Can you increase frequency resolution of FFT without increasing window size? I'd like to use STFT for multipitch analysis. 
I realise detecting the partials existing in the signal is just the beginning.
Still I have problem with it.

Let's say I have signal sampled with 'CD' frequency 44100Hz.
With window of 1024 samples I get frequency bin resolution of 22500Hz/512=43Hz.
This is enough only to discern high piano notes like:
C5 = 523.251Hz and C#5 = 554.365.

I used to think 1024 is quite a large window.
But maybe it isn't and normally larger windows are used for detecting partials?

Can frequency resolution be increased with some other method than increasing window size, which worsens time resolution?
I thought of two methods:

Method1:


Divide signal into frequency bands with bandpassfilters (for example 0-11.25Hz and 11.25-22.5Hz).
Downsample higher bands so that original high frequencies will be now low frequencies
(so do it for second band 11.25-22.5Hz -&gt; 0Hz-22.5Hz) - not sure this is possible.
Concat resulting bins sets with adjusted labels.


Method2:


Use series of lowpass filters with increasing limit.
Perform FFT on increasing frequency ranges.
For each frequency use best possible resolution (bins from the first FFT in which this frequency was included).
This will cause low frequencies to have better resolution but I think this is ok because for higher notes
the frequency difference is grater.


I will be grateful any remarks on this issues.

I also read here:
How do window size, sample rate influence FFT pitch estimation? 
about method of improving peak picking results. I think will try to use it.
If you really insist on using FFT (rather than parametric methods, which wouldn't suffer from time/frequency trade-offs), you can fake a much better resolution by using the phase information to recover the instantaneous frequency for each FFT bin. Partials can then be detected by looking for plateaus in the function giving instantaneous frequency as a function of FFT bin index. The common implementation of this technique as described in this paper will "cost" you one extra STFT (instantaneous frequency is recovered by operations on the STFT of the signal, and STFT of the derivative of the signal).

See for example the ifgram function in this Matlab implementation of sinusoidal modeling of audio signals.

Note that this won't help resolving two partials falling into adjacent FFT bins. It will just provide a much more accurate frequency estimate than just converting into a frequency the FFT bin index of a spectral peak.