How to get finer grained spectral decomposition for audio signals?
if you are not experiencing computational limits (like you have all of the CPU speed and memory that your heart desires), what i would suggest is just do a much longer DFT or FFT.  instead of 1024 points, do it with 16384 points or 65536 points or more.  you will have lotsa data up in the top two octaves that you don't need and can decide to ignore.

if you do not want to have all of that data above 5512 Hz, then what i would recommend is formally downsampling by a factor of 4 and running the FFT on the downsampled data.  to downsample legitimately, first you have to apply a good low-pass filter and kill all of the content above 5 kHz.  after that LPF, then simply pick out every 4th sample, discarding the 3 samples in between.  your data will be reduced by a factor of 4, your sample rate will be 11025 Hz and your Nyquist will be 5512.5 Hz and you can send that data into an FFT and look at those results.