In this paper, we study an underexplored, yet important and challenging problem: counting the number of distinct sounds in raw audio characterized by a high degree of polyphonicity. We do so by systematically proposing a novel end-to-end trainable neural network (which we call DyDecNet, consisting of a dyadic decomposition front-end and backbone network), and quantifying the difficulty level of counting depending on sound polyphonicity. The dyadic decomposition front-end progressively decomposes the raw waveform dyadically along the frequency axis to obtain time-frequency representation in multi-stage, coarse-to-fine manner. Each intermediate waveform convolved by a parent filter is further processed by a pair of child filters that evenly split the parent filter's carried frequency response, with the higher-half child filter encoding the detail and lower-half child filter encoding the approximation. We further introduce an energy gain normalization to normalize sound loudness variance and spectrum overlap, and apply it to each intermediate parent waveform before feeding it to the two child filters. To better quantify sound counting difficulty level, we further design three polyphony-aware metrics: polyphony ratio, max polyphony and mean polyphony. We test DyDecNet on various datasets to show its superiority, and we further show dyadic decomposition network can be used as a general front-end to tackle other acoustic tasks.