The magnetic properties of the TiNiSi-type RMnSi compounds (R = Ho, Lu, Sc 0 . 9 Lu 0 . 1 ) have been studied by magnetization and neutron diffraction measurements. In their whole ordered range, LuMnSi and Sc 0 . 9 Lu 0 . 1 MnSi exhibit flat spiral antiferromagnetic structures (T N = 255 and 125 K, respectively) whereas HoMnSi displays three different magnetic regimes. Between 300 and 55 K, a collinear antiferromagnetic structure prevails while between 55 and 15 K, a canted antiferromagnetic structure, due to a non-collinear arrangement of the Ho moments, takes place. At lower temperatures, incommensurate components appear on both Mn and Ho sublattices yielding a ''double-cone'' magnetic arrangement. Within the three compounds, the Mn moment value, at 2 K, strongly decreases with the R size from 2.67 μ B in HoMnSi to 2.18 μ B in LuMnSi and 1.53 μ B in Sc 0 . 9 Lu 0 . 1 MnSi. In HoMnSi, the magnetic structure is characterized by antiferromagnetic Mn-Mn zigzag chains aligned along [010]. In the LuMnSi and Sc 0 . 9 Lu 0 . 1 MnSi compounds, the angle between the Mn moment directions in a chain decreases with the lanthanide size from 121 o in LuMnSi to 52 o in Sc 0 . 9 Lu 0 . 1 MnSi. The magnetic interactions in the Mn-Mn zigzag chains and the evolution of the Mn magnetic moment value are discussed and compared with those observed in the related CeFeSi-type RMnSi compounds.