Copyright issues with using someone else's arrangement of old tunes

I can't say I have an answer for you but...
 
I am just now spending a fair amount (2-3 weeks) of time getting all my copyrights, publishing, ISRC codes, Soundexchange, etc all up to date and absolutely correct just "because". I figure, might as well do it right now so if anything good comes out of it I'm good to go rather than playing the "I shoulda" game.
 
It's a fair amount of work now because I let everything lapse since, like 1985 :)
 
But not so much once you are up to date.
 
As musicians and creative artists, we don't get our fair share very often, so in the event that I do, I don't want to have to tall my wife "d'oh!"
 
What you are doing sounds really neat so I would say, check into it and contact Alexander, then you're good to go.

That is the problem with folk song collections in general, and song books derived from other books or manuscripts. A work that is in the public domain can be copied (or performed) freely without any kind of acknowledgement. You can make your own arrangement of a public domain work, and you will then own the new copyright on your own arrangement, but not on the portions that are substantially similar to the original. So the melody of Greensleeves can be arranged any way you want, but no one can copyright the melody. If someone uses your new arrangement, however they are infringing your new copyright. That is the reason Project Gutenberg et al will only accept submissions that can show they have been located and correctly copied or transcribed from an original published version old enough for the copyright to have been expired given all available extensions (whether actually applied for or not).
 
This can get particularly crazy when you have an "author" who acknowledges that he did not author the work, who records someone else performing it and then claims the copyright of  the composition when he records or transcribes it. Neither of those actions is supposed to result in a copyright to the underlying composition as opposed to the sound recording itself. It is a "slavish copy" in the language of the case law, and lacks sufficient originality to qualify for protection. In addition the right to a copyright should belong only to the actual creator of the work, unless it is a work for hire in which case it belongs to his employer, which principle should bar a collector from being able to register copyright to a composition he did not compose. Copyright was never intended to be a finders keepers system, but the law absolutely hates the idea that some particular individual does not own every identifiable thing, and you are taking your chances in that system.
 
https://www.law.cornell.edu/copyright/cases/36_FSupp2d_191.htm
https://www.techdirt.com/articles/20120201/00472317614/lomax-jukebox-going-digital-is-great-news-lets-not-forget-that-he-claimed-copyright-cultural-works-that-werent-his.shtml
http://www.peterirvinelaw.com/pdf/Folk_Music_Copyright_PD.pdf
 
In your case you are going to be depending on the author's apparent statement that does not include a specific guarantee that he did not author any particular part of the work upon which you are basing your derivative work. A scholarly book should include references to the original sources, and you would be safe if you could find a copy of the original source or find the same work in an older compilation. 
 
 

Actually upon reading the blurb in the book more closely it would appear that he only had the "fragmentary melodic lines" and arranged them to the best of his knowledge of the time, style and character of the Provençal school. So what that means is that all of the pretty sounding parallel fourths, fifths and octaves the melody are harmonized with are his stab at estimating how the melodies would have been harmonized at the time, and thus his own work. So I guess I would have to get some kind of permission unless I reharmonized them myself.