Long　stay　cables　may　be　subject　to　vortex-and　rain-wind-induced　vibrations.　To　suppress　the　two　types　of　vibrations　simultaneously,　integrating　the　advantages　of　negative　stiffness　and　inerter-based　damper,　a　new　negative　stiffness　electromagnetic　tuned　inerter　damper　(NSETID)　is　proposed　in　this　study.　The　equivalent　mechanical　model　of　the　NSETID　is　presented　according　to　its　working　principle.　On　this　basis,　non-dimensional　analytical　equations　of　the　cable　with　NSETID　are　developed.　Subsequently,　the　optimum　parameters　of　the　NSETID　are　identified　by　optimizing　the　damping　ratio　of　the　tuned　cable　mode.　The　effects　of　the　cable　sag　and　flexural　stiffness　on　the　control　performance　and　optimum　parameters　of　the　NSETID　are　also　taken　into　consideration.　In　addition,　the　benefits　and　damping　enhancement　of　the　NSETID　for　mitigating　single-mode　and　multi-mode　cable　vibrations　are　illustrated　by　comparing　with　tuned　inerter　damper　(TID),　negative　stiffness　damper　(NSD),　and　negative　stiffness　inerter　damper　(NSID).　Finally,　the　damping　enhancement　mechanism　of　the　NSETID　for　cable　multi-mode　is　explored　through　its　stiffness　and　damping　properties.　The　results　highlight　the　superior　damping　enhancement　capabilities　of　the　NSETID　in　mitigating　single-mode　and　multi-mode　cable　vibrations.　The　better　control　performance　of　the　NSETID　can　be　attributed　not　only　to　its　negative　stiffness　behavior　but　also　to　the　adaptability　of　its　equivalent　damping　and　negative　stiffness　coefficients　to　the　cable　mode　order.