Gemara

123a (Mishnah): If one was ripping a Tamei cloak to be Metaher it, once he rips the majority, it becomes Tahor.

123b - Version #1 (Rav Huna): If one tears the majority of a cloak, the resulting pieces are Tehorim only if neither is big enough to make a turban. If a piece is big enough, it is Tamei.

124a - Version #2 (Rav Huna): Tearing the majority is Metaher even if a resulting piece is big enough for a turban.

Zevachim 94b (Mishnah): If blood splashed onto a garment and it left the Azarah, we return it to the Azarah and launder it there. If this garment became Tamei outside the Azarah, we tear it (to be Metaher it), return it to the Azarah, and launder it there.

Question (Ravina) Question: If we tear the garment, it is not longer called a Beged, so there is no Mitzvah to launder it!

Answer: We tear it, but we leave connected enough cloth for a turban. (For laundering, it is still called a Beged.)

Objection: Rav Huna taught that if enough cloth remains connected to make a turban, it remains Tamei!

Answer: That is mid'Rabanan. Mid'Oraisa it is Tahor.

Menachos 40b (Rava, and R. Zeira): Chachamim forbade a linen garment, lest it tear within three fingers of the corner;

Since the strings were there before the corner was fixed, they are invalid due to "Ta'aseh", v'Lo Min ha'Asuy... (one must attach Tzitzis to a garment to which the Mitzvah applies).

41a (Rav Yehudah): If a garment was torn more than three fingers from the corner, we may fix it (and the Tzitzis are Kosher). If it tore within three fingers of the corner, we may not fix it.

Support (Beraisa): If a garment was torn more than three fingers from the corner, we may fix it;

R. Meir says, if it tore within three fingers, we may not fix it;

Chachamim permit fixing it.

All forbid bringing even a large piece of material from another garment and attaching it;

All agree that one may bring strings and attach them, on condition that they were not cut.

(Beraisa): If a garment is folded over, it requires Tzitzis (on the current corners, i.e. two are on the fold);

R. Shimon exempts.

All agree that if it was sewn shut, it is obligated.

Rishonim

Rambam (Hilchos Tzitzis 1:18): If the corner with Tzitzis was torn outside of three fingers, one sews it in its place. If it tore within three fingers, he may not sew it.

Rosh (Hilchos Tzitzis 11): Rashi explains that we are concerned lest some strings used to fix it will be left hanging on the corner, and used for Tzitzis. If so, if a wool Talis was torn within three, one may sew it nowadays, for it is not common to sew with wool threads. People sew only with hemp, which is Pasul for Tzitzis. Rav Amram explains that if it tore outside three one may sew it, for three by three is considered a garment. Less than three is not considered a garment. It is as if it is not there. Even if he sewed it, it is as if it is cut. If he attached Tzitzis, they do not exempt the Talis.

Tosfos (Menachos 40b DH Shema): Rashi explains that we are concerned lest a linen garment tear within three (fingers) of the corner, and he will sew it with linen threads, and leave them for Tzitzis, and this is Ta'aseh v'Lo... He should have said that it is not Lishmah. This is not a problem of Ta'aseh.... In a Beraisa, Chachamim and R. Meir argue about whether one may sew it if it tore within three. Even Chachamim forbid only due to Kil'ayim (lest one sew it with wool). If a garment (on which Dam Chatas splashed) left the Azarah, even if three by three remained but it is not enough for a turban, it is not considered disconnected. The Gemara in Chulin proves that the Shi'ur for a turban is more than three. There is different, for the entire garment tore. Here, only a piece tore, therefore it is not Pasul.

Chazon Ish (Mo'ed 3:20 DH 40b): Tosfos learns from the Mishnah that is Metaher as long as the majority was torn. I.e. even if three remains, it is Tahor. Even though enough remains to receive Tum'ah, it is not connected. Rashi and Tosfos hold that even the remainder is not Tamei from having 'touched' the whole garment at the last moment it was whole. If enough for a turban remains, it has Tum'as Medras, even though it is less than three by three Tefachim (the Shi'ur for Medras), for the entire garment is considered whole. Tosfos proved that three by three is not considered connected. Regarding Tzitzis, the torn part is considered whole and the Tzitzis are not Batel. Tosfos did not ask that since it is connected, the Talis keep its Tum'ah. In Zevachim we say that even when enough for a turban remains, it is Tamei only mid'Rabanan. Mid'Oraisa, tearing the majority is Metaher. Regarding Tzitzis, as long as it is considered a garment now, even if it is a new entity, the Tzitzis are still Kosher. Even If you will say that one must sew it first, this is not Ta'aseh.... Rather, Tosfos asked that for laundering, it is a garment only if enough for a turban remains, but for Tzitzis, three by three fingers suffices. Tosfos answered that there, the entire garment was torn, i.e. the Kli lost its form through tearing, so less than enough for a turban is not considered a Kli. Here, one piece was torn and the rest is intact. Therefore, sewing helps, and it is not considered Ta'aseh.... If not for Tosfos, I would have said that Ta'aseh... is more lenient, and three by three precludes this Pesul. R. Yerucham says that any amount suffices for this. It seems that the Rambam and Nimukei Yosef hold that even if it was totally severed, sewing helps. Perhaps three fingers helps also for laundering, but Chachamim were stringent to require enough for a turban, since it suffices to tear that amount to be Metaher.

Poskim

Shulchan Aruch (OC 15:4): If the Talis was torn within three fingers of the corner, one may not sew it. Rashi explains that we are concerned perhaps some strings used to fix it will be left hanging on the corner and used for Tzitzis. If so, even if any amount was torn, one may not sew it. Also, if a woolen Talis was torn within three, one may sew it nowadays, for it is not common to sew with wool threads.

Magen Avraham (6): Everyone knows that other materials do not exempt a woolen Talis. It seems that in a place where people sew with wool threads, we forbid even Talesim of other materials, due to this decree. However, in Sa'if 6 the Shulchan Aruch forbids only for a silk Talis. We do not decree other materials due to silk! Perhaps we more lenient about a silk Talis for some obligate it only mid'Rabanan. However, the same applies here! (Only wool and linen are liable mid'Oraisa.) With difficulty we can say that even in a place where people sew with wool threads, one may sew Talesim of other materials.

Mishnah Berurah (10): If more than three was cut off, one may reattach it with the Tzitzis. The Taz is stringent and requires untying the Tzitzis and attaching them again. One should be stringent.

Kaf ha'Chayim (22): If it was torn within one finger, one may sew it, for this is not a place to attach Tzitzis, just like outside of three fingers.

Shulchan Aruch (ibid.): Rav Amram explains that less than three is not considered a garment. It is as if it is not there. Even if he sewed it, it is as if it is cut. If he attached Tzitzis, they do not exempt the Talis. If so, if it tore and any amount remained, it is Kosher. Some say that Rav Amram disqualified only Tzitzis that were on when it was sewed, but if he attached Tzitzis after it was sewed, it is Kosher. One who fears Shamayim will fulfill all opinions, when possible.

Beis Yosef (DH Nikre'ah): The Rif and Rosh have our text of the Beraisa (41a), in which Chachamim permit sewing. How does this support Rav Yehudah, who forbids sewing? In the Nimukei Yosef's text, R. Meir permits sewing and Chachamim forbid. Hagahos Maimoniyos answers (for our text) that the Halachah follows R. Meir's decrees (Eruvin 46b).

Beis Yosef (DH b'Soch): In a Teshuvah (2:11), the Rosh says that the concern is lest one leave a thread (used to sew) for the sake of Tzitzis. Therefore, one may not sew a tear of any size. It seems that he holds that Rashi's opinion is primary.

Beis Yosef (DH v'Rabbeinu): R. Yerucham says that if it tore and any amount remained, it is Kosher. This is like Rav Amram, unlike Rashi. He holds that Rav Amram forbids sewing only when the entire garment tore, for then it is as if it is cut off. If any amount remained, it is not considered disconnected. It seems that the Mordechai 948 agrees.

Beis Yosef (DH veha'Tosfos): Tosfos explains that we discuss when the corner tears. If three remains, it is considered properly attached. This is like Rav Amram. 'Within' and 'outside' three do not refer to the width of the garment, for it does not matter whether or not it was within three. Rather, they refer to the length. I.e. what is intact outside of three one may sew. Since three by three is intact, it is not considered detached. If it tore within three, what is intact is less than three, so it is as if it is detached. One may not sew it. The Rosh and R. Yerucham say so. Tzitzis that were already attached when it was sewn are Pasul, due to Ta'aseh.... Further, we disqualify even what was attached afterwards, for it is not considered part of this garment. The Rosh and R. Yerucham say so. The Nimukei Yosef (on Rif Hilchos Tzitzis 12a, DH v'Shavin, citing Rav Amram) connotes that it depends on whether it is within three along the width. He disqualifies what was attached when it was sewn, but he is Machshir what was attached afterwards. The Rambam connotes like this. Since he holds like Rav Amram, we rule like him. However, when possible it is proper to be concerned for Rashi's opinion. I do not know why the Tur brought Rav Yehudah's teaching and did not explain it.

Taz (4): The Beis Yosef holds that the Nimukei Yosef permits attaching Tzitzis after sewing, unlike the Rosh. The piece was entirely cut off, but if any amount remained, even the old Tzitzis remain Kosher. I say that all agree that if a piece was totally severed and sewn back, one may attach new Tzitzis. Surely, sewing makes it one garment! It is no different if it was torn and sewn, or if when making the garment a small piece was added to a big piece. A Beraisa forbids attaching a piece with Tzitzis on it, but all permit to attach a piece and then tie Tzitzis on it! Rather, the Rishonim discuss only disqualifying the initial Tzitzis. The Rambam connotes like this. The Rosh says 'we do not attach Tzitzis to it', i.e. before sewing it. The Terumas ha'Deshen (46) explains the Rosh this way. We cannot say that the Rosh forbids to attach after sewing, i.e. because the Tzitzis will not be on the garment. Ta'aseh... would not disqualify, for the Tzitzis were attached after it was sewn. The Gemara said that we decree lest it tear and he sew it, and it will be Ta'aseh...! Also, if one cannot attach Tzitzis on a piece sewn on, how can one put Tzitzis above there? They will be too far from the corner! If so, there is no way to attach Tzitzis to the garment. This cannot be. Rather, Rashi's Perush is primary. The Bach (DH Nikre'ah) wrote like I said. I think that all the Poskim hold like this.

Magen Avraham (7): The Rosh disqualifies even what is attached after it is sewn. See 10:3,6.

Levushei Serad (14): In 10:3,6, we consider what is sewn as if it is not sewn, in cases when we say that later he will untie it. This answers the Taz' objections.

Gra (6 DH v'Yesh): The Taz is correct. Sewing is always considered connected. The opinion of Tosfos and Rav Amram is primary. The Beis Yosef says that R. Yerucham explains that the Rosh is Machshir when any amount remains, and then he writes that the Rosh holds like Tosfos! Really, Tosfos says that it depends on whether three remain in the length. The Rosh holds like the Nimukei Yosef, that it depends on whether three remain in the width (but any remnant in the length suffices).

Damesek Eliezer: The Gemara 41a proves that sewing helps (if a corner was folded and sewn, all agree that it requires Tzitzis). The Rosh and Nimukei Yosef hold that if three remain in the width, it is considered a garment, so Ta'aseh v'Lo Min ha'Asuy does not apply.

Mishnah Berurah (15): Many agree with the Taz. To fulfill the first opinion, one sews a cloth three by three under the corner.

Kaf ha'Chayim (23): Ma'amar Mordechai says that the Rosh connotes like the Beis Yosef says, unlike the Taz.