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1、Fractals,Chaos,andMusicAbstract:Manyrelationshavebeendiscoveredinrecentyearsbetweenfractalsandmusic.Thispaperwillcoversomeoftheresearchthathasbeendoneontheserelations,includingsomeofthecontroversiesoverconflictingdiscoveries.Itwillshowhowsomeartistsarecurren
2、tlyusingfractalstogeneratethebasicmelodiesintheircompositions.Afewcomputerprogramswhichcanbeusedtogeneratethesemelodieswillbediscussed.Thispaperwillalsoexaminepossiblepracticalusesforsoftwarewhichcanbeusedtohearpatternsinfractals.Thealgorithmsbehindthemusic-
3、generatingalgorithmswillbediscussed,aswellassomeofthebasicrelationsbetweenmathematicsandmusic.SomeoftheworkofmusictheoristJosephSchillinger,whoseideasinthe1920'sand1930'saboutgeneratingmusicbyrecursiveandchaoticmeanswerefaraheadoftheirtime,willalsobecovered.
4、Keywords:Fractals,Chaos,noiseHistoricalperspectiveThemathematicalstudyofmusiciscertainlynothingnew,datingbacktoancientGreece.Aroundthe5thcenturyBC,thePythagoreansformulatedascientificapproachtomusic,expressingmusicalintervalsasnumericproportions.Thiswasproba
5、blydonebyobservingthetonesproducedbypluckedstringsofdifferentlengths;forexample,thetoneproducedbyastringheldatthemiddleisanoctavehigherthanthatofthewholestring.Theywentontocalculatetheintervalsforseveraldifferentscales,includingthechromaticanddiatonicscales.
6、ArchytasofTarentum,aPythagoreanmathematicianwholivedaround400-350BC,wasevenabletoworkouttherelationshipsbetweennotesintheenharmonicscale,whichincludesquartertones.(Britannica).Themostrigorousmathematicalstudyofmusicinmorerecentyearswouldbethesystemformulated
7、byJosephSchillingerinthe1920'sand1930's.Schillinger,aRussian-Americanmusictheorist,wasobsessedwithdevelopinga"scientification"ofmusicthroughmathematics.In1941hepublishedTheSchillingerSystemofMusicalComposition,amassivetwelve-bookworkthatwasyearsaheadofitstim
8、e.Thissystemhasbeendescribedas"asortofcomputermusicbeforethecomputer,"sincehisworkpresagedmanydevelopmentsofalgorithmiccompositionwhichwouldnotbeexpandeduponuntildecadeslater(Degazio).Book1ofSch
