ΔE 24.6, loose: different enough that you should pick on purpose. Gloomy Violet is a little darker, much more saturated, warmer than Tyrian Navy.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 25). Gloomy Violet is a little darker, much more saturated, warmer than Tyrian Navy.
The same difference applies the other way round: Tyrian Navy is a little lighter, much more muted, 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 Tyrian Navy on the colour wheel See Gloomy Violet on the colour wheel
All matches: Tyrian Navy equivalents · Gloomy Violet equivalents