A Folk Song can be any song sung in a Folk Context, by which I mean a Folk Club, Singaround, Folk Festival, Folk Concert, Folk Party, Ceilidh, Morris Moot, Folk Custom, or any otherwise denominated Folk Event. Such a song need not exclusively be a Folk Song - at other times, and in other contexts, the same song might be something else entirely, a Pop Song for example, or yet a Jazz Standard, or even an Operatic Aria. All of these I have heard sung in Folk Contexts and have, by dint of that context, accepted them as being Folk Songs. So what makes it a Folk Song is the context in which it is being sung.

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As for the 1954 definition:

Folk music is the product of a musical tradition that has been evolved through the process of oral transmission.

No musical tradition has ever evolved without the process of oral transmission.

The factors that shape the tradition are: (i) continuity which links the present with the past; (ii) variation which springs from the creative impulse of the individual or the group; and (iii) selection by the community, which determines the form or forms in which the music survives.

All musical traditions are thus shaped - from Hip-Hop to Free Jazz, from Karaoke to Gamelan, from Drum & Bass to Dub Reggae, from Elvis Impersonators to Crusty Didgeridoo Players, from Trad Jazzers to George Formby Enthusiasts, from Neo-Medievalists to Death Metal Headbangers. This is the very nature of musical tradition, simply to be utterly dependent on the people playing it, who, in being fully conversant with the past are nevertheless re-determining it for both themselves and thus assuring its future survival.   

The term can be applied to music that has been evolved from rudimentary beginnings by a community uninfluenced by popular and art music and it can likewise be applied to music which has originated with an individual composer and has subsequently been absorbed into the unwritten living tradition of a community.

All music has evolved from rudimentary beginnings and I very much doubt there has ever been any such an uninfluenced community except in the twisted fantasies of academics who postulate such bullshit. Otherwise - all music has thus originated and been absorbed and transformed. In the composing of a Pop Song, for example - an idea becomes a composition, which is then further interpreted by a community of arrangers, session players, engineers and producers ever before the finished product hits the shelves. There we have The Folk Process in a nutshell. Was anything ever unwritten? Hell, even The Copper Family sing from a fecking book!   

The term does not cover composed popular music that has been taken over ready-made by a community and remains unchanged, for it is the re-fashioning and re-creation of the music by the community that gives it its folk character.

No music ever remains unchanged, however so conveniently one might qualify the word change; each performance is a renewal within the expectations of its community which are further transfigured by its corporeal & empirical experience. A performance of Purcell's Dido and Aeneas in 2009 will be, out of necessity, very different from a performance of Dido & Aeneas within the same community from 40 years earlier. Ditto a rock band comprised of variously talented 14-year-olds going over Eleanor Rigby in a garage are re-fashioning a music, re-creating it, and giving it its folk-character. Likewise, a Folk Singer adapting Eleanor Rigby to their own needs and abilities for performance at his local Folk Club is effecting a transformation over a given piece of music, thus giving it its Folk Character.   A Karaoke singer singing Eleanor Rigby is doing exactly that too, likewise the worker who whistles the melody of Eleanor Rigby as they go cheerfully about their daily business, or else the schoolboy singing Eleanor Rigby as he walks to school.

So is there much of a difference between the 1954 and Horse definition after all?

As for what makes a Traditional Song, the lines are clearer with respect of a canon of a material collected, recorded, catalogued, cut and dried, sourced and analysed, numbered, indexed, with occasions, performers and variations duly noted. Sometimes Traditional Songs might be sung as Folk Songs, other times they might be sung as Classical Songs, or as Rock Songs, or Wyrd-Folk Songs, or Jazz Songs, or Pop Songs - I often sing them in the context of Free Improvisation (check out Wife of Usher's Well on my Myspace Page). Anything is possible with Traditional Song - but the song, essentially, remains the same. That said, even I get irked when I hear Child #10 sung to the melody of Child #1, but there are times I might sing it with no melody at all...