(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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If this had not been done, all \ three would have had the same volume on output (see the section above on ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Setting the PlayRange", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[").", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Decay Envelopes"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Envelopes of decay curves are exponential, meaning that the amplitude \ diminishes to half its magnitude in a constant length of time.\n\nThe \ equation for exponential decay:\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[" * E^(", Evaluatable->False, AspectRatioFixed->True], StyleBox["-lambda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[")\n\t\nwhere\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["lambda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = the decay constant\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["e", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = amplitude at time ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["A", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = maximum amplitude\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["\tE = exponential constant \n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = time\n\t\nThe duration of the decay is inversely proportional to the \ magnitude of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda.\n\n", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["For a decay of 100db in 1 second, set ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to 11.5129.\nFor a decay of 50db in 1 second, set ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to 5.75646.\nFor a decay of 10db in 1 second, set ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to 1.15129.", Evaluatable->False, 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Often they do not. Higher frequencies tend to damp out faster \ and thus the timbre of a tone may change with its decay. Therefore:\n\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["e", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = Sum[ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["/", Evaluatable->False, AspectRatioFixed->True], StyleBox["i", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * E^( - ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ) * Cos[", Evaluatable->False, AspectRatioFixed->True], StyleBox["omega", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" * ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["]\n\nwhere\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["e", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = amplitude at time ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["A", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = maximum amplitude\n\tE = exponential constant\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = the decay constant\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = time\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["omega", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = fundamental frequency in radians", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Here is a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " function to create an envelope for a sawtooth wave in which the upper \ partials decay faster than the lower ones:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ComplexDecay[Amplitude_,Frequency_,Harmonics_,\n\t\tDecay_,Duration_] := \n\t\ Play[Evaluate[\n\t\tSum[Amplitude/i E^(-Decay t) * \n\t\t\tSin[i * 2 Pi \ Frequency t],\n\t\t\t\t{i,1,Harmonics}]],\n\t\t\t\t{t,0,Duration},\n\t\t\t\t\ PlayRange->All,\n\t\t\t\tSampleRate->44100]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["And here is a 440-Hz. sawtooth tone, with 22 harmonics, a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lamda", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " value of 3, and a total duration of 2 seconds. Effectually what you are \ hearing is a sawtooth tone transforming into a sine tone.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["ComplexDecay[1, 440, 22, 3, 2]"], "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 22050 44100 16 { currentfile 1 index readhexstring pop } false 1 sampledsound FFFFDFC5E7DDDDFFDA3ED7F1CF33D000C5B0C6CABD3CBCC1B560B260ADA4A81AA59D9E4A9D04 952193C38C9889FB84787FF47C6F760774216C866B4C63A361D55B5F57D453864D944BBF437D 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186, 187}, {12, 18480, 42}, {13, 3.141592653589793, 188}, {20, 42, 189}, {36, 189, 188, 4, 190}, {49, 190, 191}, {36, 184, 187, 191, 192}, {13, 0.04545454545454546, 193}, {12, -3, 43}, {20, 43, 194}, {36, 194, 4, 195}, {47, 195, 196}, {12, 19360, 44}, {13, 3.141592653589793, 197}, {20, 44, 198}, {36, 198, 197, 4, 199}, {49, 199, 200}, {36, 193, 196, 200, 201}, {33, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93, 102, 111, 120, 129, 138, 147, 156, 165, 174, 183, 192, 201, 202}, {13, -0.0006522354096556438, 203}, {33, 202, 203, 204}, {13, 0.5645916208528568, 205}, {36, 204, 205, 206}, {8, 206}}, Function[{Play`Time4}, Block[{t = 0. + 0.0000453514739229025*Play`Time4}, (Sin[880*Pi*t]/E^(3*t) + Sin[1760*Pi*t]/(2*E^(3*t)) + Sin[2640*Pi*t]/(3*E^(3*t)) + Sin[3520*Pi*t]/(4*E^(3*t)) + Sin[4400*Pi*t]/(5*E^(3*t)) + Sin[5280*Pi*t]/(6*E^(3*t)) + Sin[6160*Pi*t]/(7*E^(3*t)) + Sin[7040*Pi*t]/(8*E^(3*t)) + Sin[7920*Pi*t]/(9*E^(3*t)) + Sin[8800*Pi*t]/(10*E^(3*t)) + Sin[9680*Pi*t]/(11*E^(3*t)) + Sin[10560*Pi*t]/(12*E^(3*t)) + Sin[11440*Pi*t]/(13*E^(3*t)) + Sin[12320*Pi*t]/(14*E^(3*t)) + Sin[13200*Pi*t]/(15*E^(3*t)) + Sin[14080*Pi*t]/(16*E^(3*t)) + Sin[14960*Pi*t]/(17*E^(3*t)) + Sin[15840*Pi*t]/(18*E^(3*t)) + Sin[16720*Pi*t]/(19*E^(3*t)) + Sin[17600*Pi*t]/(20*E^(3*t)) + Sin[18480*Pi*t]/(21*E^(3*t)) + Sin[19360*Pi*t]/(22*E^(3*t)) - 0.0006522354096556438)*0.5645916208528568]]], 44100, {22050, 16}]]\ \>", "\<\ -Sound-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "ComplexDecay2[Amplitude_,Frequency_,Harmonics_,\n\t\tDecay_,Duration_] := \n\ \tPlay[Evaluate[\n\t\tSum[Amplitude/i E^(-Decay i t) * \n\t\t\tSin[i * 2 Pi \ Frequency t],\n\t\t\t\t{i,1,Harmonics}]],\n\t\t\t\t{t,0,Duration},\n\t\t\t\t\ PlayRange->All,\n\t\t\t\tSampleRate->44100]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["ComplexDecay2[1, 440, 22, 3, 2]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica 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