Odd time signature creation

As a teaching tool, I would like to be able to create time signatures like those used by David Newell in his band method "Simple Rhythmatician," whereby

1. The denominator of the time signature can be anything;
2. The measure is exactly 1 whole note long in every measure regardless of the time signature.
3. Each whole note will play back as a note equaling the numerical value of the denominator.

I can make the first two happen. In other words I can create a time signature that SHOWS any number in the denominator. However, I cannot make the third desire happen: to play back the whole note for the duration of the number in the denominator.

For example, I want to have a time signature with three as the denominator; a whole note in each measure; and each measure last for three (and only three) counts on play back.

Is this possible?