Texture synthesis techniques based on matching the Gram matrix of feature activations in neural networks have achieved spectacular success in the image domain. In this paper we extend these techniques to the audio domain. We demonstrate that synthesizing diverse audio textures is challenging, and argue that this is because audio data is relatively low-dimensional. We therefore introduce two new terms to the original Grammian loss: an autocorrelation term that preserves rhythm, and a diversity term that encourages the optimization procedure to synthesize unique textures. We quantitatively study the impact of our design choices on the quality of the synthesized audio by introducing an audio analogue to the Inception loss which we term the VGGish loss. We show that there is a trade-off between the diversity and quality of the synthesized audio using this technique. Finally we perform a number of experiments to qualitatively study how these design choices impact the quality of the synthesized audio.