The Computation of Pitch with Vectors

Aluizio Arcela

A pitch model is proposed which is supported by a vectorrepresentation of tones. First, an algorithm capable ofperforming the vector addition of the spectral componentsof two-tone harmonic complexes is introduced which initiallyconverts the amplitude, frequency, and phase (AFP)parameters into coordinates of the here introduced quotient,distance in octaves, and loudness (QOL) tone space.As QOL is isomorphic to the hue, saturation, and value(HSV) color space, a transformation from QOL to the red,green, and blue (RGB) vector space can be formulated sothat the vector addition of two pure tones is conceivedby analogy with color mixing operations. Since the QOLto RGB transformation is invertible, the resulting RGBvector sum can be transformed back to QOL. Then, byconverting QOL coordinates back to AFP parameters, atone is found whose frequency supposedly corresponds tothe pitch evoked by the original two-tone complex. Asfor complexes having more than two components, the algorithmis to be sequentially applied to pairs of vectorsin such a way that initially the first two vector tones areadded together, then the resulting vector is added to thethird vector tone, and so on.

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