Received April 2017
“What can they have to do with one another?”:Approaches to Analysis and Performance in John Cage’s
Four
2
*
Drake Andersen
! #: he e$amples for the %te$t&only' P() version of this item are availale online at:h+p:,,wwwmtosmtorg,iss.es,mto/0123,mto/0123andersenphp
4#5W!6(7: John Cage8 indeterminate m.sic8 performanceA97 6AC : n this st.dy8 eval.ate the sonic possiilities of John Cage’s
Four
2
%/;;<' y comparinge$isting performances of the piece with alternate renditions generated comp.tationally and y handspeci=cally for this analysis
Four
2
8 one of Cage’s .mer Pieces8 is f.lly determinate with respect topitch8 instr.mentation and overall d.ration8 .t a>ords the performer the e$iility to choose thed.rations of speci=c so.nds thro.gh time&rac@et notation he di>erences in the m.sical res.lts etween vario.s performances8 oth real and virt.al8 prompt a disc.ssion of the performancepractices of the piece oth as o.tlined y Cage and as .nderstood y scholars and performersAccompanying this te$t is a comp.ter program with which readers can edit and play ac@ their owninterpretations of
Four
2

ol.me 128 .mer 38 (ecemer 1</0Copyright B 1</0 7ociety for .sic heory
1. Introduction
D//E he time&rac@et notation that John Cage employs in his .mer Pieces permits performersto prod.ce di>erent renditions of a given composition y choosing the position and d.ration ofso.nds within a e$ile time range
%/'
hese wor@s8 from late in Cage’s career8 are typically f.llydetermined with respect to pitch8 dynamics8 instr.mentation8 and total d.ration Fowever8 theirrelative e$iility and Cage’s own poetics of non&intention have inhiited analytical disc.ssion ofsonic relationships8 even tho.gh the possile variation of the so.nds themselves is highlyregimented
%1'
D/1E n this st.dy8 analyGe the sonic relationships in one s.ch piece8
Four
2
%/;;<' for mi$edchor.s8 y approaching the wor@ from three interpretive perspectives: a onte Carlo sim.lation of
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many %virt.al' performances of the wor@H real&world performancesH and fo.r imaginedperformances8 designed int.itively y the a.thor to reect distinct interpretive priorities emodied y fo.r invented cond.ctors
%2'
hese divergent strategies prod.ce characteristic res.lts thatemphasiGe competingIand sometimes contradictoryIconcerns in the performance of Cage’s.mer Pieces a@en as a whole8 they contri.te to a more complete acco.nt of the m.sicalpossiilities of
Four
2
D/2E )irst8 descrie the methodology of the onte Carlo sim.lation and s.mmariGe the dataotained therein to estalish some characteristic li@elihoods in the piece n the ne$t section8 introd.ce each of the =ctional cond.ctors’ interpretations thro.gh a score&li@e transcription and rief analytical narrative hese interpretations were generated y the a.thor to ill.strate speci=cpossiilities within the piece any aspects of them are statistically .nli@ely8 .t are incl.ded hereto provide a more comprehensive .nderstanding of what is possile in a performance of
Four
2
 hesim.lation data and =ctional cond.ctors’ performances are then eval.ated alongside real&worldperformances8 transcried from commercially availale recordingsD/3E his comparative approach responds to J.dy ochhead’s s.ggestion to “DapproachE the m.sicfrom the perspective of what performers do8” as well as concerns e$pressed y Ale$andre Popo>regarding how to acco.nt for di>erences etween comp.tational sim.lations and liveperformances of Cage’s .mer Pieces %ochhead /;;3 8 13/H Popo> 1</2 8 1<8 13' he live
performances vary in some ways from the sim.lated performancesH accordingly8 proceed with adisc.ssion of the performance practice of
Four
2
,
and the .mer Pieces in general8 in order toacco.nt for some of the di>erences he variety of performance practices in .se s.ggests roaderK.estions ao.t the e$tent and meaning of indeterminacy in the .mer Pieces8 which address inthe concl.ding sectionD/LE
Example 1
gives the soprano part of
Four
2
 he soprano part consists of three so.nds8 eachpositioned within a .niK.e time rac@et Cage’s time&rac@et notation speci=es two windows oftime for each so.nd: one in which each so.nd may egin and one in which each may end
%3'
he=rst rac@et8 consisting of an )3 s.ng on the vowel “e8” allows for the so.nd to egin anywhere etween <’<<” and /’<<” from the start of the piece8 and for the so.nd to end anywhere etween<’3<” and /’3<” his means that8 for e$ample8 an M<&second so.nd eginning at <’/<” and endingat /’2<” wo.ld e permissile8 as wo.ld an M&second so.nd eginning at <’L/” and ending at <’L;”n other words8 the time&rac@et notation allows for a wide range of d.rations
%L'
D/NE A .niK.e feat.re of these rac@ets is the overlap etween the starting and ending windowsIin this case8 etween <’3<” and /’<<” his internal overlap allows for greater sonic indeterminacy8permi+ing8 for instance8 the e$ile placement of so.nds of e$tremely short d.ration8 and avoidingan intermediary interval of oligatory so.nd etween the starting and ending intervalsD/0E
Example 2
is a transcription of the twenty time rac@ets of the fo.r parts in
Four
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into a score&li@e format8 .sing solid horiGontal lines to represent individ.al so.nds Phonemes and dynamicsare also incl.ded for each so.nd As the legend elow the score e$plains8 the starting interval ofany given time rac@et e$tends from the leftmost edge of the lac@ line to the right edge of theinternal overlap o$ in the middle of each line he ending interval of any given time rac@ete$tends from the left edge of the internal overlap o$ to the rightmost edge of the lac@ lineD/ME !ften the time rac@ets of two s.ccessive notes in the same part overlap e$ternally8 s.ch asthe =rst two notes of the tenor part n the transcription aove8 when these notes are di>erent8 it isgenerally clear vis.ally where the two rac@ets egin and end Fowever8 the last two notes of thealto part overlap e$ternally on the same pitch %('H in this case8 .se a do+ed line to indicate thee$tent of the overlap
2. Computational Simulation of
Four
2
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D1/E Cage’s .mer Pieces have rarely een analyGed in detail for two primary reasons: theirscores represent a m.ltiplicity of possile performances in which so.nds’ order and d.ration canvary widely8 and a teleological analytical narrative seems to contravene Cage’s description of hisindeterminate wor@s and what we @now of his compositional process will address the la+erconcern in the concl.sion of this articleH regarding the former8 the onte Carlo sim.lation providesa potential sol.tionD11E Osing this techniK.e8 virt.al performances of
Four
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are otained y comp.tationallydetermining the random variales %ie the starting and end points of the so.nds' corresponding toCage’s notated constraints Popo> 1</2 advocates this approach:9y averaging over a large n.mer of realiGations %which is achieved thro.gh acomp.ter program r.nning the determination of the parts repeatedly' we can accessthe proaility distri.tions of each pitch&class set over time8 th.s t.rning the .merPieces into stochastic processes 9y doing so8 we solve the prolem posed y Fas@insand Weisser of coping with all the possiilities o>ered y the .mer Pieces %Popo>1</2 8 2'D12E have carried o.t a onte Carlo sim.lation of over one tho.sand virt.al performances of
Four
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H however8 efore disc.ssing the res.lts8 will raise a few speci=c points regarding themethodology )irst8 created a specialiGed random val.e generator .sing a$ software Completeperformances of the piece were generated as lists of time val.es %corresponding to the eginningsand endings of so.nds' .sing the interval of a second as oth the generative grain and samplingrate8 since this is the smallest .nit of s.division in Cage’s notation As each performance lastse$actly seven min.tes8 each sim.lation contains 31< data points8 each corresponding to one secondof the performanceD13E r.ly random selection of val.es within the %overlapping' starting and ending rac@ets ofeach so.nd means that8 in a comp.tational environment8 a so.nd co.ld end efore it egins oavoid this parado$8 the software chooses a starting point =rst8 then tests randomly&selected endingpoints8 reecting and discarding any that precede the starting point in time he entire“performance” is then tested for any e$ternal overlaps8 and if any are fo.nd8 the entireperformance is discarded and a new performance is generated in its placeD1LE Additionally8 in the software have estalished that a so.nd is eK.ally li@ely to egin or endat any time d.ring a given rac@etIin other words8 amongst many performances of the wor@8 thedistri.tion of each rac@et approaches .niformity his random %.niform' approach may econtrasted with that of Popo> 1</28 in which a Qa.ssian c.rve is .sed to distri.te starting andending points within the time rac@ets
%N'
his approach emphasiGes the center of each starting andending rac@et and “gives less prevalence to so.nd events occ.rring at the very eginning or endof their time interval” %Popo> 1</2 8 L' he random approach am employing emphasiGes all partsof each rac@et eK.ally8 enriching the analysis with a wider variety of possile res.ltsD10E
Example 3
displays all of the pitch&class sets %pc sets'8 incl.ding m.ltisets8 otained in /<;;sim.lations of
Four
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8 organiGed y set class %sc' he raw n.mer of seconds d.ring which each setclass so.nds8 added together in all sim.lations8 is displayed in the col.mn to the right of the pc set he rightmost col.mn displays the prevalence of each set compared to the others %of anycardinality' as a percentage he percentage is otained y dividing the raw n.mer of seconds y3N/8LM<8 the total n.mer of data points amongst all sim.lations %/<;; sim.lations m.ltiplied y31< data points per sim.lation' he total n.mer of val.es and total percentage per set class isdisplayed at the o+om of each o$ he o$es display the set classes in )orte order from left toright and top to o+omD1ME he single most prevalent pitch&class set is the n.ll setIsilenceIwhich occ.rs /2NR of the
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time amongst all sim.lations he least prevalent pitch&class set is S<30;TIan instance of sc %<2LM'Iwhich occ.rs only once8 or appro$imately <<<<<<11R of the time 9etween these two e$tremeslies a varied collection of sonorities8 with 1/ of 3; possile set classes of cardinality <&3 representedFUnermann oserves that8 “DtEhe increased proaility of traditional harmonic content rests .ponthe simplicity of pitch material and the red.ction of density” %FUnermann 1</3 8 N/1' ndeed8 thereare only seven distinct pitch classes availale in the piece: the (&minor diatonic collection min.s9&at8 pl.s the raised seventh degree wri+en as oth C and ( he limited pitch material ens.resthat many of the sonorities that emerge from any given performance will e recogniGale s.sets ofthe same diatonic scale ).rther constraining the harmonic possiilities is the fact that icN is notpossile as part of any sim.ltaneity in the pieceD1;E
Example 4
presents the same data as #$ample 2 organiGed y %m.ltiset' cardinality: then.mer of active voices8 even when two voices are singing the same pitch class conc.rrently h.sS<//LT reects a cardinality of 38 even tho.gh in normal form it wo.ld e red.ced down to thecardinality&2 set S</LT Comparing the total percentages in the lower&right cells of each col.mnshows that the piece is silent appro$imately /2NR of the time %ao.t L0 seconds per sim.lation'8one voice is so.nding 2/1R of the time %ao.t 1’//”'8 two voices 223R of the time %ao.t 1’1<”'8three voices /0/R of the time %ao.t /’/1”'8 and fo.r voices .st .nder LR of the time %ao.t 1<seconds' his data helps ill.strate the sparseness of the m.sic: in a composition with fo.r parts8te$t.res of two or fewer parts are heard over 0LR of the timeD1/<E
Example 5
depicts the prevalence of each set class otained in the sim.lations of
Four
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graphically over time8 .sing lines of di>erent colors to represent each of the 1/ set classes listed in#$ample 2 n #$amples 2 and 38 prevalence is e$pressed as a proportion of the total d.ration of thepieceH the graphs in #$ample L demonstrate how prevalence changes over time d.ring aperformance of the piece he horiGontal a$is represents time %in seconds' and the vertical a$ise$presses the percentage of sim.lations in which a given set is so.nding at a given time )orinstance8 in the opening seconds of the piece8 almost ;<R of the sim.lations are silent #$ample Lais an interactive graph for which individ.al lines representing set classes can e toggled on and o> y clic@ing on the set name in the legend at the top of the graph #$ample L is a still image givingall 1/ set classesH #$amples Lc and Ld depict only sets of cardinalities <V1 and 2V38 respectively8 forlegiilityD1//E As #$ample 2 demonstrates8 many of the set classes in
Four
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may e constit.ted y severaldi>erent pitch class sets
Example 6
depicts all of the instances of a single set classIsc %<2'Iovertime n the sim.lations of
Four
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8 sc %<2' was formed y =ve di>erent pitch&class sets which8 eca.se of the presence of d.plicate pitch classes8 reect three distinct pitch&class pairings !verthe co.rse of the piece8 fo.r distinct pea@s can e ascertained: pcs %1L' and %1LL'8 pcs %30' and %300'8pcs %/3'8 and a ret.rn to pcs %1L' he vertical a$is displays the prevalence of each partic.lar pitch&class set %as a percentage' in relation to every other pitch&class set otained among the sim.lationsat each .nit time %in seconds'D1/1E n addition to the prevalence of diatonic harmony in
Four
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already disc.ssed8 writers haveremar@ed speci=cally on the strong potential for triadic harmony FUnermann descries “an.ndenialy consonant so.nd environment circling aro.nd d minor8” thro.gh which8 in relation tothe chromatic material in the =nal min.tes of the piece8 “the listener may well perceive %and Cageat least facilitates' a teleological harmonic process” %1</3 8 N/1' While a disc.ssion of “perception”is eyond the scope of this st.dy8 the sim.lation data provides a K.antitative description of theprevalence of triads within the pieceD1/2E
Example 7
depicts the prevalence of sc %<20' over time As efore8 the lines of di>erent colorsrepresent di>erent pitch&class sets that form triads in this piece here are three distinct instances ofsc %<20': pcs %1L;' and %1LL;'H pcs %<30' and %<300'H and pcs %/3;'8 corresponding to ( minor8 Cmaor and A maor triads8 respectively he =rst and third pea@s %( minor and A maor' are m.ch
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