Speech and music signals are multifractal phenomena. The time displacement profile of speech and music signal show strikingly different scaling behaviour. However, a full complexity analysis of their frequency and amplitude has not been made so far. We propose a novel complex network based approach (Visibility Graph) to study the scaling behaviour of frequency wise amplitude variation of speech and music signals over time and then extract their PSVG (Power of Scale freeness of Visibility Graph). From this analysis it emerges that the scaling behaviour of amplitude-profile of music varies a lot from frequency to frequency whereas it’s almost consistent for the speech signal. Our left auditory cortical areas are proposed to be neurocognitively specialised in speech perception and right ones in music. Hence we can conclude that human brain might have adapted to the distinctly different scaling behaviour of speech and music signals and developed different decoding mechanisms, as if following the so called Fractal Darwinism. Using this method, we can capture all non-stationary aspects of the acoustic properties of the source signal to the deepest level, which has huge neurocognitive significance. Further, we propose a novel non-invasive application to detect neurological illness (here autism spectrum disorder, ASD), using the quantitative parameters deduced from the variation of scaling behaviour for speech and music.