  The common handsaw can be converted into a bowed musical instrument capable
of producing exquisitely sustained notes when its blade is appropriately bent.
Acoustic modes localized at an inflection point are known to underlie the saw's
sonorous quality, yet the origin of localization has remained mysterious. Here
we uncover a topological basis for the existence of localized modes, that
relies on and is protected by spatial curvature. By combining experimental
demonstrations, theory and computation, we show how spatial variations in blade
curvature control the localization of these trapped states, allowing the saw to
function as a geometrically tunable high quality oscillator. Our work
establishes an unexpected connection between the dynamics of thin shells and
topological insulators, and offers a robust principle to design high quality
resonators across scales, from macroscopic instruments to nanoscale devices,
simply through geometry.
