Re Paul Burkes equation. It is totally correct but ignores the effect of purely local effects such as masses of iron, oil or coal bearing rocks. Mountains too have an effect.

So when someone uses a piston gauge to do precise measurements they call in the services of someone who uses a gravitometer to measure the local G to a reasonable degree of uncertainty. That figure is then used to correct the piston guage readings.

By the way a vibrating string can be affected by local G but to a very small degree. The frequency of vibration is proportional to the square root of the stiffness of the string (k) over the mass of the string. As mass is subject to gravity you can blame your out of tune guitar, banjo, mando, fiddle, zither, dulcimer on the local G ;)