As I have tied various Turk'shead patterns over the years, and taught them to others along the way, it's become easier and easier for me to see the relationships, both similarities and differences, between the various singleline knots that I concentrate on.
I used to refer to the basic patterns (always begun with a threelead knot) as "one up" or "one down", because there is always a difference of one between the number of bights (edge loops) and some multiple of the number of leads. Thus, a threelead Turk'shead with four loops 'round the edge is "one up", and one with two is "one down". (see the 5x4 and the 5x6 pages  the first two pictures on each show this very well.) Expanding the number of loops, a threelead with five loops is "one down" (3*21=5) and a threelead with seven is "one up" (3*2+1=7). The expansion to more leads with a knot of this sort will always result in the same numerical relationship, i.e., a threebyfive will turn into a fivebynine, then into a sevenbythirteen, and soon. (Note that I began with the statement that I used to call them that. It turns out that "one up" is the same as "two down", when you're looking at a threelead knot. More on this later.)
The next pattern, in my hierarchy of knots, is the "two up" and "two down" pair (+/2), which I exemplify with the fivelead knots, seven bights and eight bights, respectively. All of the fivelead knots are either "one up/down" or "two up/down", regardless of how many bights there are... a fivelead knot with twentythree bights is made with the exact same beginning as a fivelead with eight bights, thirteen bights, eighteen bights, etc.
The beginning of this knot looks different, because it starts out with multiple strands in the second pass before the running end is tucked through, resulting in an oddlooking threestrand braid, the second strand of which is doubled for a fivelead (tripled for a seven, quadrupled for nine, etc.) and the first strand is picked up in the interweave at every subsequent pass.
That's all one needs to know, unless wider knots are the goal. A fivelead knot can only have the four "starts", because every fifth bight along the edge is a repeat, and multiples of five selfterminate before the pattern fills in. (You can have ANY number of leads and bights, literally, but not with a "single line" Turk'shead. It just requires multiple pieces of cord or whatever material is used.)
For a sevenlead knot, there are two more possibilities, e.g., sevenbyten and sevenbyeleven. (The eights, nines, twelves and thirteens are started with the "one/two up/down" approach discussed previously.) These two, the "three up/down" (+/3) patterns are nontrivial, compared with the others, because the expansion occurs in two stages and the transition is easily mismanaged. To create a sevenlead with ten bights (7x10), start with a 3x4 and begin to double it. On the second doubling pass, interweave the running part between the beginning of the paired strands and continue from there. To create a 7x11, begin with a 3x5, start doubling, and interweave, again, on the third doubling pass. (It might be physically possible to tie a 7x4 by starting with a 3x2. Knock yourself out. :)
Past this point use of the "up/down" designation is potentially misleading, since more complex knots will have other relationships. Perhaps a better term would be "compound expansion". Consider an elevenlead with sixteen bights, for example, which is a result of expanding a sevenlead by ten bights, which results from expanding a threelead with four bights. There is no simple way to label these jumps, but I intend to make them as clear as possible for those who wish to tie them "in hand", as I do.
I don't much care for even numbers, as you may have gathered, reading my various pages here. The next pattern I'll address is the ninelead, for which there are six (not eight!) possible beginnings. Nine is not a prime number, so the possibilities exclude knots with either twelve or fifteen bights around the edge. (Or any multiple of three.) The first complementary pair (+/1) are just the natural progressions from previous knots. The second pair (+/2) are tied directly by tripling the second pass before tucking the end and beginning the interweave. The third pair, though, are interesting because they start out as fivelead knots, made with the +/2 patterns. A 5x7 or 5x8 is doubled for two passes, then the running end is interwoven to pick up the first strand and the expansion continues from there, to create the 9x13 and 9x14 patterns.
The majority of the ones pictured below were tied in hand without a pattern or guide, using various beginnings and expansions. (Well, not the Mobius strip! That's another discussion, altogether.)



Getting back to the patterns and how to understand them... the basic Turk'shead, with three leads, can be thought of as a special case of a more general class. Rather than think of a 3by2 knot as "one down", think of it as "two up". If you increase the edge loops by adding another zigzag of unders and overs, you add three loops. That makes it into a 3by5. Think: 3+2=5.
If you create a 3by4, similarly, adding a zigzag turns it into a 3by7. Here, again: 32=1, and 62=4, and 92=7. Generally, in other words, 3x+2 or 3x2 describe the number of bights 'round the edges of these threelead knots.
Now it gets interesting. In my exploration of the higher (odd) numbers of leads, I discovered that the number of times you go around after the first pass and before beginning to tuck under it determine the number of leads for the "two up/down" knots. If you tie a threelead, there is a first pass, a single crossing pass, and then a tuck under the first pass. To do a fivelead, there is a first pass, again, two crossing passes, and then a tuck. The tuck is either under the first and across the rest (two up) or across the rest and under the first (two down), and each zigzag adds as many loops 'round the edge as there will be leads in the completed knot.



