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The jewelry gold usually comes historically rated in grades 14 or 18 karats, where 24 means pure gold. The rest is mostly silver.

Find out specific heat capacity for the complementary metals like silver or copper, so you could roughly estimate via interpolation.

Another method would be pyknometry, determining the ring density by differential weighting the mass of displaced water - old good method of Archimedes. Gold has almost double density, compared to other expected metals.

Edit:  list of densities:


Gold 19.3. 
Silver 10.5.


If linearly interpolated by mass fraction:.    


14 karat gold  15.63.
18 karat gold  17.1