Yom Kippur Katan
Yom Kippur Katan – The Miniature Day of Atonement – is Erev Rosh Chodesh according to the custom of Rabbi Moshe Cordevero. (Pri Chadash; Laws of Rosh Chodesh 417)
The Shelah HaKodesh explains that since Rosh Chodesh is a time of atonement, a person should prepare with a day of repentance, meaning; use the day as a miniature Yom Kippur to fix all he can whether in monetary matters, matters of the soul or body. He should confess his sins, express his regret and commit to abandon his ways so that he can enter the new month as a new creation. (See too Otzar Hatefillot; Seder Yom Kippur Katan)
The Ya’avetz urges people who work long hours and do not have sufficient time to focus on their spiritual development to use this one day each month to remind themselves of the importance of focusing on the purity of their souls.
Others say that the final day of the month can repair the entire month, much as Elul can repair the entire year. (Otzar Tefillot; Seder Yom Kippur Katan)
The Besamim Rosh explains that fasting on Erev Rosh Chodesh can replace the sin-offering brought in the Beit Hamikdash on Rosh Chodesh.
The Tzelach (Pesachim, Chapter 4) mentions that some communities actually recite “Barush Shem kivod Malchuto” aloud as we do on Yom Kippur. Although most authorities are critical of this practice, the Tzelach finds reasons to defend it.
The Mahar MiFano explains that we fast because of the shrinking of the moon.