Function Blend actually does a great color interpolation. For simplicity assume that all your values are always in $[0,1]$ interval. Choose a specific Color Scheme, for example DeepSeaColors , and define:
cf[v_][x_] := Blend[ColorData["DeepSeaColors"] /@ v, x]
You do not have to use built in color schemes, you could just have assigned some colors to your numbers. The point is in blending them. Then using detailed Raster make:
Graphics[Raster[{Range[100]/100}, ColorFunction -> cf[t1]], AspectRatio -> .3]

Graphics[Raster[{Range[100]/100}, ColorFunction -> cf[t2]], AspectRatio -> .3]

It is not clear from your question how exactly you want to rescale / normalize. But you can always use Rescale to adjust the above aproach.
ArrayPlot[With[{n = 5}, ConstantArray[t1, n] ], ColorFunction -> "TemperatureMap"]
$\endgroup$ – b.gates.you.know.what May 30 '19 at 8:28Column[Quiet@DensityPlot[Interpolation[#][x], {x, 0, 1}, {y, 0, 1/5}, AspectRatio->Automatic,Frame -> False, PlotRangePadding->0,ImagePadding->0,ColorFunction->"Rainbow", ColorFunctionScaling->False, ImageSize->Medium]&/@{t1,t2}]
give what you need? $\endgroup$ – kglr May 30 '19 at 8:29