Measurements of two- and multi-particle angular correlations in pp collisions at View the MathML sources=5,7, and 13TeV are presented as a function of charged-particle multiplicity. The data, corresponding to integrated luminosities of View the MathML source1.0pb−1 (5 TeV), View the MathML source6.2pb−1 (7 TeV), and View the MathML source0.7pb−1 (13 TeV), were collected using the CMS detector at the LHC. The second-order (v2v2) and third-order (v3v3) azimuthal anisotropy harmonics of unidentified charged particles, as well as v2v2 of View the MathML sourceKS0 and View the MathML sourceΛ/Λ‾ 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 v2v2 values of charged hadrons (mostly pions), View the MathML sourceKS0, and View the MathML sourceΛ/Λ‾, with lighter particle species exhibiting a stronger azimuthal anisotropy signal below View the MathML sourcepT≈2GeV/c. For 13 TeV data, the v2v2 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.