(I should have been sewing, but the pins are too fiddly for the wrenched wrist. So I have been reading, instead.)
I started with Gertrud Grenander-Nyberg's Lanthemmens vävstolar (1974, ISBN 91-7108-076-7). On pages 287 f she writes about double pulleys (my translation; my comments in italics):
For plain weave, threaded on four shafts, there was a pulley type seldom used nowadays, namely two pulleys mounted on the same axle. The pulleys can be of different size. For the tying, the outer shafts were tied with the cord over the bigger pulley, which made the shaft tieup clearer (simpler?)
Important words here: CAN be of diff size. In most (all?) pictures I've seen they have been of the same (or so near the same as possible) size. (Had they been different size, they could have been called "horizontal dräll pulleys"? Has anyone seen an example of that (ie with more than two pulleys)? Edit: here.
The Finnish textile technician Helvi Pyysalo has showed that two pulleys on the same level can be used for four-shaft even-sided twill by connecting shafts one and three over one pulley and shafts two and four over the other pulley.
With this method follows that, when the shafts are lowered in pairs in four different combinations, the cords on both pulleys will move, which is (recommended, necessary? - "good") to get a clear shed.
I don't understand that: does she say that for EVERY shed on a CB loom "both cords should move"? If so: why?? Or what is it I don't understand here? With one pulley and horses that is not always the case, but sheds can be good anyway.
This can happen not only with two pulleys on two axles on the same level but also with the pulleys mounted on the same axle like on a double pulley.
Is there a difference in appearance in the two described above? (for "double pulley" she uses the Swedish "parblock", so in the text there is a distinction between the two types here.) ETA: see picture here.
It is likely that the double pulleys were used in this fashion in older times for weaving even-sided twill. With the pulleys on the same level the upper tieup was lower than if it was accomplished with several tools on different levels. Double pulleys were therefore an advantage when using the oldest looms which have low castles. This can be an explanation to the numerous double pulleys from the older times.
Nowadays, for four shafts, usually only one of the pulleys is used in combination with horses.
Same author, 25 years later, in Linnelärft i Ångermanland (2001, ISBN 91-7108-480-0), has slightly changed her ideas.
Background, about "premielärft" - roughly "prime quality linen cloth": in 18th century the "powers" divided the country and decided Ångermanland was the region to grow flax and (spin and) weave linen. To make the people more enthusiastic about it, certain standards were set, and those who fulfilled them got extra pay.
Anyway. According to Grenander, weavers of prime cloth were specialists, using special looms. Those looms were often shallower than "allround" looms - sources from the time state that an allround loom was usually about 2 1/2 aln (about 5 ft), while linen looms were only 2 aln (4 ft).
The reasoning behind that is (apart from the fact that German linen looms often were shallow) that the "free" warp is shorter, and therefore less exposed to wear (p 99).
Something I don't understand here: as linen is not elastic, shouldn't it benefit from a longer "free" warp, by distributing the stretch over a longer section?
(She goes on to say that plain weave needs less loom-depth than multi-shafted cloth, which is of course true.)
She also thinks that the double pulleys, at least those with short uprights, were used primarily by these specialist weavers.
(To me it is not clear exactly how she came to that conclusion.)
On p 103 she writes that, because of the fineness of the prime cloth, the warps had to be distributed over four shafts, and therefore the double pulleys were "the most efficient". Thus looms with double pulleys were, together with very fine reeds, the most typical tools of a "prime" weaver.
I'm still unsure of why double pulleys are "the most efficient"...
Anyway, it is apparent that she has changed her mind over the 25 years between her thesis and her last book - from thinking that double pulleys were, in effect, "old-fashioned", to thinking they were the mark of a specialist.
Just for curiosity: "prime linens" came i 8 classes. They were (at least most of them) 1 1/2 aln (3 ft, or 88 cm) wide.
For class 1 there were 2720-2920 ends ( or 30-32 ends/cm); for class 8 (the finest) there were 4120-4 320 ends (or 45-48 ends/cm). (from Uppfinningarnas bok, band 6 from 1873.
And this, my friends, was all handspun singles!