Rif Rosh HaShana 10a {Rosh haShana 33b continues; 34a}

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10a

{Rosh haShana 33b, Mishna, continues}
If he blew the first tekiah {of the first set} and prolonged the second tekiah {at the end of the first set, such that it was as long as two tekiot}, he only has in hand {of that long tekiah} one {rather than two - that is, he needs to blow another tekiah to begin the second set}.

One who has already blessed {Mussaf} and only then gets ahold of a shofar should blow tekiah teruah tekiah 3 times. Just as the shliach tzibur {agent of the congregation} is obligated, so is every individual obligated.
Rabban Gamliel says: The shliach tzibbur is motzi the public their obligation {acting on their behalf}.

Gemara:
The measure of a tekiah is three teruot? But they learnt {in a brayta}: The measure of a tekiah is equal to a teruah!

{Note: There are multiple ways to explain the following. Read through the Rishonim and Acharonim on the matter. This follows Rambam.}
Abaye said: {Assume, based on this girsa, that a teruah blowing, which is between the tekiot, is twice as long as 1 tekia. Therefore:} Our Tanna {of the Mishna} is reckoning the tekiot of all the sets against the teruot of all the sets. {Therefore, in total, there are 6 tekiahs and 3 teruot, and so they are equal.} The Tanna of the brayta is reckoning those of a single set {such that one teruah is against two tekiot, such that they are equal} and they don't argue.

"The measure of a teruah is three moans {yabevot}":
But they learnt {in a brayta}: The measure of a teruah is three broken {stoccato} tones {shevarim}!
Abaye said: In this they certainly argue. For it is written {Bemidbar 29:1}:

א וּבַחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ, מִקְרָא-קֹדֶשׁ יִהְיֶה לָכֶם--כָּל-מְלֶאכֶת עֲבֹדָה, לֹא תַעֲשׂוּ: יוֹם תְּרוּעָה, יִהְיֶה לָכֶם.

1 And in the seventh month, on the first day of the month, ye shall have a holy convocation: ye shall do no manner of servile work; it is a day of blowing the horn unto you.

which is translated {in targum} - it is a day of יבבא unto you.
And we learn from the mother of Sisra {Shoftim 5:28}:

כח בְּעַד {ר} הַחַלּוֹן נִשְׁקְפָה וַתְּיַבֵּב אֵם סִיסְרָא, בְּעַד הָאֶשְׁנָב: {ס} מַדּוּעַ, בֹּשֵׁשׁ רִכְבּוֹ {ר} לָבוֹא-- {ס} מַדּוּעַ אֶחֱרוּ, פַּעֲמֵי מַרְכְּבוֹתָיו. {ס}

28 Through the window she looked forth, and peered, the mother of Sisera, through the lattice: 'Why is his chariot so long in coming? Why tarry the wheels of his chariots?

{but ותיבב is translated as some sort of wailing, rather than peering.}

One holds she made a protracted groan and one other holds she sobbed {a series of short sobs}.

The Sages learnt {in a brayta}: From where do we know that it must be with a shofar? Therefore it informs us {Vayikra 25:9}:

ח וְסָפַרְתָּ לְךָ, שֶׁבַע שַׁבְּתֹת שָׁנִים--שֶׁבַע שָׁנִים, שֶׁבַע פְּעָמִים; וְהָיוּ לְךָ, יְמֵי שֶׁבַע שַׁבְּתֹת הַשָּׁנִים, תֵּשַׁע וְאַרְבָּעִים, שָׁנָה.	8 And thou shalt number seven sabbaths of years unto thee, seven times seven years; and there shall be unto thee the days of seven sabbaths of years, even forty and nine years.
ט וְהַעֲבַרְתָּ שׁוֹפַר תְּרוּעָה, בַּחֹדֶשׁ הַשְּׁבִעִי, בֶּעָשׂוֹר, לַחֹדֶשׁ; בְּיוֹם, הַכִּפֻּרִים, תַּעֲבִירוּ שׁוֹפָר, בְּכָל-אַרְצְכֶם.	9 Then shalt thou make proclamation with the blast of the horn on the tenth day of the seventh month; in the day of atonement shall ye make proclamation with the horn throughout all your land.

I only have that this is true by Yovel {for that is the context in the pesukim}. On Rosh haShana, from where? Therefore it informs בַּחֹדֶשׁ הַשְּׁבִעִי, for בַּחֹדֶשׁ הַשְּׁבִעִי teaches us nothing {for we already know from elsewhere on what month Yom Kippur falls}. What does בַּחֹדֶשׁ הַשְּׁבִעִי teach us? That all teruot of the seventh month shall be like this one. And how do we know that there is a peshuta {=a straight, flat sound = tekiah} before it? Therefore it informs us וְהַעֲבַרְתָּ. And how do we know that there is a peshuta after it? Therefore it teaches us {later in the same pasuk} תַּעֲבִירוּ. And this is only on Yovel. Rosh Hashana, from where? Therefore it teaches בַּחֹדֶשׁ הַשְּׁבִעִי.
{Rosh haShana 34a|
For there is no purpose in teaching בַּחֹדֶשׁ הַשְּׁבִעִי {since we know what month Yom Kippur is}. And thus what is it teaching with בַּחֹדֶשׁ הַשְּׁבִעִי? That all the teruot of the seventh month are like this. And how do we know that there should be three sets of three each? Therefore it teaches us:
{Vayikra 25:9}: וְהַעֲבַרְתָּ שׁוֹפַר תְּרוּעָה
{Vayikra 23:24}:

כד דַּבֵּר אֶל-בְּנֵי יִשְׂרָאֵל, לֵאמֹר: בַּחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ, יִהְיֶה לָכֶם שַׁבָּתוֹן--זִכְרוֹן תְּרוּעָה, מִקְרָא-קֹדֶשׁ.

24 Speak unto the children of Israel, saying: In the seventh month, in the first day of the month, shall be a solemn rest unto you, a memorial proclaimed with the blast of horns, a holy convocation.

{Bemidbar 29:1}:

א וּבַחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ, מִקְרָא-קֹדֶשׁ יִהְיֶה לָכֶם--כָּל-מְלֶאכֶת עֲבֹדָה, לֹא תַעֲשׂוּ: יוֹם תְּרוּעָה, יִהְיֶה לָכֶם.

1 And in the seventh month, on the first day of the month, ye shall have a holy convocation: ye shall do no manner of servile work; it is a day of blowing the horn unto you.

And from where do we know to apply that which was stated here to there and what was stated there to here {since two were written regarding Rosh haShana and one regarding Yovel}?
Therefore it teaches שְּׁבִיעִי - שְּׁבִיעִי for a gezera shava.
How so? Three {sets} which are nine {blasts}. The measure of tekiah is equal to that of teruah. The measure of teruah is like three shevarim.

Rabbi Abahu enacted in Caesaria {to blow} tekiah, three shevarim, teruah, tekiah.