Dear Peter: Your question IMHO has mre than one answer because there are mre than one factor involve.
If we take from the geometry point of view the 12" tonearm always will has advantage ( in theory ) over any other shorter tonearm. But unfortunately things are not so easy.
Looking to LOMC today cartridges the first " trouble " that we have is that the same cartridge, say a Titan i, will have a different resonance frequency with the SME 9" than with its " big brother ", this fact alone preclude a fair comparison ( and I'm not saying that the Titan is the best match for the tonearm, I take it only like an example. ) because that difference in the resonance frequency has its own " sound signature ".
Other subject that Dertonarm already point out: +++ the further away the cartridge is, the higher its influence on the mechanic resonance behaviour of the tonearm. +++++
and in my experience not only for the tonearm torsion resistance factor but depending on the tonearm build material, a 9" arm wand has a different " sound signature " than a 12" arm wand .
In the last three years that Guillermo and I been in our self tonearm designs we made ( and still do it. ) several and different tests and one of them was to have a shoot-out with the same cartridge, different ( same build material ) length on the arm wands, very near resonance frequencies between them ( due that we use different headshell weights ).
We do it with 9", 10", 11", 12" arm wands and we find that with the 9" is a clear overall advantage ( what we can hear )but betwen 10" and 12" seems to me that exist a " threshold "/land where it is extremly dificult to discern if there is a 12" advantage in the sound reproduction quality due to the stand alone length factor.
Our findings are very interesting because our propietary tonearm build material is almost neutral ( it does not have a " sound " ).
When the build material move away from " neutrality " then the differences are more " obvious ".
We are building a 14" and 16" other ones and we will see what happen, here the challenges are a little different.
I already posted on other thread that when you are in the audio item design ( any ) we learn a lot on the subject because we " live " every design day with the: why's, how's, where's and the like.
I hope to finish our tonearm project in the next three months.
What I learn/learned through the tonearm design help me to understand and learn too on the TT " behavior ", that's why in other thread I insist/push hard on TT build materials, this factor is definitive in tonearms and TTs quality performance more than any one can imagine.
Regards and enjoy the music.
Raul.