Looking through “The Easy Winners” by Scott Joplin (sheet music), the eighth measure confuses me:
In the right hand, there is a pair of half notes tied to notes in the previous measure, played at the same time and a string of six slurred 16th notes. This is all normal. However, why is there an eighth-note rest above the half notes? What does it do?
That eighth-note rest is a consequence of the fact that the top stave is being split into two different voices. Each different voice must fill up the same amount of time as was specified in the key signature (2/4, in this piece).
The way one can tell the two voices apart is by the direction in which the notes' stems point; the lower voice's stems point downwards no matter where on (or off) the staff the notes are, and the upper voice's notes always have their stems point upwards. Note that this is backwards from normal stem directions; one would expect a low note in any given clef to be "right-side-up", but when two or more voices share a staff, the lower of the two is upside-down. The other way is that normally, rests have fixed positions, but with multiple voices, the rests just go wherever is convenient and easy to read, like here, where a rest floats above the measure.
Another easy way to rationalise it is that the stems normally point inwards to save space on the page, but with two voices, if the stems pointed inwards like normal, the two lines would hit each other, so they face outwards.
Regardless, as you probably already figured out, the eighth-note rest is part of the upper voice (unfortunately, in this case, the "upper voice" has momentarily dipped below the lower voice, but it's easy to tell because of the rest being above the lower voice), and it is there to complete the measure in the top voice of the staff.
Actually, sometimes sheet music is printed that leaves out these rests when staves are divided. I'm not a huge fan of that convention, but I recognise that sometimes it is useful to improve legibility, and thus I mention it here in case one were to run across it and be stumped (I'm pretty sure someone's already asked that exact question on this site).
The two half notes in the same measure now form a sustained chord while the left hand continues a broken chord pattern which is a one of the signature elements of ragtime music. The sixteenth note run leads into a recap. of the rhythm in measures 5, 6 and 7.
Since there are 2 beats per measure, the first beat consists of one eighth rest on beat one(counting:one ) followed by 2 sixteenth notes (counting: and-uh) which make up beat one, and then, beat two- four sixteenth notes (two-e-and-uh) which complete the run that leads into the measures 9, 10 and 11.
