IntervalName 0 PerfectUnisonPU1 MinorSecondorhalfstepm22 MajorSecondorwholestepM23 MinorThirdm34 MajorThirdM35 PerfectFourthP46 TritoneTT7 PerfectFifthP58 MinorSixthm69 MajorSixth ID: 255139
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2ADAMKOSSDenition1.4.Anintervalisthedistancebetweentwomusicalnotes.Table1belowgivesacompletelistofmusicalintervalsandtheircommonnames.Denition1.5.Atriadisasetofthreenotessuchthattheintervalsbetweenconsecutivenotesarethirds,eithermajororminor.Table1.ListofIntervalsNumberofHalfSteps IntervalName 0 PerfectUnison(PU)1 MinorSecond,orhalf-step(m2)2 MajorSecond,orwholestep(M2)3 MinorThird(m3)4 MajorThird(M3)5 PerfectFourth(P4)6 Tritone(TT)7 PerfectFifth(P5)8 MinorSixth(m6)9 MajorSixth(M6)10 MinorSeventh(m7)11 MajorSeventh(M7)12 PerfectOctave(PO)2.BasicGroupTheoryInorderforonetounderstandanymusicintermsofmathonemustrstunder-standtheconceptofagroup.Denition2.1.AgroupisasetGwithanoperation:GG!Gsatisfying(1)(associativity)Forallx;y;z2G,(xy)z=x(yz).(2)(identity)Thereexistse2Gsuchthatforallx2G,ex=xe=x(3)(inverses)Forallx2Gthereexistsay2Gsuchthatxy=yx=ewhereeistheidentityelement.Nowwewilldenetwogroupsusedtounderstandmusic,byassociatinggroupelementswithmusicalnotes.Therstgroup,Z12,givesanumericalrepresentationofthechromaticscale.Denition2.2.Z12isthesetf0;1;2;3;4;5;6;7;8;9;10;11gwiththeoperationdenedbyxy=x+y(mod12).Remark2.3.Foranyx2Z,x(mod12)istheelementoff0;1;2;:::;11gsuchthatx(mod12)+12k=xforsomek2Z.WeuseZ12torepresentthechromaticscalebyassociatingeachelementtoanote,pairingthemsothatC=0andC]=1andfollowingthispatternuntilB=11.Proposition2.4.(Z12;)isagroup. 4ADAMKOSSExample3.4.WecanalsodenethediatonicGIS(Z7;Z7;int)withint(x;y)=y�xmod7.Thefunctionintsatisesthetwopropertiesbecause.int(x;y)+int(y;z)=(y�x)+(z�y)=y�x+z�y=z�x=int(x;z):(alloperationsabovebeingmod7).Alsoforanyx2Z7andg2Z7thereexistsoneuniquey2Z7suchthaty�x=g.Thusthereexistsonlyoney2Z7suchthatint(x;y)=gbythedenitionofint.Thus(Z7;Z7;int)isacommutativegeneralizedintervalsystem.Remark3.5.ThefunctionintdenedintheGISforZ12isthechromaticinterval,thatistosay,int(x;y)givesthenumberofhalfstepsbetweenthenotesxandy,whenweidentifynoteswithelementsofZ12.However,thefunctionintfortheGISofZ7doesnotrepresentthechromaticintervals,butthenumberofscalestepsbetweentwonotes.4.TheGeneralizedTonnetzInthissectionwewillintroducethegeneralizedTonnetz.Denition4.1.LetG0=(S;G;int)beacommutativeGISandassumethatthegroupGisgeneratedbyanitesubsetB.AgeneralizedTonnetzT(G;B)isthedirectedgraph(S;A;B;int)whereAisthecompleteinverseimageofBunderintandintistherestrictionofinttoA.ThegroupGisthesetofverticesofthisgraph,andtwoverticesxandyareconnectedbyanedgedirectedfromxtoywheneverint(x;y)isanelementofB.[2]ThecompleteinverseimageofBunderint:SS!Gisthesetofallorderpairsofvertices(x;y)suchthattheirimageint(x;y)isinB.NextaretwoexamplesofgeneralizedTonnetze.Example4.2.TherstexampleofaGISisbasedonthechromaticscale.InthisexampletheGISis(Z12;Z12;int).Denethegeneratorsasthesetf3;4;7g.ThesetZ12isgeneratedbythesetf3;4;7gbecauseanynumbercanbewrittenasasumofmultiplesof3;4and7mod12.ThegraphhasthesetofverticeswhichisthesetZ12andthesetoftheedgeslistedinTable2.ThegraphofthegeneralizedTonnetzofthechromaticscaleisembeddedonthetorusorthedonutandnottheplane.ForavisualrepresentationseeFigure1.ThemainideaofthispaperisthegraphorgeneralizedTonnetzofthediatonicscale.Example4.3.StartwiththeGIS(Z7;Z7;int).DenethegeneratorsofthetonnetzfromthisGISasthesetf2;4g.ThesetZ7isgeneratedbythesetf2;4gbecauseifwestartwiththe0elementandaddthegenerator2repeatedlywewillobtainallthedi erentelementsoftheset.ThesetofverticesisZ7andthesetofedgesisasfollows.SeeFigure2forabetterimageofthegraph. 6ADAMKOSSTable3.ThesetofedgesforT(Z7;f2;4g)(0,2)(0,4)(1,3)(1,5)(2,4)(2,6)(3,5)(3,0)(4,6)(4,1)(5,0)(5,2)(6,1)(6,3)TonnetzoftheDiatonicScale.pdf Figure2.Thelatticestructuregraphofthediatonicscale.5.ComparisonoftheChromaticScaleandDiatonicScaleThereasonthatthegraphofthechromaticscalegeneralizedTonnetzembedsonatorusisbasedonthefactthateachvertexhasdegree6,whichmeansthatthegraphis6�regular.OntheotherhandthegraphofthediatonicscalegeneralizedTonnetzembedsonaMobiusstripbecauseeachvertexisofdegree4,orthegraphis4�regular.Thatiswhythegraphbasedonthediatonicscaledoesnotembedonthesamespaceasthechromaticscale.Thedi erencesinthesegraphsindicatethedi erencesinmusicalstructureofthetwoscales.Acknowledgments.Itisapleasuretothankmymentor,KatieMannhelpingthroughthispaperandgettingabettersenseofhowtowriteaMathematicalPaper.IwouldalsoliketothankProfessorTomFioreforhelpingmeunderstandtheideasbehindthepaperandforprovidingthisinterestingtopictoexplore.IwouldalsoliketothankProfessorPeterMayfororganizingtheREUandtheNationalScienceFoundationforfundingmyparticipationintheREU. ACOMPARISONOFTHEGRAPHSOFTHECHROMATICANDDIATONICSCALE7Strip.pdf Figure3.AGreatPictureoftheMobiusStrip!References[1]Lewin,D.GeneralizedMusicalIntervalsandTransformations.OxfordUniversityPress.2007.(OriginallyYaleUniversityPress,1987)[2]Zabka,Marek.GeneralizedTonnetzandWell-FormedGTS:AScaleTheoryInspiredbytheNeo-Riemannians.Springer-VerlagBerlinHeidelberg.2009.MathematicsandComputationinMusicSecondInternationalConference,MCM2009,NewHaven,CT,USA,June19-22,2009.ProceedingsSeries:CommunicationsinComputerandInformationScience,Vol.38Chew,Elaine;Childs,Adrian;Chuan,Ching-Hua(Eds.)2009,XVI,299p.,Softcover.[3]Mazzola,G.TheToposofMusic:GeometricLogicofConcepts,Theory,andPerformance.Birkhauser,Basel.2002