OK, this one's not going to be easy. This tune is on an Ampico "B" coded reproducing piano roll discovered in the Ampico factory in Rochester, New York when the company went into receivership in the 1940s. ("Reproducing" means the dynamic performance of the recording artist was converted into a digital code and punched into the margins of the roll. When played back on a properly equipped and adjusted piano, the music plays back with the same expression, i.e., "reproducing" the original performance. Pretty sophisticated stuff for 1913 pneumatic technology!)

Anyway, this roll seems to be some sort of factory "trial" as it has no markings as to title, composer or recording artist. The playing style, however, is very distinctive and is almost certainly the work of noted Ampico recording artist Frank Milne (1887 - 1959). It is possible that the tune was never published or submitted for copyright, and could even be an original composition by Milne. It is typical of the "fox trot" style of the '40s but at a leisurely tempo. The following is a YouTube link posted by YT user ampplay1 whose channel contains many Ampico performances: https://www.youtube.com/watch?v=LlwMiJ9I9oU After a short intro and perhaps a verse, the assumed melody begins at 0:50. A close approximation of the melody is shown in the attached JPG image. Hopefully, someone can help solve this almost 70 year-old mystery. My searches here using Parsons code and rhythm contour have not helped.