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Yarns of different materials were used to weave the garment: If a Tallis was woven from yarn/threads of different materials, such as if the warp was made of wool and the woof from silk or other materials, or vice versa, then every G-d Fearing Jew is to avoid wearing this Tallis, even if it has wool Tzitzis tied to it. Likewise, even those that are lenient to tie wool strings to any material garment, and hence in this case as well they tie wool Tzitzis to the mixed material garment, nevertheless it is proper for them not to recite a blessing upon wearing such a Tallis that contains yarns of silk or other materials, even if it has wool Tzitzis tied to it. Certainly if the garment is entirely made of silk or other materials, a blessing may not be recited if it has wool fringes. Rather one is to recite a blessing over a different Tallis and have in mind to also exempt this Tallis. [Nevertheless, from the letter of the law one may say a blessing over this Tallis, if he so chooses. All the above is regarding if the actual garment is made of threads of two materials, it is however permitted to strengthen the holes and sides of the garment using other material threads, and doing so does not change its material identity, and so is the custom.]
If the yarn used to weave the Tallis is made of mixed material: If one wove yarn from different materials and used it to make a four cornered garment, then the garment is viewed in accordance to whichever material is the majority. Thus, if one mixed raw material of wool and silk or other materials, wove this material into yarn, and then used it to weave a Tallis garment, then if the majority material is wool, then it is viewed as a wool garment and one is to use wool strings for the Tzitzis. If the majority material is silk, or another material, then the Tzitzis are to be made from that material. If the garment does not have a majority material being it is made of 50% wool and 50% of another material, then one is to use wool strings as Tzitzis. Nevertheless, every G-d fearing Jew is to avoid wearing such a garment that does not have a majority material.
B. Fringes of mixed materials:
All four sets of fringes are to be made from the same material. [If the Tallis contains fringes made of different materials, it may not be worn.] For example, a silk Tallis is to have only silk Tzitzis, and not some of wool and some of silk. [Likewise, a cotton Tallis is to have all its fringes be either of cotton or wool, and not have some of wool and some of cotton. If a single string was spun or intertwined from more than one material then some write its identity follows whatever is its majority material.]
Fringes made of wool and linen: According to all opinions, if two of the four strings are of wool and the other two of linen it may Biblically be used on any material garment. Nevertheless, practically it is Rabbinically forbidden to use fringes that contain wool and linen due to Shatnez. This applies whether the string itself was woven from wool and linen, or one is using some wool strings and some linen strings to make the braid of the fringe. Regarding if one may make on a cotton garment two corners of wool strings and two corners of linen strings, see Q&A!
The Tzitzis: All the fringes of the Tzitzis are to be made of the same material. Thus by a cotton Tallis, one may use either all cotton strings, or all wool strings.
The Tallis: Initially, one is to use a Tallis that is made of a single material. If the actual yarn/thread used for the sewing of the Tallis was made of mixed material, then one follows the majority material in order to identify which material Tzitzis it requires. If the actual yarns/threads are made of a single material but one used yarns/threads different material to make the garment, then one is not to wear such a garment, irrelevant of the material used for the Tzitzis.
May two of the corners be made of wool and the other two corners of linen?
There is no Shatnez prohibition involved in having two of the four sets of corners made of only wool and the other two of the four sets of corners made of only linen and have them placed on a non wool/linen garment, as the custom is to be lenient to not consider this form of attachment as Shatnez. Nevertheless, perhaps the Tzitzis is questionably invalid if two sets of corners are made of wool, and two sets of corners are made of linen, being that all four sets are to be made of the same material, as explained above. However some Poskim explicitly write that one can make two corners of wool strings and two corners of linen strings on non wool/linen garments, and it is valid [according to the custom to allow wool strings on all material garments]. In any event, as explained in 2A, it is proper to beware to not use linen strings at all as Tzitzis, even when it does not contain a Shatnez prohibition.
Tallis material and Tzitzis material Chart