9", 10" or 12" tonearms

additional length makes the geometry better. Theoretically the arm will track better.

There some stellar 9" arms though, so it's really pretty relative.

I have always wanted to try a linear tracking arm to do away with the geometry/tracking error problems inherent in pivoted arms.

I have seen a lot of excellent reviews for the Terminator linear tracking arm from the UK.

Bottom line, the longer the arm the less distortion than that which would be created by a shorter arms geometry. The 12" arm would track in a shallower arc allowing the cartridge to trace more accurately from the beginning to the end of the record surface.

Think in terms of a 9" arm representing a circle 18" in diameter and the 12" arm would create a circle of 24" in diameter. Cut and paste (in your minds eye) a section of each on a record and you will see the difference each arc makes.

I now have a 12.7 VPI, which is a great arm, as well as shorter arms. My own opinion is that good arms of any length have more performance in them than most ever get out of them. The 12" arms require different length arm boards or even different tables than shorter ones, something to be kept in mind. Not all tables will accommodate them. I like the longer arms but I would not make it the sole or even the most important criteria in choosing an arm.

The longer tonearm results in less derivation from the tangential zero tracking error line. Sonically it gives a wider soundstage with more stable individual positioning. Furthermore you can expect to hear less distortion in climax' and critical high frequency passages. All these are direct results of a more " stable aligned" position of the polished area of the stylus seen in the horizontal plane.
All the above are only a relative advantage between tonearms of the same design (VPI, DaVinci, SME, FR etc.) but available in different total length.
Its similar to sailing boats - length gives smoother run.....
If possible ( many TTs do not offer an option to mount tonearms with a total length in excess of 10") I would always go for a 12" tonearm. The geometrical advantages do in general (with good designs..) more than make up for increased effective moving mass and (sometimes...) small decrease in stiffness.
If you look around you will notice that its always the 12" versions of vintage tonearms (SAEC, FR, SME, Ortofon, DaVinci etc.) which do command the high prices on the second hand market. They were more expensive then their shorter brothers too when new, but they tend to preserve their value much more stable and often do increase over time ( due to their rarity too).
The "kings" of pivot tonearm designs are all 12" versions.