Teshuvah Purifies the Sin
במדרש מי יתן טהור מטמא כו' כי הנה אמרו רז"ל משל לבן שפחה כו' תבוא אמו ותקנח הצואה כך פרה תכפר על העגל
In the Midrash: "Who can produce a pure thing out of an impure one?" (Iyov 14:4) — for behold, Chazal said a parable of the son of a maidservant who soiled the king's palace; "Let his mother come and wipe away the filth." So too, let the parah (the red cow) come and atone for the eigel (the Golden Calf).
The Sfas Emes opens with the Midrash on the verse in Iyov, citing Chazal's parable: just as a mother is brought to clean up the mess her child made, so the parah adumah comes to atone for the eigel — taharah emerging out of something bound up with tumah.
כי התשובה מתקן החטא ובאמת כח החטאים הוא ע"י מציאות התשובה שיש תועלת מן החטא וזה חיות החטאים בעולם לכן נק' התשובה אם לקנח ולכפר על החטא
For teshuvah repairs the sin, and in truth the very power of sins comes about through the existence of teshuvah — that there is a benefit to be drawn out of the sin; and this is what gives sins their vitality in the world. Therefore teshuvah is called the "mother" who wipes away and atones for the sin.
Teshuvah does not merely cancel the sin but actually transforms it into a source of benefit; and remarkably, it is the very possibility of teshuvah that gives sins their foothold in the world, which is why teshuvah is figured as the 'mother' who wipes away the very filth her existence made room for.
ועל כח זה של התשובה אמרו מי יתן טהור מטמא כי יש ב' מיני טהרות
And it is concerning this power of teshuvah that they said, "Who can produce a pure thing out of an impure one?" — for there are two kinds of taharah (purity).
The verse 'who can produce a pure thing out of an impure one' points to the wonder of teshuvah, and the Sfas Emes introduces the key distinction that there are two different kinds of taharah.
א' כמו שהיו בנ"י קודם החטא צדיקים גמורים והיו באמת טהורים ורחוקים מן הטומאה
The first: as Bnei Yisrael were before the sin, when they were complete tzaddikim and were truly pure and far removed from all tumah (impurity).
The first kind is the original, untouched purity that Bnei Yisrael possessed before the eigel, when they were complete tzaddikim who had no contact whatsoever with tumah.
אמנם אחר זה הי' התיקון ע"י התשובה
However, afterward the repair came about by means of teshuvah.
The second kind arises only after the sin has occurred, through the avodah of teshuvah that repairs it.
לכן אחר שתיקנו החטא ניתן להם הטהרה שבא אחר הטומאה ע"י התשובה כנ"ל
Therefore, after they repaired the sin, there was granted to them the taharah that comes after the tumah by means of teshuvah, as stated above.
Once they had repaired the sin of the eigel, Bnei Yisrael were granted this second, higher taharah — the purity that comes specifically after and out of tumah, attained through teshuvah.
וזה הי' ב' השירות שירה ראשונה הוא שהיו בנ"י ישרים באמת
And this is the matter of the two songs: the first song was sung when Bnei Yisrael were truly upright,
The Sfas Emes maps these two kinds of taharah onto the two songs: the first song, the Shirah at the Yam, was sung when Bnei Yisrael were genuinely upright and unblemished.
ישרי לב בלי שום חטא
upright of heart, without any sin at all.
This first state was one of pure-hearted innocence, before any sin had marred them at all.
ולכן היו בנ"י במדריגה אחת עם מרע"ה
And therefore at that time Bnei Yisrael stood on one and the same level together with Moshe Rabbeinu.
Because they were in that pristine, sinless state, Bnei Yisrael stood on the very same level as Moshe Rabbeinu himself, with no gap between leader and people.
אמנם שירה אחרונה הי' אחר תיקון החטא והתיישרות דרכים עקומים ולכן בשירה זו לא נזכר מרע"ה והוא ענין במקום שבעלי תשובה עומדים אין צדיקים גמורים יכולין לעמוד:
However, the last song was sung after the repair of the sin and the straightening of crooked ways; and therefore in this song Moshe Rabbeinu is not mentioned. And this is the matter of "In the place where baalei teshuvah stand, complete tzaddikim cannot stand" (Berachos 34b).
The last song, the Shirah of the Be'er in this parshah, was sung after the sin had been repaired and crooked paths straightened through teshuvah; Moshe Rabbeinu is pointedly absent from it because it belongs to the rung of the baal teshuvah, in keeping with Chazal that where baalei teshuvah stand even complete tzaddikim cannot stand.
Summary: Building on the Midrash that likens the parah adumah atoning for the eigel to a mother cleaning up her child's mess, the Sfas Emes develops a teaching about two distinct kinds of taharah. The first is the original, untouched purity of Bnei Yisrael before they sinned, when they were complete tzaddikim and stood on the very same level as Moshe Rabbeinu — expressed in the first song, the Shirah at the Yam. The second, higher purity emerges only out of tumah, through teshuvah, which not only repairs the sin but retroactively draws benefit from it and gives sins their very vitality in the world. This second state is reflected in the last song, the Shirah of the Be'er, sung after the sin was repaired and crooked ways made straight; Moshe Rabbeinu is absent from it because it belongs to the rung of the baal teshuvah. The piece thus teaches that the baal teshuvah reaches a height the complete tzaddik cannot attain, for "in the place where baalei teshuvah stand, complete tzaddikim cannot stand."