I am a fledgling member of the Scandinavian Weavers Interest Group of the Weaver’s Guild of Minnesota (a mouthful, thankfully shortened to “Scanweavers”). We meet once a month for show-and-tell, discussion, and a generally good time brightened lots of laughter and Jane’s unfailing gift of Dove dark chocolate.
Last month Judy showed us a weft-faced placemat that she warped with 5/2 cotton sleyed at 30 EPI. She was not happy with it — at that sett the warp did not completely cover the weft. To fix it before starting the next placemat, she tied two 10/2 cotton threads to each 5/2 warp thread, increasing the effective sett to 60 EPI.
The effect? The doubled 10/2 warp now hid the weft very well.
This result was counter-intuitive, or so I thought. Wouldn’t two 10/2 threads cover the same width as a single 5/2 thread? I knew that 5/2 yardage is half of 10/2 and thus yields twice the width.
The group responded with answers that undoubtedly would have been helpful had I been listening, but I was disoriented and confused. It seemed to be a simple geometrical problem: threads are just fibrous cylinders that ought to obey geometrical rules. I am a Ph.D. mathematician for goodness sake, why couldn’t I figure this out?
On reflection later, the confusion stemmed from two causes: First, after a certain age, brain calcification sets in; I had forgotten the fundamental relationship between volumetric and linear dimensions of similar figures: Volume grows as the square of its linear dimensions. For cylinders with circular cross-section (an idealized shape of a thread), volume grows as the square of the diameter. Second, I practice the craft of weaving in its byways — I’m a tablet weaver. Until recently I wallowed in ignorance of the fundamentals well known to most weavers, in particular the need to determine the sett when warping and before weaving. Tablet-woven bands are always tightly warp-faced by virtue of how they are woven. I just pull the weft after each pick and the warps scrunch together. The sett is always “right”, and therefore of not much concern beyond making sure that the combination of yarn and tablet count would produce a band of roughly the desired width.
Later, I cleared my brain and set to work analyzing. Here’s a summary.
The conventional sizing of cotton and linen yarn is expressed as a fraction such as 5/2. The top number, multiplied by 840, gives the yardage of a single ply in a pound. The bottom number is the number of plies. The higher the ply, the shorter the yardage. For example, 5/2 cotton yields 840 * 5 ÷ 2, or 2100 yards in a pound.
A thread is an idealized cylinder whose volume is length times $\pi$ times the square of the radius (half the diameter). I assume that the diameter of the thread cylinder is what determines the warp thread’s coverage. The volume in a pound is the same for all sizes. The key is this: To keep the volume the same, the diameter must decrease as the square root of the size. For example, 5/2 cotton must be $\sqrt{2}$, or 1.41, times the diameter of 10/2 cotton.
I reasoned similarly for other cotton sizes. The first line of this table shows the relative diameters of various sizes of cotton. To make it easier to compare with 10/2 cotton, I’ve normalized the diameter of 10/2 cotton to be 1.
The table confirms what Judy observed. A doubled 10/2 warp covers significantly more than single 5/2 warp (42 percent more, in fact). Even a single 10/2 warp would not cover as well. (I observed this when weaving Sami bands. In one case I used doubled 3/2 cotton pattern threads on top of single 3/2 background. The Sami weaving technique lays a pattern thread over two background threads. The pattern threads covered the background very well. But in another band a single 3/2 thread covered 10/2 background a bit sparsely.)
I wondered whether these calculations would be confirmed by examining my tablet-woven bands. I measured one band woven with each of the four sizes. The second line in the table shows the number of cards needed to weave one inch of width. The next line shows hypothetically the width resulting from a 100-tablet band, and the final line normalizes these widths similar to the first line. The actual results are remarkably close to, and thus confirm, my theoretical calculations.
So there you have it. Thanks, Scanweavers, for briefly re-kindling mathematical embers.