Rosh Hashanah as the avodah of prayer

ראש השנה הוא בחי' תפלה.

Rosh Hashanah is the aspect of tefillah (prayer).

The Sefas Emes opens by identifying the essential character of Rosh Hashanah: among the modes of avodah, it corresponds specifically to prayer — the act of standing before Hashem and crowning Him as King.

דיש ד' מצות שמקיימין בכל יום ק"ש תפילין ציצית תפלה כמ"ש בספרי קודש שהם נגד ד' אותיות הוי'.

For there are four mitzvos that we fulfill each day — Krias Shema, tefillin, tzitzis, and tefillah — as is written in the holy seforim that they correspond to the four letters of the Shem Havayah (the four-letter Name of Hashem).

The daily avodah is built on four pillars, and each one stands opposite one of the four letters of Hashem's Name, so that a Yid's everyday service mirrors the full Name of Hashem.

וכן המועדות ג' רגלים.

And so too the festivals — the three Regalim (pilgrimage festivals).

The same fourfold structure is reflected in the yearly cycle of Yom Tov, with the three Regalim paralleling three of these mitzvos.

בפסח בחי' תפילין זרוע הנטויה.

On Pesach — the aspect of tefillin: "the outstretched arm" (Devarim 26:8).

Pesach corresponds to tefillin, which are bound on the arm, paralleling the "outstretched arm" with which Hashem took us out of Mitzrayim and revealed His power.

ובשבועות ק"ש אנכי מצוך היום כו'.

And on Shavuos — Krias Shema: "I command you this day…" (Devarim, cf. the language of acceptance of the yoke).

Shavuos, the time of Mattan Torah, corresponds to Shema — the acceptance of ol malchus Shomayim (the yoke of Heaven) that is the heart of receiving the Torah.

וסוכות ציצית בצל כנפיך יחסיון.

And Sukkos — tzitzis: "in the shadow of Your wings they take refuge" (Tehillim 36:8).

Sukkos corresponds to tzitzis, for both surround and shelter a person — like sitting beneath the wings of the Shechinah, taking refuge in the shade of Hashem.

וימים הנוראים הם בחי' תפלה:

And the Yamim Noraim (the Days of Awe) are the aspect of tefillah.

Completing the parallel, the Days of Awe — Rosh Hashanah and Yom Kippur — correspond to the fourth pillar, tefillah, the avodah of pure prayer and crowning Hashem as King.

Summary: Rosh Hashanah is fundamentally an avodah of tefillah. Just as the four daily mitzvos — Shema, tefillin, tzitzis, and tefillah — correspond to the four letters of Hashem's Name, so do the festivals: Pesach to tefillin, Shavuos to Shema, Sukkos to tzitzis, and the Yamim Noraim to tefillah.