constantQ

The constant-Q transform (CQT) is a time–frequency representation designed for musical signals in which the ratio of center frequency to bandwidth (the Q factor) remains approximately constant across analysis bins. This results in a set of logarithmically spaced frequency bins, so that equal steps in bin index correspond to equal musical intervals, such as semitones.

In a CQT, each bin k has center frequency f_k and bandwidth Δf_k chosen so that Q = f_k / Δf_k is constant. As a consequence, lower-frequency bins use longer analysis windows with narrower relative bandwidths, while higher-frequency bins use shorter windows with broader relative bandwidths. The transform can be computed either as a constant-Q filter bank or by convolving with a bank of kernels derived from the Fourier transform.

Applications include musical pitch estimation, chord and key detection, timbre analysis, audio similarity, and music transcription. The CQT is often preferred to the short-time Fourier transform when analysis aligned to musical pitch is desired, because it provides higher resolution at low frequencies and scales with musical notes.

Compared with STFT, CQT offers perceptually meaningful frequency resolution at the cost of greater computational complexity. Real-time implementations exist using FFT-based kernels, downsampling, or pruning techniques to balance accuracy and speed.

Variants and related approaches include log-frequency CQT implementations, octave-band filter banks, and wavelet-like transforms that approximate constant-Q behavior. Common considerations include handling of edge effects at the low end, normalization across bins, and selection of the Q value and range to cover a desired musical spectrum.