Danny: The first similarity that strikes me is that many Jews and Muslims who are not usually so observant are more concerned with these holy days than many others in the calendar - it seems that something about them has a broader appeal than many other festivals or rituals.
Mohammed: Maybe there's something in both that resonates deeply with people, which they recognise they need even if they don't consider themselves to be super-pious. Both Rosh Hashanah and Ramadan are opportunities for thauba or teshuvah, improving one's character through introspection, increasing one's commitment to doing good deeds, religious practice etc. I know many Muslims who don't usually go to the mosque, but they try to go more during Ramadan, and also important is making an effort with how we treat other people.
Danny: An important part of both is giving more charity, as well as trying to pray more and striving to right any wrongs that we've made during the past year. Also, both Ramadan and the Jewish New Year are connected with ten days of particular importance and holiness - the last ten days of Ramadan and the first ten days of the Jewish year, which culminate in Yom Kippur.
Mohammed: The connection between Ramadan and Yom Kippur is actually explicit in Islam: in the Quran, God proclaims that fasting is obligatory upon Muslims, "as it was upon those before you". According to one of the earliest traditional sources, or hadith, this refers to the Jewish practice of fasting on Yom Kippur (Sunan al-Tirmidhi I.145).
Danny: I think another big similarity might be that both Ramadan and our High Holy-Days are "awful", as in full of awe, but also bring us tremendous joy if we put some healthy effort into them.
Mohammed: Ramadan is very serious but also we really look forward to it and appreciate the opportunity it brings us to attain a special closeness to God, and also gain forgiveness for our wrongdoings.
Danny: Serious doesn't necessarily mean sad. There's an amazing story in the Bible about Rosh Hashanah when the Jews have just returned from exile in Babylon. When the people hear the Torah being read, they start weeping and wailing, because they suddenly realise how much they've lost touch with who they are. The prophet Nehemiah tells the people not to mourn but, on the contrary, to feast and to give gifts to one another, and to make "great rejoicing", because the day's holiness actually demanded this (Nehemiah 8:10-12).
Mohammed: Rejoicing is an important part of Ramadan as well: in most Islamic countries on each day of Ramadan, the day's fasting ends with a communal celebration, which can be quite exuberant, even including huge fireworks displays, building up to an enormous party for Eid at the end of the month, which is one of the biggest festivals of the year.
Danny: It's funny that most people seem to associate these days with mourning. Even Yom Kippur, which we tend to think of as a sad day because we're fasting, is actually one of the two happiest days of the year, according to the Mishnah, because God is forgiving us, and therefore we are "re-marrying" the Divine after a period of estrangement. Also our tradition weaves together Rosh Hashanah and Yom Kippur with Succot, the next festival, which begins a few days after Yom Kippur, and which is also called "the time of rejoicing". It's as if Rosh Hashanah and Yom Kippur are necessary steps to being able to appreciate the joyful celebration of Succot.
Mohammed: Likewise, Ramadan is definitely not a burden, but a great gift for us, because in the Quran, Muhammad says that during Ramadan the gates of Heaven are open, and the gates of Hell are closed.
Danny: That's fascinating - we learn from our rabbis that on Yom Kippur the accusing angel (sometimes called Satan) is powerless over us, and that the gates of Heaven are wide open to receive our prayers. There's always a dramatic tension in the air towards the end of the day when the gates are said to be closing - people really pour their hearts out and you can feel how sincere people are being by that point.
Mohammed: It's the same at the end of the ten final days of Ramadan. By then people have had time to work on themselves and to strip away the layers of ego we usually hide behind. When the climax of Ramadan comes, we're naked before our Creator, and instead of trying to hide our true selves, we commit to working on them, to making this coming year better than the one before.