Hebrew, then English. Open "What it means" under any line for the explanation.
Start of the fourth of the seven blessings of the Rosh Chodesh Musaf Amidah. (On Shabbat Rosh Chodesh, say Atah Yatzarta instead.)
רָאשֵׁי חֳדָשִׁים לְעַמְּךָ נָתַתָּ, זְמַן כַּפָּרָה לְכָל תּוֹלְדוֹתָם,
The start of each month You gave to Your people, a time of atonement for all their generations,
What it means
You gave Your people the start of each month, a time of atonement for all their generations,
Shibbolei HaLeket (a medieval book of Jewish law and custom) quotes the author's brother on "to Your people." He ties it to Shemot 12:2, "This month is for you." The secret of the moon's cycle, he says, was handed to Israel and to no other nation.
Shibbolei HaLeket 176:2; Shemot 12:2בִּהְיוֹתָם מַקְרִיבִים לְפָנֶיךָ זִבְחֵי רָצוֹן, וּשְׂעִירֵי חַטָּאת לְכַפֵּר בַּעֲדָם,
when they would bring before You offerings of favor, and sin-offering goats to atone for them.
What it means
The Gemara (the rabbis' long discussions of the Mishnah, the early collection of the Sages' rulings) has God say about the Rosh Chodesh goat: "Bring an atonement for Me, because I made the moon smaller." Shibbolei HaLeket calls this line a rewording of that, which is why it says "to atone for them." The Ribon HaOlamim prayer tells the moon story.
Shibbolei HaLeket 176:5; Chullin 60bזִכָּרוֹן לְכֻלָּם יִהְיוּ, וּתְּשׁוּעַת נַפְשָׁם מִיַּד שׂוֹנֵא.
They would be a reminder for all of them, and the rescue of their souls from the hand of an enemy.
What it means
Kol Bo (a medieval collection of laws and customs) quotes a reading of "an enemy": the urge to do wrong, the foe that lies in wait to make a person sin. On that reading, the goat keeps Israel remembered before God so they can be saved from it.
Kol Bo 43:9מִזְבֵּחַ חָדָשׁ בְּצִיּוֹן תָּכִין, וְעוֹלַת רֹאשׁ חֹדֶשׁ נַעֲלֶה עָלָיו,
A new altar in Zion You will set up, and the burnt offering of Rosh Chodesh we will bring up on it,
What it means
You will set up a new altar in Zion, and we will bring up on it the burnt offering of Rosh Chodesh,
Shibbolei HaLeket says this line is spelled out in the vision of Yechezkel (Ezekiel), in the reading from the Prophets that includes "In the first month, on the first of the month." There the new-moon offering is one bull and six lambs (46:6), not the Torah's two and seven. The Gemara reads that as the amount to bring when you cannot find the full number.
Shibbolei HaLeket 176:8; Yechezkel 46:6; Menachot 45aוּשְׂעִירֵי עִזִּים נַעֲשֶׂה בְרָצוֹן, וּבַעֲבוֹדַת בֵּית הַמִּקְדָּשׁ נִשְׂמַח כֻּלָּנוּ,
and goats we will offer with favor, and in the service of the Temple we will all rejoice,
What it means
and we will offer goats with favor, and in the service of the Temple we will all rejoice,
Rabbi Chaim Yosef David Azulai (1700s), in his book Petach Einayim, suggests "all of us" means men and women: in the future, he says, Rosh Chodesh will be a festival for the men too. The hard question on women and Rosh Chodesh tells why the day went to the women.
Petach Einayim on Shevuot 9a; Tur, Orach Chaim 417וּבְשִׁירֵי דָּוִד עַבְדָּךְ הַנִשְׁמָעִים בְּעִירָךְ הָאֲמוּרִים לִפְנֵי מִזְבָּחָךְ.
and in the songs of David, Your servant, that are heard in Your city and said before Your altar.
What it means
In King Chizkiyahu's day, "when the burnt offering began, the song of Hashem began" (Divrei HaYamim, also called Chronicles, II 29:27). Songs went with the offering. Shibbolei HaLeket adds: having mentioned the Temple, the prayer now mentions David, who gave himself for it.
Divrei HaYamim II 29:27; Shibbolei HaLeket 176:9אַהֲבַת עוֹלָם תָּבִיא לָהֶם, וּבְרִית אָבוֹת לְבָּנִים תִּזְכּוֹר.
You will bring them everlasting love, and You will remember the covenant of the fathers for their children.
יְהִי רָצוֹן מִלְּפָנֶיךָ יְיָ אֱלֹהֵינוּ וֵאלֹהֵי אֲבוֹתֵינוּ, שֶׁתַּעֲלֵנוּ בְשִׂמְחָה לְאַרְצֵנוּ וְתִטָּעֵנוּ בִּגְבוּלֵנוּ,
May it be Your will, Hashem, our God and God of our ancestors, that You bring us up with joy to our land and plant us within our borders,
וְשָׁם נַעֲשֶׂה לְפָנֶיךָ אֶת קָרְבְּנוֹת חוֹבוֹתֵינוּ, תְּמִידִים כְּסִדְרָם וּמוּסָפִים כְּהִלְכָתָם.
and there we will make before You the offerings we owe, the daily offerings in their order, and the extra offerings by their law.
What it means
The prayer said just before the daily offering passage in the morning makes the same kind of promise: "we will bring before You the daily offering to atone for us, as You wrote for us in Your Torah." Both then lead into the Torah's own verses.
Siddur Sefard, the prayer before Parashat HaTamidאֶת מוּסַף יוֹם רֹאשׁ הַחֹדֶשׁ הַזֶּה, נַעֲשֶׂה וְנַקְרִיב לְפָנֶיךָ בְּאַהֲבָה כְּמִצְוַת רְצוֹנָךְ,
The Musaf offering of this day of Rosh Chodesh we will make and bring before You with love, as the commandment of Your will,
What it means
Why can Musaf be said any time that day? The Gemara answers: the Musaf offering was brought all day. Rabbi Yehudah, one of the Sages of the Mishnah, limits it to the first seven hours of the day, since he holds that is how long the offering could be brought.
Berakhot 26bכְּמוֹ שֶׁכָּתַבְתָּ עָלֵינוּ בְּתוֹרָתֶךָ עַל יְדֵי מֹשֶׁה עַבְדָּךְ מִפִּי כְבוֹדֶךָ כָּאָמוּר.
as You wrote for us in Your Torah through Moshe, Your servant, from the mouth of Your glory, as it is said:
What it means
Shibbolei HaLeket says that once you have said "from the mouth of Your glory," you have done your duty even if you skip the verses below. He bases it on a rule in the Gemara (Rosh Hashanah 35a): once you say "and in Your Torah it is written," you no longer need to say the verses themselves. He adds: our custom is to say them.
Shibbolei HaLeket 176:10; Rosh Hashanah 35aBamidbar 28:11:
וּבְרָאשֵׁי חָדְשֵׁיכֶם תַּקְרִיבוּ עֹלָה לַייָ, פָּרִים בְּנֵי בָקָר שְׁנַיִם וְאַיִל אֶחָד, כְּבָשִׂים בְּנֵי שָׁנָה שִׁבְעָה תְּמִימִם.
And on the heads of your months you shall bring a burnt offering to Hashem: two bulls from the herd, and one ram, lambs in their first year, seven, without blemish.
What it means
And on the first days of your months you shall bring a burnt offering to Hashem: two bulls from the herd, one ram, and seven lambs in their first year, without blemish.
The Ribon HaOlamim prayer tells why some also add these verses to the morning offering passage and others do not.
Shulchan Aruch and Rema, Orach Chaim 48:1A summary, in the prayer's own words, of what Bamidbar 28:12-15 requires:
וּמִנְחָתָם וְנִסְכֵּיהֶם כַּמְדֻבָּר. שְׁלֹשָׁה עֶשְׂרֹנִים לַפָּר וּשְׁנֵי עֶשְׂרֹנִים לָאָיִל וְעִשָּׂרוֹן לַכֶּבֶשׂ, וְיַיִן כְּנִסְכּוֹ, וְשָׂעִיר לְכַפֵּר, וּשְׁנֵי תְמִידִים כּהִלְכָתָם.
And their grain offerings and drink offerings as specified: three tenths for the bull, and two tenths for the ram, and a tenth for the lamb, and wine like its drink offering, and a goat to atone, and the two daily offerings by their law.
What it means
And their grain offerings and drink offerings as specified: three tenths for the bull, two tenths for the ram, a tenth for the lamb, and wine like its drink offering, and a goat to atone, and the two daily offerings by their law.
"As specified" points to Parashat Pinchas, where each amount is written out. Shibbolei HaLeket reads it as "as it was spoken from God's own mouth." The prayer says only "wine like its drink offering." The Torah gives the wine: half a hin (a liquid measure) per bull, a third per ram, a quarter per lamb.
Shibbolei HaLeket 176:11; Bamidbar 28:14