Calculated setts and real setts
Sett, in its essence, is the number of warp threads per unit of distance. You can pick the unit of distance of your choice; here we will stick with inches.
Sett = # warp threads / inch
One can measure the sett of a piece of cloth with a magnifying glass (to see and count the threads) and a ruler (to measure distance), or a linen counter, a tool designed for the job.
Here is the view of a piece of cloth though a linen counter. Counting the warp (vertical) threads or ends, one gets about 22 threads in 1 inch, or 22 epi. One can also determine the weft sett, by counting the horizontal (weft) threads, or picks, and get about 18 threads in 1 inch, or 18 ppi.
While so doing, one notices that each pick goes above 2 and below 1 end: this is a 2/1 twill. Furthermore, the twill diagonal starting from the top left corner is a little too steep to reach the bottom right corner, confirming that this cloth was beaten a little looser than square (ppi < epi).
Note that to get a more accurate measure of the sett, one would count the threads over a couple of inches or more.
If one doesn't want to count so many threads, one could figure out how many ends there are in one repeat of the cloth and measure the length of the repeat. With these measurements,
Sett = (# ends/repeat) / (length of 1 repeat)
On the picture above, the repeat has 3 threads. As indicated by the pink lines, it is 1/8 inch long. With these numbers, we get a sett of 24 epi, which is pretty close to the first sett measure (22 epi). With the first measure, we get the average sett over 1 inch of cloth. With the second measure, we get the sett over 3 threads.
Sett calculations
Let's start with the latest sett formula, and express the length of the repeat as
length of repeat in inches = (length of repeat in # of yarn diameters) x (diameter of the yarn in inches)
Measuring the diameter of the yarn as such is not easy with standard home equipment because it is usually a small number. We can get to it indirectly by measuring how many yarn diameters fit in 1 inch (=diameters/inch, or dpi) and take the inverse. The sett formula becomes
Sett = (# ends/repeat) / (length of repeat in inches)
= (# ends/repeat) / [(length of repeat in # of yarn diameters) / (diameters/inch)]
Sett = (diameters/inch) x (# ends/repeat) / (length of repeat in # yarn diameters)
This formula is not particularly useful for measuring setts; however it is the basis for calculating setts. It neatly separates the size of the yarn, expressed as diameters per inch, or dpi, from characteristics of the woven structure (number of ends in the repeat, length of repeat expressed in number of yarn diameters). Let's call STR the number characterizing the woven structure:
STR = (# ends/repeat) / (length of repeat in # yarn diameters)
Standard sett calculating formula
This is the formula from the industrial textile designer playbook. It assumes that the same yarn is used as weft, and at the same sett (ppi=epi). Let's have R = number of ends per repeat and C = number of times a pick crosses from face to back or back to face in one repeat.
Sett = (diameters/inch) x STR, with STR = R / (R + C)
In other words, the standard sett calculating formula provides a way to calculate the length of a repeat in yarn diameters: R + C.
Can we make sense of this formula?
Let's represent a yarn with 10 diameters per inch, or 10 dpi.
Let's weave plain weave with this yarn, with black yarn in the warp and orange yarn in the weft.
Let's take a cross section of 1 inch of the cloth through the exact middle of a weft thread.
Warp threads will appear as 1/10 inch black circles and the weft thread will appear as a 1/10 inch wide orange ribbon.
It is easy to see from the diagram that a repeat is 4 diameters long in plain weave, 2 for the warp threads and 2 for the weft thread interlacing with the warp.
STR = R / (R + C)
= 2 / 4
STR = 0.5
This makes good sense when the cloth is represented as above, with the warp treads not bending at all and the weft thread doing all the bending.
However, in balanced cloth—which this is supposed to be, as per the starting assumptions—one would expect warp and weft to be equally flexible.
Here is a more realistic representation of the same cross section:
With this representation, it is less obvious why a repeat would have to be 4 diameter long.
In particular, could the warp threads be packed tighter? Well, yes!
On this diagram, the top cross section shows the maximum packing with stiff warp threads.
The bottom cross section shows flexible warp threads spaced to allow only the thickness of the weft thread between them.
What sett is represented on the bottom cross section? Let's count how wide is a repeat, measured in yarn diameters. Eyeballing the grey circles, it's about 3.5 diameters.
Can we calculate this number?
The pink triangle indicates the geometric relationship between the threads.
By construction, the short side of the right angle, a, is 1 diameter, the hypotenuse, c, is 2 diameters, so we can calculate b, the length of the third side (thank you Pythagoras):
a^2 + b^2 = c^2;
a= √(2^2 - 1^1 )
=√3
The width of a repeat in the bottom cross section is thus 2xb = 2√3 = 3.46 diameters.
STR = R / (R + C)
= 2 / 3.46
STR = 0.577
How does these calculated setts compare with commonly used setts?
Let’s look at an example, say, 8/2 cotton woven in plain weave.
8/2 has a grist of 3360 ypp, which corresponds to 53 diameters per inch. The calculated sett for 8/2 in plain weave is 53/2 = 26.5 epi.
However, for sturdy napkins, I sett 8/2 in plain weave (using 8/2 as weft) at 20 epi, a fair bit looser than 26.5 epi.
Looking at a close-up view of the theoretical cloth at the calculated sett, it is not surprising that the real world sett is looser than the calculated sett. The question is, how much looser?
For my sturdy napkins in 8/2 in plain weave sett at 20 epi, the real life sett is 20/26.5=0.75 or 75% of the calculated sett.
We can also use the calculated sett to compare setts from different sources, and for yarn substitutions.