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The amplitude of each overtones \ is the inverse of its number: the amplitude of the second overtone is \ one-half that of the fundamental, the amplitude of the third overtone is \ one-third that of the fundamental, and so on.\n\nThe approximate equation \ for a sawtooth wave is:\n\n\te = A * sin(omega * t) + A/2 * sin(2 * omega * \ t) + A/3 * sin(3 * omega * t) +...+\n\t\tA/n * sin(n* omega * t)\n\t\nor in \ Mathematica:\n\n\te := Sum[A/i * Sin[ i * omega * t ], {i,1,n}]\n\nwhere \n\t\ e = amplitude at time t\n\tomega = frequency in radians per second ( 2 * Pi * \ frequency in Hz)\n\tt = time in seconds\n\tA = maximum amplitude\n\ti = 1, 2, \ 3,..., n\n\tn = highest harmonic present\n\t\nPractically, n * f should be \ less than 20,000 Hz, the upper limit of human hearing.\n\nThe sawtooth is a \ basic waveform used for sound synthesis because it contains all the harmonics \ of a given fundamental, and many musical sounds derive from sawtooth sound, \ such as string tone."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Plotting Sawtooth Waves"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Below is a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " function to plot sawtooth waves. It takes three arguments:\n \n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Amplitude", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tmaximum amplitude of the wave\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Frequency", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[":\tfrequency of the fundamental in hertz\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Harmonics", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[":\tnumber of the highest overtone ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SawtoothWavePlot[Amplitude_, Frequency_, Harmonics_] := \n\tPlot[Evaluate[\n\ \t\tSum[Amplitude/i * Sin[i * 2 Pi Frequency t],\n\t\t{i,1,Harmonics}]],\n\t\t\ {t,0,3/Frequency}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Here is a plot of a sawtooth wave at 440 Hz with five 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AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " function to play a sawtooth wave, using the above formula. The variables \ are the same with the addition of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".\n\t\n\t\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": duration of sound\n\nThe ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SampleRa", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "te is set to 44100, so that the highest harmonics can be heard.\t", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SawtoothWavePlay[Amplitude_,Frequency_,Harmonics_,Duration_] := \n\t\ Play[Evaluate[\n\t\tSum[Amplitude/i * Sin[i * 2 Pi Frequency t],\n\t\t\ {i,1,Harmonics}]],\n\t\t{t,0,Duration},\n\t\tSampleRate->44100]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ 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The amplitude of each overtone is \ proportional to the inverse of its number. Thus, a square wave has half as \ many overtones present as a sawtooth wave.\n\nThe equation for a square wave \ is:\n\n\te = A * sin(omega * t) + A/3 sin(3 *omega * t) + A/5 sin(5 * omega * \ t) + ... \t\n\nAn approximation with Mathematica:\n\n\tSum[A/(2 i + 1) * \ Sin[ (2 i + 1) * omega * t],{i,0,n}]\n\nwhere \t\n\te = amplitude at time t\n\ \tA = maximum amplitude of the fundamental\n\tomega = frequency of the \ fundamental in radians\n\tt = time\n\tn = number of harmonics present\n\t\n\ Many instruments (for example., the clarinet) have only odd harmonics \ present."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Plotting Square Waves"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Below is a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " function to plot square waves. It takes three arguments:\n\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Amplitude", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tthe amplitude of the fundamental\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Frequency", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tthe frequency of the fundamental\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Harmonics", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tthe number of harmonics present", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SquareWavePlot[Amplitude_, Frequency_, Harmonics_] := \n\tPlot[Evaluate[Sum[ \ \n\t\t(Amplitude/(2 i + 1)) * \n\t\tSin[(2 * i + 1) * 2 Pi Frequency t],\n\t\t\ \t{i,0,Harmonics}]],\n\t\t\t{t,0,3/Frequency}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Here is a plot of a square wave with 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It adds ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to the three arguments above:\n\n\t\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": duration of the total sound\n\t\nThe ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SampleRate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is set to 44100, so that the highest harmonics can be heard.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SquareWavePlay[Amplitude_, Frequency_, Harmonics_,Duration_] := \n\t\ Play[Evaluate[Sum[ \n\t\t(Amplitude/(2 i + 1)) * \n\t\tSin[(2 * i + 1) * 2 Pi \ Frequency t],\n\t\t\t{i,0,Harmonics}]],\n\t\t\t{t,0,Duration},\n\t\t\t\ SampleRate->44100]"], "Input", 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The amplitude of each overtone \ is proportional to the inverse of its number squared. Notice that this \ results in the upper harmonics being much weaker than in a square wave, and, \ correspondingly, makes the sound of a triangle wave close to that of a sine \ wave.\n\nThe equation for a square wave is:\n\n\te = A * cos(omega * t) + A/9 \ *cos(3 * omega * t) + A/25 *cos(5 * omega * t) + ...\n\t\nor an approximation \ with Mathematica:\n\n\te = Sum[ A/(2 i + 1)^2 * Cos[ (2 * i + 1) * omega * \ t],{i,0,n}]\n\nwhere \n\te = amplitude at time t\n\tA = amplitude of \ fundemental\n\tomega = freqeuncy of fundamental in radians\n\tn = number of \ harmonics\n\tt = time"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Plotting Triangle Waves"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Below is a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " function to plot triangle waves. It take three arguments:\n\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Amplitude", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tamplitude of fundamental\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Frequency", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tfrequency of fundamental\n\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Harmonics", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": \tnumber of harmonics", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "TriangleWavePlot[Amplitude_, Frequency_, Harmonics_] := \n\t\ Plot[Evaluate[Sum[ \n\t\tAmplitude/(2 i + 1)^2 * \n\t\tCos[(2 i + 1) * 2 Pi \ Frequency t ],\n\t\t\t{i, 0, Harmonics}]],\n\t\t\t{t,0,3/Frequency}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Here is a plot of a triangle wave with a frequency of 440 Hz 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Triangle Waves"], "Subsubsection", 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 play a triange wave. It adds ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" to the three arguments above:\n\n\t\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["Duration", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[": duration of the total sound\n\t\nThe ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SampleRate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is set to 44100, so that the highest harmonics can be heard.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "TriangleWavePlay[Amplitude_,Frequency_,Harmonics_,Duration_] := \n\t\ Play[Evaluate[Sum[ \n\t\tAmplitude / (2 i + 1)^2 * \n\t\tCos[(2 i + 1) * 2 Pi \ Frequency t],\n\t\t\t{i, 0, Harmonics}]],\n\t\t\t{t,0,Duration},\n\t\t\t\ SampleRate->44100]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Here is the sound of a triangle wave with a fundamental of 440 Hz and 22 \ harmonics:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["TriangleWavePlay[1, 440, 22, 0.5]"], "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 44100 22050 16 { currentfile 1 index readhexstring pop } false 1 sampledsound FC48F6C8F1DCEC82E786E239DD30D7EFD2DACDA5C884C35ABE30B90EB3DDAEC1A98BA4719F3C 9A2194EE8FCE8AA1857B80547B28760770D66BB96685616A5C35571951E74CC7479A42733D4F 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