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RRR EEEE CC U U RRR SSS IIII OO NN N R R E C U U R R S I O O NN N RRR EEE C U U RRR SS I O O N NN R R E C U U R R S I O O N NN R R EEEE CC UU R R SSS IIII OO N N In general terms, the concept of recursion involves the recurrence of an idea at different levels. In set theory, recursion involves transpositions of a set class (i.e., a single prime form) by specific intervals. These intervals of transposition are determined by the structure of the set class itself. For example, take a specific form of the set class (012) with pitch classes D, D#, and E. A recursive statement would include transpositions of the set so that the second occurrence of (012) consists of pitch classes D#, E, F. The third transposition would have pitch classes E, F, and F#. • D D# E • D# EF • E F F# The original set has been transposed by consecutive semitones because the set class (012) consists of two consecutive semitones. In this way, the interval structure of the set—two successive semitones in the same direction, spanning a whole tone—is projected onto a higher level. As a result, the set class (012) has undergone a recursive statement. It sounds three times at the musical surface, and it sounds once more at a higher level that involves all transpositions of the set. To take a slightly more complicated example, consider set class (015). Remember: the intervals within the set determine the levels of transposition that will create a recursive statement. At a glance, we can see that prime form (015) consists of a semitone (interval class 1) between elements 0 and 1, and a major third (interval class 4) between elements 1 and 5. So, transposition by interval classes 1 and 4 should allow us to create a recursive statement of (015). Let’s give it a try: • A# B D# transposed up ic 1 gives us… • B CE transposed up ic 4 gives us… • D# E G# When a set class has been stated recursively, each element should create another large statement of the set with the analogous elements from other sets. Put another way, all of the initial, or ‘zero’‐elements of the three (015)s above (A#, B, and D#) should create another (015). Similarly, all of the 1‐elements (B, C, and E) should create another (015). Finally, all of the 5‐elements (D#, E, G#) should create another (015). You may have noticed that in both of the recursive statements above, each of the pitch classes in the first set also serves as the initial pitch classes in one of the sets. That is only one way to do recursion (a very obvious way, as in Schoenberg’s “Nacht”). However, only the intervals of transposition need to be consistent with the structure of the set class—not the overall direction. For this reason, the following group of sets also makes a recursive statement of (015). • A# B D# transposed down ic 1 gives us… • A Bb D transposed down ic 4 gives us… • E# F# A# To recap using a more familiar set (one not common to free atonality), take the major triad. • CEG • E G# B • GB D Or, one could go the other direction. • CEG • A C# E • FAC It is important to know that the groups of sets above are pre‐compositional sketches. They show the derivations of specific sets (of pitch classes) that can be used to create recursive statements. A composer may decide to state those sets in any order. For instance, given the group of sets immediately above, a composer may start with A‐C#‐E, then write C‐E‐G in a motivically similar fashion, and finish with F‐A‐C.

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RRR EEEECC UU RRR SSS IIII OO NNNRR E C UU RR S IOO NNNRRR EEE C UU RRR SSIOO NNNRR E C UU RR S IOO NNNRR EEEECC UU RR SSS IIII OO NNIngeneralterms,theconceptofrecursioninvolvestherecurrenceofanideaatdifferentlevels.Insettheory,recursioninvolvestranspositionsofasetclass(i.e.,asingleprimeform)byspecificintervals.Theseintervalsoftranspositionaredeterminedbythestructureofthesetclassitself.Forexample,takeaspecificformofthesetclass(012)withpitchclassesD,D#,andE.Arecursivestatementwouldincludetranspositionsofthesetsothatthesecondoccurrenceof(012)consistsofpitchclassesD#,E,F.ThethirdtranspositionwouldhavepitchclassesE,F,andF#.

• DD#E• D#EF• EFF#

Theoriginalsethasbeentransposedbyconsecutivesemitonesbecausethesetclass(012)consistsoftwoconsecutivesemitones.Inthisway,theintervalstructureoftheset—twosuccessivesemitonesinthesamedirection,spanningawholetone—isprojectedontoahigherlevel.Asaresult,thesetclass(012)hasundergonearecursivestatement.Itsoundsthreetimesatthemusicalsurface,anditsoundsoncemoreatahigherlevelthatinvolvesalltranspositionsoftheset.Totakeaslightlymorecomplicatedexample,considersetclass(015).Remember:theintervalswithinthesetdeterminethelevelsoftranspositionthatwillcreatearecursivestatement.Ataglance,wecanseethatprimeform(015)consistsofasemitone(intervalclass1)betweenelements0and1,andamajorthird(intervalclass4)betweenelements1and5.So,transpositionbyintervalclasses1and4shouldallowustocreatearecursivestatementof(015).Let’sgiveitatry:

• A#BD# transposedupic1givesus…• BCE transposedupic4givesus…• D#EG#

Whenasetclasshasbeenstatedrecursively,eachelementshouldcreateanotherlargestatementofthesetwiththeanalogouselementsfromothersets.Putanotherway,alloftheinitial,or‘zero’‐elementsofthethree(015)sabove(A#,B,andD#)shouldcreateanother(015).Similarly,allofthe1‐elements(B,C,andE)shouldcreateanother(015).Finally,allofthe5‐elements(D#,E,G#)shouldcreateanother(015).Youmayhavenoticedthatinbothoftherecursivestatementsabove,eachofthepitchclassesinthefirstsetalsoservesastheinitialpitchclassesinoneofthesets.Thatisonlyonewaytodorecursion(averyobviousway,asinSchoenberg’s“Nacht”).However,onlytheintervalsoftranspositionneedtobeconsistentwiththestructureofthesetclass—nottheoveralldirection.Forthisreason,thefollowinggroupofsetsalsomakesarecursivestatementof(015).

• A#BD# transposeddownic1givesus…• ABbD transposeddownic4givesus…• E#F#A#

Torecapusingamorefamiliarset(onenotcommontofreeatonality),takethemajortriad.• CEG• EG#B• GBD

Or,onecouldgotheotherdirection.• CEG• AC#E• FAC

Itisimportanttoknowthatthegroupsofsetsabovearepre‐compositionalsketches.Theyshowthederivationsofspecificsets(ofpitchclasses)thatcanbeusedtocreaterecursivestatements.Acomposermaydecidetostatethosesetsinanyorder.Forinstance,giventhegroupofsetsimmediatelyabove,acomposermaystartwithA‐C#‐E,thenwriteC‐E‐Ginamotivicallysimilarfashion,andfinishwithF‐A‐C.

Page 2: recursion.pdf

PartIISofar,theinitialsetsinourrecursivestatementhavealwaysservedasthe‘zero’‐elements.Let’sreturntoourfirst(015)withthepitchclassesA#,B,andD#.Wecanderiveagroupofsetswherethegivensetisthe1‐element—wejusthavetoconsiderthestructureof(015)fromthatperspective.Fromthepositionofthe1‐elementwithinan(015),anotherelementisasemitonedown,andanotherelementisaboveattheintervalofic4.Giventheinitial(015)withA#,B,andD#,onerecursivestatementwouldread:

• A#BD# transposeddownasemitonegivesus…• ABbD • DEbG (transposedupic4fromtheoriginal)

Inthegroupofsetsabove,thefirstsetisthe1‐element,thesecondisthe‘zero’‐element,andthethirdisthe5‐element.Goingtheotherdirection,wecanproduceyetanothergroupofsetsthatwillcreatearecursivestatementof(015).

• A#BD# transposedupasemitonegivesus…• BCE• F#GB (transposeddownic4formtheoriginal)

Inthegroupofsetsabove,thefirstsetisthe1‐element,thesecondisthe‘zero’‐element,andthethirdisthe5‐element.Here’sanexampleofrecursioninliterature:Children,ifyoudaretothinkOfthegreatness,rareness,muchnessFewnessofthispreciousonlyEndlessworldinwhichyousayYoulive,youthinkofthingslikethis:BlocksofslateenclosingdappledRedandgreen,enclosingtawnyYellownets,enclosingwhiteAndblackacresofdominoes,WhereaneatbrownpaperparcelTemptsyoutountiethestring.Intheparcelasmallisland,Ontheislandalargetree,Onthetreeahuskyfruit.Stripthehuskandparetherindoff:InthekernelyouwillseeBlocksofslateenclosedbydappledRedandgreen,enclosedbytawnyYellownets,enclosedbywhiteAndblackacresofdominoes,Wherethesamebrownpaperparcel‐Children,leavethestringalone!ForwhodaresundotheparcelFindshimselfatonceinsideit,Ontheisland,inthefruit,Blocksofslateabouthishead,FindshimselfenclosedbydappledGreenandred,enclosedbyyellowTawnynets,enclosedbyblackAndwhiteacresofdominoes,WiththesamebrownpaperparcelStilluntieduponhisknee.And,ifhethenshoulddaretothinkOfthefewness,muchness,rareness,GreatnessofthisendlessonlyPreciousworldinwhichhesayshelives‐hethenuntiesthestring.“AWarningtoChildren”‐RobertGraves


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