What is the א"ת ב"ש ג"ר calendar mnemonic?

Question

The Gemara Arakhin 9b discusses certain rules of the calendar. Specifically, it mentions that Rosh Hashana will always be on the same day of the week as the day of the Omer (the 2nd day of Pesach), or the day later.

Rashi there comments that with today's fixed calendar, this is no longer in flux; rather, Rosh Hashana is always on the same day of the week as the day after the Omer (the 3rd day of Pesach). This is because the latter half of the year is always kesidran (i.e. a malei, 30 day month, always follows a chaser and vice versa), so there are exactly 161 days, which is divisible by 7, between the 3rd day of Pesach and Rosh Hashana.

Now, here's the quote from Rashi that I don't understand

האידנא עבדינן לעולם חדשי ימות החמה כסדרן ויש לנו סימן קבוע א"ת ב"ש ג"ר לשלישי של פסח הוי ר"ה

As I explained, I understand Rashi's point; I just have no idea what this א"ת ב"ש ג"ר mnemonic is.

Can any calendar experts help me out?

Answer 1

The Shulchan Aruch 428:3 explains the mnemonic:

סימן לקביעת המועדים א"ת ב"ש ג"ר ד"ק ה"ץ ו"ף פירוש ביום א' של פסח יהיה לעולם ת"ב וסימן על מצות ומרורים יאכלוהו ביום ב' בו שבועו' וביום ג' בו ר"ה ביום ד' בו קריאת התורה שהוא שמחת תורה ביום ה' בו צום כיפור ביום ו' בו פורים שעבר:

The way it works is the first letter of the alef beis is matched with the last letter, the second to the second last, and so on. The first letter is referring to which day of Pesach, and the second letter is referring to a festival. The day of the week that that day of Pesach falls on, is the same day of the week that that festival falls on.

1. א"ת - Tisha BAv
2. ב"ש - Shavuos
3. ג"ר - Rosh Hashanah
4. ד"ק - Kriyas HaTorah (Simchas Torah in Chutz Laaretz)
5. ה"ץ - Tzom Kippur (Yom Kippur)
6. ו"ף - Purim (the previous one)