The method of tying 7,8,11,13 is one which was introduced after the loss of tekhelet (i.e., there is no mention of it in the Gemara). Many “reminders” were introduced into tying tzitzit to compensate for the lack of tekhelet – what the Ramban calls “the essential reminder”. So for example, we have the 5 knots, which is something introduced by the Midrash Tanhuma (closed circa 750 BCE – approximately 200 years after the redaction of the Gemara which makes no mention of 5 knots). This Midrash is brought by Rashi on the Humash where he explains that 5 knots, along with the 8 strings and the 600 gematria value of the word “tzitzit” serves as a reminder of the 613 mitzvot (see also Tosafot [Men. 39a, lo]). (The Radvaz [6:2] explains the 5 knots along with the four strings add to 13, which is gematria “echad”). Again this is something necessary when one doesn’t have the color of God’s throne (kisa hakavod) – though others hold they are of necessity even with tekhelet

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(see Radzyner, Sifrei HaTekhelet, Ptil Tekhelet, pp. 130-2). A similar argument can be made for the method of winding. The Gemara states that one must make chulyot (groupings) of 3 winds each; and that one should make between 7 and 13 chulyot. R. H. Schachter writes that the method of tying 7,8,11,13 was merely a reminder of how we used to tie when we had tekhelet. Also interesting to note is that if one ties according to the Gemara – 13 chulyot of 3 winds each – one arrives at 39 windings in total, precisely the same as tying 7+8+11+13 (=39).

The Radzyner believed that the requirement for 5 knots was of ancient origin as is the requirement to wrap chulyot of three winds. As such, he (as well as the Baal HaTanya) integrated the two requirements which he states should be done even without tekhelet. What they did was to loop the shamash around (ideally) every three winds, thus demarcating “chulyot”. Of course one runs into a problem when needing to tie a knot after 7 winds (seven not being evenly divisible by 3). Consequently the method requires one to break up the groupings of three winds when going around one of the 5 knots. (For simplicity’s sake we will discount the issue of wrapping the whiteshamash for the moment, which will be discussed in the next paragraph). As such, you start off with a double knot, you wrap 3 winds and loop the shamash [1st chulya], then wrap another 3 winds and loop the shamash [2nd chulya], then wrap 1 wind [to complete the requisite 7 winds] and tie a double knot. Now you wrap 2 winds [to complete the “3rd” chulya that was started with the 1 wind before the double knot]. This continues for the reset of the windings, breaking up the triplets around the double knots (as shown in the “Radzyn/Chabad” diagram).

Another idiosyncrasy of the method is the use of white for the initial and final windings. The Gemara states that one is to begin with white and end with white. This has been variously interpreted as can be seen from all the differing methods of tying. The Radzyner adopted the Rambam’s approach which calls for all the windings to be tekhelet except for the very first and very last winding. Thus, after the initial double knot, one make a single wind of white, looping this white shamash though itself. Then two winds of blue are made wrapping the blue shamashthrough the loops. These three winds together are considered the first “chulya”. (See the “Radzyn/Chabad” diagram for details).