Answer: Based on your description this might be a disease problem such as mildew or it might be related to spider mites.
Rosemary prefers an arid Mediterranean type climate (dry air, lots of sunshine) and is often subject to mildew infections in humid areas such as Memphis.
Make sure your plant is in a location with bright light and good air circulation. If it is indoors, keep the temperature on the cool side (say 60 degrees) and if necessary run a fan in the room for several hours a day. You may also need to thin your plant to allow air to circulate through it. Also remove the infected foliage and clean up any dropped foliage to try to limit sources of reinfection, you could also trim it back.
Since it is related to the weather conditions, there may not be a lot you can do to prevent it. You might find however that a superfine horticultural oil helps, read and follow the label instructions. Your local county extension may also have some suggestions as to how to treat it, especially if you maintain the plant as an ornamental rather than growing it for the kitchen.
If it is spidermites you should be able to see fine webbing along the stems and undersides of the foliage. If you find those, you can treat the spidermites with commercially formulated insecticidal soap (read and follow the label directions), or by rinsing the plant with a strong spray of plain water twice a week.
Here is some general cultural information you may also find helpful.
Good luck with your rosemary!
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