Effective Arm Mass: What's the right way to guess?

I have a TT w/ arm where the manual says the appropriate cartridge weight is 2.4-12g. There is no mention of effective arm mass. Solving for a resonant frequency of something in the 8-10Hz area, and a cart weight of 12g, I can move around the compliance quite a bit as long as I also move the arm mass around as well:

at F=8, 12g of cart weight, 25g of effective arm mass = compliance of 10
or, 12g cart, 18g effect. arm mass gets me compliance at 13.
If I move F to 12, my 25g +12g gets me a compliance of 5 or so.

Is there a compliance number that manufacturers took for granted in these recommendations?

What does one do in this situation? Any thoughts?

The table in question is a Pioneer PL-90 which was brought out in the mid 80s. A friend would like to borrow it to see if he is interested in vinyl. I have been playing around with it but have not found what I consider to be the best match - I do not have many high compliance carts and I am wondering if that is the main problem.
Dear T_bone, there is a formula to calculate the unknown effective mass of a given toneram, but it only works if you have the known effective mass of a reference tonearm and do know the resonance frequency of both tonearms with a given cartridge.

I do not have any other formula at hand.

With the Pioneer PL-90 and its tonearm you should get good results with cartridges having a compliance of 10 - 15 mm/N. Not lower. The Pioneer tonearm is not that heavy in effective moving mass.

Hope this is of any help.

Best regards.
Dear T_bone, if you have the resonance frequency with a given cartridge and know its compliance this formula will lead you to the effective mass:

F(r) = 1000 / 2 x P x ((M(t) + M(c)) x compliance)

F(r) = resonance frequency
P = Phi = 3,1482
M(t) = mass (effective) of tonearm
M(c)= mass (weight) of cartridge

Best regards.
Dertonarm, I think the formula you meant to write is the one I am using:

I use F=159 / sqrt[(M(t) + M(a)) x Compliance].
My problem is that I have two unknowns because the mfr does not give either F(r) with a given compliance or a make reference to a recommended cartridge, and as far as I know, it was not sold with one in the first place.

But the guidance for the 10-15 mm/N is helpful, and that's probably what I can look to change.
The Hi-Fi news test record will work. Tracks 2 and 3 on side two have a frequency sweep from 20 to 5 Hz. The tonearm will start vibrating quite heavily at the resonant frequency.
Thanks for that. That's not a bad idea. Will it tell me what the frequency is? I am not sure that I can tell the difference between 8 and 12 by myself.