Here is perhaps the best known musical version of Avinu Malkeinu, often called "A Folk Melody"
Introduction
Avinu Malkeinu is a central prayer of the ten days of repentance, which begin on Rosh Hashanah and conclude on Yom Kippur. Both the words and the melody evoke deep emotion as the congregation connects with God as a parent and monarch. On Yom Kippur, it is recited throughout the day, culminating with its recitation as part of the Neilah service in the final moments of the holiday. The earliest known version is in Seder Rav Amram (10th c) and the number of verses was 25. In various Sephardic rites, the verses number between 29-32; in German versions, 38; Polish, 44; and Salonika, 53.
ת"ר מעשה ברבי אליעזר שגזר שלש עשרה תעניות על הצבור ולא ירדו גשמים באחרונה התחילו הצבור לצאת אמר להם תקנתם קברים לעצמכם געו כל העם בבכיה וירדו גשמים שוב מעשה בר' אליעזר שירד לפני התיבה ואמר עשרים וארבע ברכות ולא נענה ירד רבי עקיבא אחריו ואמר אבינו מלכנו אין לנו מלך אלא אתה אבינו מלכנו למענך רחם עלינו וירדו גשמים הוו מרנני רבנן יצתה בת קול ואמרה לא מפני שזה גדול מזה אלא שזה מעביר על מדותיו וזה אינו מעביר על מדותיו
The Sages taught: An incident occurred involving Rabbi Eliezer, who decreed a complete cycle of thirteen fasts upon the congregation, but rain did not fall. At the end of the last fast, the congregation began to exit the synagogue. He said to them: Have you prepared graves for yourselves? If rain does not fall, we will all die of hunger. All the people burst into tears, and rain fell. There was another incident involving Rabbi Eliezer, who descended to serve as prayer leader before the ark on a fast day. And he recited twenty-four blessings, but he was not answered. Rabbi Akiva descended before the ark after him and said: Our Father, our King, we have no king other than You. Our Father, our King, for Your sake, have mercy on us. And rain immediately fell. The Sages were whispering among themselves that Rabbi Akiva was answered while his teacher, Rabbi Eliezer, was not. A Divine Voice emerged and said: It is not because this Sage, Rabbi Akiva, is greater than that one, Rabbi Eliezer, but that this one is forgiving, and that one is not forgiving. God responded to Rabbi Akiva’s forgiving nature in kind by sending rain.
Context: This is from the Babylonian Talmud, Masechet (Tractate) Ta’anit, which is about fast days. Fast days were often declared when there wasn’t rain, as a way to show God that we truly were deserving of rain for crops to grow. Usually there were heartfelt prayers said. This text is the origin of the prayer “Avinu Malkeinu”. The words “Avinu” and “Malkeinu” as references to God come from Isaiah 63:16 and Isaiah 33:22, respectively.
Here's another musical version of Avinu Malkeinu
Why don't we say Avinu Malkeinu on Shabbat? Because Rabbi Akiva first said it on a public fast and no public fast can be ordered to fall on Shabbat. (Orchot Hayyim). It is also seen as a petitionary prayer.
The five repetitions of 'Avinu Malkeinu inscribe us in a book..' corresponds to the five Books of Moses. Each version relates to a different book. (Days of Awe, S.Y. Agnon pg.62)
For Those Who Are Uncomfortable With the Patriarchal Images of Avinu (Our Father) Malkeinu (Our King), Some Options...
Mkoreinu Eloheinu, Our Source, our God, inscribe us in the book of merit.
Mkoreinu Eloheinu, Our Source, our God, inscribe us in the book of forgiveness and atonement.
Mkoreinu Eloheinu, Our Source, our God, we have done wrong in Your presence.
Mkoreinu Eloheinu, Our Source, our God, renew us for a good year.
Mkoreinu Eloheinu, Our Source, our God, nullify the plans of any who may seek to do us harm. (Kol Haneshama, Reconstructing Judaism)
Loving Father, Infinite Power, Gentle, Forgiving, Lofty, Inscrutable, Avinu Malkeinu
Compassionate Mother, Omnipotent Lord, Comforting Presence, Fathomless Mystery, Avinu Malkeinu
Our Rock and Redeemer, Life of the Universe, Close to us Always, Impossibly Far, Avinu Malkeinu
Embracing, Confounding, Accepter of our frailty, Decreeing our end, Avinu Malkeinu
Avinu Malkeinu:
Shema Koleinu, We speak this sacred truth aloud
Chatanu Lefanecha, We stand as one, accountable for our sins
Hamol Aleinu v'al olaleinu v'tapeinu. We yearn for true compassion--for our children most of all.
Kaleighs dever, herev, v'ra'v mei'aleinu. May we resist the ravages of illness fear and despair,
Chadeish yameinu Shanah Tovah. May we taste anew the sweetness of each day.
Haneiniu v'aneinu, ki ein banu ma'asim... However small our deeds, let us see their power to heal. May we save lives through compassion, generosity and justice. (Mishkan HaNefesh)
Mkoreinu Eloheinu, Our Source, our God, inscribe us in the book of merit.
Mkoreinu Eloheinu, Our Source, our God, inscribe us in the book of forgiveness and atonement.
Mkoreinu Eloheinu, Our Source, our God, we have done wrong in Your presence.
Mkoreinu Eloheinu, Our Source, our God, renew us for a good year.
Mkoreinu Eloheinu, Our Source, our God, nullify the plans of any who may seek to do us harm. (Kol Haneshama, Reconstructing Judaism)
Loving Father, Infinite Power, Gentle, Forgiving, Lofty, Inscrutable, Avinu Malkeinu
Compassionate Mother, Omnipotent Lord, Comforting Presence, Fathomless Mystery, Avinu Malkeinu
Our Rock and Redeemer, Life of the Universe, Close to us Always, Impossibly Far, Avinu Malkeinu
Embracing, Confounding, Accepter of our frailty, Decreeing our end, Avinu Malkeinu
Avinu Malkeinu:
Shema Koleinu, We speak this sacred truth aloud
Chatanu Lefanecha, We stand as one, accountable for our sins
Hamol Aleinu v'al olaleinu v'tapeinu. We yearn for true compassion--for our children most of all.
Kaleighs dever, herev, v'ra'v mei'aleinu. May we resist the ravages of illness fear and despair,
Chadeish yameinu Shanah Tovah. May we taste anew the sweetness of each day.
Haneiniu v'aneinu, ki ein banu ma'asim... However small our deeds, let us see their power to heal. May we save lives through compassion, generosity and justice. (Mishkan HaNefesh)
Here's a third musical version
This is Real and You are Completely Unprepared, by Rabbi Alan Lew (2003, p. 107)
Heartbreak is precisely the feeling that we have done our best, we have given it our all, but it hasn’t been enough. Not nearly enough. And this is what we mean when we say, “God is close to the broken-hearted.” And this is what we mean when we say ain banu ma’asim — we have no good deeds. This is what it says in “Avinu Malkeinu”, that lovely song with the haunting, heartbreaking melody we all love to sing so much on Rosh Hashanah. Avinu malkenu, chanenu v’anenu — Our Father, our King, be gracious to us, be gratuitously loving, and answer us even though we don’t deserve it — ki ain banu ma’asim — because we have no good deeds to invoke in our own defense. In other words, now that we are standing in the face of this immense reality, we realize that there is nothing we can do about it; that all our deeds are as nothing. We realize that we have greatly overestimated our cleverness and our potency; we have overestimated the efficacy of our conscious behavior, and we have underestimated the persistence and the depth of our destructive tendencies. We realize that our attempts to do good are very small next to the unconscious havoc we constantly wreak to our right and to our left.
Heartbreak is precisely the feeling that we have done our best, we have given it our all, but it hasn’t been enough. Not nearly enough. And this is what we mean when we say, “God is close to the broken-hearted.” And this is what we mean when we say ain banu ma’asim — we have no good deeds. This is what it says in “Avinu Malkeinu”, that lovely song with the haunting, heartbreaking melody we all love to sing so much on Rosh Hashanah. Avinu malkenu, chanenu v’anenu — Our Father, our King, be gracious to us, be gratuitously loving, and answer us even though we don’t deserve it — ki ain banu ma’asim — because we have no good deeds to invoke in our own defense. In other words, now that we are standing in the face of this immense reality, we realize that there is nothing we can do about it; that all our deeds are as nothing. We realize that we have greatly overestimated our cleverness and our potency; we have overestimated the efficacy of our conscious behavior, and we have underestimated the persistence and the depth of our destructive tendencies. We realize that our attempts to do good are very small next to the unconscious havoc we constantly wreak to our right and to our left.