logfrequency

Logfrequency refers to a representation of a signal’s spectrum in which frequency bands are spaced logarithmically rather than linearly. This arrangement aligns with human auditory perception and musical pitch relationships, where equal perceptual steps correspond to multiplicative changes in frequency rather than additive ones.

In a log-frequency analysis, the center frequencies are typically arranged as f_k = f0 × r^k, with f0 the lowest frequency and r > 1 the spacing ratio. If twelve steps span an octave, r is the twelfth root of 2 (about 1.0595). This leads to bands that get narrower in absolute width at higher frequencies and wider in relative terms at lower frequencies, mirroring how ears perceive pitch differences more distinctly at lower frequencies.

The most common practical realization is the constant-Q transform (CQT), which maintains a constant quality factor Q = f_k / B_k across bands, yielding proportional bandwidths relative to their center frequencies. Log-frequency representations are also produced by mapping the linear-Fourier spectrum onto a log-spaced grid or by using perceptual scales such as the mel scale, which emphasizes perceptual rather than purely geometric spacing.

Applications include music information retrieval, pitch detection, instrument recognition, audio equalization, and speech analysis. They provide improved resolution for lower frequencies and better alignment with musical intervals and octave relationships compared with linear-frequency spectrograms. Limitations include potential computational complexity and nonuniform time resolution across the log-frequency grid if not carefully managed.

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