Various customs exist regarding the amount and the type of sounds blown during Musaf. The [widespread Ashkenazi] custom is to blow a single set of תשר“ת [4 blows] at the end of each blessing. Others are accustomed to blow a single different set at the end each blessing. They blow one set of תשר“ת for the Kingship blessing, one set of תש“ת for the Remembrance blessing, and one set of תר“ת for the Shofros blessing. Others are accustomed to blow three sets of one of the three groupings by each blessing. Thus they blow three times תשר“ת for the Kingship blessing, three times תש“ת for the Remembrance blessing, and three times תר“ת for the Shofros blessing. Others are accustomed to blow one set of each of the three groupings [for a total of three sets] by each blessing. Thus they blow תשר“ת/תש“ת/תר“ת by each one of the three blessings. Practically the best of the above customs is like the last custom mentioned. [In the Siddur Admur rules like this latter opinion and so is the Chabad custom.]
Various customs exist in this regard. The best custom to follow is to blow תשר“ת/תש“ת/תר“ת by each one of the three blessings and so is the Chabad custom.
The Sages instituted that only one set of תר“ת be sounded after each of the three middle blessings. [592/1] However now [that we have doubt as to the correct sound of the Teruah and therefore] the Sages instituted to blow all the possible sounds of the Teruah in order to avoid the doubt it should have been proper to blow one set of Tashrat Tashat and Tarat by the end of each of the three blessings in order to fulfill one’s obligation according to all the sounds of the Teruah. Nevertheless being that we do not overburden the congregation therefore many are not accustomed to do so. [592/2; See Rambam Shofar 3/12]
 This is based on the background above that one is not required to blow all three sounds of the Teruah three times. Hence various customs evolved regarding which sound of the Teruah is to be preferred when.
 Rama 592/1
 Admur 592/3 [first and Stam opinion]; Rama 592/1; Tur in name of Rabbeinu Tam
The reason: The reason behind this custom is because by blowing the set of Tashrat one sounds every possible sound of the Teruah and hence the only applicable worry is that (perhaps the Teruah is only a Shevarim or only a Teruah and thus) perhaps he is making an interval between the Teruah and the Tekiah that precedes it or proceeds it. Now this alone is of no worry to us being that we already fulfilled our obligation with the Tekios Meyushav, as with those Tekios we fulfilled the obligation according to all the possible doubts. [Admur ibid]
 Admur 592/4; Michaber 592/1 [first opinion]; Rif; Rambam 3/10
 The reason: The reason behind this custom is because [they hold that] Biblically all the sounds of the Teruah, whether it be a Teruah or a Shevarim, are considered a valid Teruah, and thus whatever a person sounds, whether only a Shevarim or only a Teruah, he fulfills his [Biblical] obligation. Thus in earlier generations some communities sounded only the Shevarim and some sounded only the Teruah, and they both fulfilled their Biblical obligation. However since this matter seemed like a dispute in the eyes of the masses therefore the Sages instituted that everyone should blow the same sounds and they should not blow sounds that are under dispute in the eyes of the masses. Hence they instituted to blow Tashrat; Tashat and Tarat three times each in order to remove the masses from questioning the sounds based on the differences they witness in the customs. Accordingly [since blowing all the sounds of the Teruah is only a Rabbinical institution to satisfy the masses] they did not trouble the community to blow all the sounds also by the Tekios of Musaf. On the contrary, with the blows in Musaf we show the public that all the sounds of the Teruah are valid Teruos and a person fulfills his obligation with it. Thus they blow one time Tashrat and one time Tarat and one time Tashat to show that each one of these sets are proper and Kosher. [Admur ibid; based on Rav Haiy Gaon that all the sounds are valid]
 Admur 592/5; Michaber 592/1 [that so is the custom today]; This custom was not mentioned in the Beis Yosef. The Mamar Mordechai 592/1 says that perhaps it came to the Michaber’s attention only after first writing the Beis Yosef.
 Admur 592/5; Aruch; Hamanhig; Ridbaz 29; Riaz; Shalah p. 314; Erech Hashulchan 592/1; Kaf Hachaim 592/4
 Admur ibid; Shlah ibid “praised be the holy congregation that chooses this custom”
The reason: As explained above there is doubt as to which sound is the Teruah, and only in the fourth custom is one for certain blowing a correct set of a Teruah by each blessing. [Shelah ibid]