A Riemannian distance is defined which is appropriate for clustering multivariate data. This distance requires that data be first fitted with a differentiable density model allowing the definition of an appropriate Riemannian metric. A tractable approximation is developed for the case of a Gaussian mixture model and the distance is tested on artificial data, demonstrating an ability to deal with differing length scales and linearly inseparable data clusters. Further work is required to investigate performance on larger data sets.