Constant-Q transform

[3] The simplest way to implement a variable-Q transform is add a bandwidth offset called γ like this one:[citation needed] This formula can be modified to have extra parameters to adjust sharpness of the transition between constant-Q and constant-bandwidth like this:[citation needed] with α as a parameter for transition sharpness and where α of 2 is equals to hyperbolic sine frequency scale, in terms of frequency resolution.

[4] An approximate inverse to such an implementation was proposed in 2006; it works by going back to the discrete Fourier transform, and is only suitable for pitch instruments.

[5] A development on this method with improved invertibility involves performing CQT (via fast Fourier transform) octave-by-octave, using lowpass filtered and downsampled results for consecutively lower pitches.

As the range of human hearing covers approximately ten octaves from 20 Hz to around 20 kHz, this reduction in output data is significant.

The constant Q transform can also be used for automatic recognition of musical keys based on accumulated chroma content.

This is due to the varying number of samples used in the calculation of each frequency bin, which also affects the length of any windowing function implemented.

Constant-Q transform applied to the waveform of a C major piano chord . The x-axis is frequency , mapped to standard musical pitches , from low (left) to high (right). The y-axis is time, starting from pressing the piano chord at the bottom, and releasing the piano chord at the top, 8 seconds later. Darker pixels correspond to higher values of the Constant-Q transform. The peaks correspond closely to the precise frequencies of the vibrating piano strings. Thus the peaks can be used to detect the notes played on the piano. The lowest 3 peaks are the fundamental frequencies of the C major chord (C, E, G). Each string also vibrates at multiples of the fundamental, known as overtones , which correspond to the remaining smaller peaks to the right of the fundamental pitches. The overtones are smaller in intensity than the fundamental pitch.
Audio of the C Major piano chord used to generate the Constant-Q transform above.
Its waveform does not visually communicate pitch information like the Constant-Q transform is able to do.
Audio of the C Major piano chord used to generate the Constant-Q transform above.
Its waveform does not visually communicate pitch information like the Constant-Q transform is able to do.