  In this paper, we propose a musical instrument sound synthesis (MISS) method
based on a variational autoencoder (VAE) that has a hierarchy-inducing latent
space for timbre. VAE-based MISS methods embed an input signal into a
low-dimensional latent representation that captures the characteristics of the
input. Adequately manipulating this representation leads to sound morphing and
timbre replacement. Although most VAE-based MISS methods seek a disentangled
representation of pitch and timbre, how to capture an underlying structure in
timbre remains an open problem. To address this problem, we focus on the fact
that musical instruments can be hierarchically classified on the basis of their
physical mechanisms. Motivated by this hierarchy, we propose a VAE-based MISS
method by introducing a hyperbolic space for timbre. The hyperbolic space can
represent treelike data more efficiently than the Euclidean space owing to its
exponential growth property. Results of experiments show that the proposed
method provides an efficient latent representation of timbre compared with the
method using the Euclidean space.
