Why is it that on 16b the gemarah states on alternate brachos when the Shofar is blown it is Teruah, Tikah,Teruah (and the Rambam brings this down lhalachah). Rashi stated in 14a Shofros- stam teruah peshutah lefanehah vachrechah (Rosh Hashanah 34a) to explain why there are 18 kollos for the 6 additional brachos on the last 7 taanisim. Is this gemorah going according to those who limit this principle to only teruahs of Tishrai (top of Rosh Hashana 34a)? Even if it is ,once 3 kollos are being blown for each bracha, why change the order from Tikiyah Teruah Tikiya - kal d'tikun kain d'oritah tikun?
TIFERES YISRAEL #55 explains that even though for every alternate blowing Teruah came before Tekiah, nevertheless this does not contradict the principle, stated by RASHI 14a DH BE-SHOFROS, that there is always a Tekiah before every Teruah.
Tiferes Yisrael explains that since there was a Tekiah in the first Berachah, this constitutes the Tekiah required to precede the Teruah which opens the blowing for the second Berachah. TI asserts that the Berachah between the Tekiah at the end of the blowing of the first Berachah, and between the Teruah of the second Berachah, does not constitute a "Hefsek"(interruption), because the Gemara states (Rosh Hashanah top 34b) that if one heard 9 Tekios at 9 different hours of Rosh Hashanah one thereby performs the Mitzvah, so a Berachah in the middle of the blowing certainly should not represent a Hefsek.
It may be possible to suggest an explanation why Chazal decided that the first blowing of each set should alternate between Tekiah and Teruah. This explanation is based on a Sevara mentioned by BEIS YOSEF OC 579:2 DH VE-KASAV Ha-RAV Ha-MAGID, cited by TOSFOS YOM TOV. Beis Yosef writes that Teruah was announced first, on every second occasion, in order to give the same honor to Teruah as was given to Tekiah. See also MAHARSHA 16b DH U-BE-A who writes that Tekiah corresponds to mercy, whilst Teruah corresponds to judgement. Therefore it is possible that even though on Rosh Hashanah Tekiah always comes first, this is because on Rosh Hashanah we are confident only to mention mercy to Hash-m since we are in the middle of the 10 days of Teshuvah, when Hash-m judges the world with a special mercy. In contrast the rest of the year we have to concede that the mercy is also mixed with judgement, and therefore we wish to pay respect also to Teruah, and blow it first sometimes.
I found a very different approach to our Sugya in RITVA (above 15a DH VE-OD NECHLEKU on p.123 in Mosad HaRav Kook edition). Ritva writes that according to Chachamim the Shofar is only blown once (not 3 times) for each Berachah i.e. after the first Berachah the Chazan HaKnesses says "Blow! (Tiku!) the sons of Ahaton, Tiku!" and they blew only one Tekiah. The next time he said "Horiu! the sons of Aharon, Horiu!" and they blew only one Teruah, and so this continued alternatively.
Ritva writes that the reason for this is because "Tekiah - Teruah - Tekiah" should only be blown on Rosh Hashanah and Yom Kippur as Gemara Rosh Hashanah 34a derives from verses, as you mentioned in your question. It is only on R.H. and Y.K. that the rule of RASHI 14a DH BE-SHOFROS applies, that every Teruah must have a Tekiah before and after it. In contrast on fast days one learns from Bamidbar 10:9 "Ve-harayoschem Be-chatzotzros" that the prayer of the community should ascend through Teruah. Therefore it is sufficient to have the smallest blowing - whether Tekiah or Teruah - to achieve this.
KOL TUV
D. Bloom