Classical Music Prediction and Composition by Means of Variational Autoencoders

This paper proposes a new model for music prediction based on Variational Autoencoders (VAEs). In this work, VAEs are used in a novel way to address two different issues: music representation into the latent space, and using this representation to make predictions of the future note events of the musical piece. This approach was trained with different songs of Handel. As a result, the system can represent the music in the latent space, and make accurate predictions. Therefore, the system can be used to compose new music either from an existing piece or from a random starting point. An additional feature of this system is that a small dataset was used for training. However, results show that the system is able to return accurate representations and predictions on unseen data.

An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability, namely 256/331776^t for t iterations of the test in the worst case. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order dividing 24. We also give bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^{-143} for k=500, t = 2. Compared to earlier similar results for the Miller-Rabin test, the results indicate that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point.