This　paper　investigates　Gabor　frame　sets　in　a　periodic　subset　S　of　R.　We　characterize　tight　Gabor　sets　in　S,　and　obtain　some　necessary/sufficient　conditions　for　a　measurable　subset　of　S　to　be　a　Gabor　frame　set　in　S.　We　also　characterize　those　sets　S　admitting　tight　Gabor　sets,　and　obtain　an　explicit　construction　of　a　class　of　tight　Gabor　sets　in　S　for　the　case　that　the　product　of　time-frequency　shift　parameters　is　a　rational　number.　Our　results　are　new　even　if　S　=　R.