CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL

MATHEMATICS AND MUSIC COMPOSITION, PERCEPTION, AND PERFORMANCE

MATHEMATICS AND MUSIC COMPOSITION, PERCEPTION, AND PERFORMANCE JAMES S. WALKER GARY W. DON UNIVERSITY OF WISCONSIN EAU CLAIRE, USA

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130410 International Standard Book Number-13: 978-1-4822-0850-4 (ebook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Preface Abouttheauthors 1 Pitch, Frequency, and Musical Scales 1 1.1 PitchandFrequency... 1 1.1.1 InstrumentalTones... 2 1.1.2 PureTonesCombiningtoCreateanInstrumentalTone... 4 1.2 Overtones,PitchEquivalence,andMusicalScales... 6 1.2.1 PitchEquivalence... 7 1.2.2 MusicalScales... 8 1.3 The12-ToneEqual-TemperedScale... 12 1.4 MusicalScaleswithintheChromaticScale... 15 1.4.1 TheC-MajorScale... 15 1.4.2 OtherMajorScales... 17 1.4.3 ScalesandClockArithmetic... 18 1.4.4 RelationbetweenJustandEqual-TemperedTunings... 20 1.5 Logarithms... 23 1.5.1 HalfStepsandLogarithms... 23 1.5.2 Cents... 27 2 BasicMusicalNotation 33 2.1 StaffNotation,Clefs,andNotePositions... 33 2.1.1 TrebleClefStaff... 33 2.1.2 BassClefStaff... 34 2.1.3 GrandStaff... 35 2.2 TimeSignaturesandTempo... 37 2.2.1 TimeSignatures... 39 2.2.2 Tempo... 42 2.2.3 RhythmicEmphasis... 42 2.3 KeySignaturesandtheCircleofFifths... 45 2.3.1 CircleofFifths... 46 2.3.2 CircleofFifthsforNaturalMinorScales... 48 3 SomeMusicTheory 51 3.1 IntervalsandChords... 51 3.1.1 MelodicandHarmonicIntervals... 51 3.1.2 Chords... 53 3.2 DiatonicMusic... 58 3.2.1 LevelsofImportanceofNotesandChords... 58 3.2.2 MajorKeys... 58 3.2.3 ChordProgressions... 60 3.2.4 RelationofMelodytoChords... 61 3.2.5 MinorKeys... 62 3.2.6 Chromaticism... 64 3.3 DiatonicTransformations ScaleShifts... 68

3.4 DiatonicTransformations Inversions,Retrograde... 76 3.4.1 DiatonicScaleInversions... 76 3.4.2 Retrograde... 78 3.5 ChromaticTransformations... 81 3.5.1 Transpositions... 81 3.5.2 ChromaticInversionsandRetrograde... 85 3.5.3 ChromaticTransformationsandChords... 87 3.6 WebResources... 90 4 SpectrogramsandMusicalTones 93 4.1 MusicalGesturesinSpectrograms... 93 4.2 MathematicalModelforMusicalTones... 97 4.2.1 BasicTrigonometry.... 98 4.2.2 ModelingPureTones... 99 4.3 ModelingInstrumentalTones...103 4.3.1 Beating...104 4.4 BeatingandDissonance...107 4.4.1 SomeUsesofDissonanceinMusic...108 4.5 EstimatingAmplitudeandFrequency...111 4.5.1 HowtheEstimatingIsDone...112 4.6 WindowingtheWaveform:Spectrograms...118 4.7 ADeeperStudyofAmplitudeEstimation...122 4.7.1 MoreonRectangularWindowing.......126 4.7.2 MoreonBlackmanWindowing...128 5 SpectrogramsandMusic 131 5.1 Singing...131 5.1.1 AnOperaticPerformancebyRenéeFleming...131 5.1.2 AnOperaticPerformancebyLucianoPavarotti...135 5.1.3 ABluesPerformancebyAliciaKeys...135 5.1.4 AChoralPerformancebySweetHoneyintheRock...136 5.1.5 Summary...137 5.2 Instrumentals...140 5.2.1 JazzTrumpet:LouisArmstrong...141 5.2.2 Beethoven,Goodman,andHendrix......143 5.2.3 HarmonicsinStringedInstruments...145 5.3 Compositions...153 5.3.1 Roy Hargrove s Strasbourg/St.Denis...153 5.3.2 TheBeatles TomorrowNeverKnows...154 5.3.3 APortionofMortonFeldman stherothkochapel...155 5.3.4 APortionofaRaviShankarComposition...156 5.3.5 MusicalIllusions:Little BoyandThe Devil s Staircase...157 5.3.6 TheFinaleofStravinsky sfirebird Suite...161 5.3.7 DukeEllington sjackthebear...162 5.3.8 ConcludingRemarks...165 6 Analyzing Pitch and Rhythm 167 6.1 GeometryofPitchOrganizationandTranspositions...167 6.1.1 PitchClasses,Intervals,Chords,andScales...167 6.1.2 Transpositions...169 6.1.3 Clock Arithmetic Formally Defined...170 6.2 GeometryofChromaticInversions...173

6.3 CyclicRhythms...177 6.3.1 CyclicRhythminDownintheValley...177 6.3.2 CyclicRhythminDrumming...178 6.3.3 TimeTranspositionsofCyclicRhythms...178 6.3.4 Afro-LatinClaveRhythms...179 6.3.5 Phasing...182 6.4 RhythmicInversion...188 6.4.1 InversionofCyclicRhythms...188 6.4.2 RhythmicandPitchTransformationGroups......189 6.5 ConstructionofScalesandCyclicRhythms...192 6.5.1 TheEuclideanAlgorithm...192 6.5.2 ConstructingMusicalScales...193 6.5.3 ConstructingCyclicRhythms...195 6.6 ComparingMusicalScalesandCyclicRhythms...198 6.6.1 IntervalFrequenciesandMusicalScales...198 6.6.2 MeasuringDissonanceforaScale...201 6.6.3 IntervalFrequenciesforCyclicRhythms...202 6.7 Serialism...206 6.7.1 PitchSerialism...206 6.7.2 MusicalMatrices...208 6.7.3 TotalSerialism...212 7 AGeometryofHarmony 217 7.1 Riemann schromaticinversions...217 7.2 ANetworkofTriadicChords...223 7.2.1 MusicalExamples...225 7.3 Embedding Pitch Classes within the Tonnetz...228 7.3.1 GeometryofAcousticConsonanceandDissonance...229 7.3.2 AnalyzingScales...230 7.4 OtherChordalTransformations...233 7.4.1 UsingtheTonnetztoDefineMoreTransformations...233 7.4.2 ModelingChordProgressionsinDiatonicMusic...235 7.4.3 ExplainingtheQualitativeDifferenceinModes....237 8 AudioSynthesisinMusic 241 8.1 CreatingNewMusicfromSpectrograms...241 8.2 PhaseVocoding...245 8.2.1 ABasicExample...245 8.2.2 ImogenHeap shideandseek...247 8.3 TimeStretchingandTimeShrinking...249 8.4 MIDISynthesis...253 A ExerciseSolutions 257 B MusicSoftware 295 B.1 AUDACITY...295 B.1.1 ConfiguringAUDACITY...295 B.1.2 LoadingandDisplayingaMusicFile...295 B.1.3 MusicFilesfromCDs...296 B.2 MUSESCORE...296 B.2.1 DifferentSoundfontsforMUSESCORE...296

C Amplitude and Frequency Results 299 C.1 ProofofTheorem4.7.1...299 C.2 ProofofExactAmplitudeandFrequencyEstimates...300 D Glossary 303 E Permissions 305 Bibliography 307 Index 309

Preface No school would eliminate the study of language, mathematics, or history from its curriculum, yet the studyofmusic,whichencompassessomanyaspectsofthese fieldsandcanevencontributetoabetter understanding of them, is entirely ignored. Daniel Barenboim Thepurposeofthisbookistoexploretheconnectionsbetweenmathematicsandmusic. Thismay seem to be a curious task. Aren t mathematics and music from separate worlds, mathematics from theworldofscienceandmusicfromtheworldofart? Whilemathematicsdoesbelongtotheworld ofscience,oneofthegoalsofscienceistounderstandeverythingthatweexperience,andmusicis nodoubtanessentialpartofhumanexperience. Mathematicshasbeendescribedasthescienceof patterns,andweshallseethattherearemanypatternsinmusicthatcanbedescribedwithmathematics. Mathematicshasalsobeendescribedasthelanguageoftheuniverse,andmusicitselfhasbeen describedinsuchapoeticway. Infact,connectionsbetweenthesetwosubjectsgobackthousands ofyears.forexample,theclassicalgreekmathematician,pythagoras,contributedtheessentialideas forhowwequantifychangesofpitchformusicaltones(musicalintervals).theconnectionsbetween mathematicsandmusichavegrownenormouslysincethoseancientdays. Wewilltrytoexploreas manyoftheseconnectionsaspossible,inawaythatpresentsboththemathematicsandthemusicto aswideanaudienceaspossible. Summary of Chapters Hereisabriefdescriptionofthemaintopicscoveredinthebook.Formoredetails,pleaseconsultthe TableofContents. Chapter1describesthescientificapproachtomusicalpitch, firstworkedoutbyhelmholtzin the 19 th century. Helmholtz stheory,whichrelatespitch to frequency,providesafoundationfor understandingdifferentmusicalscales. Oneverydistinctiveaspectofourtreatmentofthismaterial, isthatweusethemethodofspectrograms.aspectrogramisagraphicalportraitofthetoneswithin a musical passage, plotting these tones in terms of their frequencies and the time during which they aresounding.webelievethatspectrogramsareanimportanttoolforunderstandingandappreciating music,andthattheyarenotdifficulttointerpretcorrectly.soweintroducethembeforewedescribe the mathematics used to create them; we postponethat discussion to Chapter 4. Althoughsome mightobjecttousingamathematicaltechniquebeforedescribingthedetailsunderlyingit,webelieve that the spectrogram examples described here are so compelling, and so dramatically illustrate this material,thatwesimplyhadtoincludethem. Inanycase,theyshouldprovideastrongmotivation for learningthemathematicsof spectrogramsdescribedin Chapter4. Chapter 2 providesa brief introductionto musical notation. It describesjustenoughnotationso that allreaders, eventhose whoarenotmusicians,shouldbeabletoreadthebriefscoreexcerptsthatweincludeinthebook. ThereareanumberofsuchscoreexcerptsinChapter3,whereweprovidesomebackgroundinbasic music theory. This basic music theoryis surprisinglymathematical. We emphasizethe different musicaltransformations scaleshiftings,transpositions,inversions thatcomposershaveemployed for centuries. These transformations do have a clear mathematical interpretation. Asdescribedinthelastparagraph,inChapter4wediscussthemathematicsofspectrograms. In addition to the mathematics, we also provide some interesting musical illustrations, such as the

phenomenonknown asbeating and its relation to musicalconsonanceand dissonance. In Chapter5wedemonstratehowspectrogramsproviderevealinginsightsintomusicalstructure. Theseinsightswouldbedifficultifnotimpossibletoobtainthroughlisteningalone,becauselisteninginvolves mostlyshort-termmemory,whilespectrogramscandisplayananalysisofseveralminutesofmusic. Furthermore,whenvideosofspectrogramsaretracedoutasthemusicisplayedtheyallowustosee aheadwhattonesaretobeplayed,therebyenhancingouranticipationofthemusic sdevelopment. Spectrogramsalsoallowustodetect,andmoredeeplyappreciate,subtleaspectsofmusicalsound qualitysuchasvibrato,dynamicemphasis,andpercussion. Alloftheseinsightswouldbedifficult, ifnotimpossible,togainifoneonlyanalyzedscores.spectrogramsprovideapowerfultoolforanalyzingthemusicthatwehear,ratherthanthenotesprescribedformusicianstoplay.havinganother toolforanalyzingmusic,inadditiontomusicalscores,isveryvaluable.onewaythatspectrograms andscoresworktogetheristhatspectrogramsrevealtheovertonestructureofthenotesplayedfrom amusicalscore.thisovertonestructureisveryimportantforunderstandingmusicalintervals,which arethebuildingblocksofmelodyandharmony. Wehavedescribedsomeofthemanyvaluablecontributionsthatspectrogramsmaketothestudy andappreciationofmusic.ourstudentsgenerallyconsiderthematerialonspectrogramsinchapters4 and5tobethehighlightofthebook.followingthesechapters,weincorporaterhythmintoourstudy ofthemathematicalaspectsofmusic. InChapter6wedescribehowpitchandrhythmsharemany ofthesamemathematicalfeatures.mostbooksonmusic,bothinmusictheoryandinmathematical treatments,focusexclusivelyonpitchandharmony.webelieveourtreatmentofrhythmprovidesour bookwithamorecompletedescriptionofmusic. Thesixchaptersjustdescribedformthecorematerialofthebook. Thetwochaptersthatfollow themdescribemoreadvancedmathematicalaspectsofmusic.throughoutthebook,wemakeuseof geometricaldiagramstoaidusinunderstandingthebasiclogicofpitchorganizationandharmony. Chapter 7 explores this connection of geometry with music theory more deeply.chapter 8 describes some of the ways thatcomputerscan be used for synthesizingmusic. Electronicallysynthesized musiciswidelyused,andwehavetriedtoexplainhowitworkswithoutgettingoverwhelmedby technicalities. Web site Toaidinthestudyofthisbook,thereisanaccompanyingwebsite.Toaccessthissite,gototheCRC website: www.crcpress.com anddoasearchusingthebooktitle: Mathematics and Music: Composition,Perception,and Performance. Youwillthenarriveatthewebpageforourbook,whereyoucanclickonalinktoaccessthesupportingwebsite.Therearelinksatthebook swebsiteforvideosofmanyofthespectrogramswediscuss inthebook. Youcanalsodownloadthemusicalscoresweexamineinthebook,playablewiththe freemusicsoftwaremusescore. Wehavesuppliedanonlinebibliographywithmanylinkstofree downloadablearticlesonmathandmusic. Finally,therearelinkstootherwebsitesrelatedtomath andmusic,includingalltheonesmentionedinthebook. Prerequisites Toreadthisbook,oneneedstohaveagoodbackgroundinhighschoolmathematics. Wewillnot assume,however,theabilitytoreadmusic. Thebookaimstoteachsomemathematics,sothereare exercisesattheendofeachsection. Italsoaimstoteachhowthemathematicsrelatestomusic,so manyoftheexercisesinvolvemusicalexamples. Attheendwehopethereaderwillhaveagreater masteryofsomefundamentalmathematics,andadeeperappreciationofmusic. Anappreciationof musicmadedeeperbecauseitisinformedbybothitsmathematicalandaestheticstructure.

Music Software Theworldofrecordedmusichasbeenenormouslychangedinthelastthreedecadesorsowiththe introductionofcomputertechnology.inthisbook,weusecomputerstoaidinapplyingmathematics totheanalysisofmusic,andalsotothecreationofnewmusic. Mostly,weusetwo freesoftware programs.thesetwofreeprogramsare 1. AUDACITY.Anaudioeditor.Wehaveuseditforcreatingandplayingspectrograms. 2. MUSESCORE. Amusicalscoringprogram.Wehaveusedittocreatebriefpassagesofmusical scores,whichyoucanplayonmusescorewhenstudyingthesepassagesinthetext. Thebookcanbestudiedwithoutworkingwiththeseprograms,althoughweencourageyoutotry them.weprovidesometutorialsonusingtheseprogramsinappendixb. Order of Chapters Chaptersaremostlyorganizedsequentially. Eachchapteruses,toadegree,materialfrompreceding chapters. Chapter8isanexception,asitcanbereadimmediatelyfollowingChapter4. Although chaptersproceedsequentially,thereissome flexibilityinhowtheycanbecoveredinaclassroom setting.forexample,inourmathematicsandmusiccourseatuw-eauclaire,wehavesuccessfully taughtthematerialusingthefollowingsequence: Chapter1, Chapter4, Chapter5, Chapter3, Chapter6, Chapter8, Chapter7. Sincetypicallyatleasthalfoftheclasscanplayandreadmusic,Chapter2isgivenasoptionalreading atthestartoftheclassforthosestudentswhoneedtolearnbasicmusicnotation. Havingstudents workingroupsonmaterial,suchaschapter3withitsemphasisonmusictheory,canbeveryhelpful forthosestudentswhohaveagreatinterestinunderstandingmusicbutlackperformanceability.we havefound,however,thatevenstudentswhoarenotmusicianscanmastertheelementarymaterialin Chapter2ontheirown,andthenreadthemusictheoryinChapter3withunderstanding. Acknowledgments It is a pleasure to acknowledgeas many people as I can, who have helped me with this project. GaryDon,associateprofessorofmusicatUWEC,hasbeenaconstantsupportivecolleaguefromthe worldofmusic. Simplylistinghimasmusicalconsultantforthisbookdoesnotreallydojusticeto theenjoyableinteractionsandcollaborationsthatwehavebeenengagedinforoveradecade. My MathematicsDepartmentChairatUWEC,AlexSmith,hasdoneeverythinginhispowertohelpme teach mymathematicsandmusiccourse. Withouthis hardworkonmy behalf, thisbookwould simplynotexist. StevenKrantz,ProfessorofMathematicsatWashingtonUniversity-St.Louis,has givenmealotofencouragementandhelpinpublishingmypapersonthissubject. Thescholarly supportprogramsat UWEC the Center forexcellencein Teachingand LearningandtheOffice ofresearchandsponsoredprograms havegenerouslyprovidedmewithgrantsforpursuingthe researchandwritingactivitiesneededforproducingthisbook. Oneextremelyimportantgrantwas forfundingmysabbaticalleaveatmacalestercollegeintheacademicyear2011-12.whileiwasat Macalester,IwasabletoteachacourseinMathematicsandMusic.IparticularlywanttothankKaren Saxe, Chair of the Department of Mathematics, Statistics, and Computer Science, and Mark Mazullo, ChairoftheDepartmentofMusic,atMacalesterforarrangingmypositionasVisitingProfessorin thosedepartments.mystudentshavegivenmealotofhelpaswell.iwouldespeciallyliketothank LaraConrad, HannahStoelze, MichaelJacobs, Stewart Wallace, JeanneKnauf, AndrewJannsen, GaryBaier,AndrewDetra,KaitlynJohnstone,JoshuaFuchs,AbigailDoering,ThomasKokemoor, CarmenWhitehead,AndrewHanson,XiaowenCheng,JarodHart,KarynMuir,BrentMcKain,Yeng Chang,andMarisaBerseth.Finally,aheartfeltthankstomywife,AngelaHuang.Iamverygrateful forherpatientsupport,andidedicatethisbooktoher.

About the authors PrincipalAuthor JamesS.WalkerreceivedhisdoctoratefromtheUniversityofIllinois,Chicago,in1982. Hehasbeenaprofessor ofmathematicsattheuniversityofwisconsin-eauclaire since1982.hispublicationsincludepapersonfourieranalysis,waveletanalysis,complexvariables,logic,andmathematicsandmusic. Heisalsotheauthorof fivebookson Fourieranalysis,FFTs,andwaveletanalysis. MusicalConsultant GaryW.Donreceivedhisdoctorateinmusictheoryfrom theuniversityofwashingtonin 1991. Heteachestheory andauralskills, 20th-centurytechniques,counterpoint,and formandanalysisasanassociateprofessorattheuniversityofwisconsin-eauclaire. Thetopicsofhispublished articlesincludegoethe sinfluenceonmusictheoristsofthe 19th and 20th centuries, overtone structures in the music of Debussy,andtheorypedagogy.

Chapter 1 Pitch,Frequency,andMusicalScales...innatureitself,asinglenotesetsupaharmonyofitsown;andthisharmonicserieshasbeenthe (unconscious) basis of Western European harmony, and the tonal system. Deryk Cooke The scientific study of music was put on a firm footing with the seminal work of Hermann von Helmholtzinthemiddleofthe 19thcentury. Hismasterpiece, On the Sensations of Tone, isstill worthstudyingtoday. InthischapterwedescribeHelmholtz sideasandshowhowtheyprovidea rationaleformusicalscales. 1.1 Pitch and Frequency Thereisacloseconnectionbetweenpitchesinmusicaltonesandthemathematicalconceptof frequency. Inthe 19thcentury,Helmholtzdidanexperimentwithtuningforks. Heattachedapento oneofthetinesofatuningforkanddrewtheforkacrossapieceofpaperwhileitwassoundinga specificpitch. Thevibrationofthepentracedoutasimplewaveform. Wewillrefertoitasapure tone waveform.seefigure1.1. Themostfundamentalaspectofapuretonewaveformisthatitrepeatsitself periodically. In physics,thedistancefromonepeakofthewavetothenextiscalleditswavelength. Anotherterm forwavelengthiscycle. WehavemarkedonecycleforthepuretonewaveforminFigure1.1. The numberofcyclesinthepuretonewaveformthatoccurin 1secondiscalleditsfrequency.Wehave thisformulaforfrequency: frequency = numberofcycles. second Theunitofcycles/secformeasuringfrequencyisalsocalledHz,whichisshortforHertz(another Germanphysicistwhodid fundamentalwork in thestudyoffrequency). Forexample, ifthecycleshowninfigure1.1hasatimedurationof 0.025seconds,thenthefrequencyofthepuretone waveformis1/0.025= 40Hz. Nowadays, with digital technology,we can recordarepresentationofthesound wavefrom a tuningforkasafurtherdemonstrationofhelmholtz sidea.infigure1.2weshowtheplotofsucha digitalwaveformrecordedfromatuningfork. Thistuningforkwasdesignedtomatchthepitchfor thekeyofmiddleconapiano. 1 AsdescribedinthecaptionofFigure1.2,thefrequencyforthepure tonewaveformproducedbythetuningforkis 262Hz. Themaximumy-valuesofthewaveformare all approximately 5000(and the minimum values are all approximately 5000).This number,5000, iscalledtheamplitude ofthepuretone.thelargerapuretone samplitude,thelouderthevolumeof itssound. To concludeouranalysisofpuretones fromtuningforks, we lookat an extremelyimportant methodfordisplayingthesingle, constantfrequencyofthepuretoneovertime. Thismethodof 1 SeeFigure1.12onp.15. 1

2 1.Pitch,Frequency,andMusicalScales 1cycle Figure1.1. Illustrationof the famous experiment of Helmholtz. A pen is attached toatine of tuning fork. Astuningforkisstruckanddrawnacrossapieceofpaperatauniformspeed,thepentracesoutapuretone waveform. Distance marked between two peaks of waveform is called a cycle. p 8000 4000 y 0 Amp. 5000 7500 5000 Amp. 2500 4000 0 8000 0 0.012 0.023 0.035 0.046 Time(sec) 2500 0 524 1048 1572 2096 Frequency(Hz) Figure 1.2. Left: Waveform from recording of tuning fork with one cycle marked, p = 0.00382 seconds. Frequency is about 1cycle/0.00382 sec = 262 Hz, so note from tuning fork is middle C (c.f. Table 1.2, p. 13). Right: Amplitude for waveform. The height of the spike at frequency 262 Hz is approximately 5000. In Chapter 4 we describe how this amplitude plot was obtained. displayiscalledaspectrogram.infigure1.3weshowaspectrogramofarecordingofthesoundfrom thetuningforkdiscussedabove.wewilldescribehowthisspectrogramwasproducedinchapter4. Weareusingitnowbecauseitprovidessuchaneasilyinterpretableandcompellingpictureofthe frequencycontentofthispuretone.thesinglebright,horizontalbandinthespectrogramiscentered onthesinglefrequencyof262hzthatwefoundforthistuningfork spuretone. Wehavenowshownthatthereisadefiniteconnectionbetweenfrequencyandpitch. Forapure tonefromatuningfork,therewillbeasingleprecisefrequencyforitswaveform.music,ofcourse,is playedonmusicalinstrumentsratherthantuningforks.sowenowturntothequestionoffrequency and pitch for musical instruments. 1.1.1 InstrumentalTones Thereisahugevarietyofmusicalinstruments,includinghumanvoices,violins,pianos,clarinets, and manyothers. Thetonesproducedby theseinstrumentsarefar morecomplex, and musically interesting,thanthetonesproducedbytuningforks. SeetheleftsidesofFigures1.4and1.5for examplesofportionsofwaveformsfromafluteandpianoplayingthesamenote. Thesewaveforms haveafundamentalcycle,atleastapproximately.weshowthisapproximatefundamentalcycleonthe leftsideoffigure1.4forthe flutetone,whereitiseasiertoseeinthegraph.thisfundamentalcycle determinesthefrequencyofapproximately329hzforthenotebeingplayed. Wecansee,however, thatthesewaveformsarenotcyclingin nearlyso uniformamannerasthetuningforkwaveform

1.1 PitchandFrequency 3 262Hz Figure 1.3. Spectrogram of recording of tone from tuning fork. Horizontal axis is time axis, labeled in units of secondsalongtop. Verticalaxisisfrequencyaxis,labeledinunitsofkHz(1 khz = 1000 Hz)alongleftside. Bright white band is centered on frequency 262 Hz, the frequency for tuning fork tone(c.f. Figure 1.2). Background features of spectrogram come from faint background noise, largely inaudible, present during recording process. Towatchavideoofthisspectrogramtracingoutintime,asthesoundfromthetuningforkisplayed, pleasevisitthebook swebsiteandclickonthelinkvideos. showninfigure1.2. Infact,asweshallexaminemorecloselyinChapter4,theyarecombinations ofseveralpuretonewaveformsofdifferingfrequencyandloudness. Forbothoftheseinstruments, these combined pure tone waveforms have frequencies that are positive integer multiples of 329Hz: 329Hz, 2 329Hz, 3 329Hz, 4 329Hz, 5 329Hz, 6 329Hz,... (1.1) Thesefrequenciesarecalledtheharmonics fortheseinstrumentaltones. 16000 3600 p 8000 0 y 2400 Amp. 1200 8000 0 16000 0 0.012 0.023 0.035 0.046 Time(sec) 30 0 658 1316 1974 2632 Frequency(Hz) Figure1.4. Left: Waveformforarecordingof fluteplayingsinglenote. Time pforanapproximatecycleis p = 0.00304 seconds. Fundamental frequency is approximately 329 Hz. Right: Amplitudes of harmonics for waveform. Harmonicsof 329Hz, 2 329Hz,and 3 329Hz,areclearlymarkedbyspikes. Heightsofspikes correspond with amplitudes of each pure tone within the complete tone. In Chapter 4 we describe how this plot of amplitudes for harmonics was obtained. Forthe flutenote,the firsttwoharmonicsof 329Hzand 2 329 = 658Hzaretheloudest,the thirdharmonic3 329 = 987Hzisfainter,andthehighermultiplesof 329Hzarefainterstill. See therightsideoffigure1.4. Inthegraphshownthere,theheightsofthepeaksat 329Hz, 658Hz, and 987Hz,correspondtohowloudthosepitcheswouldbeifthosepitcheswereheardseparately. Theyarenotheardseparately,however. Itistheircombinationthatproducesthecomplexsoundof the flute stone.wewilldiscussthispointinmoredetailattheendofthissection.

4 1.Pitch,Frequency,andMusicalScales Similarly,thenoteplayedbythepianoisacombinationofharmonics. Forthepiano,however, the graph on the right of Figure 1.5 shows that the amplitudes of the harmonics are much more equallydistributedinsize.aninterestingfeatureofthisgraphisthatthemagnitudefortheharmonic 2 329Hzisactuallylargerthanthemagnitudefor329Hz.Nevertheless,wereferto 329Hzasthe fundamentalforthispianonotesinceitcorrespondstothepitchthatthenoteissoundingat(which isthesamepitchasthe flutenote).weshallemphasizethispointagainlater,asitisasubtleone:the fundamental harmonic for a note is the frequency that determines the note s pitch, and that may ormaynotbetheloudestharmonicproducedwhenthenoteisplayed. 16000 3000 8000 0 y 2400 Amp. 1200 8000 0 16000 0 0.012 0.023 0.035 0.046 Time(sec) 30 0 658 1316 1974 2632 Frequency(Hz) Figure 1.5. Left: Waveform for recording of piano playing single note. Right: Amplitudes of harmonics within pianonote,harmonicsmarkedbyspikesatmultiplesof329hz.firstspikeisat329hz,secondspikeat2 329 Hz,thirdspikeat 3 329Hz,uptoseventhspikeat7 329Hz. Thegraphsofamplitudesofharmonicsof fluteandpianotonesshownontherightoffigures1.4 and 1.5 wereobtained by computerprocessingof recordingsof these tones. We shall explainin Chapter4howthisprocessingisdone. We summarize this discussion with a definition of these harmonics in instrumental tones. Definition1.1.1. Foraninstrumentaltonethatcontainsfrequenciesoftheform: ν o, 2ν o, 3ν o, 4ν o, 5ν o, 6ν o,... Thesmallestfrequency, ν o,iscalledthefundamental. Theotherfrequenciesarecalledovertones. Allofthefrequenciesarecalledharmonics.Thefirstharmonicis ν o,thesecondharmonicis 2ν o, thethirdharmonicis3ν o,andsoon. Remark1.1.1. Thephysicalexplanationforwhytonesfrommusicalinstrumentscontainmultiple harmonicsisbeyondthescopeofthisbook.seechapter3offourieranalysis,byjamess.walker (Oxford University Press, 1988) for a discussion of stringed instruments. For other instruments, consultthebookmusic,physicsandengineering,byharryf.olson(dover,1967). 1.1.2 PureTonesCombiningtoCreateanInstrumentalTone Wehavedescribedhowinstrumentaltonesarecombinationsofpuretones.Weshallnowdemonstrate thisforasingletrumpettone. InFigure1.6, weshowaspectrogramofarecordingofatrumpet playing the note of middle C, with fundamental ν o = 262 Hz, and of the individualpuretones that combine to create this instrumentaltone. As a static picture, this spectrogramshows single brightbandscorrespondingtotheindividualharmonicsofthetrumpettone,andhowtheycombineto

1.1 PitchandFrequency 5 producethecompletetone.however,tofullyappreciatethisspectrogram,itisabsolutelynecessaryto watchavideoofit.pleasevisitthebookwebpagelistedinthepreface,andclickonthelink,videos. Asthespectrogramtracesout,youwillheartheindividualharmonicssoundingjustlikeindividual tuningforks,astheyshouldbecausetheycorrespondtothepuretonesmakingupthetrumpettone. Attheendofthespectrogram,youwillhearallofthesepuretonesplayingtogether,creatingthe completetrumpetsound. Figure1.6. Spectrogramillustratingcombinationofharmonicsintrumpettoneforsinglenote. From 0.00to 1.5 seconds: tone s fundamental of 262 Hz displayed as bright band. From 1.5 to3.0 seconds: the tone s second harmonic of 524Hz(=2 262Hz)displayedasanother brightband. From 3.0to 4.5seconds, tone sthird harmonicof 786Hz(=3 262Hz)displayedasthirdbrightband. Eachof first 8harmonicsaredisplayedin ascending order from left to right, finishing at 12.0 seconds. Sounds from these harmonics are indistinguishable from tuning fork tones. (Thin vertical bars at start and end of individual harmonics heard as clicking noises in the playback are artifacts of process of clipping these harmonics out from original trumpet note recording.) Variations in brightness in these harmonic bands correspond to variations in loudness of sound from the harmonics: brighter parts of bands corresponding to louder parts of sound from the harmonics. From 12.0 seconds onward,these 8harmonicscombinetocreateonetone,thetoneofatrumpet. Thereadermayhavenoticedthattheseparateeightpuretonesdisplayedinthislastexamplesound likeaportionofanascendingscale.inthenextsection,wemakethisideaprecisebydiscussingthe connectionbetweenharmonicsofinstrumentaltonesandthenotesusedonmusicalscales. Exercises 1.1.1. Apuretonehasduration p =0.02secondsforonecycle.Whatisitsfrequency? 1.1.2. Apuretonehasduration p =0.004secondsforonecycle.Whatisitsfrequency? 1.1.3. ForthegraphofamplitudesofharmonicsshownontheleftofFigure1.7,estimatethefrequenciesofthe harmonics and find the fundamental frequency. 1.1.4. For the graph of amplitudes of harmonics shown on the right of Figure 1.7, estimate the frequencies of the harmonics and find the fundamental frequency. 1.1.5. OntheleftofFigure1.8thereisagraphofthemagnitudesoftheharmonicsfromafemalepronouncing the vowel a(long a). Estimate the frequencies of the harmonics and find the fundamental frequency. 1.1.6. OntherightofFigure1.8thereisagraphoftheamplitudesoftheharmonicsfromamalepronouncing the vowel a (long a). Estimate the frequencies of the harmonics and find the fundamental frequency. What difference do you observe compared to the previous exercise with the female speaker, and how do you explain this difference?

6 1.Pitch,Frequency,andMusicalScales 6 4.5 4 Amp. 3 Amp. 2 1.5 0 0 0 110 220 330 440 2 Frequency(Hz) 0 200 400 600 800 1.5 Frequency(Hz) Figure 1.7. Left: Graph of amplitudes of harmonics for Exercise 1.1.3. Right: Graph of amplitudes of harmonics forexercise1.1.4. 600 180 400 200 Amp. 120 60 Amp. 0 0 200 0 430 860 1290 1720 Frequency(Hz) 0 216 432 648 864 60 Frequency(Hz) Figure 1.8. Left: Graph of amplitudes of harmonics for Exercise 1.1.5, female pronouncing vowel a(long a). Right: Graph of amplitudes of harmonics for Exercise 1.1.6, male pronouncing same vowel. 1.1.7. Whydoesthetonefromthe flute(withmagnitudeofitsharmonicsgraphedinfigure1.4),soundmore purethanthetonefromthepiano(withmagnitudeofitsharmonicsgraphedinfigure1.5)? Ontheotherhand, whydoesthetonefromthepianosoundmorerich(morecomplex)thanthetonefromthe flute? 1.1.8. OntheleftofFigure1.9thereisagraphoftheamplitudesoftheharmonicsfromamalepronouncingthe vowel e(long e). Estimate the frequencies of the harmonics and find the fundamental frequency. 1.1.9. OntherightofFigure1.9thereisagraphoftheamplitudesoftheharmonicsfromatrumpetplayingone note. Estimate the frequencies of the harmonics and find the fundamental frequency. Using Table 1.2 on page 13, determine which note is being played. 1.2 Overtones, Pitch Equivalence, and Musical Scales Wehaveseenthatthetonesfrommusicalinstrumentscontainharmonicsthatareallmultiplesofa fundamentalfrequency.thisfactprovidesabasisforanequivalenceoftwotones,whenthefrequency foronetoneistwicethefrequencyoftheothertone.forexample,supposeonetonehasfundamental

1.2 Overtones,PitchEquivalence,andMusicalScales 7 120 1200 80 40 Amp. 800 400 Amp. 0 0 0 210 420 630 840 40 Frequency(Hz) 400 0 880 1760 2640 3520 Frequency(Hz) Figure 1.9. Left: Graph of amplitudes of harmonics for Exercise 1.1.8, male pronouncing vowel e(long e). Right: Graph of amplitudes of harmonics for Exercise 1.1.9, trumpet playing one note. 110Hz,andasecondtonehasfundamental220Hz.Thenthe firsttone sharmonicsare(inhz): 110, 220, 330, 440, 550, 660, 770, 880, 990, 1100, 1210,... andthesecondtone sharmonicsare 220, 440, 660, 880, 1100,... Alloftheharmonicsforthetonewithfundamental220Hzarealsoharmonicsforthetonewith 110 Hz.Clearly,thiswillhappenwhenevertwotoneshavefundamentalsofν o and2ν o.wewillsaythat thetonewithfrequency2ν o isanoctavehigherinpitchthanthetonewithfrequencyν o.theterm octavecomesfromtheuseof8notesonthescalescommonlyusedinwesternmusic.onsuchscales, theeighthtonehasdoublethefundamentalofthe firsttone. 1.2.1 Pitch Equivalence Whenonetoneisanoctavehigherinpitch,thenthetwotoneswillbenearlyindistinguishablewhen they are sounded together.theyareharmonically equivalent.thisharmonicequivalenceisusually referredtoinmusictheoryasoctaveequivalence. Harmonicequivalence(oroctaveequivalence) canbeshownbyplayingtwonotesthatareanoctaveapart, firstseparatelyandthentogether. In Figure1.10weshowaspectrogramoftheresultingsound. Thisspectrogramshowstheharmonics fromthetonestracedoutovertime,asbrighthorizontalbands. Onecanseehowthesecondtone, with pitch an octavehigher, has all of its harmonicscontainedwithin thoseof the first tone. So when the two tones are sounded together, the horizontalbandsin the spectrogramappear almost indistinguishablefromthoseofthe firsttone. Thesoundofthetwotonestogethersoundsmuchlike the first tone, almostasif that firsttone was playedby strikingits key onthe pianoin a slightly differentmanner,ratherthanwhatwasactuallydone(strikingtwokeystogether). Sincethetermoctaveequivalenceisstandardinmusictheoryweshallemployitfromnowon. Itshouldberemembered,however,thatoctaveequivalencerefersexclusivelytonotesplayedinharmony. When notesan octaveapartare playedseparately, theyare easily distinguishableby their differencesinpitch.itisonlywhentheyareplayedtogetherinharmonythattheybecomeequivalent. Ontheotherhand,evenwithnotesplayedseparatelyinmelody,thereisoftenanunderlyingharmonic schemeforwhichoctaveequivalencedoesplayaroleinanalyzingandappreciatingthemusic. For thisreason,musicianstrainthemselvestoheartheequivalencyofseparatenotesanoctaveapartin pitch.wewilldiscusstheideaofanunderlyingharmonicschemeforamelodyinchapter3.

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