Flute acoustics PDF
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Flute acoustics: measurement, modelling and design by Paul A. Dickens BSc(Hons),Syd Infulfillment oftherequirementsforthedegree DoctorofPhilosophy SchoolofPhysics UniversityofNewSouthWales November2007 Copyright©2007byPaulA.Dickens ii Iherebydeclarethatthissubmissionismyownworkandtothebestofmyknowledgeit containsnomaterialspreviouslypublishedorwrittenbyanotherperson, orsubstantialpro- portionsofmaterialwhichhavebeenacceptedfortheawardofanyotherdegreeordiploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSWorelsewhere,isexplicitlyacknowledgedinthethesis.Ialsodeclarethattheintellectual contentofthisthesisistheproductofmyownwork,excepttotheextentthatassistancefrom othersintheproject’sdesignandconceptionorinstyle,presentationandlinguisticexpression isacknowledged. IherebygranttheUniversityofNewSouthWalesoritsagentstherighttoarchiveandto makeavailablemythesisordissertationinwholeorpartintheUniversitylibrariesinallforms ofmedia,noworhereafterknown,subjecttotheprovisionsoftheCopyrightAct1968.Iretain allproprietaryrights,suchaspatentrights.Ialsoretaintherighttouseinfutureworks(suchas articlesorbooks)allorpartofthisthesisordissertation.IalsoauthoriseUniversityMicrofilms tousethe350wordabstractofmythesisinDissertationAbstractInternational(thisisapplica- bletodoctoralthesesonly). Ihaveeitherusednosubstantialportionsofcopyrightmaterialin mythesisorIhaveobtainedpermissiontousecopyrightmaterial; wherepermissionhasnot beengrantedIhaveapplied/willapplyforapartialrestrictionofthedigitalcopyofmythesis ordissertation. IcertifythattheLibrarydepositdigitalcopyisadirectequivalentofthefinalofficiallyap- provedversionofmythesis.Noemendationofcontenthasoccurredandifthereareanyminor variationsinformatting,theyaretheresultoftheconversiontodigitalformat. iii ToRenée iv Acknowledgements Thisthesiswouldnotbepossiblewithoutthecollaborationandassistanceofmanypeople. Firstly, many thanks to my supervisor Joe Wolfe, for well-timed encouragement, motiva- tionandunderstandingthroughoutmycandidature,especiallyregardingthevarieddemands ofhavingayoungfamily.Joe’scommitmenttointegrityinresearchisapowerfulexample,sup- portedbyhiswillingcomradeshipthroughmanylaboriouslate-nightsexperiments. Joe(with Andrew Botros) also conducted the flute tuning measurements of Chapter 7 in my absence. Thankstomyco-supervisorJohnSmithforhismanyhelpfulinsightsandforsharinghisideas astheprojectprogressed.JoeandJohncollaboratedwithmeonthewritingofpapersthatpub- lishedthematerialreportedinChapters3and5. Theircommentsonearlydraftsofthethesis aremuchappreciated. ThankstoJohnTannforinvaluableandvariedtechnicalassistance,forapositive,‘can-do’ approachtothemanyproblemsIbroughttohim,andforthemanyilluminatingdiscussions resultingfrommysometimeserraticandill-formedquestions. MuchofthesoftwarecodefortheflutemodelisderivedfromtheexcellentworkofAndrew Botros. Andrewgaveuphisowntimetohelpwithissuesofsoftwaredesignandhisgenerosity inthisisappreciated. ThankstoTerryMcGeeandMarkO’Connerforgraciouslyimpartingtheirpracticalknowl- edgeintheartandscienceofflutemaking, toTerryfortakingmeonasapprenticeforaday and for being a guinea-pig in the development of the flute design software. Terry’s insights aboutclassicalflutetuningformpartofthediscussioninChapter9.ThankstothePowerhouse Museum, Sydney and the museum’s curator of musical instruments, Michael Lee, for access totheircollectionofinstrumentsaswellasfinancialsupportoftheproject. Thecollaboration withTerryandMichaelwasmadepossiblethroughanAustralianPostgraduateAward(Indus- try)grantedbytheAustralianResearchCouncil. FlutistsJaneCavanagh,CécilevanderBurgh,CaoimheMcMillanandclarinettistCatherine Youngaddedamuch-neededhumandimensiontowoodwindmodelling.Iamgratefulforthe wayinwhicheachonecheerfullycooperatedwithuncomfortableandrepetitiveexperiments. MythankstofellowstudentsintheAcousticslab,particularlyClaudiaFritzandRaInta,for helpingtokeepmyspiritsupandformanyfascinatingdiscussions(someofwhichwereabout acoustics). Ken Jackson and Pritipal Baweja in the Physics workshop turned my hasty sketches into beautifulcreationsinbrassandaluminium—theiringenuityandcraftsmanshipisappreciated. IamsupportedbyadiversecommunityofpeoplewithwhomIamlinkedbyourcommon Christianfaith. IparticularlywishtothankIan,GinaandAmandafortheirhopeful,frequent andsincereencouragementinthisendeavour, andforhelpingprovideperspectivewhenthe treesgotinthewayoftheforest. v Iamgratefultomyparents,RossandValerie,andtomysiblingsandsiblings-in-lawforthe giftofconstantlove,whichfrequentlytooktheformofweeknightdinners. Finally,myhumblethankstomywifeRenée. Hersupportandtrustthroughmycandida- turehasbeenagreatblessingandsourceofstrength. AndtomysonOliver,forhisone-year- old’ssenseoffun. vi Abstract A well-made flute is always a compromise and the job of flute makers is to achieve a musically and aesthetically satisfying compromise; a task thatinvolves muchtrial-and-error. Thepracticalaimofthisthesisistodevelopamathematicalmodelofthefluteandacomputer programthatassistsintheflutedesignprocess. Manymusicalqualitiesofawoodwindinstrumentmaybecalculatedfromtheacousticim- pedancespectrumoftheinstrument. Atechniqueforfastandaccuratemeasurementofthis quantity is developed. The technique is based on the multiple-microphone technique, and usesresonance-freeimpedanceloadstocalibratethesystemandspectralshapingtoimprove theprecisionatimpedanceextrema.Theimpedancespectraofthefluteandclarinetaremea- sured over a wide range of fingerings, yielding a comprehensive and accurate database. The impedance properties of single finger holes are measured using a related technique, and fit- formulaearederivedforthelengthcorrectionsofclosedfingerholesforatypicalrangeofhole sizesandlengths. Theboresurfaceofwoodeninstrumentscanchangeovertimewithplayingandthiscan affecttheacousticimpedance,andthereforetheplayingquality. Suchchangesinacousticim- pedanceareexploredusingwoodentestpipes. Toaccountfortheeffectofatypicalplayeron flutetuning,anempiricalcorrectionisdeterminedfromthemeasuredtuningofbothmodern andclassicalflutesasplayedbyseveralprofessionalandsemi-professionalplayers. Bycom- biningthemeasuredimpedancedatabasewiththeplayereffectsandvariousresultsinthelit- eratureamathematicalmodeloftheinputimpedanceofflutesisdevelopedandimplemented incommand-lineprogramswritteninthesoftwarelanguageC. Auser-friendlygraphicalinterfaceiscreatedusingthefluteimpedancemodelforthepur- poses of flute acoustical design and analysis. The program calculates the tuning and other acousticalpropertiesforanygivengeometry. Theprogramisappliedtoamodernfluteanda classicalflute. Thecapabilitiesandlimitationsofthesoftwarearetherebyillustratedandpos- siblecontributionsoftheprogramtocontemporaryflutedesignareexplored. vii Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTERS I Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introductiontofluteacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Theflutemaker,thecuratorandthesoftwareengineer . . . . . . . . . . . . . . . . . . 2 1.3 ‘FluteCAD’:takingtheguessworkoutofflutedesign . . . . . . . . . . . . . . . . . . . . 3 1.4 Howfluteswork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Acousticimpedance: predictingtheplayingqualitiesofaflute . . . . . . . . . . . . . . 6 1.6 Tuning,traditionandthe‘standardflutist’ . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Softwareimplementation: goalsandlimitations. . . . . . . . . . . . . . . . . . . . . . . 8 1.8 Guidetothethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 II Theoryandliteraturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Flutesandflutemaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Acousticsofwoodwindinstruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Computermodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 III Measuringacousticimpedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Reviewofmeasurementtechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Theoryofacousticimpedancemeasurements . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Calibrationofimpedanceheads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Optimisationoftheoutputsignal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Materialsandmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.8 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 IV Fingerholeimpedancespectraandlengthcorrections . . . . . . . . . . . . . . . . . . 51 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Materialsandmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 viii 4.4 Conclusionsandfurtherdirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 V Impedancespectraofthefluteandclarinet . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 Materialsandmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 VI Materialandsurfaceeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Materialsandmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 Furtherinvestigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 VII Theembouchureholeandplayercorrections . . . . . . . . . . . . . . . . . . . . . . . . 100 7.1 Playerimpedancecorrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100 7.2 Modernflutetuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 7.3 Correctionfactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 7.4 Applicationtotheclassicalflute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 7.5 Conclusionsandfurtherdirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 VIII Softwareimplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.1 Theimpedancemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 8.2 Theuserinterface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 IX Applicationsandfurtherdirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.1 Addinganewholetothemodernflute . . . . . . . . . . . . . . . . . . . . . . . . . . .122 9.2 Eight-keyflutebyRudall&Rose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 9.3 Conclusionsandfurtherdirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 APPENDICES AppendixA—Impedancespectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 AppendixB—Programlistings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 AppendixC—Quantifyingmusic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 1 Chapter I Introduction Theworldwillneverstarveforwantofwonders;butonlyforwantofwonder. G.K.Chesterton 1.1 INTRODUCTIONTOFLUTEACOUSTICS Musicandmusicalinstrumentshavelongfascinatedphysicistsandphilosophers. Thesimple numericalbasisofmusicalharmoniesisastrikingexampleoftheconcordancebetweenmath- ematicsandaesthetics, andformanyphilosophersthemathematicaltheoryofharmonyhas servedasametaphorforthequantitativeunderstandingoftheuniverse.ForPythagoras(c.6th centuryBC)musicalharmonieswerethekeytounderstandingthecosmos. ThePythagoreans likened planetary orbits to huge strings on which the universe itself made a kind of music— the‘HarmonyoftheSpheres’. Theorbitsofeachplanetwererelatedtothoseofitsneighbours byasimplemusicalinterval. Thisconcept,althoughsupersededbymoderncosmology,cap- turedtheimaginationofpoetsandscientistsalike. Thecelestialmusicfeaturedintheworkof Shakespeare,MiltonandDonne: Thesphereshavemusic,buttheyhavenotongue, Theirharmonyisratherdancedthansung... (Donne1971,p.333)andinspiredDonne’scontemporaryKeplertodevisehisfamouslawsof planetarymotion(Koestler1959). MusicalanalogiesalsofeatureinBohr’smodeloftheatom whichledtomodernquantumphysics. Quite apart from the influence of music in physics and philosophy, many scientists have been interested in music and musical instruments in their own right. In the 19th century HelmholtzandRayleighlaidthefoundationsofmusicalacousticsandpsychoacoustics. Many problems in musical acoustics are surprisingly complex and involve the interplay of physics, cultureandpsychology. Somemusicalinstrumentsareadirectcreativedevelopmentofasci- entificdiscovery—onethinksoftheelectricguitarandFaraday’slaw—andinstrumentmakers havehistoricallymadeactiveuseofscientificandtechnologicaldevelopments. Mostmusicalinstrumentsaremadewithmanydifferent(andsometimesconflicting)goals in mind. Instruments are usually required to play in tune over a wide range of pitches with reasonabletimbaluniformityovertherange(althoughinsomemusicaltraditionsnoteswith differenttimbreareusedformusicaleffect). Inthecaseofthefluteandotherwoodwindin- strumentstheseveraldozennotesplayableontheinstrumentaremadewhentheplayercovers various holes in the instrument or operates various keys and the design of such instruments is constrained by the necessity for a relatively simple, responsive fingering mechanism. In- evitably, singleholesontheinstrumentcontributetotheproductionofmorethanonenote, CHAPTER1. INTRODUCTION 2 and the position and size of such holes is a compromise. Existing instruments are therefore notperfectandeventheideaofperfectioninthiscontextiscontestable. Instrumentsusually change gradually in response to musical tastes and performance demands. There are some notable exceptions—such as Boehm’s development of the modern flute and the invention of thesaxophonebyAdolphSaxinthe19thcentury.Instrumentmakersvaryintheirapproachto instrumentdesign,someguidedprimarilybytheworkofearliermakersandsimplerules-of- thumb,whileothersaremorerigourousandacousticallyinformed.Inmostcases,however,an elementoftrial-and-errorisinevitablyinvolved. Inthelightofthis,oneoftheaimsofmusicacousticsistohelpmakers:tobetterunderstand thephysicsofmusicalinstrumentsandtousethisknowledgetomakebetterinstruments,or evenjusttomakegoodinstrumentsmorequicklyorwithlesscost.Suchisthegoalofthisthe- sis. Thetaskissubtleandrequiresahighdegreeofprecision,sincethehumanearissensitive tominutechangesinpitchandtimbre,andanyphysicalmodelneedstomatchthatsensitiv- ity.Therearealsomanyconstraintsonthedesignofmusicalinstruments(determinedbysuch thingsasthesizeofaplayer’shands)andnewdesignsneedtofitwithintheseconstraints. 1.2 THEFLUTEMAKER,THECURATORANDTHESOFTWAREENGINEER Thisprojectpresentedtheopportunitytobringtogetherthreediversefields,expertsinwhich mightnottypicallyfindthemselvesincollaboration. Terry McGee (shown at his mill in Figure 1.1) is an Australian maker of wooden flutes in theclassicalstyle. Hisflutesareshippedworldwide,particularlytoplayersofIrishtraditional music, of which he is one. Terry’s designs are based on popular historical instruments from theclassicalperiod(late18thtoearly19thcenturies),butheregularlymodifieshisdesignsto http://www. improveintonationandbettertosuitcustomers’requirements.Terry’swebsite< mcgee-flutes.com/ ∗ > hasextensiveinformationonthehistoryofflutemaking,hisdesigns andinnovations,andfluteperformance. ThePowerhouseMuseuminSydneyhousesalargecollectionofmusicalinstruments. The museumhasanextensiverangeofearlyflutes,manyofwhicharemadeofwoodandarenow unplayableduetocrackingorothermechanicaldefects. Themuseum’scuratorofmusicalin- struments,MichaelLee,isinterestedintheplayingcharacteristicsandintonationoftheseold instruments. Thismightpotentiallybeusedtotracethehistoryofflutedevelopment,aswell aschangesintuningandtemperamentoverthelastfewhundredyears. TheVirtualFluteisawebservicebuiltandmaintainedbysoftwareengineerAndrewBotros that uses a semi-empirical model of the impedance of the modern flute, based on measure- mentsperformedattheMusicAcousticsLaboratoryattheUniversityofNewSouthWales.The webservicecalculatestheimpedanceofthemodernfluteforanyofits39744possibleacoustic configurationsandprovidesplayerswithtuning, timbreandplayabilitypredictionsforthese fingerings(Botrosetal.2006).Thewebservicehasmanyanddiverseapplicationsamongplay- ers and composers such as finding new fingering combinations for difficult-to-play phrases, ∗ AcopyofthiswebsitearchivedbytheNationalLibraryofAustraliaisavailableat<http://nla.gov.au/nla. arc-24785>.
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