Four expressions involving sums of position and velocity coordinates bounding the total angular momentum of particle systems, and by extension of any continuous or discontinuous material systems, are derived which are tighter for any particle configuration than similar inequalities derived by Sundman (Acta Math 36:105, 1913), Saari (Conference board of the mathematical sciences, No. 104, 2005. http://books.google.ch/books?id=uNXgf2X1GeYC) and Scheeres (Celest Mech Dyn Astron 113:291–320, 2012. https://doi.org/10.1007/s10569-012-9416-0). Eight distinct inequalities can thus be ordered according to their tightness to angular momentum.