Filter Design in Matlab
You just need to make a simple modification:

Send the original signal through each filter and sum the output of each filter together to get your final result.

At the moment you are passing the filtered data on to each subsequent filter. If you think carefully about that, the output will be silent i.e. you filter out everything but 523 Hz  +/- 25 which leaves you just frequencies around 523 Hz, then you take that and filter out everything but 1046 Hz  +/- 25 which will strip out the frequencies around 523 Hz and then you have silence after just two iterations of your loop.



s = s(:).';
sNew = zeros(1,length(s)); % The eventual output

for N=2:1:(Fs/(2*(523+25)))
    % High-pass filter
    FsNorm =  (523-25).*N / (Fs/2);
    [b,a] = butter(10, FsNorm, 'high');
    sHigh = filter(b, a, s);

    % Low pass
    FsNorm = (523+25).*N / (Fs/2);
    [b,a] = butter(10, FsNorm, 'low'); 
    sLow = filter(b, a, sHigh); % Note we are filtering sHigh, not 's'

    % Now sum the results
    sNew = sNew + sLow;
end