Quantifying the variability in Mansour nay: a stochastic approach combining vibroacoustic modelling and PCE-Kriging

The Mansour nay is a woodwind instrument made from a single piece of giant reed (Arundo donax L.). Because the body is natural and cannot be modified after construction, geometric variability across specimens is inevitable and directly affects the pitch frequencies produced by the instrument. Identifying which geometric parameters contribute most to this variability is essential for instrument makers selecting raw material and determining finger hole locations. In this study, 26 Mansour nay samples manufactured from giant reeds of different ages and geographic origins are investigated through audio-based pitch measurements, vibroacoustic finite element modelling, and uncertainty quantification. An Average Mansour Nay Model (AMNM) is constructed from the measured geometric properties of all samples and used as a reference for quantifying specimen-to-specimen deviations. Polynomial chaos expansion combined with Kriging (PCE-Kriging) is employed as a surrogate modelling technique suited to small datasets, and the expansion coefficients are used to rank the contribution of each geometric variable to the observed variability. The analysis identifies the top and bottom diameters as the dominant sources of pitch-frequency variation, with comparable influence, followed by the distances between finger holes, and then the total length. As a complementary methodological demonstration, the same PCE-Kriging framework is applied to the structural eigenfrequencies of the instrument measured under free boundary conditions, where the bottom diameter and mass emerge as the statistically significant variables. The acoustic findings provide quantitative guidance for giant reed selection and finger hole placement in nay making, and the overall methodology is transferable to other instruments made from natural materials where specimen variability is inherent.