In the early visual and auditory system neurons are sensitive to a variety of parameters including orientation, contrast, and spatial and temporal frequencies, amplitude, timing, and spectral variables. There are theoretical reasons to believe that neural tuning for these particular parameters is fundamental to the information processing in each area. In contrast, we argue on both principled and empirical grounds that the idea of parametric encoding that has been so fruitfully applied to processing in early sensory systems does not have the potential to achieve more than heuristic or operational status in explanations of the motor system. In the motor system, inherent correlations among parameters of motion that occur in natural movements will necessarily make a neuron that is tuned to one variable also appear to be sensitive to other variables at different time lags. Similarly, depending on the nature of the task, neurons that appear to be tuned to parameters in one coordinate frame will often appear to be tuned to correlated variables in other coordinate frames. Finally, we point out that the tuning for any parameter can vary significantly with time lag. For all these reasons, we suggest that it may not be particularly meaningful to ask whether one or another movement parameter is represented in motor cortex. Instead, we propose that the tuning of any movement-sensitive cortical neuron is best envisioned as carving out a specific hyper-volume in a high-dimensional movement space. When one considers the way this tuning space changes over time, the time-varying preferred parameter values of the neuron describe a small segment of movement that we call a "movement fragment".