Extraction of symbolic information from signals is an active field of research enabling numerous applications especially in the Musical Information Retrieval domain. This complex task, that is also related to other topics such as pitch extraction or instrument recognition, is a demanding subject that gave birth to numerous approaches, mostly based on advanced signal processing-based algorithms. However, these techniques are often non-generic, allowing the extraction of definite physical properties of the signal (pitch, octave), but not allowing arbitrary vocabularies or more general annotations. On top of that, these techniques are one-sided, meaning that they can extract symbolic data from an audio signal, but cannot perform the reverse process and make symbol-to-signal generation. In this paper, we propose an bijective approach for signal/symbol translation by turning this problem into a density estimation task over signal and symbolic domains, considered both as related random variables. We estimate this joint distribution with two different variational auto-encoders, one for each domain, whose inner representations are forced to match with an additive constraint, allowing both models to learn and generate separately while allowing signal-to-symbol and symbol-to-signal inference. In this article, we test our models on pitch, octave and dynamics symbols, which comprise a fundamental step towards music transcription and label-constrained audio generation. In addition to its versatility, this system is rather light during training and generation while allowing several interesting creative uses that we outline at the end of the article.