Dogpile,
The formula works in both planes of course. If your cartridge has different V and H compliance, like a ZYX, run the formula twice.
The usual reason to check system resonance frequencies, whether by math or by test record, is simply to expose gross mis-matches - though frankly those are rare these days. Resonance frequencies won't tell you how a cartridge and arm will synergize and make music. Only listening will reveal that.
You do raise an interesting question though. I wouldn't assume either direction is less (or more) critical, but they are different. It might be theoretically desirable to have a higher R on the vertical axis and a lower R on the horizontal one.
Warps and floor-borne vibrations are largely vertical and fall mostly below 6Hz, so a slightly higher vertical R would help us avoid those. Some musical frequencies go as low as 16Hz and they are somewhat more horizontal, so we want horizontal R high enough to avoid them. I suppose a vertical R of 12-14Hz and a horizontal R of 8-10 Hz might be "ideal".
This would require either that the cartridge have a very non-linear suspension or that the arm have much higher effective mass on the horizontal axis than on the vertical. There are a few arms like that: air-bearing tangential arms and a few pivoted arms like the Dynavector 505/507 or a Twl-modded OL or Rega. Thanks to its relatively high horizontal inertia, my Twl-modded OL Silver was the most dynamic arm I've heard. Leading edge transient speed and extension were better than a TriPlanar, a Schroeder Reference or a Graham 2.2. I'm still impressed by Twl's brilliantly simple invention.
The formula works in both planes of course. If your cartridge has different V and H compliance, like a ZYX, run the formula twice.
The usual reason to check system resonance frequencies, whether by math or by test record, is simply to expose gross mis-matches - though frankly those are rare these days. Resonance frequencies won't tell you how a cartridge and arm will synergize and make music. Only listening will reveal that.
You do raise an interesting question though. I wouldn't assume either direction is less (or more) critical, but they are different. It might be theoretically desirable to have a higher R on the vertical axis and a lower R on the horizontal one.
Warps and floor-borne vibrations are largely vertical and fall mostly below 6Hz, so a slightly higher vertical R would help us avoid those. Some musical frequencies go as low as 16Hz and they are somewhat more horizontal, so we want horizontal R high enough to avoid them. I suppose a vertical R of 12-14Hz and a horizontal R of 8-10 Hz might be "ideal".
This would require either that the cartridge have a very non-linear suspension or that the arm have much higher effective mass on the horizontal axis than on the vertical. There are a few arms like that: air-bearing tangential arms and a few pivoted arms like the Dynavector 505/507 or a Twl-modded OL or Rega. Thanks to its relatively high horizontal inertia, my Twl-modded OL Silver was the most dynamic arm I've heard. Leading edge transient speed and extension were better than a TriPlanar, a Schroeder Reference or a Graham 2.2. I'm still impressed by Twl's brilliantly simple invention.