Unnaturally sounding speech prevents the listeners from recognizing the message of the signal. In this paper we demonstrate how a precise initial phase approximation can improve the naturalness of artificially generated speech. Using the Harmonic plus Noise Model provided by Stylianou as a framework for a Hungarian speech synthesis, the exact initial phase extension of the system can be easily performed. The proposed method turns out to be more effective in preserving the sound characteristics and quality than the original one.