music the inevitable is what serial premeditation ordains. The unexpected, however, is not a result of the composer's kicking against the self-imposed limitations,but of the built-in surprise mechanism, as we shall see later on. In my article Is the Twelve-Tone Technique on the Decline? (in The Musical Quarterly, Oct. 1953, p. 523 ff.) I indicated that Boulez in his Second Piano Sonata probably applied the principle of rotation. 6 Die Reihe, No. 4, p. 38 ff. Universal Edition, Vienna.

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Extents and Limits of Serial Techniques

217

The elementsof the tone seriesare numberedfrom 1 to 12: Ex. 7t I

2

3

4

5

6

7

9

8

10

1

12

To this a seriesof time values corresponds,expressedin termsof I

2

3

4

5

6

7

8

9

9

12

0

12

II

If we transposethe tone row, for instance,a major third higher,the originalorderof the tones is changed into: Ex. 8 6

5

8

10

II

4

7

2

3

I

the time serieswould take on the shae: Correspondingly

4 .).. fl! %)J InJ-J. fact, the whole work consistsof manifoldcombinations U,. , of the toneand time-setsthus obtained. 5

6

4

8

9

12

SERIALISM

10

4

IN THE

i-

7

ELECTRONIC

2

3

I

MEDIUM

Karlheinz Stockhausen'sworkdescribedalternatingly as Komposition 1953 No. 2 and ElektronischeStudie 18 is based on a six-toneseries which accordingto the composer'sown elaborate analysis9is an expression of thisseriesof ratiosof frequencies: 8 4 5 5 12 5 5 4 5 12 in vibration numbers,or cycles,persecond,thefirstseriesreads: Expressed 800 1000 1500 1920 625 1200 8 : 5 4 5 : 12 : 5 5 : 12 4 : 5 In notesit reads approximately: Ex. 9 Five more seriesare derivedby makingthe consecutivetonesof the first series points of departurefor new series identicallybuilt (a procedure somewhatreminiscentof my Lamentatio rotation): 800 333 417 625 500 260 1000 417 781 521 325 625 625 325 488 260 203 390 1500 781 625 488 1170 937 500 1200 390 625 937 750 8Universal Edition, Vienna. Recorded by the Deutsche Grammophon Gesellschaft. STechnische Hausmitteilungen des Nordwestdeutschen Rundfunks, Vol. VI, No. 1/2, Cologne, 1954, Item 10, p. 46 ff.

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The Musical Quarterly

218

A second set of six seriesis obtained by making the second line of the firstset the top line of the new set,then the third,and so on. All parametersare seriallyordered in termsof some variantsof the numericalsequence 1 to 6. For instance,the combinationsof the above frequenciesfollow from the series 4 5 3 6 2 1 in that the firsttonecombination("Tongemisch") has fourtones,the second five,and so on. There are foursuch "Gemische" in "sequence 1" (a "sequence" being a grouping of consecutiveelements), and four "sequences" in the first "structure,"which is the next highercompound, "horizontal" or "vertical." (It does not become quite clear on what groundsone or the other of thesetwo dimensionswas chosen.) There are six dynamiclevelswhich are assigned to the various frequenciesin proportionto their relative positionsin the groupsand columnsof the entiresystem.The seriesthat ordersthe successionof dynamiclevels withinthis frameof referenceis 3 4 2 1 6 5. Finally,the time factoris determinedby relatingthe durationsof the individualsoundingelementsto the pitch levels and degrees of loudnessof those elementsas orderedby the previousrules.The governingseriesin this parameteris 2 4 6 3 5 1. The details of this organizationare far more complex than what we the preare able to indicate here in an abridged sketch.Unfortunately sentationby the author is not always felicitous,so that some of the intricaciesof his work remain obscure. At any rate, the characterof his reasoningseemsto reveal a desireto derivethe rulesof serialorganization fromthe natureof the chosen materialand its intervallictexture.In this respect Stockhausen differssomewhat from Boulez, who has a rather mechanisticapproach in assigningnumericalvalues to the various magnitudes manipulated in his work. While this procedureof Boulez's has it has nevertheless produceda fascinating been criticizedas "anorganic,"o piece of music. On the otherhand, Stockhausen'sStudie, althoughmuch shorterthan the Structures,suffersfromconsiderablemonotonyof harmonic flavor,which is due to the prevalenceof augmentedtriadsin the subtletiesof combinations originalseries (see Ex. 9). The extraordinary of dynamicshadings,timevalues, echo effects,and the like cannot overcome thisinitialhandicap. The objection was raised that music here becomes the victimof an abstractnumbersgame which is contraryto the nature of music. While thereundoubtedlyis room formore than one definitionof the nature of music, we did not extend our inquiryinto this field.The numbersused 1o Ligeti, loc. cit., p. 41.

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Extents and Limits of Serial Techniques

219

in the orderingof the parametersof serial music are almost always derivedfromproportionsand measurementsof the basic musical substance. Of course,thesenumbersdetach themselvesfromthe objects withwhich theywere associatedand take on a lifeof theirown in the variousoperationsperformed.The resultsof theseoperationsare, however,retranslated into musical termsand applied to the soundingmaterial.In thisrelation of numberand realityone may see a vague analogy to the connectionof contemporarymathematicsand physics. PREMEDITATED,

BUT

UNPREDICTABLE

In myoratorioforvoicesand electronicsounds,Spiritusintelligentiae, sanctus,"thereis a sectionwithoutvoices (so to speak an "instrumental" interlude). The material of this sectionis a temperedscale of thirteen tones.From the continuumof thisscale, groupsof toneswere selectedto form alternatinglydisjunct and conjunct heptachords of equal and structure(see leftside of Diagram 1). A seven-tonepattern symmetrical (seven-tonerow) meanders throughthis systemof pitches constantly retainingits principleof progress: from any tone on which it startsit goes up to the thirdand fourth,thenback to the second, up to the sixth, back to the fifth,and it stops on the seventhtone of the networkof pitches.Since the patternalwaysprogressesconjunctly(which means that the firsttone of its next appearance is identicalwith the last of the preceding) while the pitch systemis based on the alternationof conjunct of the pattern and disjunctshapes, the internalintervallicconfiguration is always different, although its general outline remains the same (see rightside of Diagram 1). Afterthirteenappearances the patternlands again on the tone fromwhich it started,and the "rotation"has come to an end. The interludein question may technicallybe called a double canon. One of the two elementssubject to imitationis a tone-lineconsistingof the chain of the thirteenpossiblevariantsof the seven-tonepatternjust described,the otheris an analogous line presentingthe chain of the inverted formsof the pattern.The firsttone-lineis so designed that it begins on the centraltone of the entiregamut (330 cycles), risesto its highestlevel (4754 c) in the firstthirdof itslength,returnsto the center in the second third,and descends to the lowest level (26 c) in its last portion.The second line beginson the lowestpointwhen the firstreaches its apex, risesto crossthe firstline whereit passes on its descentthe central tone, goes up to its own high point which it reaches approximately when the firstline ends, and returnsto the center. 11Recorded by the Deutsche Grammophon Gesellschaft (LP

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16134 Hi-Fi).

220

The MusicalQuarterly

Diagram1 Read frombottomup Heavy linesindicateoctaves Chain of disjunctand conjunct heptachords

Progressof the seven-tonepattern

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Extents and Limits of Serial Techniques

221

The canonic imitationswere obtained by rerecordingthe original material at a higherand a lower speed, in which procedure the pitch level of the originaltape was automaticallyraised or loweredin the same proportion.These imitationswere so synchronizedwiththe originallines that the slowed-downversionof the ascendingbranch of the firsttonelowerthan thesummit line would reach itshighestpoint (proportionately of the original) when the originalline had returnedto the center.It was followedby the slowed-downimitationof the descendingbranch of the second line. The above-centerarcs of both lines were imitatedin accelerated versionsreaching their (proportionatelyhigher) apices shortly afteror beforethose of the original lines. Finally, a veryhighlyaccelerated imitationof the below-centerbranchesof both lines was inserted shortlybeforethe end of the section. To determinethe timevalues of thesingleelementsthe whole expanse of the piece was viewed as one unit. Through measuringthe linear distances of the importantpointsof articulation- entrancesof imitations, turningpointsand such - a seriesof elevenspans was established,a sort of macro-rhythm articulatingthe over-all structure.It was reduced in scale to a micro-rhythm In order to determinethe durations of the individual tones in each tone-line.Since each line takes approximately three quarters of the entirelengthof the piece and each line contains of eleven ninety-onetones (seven times thirteen), the micro-rhythm values has to be repeatedeighttimes,leavingthreetonesfreeat the end. had to be This conceptdeterminedthe ratio by which the macro-rhythm reduced. Since the rhythmicseries thus establishedhas eleven terms whereas the tone-serieshas onlyseven tones,it followsthat the last four termsof the firsttimeserieswill apply to the firstfourtonesof the second tone series, and so forth,so that here again mechanical repetitionis in a highersense is maintained. (See Diagram avoided while uniformity 2.) It may be statedthatwhateveroccursin thispiece at any givenpoint is premeditatedand therefore technicallypredictable.However,whilethe of and the layout the materialas well as the operationsperpreparation formedthereinare the consequence of serial premeditation,the audible resultsof these procedureswere not visualized as the purpose of the procedures.Seen from this angle, the resultsare incidental. They are also practicallyunpredictablebecause the simultaneousprogressof highly complex rhythmicpatternsat various relativespeeds togetherwith the correspondingtranspositionsof equally complex pitch patternscreates situationsthat defyprecisevisualization.

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Diagram2 Fc

a"

F

II

IB lie

t

,

I I

I

I I

c

I

L'ne

, M&re-

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End

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a

|

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s

Extents and Limits of Serial Techniques THE

TIME

MECHANISM

223

OF MY "SESTINA'12

The Sestina is one of the poetic formsdeveloped by the Provencal poets of the twelfthcentury,its original specimen being ascribed to Arnaut Daniel. It may well be called a serial formof poetry,and its essentialformativeprincipleis rotation. The poem consistsof six stanzas of six blank verseseach. It hingesupon six keywordswhich appear at the endings of the individual lines. If in the first stanza the order of these words is 1 2 3 4 5 6, the words will appear in the second stanza in the order 6 1 5 2 4 3. The principle of rotationwhich is applied here consistsin switchingthe positionof everytwo keywordsequidistant from the center of the series, proceeding from the end toward the middle. Accordingto the same principle,the positionsof the keywordsin the subsequent stanzas are 3 6 4 1 2 5; 5 3 2 6 1 4; 4 5 1 3 6 2; 2 4 6 5 3 1. The process ends here, since the next rotationwould produce the original series. The six stanzas are followedby a Tornada of threelines in which the keywords,one of each pair in the middle and the other at the end of the line, appear in the order 2 5, 4 3, 6 1. The contentof the Sestina which I wrote (in German) as textforthe present compositionis a contemplationof the implicationsof the idea governingthe of the work.13 musical construction

The firsttwo stanzas may sufficeto indicate the characterand form of the poem: 1. VergangenKlang und Klage, sanfterStrom. Die Schwingungder Sekunde wird zum Mass. Was in Geschichtelebt, war's nur ein Zufall? Verfall,Verhall, zerronneneGestalt? Die Stunde zeitigtWandel, wendetZeit. Das Vorgeschrittne ordnetsich der Zahl. 2. In Schrittenvorgeordnetdurch die Zahl gestaltetsich Gedanke,doch zum Strom wird strengeTeilung,uhr-genaueZeit. Ist es vermessen,solches Mass von Mass dem Leben aufzuzwingen, der Gestalt? Der Zwang zerrinnt,erzeugtden neuen Zufall.14 12Birenreiter-Verlag,Kassel. Epic Records, LC 3509. 13 Quoted frommy noteson

the jacket of the recordcited in note 12.

14In a nearly literal translationwhich reproduces the positions of the key words: Bygone are sound and mourning,tender stream. Vibration of the second becomes the measure.

What lives in history, was it only chance?

Decline, fading sound, vanished shape? The hour causes change, turns the time. What looks ahead subordinates itself to number.

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The Musical Quarterly

224

The music of my Sestina is based on a twelve-tonerow divided into two groupsof six toneseach: B A Ex. Io

,

4

3

1

a

6

I

6

2

4

6

s

(2)

The figuresindicate the size of the intervalsmeasured in half-steps. These tone-rowsare rotatedaccordingto the principleof the sestinaso that the second A- and B-groupsread: Ex.

rI ;I ,'-I"

,.-

4

4

6

,

"

3

5

4

z 2

(1)

The thirdline is: Ex.

12 S

4

2

44

I

3

I

4

6

4

and so forth.The tones are always placed so that theywill not exceed the ambitusof the originalrow and the intervals(indicatedby the numbersbelow the staff) are so measuredup or down that theirmagnitudes will not exceed the figure6. Obviouslythe sequence of these intervallic magnitudesconstantlychanges as a resultof the rotationof the tones prescribedby the sestina pattern,but these changes are of a different order. The durationsof the tonesof the whole compositionare derivedfrom these magnitudesin the followingmanner: each intervallicmagnitude correspondsto a time segmentwhich containsas many basic time units as the intervalfigureindicates.Consequentlythe firsttime segmenthas four units,the second three,etc. Each segmenthas as many tones as it has units (4, 3, etc.). The durationof the individualtonesis determined by a subdivisionbased on the same serial sequence of magnitudes.If the firstsegmentcontainsfour units and fourtones,its subdivisionis based on the firstfour values of the original series: 4 3 1 6. The sum of these being 14, the subdivisionunit within the firstsegmentis 4/14. The durations of the individual tones within the firstsegment are determinedby multiplying4/14 consecutivelyby 4, 3, 1, and 6. The In stages preordained by number thought takes shape, but a stream is (the result of) strict division, of clocklike, precise time. Is it presuming to force such an extent of measure on life, on shape? Force vanishes, brings forth new chance.

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Extents and Limits of Serial Techniques

225

durations,then, are 16/14, 12/14, and 24/14, or 8/7, 6/7, 2/7, and 12/7 of the basic value. Actually the determinationof the durationsis due to much more complicatedcomputationbecause it is influencedby serial organization of other parameters.In order to achieve higherrhythmicdiversity,the concept of "internalspeed" was introduced.It is derived fromthe assumptionthat in every group of six tones one to five tones might be sounded an octave higherso that the magnitudeof the affectedintervals would be augmentedby twelve. The successionof "internalspeeds" is derived fromthe position of the tones in group B (see Ex. 10). The lowest (A) is designatedas 1, the highest(F) as 6. The initialrow of internalspeeds is therefore5 1 4 3 6 2. The firstsegment,then,has the internalspeed 5 so that 12 is added to fiveout of six subdivisionnumbers. Thus these numbersread 16 15 13 18 14 instead of 4 3 1 6 2. The followingnumber- 1 - remainsunaltered.The sum of the numbers attached to the firstsegmentis therefore62, instead of 14. Consequentlythe durationsof the individualtoneswill be considerablyshorter than if the "internalspeed" were, for instance,1 or 2. To facilitatecomputationseach basic unit is assumed to contain ten micro-units.We arriveat the subdivisionof the firstsegmentby dividing 40 (fourtimesten) by 62. The resultis 0.645. This numberis multiplied consecutivelyby 4, 3, 1, 6. The resultsare 2.58, 1.935, 0.645, 3.87. If the work had been realized by electronicmeans on tape, these values could be produced with utmost accuracy. Since it was conceived for conventionalmannersof rendition,the time values had to be adjusted as follows: 2.5, 2, 0.5, 4. If the smallestnumericalunit is expressedby ~ therhythmic shape of the firstfourtonesis , __, S9/16. "Density" is the next parameter to be determinedserially.There are six degreesof densitywhose successionis determinedby the position of the pitchesin group A (Ex. 10). Again the lowest (C) is called 1, the highest (G#) 6. Consequentlythe initial seriesof densitiesis 6 3 5 4 1 2. In "density1" the two tone-groupsA and B run offsimultaneously in a sortof two-partsettingin whichthe durationof the individualtones is determinedby the mechanism described above. In "density2" the firstand second time segmentsof group A run concurrentlywith the firstsegmentof group B. In "density3" two segmentsof each group are and so forth,untilin "density6" six segments developed simultaneously, of each group,i.e. twelveall together,run offat the same time.

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Anotherparameteris the location of the tones withinthe gamut of six octaves designatedas the ambitusof the work.The serial statement adopted for this area reads that the tones of each segmentshould run throughas many octaves as thereare tones.The directionof the motion is determinedby the direction of the correspondinginterval in the original series. Since many segmentscontain less than six tones, they coverless than six octaves and thereforecould extendover variousbands of the complete ambitus. This, too, is regulatedby special serial statements.Needless to say that all theseserial organismsare subject to rotation accordingto the sestinapattern,whichis the supremelaw governing everymove of everyvariable withinthe whole composition. The structurallayout is designedto combine each "rotated" version of any six-tonegroup witheveryother.Thus the musicof the firststanza is based on the firststatementof the A-group,in each consecutiveline of the poem combined with one of the formsof the B-group rotated fromB 1 to B 6. The second stanza has A 2, combinedagain with all six B-groups,but now in a different sequence, accordingto the sestina B 3. B B B B B 1, 5, 2, 4, pattern: 6, Paralleling the arrangementof the key words in the tornada, the tone series assigned to it reads 2 5 4 3 6 1. The music of the tornada consistsof six sections,the firstfourand the last of which are givenover alone. While the tone row of the tornada undergoes to the instruments the densityincreasesfrom the now familiarsix sestina transformations, tornada of the section onlyone each of the A1 to 6 so that in the first and B-rows are employed,while in the last sectionsix of each, that is twelve,or all available formsare used simultaneously. The parameterof "externalspeed" has six stepsalso, the lowestbeing 180. The formeris associated with the M 90, the highest ) ) highestdegree of density,the latterwith the lowest. Example 13 shows the firstten sixteenths(micro-units)which form the firstbasic time unit of the Sestina. On the leftside one may see the distributionof the tones of the A- and B-groupsover the twelve layers timesegments,each tone enterprogressing (density6) of simultaneously ing at the point assignedto it by the time mechanismexplained above. The tones occupy theirplaces fromtop to bottomlayerin theirorderof successionin the row. The "internalspeed" for the A-layers(top six) is 5, forthe B-layers (bottomsix) 1 (no acceleration). Encirclednumbers indicate the number of tones allotted to the particularsegments. Arrowsindicate the directionof the tone lines. The figuresabove the

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Extentsand Limits of Serial Techniques

227

top staffgive the durationsof the firstfourtonesin , as computedon p. 225. The rightside of the example shows how these tones are represented in the actual score, and a few connectinglines were drawn to demonstratewheresome particulartonesmay be found. Ex. '3

A

0.5

)2.5

.. .. _ .----,--:.=--,

A,

'-

4 -

-FkA 14 1

\'

II

_-

B~

_-

?~ ~

,-=

1,-

,

: i

6

,Gu.

10 s

?~,L

irr

It is easy to see that the parameterof timbrelies beyond the limits of the presentserial arrangement.If this parametertoo were organized seriallyand this procedurewould, for instance,require the firsttone of the top layer (G.) to be played by the trumpet,it would obviouslybe

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The Musical Quarterly

at variance with the octave registerdemanded by the serial regulation of spacing,since the trumpetcannot play the G$ in question. THE

ELEMENT

OF CHANCE

Other parametersmay be affectedin the same way. If the succession of tonesis determinedby serial regulation(as is the case in the classical twelve-tonetechnique) and, in addition to this, the timing of the entranceinto the musical processof thesetonesis also predetermined by serial calculation (as, for example, in the case of the Sestina), it is no longer possible to decide freely(that is, by "inspiration") which tones should sound simultaneouslyat any given point. In otherwords,the socalled harmonicaspect of the piece will be entirelythe resultof operations performedon premisesthat have nothingto do with concepts of "harmony,"be it on the assumptionof tonalityor atonalityor anything else. Whatever happens at any given point is a product of the preconceivedserialorganization,but by the same tokenit is a chance occurrence because it is as such not anticipatedby the mind that inventedthe mechanismand set it in motion. Generallyand traditionally"inspiration"is held in great respectas source of the creativeprocessin art. It should be the most distinguished rememberedthat inspirationby definitionis closely related to chance, forit is the verythingthat cannot be controlled,manufactured,or premeditatedin any way. It is what falls into the mind (according to the German term Einfall), unsolicited,unprepared, unrehearsed,coming fromnowhere.This obviouslyanswersthe definitionof chance as "the absence of any known reason why an event should turn out one way ratherthan another."'5Actuallythe composerhas come to distrusthis inspirationbecause it is not reallyas innocentas it was supposed to be, but ratherconditionedby a tremendousbody of recollection,tradition, training,and experience.In orderto avoid the dictationsof such ghosts, he prefersto set up an impersonalmechanismwhichwill furnish,according to premeditatedpatterns,unpredictablesituations.Ligeti characterizes this state of affairsverywell: "We stand in frontof a row of slot machines ["Automaten"] and we can choose freelyinto which one we want to drop our coin, but at the same time we are forced to choose one of them. One constructshis own prisonaccordingto his wishesand is afterwardsequally freelyactive within those walls - that is: not entirelyfree,but not totallyconstrainedeither.Thus automation does not functionas the opposite of free decision: ratherfreeselectionand 15 The American College Dictionary, New York and London, 1948, p. 200.

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Extents and Limits of Serial Techniques

229

mechanizationare united in the process of selectingthe mechanism.""' In other words,the creativeact takes place in an area in which it has so far been entirelyunsuspected,namely in settingup the serial statements (selectingthe slot machines). What happens afterwardsis predeterminedby the selection of the mechanism,but not premeditated exceptas an unconsciousresultof the predetermined operations.The unexpectedhappens by necessity.The surpriseis built in. LAYERS AND DENSITIES

A later serial work of mine is a set of six piano pieces, called Sechs Vermessene.This German title is a play on words, since vermessenin German means "completelymeasured" as well as "presuming,"a pun that cannot be reproduced in English. While the time mechanism is similarto that of the Sestina, the constructiondiffersfromit in that for the firstthree pieces a systemof fivelayersis set up in which the first has "density 1" (i.e. one tone at a time), the next has two tones together,the third three,the fourthfour, and the fifthsix tones. The time measurementsforthe variouslayersare a resultof summingup the intervalmagnitudesinvolved in the consecutivetone combinations.For example, the tone seriesof this compositionbeing: .

.... _ 1_

_ ,

..

,

the firstcombinationof tonesin "density2" is: Lx.

15

The numerical values derived from this progressionare 3 (a minor thirdfromG to Bb) and 1 (a half-stepfromE to F). Consequentlythe firsttimesegmentof the firstlayerhas threeunits,the firstof the second has four (3 + 1). As the densityof the layersincreases,the numberof simultaneously soundingintervalsand thus the numericalvalues of their sums become higher.Thereforethe timesegmentsbecome longer,which with increasingthicknessare means that the chords, or tone-clusters, spaced fartherapart, while the singletones of the firstlayerfolloweach other more rapidly. Computationsof this kind form the basis of the whole composition. As explained before,phenomenain the parameterof harmonymust be accepted as resultsof the operationsin the sectorsof pitchsuccession and time. In the fourthof the piano pieces an attemptwas made to 16

Loc. cit., p. 38 (translated from the German by this writer).

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begin with a selectionof sound elements.From the tone row we developed twelve sets of four elementseach (consistingof one, or of two, three,or four tones played simultaneously)plus two six-tonechords. These fiftyelementswere numberedfrom 1 to 50 and theirsuccession was determinedby progressing along thisseriesby the distancesindicated in the numericalvalues of the intervalsof the basic row: 1 2 3 4.5 6 7 8 9 10 11 12 13 14 15... seriesof elements: 4 2 5 intervalsof the tone row: 3 15 ... 11 4 6 1 selectedelements: In the fifthpiece the five degrees of thicknesses(see above) are distributedover fivelayerswhich progressat various speeds so that the timemeasurementsof the slowestlayerare reducedto 1/2 in the second, to 1/3 in the third,to 1/4 in the fourth,and to 1/6 in the fastestlayer. PROGRESSIVELY

VARYING

SERIES

In the field of serial music one may observe a tendencytowards vary according to some using series of magnitudesthat progressively serial orderingof their own. The speed levels of the Sestina are an example. Anothertimeseriesof thisnaturewas establishedforthe voice line of this work. It is based on the successionof 1 2 3 5 7 and 10 ) for the accented syllables.The openingsuccessionis 2 3 10 5 7 1 and the followingforms are obtained throughthe sestina rotation. Since situaeach line of the poem has only fiveaccented syllables,interesting tions of overlaps occur. It may be seen that the serieshere applied is a modificationof the so-called Fibonacci seriesin which each termis equal to the sum of the two precedingterms: 1 2 3 5 8 13 21 34 55 etc.1'Luigi Nono has used the firstsix termsof this seriesas factorswith which he multipliesthe basic time values of his II Canto sospeso in order to obtain the actual I have used the termsof the Fibonacci durationsof the individualtones.18 seriesfrom2 to 21 to determinethe speed zones in a recentorchestral compositionentitledQuaestio temporis(A Questionof Time). This work is based on a twelve-tonerow that containsall eleven intervalsin this order (measured in half-steps): 3851011612749 The entireexpanse of the compositionis thoughtof as consistingof 66 of Art and Life, New York, 1946, p. 13 f. 1' Cf. Matila Ghyka, The Geometry

' Cf. Karlheinz Stockhausen's analysis of the work in Darmstiidter Beitrage zur neuen Musik, Mainz, 1958, p. 70.

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Extentsand Limits of Serial Techniques

231

time units (the sum of the above figures),which formeleven sections of varyinglengthsaccording to the magnitudesof the basic series.To thesesectionssix different speeds are assigned: M -= 20, 30, 50, 80, 130, and 210 THE

CONCEPT

OF DENSITY

GENERALIZED

It appears thatdensityis a functionof speed and thicknessof texture. If the lattermay be called the verticalcomponentof densitybecause it depends on how many layersare in operationat the same time, speed is the horizontalcomponentof densitysince the tones followeach other more closelythe fasterthe tempo of the music is. If both parameters approach maximumvalues, a degree of saturationis reached at which accurate computationsof time points and durationsbecome irrelevant. When in the finalsectionof Quaestio twelvelayers (maximum vertical density) progressat a speed of J = 210 per minute,the tones come so close togetherthat nearly everysixteenthis sounded, frequentlyby to The velocityof the music causes 14 several tonessimultaneously. run offper second. At thisrate even the successionof pitchesis not any to determineby experiment longerof greatsignificance.It seemssufficient withina limitedarea the average numberof time unitsneeded for running throughthe twelve-toneseries.The resultsof thisstatisticalexamination are then used in order to fill this area of highestdensitywith actual musical sounds. WHAT

DOES

SERIAL

MUSIC

"MEAN,"

IF

ANYTHING?

One of the parametersthat obviouslycannot be controlledby premeditationwhen those so far discussedare subjected to serial ordering is the expressive,or communicative,aspect of music. If a serial composer were concernedwith this problem,he would have to set up a seriesof "moods," or "ideas," or somethingof thissort,to begin with,and then let the otherparametersfall in line. It so happens that serial composers are not thinkingin such terms. In a more pessimisticattitudethan he now seems to entertain,the German composer and philosopher,T. W. Adorno, has criticizedthe recentdevelopmentsof serial music'9because in these the (according to of music and speech him) deep-rootedand essentialanalogy and affinity is abandoned. While it may be true that music fromthe time of plainchant has been orientedtowards speech-likearticulation,diction, and over-allstructure,and while especiallythe exploitsof Expressionismand 19Das Altern der neuen Musik, in Der Monat, May 1955.

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atonalitypoint to a veryclose associationwith the free articulationof prose,we have to face the fact that under the influenceof the constructive rigorthatwas the veryconsequenceof Expressionistic roamingserial music has turned away fromits rhetoricalpast. Since whatevermusic seemsto communicateis not so much the supposedcontentof the audible matteras it is the product of the listener'sreactiontouched offby his auditory experience,there is no reason to assume that the nature of serial music excludes the possibilityof interpreting it as a medium of some sortof communication.The interestit may evoke is similarto that elicited by the process of life, to which serial music is related in the traceable paradox of the chaotic appearance of totallyand systematically itself. as or as as life It mean much little causality. may

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