Abstract Collinear (small-angle) and large-angle, as well as soft and hard radiations are investigated in three-jet and $${\text {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Z</mml:mtext> </mml:math> + two-jet events collected in proton-proton collisions at the LHC. The normalized production cross sections are measured as a function of the ratio of transverse momenta of two jets and their angular separation. The measurements in the three-jet and $${\text {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Z</mml:mtext> </mml:math> + two-jet events are based on data collected at a center-of-mass energy of 8 $$\,{\text {TeV}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace /> <mml:mtext>TeV</mml:mtext> </mml:mrow> </mml:math> , corresponding to an integrated luminosity of 19.8 $$\,\text {fb}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace /> <mml:msup> <mml:mtext>fb</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The $${\text {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Z</mml:mtext> </mml:math> + two-jet events are reconstructed in the dimuon decay channel of the $${\text {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Z</mml:mtext> </mml:math> boson. The three-jet measurement is extended to include $$\sqrt{s} = 13\,{\text {TeV}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> <mml:mspace /> <mml:mtext>TeV</mml:mtext> </mml:mrow> </mml:math> data corresponding to an integrated luminosity of 2.3 $$\,\text {fb}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace /> <mml:msup> <mml:mtext>fb</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The results are compared to predictions from event generators that include parton showers, multiple parton interactions, and hadronization. The collinear and soft regions are in general well described by parton showers, whereas the regions of large angular separation are often best described by calculations using higher-order matrix elements.