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Senior Theses
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5-24-2016
Nonlinear Harmonic Modes of Steel Strings on an
Electric Guitar
Joel Wenrich
Linfield College
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Recommended Citation
Wenrich, Joel, "Nonlinear Harmonic Modes of Steel Strings on an Electric Guitar" (2016). Senior Theses. Paper 21.
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NonlinearHarmonicModesofSteelStringsonanElectric
Guitar
JoelWenrich
ATHESIS
PresentedtotheDepartmentofPhysics
LinfieldCollege
McMinnville,Oregon
Inpartialfulfillmentoftherequirements
fortheDegreeof
BACHELOROFSCIENCE
May,2016
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Abstract
NonlinearHarmonicModesofSteelStringsonanElectricGuitar
Steelstringsusedonelectricandacousticguitarsarenon-ideal
oscillatorsthatcanproduceimperfectintonation.Accordingtotheory,
thisintonationshouldbeafunctionofthebendingstiffnessofthestring,
whichisrelatedtothedimensionsoflengthandthicknessofthestring.
Totestthistheory,solidsteelstringsofthreedifferentlineardensities
wereanalyzedusinganoscilloscopeandaFastFourierTransform
function.Wefoundthatstringsexhibitedmoredrasticnonlinear
harmonicbehaviorastheireffectivelengthwasshortenedandaslinear
densityincreased.
3
Contents:
Introduction....................................................................................................5
Theory...............................................................................................................8
TheIdealString.........................................................................................................8
BendingStiffness...................................................................................................10
MusicTheory...........................................................................................................11
ExperimentalMethods..............................................................................13
Results&Analysis......................................................................................17
Conclusion.....................................................................................................25
Bibliography.................................................................................................26
4
Introduction
Theguitarisastringedinstrumentthatcanbeusedtoproduceawiderange
ofnotesbyeffectivelychangingthelengthofthesetofstrings,eachofadifferent
lineardensity.Thestringsarestretchedoverastrongpieceofwoodwithoneside
roundandtheotherflat,calledtheneckandfretboardrespectively,withstripsof
metalwiresetintothefretboard,calledfrets.Onanelectricguitar,vibrationsofthe
metalstringsaresensedbypickupcoils,amplifiedbyasimplecircuit,andfinally
outputted.Belowisadiagramhighlightingthebasicpartsofanelectricguitar.(See
Fig.1)
Fig. 1: Labeled diagram of an electric guitar [15].
Likeanymusicalinstrument,guitarsmustbekeptintunesothattheycanbe
usedtoproducethequalityofsoundforwhichtheyweredesigned.Possible
contributingfactorstopoorintonationhavebeenstudiedsuchasdeadspots[5],tone
decay[10],andfret-stringinteractions[1][13].Otherfactorsincludetheharmonic
5
behaviorofthevibratingstrings[3][11][12][14].Inthispaper,wewillcontinuetobuild
offoftheworkdonewithharmonics.Harmonicsareparticularfrequenciesthat
yieldzero-amplitudepointsalongavibratingstring.Theyareoftenanalyzedasa
standingwave(asnapshotofthewaveatitsgreatestamplitude).
Westudytheharmonicvibrationsofguitarstringsandwebelievethereisa
relationshipbetweenthenonlinearnatureofharmonicsonastringanditsbending
stiffness.Thismatterstoguitarplayersandlistenersalikebecausethepresenceof
sharpeningharmonicmodesmaybecomeaudiblysharporflatwhenplayedin
conjunctionwithothernotes[7].
Thisisastudyofsteelstrings.Whilemostsetsofstringsaremadeofsteel,
otheralloyssuchasbronze,copper,andnickelareusedintheindustryforthe
varietyoftonalqualitiestheyoffer.Thethickerstringsresponsibleforproducing
lowerfrequencynotesarenotmadefromasinglecoreofsteel;rather,theyconsist
ofasoftermetal–usuallynickelorcopper–woundaroundasteelcore.Fig.2shows
SEMimagesofthesetofguitarstringsusedinthisexperiment:
Fig.2:SEMimagesofastandardsetofguitarstrings.
6
Usinganelectricguitarandadigitaloscilloscopewetesttheharmonic
behaviorofdifferentsteelstringsandcompareittotheoreticalresultsyielded
throughknownrelationshipsbetweenharmonicsandbendingstiffness.Because
woundstrings(seeFig.2:LowE,A,D)werenotcomprisedofasinglematerial,they
wereexcludedfromthisstudytoavoidfurthercomplexity.
7
Theory
TheIdealString
Toanalyzethevibrationofasteelstringwemustfirstunderstandthesimplestcase,
anidealstring.Anidealstringisinfinitelythin(withoutbendingstiffness)andhas
uniformlineardensity.Whenplucked,twowavetrainspropagateinopposite
directions,reflectingoffofthetwofixedends,whichisthebridgeandeitherthefret
orthenut.Thesetwowavetrainscontinuetoreflectbackandforth,creatingan
interferencecalledastandingwave.BythePrincipleofSuperposition,the
interferingwavesarethesum
𝑦 𝑥, 𝑡 = sin 𝑘𝑥 − 𝜔𝑡 + sin 𝑘𝑥 + 𝜔𝑡 (1)
2𝜋
Here,𝑦isthewaveamplitude,𝜔 = 2𝜋𝑓isangularvelocity,and𝑘 = 𝜆 isthewave
numberwith𝜆beingwavelength.Bywayoftrigonometricidentities,Eq.(1)canbe
simplifiedto
y x, t = 2sin 𝑘𝑥 cos 𝜔𝑡 (2)
Ifwedefinethefixedendstobeat𝑥 = 0 and 𝑥 = 𝐿,whenastringisplucked,the
onlywavesthatcontinueanylongerthanafractionofasecondaretheoneswhich
yieldanamplitudeofzeroatthosepositions.NotethatinEq.(2)thefirstterm
describesamplitudeandthesecondtermdescribesfrequency.Here,itisimportant
tonotethattheresultofEq.(2)iszeroatbothfixedends,say𝑥 = 0, 𝑥 = 𝐿.What
8
arisesisaninfiniteamountofparticularfrequenciesthatyieldpositionsalongthe
string,𝑥,thatexperienceanamplitudeofzeroatalltimes;thesepointsarecalled
nodes.The𝑛!" harmonic,denoted𝑓! , isafrequencythatproduces𝑛numberof
nodes.Oftenindiscussionofthistopic,theterm“harmonic”issynonymouswith
“mode.”Thefirstthreeharmonicsforanidealstringfixedatbothendsareshownin
thefigurebelow.Alsointhefigureisthesumofthefirstthreeharmonicswhereitis
importanttonotethattheamplitudesaredescendingfromthefirsttothird
harmonic.(SeeFig.3)
Figure3:Standingwavesofthefirst,second,andthirdharmonics(left);sumofthe
firstthreeharmonicsforonefullwavelengthplottedontimevs.amplitude(right).
Motivatedtogeneratearelationshipbetweenharmonicsandtheircorresponding
modes,recallthespeedofatransversewavetravelingonastring[4],
𝑣=
𝑇
𝜇
(3)
where𝑣iswavespeed,𝑇istensionforce,and𝜇isthestring’slineardensity.For
suchastringstretchedbetweentwoendsupports,naturalmodesofvibrationare
givenby
𝜆! = 2𝐿
𝑛
9
𝑛 = 1,2,3 … (4)
where𝜆! iswavelengthofmode𝑛withstringlength𝐿.Fromfigure3andEq.(4)a
criticalconceptarises:allharmonicsoccurwhenastringisinitiallyplucked.Again,
thereareinfinitelymanyfrequenciesthatappearwhenastringisfirstpluckedbut
mostofthemaredampedbythebridge–fret/nutinteraction.Speedandwavelength
arebothrelatedtofrequency, 𝑓,through
𝑓! =
𝑣
𝜆!
𝑛 = 1,2,3 …
(5)
BysubstitutionofEqs.(3)and(4)into(5),
𝑛
𝑇
𝑓! = 2𝐿 𝜇 = 𝑛𝑓!
𝑛 = 1,2,3 …
(6)
Herewehaveawayofcalculatingthefrequencyofeachmodeofvibrationfora
stringwithspecificproperties.Wereferto𝑓! asthefundamentalfrequency.Notice
thepositivelinearrelationshipbetweenfrequencyandmode.Intheidealcase
length,tension,andlineardensityareassumedtobeconstant.
BendingStiffness
Theidealstringmodelbreaksdownwhenwenolongermaketheassumptionthat
thestringisinfinitelythin.Asaresult,harmonicsdonotaddtogethertoproducea
wavethatisconsistent.Complicationensueswhenweincorporatethicknessand
materialpropertiesofthestring[4]:
(7)
10
𝑓! = 𝑛𝑓! 1 + 𝐵𝑛! 1 +
with
2
4
𝐵+ 𝐵 𝜋
𝜋
𝐵 = 𝐸𝑆𝐾 ! 𝑇𝐿! (8)
beingbendingstiffness.Here,bendingstiffnessisapropertyofastringofgiven
materialwith𝐸,modulusofelasticity,𝑆,cross-sectionalareaandradiusofgyration,
whichforacylinderofradius𝑎is𝐾 = 𝑎 2.Clearly,Eq.(7)isadeparturefromthe
linearrelationshipbetweenfrequencyandmode.Itisthisnonlinearequationthat
wewilluselatertocomparewithresultsobtainedfromourexperiment.
MusicTheory
Wecareaboutthesharpeningofharmonicsbecausetherearepotential
ramificationsforthepurityofnotesastheyareplayedtogether.Considertwo
separatestringsoflengthLfixedatbothends.Ifwetakeonestringandstopit
halfwaysothatitsnewlengthis1/2L,wewouldbeabletoproduceanotethatis
twicethefrequencyofthenoteproducedbyastringoflengthL.Thisnoteiscalled
an‘octave.’Noticethatthefrequencyoftheoctaveisthefirstharmonicofthe
originalnote.Ifweweretotakeastringofthesamelengthandstopitat2/3Lwe
wouldproduceanotethatiscalleda‘fifth.’
Octavesarenotesseparatedbyanintervaloftwelvehalfsteps.Theinterval
ofperfectfifths,asit’scalled,consistsofnotesseparatedbysevenhalfsteps.Using
thisinterval,chordscanbemadebyplayingtwoormoreofthesefifthstogether.
Hence,itiscrucialthatchordsarecomprisedofnotesthatareintuneortheywill
11
sufferabeatfrequency,whichistheperiodicoscillationintotalamplitudeofa
compoundwavecomprisedofmultiplecomponentfrequencies[8].
12
ExperimentalMethods
Inthisexperiment,themeasurementprocedureconsistedofpluckingaguitarstring
andobtainingthecomponentfrequenciesofthesignalusingtheTektronixTDS
2002Boscilloscope.Figure4showsaschematicoftheexperimentalsetup.
(a)
(c1)
(b)
+V
(d)
(c2)
Ground
Fig.4Schematicofexperimentalsetup.(a)TektronixTDS2002BOscilloscope.(b)
SchecterC-1ElectricGuitar.(c1)ConnectionbetweenguitarcableandChannel1.
(c2)Expandeddiagramofconnection(d)Connectionbetweenguitarandguitar
cable.
TheTektronixTDS2002BOscilloscopeservedthepurposeofthisexperiment
wellbecauseofitsFastFourierTransform(FFT)andUSBfunctions.Three
parametersoftheFFTareadjustable:source,windowandzoom.Source
13
determineswhichchannelissampled.Onlyonechannelwasusedinthis
experiment.Threetypesofwindowsettingsdeterminedfrequencyresolutionand
amplitudeaccuracy.Bothmeasuringperiodicwaveforms,the“Hanning”and
“flattop”windowmodesmostaccuratelyjudgedfrequencyandamplitude,
respectively.Thethirdwindowoptionwas“rectangular”,lendingspecialattention
towaveformswithoutdiscontinuities.Cateringtothefocusoftheexperiment,
“Hanning”wastheoptimalchoice.Zoomdetermineshorizontalmagnificationofthe
datawithoptionsofX1,X2,X5,X10.Onceset,windowandzoomwereheldconstant
whereassamplingratewascycledthroughwhencollectingasetofdata.The
oscilloscope’srangewasbetween5.0samplespersecond(S/s)and2.0GS/sbutfor
thepurposesofmeasuringfrequenciesproducedbyaguitararangeof1.0kS/sto
50.0kS/swassufficient.Eachsamplingratescanned0Hz.totheNyquistfrequency,
whichishalfthesamplingrate,toavoidaliasing.Moredetailedinformationcanbe
foundintheoscilloscope’smanual.
Withtheoscilloscopetuned,specialattentionhadtobepaidtotheguitarand
howeachstringwasplucked.Astandard1.0mmguitarpickwasusedtoexcitethe
stringsduringtheexperiment.AsPolitzersuggests[11],theangleatwhichastringis
pluckedrelativetothefretboardaffectstheintensityofvibrationimmediatelyafter
itisplucked.Itturnsoutthatplucksparalleltothefretboardproducethehighest
intensityvibrationswiththeleastdecay.Thus,inordertostayconsistentwiththe
pluckinghandasbestaspossiblewithoutsomesortofautomatedprocess,the
experimenterdidentiresetsofdataatatimebeingcarefultomaintainsteadyhand
position,pluckingforceandpluckingpositionalongthestring.
14
Determiningthemomentatwhichthesamplewastakenrequiredsome
techniqueaswell.Immediatelyafterastringispluckedthereisa“twang,”asitis
referredto,thatistheaudiblypositiveshiftinfrequencyduetowhatErrede
suggestsistheextremeinitialamplitudegeneratedbythepluck[3].Becauseofthis
effect,whichlastslessthanatenthofasecond,thepropersampletotakewasjust
afterthepluck.Thisentailedsettingtheoscilloscopeautomatictriggerandpressing
STOPonthesecondorthirdsamplingdependingontherate.Fortheslowerrates,
suchasa1kS/sand2.5kS/s,thesamplingtimewastoolongtotakethesecond
framebecausebythentheintensityofthevibratingstringdecayedbeyond
measurement.
Withtheoscilloscopeandguitarpreparedfordata,samplescouldthenbe
taken.Aspreviouslydescribed,thesamplingoftheoscilloscopewaspausedjust
aftertheinitialpluckofastring.UtilizingtheUSBfunctionoftheoscilloscope,the
waveformdatawassavedontoaflashdrive.Organizationwascriticalduetothe
volumesofdatacollected.Datawasseparatedfirstintofoldersdifferentiatedby
stringthenintofoldersaccordingtofret.Datawasnotseparatedintotheirown
subfoldersbasedonsamplingratebecausesuchinformationwasincludedinthe
waveformfile.
TheoscilloscopesavedthedataasaCommaSeparatedDocument(CSD)
whereeachpointwasacoordinateoffrequencyandintensity.Thesedatawere
thenputintothecomputerprogram,Origin,inordertobetterfitthepoints.Excel
wasusedtokeepalistofthefittedpoints.Tobringitalltogether,Mathematica
15
provedusefulincalculatingtheoreticaldata,whichwascompareddirectlytothe
experimentaldata.Greaterdetailoftheanalysisisdescribedinthenextsection.
16
Results&Analysis
Inthisexperimentonlythethreethinneststringsontheguitar(G,B,HighE)
wereanalyzedbecausetheyweresolidsteelwhereastheotherthreestrings(LowE,
A,D)havecopperornickelwindingaroundsolidsteelcores.Thephysical
propertiesofthestringsusedareprovidedinTable1below.
Table1PhysicalpropertiesofG,B,andHighEstringsusedintheexperiment.Diameterwas
obtainedusingveneercalipers,Tensionthroughmanufacturerdata,andf0fromtheoscilloscope.
String
G
B
HighE
Diameter(mm)
0.406
0.279
0.228
EstimatedTension*(N)
65.3
49.0
58.4
Measuredf0(Hz.)
194.8
246.2
327.6
*Tensionwhileinstandardtuningandonfretzero,asperthemanufacturerofthestringsused,
D’AddarioXL120.
Eachsetofdatabeganasareadingfromtheoscilloscopeintheformof1024
pointsspanningatunablerangeoffrequency.Samplingratesusedinthis
experimentvariedfrom1.0kS/sto50.0kS/swithrespectiveintervalsbetweendata
of0.488Hz.to24.41Hz.Figure5isanexampleofrawdatacollectedfromthe
oscilloscope.Notethatalthoughtherearespikesinthedatatheyarenotreliable
duetothesamplinginterval.Generally,thepeaksclosetothemaximumthreshold
wereignoredbecausedatafromfastersamplingratessuggestedtheactualpeak
wasslightlybeyondthethreshold.
17
0
-10
Intensity (dB)
-20
-30
-40
-50
-60
-70
-80
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Frequency (Hz.)
Fig.5Rawdatacollectedfromtheoscilloscope.HighEString,fret1,10.0kS/s
BetteraccuracycouldbeobtainedfromtherawdatabyusingOrigintofit
Gaussiancurvestoeachpeak.Usingthe“FitMultiplePeaks”function,upto12
peaksatatimecouldbemodeled.Forsomeofthepeaksthatwereoflowintensity
buthigherthanthatofnoise,thefunction“FitSinglePeak”wasrequired.Figure6
showsasnapshotofbothamultiplepeakanalysisandsinglepeakanalysis.
Fig.6Experimentaldatafitted
usingOrigin.HighEString,fret1,
10.0kS/s
18
Aftereachfitoftherawdata,thevaluesofthepeakswererecordedinExcel
spreadsheets.Itwasthesedatathatweconsideredtobeourexperimentaldata.
Thefollowingfiguresexhibitexperimentaldataandaccompanying
theoreticalcurvescalculatedusingEq.(7)alongwiththeinformationfromTable1.
Foreachstring,resultsforfrets0,12,and24aregiven.Thethreefretswerechosen
becausetheycorrespondtothefirstthreeharmonics—𝑓! atthefulllength,𝑓! at
half-length,and𝑓! atone-thirdlength.Thex-axisisthemode,n,andthey-axisis
frequencyofmode,fn,dividedbythemode.Themotivationforpresentingthedata
insuchafashionisthatitmakesmorevisibleanincreasingshiftinharmonics,
whichiswhatwewouldexpecttoseeinastringwithhighbendingstiffness.Note
thatanidealstringwouldbeahorizontallinewiththeonlyy-valuebeingthe
fundamentalfrequency.ByEq.(7)wewouldexpecttoseethemostdrasticpositive
shiftsforfret24,thenfret12,thenfret0,regardlessofstring.Additionally,the
sameequationsuggeststhattherewouldbegreatershiftsinthethickerstringson
anygivenfret.Notethatinthediscussionofthesefiguresanyshiftinfrequency
describesadeviationfromthefundamentalfrequencywhendividedbythe
correspondingharmonicnumber,𝑛.Therefore,eachpointrepresentsafrequency
shifted𝑛timesawayfromwhatthefrequencywouldideallybeatmoden.Graphs
canbecomparedinthismannerbecausethex-scaleisthesameineverygraphfor
theFigs.7,8and9below.
19
Fig.7Deviationfrominteger-multiple
harmonicnatureoftheG-stringonfrets0,
12,and24.Dotsdenoteexperimentaldata,
dashedlinesdenoteidealpredictionsfrom
Eq.(6),andsolidlinesdenotepredictions
fromEq.(7).
FromFig.7wecanseethatasthestringlengthdecreases(fretnumberincreases)the
harmonicsshiftmoresignificantly.Onthe0thfretweseea+5Hz.shiftat𝑛 = 20.Forthe
12thand24thfretsweseea+5Hz.shiftreachedatroughly𝑛 = 8and𝑛 = 3respectively.
20
Fig.8Deviationfrominteger-multiple
harmonicnatureoftheB-stringonfrets0,
12,and24.Dotsdenoteexperimentaldata,
dashedlinesdenoteidealpredictionsfrom
Eq.(6),andsolidlinesdenotepredictions
fromEq.(7).
Figure8continuesthetrendobservedinFig.7butfortheB-string.Comparingthetwo,
weseethatineachcasebetweenthe0th,12thand24thfrets,thethickerG-stringdisplayed
amoredrasticshift.
21
Fig.9Deviationfrominteger-multiple
harmonicnatureoftheHighE-stringon
frets0,12,and24.Dotsdenote
experimentaldata,dashedlinesdenote
idealpredictionsfromEq.(6),andsolid
linesdenotepredictionsfromEq.(7).
Withthethirdandfinalsetofdata,Fig.9completesthetrendobservedintheprevious
twofigures.Asthethinnestofthestrings,theHighE-stringexhibitstheleastdrasticof
theshiftsforeachfret.Fromthesefiguresweobserveproportionalitybetween
nonlinearharmonicbehaviorandboththeinverseoflengthsquaredandcrosssectional
areaassuggestedbyEq.(7).Tohighlighttheinverseoflengthsquareddependencywe
canlookateachstringasanindividualcaseandseethatasthelengthofthestring
decreasesweseeagreaterpositiveshift.Asforthecrosssectionalareadependency,
wecanlookateachfretasanindividualcaseandcomparetheharmonicshiftsforeach
string.Indeed,weseethatforeachfrettheG-stringhadthemostsignificantshift,
22
followedbytheB-string,andlastlybytheHighE-string(descendingorderof
thickness).
Whenbothcontributingfactorsaretakenintoaccount,directcomparisons
betweenstringscanbemadeforagivennote.InFig.10,thesamenote(HighE)was
playedoneachofthethreestrings.Bothlengthandlineardensityarebeingvariedin
thisexample.
Fig.10Deviationfrominteger-multiple
harmonicnatureoftheG,B,andHighE
stringsplayedonthesamenote(roughly
thatoftheopenHighE-string).Dots
denoteexperimentaldata,dashedlines
denoteidealpredictionsfromEq.(6),and
solidlinesdenotepredictionsfromEq.(7).
Figure10concurswiththebehaviorsobservedinthepreviousthreefigures.
Theseresultssuggestthatonecanoptimizetheintonationoftheirplay,giventhe
optiontoplaythesamenoteondifferentstrings.
Errorinthedatastemsfromthepotentiallackinlow-frequencysensitivityof
theoscilloscopeaswellasthepossibilitythatthepickups[6]andcircuitryoftheelectric
23
guitararenotbuilttobesensitivetofrequenciesfarabovethethirdharmonicofthe
HighEstring(roughly1318Hz.).
24
Conclusion
NonlinearharmonicbehaviorwasobservedintheG,B,andHighEstringsofan
electricguitar.InaccordancewiththepredictionofEq.(7),resultsshowthatthe
degreeofnonlinearharmonicbehaviorisgreaterasstringthicknessincreasesand
lengthdecreases.OurobservationsagreewiththatofPolitzer[11].Fromtheresults,it
ispossiblethatobserveddeviationsinharmonicswouldbeperceivable(see[9]).
Furtherresearchcouldincludetestingwoundstringsforsimilarbehavior.
Sourcesoferrorinthisstudyvaryfromhumanmeasurementtoinstrumental
limitations.Itwasnotpossibletomaintainperfectconsistencyinpluckingthestrings
sotheremayhavebeenharmonicsthatweretoofainttoobserve.Betterequipment
couldhavebeenusedbecauseouroscilloscopewasonlyabletoachieveaprecisionof
+/-24.41Hz.at50.0kS/s.Thissortoferrorwasonlysignificantinlow-frequency
measurement,whichwascriticalbecausethetheoreticallinesinFigs.7,8,and9were
calculatedbasedonthefundamentalfrequency.Ideally,wewouldbeabletoscanthat
rangeofdatawithaprecisionofcloserto+/-0.5Hz.Becausewechosetoanalyzethe
vibrationofthestringsthroughelectronicmeans(guitarandpickupsystem),itmay
justbethatelectricguitarsarenotmadetoregisterharmonicsfarbeyondthefifthor
sixth.Itwouldbeinterestingtoreplicatetheexperimentwithanacousticanalysis
insteadandcompareresults.
Itisnottangibletoeliminatesteelasthemainmetalusedinguitarstringsbut
outofstudiessuchasthis,theremaydevelopnewtechniquesforcompensatingthis
effect.Perhapsstringsofnon-uniformdensitycouldbedesignedspecificallyto
minimizeskewinharmonicsallthroughoutthelengthofthefretboard.
25
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[2]ConklinJr.,HaroldA.“Generationofpartialsduetononlinearpartialsina
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