This study investigates the use of non-linear unsupervised dimensionality reduction techniques to compress a music dataset into a low-dimensional representation which can be used in turn for the synthesis of new sounds. We systematically compare (shallow) autoencoders (AEs), deep autoencoders (DAEs), recurrent autoencoders (with Long Short-Term Memory cells -- LSTM-AEs) and variational autoencoders (VAEs) with principal component analysis (PCA) for representing the high-resolution short-term magnitude spectrum of a large and dense dataset of music notes into a lower-dimensional vector (and then convert it back to a magnitude spectrum used for sound resynthesis). Our experiments were conducted on the publicly available multi-instrument and multi-pitch database NSynth. Interestingly and contrary to the recent literature on image processing, we can show that PCA systematically outperforms shallow AE. Only deep and recurrent architectures (DAEs and LSTM-AEs) lead to a lower reconstruction error. The optimization criterion in VAEs being the sum of the reconstruction error and a regularization term, it naturally leads to a lower reconstruction accuracy than DAEs but we show that VAEs are still able to outperform PCA while providing a low-dimensional latent space with nice "usability" properties. We also provide corresponding objective measures of perceptual audio quality (PEMO-Q scores), which generally correlate well with the reconstruction error.