It's easy for island owners to manage their sound. By tapping on your Season number in the top left, then tapping on the Settings cog, you will find two sliders in your settings menu. These sliders are Music and SFX.

Turning down the Music slider will reduce the music volume individually from the sound effects in the game. To turn down the sound effects, use the SFX slider.

A note for the savvy Island entrepreneur: ads can sometimes use your devices sound settings as opposed to the games. If you want to play silently and still get the full benefit of ad boosts, make sure to turn your device's media volume to 0!