ΔE 48.3, loose: different enough that you should pick on purpose. Mahogany Ink is a little lighter, much more muted, warmer than Transparent Purple.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 48). Mahogany Ink is a little lighter, much more muted, warmer than Transparent Purple.
The same difference applies the other way round: Transparent Purple is a little darker, much more saturated, cooler.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Transparent Purple on the colour wheel See Mahogany Ink on the colour wheel
All matches: Transparent Purple equivalents · Mahogany Ink equivalents