Each of the following passages from Parshat Noach represents a phase of spiritual cleansing that we see both on Rosh Hashana and, on a smaller scale, on Rosh Chodesh.
(יא) בִּשְׁנַ֨ת שֵׁשׁ־מֵא֤וֹת שָׁנָה֙ לְחַיֵּי־נֹ֔חַ בַּחֹ֙דֶשׁ֙ הַשֵּׁנִ֔י בְּשִׁבְעָֽה־עָשָׂ֥ר י֖וֹם לַחֹ֑דֶשׁ בַּיּ֣וֹם הַזֶּ֗ה נִבְקְעוּ֙ כָּֽל־מַעְיְנֹת֙ תְּה֣וֹם רַבָּ֔ה וַאֲרֻבֹּ֥ת הַשָּׁמַ֖יִם נִפְתָּֽחוּ׃ (יב) וַֽיְהִ֥י הַגֶּ֖שֶׁם עַל־הָאָ֑רֶץ אַרְבָּעִ֣ים י֔וֹם וְאַרְבָּעִ֖ים לָֽיְלָה׃
(11) In the six hundredth year of Noah’s life, in the second month, on the seventeenth day of the month, on the same day were all the fountains of the great deep broken up, and the windows of heaven were opened. (12) And the rain was upon the earth forty days and forty nights.
(כ) וַיִּ֥בֶן נֹ֛חַ מִזְבֵּ֖חַ לַֽיהוָ֑ה וַיִּקַּ֞ח מִכֹּ֣ל ׀ הַבְּהֵמָ֣ה הַטְּהוֹרָ֗ה וּמִכֹּל֙ הָע֣וֹף הַטָּהֹ֔ר וַיַּ֥עַל עֹלֹ֖ת בַּמִּזְבֵּֽחַ׃ (כא) וַיָּ֣רַח יְהוָה֮ אֶת־רֵ֣יחַ הַנִּיחֹחַ֒ וַיֹּ֨אמֶר יְהוָ֜ה אֶל־לִבּ֗וֹ לֹֽא־אֹ֠סִף לְקַלֵּ֨ל ע֤וֹד אֶת־הָֽאֲדָמָה֙ בַּעֲב֣וּר הָֽאָדָ֔ם כִּ֠י יֵ֣צֶר לֵ֧ב הָאָדָ֛ם רַ֖ע מִנְּעֻרָ֑יו וְלֹֽא־אֹסִ֥ף ע֛וֹד לְהַכּ֥וֹת אֶת־כָּל־חַ֖י כַּֽאֲשֶׁ֥ר עָשִֽׂיתִי׃
(20) And Noah built an altar unto the LORD; and took of every clean beast, and of every clean fowl, and offered burnt-offerings on the altar. (21) And the LORD smelled the sweet savour; and the LORD said in His heart: ‘I will not again curse the ground any more for man’s sake; for the imagination of man’s heart is evil from his youth; neither will I again smite any more every thing living, as I have done.
(יא) וּבְרָאשֵׁי֙ חָדְשֵׁיכֶ֔ם תַּקְרִ֥יבוּ עֹלָ֖ה לַיהוָ֑ה פָּרִ֨ים בְּנֵֽי־בָקָ֤ר שְׁנַ֙יִם֙ וְאַ֣יִל אֶחָ֔ד כְּבָשִׂ֧ים בְּנֵי־שָׁנָ֛ה שִׁבְעָ֖ה תְּמִימִֽם׃ (יב) וּשְׁלֹשָׁ֣ה עֶשְׂרֹנִ֗ים סֹ֤לֶת מִנְחָה֙ בְּלוּלָ֣ה בַשֶּׁ֔מֶן לַפָּ֖ר הָאֶחָ֑ד וּשְׁנֵ֣י עֶשְׂרֹנִ֗ים סֹ֤לֶת מִנְחָה֙ בְּלוּלָ֣ה בַשֶּׁ֔מֶן לָאַ֖יִל הָֽאֶחָֽד׃ (יג) וְעִשָּׂרֹ֣ן עִשָּׂר֗וֹן סֹ֤לֶת מִנְחָה֙ בְּלוּלָ֣ה בַשֶּׁ֔מֶן לַכֶּ֖בֶשׂ הָאֶחָ֑ד עֹלָה֙ רֵ֣יחַ נִיחֹ֔חַ אִשֶּׁ֖ה לַיהוָֽה׃
(יד) וְהָיָ֕ה בְּעַֽנְנִ֥י עָנָ֖ן עַל־הָאָ֑רֶץ וְנִרְאֲתָ֥ה הַקֶּ֖שֶׁת בֶּעָנָֽן׃ (טו) וְזָכַרְתִּ֣י אֶת־בְּרִיתִ֗י אֲשֶׁ֤ר בֵּינִי֙ וּבֵ֣ינֵיכֶ֔ם וּבֵ֛ין כָּל־נֶ֥פֶשׁ חַיָּ֖ה בְּכָל־בָּשָׂ֑ר וְלֹֽא־יִֽהְיֶ֨ה ע֤וֹד הַמַּ֙יִם֙ לְמַבּ֔וּל לְשַׁחֵ֖ת כָּל־בָּשָֽׂר׃
(14) And it shall come to pass, when I bring clouds over the earth, and the bow is seen in the cloud, (15) that I will remember My covenant, which is between Me and you and every living creature of all flesh; and the waters shall no more become a flood to destroy all flesh.
According to R. Munk, Rosh Chodesh is a time of spiritual renewal, one-twelfth of the amount of Rosh Hashana. Shacharit is like a festival (ya’ale v’yavo, Hallel), and Mussaf is a time of “self-evaluation and spiritual rebirth.” This mirrors the Tishrei-Cheshvan, Flood-Rainbow analogy.
What is it about Rosh Chodesh that it acts as a renewal, a mini Rosh Hashana?
Sforno comments on the Torah reading for Rosh Chodesh and describes the relationship between the Jewish people and the moon. Just as the moon is not its own source of life, instead depending entirely on the sun, so too the Jewish people do not have their own “Mazal.” They are not masters of their own fate; rather, they are entirely dependent on shefa and brachos from HaShem. When do they get these brachos? When they do the Will of HaShem. Thus, Rosh Chodesh is a monthly reminder to self-assess and to ask ourselves: am I acting in accordance with the Will of HaShem? Am I keeping my end of the bargain?
Therefore, as Sforno points out, Rosh Chodesh is the only Mo’ed referred to as “chodsheichem,” i.e. your months, and the sin offering brought is the only one referred to as “chata’as L’HaShem,” for HaShem. This distinction alludes to the fact that Rosh Chodesh is celebrated to remind us of the brit, the covenant, the pact, בֵּינִי֙ וּבֵ֣ינֵיכֶ֔ם.
How do we engage in this self-reflection? For that, we end Rosh Chodesh davening with mussaf, the silent, personal prayer that speaks of atonement and renewal. Rav Hirsch explains that "Hitpallel, from which “Tefillah” is derived, originally meant to deliver an opinion about oneself, to judge oneself or an inner attempt at so doing such as the hitpael (reflexive) form of the Hebrew verb frequently denotes … Thus it denotes to step out of active life in order to attempt to gain a true judgment about oneself … about one’s relationship to God and the world, and the world to oneself … In English we call Tefillah “prayer,” but this word only incompletely expresses the concept “to pray,” i.e., to ask for something is only a minor section of Tefillah."