This　paper　develops　a　root-locus　optimal　formula　for　seismic　control　of　multi-story　structures　equipped　with　a　tuned　mass　damper　inerter　enhanced　by　negative　stiffness　(TMDI-NS),　accounting　for　non-resonant　mode　contributions.　Closed-form　optimal　solutions　are　derived　for　the　optimal　parameters　of　TMDI-NS.　These　parameters　differ　from　those　obtained　using　the　classical　fixed-point　method　when　extra　negative　stiffness　is　introduced　in　TMDI,　despite　the　two　methods　yielding　identical　optimal　tuning　frequency　for　TMDI　in　the　absence　of　negative　stiffness.　When　the　TMDI-NS　is　applied　to　multi-story　structures　and　tuned　solely　for　the　resonant　mode,　its　control　effectiveness　diminishes.　Consequently,　the　background　flexibility　caused　by　nonresonant　mode　in　multi-degree-of-freedom　(MDOF)　system　is　further　analyzed,　with　discussion　on　how　the　mounting　position　affects　the　non-resonant　contributions　and　the　optimal　parameters　of　TMDI-NS.　Seismic　performance　evaluations　of　two　typical　multi-story　buildings　via　frequency/　time-domain　analysis　demonstrate　that　non-resonant　mode　contributions　are　critical　for　the　optimal　design　of　TMDI-NS,　particularly　when　installed　on　lower　stories.　In　this　case,　a　further　reduction　of　10　%　in　roof　displacement　is　achieved　by　incorporating　non-resonant　modes　into　the　design,　as　the　dissipation　capability　of　TMDI-NS　is　significantly　enhanced.　Conversely,　installing　TMDI-NS　on　the　upper　floor　or　on　＂zero　vibration＂　floors　can　eliminate　the　non-resonant　mode　contributions,　provided　the　relative　displacement　of　the　TMDI-NS　is　not　excessively　constrained.