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In this paper, a new generalized cyclotomy with respect to a positive odd integer is introduced, and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications. The frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the Peng-Fan bound and Lempel-Greenberger bound, respectively. Further, the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3. Some of them have new parameters.