If you cut a circle from your fabric the pattern will be the right way up, sideways and upside-down on different parts of your circle. If you cut two semi-circles with right-way-up-fabric in the middle of each one it will still be sideways on the sides.

You could piece the skirt together out of several narrower trapezoid panels instead. I usually do this by alternating right-way-up and upside-down panels, in order not to waste any fabric, but since your pattern is completely asymmetric (I assume, if it's sheet music) you'll need to cut out all the panels laid out in the same direction with waste fabric around them, unless you don't mind the sheet music being upside-down on half of them. This method will give you an approximation of a circle or semicircle skirt (depending on the number of trapezoids). Don't do this; it's a worse fit for your fabric than my next suggestion.

It would be simpler to use this fabric for a skirt design which is based on a cylinder, like one which is pleated or gathered at the top. A single cylinder would give you a very different shape to a circle skirt, but you could cut three cylindrical tiers which are about 1/3 of the height each (or, if your fabric is half the height you want for the skirt, maybe 4 tiers 1/4 of the height each) and get wider as you get further down the skirt.

The top one would be just wide enough to be comfortable around your hips, and would be gathered at the top to fit the waistband. The next tier would be gathered at the top and attached to the bottom of the top one, and the bottom layer would be gathered at the top and attached to the bottom of the middle one. I have a skirt with this kind of construction, and it has a shape which is quite close to a circle skirt.

I think that this approach may be your best bet. You can piece together the longer tiers out of shorter rectangles; if you match up the lines of the fabric it shouldn't be obvious.