It didn’t take Gil Rose long to fill his schedule. Just before Christmas, Rose’s position as artistic director of Opera Boston disappeared when the company closed. Today, Monadnock Music, based in Peterborough, N.H., and known for its annual summer music festival, appointed Rose as its new artistic director. Rose, who also leads the Boston Modern Orchestra Project, replaces Jonathan Bagg and Laura Gilbert, the former co-artistic directors who last year were told that they would not be invited back as part of changes being made at Monadnock Music. Though he doesn’t start his job until Feb. 1, Rose should already be familiar with Monadnock Music executive director Will Chapman. Chapman is the former director of development and marketing at Opera Boston.