Which finish should you choose, matte or gloss?
Matte if the menu is read under a spot of light and you do not want glare on the price. Gloss only on a cover that carries no small text, because the reflection eats the line.
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Ask the printer for two scraps, matte and gloss, and look at them on one of your tables at night. Gloss is chosen at the counter in full light and in the dining room the price disappears. The print sheet with the chosen finish written down, the one that survived the test on the table.