One of the simplest ways of designing allpass fractional-delay filters with maximally fiat group delays is by using the Thiran approximation by which the filter coefficients are calculated using a closed-form equation. However, due to the number of multiplications and divisions involved, the calculation of these coefficients is a computationally costly task and is not suitable for real-time applications. The analysis of a root-displacement-based interpolation method used in allpass tunable fractional delays is presented in this paper. The method allows continuous adjustments of the approximated fractional delay without the explicit calculation of a new set of filter coefficients. The transient error observed at the output due to the change of filter coefficients is analyzed. The direct and cascade implementations are compared with respect to their transient errors. An example application of the proposed method from the field of model-based sound synthesis is given.