Why are there rising/falling tones in English-language chanting? I gather from reading around that the rising/falling pitches that appear in Pali chanting by Thais
is due to the tones in the Thai language.

But when I see English-language chanting by monks from a Thai tradition,
they still have rising/falling tones.
See excerpt from the Abhayagiri Chanting Book,
with the tones marked:



It doesn't seem to match the equivalent Pali.

Is there a system that they're using,
such as shoehorning the Thai tone rules into English syllables?

Or are they just making it up entirely?
Peter Harvey in an Introduction to Buddhism says of chanting


  [chanting] has a rhythm which encourages the mind to flow from word to
  word [but ..] usually lacks melody which might encourage some words to
  become distorted


Also


  chanting is neither singing or a monotonous dirge [...] it holds
  interest with it's minor variations of pitch and rhythm


So from this I would suggest that the tone of chanting is an aid to memory and interest and doesn't, in English, have a semantic content deriving from the original language. Of course Thai being a tonal language (i am not any kind of expert here) the tones may well have a more semantic content.

I guess less charitably you could say we are making it up a bit. Certainly from my own experience chanting within the Triratna community we have a tone they we vary widely from  and add harmonies to on occasion. The the Padmasambhava chant we actually got the words wrong too initially (50 years ago) but we have just stuck to our original bad interpretation. We kind of like it.