I think that if you wish to live happily ever after with those colors, you will see that they get married to each other... or perhaps it will be a menage a trois. You'll do this by introducing purple to the dining room. Maybe by placemats or napkins or flowers. And you'll add orange, quite subtly (if orange can be subtle) via candlesticks, or fruit, or flowers.

Then you'll do the reverse in the living room, bringing in turquoise and orange as accents to the purple. Don't let the supporting players overwhelm the star of each room. Turquoise will star in the dining room and purple in the living room, but they will be happily supported by the other colors.