This chapter talks about John Birchensha, John Wallis and Thomas Salmon. These men took broadly mathematical approaches to music, attempting to imagine, construct or enact a mathematical practice which could legitimately be called musical and which, in salmon's case, also incorporated an element of musical practice. They were engaged, in a sense, in a 're-mathematisation' of a subject whose traditional mathematical basis had lost some of its credibility, eroded both by developments in musical practice and by the problems with the coincidence theory of consonance. The chapter deals with the definition and manipulation of musical ratios, introducing and discussing the correspondence between musical intervals and numerical ratios through string lengths. It deals with terminology, intervals and scales. The chapter discusses Birchensha's insistence on a diatonic scale which was pythagorean in the sense that every octave was pure, with ratios 2:1, 4:3 and 3:2 respectively.