This paper presents a novel training algorithm for Hidden Markov Model (HMM)-based speech synthesis. One of the biggest issues causing significant quality degradation in synthetic speech is the over-smoothing effect often observed in generated speech parameter trajectories. Recently, we have found that a Modulation Spectrum (MS) of the generated speech parameters is sensitively correlated with the over-smoothing effect, and have proposed the parameter generation algorithm considering the MS. The over-smoothing effect is effectively alleviated by the proposed parameter generation algorithm. On the other hand, it loses the computationally-efficient generation processing of the conventional generation algorithm. In this paper, the MS is integrated into the training stage instead of the parameter generation stage in a similar manner as our previous work on Gaussian Mixture Model (GMM)-based spectral parameter trajectory conversion. The trajectory HMM is trained with a novel objective function consisting of both the conventional trajectory HMM likelihood and a newly implemented MS likelihood. This training framework is further extended to the F0 component. The experimental results demonstrate that the proposed algorithm yields improvements in synthetic speech quality while preserving a capability of the computationally-efficient generation processing.