We propose a new non-intrusive (reference-free) objective measure of speech intelligibility that is inspired from previous works on image sharpness. We define the audio Sharpness Index (aSI) as the sensitivity of the spectrogram sparsity to the convolution of the signal with a white noise, and we calculate a closed-form formula of the aSI. Experiments with various speakers, noise and reverberation conditions show a high correlation between the aSI and the well-established Speech Transmission Index (STI), which is intrusive (full-reference). Additionally, the aSI can be used as an intelligibility or clarity criterion to drive sound enhancement algorithms. Experimental results on stereo mixtures of two sounds show that blind source separation based on aSI maximization performs well for speech and for music.