I need a new and improved DFT The traditional Discrete Fourier Transform (DFT) and its cousin, the FFT, produce bins that are spaced equally. In other words, you get something like the first 10 hertz in the first bin, 10.1 through 20 in the second, etc. However, I need something a little different. I want the range of frequency covered by each bin to increase geometrically. Suppose I select a multiplier of 1.5. Then we have 0 through 10 in the first bin, I want 11 through 25 in the second bin, 26 through 48 in the third, etc. Is it possible to modify the DFT algorithm to behave in this fashion?
 A: To quote my dissertation:

A collection of transforms are given the name constant Q and are
  similar to the Fourier transform.
Computation of the discrete Fourier transform can be very efficient
  when employing the use of the fast Fourier transform. However we
  notice that energy of a signal is divided into uniformly sized
  frequency buckets across the spectrum. While in many cases this is
  useful, we notice situations where this uniform distribution is
  sub-optimal. An important example of such a case is observed with the
  analysis of musical frequencies. In Western music, the frequencies
  that make up the musical scales are geometrically spaced. We therefore
  see that the map between frequency bins of the discrete Fourier
  transform and the frequencies of musical scales is insufficient in the
  sense that the bins match poorly. The constant Q transform addresses
  this issue. 
The aim of the constant Q is to produce a set of logarithmically
  spaced frequency bins in which the width of the frequency bin is a
  product of the previous. As a result we may produce an identical
  number of bins per musical note across the audible spectrum, thus
  maintaining a constant level of accuracy for each musical note. The
  frequency bins become wider towards the higher frequencies and
  narrower towards the lower frequencies. This spread in the accuracy of
  frequency detection closely imitates the manner in which the
  human-auditory system responds to frequencies. 
Additionally, the close matching of notes in western scales renders
  the constant-Q particularly useful in note detection; identifying a
  musical note value rather than an explicit frequency value.
  Furthermore the constant Q simplifies the process of timbre analysis.
  The frequencies of a musical note played by an instrument are often
  comprised of harmonically related partials. The timbre of the
  instrument can be characterised by the ratios of the harmonics. With
  the constant Q transform, the harmonics are equally spaced across the
  bins regardless of the fundamental frequency. This greatly simplifies
  the process of identifying an instrument playing a note anywhere in
  the scale simply by shifting the characterisation across the bins. A
  potential downside to using the constant Q transform is that it
  demands more computation than the Fourier transform.
An efficient algorithm for transforming a discrete Fourier transform
  (which may be computed with the FFT) into a Constant Q transform is
  detailed in [Brown and Puckette (1992).][1]
[1]:
  http://www.wellesley.edu/Physics/brown/pubs/effalgV92P2698-P2701.pdf

