Continuous, frequency-domain digital equalizer
Your question is better suited for the signal processing stack exchange, and this is perhaps the answer you are looking for. However, the basic issues are:


FFTs/DFTs are filters, however (1) their frequency response is not ideal as these overlap each other on the continuous domain, (2) are not spaced in a way that best matches our auditory response leading to wasteful processing, and (3) are circular convolutions which can lead to sample-to-sample discontinuities. 
What you are looking for is an efficient orthogonal filter bank that is logarithmically spaced in the frequency domain. There exist several approaches to these. 
An appropriate wavelet transform could be more efficient, as decimation is introduced on every step which bypasses the fixed bin resolution of FFTs. 
In particular the Invertible Constant-Q Transform was designed with these problems in mind. 


Although I am not very familiar with it, I can come-up with a methodology that might serve your purpose based on the wavelet approach. 


Design a filter that extracts the top half of the frequency domain to the desired order. 
Divide the domain in two orthogonal sections, by applying the filter and subtracting the filter output from the input. 
Keep the top component for processing. This is the detail coefficient. 
Decimate the lower frequency component by throwing away half the samples. This is the approximation coefficient. 
Apply the filter again to the decimated half. 
Repeat the process, successively dividing the spectrum in bands that are logarithmically spaced. 
Apply the desired scaling. 
Successively upsample (which includes low-pass filtering) each low band and add it to its corresponding high-band to reconstruct the filtered signal. 
As the detail coefficients are band-limited as well, these can also be decimated and processed the same way to provide a higher frequency resolution. 


Note that each lower-band takes twice the number of samples to process so this is an instance of variable-rate filtering characteristic of wavelet transforms. However, the total sampling rate remains constant due to the intervening sub-sampling. 

There are some phase issues that would need to be addressed and would lead to practical limitations in the filters. Wikipedia goes at length on the Discrete Wavelet Transform.