Say you're sitting on a comfy couch at home, sipping a nice cup of something, gazing at your trusty but inevitably worn copy of Davison's "A Handweaver's Pattern Book" and come across a lovely twill draft. Except you have several more shafts than the draft asks for. What do you do?
I used to modify the draft so I would use strange number of shafts, or just try to use all 16; sometimes the twill became more intricate, sometimes I just wasted time and went back to the original draft, but now I have another option, to turn it into a tied weave.
Davison's twills don't have long floats, so structurally there is little advantage. (Read more about floats below.) But if you imagine one warp end to be the smallest vertical unit and one weft end the smallest horizontal unit in your twill, i.e. 1 end x1 pick is the smallest square on your graph paper, in Summer & Winter the smallest unit is 4x4, (I also read 8x8 in elsewhere,) Bergman, 8x8, Quigley in diamonds, 6x6. This means if I use tied weaves, I can weave with fine yarns and still show the lovely twill "lines" in a reasonable size, or you can weave a reasonably-sized piece and show a whopper of an undulation, for example.
Let me show you.
This is an Undulating Herringbone draft, top left, Treadling III, on page 53. To weave this as a twill, it requires 86 warp ends in one repeat; with my usual 18EPI merino, one repeat is 4.8 inches; in my 2/20 mercerized cotton at 36EPI, a mere 2.4 inches.
Top right quadrant of draft converted to Summer & Winter
Bergman, as I understand it, is 3-tied single unit weave threaded 1-P-2-P-3-P-1-P-3-P-2-P-1-P-3-P where P is the pattern block shaft, and treadled in a similar pattern; 1-2-3-1-3-2-1-3, if you get my meaning. This draft requires 1376 warp ends.
Top right quadrant of draft converted to Bergman
Quigley, again as I understand it, is a 4-tied single unit weave threaded 1-P-2-P-3-P-4-P-3-P-2-P, and to get diamond-shaped tie-down, is treadled twice, in points. This requires 1032 ends.
Top right quadrant of draft converted to Quigley
You can also split the threading and treadling units to create a smaller shapes, e.g. in Quigley by threading 1-P-2-P-3-P-4-p-3-p-2-p where p is another pattern shaft different from P. In my P2P piece, I completely ignored everything and switched pattern shafts and colors in the warp and weft at whim.
So the best argument for tied weaves is you can enlarge the visual element. Theoretically, it is structurally sound as well, allowing adapt my "pattern" for different purposes. (But read on about floats.)
Now the cons and issues.
In Summer & Winter, the tie-down is in plain weave, so A-side colors are the opposite of B-side. In Bergman, the tie-down is a 1/2 twill, so A-side is not exactly the opposite of B-side, and to me, the side with the longer float definitely looks nicer. (Though Pat disagrees.) To suit my taste, I need to balance the number of lifted pattern block ends roughly the same as those not lifted. In Quigley, my software automatically selects a 1/3 twill for tie-down, but I could convert this to 2/2; in theory I like this better, but depending on the pattern, this may make the two sides too similar and render the finished piece boring. Further investigation is needed.
If the blocks are too narrow or color changes too frequent, one's eyes are distracted and the tie-down shapes are hard to see. So this balance also requires further investigation.
In Bergman, the longest float is seven threads in the weft; in Quigley with 1/3 twill diamond tie-down it's a whopping 11 threads; Quigley with 2/2 twill diamonds is 7. To me, this, and the two or more shuttles required to weave these structures scream out, "Turn draft!", particularly if I am considering weaving a scarf. So now I'm looking into a turned Bergman and using thick and thin warps vs uniform warp ends to study the hand. (Whereas Bonnie Inouye would shout, "Don't bother, go to turned taquete!" and I might.)
Oh, if you want to weave that set of lovely napkins/serviettes, turn the draft; it would go so much faster. And as long as you're at it, the original Davison twill draft, woven as drawn in, would make a spectacular set, don't you think? (And by that, I mean, turn it into a tied weave first.)