  The main purpose of this work is to generalize the $S^3_\bfw$ Sasaki join construction $M\star_\bfl S^3_\bfw$ described in \cite{BoTo14a} when the Sasakian structure on $M$ is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case ($c_1(\cald)=0$) we construct a polynomial whose coeffients are linear in the components of $\bfw$ and whose unique root in the interval $(1,\infty)$ completely determines the Sasaki-Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper \cite{ApCa18} of Apostolov and Calderbank as well as the relation with K-stability. 
The $S^3_\bfw$ Sasaki Join Construction