I guess it depend on how the vco is built. I built one a time ago by just four transistors, and a small number of resistors and two caps, and I guess the scale where near linear, exept from low voltage input.
I don't know the definition of "tuning ring" either.
If you have the posibillity to test one in a lab environoment and plot the input voltage versus the frequenzy, I guess you'll have a fair idea about how a formula would look like.
Like I say before, if the vco consist of a transistor pair that controls the voltage (over a pair of fixed resistors) and then also the current, there will be nearly like a linear formula. That is F=a+bX where F is frequenzy, a is some variable you have to find out yoursel by experiments, and b is the difference in F for one volts in differece.
To get it even more correct you have to compute the voltage loss over the base-emitter on the transistor pair, wich I slightly remember is a logaritmic scale.