Measurements of two- and multi-particle angular correlations in pp collisions at (\sqrt{s}) = 5,7, and 13 TeV are presented as a function of charged-particle multiplicity. The data, corresponding to integrated luminosities of 1.0 pb(^{-1}) (5 TeV), 6.2 pb(^{-1}) (7 TeV), and 0.7 pb(^{-1}) (13 TeV), were collected using the CMS detector at the LHC. The second-order ((v_{2})) and third-order ((v_{3})) azimuthal anisotropy harmonics of unidentified charged particles, as well as (v_{2}) of (K_{S}^{0}) and (\Lambda / \overline{\Lambda}) particles, are extracted from long-range two-particle correlations as functions of particle multiplicity and transverse momentum. For high-multiplicity pp events, a mass ordering is observed for the v2 values of charged hadrons (mostly pions), (K_{S}^{0}), and (\Lambda / \overline{\Lambda}), with lighter particle species exhibiting a stronger azimuthal anisotropy signal below (p_T \approx 2GeV/c). For 13 TeV data, the (v_{2}) signals are also extracted from four- and six-particle correlations for the first time in pp collisions, with comparable magnitude to those from two-particle correlations. These observations are similar to those seen in pPb and PbPb collisions, and support the interpretation of a collective origin for the observed long-range correlations in high-multiplicity pp collisions.