Kullback-Leibler Distance of Spectral Data
The expression in your question seems to have been written for univariate Gaussians. For multivariate Gaussians, 
$KL(A,B) = \frac{1}{2}\left[\log\frac{|\Sigma_B|}{|\Sigma_A|} - d + Tr(\Sigma_B^{-1}\Sigma_A) + (\mu_B - \mu_A)^T \Sigma_B^{-1}(\mu_B - \mu_A)\right]$
where $d$ is the dimensionality of the feature vectors. 

(https://stats.stackexchange.com/questions/60680/kl-divergence-between-two-multivariate-gaussians)

So 

\begin{eqnarray*}
KL2(A,B) &amp;:=&amp; KL(A,B)+KL(B,A)\\ 
&amp;=&amp; \frac{1}{2}\left[ - 2d + Tr(\Sigma_B^{-1}\Sigma_A) + Tr(\Sigma_A^{-1}\Sigma_B) + (\mu_B - \mu_A)^T (\Sigma_A^{-1}+\Sigma_B^{-1})(\mu_B - \mu_A)\right].
\end{eqnarray*}

I don't think you compute this for the spectrograms themselves but for feature vectors (some of the features may be derived from the spectrogram).