The recently discovered shear modulus anomaly in solid 4He bears a strong similarity to the phenomenon of supersolidity in solid 4He and can lead to the period shift and dissipative signals in torsional oscillator experiments that are nearly identical to the classic NCRI signals observed by Kim and Chan. In the experiments described here, we attempt to isolate the effects of these two phenomena on the resonance periods of torsion oscillators. We have constructed a triple compound oscillator with distinct normal modes. We are able to demonstrate that, for this oscillator, the period shifts observed below 200 mK have their primary origin in the temperature dependence of the shear modulus of the solid 4He sample rather than the formation of a supersolid state. © Published under licence by IOP Publishing Ltd.