The Mishna (Sukkah 53) tells is that no fewer that 21 chatzotzros blasts were blown daily in the Mikdash as part of the process of offering nesachim, but the gemara cites R' Yehudah's opinion that the minimum number of blasts was 7. Explains the gemara that there is no argument. The blowing had to be done in units of three: tekiya, teru'ah, tekiya. The Tanna of the Mishna countes 21 blasts in total; R' Yehudah counts 7 units of three.
The Rogatchover explains the machlokes as follows -- for any series of items that must go together, e.g. 4 minim on Sukkos, 4 kosos on Pesach, tekiya-teru'ah-tekiya, one can make the following chakira: are these seperate independent units which must go together, or does that fact that they must always go together prove that they are not independent units but are parts of some larger whole? Is there a single note called tekiya-teru'ah-tekiya, or are these three seperate notes which must always be played together?
The gemara explains that R' Yehudah learned from the pasuk "u'tekatem teru'a" that the terms tekiya and teru'ah are interchangable; they are one and the same unit and go together. The Tanna of the Mishna holds that the blasts are seperate units based on the pasuk which describes the blowing done to gather the camp in the desert, "tiski'u v'lo tari'u" -- if tikiya and teru'ah together form one unit, says the gemara, how could the Torah tell Moshe to do half a mitzvah?
The Aruch laNer is bothered by the gemara's logic and develops his question through an analogy to the mitzvah of zerikas hadam by korbanos. Some korbanos require 4 sprinklings of blood; some require only one. If a korban requires only one sprinkling of blood, we wouldn't call that 1/4 of the mitzvah of zerikas hadam -- viz a viz that korban, 1 sprinkling of blood is 100% of the mitzvah, not 25%! So too with respect to blowing chatzotzros. If the Torah requires a single tekiya in some circumstances, in those circumstances that single tekiya is 100% of the mitzvah, not 50% of the mitzvah that appears in other circumstances. What does the gemara mean?
Based on the Rogatchover, we can perhaps distinguish between the case of zeikas hadam and chatzotzros. With respect to korbanos, how many sprinkles should be are done is a function of the chovas hagavra, i.e. what action the person must take to fulfill the mitzvah. R' Yehudah and the Tanna are not arguing about what the act of blowing shofar entails or how many sounds must be blown. Their argument is regarding the cheftza shel mitzvah of a tekiya -- what is a shofar note in its simplest form: a three note tekiya-teru'ah-tekiya, or 1/3 of that series. What a person must do can change in different contexts depending on the requirements of the mitzvah, but the definition of an object is something that remains constant -- either a tekiya (or teru'ah) alone is a note, or it is not. But something can't both be a note and not be a note at the same time.
If we consider notes as one unit, it makes sense that they should be blown together, without even a breath in between. Some Rishonim suggest this nafka minah between R' Yehudah and the Tanna. This brings us to the practical question of the shevarim-teru'ah which we blow - is it one note or two notes that go hand in hand; one breath or not? Fortunately there are enough kolos to cover all the bases and not have a safeik.
Kesiva v'chasmima tova to all!