Word for fitting a tune to lyrics

Solution 1:

The expression you are looking for is set something to music:

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to write or provide music for a poem, story, or other words so that it can be performed:

• The poem was set to music in 1888.

(Cambridge Dictionary)

Solution 2:

You can use a variety of phrases to describe the act of composing a tune to match a poem. This Wikipedia article about "In the Bleak Midwinter" uses "set to" as in:

The album consists of Emily Dickinson poems set to music by (Iron Maiden?!).

In the case of "In the Bleak Midwinter" there are two settings. Each of these tunes may have numerous "orchestrations" or arrangements for different groups of instruments.