Shepard's Tone: An Auditory Illusion

Create an auditory illusion.

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range = 12; duration = 10; lfoFreq = .05; basePitch = 30; numOsc = 8; midiToFreq[m_] := 2^((m - 69)/12)*440.

Generate ramps to control frequencies and amplitudes of the oscillators.

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phasors = Table[AudioGenerator[{"Sawtooth", lfoFreq, 2. Pi/numOsc (i - 1.)}, duration, SampleRate -> 500]/2. + .5, {i, numOsc}]; AudioPlot[phasors, PlotLayout -> "Overlaid", PlotRange -> All]

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Create frequencies that will control the oscillators. The frequencies are exactly one octave apart from each other and increase exponentially, so that pitch increases linearly.

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freqs = Table[ midiToFreq[(phasors[[i]])*12*numOsc + basePitch], {i, numOsc}]; AudioPlot[freqs, PlotLayout -> "Overlaid", PlotRange -> All]

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Create amplitudes that will control the oscillators. Amplitudes go to 0 when the frequencies drop to the minimum value.

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amps = Cos[#*Pi - Pi/2.] & /@ phasors; AudioPlot[amps, PlotLayout -> "Overlaid", PlotRange -> All]

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Show the relation between frequency and amplitude of one oscillator.

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AudioPlot[AudioNormalize /@ {freqs[[2]], amps[[2]]}, PlotRange -> All]

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Combine a bank of oscillators using the created frequencies and amplitudes.

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res = Mean[ Table[amps[[i]] AudioGenerator[{"Sin", freqs[[i]]}], {i, numOsc}]]

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Spectrogram[res, 8192, 4096, HannWindow, PlotRange -> {All, {0, 10000}}, ImageSize -> Medium]

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