Abstract
Abstract A pitch model is proposed which is supported by a vector representation of tones. First, an algorithm capable of performing the vector addition of the spectral components of two-tone harmonic complexes is introduced which initially converts 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 so that the vector addition of two pure tones is conceived by analogy with color mixing operations. Since the QOL to RGB transformation is invertible, the resulting RGB vector sum can be transformed back to QOL. Then, by converting QOL coordinates back to AFP parameters, a tone is found whose frequency supposedly corresponds to the pitch evoked by the original two-tone complex. As for complexes having more than two components, the algorithm is to be sequentially applied to pairs of vectors in such a way that initially the first two vector tones are added together, then the resulting vector is added to the third vector tone, and so on.
Highlights
Seebeck [21] proposed a relationship between pitch and periodicity after having observed that the waveform’s repetition period could be perceived as pitch even if there is no spectral component at the corresponding frequency—a consideration which gave rise to the concept of “missing fundamental”
The results found with vector addition tones show that the pitch of harmonic complexes corresponds to the frequency of the fundamental only in some special cases
The pitch of a 2: 3-complex in equilibrium has a bipolar response to the phase relationship, since it can assume just one of two values, i.e., according to Equation (61), fminqx, when the transposed resulting vector is on the red sail, and 1.5fminqx, when it is on the cyan sail
Summary
Seebeck [21] proposed a relationship between pitch and periodicity after having observed that the waveform’s repetition period could be perceived as pitch even if there is no spectral component at the corresponding frequency—a consideration which gave rise to the concept of “missing fundamental”. They are presented in a sequence which intends to illustrate the main points of the mentioned conditions, being the first three complexes taken from the literature so as to compare the results found in this study with those of some important pitch models (e.g., [5, 6, 8, 17, 18, 29, 32]) either from the “temporal” view, which is based on the “autocorrelation” hypothesis raised by Licklider [13], or from the “pattern matching” models which was first described by de Boer [6]. The results found with vector addition tones show that the pitch of harmonic complexes corresponds to the frequency of the fundamental only in some special cases
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