Detailed calculations on the K_beta X-ray emission spectrum of atomic argon are reported, using CI wavefunctions based on MCDF-EAL calculations for a number of configurations of varying degrees of excitation and ionisation. Crude approximations to the total widths of states, initial-state populations from shake theory and radiative transition probabilities calculated in a transition orbitals approach were used to obtain line intensities which were synthesised into a predicted spectrum which could be compared with experiment. It is shown that the Rydberg spectator transition (1s)((3p)4p ¹S) ²S-(3p² ¹D)34p ²P⁰ contributes substantially to the K_beta^v satellite, altering its shape from that predicted in a simple model which includes only primary hole states, and bringing it into agreement with the experimentally determined shape. It was not possible to identify the major contributions the the K_{beta "} satellite, but the (1s3s) transitions could be excluded by energy considerations, and the most likely contributions are considered to be the (1s3p²)-(3p³) transitions with or without Rydberg spectator satellites.