The charged particle multiplicity in hadronic three-jet events from Z decays is investigated. The topology dependence of the event multiplicity is found to be well described by a modified leading logarithmic prediction. A parameter fit of the prediction to the data yields a measurement of the colour factor ratio CA/CF with the result CA/CF = 2.261 ± 0.014stat. ± 0.036exp. ± 0.066theo. in agreement with the SU(3) expectation of QCD. The quark-related contribution to the event multiplicity is subtracted from the three-jet event multiplicity resulting in a measurement of the multiplicity of twogluon colour-singlet states over a wide energy range. The ratios r = Ngg(s)/Nq¯q(s) of the gluon and quark multiplicities and r(1) = N gg(s)/Nq ¯q(s) of their derivatives are compared with perturbative calculations. While a good agreement between calculations and data is observed for r(1), larger deviations are found for r indicating that non-perturbative effects are more important for r than for r(1).