Thank you to those who responded.  Rather than drag this out, I'll just
give you my answers and maybe raise issues thereby for more discussion.

1. Who wrote it, when?

The melody is from Wllhelm Stenhammar's 1907 second piano concerto in D
minor, Adagio (3rd section), By removing short answering calls from the
strings, I made the melody sound even more four-squared, unfortunately.
The three-against-two accompaniment is very Brahmsian, but the major/minor
switches seem to me to be closer to Grieg (and Dvorak).  The clarity is
certainly Schubertian.

2. Which interpretation do you like the best?

The first is played by Janos Solyom (EMI), the second by Cristina Ortiz
(BIS).  I can't decide between the two.  I agree the accompaniment in
the second is more accomplished.  It depends upon whether you want to
emphasize the nobility or the romance of the theme.  Both are valid
approaches in the context of the work.  I don't know how others of you
feel out there about this, and it's certainly not politically correct,
but it seems to me that female pianists are more likely to not observe
strict rhythm than male pianists.

My preference would be for strict rhythm at first, with some rubato at
the end.  I particularly like the way Ortiz shifts the climax of the
melody more to the end compared to Solyom.

3. Why or why not does the melody speak to you?

Of course no musical element can be isolated without damage.  Even so,
matters of taste play a big role here.  I'm one of those folks who is
passionate about melody, and will hum a beloved tune in my head for
weeks at a time. But the thrill in reliving it in my head comes from my
background knowledge of its context as well - missing in these excerpts.

Those of you who list many tunes or say there are too many to list may
not have loved tunes as intensely as I, and I probably have not experienced
other musical elements as intensely as you.

The context of this particular melody, however, is interesting and worth
describing in detail.  The concerto is one of the strongest I know in
"narrative" form without a program or even a subtitle.  To me, it is,
even more than many other romantic concertos, an argument between two
personalities, the orchestra and the piano - sharply drawn.

In the first section (there are four, played without pause for close
to 30 minutes), the orchestra has no true melodies, just fragments and
motivs in an angry "sturm und drang" manner.  The piano, besides the
conventional flourishes, mostly plays melody, beginning with a very
Grieg-like tune in a major key.  Forgive me for personifying, but the
orchestra will have none of it, continuing to rant.  It gets so bad at
the end of the first movement that the two "personalities" are playing
diminished chords a half-step apart, each trying to resolve to a different
key.

In the second section, the piano tries to "cheer up" the orchestra with
a sprightly scherzo, again in a major key, but again it falls on deaf
ears (so to speak).  Again no melodies in the orchestra.

It is only in the third section, after the piano quiets down and plays
some hymnlike passages, that it seems that the orchestra seems to "listen."
And that's when the excerpted melody starts.  In my silly head, because
the piano "empathizes" with the orchestra's angst, and finally plays
something in a minor key, the orchestra can at last respond, which it
does by echoing the ends of the melody's phrases (cut off in my excerpts).
Later, for the first time, the orchestra plays the melody complete in
solo while the piano accompanies.  An understanding has been obtained.

In the last movement, the two personalities play together, and just
before the end, the two do a peroration of the excerpted tune, this time
in D major.

So even if you don't particularly like the tune, in hearing the whole
work, it's interesting what Stenhammar does with it - that is, if you
like symphonic poems, for of course the "narrative" approach perforce
distorts formal structures.

4. Is melody even that important or necessary nowadays?

In my view, melody is generally neglected in today's music.  But I'm
probably lamenting an "old fashioned kind" that went out like with Richard
Rogers when Sondheim came along.  I'll tell you one thing: a good
old-fashioned classical melody is VERY hard to write!

There are so many kinds of great melodies.  This species hits a soft
spot for me with a characteristic that is often criticized by folks with
different sensibilities: the presence of sequences.  For some reason, I
like repetition, but not the minimalist variety.  It fits into my obsessive
earworm predilections.  In this regard, Elgar's melodies, with the
constant repetition of melodic cells under different harmonic guises,
really do it for me.  Take his third symphony, for instance.  Anyone who
called its main theme a stupid saw going back and forth would be right.
But for me, it came, it sawed, and conquered!

5. What are your all time favorite melodies?

I love the medieval tune brilliantly utilized by Hindemith in Choir of
Angels, Hindemith's and Ron Nelson's Passacaglias, De Sabata's theme to
"Jeventus," the love sequence from Act II of Puccini's "Fanciulla,"
Wagner's "Vahalla" leitmotif, Tveitt's "Prillar," Barber's "Sure on this
Shining Night," Elgar's "Enigma" theme, V-W's "Dives and Lazarus," and
the most hummable theme from Piston's "Incredible Flutist," for starters.

Jeff Dunn