How do you explain the Sefer HaChinuch method? And how can he be talking about tying with tekhelet when he says to tie a knot after 3 chulyot, whereas the Gemara states that one should tie on every chulya? It seems that the Chinuch only intended a method for white.
The Chinuch’s words are most sparing and thus leave quite a lot of room for interpretation.
Your argument (that the Chinuch is not as we currently explain) is based on the Gemara
(Men. 38b) in which Rava says to knot every chulya – likshor al kol chulya v’chulya. The fact is that the Gemara is inconclusive as to whether knots are required. The opinion that knots are required is based on the presumption that without knots one could not distinguish between the chulyot (i.e., they would be abutted and simply look like many wraps – see Tosafot [Men. 39a, s.v. lo yifchot]). Based on this, the Rambam, who holds that indeed all the chulyot are blue, does require a knot on every chulya; in contradistinction, R. Amram HaGaon, who holds that the chulyot are of alternating color (white chulya, blue chulya, etc.) determines that there is no need for knots on the chulyot – and only a knot at the beginning (what he calls “kesher tachton”) and a knot at the end (what he calls “kesher elyon”) – presumably to hold the wraps in place. (The above explanation is based on R. Y. Rak’s article in Techumin, see: https://www.tekhelet.com/pdf/rak.pdf, p.21).
From all this I believe one answer to how the Chinuch (or anyone for that matter) could have a method without a knot on every chulya is to say that the knot on every chulya is simply not a requirement and, like R. Amram Gaon, the Chinuch does not hold by the requirement.
However, there is a way we can explain the Chinuch, and still uphold Rava’s statement. Returning to the Tosafot, they explain their method by noting the exact language of Rava when he said “al kol chulya v’chulya” – that is to say on every 2 chulyot (chulya v’chulya). The Gra then explains that this applies to the minimum amount of 7 chulyot, but if we went to the maximum amount of 13 chulyot then we would still keep the 5 knots but merely increase the number of chulyot between knots. The Gra then expands the chulya count to 4-4-4-1 (from Tosafot’s 2-2-2-1). (This explanation of Tosafot and the Gra is based on R. Tavger’s write-up found at:
http://www.tekhelet.net/diagrams/GraTyingSource.htm). In my humble opinion, the Chinuch is simply a variation on this theme, expanding to 3-3-3-4 (instead of 4-4-4-1). One could justify the Chinuch’s method in that there is an issue of keeping the wraps between the knots as (close to) equal as possible (Rosh, Hil. Tzitzit 15 – see R. Rak p.22, text of n.108).
You state that the Chinuch is providing the method for tying white – indeed R. Rak also questions whether the Chinuch’s method is applicable to tekhelet (see R. Rak, n.105). Nevertheless, if it was only for white, and not for when tekhelet is available, why go to the trouble of telling us in terms of chulyot, which with white only can not be recognized? (Of course you may answer “siman de’alma”, a “remembrance”, however, what kind of a siman or remembrance is there when you can’t see any chulya!) Now this is a question that has long bothered me, and seems to imply that the Chinuch made some kind of demarcation on each chulya, either like the Radziner, or perhaps like the R. Amram Gaon’s knot which has the shamash go through itself. However this is really squeezing a lot into the words (or lack thereof) of the Chinuch. I will leave it as “tzarich iyun”.
– Mois Navon