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In this paper, we propose a quasi-periodic neural network (QPNet) vocoder with a novel network architecture named pitch-dependent dilated convolution (PDCNN) to improve the pitch controllability of WaveNet (WN) vocoder. The effectiveness of the WN vocoder to generate high-fidelity speech samples from given acoustic features has been proved recently. However, because of the fixed dilated convolution and generic network architecture, the WN vocoder hardly generates speech with given F0 values which are outside the range observed in training data. Consequently, the WN vocoder lacks the pitch controllability which is one of the essential capabilities of conventional vocoders. To address this limitation, we propose the PDCNN component which has the time-variant adaptive dilation size related to the given F0 values and a cascade network structure of the QPNet vocoder to generate quasi-periodic signals such as speech. Both objective and subjective tests are conducted, and the experimental results demonstrate the better pitch controllability of the QPNet vocoder compared to the same and double sized WN vocoders while attaining comparable speech qualities.