-:" "...... -.""-.I-. \ ;- ----- - - ---- - J \ J {. ,, ----- -- .,.. -.-ORCUSTECHNICAL . PUBLICATION-TP- 3003 .SECONDEDITION Manford L.Eaton j- - '.,. '.. - .., ..; ->,,' - '... ':... ..... l : ""'..... ...ELECTRONICMUSIC-. , . --. ; .AHANDBOOK: 9F SOUNDSYNTHESIS&CONTROL ," ,...

-, . ORCUS OPERATI ONALRESEARCHCOMPANYUNIVERSAL.SYSTEMS .' . Box16022- KansasCity,Mo.64112- U.S. A . i - -..-. r '.. _..... ..__.._-_ _____..:.:.. '. .. _ _.:.... ___ _"".4" ____ ___ I \ , ,- -i I Ii II ! 1 I I I ! i , , I : I !92-:27183 ! ..-ORCUSTECHNICAL PUBUCATION-TP- 3003 SECONDEDITION ManfordL.Eaton ELECTRONICMUSIC-.AHANDBooK.9F SOUNDSYNTHESIS&CONTROL . ORCUS OPERATIONALRESEARCHCOMPANYUNIVERSALSYSTEMS Box16022- KansasCity,Mo ..64112- U.S. A . 0"\'..... :" " " ,t. . . ' I \ 11 ! I ;i II ;\ !' I i I : I ,I jl Ii 1' I ,i I' \ I! II. I I / ~....-----------ELECTRONICMUSIC ,/ AHANDBOOKOF SOUNDSYNTHESIS&CONTROL ManfordL.Eaton ~ / SECONDEDITION 1971 ORCUSRESEARCH Allrightsreserved Noportionofthisbookmaybereproducedwithoutprior written permission ofORCUSResearch.Thecircuit dia-grams includedinthisbookare forillustration oftypical applicationsandarenot intendedas constructionalinfor-mation.Althoughreasonablecarehasbeentakenin their preparation toassuretheirtechnicalcorrectness,nore-sponsibilityis assumed byORCUSResearch foranycon-sequencesoftheir use. ,. I '-/------ORCUSTECHNICAL PUBLICATION-M.L.Eaton TP- 3003 ELECTRONICMUSIC AHANDBOOKOF SOUNDSYNTHESIS& CONTROL Note:Electrical,ElectroniC,andLogicDiagramshavebeen prepared inconformancewithUSAStandards InstitutePublications Y32. 2 andY32.l4 ELE C TRONICMUSIC-AHANDBOOKOF SOUNDSYNTHESIS&CONTROL M.L.Eaton,ORCUSResearch 1.SOUND,ELECTRONICS&TRANSDUCERS-- 1 2.SOUNDWAVES&MUSIC-- 3 3.PARAMETERSOF SOUND-- 8 4.SOMEPROPERTIESOFHEARING-- 11 5.ELECTRONICS-- 13 6.BLACKBOX SYNTHESISOFELECTRONICSYSTEMS-- 22 7.MICROPHONES,LOUDSPEAKERSANDTAPERECORDERS.:..- 31 8.TECHNIQUESOFGENERATINGCOMPLEX WAVES-- 36 9.CONTROLTECHNIQUES-- 52 10.COMPOSITIONALPROCEDURES-- 59 ll.ELECTRONICMUSICINTHEFUTURE-- 63 12.CHARTS-- 65 13.SUGGESTIONSFORFURTHER STUDY-- 72 14.GLOSSARYOF TERMS/INDEX-- 74 15.BIO-MUSIC-APPENDIX A FIRSTEDITION First printing--July,1969 Secondprinting--September,1969 Thirdprinting--April,1970 Fourth printing--July, 1970 SECONDEDITION First printing--January, 1971 Secondprinting--September,1971 Thirdprinting--October,1972 INTRODUCTION Thisbookcontainsinformation whichwillenablethe readerto design and eval-uateelectronicmusicsystems.It isthefirst authoritativepresentation ofthe basictheories ofsoundgeneration andcontrol.Thefewbooksandmanualsthat doexist deal either withvarious advanced technicalproblems or areinstruction manuals forspecificelectronicmusicinstruments. The materialcontainedinthe followingpages is drawn from manyfields;music, mathematics, acoustics,biology, electronics and psychology .Someof thesub-jectmatter of eachof these fieldsis relevantto electronicmusicbut muchof it is not.It is thepurposeofthisbookto helpthestudent determinewhat is of importance.Its purposeis togivethe reader theperspectivehe needs in order toorganizehis furtherstudies. If eachofthesubjectareasmentionedabovewas givenathoroughtreatment, thepresent book wouldconsist ofmanyvolumes.This however,is not our aim; ratherisis to present anoutlineofthebasicareasandtechnologieswhichare integralto electronic music.Thus, few of the topics are pursuedto great length. Theonearea whichhasbeen given acertain specialtreatmentis electronics. There are at present fewbooks which approach electronics from a systemspoint of view.For thestudentstudyingelectronics forthefirsttimeor forthestu-dentwhohas been previously acquainted withbasic electronicsthe material pre-sented herewill quicklyplace himin aposition tounderstandthefunctiOningof electronicsystems withoutthenecessity forlongstudyof circuit operation and design.Thisis a quiteSignificant approachtoelectronics forpersons in fields such as electronic music since without knowledge ofelectronic systems the com-poser andtheorist is at themercyof commercial producers of electronic music equipment whoseinstrumentsare frequentlyincapableofthedesiredperform-ance.Thesystemsapproachto electronicsdesignis alsoconsonantwiththe changes whichhavebeen occurring in engineering thepast ten years.Withthe adventofintegratedanalogand digitalcircuits andtheproductionof discrete component modules andlogic cards it is practicalto approach electronics at the systemslevel.Thereader is,ofcourse,encouragedtopursuehisstudyof electronics beyondthat which is presentedinthisbook.However,the material hereshouldgivehimthe abilitytouseelectronics effectivelytoaccomplishhis designaims. Thesuggestionsforfurtherreadingin theback ofthebookarerecommended tothe reader withthe idea ofleading him intobasicbooks ofthefields mentioned aboveandtoenablehimtomakeasmoothtransition fromthematerialin this bookto the. more advancedliterature.The studentwill findthatthe present book is in factaconcentratedsourceandreferencebook enablinghimtocharthis ownpaththroughthe myriad books and articles which relateto electronic music. , Thoseworkers inthis fieldwhohavehadtosiftthroughhundredsofbooksand articlestoarriveatanunderstandingoftheproblemsandpotentialsofelec-tronicmusic willappreciatethe valueof theapproachexemplifiedbythisbook. Armed withthe informationpresented here andspecifically withthe abilityto de-sign andevaluateblackboxelectronicgeneration andcontrolsystemsthecom-poser or theorist is inan excellent position to make aworthwhile contribution to thefield .. Alltoofrequently,persons interested in electronicmusic,but without thisknowledge,invest incommercial electronicmusicequipment whichisinca-pable of fulfillingtheir needs.Theyfindthemselve s inventing way s of composing musicwhichtheinstrumentis capableofgeneratinginsteadofspecifyingthe characteristics ofa system whichwillgeneratethe musictheywish tocompose. Thislatter is theprinCipaladvantageof electronicmusic;it is possibletogen-erateandcontrolsoundswithahigh degreeof preCision andtoimplement com-positionalprocedures whichare quiteimpossible withconventionalinstrumen-tation. It is a rather unfortunatefactthat the role of the musictheorist in contemporary musical life is aspurveyor of harmonic andcompositionalrules ofabygone age. It is tragicthat thetheorist as adesigner of possible and desirable software and hardwarefor musicalsystemsis a virtually non-existent entityinmusicallife. ElectronicmusiC,fortunately,is changingthissituationsinceitissodepen-dentuponnewprocedures andequipment that it is virtuallyimpossibletomake anySignificantcontribution withoutanunderstandingofthevariousfacetsof soundsynthesis andcontrol. Untilsuchtimeas composers andtheoristslearn todesignandevaluateboth theproceduresandtheequipment neededtoimplement theirideas,electronic music willremain inits present adolescentstate.It is ourhopethatthisbook mayinsomesmall wayassist ineasing thegrowing pains. ORCUSRESEARCH INTRODUCTIONTOTHESECONDEDITION WeatORCUSResearchwishtotakethisopportunitytothank you,theuser ofthisbook formakingitthestandardEnglish languagetext onelectronic music.Atthetimeofpublicationofthefirst edition ofElectronicMusic-AHandbookOfSoundSynthesis AndControl wefelttherewasaneedfora textpresenting themorebasicprinciplesofsoundgeneration,controland applications ofelectronicstothe art ofmusic.It was also agoalofthebook tokindlethestudentsinterestin thepossibilitiesforinteractionsbetween music,electronics,andthelifesciences.Thenumber ofrequests wehave received fromindividuals andorganizations concerningBio-Musicandsen-sorystimulationsystemshasinducedustopublishmuchadditionalinfor-mationonourresearchintheseareas.Theinformationcontainedinthe present editionprovidesabasicfoundationforstudyinthemorecomplex interdisciplinaryareas ofBio-Music,hallucinogenic,andelectronicsen-sorystimulationsystems. Iampersonallyindebtedtomycolleagues,fellowconsultantsandothers whohavecontributedtothesuccessofthispublication.Iwishtothank CarolHodgesforherconsiderableeffortsintypingand editingthemanu-script,HolcombMcKinleyforhismeticulousproofreadingof thisedition, Viera Jagosova forpreparation ofthedrawingsas wellas fortheconstruc-tion of engineering models ofsound generation andcontrolcircuitry for elec-tronicandBio-Musicexperimentation.SpecialthanksarealsoduetoRay Stellhorn for suggestions concerning some errors present inthefirst edition andtoMr.JohnTsividisforpointing outsome errors andoversightsinthe electroniCSsectionofthat edition. Butmost ofallthankyou,thestudentsof the art ofmusic whohaverespon-dedtotheneedforincreased qualityof artisticcommunicationthatispos-siblethroughthe application of electronictechniquestothearts.Theelec-tronic interactionsbetween thelife sciences such aspsychology,psychiatry, neurology,phy siologyandthetechnologicalarts ofmusic,television,film, radio,kineticart, formanimportant part of theeffortsoftheORCUSCon-sulting Group.Yourinterest inthesepossibilitiesisgreeted withsincere appreciation. KansasCity,?vIissouri"USA February,1971 ORCUSResearch ManfordL.Eaton, Consultant .., , ] SOUND,ELECTRONICS&TRANSDUCERS SOUND,as perceivedbytheear,isaVIBRATIONofairor,forthatmatter,of anyother gas.Vibrationsinthegasare causedbyvibrationsofsolidmatter. Thesolidmatter vibratinginthegasmaybeintheformofanessentiallyONE-DIMENSIONALbodysuchasastring,aTWO-DIMENSIONALobjectsuchas a steelplate, or a THREE-DIMENSIONAL object such as a blockofwood.Of course vibratingsolidmatter mayalso have such physicalstructures as vocalcordsand thespecialdevicesformakinggas vibratecalledmusicalinstruments.There can benosoundinavacuumsincethereisnogasto vibrate. Whenthe air is madetovibratebackandforthat any ratebetween20times a sec-ondand20,000timesa second,the vibrationsoftheair areperceived bythe ear as sound.Thefasterthe air ismade to vibrate,thehigherthepitchofthe sound. Vibration isarather vagueterm,however;andeventhoughmostpeopleintui-tivelyfeelthattheyknowwhat vibrations are,itis perhaps bettertobemore ex-plicit concerningthemovement oftheair whichcausessound.Supposethatwe strike asteelplate whichis hangingfroma string.At themoment ofimpact the platebendsinadirectionawayfromtheforceoftheblow.If theplateis in a medium of gas,thegasispushedin frontoftheplateandcompressed.Theair inbackof theplatecannow expand.Aftertheplatehastraveledas faras it can inthedirectionawayfromthestriking force,itbeginstoreturntoitsoriginal position.Itsmomentum,however,carries itback pastthispositionandbends theplatein theotherdirection.Theplatecontinuesthisbackand forthmotion untilitagaincomestorest.This vibrationoftheplate(ormoreproperlyOS-CILLA TION)alternatelycompressesandrarefiestheairor othergasoneach sideof theplate,andcausesalistener,whoisinthesamegaseousmedium,to perceivesound. Therateatwhichan objectvibrates(thenumberoftimesaseconditmakesa completeoscillation fromoneextreme position back tothatextremeposition)is its FREQUENCY.Each such oscillation will produce one soundwave;that is,one area of compressed airandone area ofrarefiedair.The numberof soundwaves emanating fromavibratingbodyeachsecondis thefrequencyofthesound. The tendencyof abodytoreturn toan equilibriumposition afterhavingbeen for-cedor deformedoutof position is a function ofthe ELASTICITYofthebody.Ac-tually,suchmaterialsassteelandglassare far better examples of elasticsub-stancesthanisrubber,but theyaresohardthatdeformationistooslightand recoverytoorapidtobeeasilynoticeable.Nevertheless,steelor glassballs willbounceif allowedtofallonahardsurface.Substancessuchas wax,putty, andlead,ontheotherhandarerelativelyinelasticandwhen deformed,as bya fallagainst a hard floor,willnothavemuchtendencytopushbackintotheir or-iginalshape. 1 Thespeed withwhichportionsof air oscillatebackandforthaboutsomepoint of equilibrium dependsinpartupontheelasticityof air Ijustas thespeed of oscill-ation ofa spring depends in part upontheelasticityofthemetals composingit.Up-onthenaturalspeedof oscillation oftheair depends,in turn,thevelocityof pro-pagationof the soundwave.Thevelocity ofsoundinairisequaltothesquareroot ofthe elasticity of air dividedbythedensityof air.Thevelocityofsoundin air at 00 Centigradeis331meterspersecondor740miles perhour.Theelasticity of air increases withtemperatureandso,consequently,doesthevelocityofsound. Theincreaseinthevelocityofsoundis roughlyhalfameterpersecondforeach Centigradedegreerise.Speedsofsoundinsomecommonmaterialsare:Water 3240mph;Hydrogen2840mph;Steel11,200mph;Glassupto13,500mph.In the exampleof thesteelplatementionedabove,theoscillationsbecomeweakerafter theinitialblowuntiltheplate finallycomesagain torest.Thisisproperlycalled DAMPEDOSCILLATION.However,it is possible tomake theplatecontinue toos-cillate foran indefinitelylongtime if we applyenergytoit in an appropriate fashion. Supposethat wehiredtwoverynimble elvestohelpus;we couldstation one on each side ofthe steelplatetowait untilit carne asclosetohimasit wasgoingtobefore starting itstravelin theoppositedirection.Atthat veryinstant hecould pushthe plateintheoppositedirection veryeasily.Theotherelf woulddothesamething ontheothersideof theplate.This,then,wouldconstituteSUSTAINEDOSCILL-A TIONforas longaswekept theelvesat work.A quitefamiliarexampleofsus-tainedoscillationisachildreIfsswingwithsomeonetopushit.It ismucheasier topushthe swing whenit is at the highpointof its travelthan it is,eithertooppose it when it is travelingtoward you,ortopush it away fasterthan it will gonaturally. It is notonlywholesolidbodiesthat can be madetooscillate;electrons withinmat-erials can alsobe madetoexhibitsuchbehavior.WhenelectrOniccomponentsare arrangedsothattheymakeelectronsoscillatecontinuously,theresultingdevice is calledanOSCILLATOR.Arrangementsof electroniccomponentswhichenlarge the numbers of electrons participating inthese oscillations are calledAMPLIFIERS. Therearemanydifferentkindsofoscillatorsandamplifiers,buttheabovede-scribes thebasicjobofeach.Sometimes onefindsthat hehastoomanyelectrons partiCipatingin oscillations andit is desiredtoexclude some ofthem.This is done withanarrangementof electroniccomponents,calledanA TTE NUA TOR. Anaturalquestionat thispointis,"Howdoestheoscillation ofelectronsrelateto sound?"TheansweristhattherearedevicescalledTRANSDUCERS.Theseare deviceswhichtransform energyfromoneformtoanother.Forexample,ifwe want tochangesunlighttoelectriCity,we needatransducercalledasolar cell.If we wantedtochangeburning fueloilintorotationalmotion,wewouldwant atrans-ducercalledadieselengine.Andif wewanttochangesound vibrationsintoelec-tronic vibrations,weneedatransducercalledaMICROPHONEand,conversely, if wehadsomeelectronicvibrations whichwewantedtochangeintosoundvibra-tionswewouldwantaLOUDSPEAKER.Therearemanydifferent typesofmicro-phones andloudspeakers,butthe basicfunction of allof themis as describedabove. It is difficulttoconstruct transducers whichconvert energyefficientlyandaccur-ately.Inanysort of soundsystemthe microphones andspeakersaretheelements 2 'O} - ~ " >:'..[ ( I I I I I I I I I I I 1 I - ~ - . whichlimit theperformance of thesystem.Ideally,theformoftheoscillations in theairandtheoscillationsof electrons atthemicrophone'soutputshouldbe identical;andtheelectricaloscillationsbeingfedintotheloudspeakerandthe soundcoming outshouldalsobeidentical.Anydifferencebetween whatgoesin and what comes outis calledDISTORTION. SOUNDWAVES&MUSIC Oneof the commonest typesofmotionin natureistheso-called SINEWAVE.It is alsocalledSIMPLEHARMONICMOTION.If apendulumlikethatin aclock hasapenattachedtothebottomof it,suchthatit willdrawalineonapieceof paperplacedunderneath,andif thispaperis madetomove at right anglestothe direction of thependulurnmotion,asine wave willbedrawn onthepaper.Asine waveis shownbelow. + o The"0"linerunningthroughthemiddleof thesinewaveformindicatesthein-stantsatwhichthependulumisatitsrestposition.InthecaseofOSCillating electrons,theintersection ofthesinewaveandthe"0"lineindicatesthe pOints wherethe electrons are at rest.Point" Bttindicatesthetime whenthemaximum number of electrons are flowingthroughtheoscillatorin onedirection andPoint "D",thetime whenthe maximum number ofelectrons are flowingthroughtheos-cillatorintheoppositedirection.If thiselectronicenergyisconvertedintoa-cousticalenergythecompression andrarefactionoftheair at eachinstant will bethesameastheamplitudesofthe sine waveshownabove.Agoodexampleof amusical devicewhichproducesasine waveis thetuning fork.Itsoutputisan almost puresine wave. Thetimebetweensimilar pointsonthewaverepresentsthetimeforonecom-pleteoscillation.For example,fromBtoF.orCtoG.or AtoErepresents onecompleteoscillation.Theproper nameforoneoscillationistheHERTZor itsabbreviationHz.(ThenamecomesfromtheGerman researcher,Heinrich Hertz,whodiscoveredelectromagneticwaves inthenineteenthcentury).Inold-er books youwillfindthetermCYCLEinsteadofHertz,butthemeaningis the same. Inthe illustration above,if the time for1Hz(forexample from BtoF)is O.5sec-0nd'thenthefrequencyofoscillationis_1_or 2Hz.Inengineeringunits,if 0.5 wehave a frequencyof 1, OOOHzwe wouldwrite1KHz.(TheKstands forthepre-fix"Kilo"whichmeansthousands).Thus,1KHzmeansI, OOOHz.Thatis,to 3 obtainthe actual valuethedecimalpointmust be movedthree placestothe right. Standard prefixesandtheirmeaningsare presented below. SYMBOLABBREVIA TIONFOR P pico umicro mmilli Kkilo Mmega Ggiga Numberofplaces decimal pointmust bemoved;+= movetoright/- means movetoleft. -12 -6 -3 +3 +6 +9 The sine wavegetsits namefromthe fact that acertain mathematicalfunction call-edtheSINEOF ANANGLEgivestheproper valueforeachinstantto describe the wave illustrated above.However;foryourpresent purposes it isnotnecessaryto mathematically generateasine wave. TheAMPLITUDEof asine wave is ameasure oftheheightofthepeaks(BandFin theaboveillustration)andthevalleys(D)ofthesinewave.Thehigherthe peaks andthelowerthevalleys,thegreater theamplitude. Everysinewavehasthree variablequantities.ThesearecalledPARAMETERS. Thethree parameters ofasinewaveareitsFREQUENCY,itsAMPLITlJDEand itsDURATION.Duration is merelythelengthof time insecondsthatthe sinewave continuesatan unchangingamplitude.Thislimiting ofourdiscussiontounvary-ingor STEADY-STATEwaveformsis traditionalinacousticaltheory.Problems whicharise fromthissimplification willbediscussedinamoment. HARMONICSis the name given tosine waveswhich areexact multiplesof anygiven frequency.Musicians frequentlycall harmonics OVERTONES.However;harmon-icsis amuchmoreuniversallyunderstoodterm.Inconventionalmusicalinstru-mentsthelowestfrequencysinewavecomponentalmostalwayshasthegreatest amplitude.In traditionalacousticaltheorythis lowest frequencyis calledthe FUN-DAME NT AL.Thefrequencythatistwiceas highasthefundamentalis calledthe SECONDHARMONIC,andthefrequencywhichisthreetimeshigher,theTHIRD HARMONIC,etc. Non-harmonicsinewavesarethosewhicharenot directmultiplesof agiven fun-damentalfrequency.For example,ifthegivenfrequencyis100 Hz,anothersine waveof167Hzis notharmonicallyrelated.Generally.sine waves whichare not harmonicallyrelatedtothefundamentalfrequencyare calledPARTIALStodistin-gUishthem fromharmonics.Anysoundwhichcontainsmorethanonesine wave is called aCOMPLEX WAVE.Complex waves,in whichallof thesinewavesare harmonicallyrelatedto the fundamentalare calledHARMONICWAVEFORMS.All soundsin whichoneor moreofthesine wavesare notrelatedtothefundamental frequencyare calledNON-HARMONICWAVEFORMS. 4 J I It might bewellto point out herethat thesoundsutilizedin electronicmusicoften have quitedifferentstructures fromthosegeneratedbytraditionalinstruments. Thedifferences between thesoundsandthereasons are outlined below.Asmost musicstudents alreadyknow,traditionalmusicalinstruments fallintoone of four classes:string, woodwind,brass, andpercussion instruments.Allof these types ofinstruments arerather limited bythefactthat thereisrelativelylittle control overthe frequencycontentofthesound.Asamatterof fact,tomostmusicians there is an idealtrumpetsoundor ideal violin tone forexample.Even iftheplay-er wishedtochangethestructure ofthesoundshewasproducingonaparticular instrument, he wouldfindit a rather discouraging job.Electroniccircuitry, since itdoesnotrelyonthephysicalproperties ofstrings,vibratingair columns,or other phYSicallyfixedobjects,is capableof producinganysound. Acousticalmusicalinstruments,thatistosay,allinstrumentswhichproduce sounddirectly(exceptpercussion instruments)havethefollowinggeneralchar-acteristics: 1.Thesoundsare composedsolelyofsinewavecomponents. 2.Afundamentalsinewaveisproduced.Thisfundamentalis,as men-tionedabove,the lowest frequencyinthesound andhas the greatest am-plitude. 3.Harmonicsofthisfundamentalareinvariablyproducedandarealways ofsmaller amplitudethanthefundamental. 4.Wheneveran acousticalmusicalinstrumentproducesasteadymusical tone,thereareaconsiderablenumberof frequencies present whichare rapidlychanginginamplitude,duration,andfrequency.(Besureto notethatthesecomponentsofthesoundare not objectsofstudyintra-ditionalmusicalacoustics.)Inourpreviousdescription,frequencyand amplitudewereconsidered tobeconstant forthe duration of each sound. Thesevarying componentsofthesoundarecalledTRANSIE NTS. Percussion instrumentsdonotfallintothecategoryofsoundswhichhavesine wavesofsteadyamplitudeformostoftheirduration.Percussion instruments' soundsareprimarilycomposedoftransients.Asyouwillprobablyrealizeby now,thetheoryofmusicalsound;thetheoryofsteadysine wavesis quiteinad-equatetodescribethe varietyofsounds used,even intraditionalmusic.Percus-sioninstruments,whicharealmost totallyoutsidetherealmoftraditionalmusi-calacoustic s, contain alarge numberof partials whicharetransientin character. Theamplitudes andfrequencies of thesepartials are constantlychanging withres-pect to eachother.It is quite interestingtonote thatnon-percussive instruments, whichgenerateonlyarelativelysmallamountoftransients,aredistinguished fromoneanotherbylisteners bytheirtransientcharacteristics.Forexample, if we generate allofthe constant frequency andamplitude sine components of trum-pet andviolinsounds andleave out thetransients,most listeners cannottellthem apart.Thus we have a rather difficultsituation; traditionalmusicalacoustics deals withcollectionsofsinewaves(generallyhavingasimpleharmonicstructure), whichhave constant frequency andamplitude values fortheirentire duration.Yet the most important characteristic s of musicalsounds donot behave in this manner. 5 f .-- In the nineteenth century Herman Helmholtzdeveloped the theory of musicalacous-ticsin his book,DieLehre von demTonempfindungen(1862)whichwas translated intoEnglishbyJ.A.Ellis underthetitle,SensationsofToneasaPhysiological Basis fortheTheoryofMusic.The bookis amasterpieceofsCientificinquiry. But the measuringinstrumentsof his daywere not capableof detecting andmea-suring transientsaccurately.Therefore,hetendedto neglecttheirimportance and until quite recently,acoustical theory virtuallyignoredthe existence of tran-sient soundsinmusic.Theimportanceof transientsis outlined below: 1.Allmusical instruments generatesome transients which are outside tra-ditionalmusicalandacousticaltheory. 2.Percussion instruments, which formanextremelybroad andimportant collection ofmusicalinstruments,producesoundswhicharealmost completelymade upoftransients. 3.Mostlistenersidentifyinstruments,includingthosewithonlyasmall transient content bytheirtransients.Youcan demonstratetheimpor-tanceoftransientsinconventionalmusicalinstrumentsveryeasily. Piano toneshaveahightransient content when the hammer firststrikes thestring.If yourecord apianotoneontape,andthen cut offthe first part of thetone,it is quitedifficult totell whatinstrument is producing thesound.In most conventionalmusicalinstruments thetransient con-tentis muchgreater atthebeginningandtheendofthesoundthan in anyother part. AllCOMPLEX WAVES(thatis,allsoundscontaining morethanonefrequency component)arecomposedofsimpleharmonicmotions,(sinewaves).Thisis trueofallsounds(musicalorotherwise)withoutexception.Thiswasshown mathematicallyin 1807bythe Frenchmathematician andphysicist,Jean Baptiste JosephFourier.Hewasableto provethatanywaveformcanbeanalyzed into component sine waves.Each of thesesinewavesdiffers fromtheothersin am-plitudeand frequencyandduration insuchawaythatwhen thesesinewavesare addedtogether,theywillreproduce the originalcomplex wave form.One can see that if it is possible toanalyze anycomplex wave intoa groupof simple sine waves then,conversely,one can synthesizeanycomplex wavebyaddingsinewaves to-gether.Let'sseewhat happenswhenweaddtwosinewavestogether. Sinewave#1 --""'---.....-+-......--+--- Sinewave#2 6 M ----a.-------1Wt----r-- Resultant Theresultant is merelythesumof thetwo componentsine waves at each instant. Whenthetwowavesstartatthesame time,astheydoin theaboveillustration, theyaresaid tobeinPHASE.Whentheydonotstartat thesametimetheyare saidtobeoutofphase.In theaboveillustration thetwosinecomponentsare in phasewitheach other.Suppose,however,thatthesecondwaveis outof phase withthefirst,butthatwestillhavethesametwofrequenciesandamplitudes. Nowtheresultantwavewilllookdifferent,butitwillsoundthesamebecause theear is notsensitivetophasedifference.Thepossibilityof constructing any complex wave formfromsine wavesis offundamentalimportancein electronic music,andmanysystems forgenerating complex waves rely uponthis discovery byFourier. Electronicengineers haveknownhowtoconstruct circuitstogenerate sine waves formany years.Thesesine waveoscillatorshavelongbeen used foravariety of research andtesting in the electronic laboratory.One of the earliest approaches toelectronicmusicwassimplytoborrowsinewaveoscillatorsfromtheelec-tronics'laboratoriesand usethemtocreate complex waves. Thereare severalotherspecialelectronicscircuits whichgenerate certain spe-cialwaveforms.It isimportanttorealizethat it isimpracticaltoconstruct a specialcircuitforeachcomplex wavedesired.This wouldrequirethedesign ofanastronomicalnumberof circuits.Thecircuits whichwewilldiscuss gen-erate SQUAREWAVE,RAMP,TRIANGULAR,STAIRCASEandPULSEWAVE-FORMS.EachoftheserepresentsaratherspeCialtypeofcomplex wave;they allhaveratherobviousgeometricshapes.It is importanttorealizethatmost complex waveformsdonothave obviousgeometriCshapes,andthat thegeomet-ricshapesof theabove-mentioned waveformsarethereasonthatit isrelative l'(fA$,y todesigncircuitstogeneratethem directly. GEOMETRICWAVEFORMS SQUARETRIANGULARRAMPPULSE Thesegeometricalwaveformsallhavethecharacteristicofbeingcomposedof verysimpleharmonicrelationships.For example,the square waveis composed of afundamentalplus allofthe oddharmonic s ofthis fundamentalsineOnecouldconstructasquarewavebygeneratingafundamentalsine waveand generatingsine wavesatalltheoddharmonicfrequenciesofthisfundamental. Itis interesting tonote herethat in order forthefundamentalandoddharmonics tolooksquare,thefrequencycomponents must havedefinitephase relationships. Butsincetheear isinsensitivetophaserelationships,theresultantwaveform maynot looksquare,but yet willsoundexactlythe same asasquare wave.How-ever.sinceacircuitcanbebuilttogeneratesquarewavesthatrequirefewer 7 -componentsthan manysine waveoscillators,it wouldberather foolishtogen-eratesquare wavesbyusingseveralsinewaveoscillators.Thesamesortof thing is trueof theotherspeCialwave formsabove. PARAMETERSOFSOUND(FREQUENCY) As alreadyindicated,thefrequencyof asound(orits representationin electrons in acircuit)is thenumberof times it vibrates backandfortheachsecond.The moretimes persecondthatitvibrates,thehigherthepitchof thesoundseems tobe.When we lowerthe numberof vibrations persecond,the soundseems low-er in pitchand,belowabout20cyclesper second,thevibrationsare nolonger heard as continuoussoundbutasseparate pulses.Itis interestingtonotethat if wehaveacomplexwavewithafundamentalfrequencyof 100Hzand harmonics extending up to 10, OOOHz,then if we change thefundamentalfrequencyofthis same complex wavetoafrequencyof1,000Hz,thepartials whichwere previouslyin the 10, OOOHzrange willnowbe 100, OOOHz,andthus falloutside the range ofhuman hearing.Thus,onecannotgenerateacomplex waveandthenproduceit at var-iousfundamentalfrequencies andexpect it tohavethesameoverall characteris-tics inall cases. Itmightbewellheretomentiontherangeof conventionalmusicalinstruments. Thefundamentaltonesonthepiano,forexample,havearangefromabout 25Hz to4900Hz.Theharmonics ofthese fundamentaltonesextend upwards toinfinity. However,we can hearonlythosethat are 20, OOOHzor lower.This is thereason the highest notes on the pianohaveaverythin sound;the harmoniccontent ofthese highpitchnotessoongoesoutsidetherangeof humanhearing. Even thoughthe range ofhuman hearingis consideredtoextendto20, OOOHz,most peoplecannot hearasine waveat this frequency.Onlyduring childhoodcan one hear frequencies as highas20, OOOHz.Bythe age21orsomanypeoples 1hearing hasan upper frequencylimit ofabout15or 16, OOOHz.Byage65theupper limit generally decreases to10 or 12, OOOHz.Thus the subtle nuances of complex waves, havingpartials between 12 and20, OOOHz,are perceptible only to younger listeners. Anotherinteresting andimportantproperty ofthefrequencyofsoundis thatthe human ear isnotequallysensitive toall frequencies.In order forallfrequencies toseem tobeof equalloudness,somefrequenciesmustbeproducedatamuch higheramplitudethanothers.Theear isgenerallymuchmoresensitivetofre-quenciesintherangeof400Hzto4KHz. 8 HCllNG

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