In a small bowl, combine vinegar, water and salt to make a brine. Let sit for a few minutes and stir to mix until the salt is dissolved.
Cut the turnips and beet from stem to blossom end, then in half again to make small bite-sized wedges.
Cut the jalapeno in half lengthwise to expose the ribs and seeds.
Pack the turnips and beet wedges into a clean wide-mouth quart-sized jar, layering along with the garlic cloves and jalapeno (if using). Sprinkle in the pickling spices.
Pour the brine into jars and weight down all ingredients to be sure they are covered with brine. Put a lid on top and refrigerate for at least 3 days and up to a couple months. Shake the jar occasionally to distribute the pink hue of the beet.