Latin, Jim, but not as we know it

Here's one for the latinists among you. It's from the Sarum Tonal.

Omnis antiphona primi toni que incipit in desolre descendendo in cefaut et statim saltat ad gesolreut per efaut, et postea ab efaut per gesolreut vel sine gesolreut ascendit ad alamire.

Well I have wasted a lot of time trying to figure this out. Though it has been fascinating in a rather frustrating way. It didn't take that long to discover that all those strange terms are names for the degrees of the musical scale.

It is common knowledge that what we call our 'major' scale derives from the work of a Benedictine monk, Guido d'Arezzo (d.about 1050), who worked out that each half verse of the hymn to St John the Baptist ascends a step, and thus might provide a sort of musical mnemonic.

UT queant laxis
REsonare fibris
MIra gestorum
FAmuli tuorum
SOLve polluti
LAbii reatum, Sancte Joannes.

So, Ut, Re, Mi, Fa, Sol, La. Which ought to ring a bell. If you substitute Do for Ut, and add a Ti (to drink with jam and bread), it will bring us back to Do.

But d'Arezzo's scale (and 'scale' means ladder) didn't include the Ti, so only had six notes. This is sometimes known as the Hexachord, or the Scala Aretina (Arezzo's ladder). Long before Messrs. Rodgers and Hammerstein got to work with Do a Deer (which seems to suggest granny-bashing), many composers had used the simple Hexachord as a cantus firmus for very splendid music—Palestrina, for instance, wrote a Hexachord Mass. Indeed, the challenge was precisely to make something extraordinary from what is really quite a dull motif. But this dull motif was recognized by all as the most important building-block of music, and therefore composing explicitly to it was an act of homage to d'Arezzo.

The point of the hexachord is that it is not necessarily pitch-related: what we call 'keys' are all applications of the same hexachord with Do (or Ut) being really at any pitch you want. As time went on, the seventh degree of the scale (which we call Ti) was added, and so the hexachord ('six notes') became an Octave ('eight notes'), with a Do (or Ut) at each end. And, voila, we have modern music.

Once you have musical instruments, though, then fixed pitch, a common agreement as to how to judge a note, is going to be important. A wind instrument, still less an organ, cannot adjust as the human voice can, to any pitch. They are strictly limited by physical constraints of holes along a tube. And, since the West has liked to use musical instruments in church for a very long time, this is no doubt why the quarter tones so typical of, say, Greek chant, are not a feature of Western music. If an organ is going to play with a crumhorn, then both need to agree as to pitch, or they will play out of tune with each other. Ut re me fa sol la will not help them here.

And so the mediævals came up with names for the various degrees of pitch, which were really concretized (sorry) elements of d'Arezzo's scale. To wit:
Alamire
Bemi
Cesolfaut
Delasolre
Elami
Faut
Gesolreut.

And it won't take a genius to work out that this corresponds to our modern and simpler pitch names of A, B, C, D, E, F and G. No doubt the reason that Alamire does not sit where Ut does, but rather on La, is that so much liturgical music, Gregorian Chant, preceded the Scala Aretina in composition—though not notation—and tends more commonly to 'home' on what we call A, just as our 'minor' scale does. But that is to get into the question of 'modes', which is another matter.

What intrigues me is the names the mediævals gave to the pitches. Perhaps here one of you can help. If you subtract the first letter or two, then the names actually have elements from the Scala Aretina:-
(A-)la-mi-re
(Ce-)sol-fa-ut and so forth.
But what do they refer to? I wondered whether they might represent the corresponding snatches of the hymn Ut queant laxis, but I can't make them fit. In fact, they don't seem to have much shape at all.

But we can at least now translate that puzzling bit of Latin at the top of this post:
All antiphons of the first tone which begin on D, then go down to C and straight away jump up to G by way of E &c.&c.