resonance and standing waves
A first analysis to do for your problem is to view your mast as a long cylinder. Draw your mast according to scale including the mast step, the deck partners, and any fore and aft rigging attachments, ignoring any lateral rigging attachments (this first step is in assuming that your real problem is as you describe in only a fore and aft vibration mode).
Using dividers mark 1/2, 1/4, 1/8 of your mast length for both the length conditions for mast head to deck and masthead to base step. Your mast pumping is not only from a resonant multiple of such even fractional denominators as noted it is also caused from a reflection of such multiples against the deck and/or step which causes standing waves in the mast.
If your fore and aft rigging attachments (like your lowers and running backs) are placed at one of these even fractional multiples then there is NOTHING that you can do with the rigging tension at those points to stop the pumping. In addition, there is nothing that you can do with the headstay and backstay tension to stop the pumping. ONLY attachments located off of the fractional multiples can damp the pumping unless you can make the mast cross-section full of damping material, not normally a viable mechanical solution.
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