What is (Fourier frame length/2 + 1)? If $N$ is discrete Fourier transform's frame length, and $N/2$ is half the frame length, how would you call $\frac N2 + 1$? 

frame_length      = N      ;
frame_length_half = N/2    ;
?                 = N/2 + 1;
Using your notations: number_frequency_bin, i.e. the number of "non-redundant" frequency bins, the first (corresponding to the DC) and last (corresponding to Nyquist) ones being always real for a real signal. The first is a (normalized) sum of samples, the last a sum of samples with alternated signs (since  $e^{i\pi n }=(-1)^n$).