The Finite Element method, widely used for solving Partial Differential Equations, may result in suboptimal computational costs when computing smooth fields within complex geometries. In such situations, IsoGeometric Analysis often offers improved per degree-of-freedom accuracy but building analysis-suitable representation of complex shapes is generally not obvious. This paper introduces a non-invasive, spline-based fictitious domain method using Free-Form Deformation to efficiently solve the Helmholtz equation in complex domains, such as in musical instruments. By immersing a fine FE mesh into a simple B-spline box, the approximation subspace size is significantly reduced without compromising accuracy. Accompanied by specific conditioning treatment, the method not only proves to be efficient, but also robust and easy to implement in existing FE software. Applied to an alto saxophone, the method reduces the number of degrees of freedom by over two orders of magnitude and the computation time by more than one compared to standard FE methods with comparable accuracy when compared to experimental tests.