(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 240489, 4387] NotebookOptionsPosition[ 226263, 4037] NotebookOutlinePosition[ 233918, 4218] CellTagsIndexPosition[ 232339, 4183] WindowTitle->Lecture 17 MIT 3.016 (Fall 2009) \251 W. Craig Carter \ 2003--2009 WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Periodic Phenomena and Fourier Series", "Title"], Cell[CellGroupData[{ Cell["Playing with Familiar Periodic Phenomena: Sound", "Subtitle", CellChangeTimes->{{3.40050416309914*^9, 3.40050416759156*^9}}], Cell[TextData[{ StyleBox["Sounds will not be available on PDF or HTML versions", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nLet's begin by \"looking\" at a familiar periodic phenomena:" }], "Text"], Cell["\<\ We index the notes and write an indexed set of frequency (in Hertz) for each \ of the notes for one octave above middle-c. We write a function to create a \ Sound for each note.\ \>", "Text", CellChangeTimes->{{3.400504280374208*^9, 3.400504412968503*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"c", " ", "=", "1"}], ";", " ", RowBox[{"d", " ", "=", "2"}], ";", " ", RowBox[{"e", "=", "3"}], ";", " ", RowBox[{"f", "=", "4"}], ";", " ", RowBox[{"g", "=", "5"}], ";", " ", RowBox[{"a", "=", "6"}], ";", " ", RowBox[{"b", "=", "7"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "c", "]"}], " ", "=", "261.6"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "d", "]"}], " ", "=", "293.7"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "e", "]"}], " ", "=", "329.6"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "f", "]"}], " ", "=", " ", "349.2"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "g", "]"}], " ", "=", "392.0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "a", "]"}], " ", "=", " ", "440.0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"freq", "[", "b", "]"}], " ", "=", " ", "493.9"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"purenote", "[", "note_Integer", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"purenote", "[", "note", "]"}], "=", RowBox[{"Play", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Pi]", " ", RowBox[{"freq", "[", "note", "]"}], "t"}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.400401346423629*^9, 3.400401360508939*^9}, { 3.400401403934223*^9, 3.40040148668467*^9}, {3.4004016963784733`*^9, 3.400401769329595*^9}, {3.4004021331485567`*^9, 3.400402153147293*^9}, { 3.4004049511999483`*^9, 3.4004049590532084`*^9}, 3.4004049971599073`*^9, 3.4004050469062347`*^9, {3.4005043778134623`*^9, 3.400504381484967*^9}, { 3.402141559164178*^9, 3.4021416062580442`*^9}, {3.402141969234612*^9, 3.402142020935495*^9}, {3.402143825600788*^9, 3.40214383329145*^9}, { 3.4021445263818808`*^9, 3.402144528161063*^9}}, CellTags->{ "mmtag:17:notes_and_frequencies", "mmtag:17:middle_c", "mmtag:17:Play[]", "mmtag:17:sound_generation"}], Cell["\<\ We extend the function to get simultaneous notes from a List. We use Thread \ which takes f[{a,b,c}] to {f[a],f[b],f[c]}\ \>", "Text", CellChangeTimes->{{3.400504280374208*^9, 3.400504412968503*^9}, { 3.400504444410273*^9, 3.40050445779214*^9}, {3.400504736072318*^9, 3.400504754496146*^9}, {3.4021476948316803`*^9, 3.40214773150391*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"notes", "[", "note_List", "]"}], ":=", " ", RowBox[{"Sound", "[", RowBox[{"Thread", "[", RowBox[{"purenote", "[", "note", "]"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.400402540350503*^9, 3.400402555100891*^9}, { 3.400402683039413*^9, 3.400402732606882*^9}, {3.400403424743939*^9, 3.4004034377665854`*^9}, {3.4004034852625933`*^9, 3.400403508558529*^9}, { 3.40040487840408*^9, 3.400404884602125*^9}, {3.400404966375834*^9, 3.40040497619512*^9}, {3.400405030669916*^9, 3.400405040650558*^9}, { 3.400405911579562*^9, 3.400405917536559*^9}, {3.400504810077655*^9, 3.4005048300839853`*^9}, {3.40214161803758*^9, 3.402141668342083*^9}}, CellTags->{ "mmtag:17:notes_and_frequencies", "mmtag:17:middle_c", "mmtag:17:Play[]", "mmtag:17:sound_generation"}], Cell["Here are examples of their use.", "Text", CellChangeTimes->{{3.4005048373862057`*^9, 3.400504843328076*^9}}], Cell[BoxData[{ RowBox[{"cnote", "=", RowBox[{"purenote", "[", "c", "]"}]}], "\[IndentingNewLine]", RowBox[{"notes", "[", RowBox[{"{", RowBox[{"a", ",", "c", ",", "e"}], "}"}], "]"}]}], "Input", CellChangeTimes->{{3.400504824015538*^9, 3.400504825634541*^9}, { 3.402143862048305*^9, 3.402143863887435*^9}, 3.4021441889199247`*^9}], Cell["\<\ We can play with variable amplitudes for a fixed frequency, here we can hear \ the increased, but non-singular amplitude through zero.\ \>", "Text", CellChangeTimes->{{3.4005048373862057`*^9, 3.400504917792182*^9}, { 3.400505089946382*^9, 3.400505110344103*^9}}], Cell[BoxData[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{"540", "x"}], "]"}], "/", "x"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", ".1"}], ",", ".1"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"Filling", "\[Rule]", " ", "Axis"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Play", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{"540", "x"}], "]"}], "/", "x"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.4004199782528753`*^9, 3.4004201208789473`*^9}, { 3.400420152058116*^9, 3.400420172343429*^9}, {3.400504933245625*^9, 3.40050498310658*^9}, {3.4005050313888407`*^9, 3.400505074554523*^9}}], Cell["\<\ We can vary amplitudes and frequencies. 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here is a bagpipe", "Text", CellChangeTimes->{{3.402148153728006*^9, 3.402148182528138*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"bagpipe", "[", "note_String", "]"}], ":=", " ", RowBox[{"SoundNote", "[", RowBox[{"note", ",", ".6", ",", "\"\\""}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Sound", "[", RowBox[{"Thread", "[", RowBox[{"bagpipe", "[", "TwoFramesLower", "]"}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.400406271050746*^9, 3.400406286962864*^9}, { 3.400406334370977*^9, 3.400406337168659*^9}, {3.400406369764358*^9, 3.400406433129799*^9}, {3.400406507372889*^9, 3.400406564840822*^9}, { 3.4021462200394278`*^9, 3.402146250413541*^9}, {3.4021462813996696`*^9, 3.402146322253899*^9}, {3.402146367114049*^9, 3.4021463903051243`*^9}, { 3.402148002306745*^9, 3.40214800323429*^9}}], Cell["And, now for the birds.", "Text", CellChangeTimes->{{3.402148187890646*^9, 3.402148193719142*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"avian", "[", "note_String", "]"}], ":=", " ", RowBox[{"SoundNote", "[", RowBox[{"note", ",", ".2", ",", "\"\\""}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"avian", "[", "rest", "]"}], " ", "=", " ", RowBox[{"SoundNote", "[", RowBox[{"None", ",", ".4"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Sound", "[", RowBox[{"Thread", "[", RowBox[{"avian", "[", RowBox[{"Riffle", "[", RowBox[{"TwoFramesLower", ",", "rest"}], "]"}], "]"}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.400406576172925*^9, 3.4004066790887547`*^9}, { 3.402146417695945*^9, 3.40214648512362*^9}, {3.4021465290314913`*^9, 3.40214665579001*^9}, {3.4021480091073847`*^9, 3.402148009866273*^9}}], Cell["Now, a little Bach meets Beethoven", "Text", CellChangeTimes->{{3.4021588414846773`*^9, 3.402158855374612*^9}}], Cell[BoxData[ RowBox[{"Sound", "[", RowBox[{"Riffle", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{ RowBox[{ RowBox[{"SoundNote", "[", RowBox[{"#", ",", ".2"}], "]"}], "&"}], "[", "TwoFramesLower", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"Thread", "[", RowBox[{ RowBox[{ RowBox[{"SoundNote", "[", RowBox[{"#", ",", ".1"}], "]"}], "&"}], "[", "TwoFramesLower", "]"}], "]"}], "}"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.402158637945074*^9, 3.402158640822895*^9}, 3.402158705124662*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Roll over Beethoven: Random Notes and Instruments", "Subtitle", CellChangeTimes->{{3.4005054880744953`*^9, 3.4005054937682457`*^9}, { 3.402140246809938*^9, 3.402140255293695*^9}, {3.402140974646674*^9, 3.402140982188442*^9}}], Cell["\<\ Let's hear what random notes sound like: SoundNote[n] will play n semitones \ above middle C. 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Because it \ asks FourierTrigSeries to calculate one more term each time, it is doing some \ redundant work. We can fix this up by having it calculate one new term and adding to the sum calculated previously. Here \ it is:\ \>", "Text", CellTags-> "mmtag:17:Fourier_series__example_function_for_calculating_and_efficiency"], Cell[BoxData[{ RowBox[{ RowBox[{"costerm", "[", RowBox[{"function_", ",", "x_", ",", " ", "n_Integer"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"FourierCosCoefficient", "[", RowBox[{"function", ",", "x", ",", "n"}], "]"}], "]"}], " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "\[Pi]", " ", "n", " ", "x"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"sinterm", "[", RowBox[{"function_", ",", "x_", ",", " ", "n_Integer"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"FourierSinCoefficient", "[", RowBox[{"function", ",", "x", ",", "n"}], "]"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Pi]", " ", "n", " ", "x"}], "]"}]}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"TruncatedFourierSeries", "[", RowBox[{"function_", ",", "x_", ",", " ", "0"}], "]"}], " ", ":=", " ", RowBox[{ 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