In this paper, we investigate the use of the Gaussian mixture model-based block quantiser for coding line spectral frequencies that uses multiple frames and mean squared error as the quantiser selection criterion. As a viable alternative to vector quantisers, the GMM-based block quantiser encompasses both low computational and memory requirements as well as bitrate scalability. Jointly quantising multiple frames allows the exploitation of correlation across successive frames which leads to more efficient block quantisation. The efficiency gained from joint quantisation permits the use of the mean squared error distortion criterion for cluster quantiser selection, rather than the computationally expensive spectral distortion. The distortion performance gains come at the cost of an increase in computational complexity and memory. Experiments on narrowband speech from the TIMIT database demonstrate that the multi-frame GMM-based block quantiser can achieve a spectral distortion of 1 dB at 22 bits/frame, or 21 bits/frame with some added complexity.