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Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * Sin[2 Pi ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Fm", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["]]\n\nwhere\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["e", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = amplitude at time ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["A", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = peak amplitude\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Fc", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = carrier frequency in hertz\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Mi", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = modulation index \t\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Fm", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = modulating frequency in hertz\n\t\nHere is the ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" function for the above equation:", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "Text", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "FM[Amplitude_,Carrier_,PeakDeviation_,Modulator_,Duration_] :=\n\t\ Play[Evaluate[\n\t\tAmplitude * Sin[2 Pi Carrier t +\n\t\t\ PeakDeviation/Modulator * \n\t\tSin[2 Pi Modulator t]]],\n\t\t\t\t\ {t,0,Duration}]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["The Modulation Index"], "Subsubsection", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The modulation index is the ratio of the peak deviation to the modulating \ frequency:\n\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["PeakDeviation", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" / ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Fm", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\n\t\n(The ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["PeakDeviation", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" is frequently called ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["DeltaF", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " in the standard literature.)\n\nHigher values of the index result in more \ partials, a \"brighter\" sound. \n\nVery high values of the index result in \ \"foldover\", or \"aliasing\", adding low-frequency components to the sound.", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "Text", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "The Modulation Ratio (Harmonic and Inharmonic Partials)"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Modulation", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Ratio", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " is the ratio of the modulating frequency to the carrier frequency.\n\nIf \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "is an integer greater than zero, the resulting overtones and sidebands \ (the spectrum) will have ha", Evaluatable->False, AspectRatioFixed->True], StyleBox["rmonic", Evaluatable->False, AspectRatioFixed->True], StyleBox[" r", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "elationships to each other, analogous to those of standard orchestral \ instruments.\n\nIf ", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "is not an integer value (if it has a fractional part greater than zero) \ the resulting overtones and sidebands will have an inharmonic", Evaluatable->False, AspectRatioFixed->True], StyleBox[" r", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "elationship to each other, analogous to percussion instruments and bells.\n\ \nOne speaks of a harmonic relationship when the overtones and sidebands have \ frequencies that coincide with the harmonics of some fundamental, not \ necessarily the carrier.\n\nWhen \t", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR =", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " 1\tthe spectrum contains all harmonics, and the fundamental is at the \t\t\ \tmodulating frequency\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR =", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " 2\tthe spectrum contains only odd-numbered harmonics, analogous to \t\t\t\ a clarinet\n \t ", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR =", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" 3\tevery 3rd harmonic is missing from the series\n \t ", Evaluatable->False, AspectRatioFixed->True], StyleBox["MR = n\tevery n harmonic is missing from the series", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Simple Frequency Modulation (Vibrato)"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "When the modulating frequency is in the sub-audio range (below 20 Hz.), \ the resulting frequency modualtion is called \"vibrato\" by musicians.\n\nIn \ simple frequency modulation, the magnitude of the ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Peak Deviation", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " controls the \"depth\" (or \"intensity\") of the vibrato, and the \ magnitude of the ", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Modulator", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " controls the rate of the vibrato.\n\nTo achieve a \"standard\" vibrato, \ the modulating frequency should be in the range of 5 to 10 Hz.\n\nHere is an \ example of vibrato, using the following variables:\n\n\t", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Amplitude", CellMargins->{{Inherited, 111}, {Inherited, 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AspectRatioFixed->True], StyleBox["Duration", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 1", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "Text", CellMargins->{{Inherited, 111}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["FM[1, 440, 36, 6, 1]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.588604 [ [ 0 0 0 0 ] [ 1 0.618034 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath % Start of sampled sound 1 string 8192 8192 8 { currentfile 1 index 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StyleBox["PeakDeviation", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 4400 * Sin[", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["]\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Modulator", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 440\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["ModulationIndex", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PeakDeviation", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" / ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Modulator", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 10 * Sin[", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, 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Cell[OutputFormData[ "\<\ Sound[SampledSoundFunction[CompiledFunction[{_Integer}, {0, 4, 21, 0}, {{1, 17}, {3, 1, 0}, {13, 0., 0}, {13, 0.0000453514739229025, 1}, {20, 0, 2}, {36, 1, 2, 3}, {33, 0, 3, 4}, {12, 880, 1}, {13, 3.141592653589793, 5}, {20, 1, 6}, {36, 6, 5, 4, 7}, {12, 10, 2}, {49, 4, 8}, {12, 880, 3}, {13, 3.141592653589793, 9}, {20, 3, 10}, {36, 10, 9, 4, 11}, {49, 11, 12}, {20, 2, 13}, {36, 13, 8, 12, 14}, {33, 7, 14, 15}, {49, 15, 16}, {13, -(9.10043735369293*10^-8), 17}, {33, 16, 17, 18}, {13, 1.000000106374658, 19}, {36, 18, 19, 20}, {8, 20}}, Function[{Play`Time54}, Block[{t = 0. + 0.0000453514739229025*Play`Time54}, (Sin[880*Pi*t + 10*Sin[t]*Sin[880*Pi*t]] - 9.10043735369293*10^-8)*1.000000106374658]]], 207816, {22050, 16}]]\ \>", "\<\ -Sound-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Serial Frequency Modulation (\"Cascade\" FM)"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The equation for frequency modulation can be set up so that the modulating \ frequency is itself modulated. This equation is sometimes called Cascade \ frequency modulation.\n\nThis requires that there be a Modulation Index for \ each modulating frequency.\n\nHere is an equation for Serial FM:\n\n\te = A * \ Sin[OmegaC * t + MI1 * Sin[OmegaM1 * t + MI2 * Sin[OmegaM2 * t]]]\n\nwhere\n\t\ e = amplitude at time t\n\tA = maximum amplitude\n\tOmegaC = carrier \ frequency in radians\n\tMI1 = modulation index for modulating frequency 1\n\t\ OmegaM1 = modulating frequency 1 in radians\n\tMI2 = modulation index for \ modulating frequncy 2 \n\tOmegaM2 = modulating frequncy 2 in radians\n\t\n\ Here is an example where\n\n\t OmegaC = 440 Hz.\n\t OmegaM1 =100 Hz.\n\t \ OmegaM2 = 50 Hz.\n\t MI1 = 10 \n\t MI2 = 5\n\t Duration = 2"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "Play[Evaluate[\nSin[2 Pi 440 t +\n\t10 * Sin[2 Pi 100 t +\n\t\t5 * Sin[2 Pi \ 50 t]]]],{t,0,2}]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Sound-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[TextData[ "This process can be extended to any number of modulating waves."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Parallel Frequency Modulation "], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "It is possible to have two or more modulating frequncies modulating a \ carrier, but (in contradistinction to \"Cascade\" FM) the modulating \ frequncies do no affect each other. \n\nAnother possibility is to have one \ modulating frequency modulating two carriers, that are otherwise not \ affecting each other. Both possibilities are described below.\n\nThe \ examples utilize three frequencies. The process can be extended to greater \ numbers of frequencies."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Double Modulator FM"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Here is an equation for Double Modulator FM using three frequencies:\n\n\t\ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["e", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * Sin[", Evaluatable->False, AspectRatioFixed->True], StyleBox["OmegaC", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" + ", Evaluatable->False, AspectRatioFixed->True], StyleBox["MI1", Evaluatable->False, 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