Logarithmically-spaced Dirac comb

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Like the standard Dirac comb, the logarithmically-spaced Dirac comb consists of an infinite sequence of Dirac delta functions. In the case of the logarithmically-spaced comb, these are spaced in octave intervals, i.e., the delta functions are placed at positions [2^k f_0], for all integers [k].

In musical applications, this type of Dirac comb can be used to create a 12-note scale that spans the infinite space of octaves. Such a log-spaced Dirac comb, with perhaps a Gaussian weighting, when ascending or descending, gives the well known Shepard tones effect of a sound which is perceived as though it is forever going up or down in pitch.

 

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