Calculation of a constant <i>Q</i> spectral transform

The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24-oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventional DFT that yields a constant spacing between frequency components. In addition to advantages for resolution, representation with a constant pattern has the advantage that note identification (‘‘note identification’’ rather than the term ‘‘pitch tracking,’’ which is widely used in the signal processing community, is being used since the editor has correctly pointed out that ‘‘pitch’’ should be reserved for a perceptual context), instrument recognition, and signal separation can be done elegantly by a straightforward pattern recognition algorithm.