Is there any merit to the idea of the so-called "melodic arch"
     -- where phrases tend to ascend and then descend in pitch?

     More specifically, are such "melodic arches" evident in
     traditional folk melodies?

Approach:

     For this demonstration we will use a sample of 20 traditional
     British folk ballads from the "Child" collection.

     A simple approach to this question is to compare the first and
     last pitches in a given phrase with all other pitches in the
     same phrase.

     For phrases conforming to the arch shape, the first and last
     pitches should typically have a lower average pitch than the middle
     pitches in the phrase.

     For each ballad, we will split-off a new spine containing just
     phrasing information.  We can do this using the REND command:

          rend -f reassign -i '**kern' ballads > temp1

     We will then translate the ballads from **kern notation
     to a numerical semitone representation (**semits).
     (The -x option eliminates non-pitch-related information.)

          semits -x temp1 > temp2

     Now we can rejoin the semitone and phrasing information
     so only the data of concern is present.

     The **semits and **phrase information are rejoined using
     the ASSEMBLE and CLEAVE commands:

     assemble temp1 temp2 | cleave -i '**phrase,**semits' > temp3

     Next we will use the RID and HUMSED commands to eliminate
     barlines, interpretations, and global and local comments.

          rid -GLI temp3 | humsed '/[=r]/d' > temp4

     Finally, we will average the first and last pitches in each
     phrase, and compare this to the average pitch for all mid-phrase
     pitches.  We will count the number of phrases that arch
     either upward or downward:

          Number of phrases that arch upward:   132
          Number of phrases that arch downward:  52

Conclusion:

     In this sample of British folk ballads, the number of phrases
     that arch upward is more than double the number of phrases
     that arch downward.

     The results are consistent with the "melodic arch" hypothesis.