Why does the Mishnah record the significance of the fifteen steps but not the twelve steps?

Question

Middos 2:3 relates that there were twelve steps in the Cheil:

וּשְׁתֵּים עֶשְׂרֵה מַעֲלוֹת הָיוּ שָׁם

There were twelve steps there.

Later, in Mishnah 5, the Mishnah relates that there are fifteen steps connecting the Ezras Nashim to the Ezras Yisrael, but it also adds the significance of this number:

וַחֲמֵשׁ עֶשְׂרֵה מַעֲלוֹת עוֹלוֹת מִתּוֹכָהּ לְעֶזְרַת יִשְׂרָאֵל, כְּנֶגֶד חֲמֵשׁ עֶשְׂרֵה מַעֲלוֹת שֶׁבַּתְּהִלִּים

There were fifteen steps going up from in [the Ezras Nashim] to the Ezras Yisrael, corresponding to the fifteen ascensions in Tehillim [120-134].

Why does the Mishnah relate special significance to these fifteen steps, but not the earlier twelve steps?

Answer 1

Tiferes yisroel on mishna 5, the words "שעליהן הלוים אומרים בשיר":

בשמחת בית השואבה עמדו שם בט"ו שורות על הט"ו מעלות, ובכל שורה שררו שיר המעלות אחר, שורה אחר שורה. ולהכי נקראין המזמורים ההם שיר המעלות, וסיימו העומדים וגו' בלילות של סוכות

Meaning that on simchas beis hashoevah a row of people would stand on each step and each row would sing one of the 15 shir hamaalos - one row after the next and because of this these 15 psalms were called "shir hamaalos- the song of the steps.

Based on this I am suggesting that everything had special significance but the mishna only brought down the significance that was clearly seen on a practical level(that a row on each step would sing one of the shir hamaalos and thereby completing all 15). And it also hints to the reason why these 15 psalms were called "shir hamaalos"