We consider bandwidth-efficient modulation schemes for use on non-linear channels, such as that due to a non-linear high power amplifier (HPA) in a wireless system. Continuous Phase Modulation (CPM) schemes are known to perform well on such channels, because they have constant amplitude, but their bandwidth efficiency is low. N-MSK improves this by superposing two or more such signals, but this results in a non-constant amplitude. In this paper we investigate the performance of N-MSK on a non-linear channel, modelled using a travelling wave tube (TWT) non-linearity. We first consider the spectrum and the BER performance of N-MSK on a non-linear channel, making use of the Euclidean distance spectrum of the modulated signal. We then consider the effect of non-linear amplification on these properties. Signal spectrum was determined by simulation, since no closed-form expression is available when the effect of AM-PM conversion are included. We find that the spectrum is remarkably little affected, being only slightly broadened. BER is also evaluated by direct simulation, as well as from the Euclidean distance spectrum. The latter now exhibits a series of clusters, instead of discrete lines, and we find that at least the whole of the first cluster must be considered in calculating the BER, not just the minimum distance or the centroid of the cluster. The detector used in the simulation applies an inverse distortion function, then uses maximum-likelihood sequence estimation (MLSE) set up for the linear channel. This is no longer optimum, because the noise is distorted, and therefore it is also compared with a true MLSE detector. We find that the BER performance is, however, somewhat degraded compared to the linear channel. We determine back-off levels from saturation to optimise overall power efficiency.