Pierre Moulu’s Missa Alma Redemptoris Mater can be performed in two ways: with or without rests greater than a minim (half-note). Notwithstanding that Moulu’s mass was cited in 16th-century sources, little is known to modern scholars about the structural mechanisms whereby two different versions of a musical work can be generated by the presence or absence of rests of certain values in contrapuntal combinations for two or more parts. Yet there is one treatise on counterpoint, hitherto generally neglected by Western scholars, where Moulu’s mass is singled out for attention. Sergei Ivanovic Taneyev’s monumental Podvizhnoi kontrapunkt strogogo pis’ma, published in Moscow in 1909, is well known to generations of Russian music students and scholars for its lengthy exposition of an overall theory of certain kinds of contrapuntal processes known collectively as ‘moveable counterpoint’. Towards the end of the book, Taneyev turned his attention to Moulu’s mass as a special case of what he termed ‘horizontal-shifting counterpoint’, which is a subtype of ‘moveable counterpoint’ whereby the numbers of rests separating voice entries are adjusted so that a second contrapuntally correct version of the same musical materials is obtained. It is worthwhile to investigate Moulu’s mass as a work that demonstrates many of the core features of Taneyev’s theory of horizontal-shifting counterpoint. This enables not only identification of an analytical approach to the specific compositional problem posed by Moulu’s mass but also assessment of the utility of Taneyev’s methodology in further study of Renaissance repertoire, most especially in works employing fuga, a compositional technique that includes horizontal-shifting counterpoint amongst its range of procedural devices. A key question of this study, therefore, is how Taneyev’s approach to a specific work such as Moulu’s mass can yield analytical insights of use to present-day scholars engaged in analytical studies of Renaissance counterpoint.