Concerning walking (and other)songs...I had another idea about "discovering" tunes that fit some of the songs Tolkien composed. It has to do with the meter of the poetry--the number of syllables per line, basically. An example to illustrate:
The / road / goes / e / ver / on / and / on
Down / from / the / door / where / it / be / gan;
Now / far / a / head / the / road / has / gone
And / I / must / fol / low / if / I / can.
There are eight syllables per line x 4 lines= 88.88, also called Long Meter.
If you are someone with access to a hymnal that has a "metrical index" you can discover tunes which fit the words above. Probably best to use less well known tunes, to avoid unintentional humor! ("I'm Sorry, I Haven't a Clue" on BBC has a hysterical routine called "One Song to the Tune of Another," e.g. "My Old Man's a Dustman" sung to the tune of "Heartbreak Hotel.")
There is also an online resource at The Cyber Hymnal with MIDI files, metrical indices, etc.
I have only classified the following by meter so far. See appendices to ROTK for reference to page numbers.
Common Meter: (86.86)
"In western lands"
"I sang of leaves, of leaves of gold"
"I sit beside the fire and think"
Long Meter: (88.88)
Song of Berien and Luthien
"Snow-white, snow white, O Lady clear"
"Farewell we call to hearth and hall"
Frodo's Lament for Gandalf
My resource is the Hymnal 1982 of the Episcopal Church USA, and some of the tunes for Common Meter I've tried out include "Forest Green," "Kingsfold," Noel," and "The Third Tune." For Long Meter: "Dunedin," "Deo Gracias," "St Patrick's Breastplate," "When Jesus Wept" and "Danby."
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The Cyber Hymnal