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The Mishna in Rosh Hashono 4 (4) says that originally in the Beis Hamikdosh, evidence of the new moon (of Tishrei) was accepted all the day of the 30th of Elul (which was made holy as a Yom Tov). On one occasion, the witnesses came late and the the Leviim did not say the correct psalm for the afternoon sacrifice. After that event, it was instituted that witnesses would be accepted only up to the time of the afternoon sacrifice and if the witnesses came later, that day and the following day would be treated as Yom Tov.

It seems to me from the Mishna that they did not offer a Korban Mussaf on the 30th of Elul. Is that right?

We observe some two-day Roshei Chodoshim. Are they modelled on the observance described in the above Mishna?

If so, why do we pray Mussaf on the first day of Rosh Chodesh?

(This is different to the observance of 2-day yomim tovim. The second day was originally instituted because of a doubt and remains as a minhag of our forefathers. We do not know on which day the YT fell and so we have all the observances of the YT for both days. But here, the original practice was not to bring a korban Musaf on the first day.)

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R. E. Melamed addresses the issue of the reason for two days of Rosh Chodesh in a footnote here:

לכאורה יש לשאול, הרי ר"ח הוא היום הראשון של החודש, ומדוע כשהחודש מלא גם יום השלושים לחודש הקודם נחשב לר"ח? באר בשבולי הלקט קסח, בשם רבנו שלמה והרי"ד (רבי ישעיה הראשון), וכך מובא בברכ"י תכז, שכאשר החודש מלא, חידושה של הלבנה חל באמצע היום השלושים (לאחר עשרים ותשעה ימים וחצי), לפיכך, אף שא' לחודש חל למחרת (כדי לאזן את החודשים, כמבואר בהלכה א), מ"מ ראוי לעשות ר"ח גם ביום חידוש הלבנה, לפיכך עושים יומיים ר"ח. והתשב"ץ ג, רמד, כתב, שהיו נוהגים להימנע ממלאכה בר"ח ולערוך סעודות, וכבר מתחילת יום שלושים היו נוהגים כן, שאולי יבואו עדים ואותו יום יתקדש כר"ח, ואם לא היו באים עדים, ממילא נהגו גם ביום שלמחרת ר"ח. נמצא שכאשר החודש מלא נהגו ר"ח יומיים (בדומה לר"ה). ואף שקרבנות מוסף הקריבו רק בא' לחודש, מתוך הטעמים הללו יוצא שבחודש מלא יש קדושה בשני הימים, ולכן אומרים בשניהם הלל ומוסף ויעלה ויבוא.

Seemingly, one could ask: Rosh Chodesh means the first day of month; why then is the thirtieth day of a full month also considered Rosh Chodesh? The Shibolei HaLeket (168) quoting Rabbeinu Shlomo and the Rid (the earlier R. Yeshayah), as well as the author of Birkei Yosef (427), explain that when a month is full, the new moon appears in the middle of the thirtieth day (after 29½ days). Therefore, even though the next day will be the first of the month (in order to balance out the months, as explained above – 1.1), nonetheless, it is fitting to also treat the day on which the moon reappears as Rosh Chodesh. This is why we keep two days of Rosh Chodesh.

The Tashbetz (3:244) writes that the Jews in ancient times used to refrain from work, and prepare festive meals in honor of Rosh Chodesh, and they would begin doing so already on the thirtieth day of the month, in the event that witnesses arrive and the Beit Din declares that day the first of the month. And if no one came to testify, they treated the next day as Rosh Chodesh as well. Thus, whenever a month was full, they kept two days of Rosh Chodesh (similar to Rosh HaShanah). Even though the Musaf sacrifices were offered up exclusively on the first of the month, these two explanations indicate that both days are holy, and we therefore say Hallel, Musaf, and Ya’aleh VeYavo on both days.

(English translation taken from here)

So there are two possible reasons for treating the thirtieth of a month as Rosh Chodesh. Either because the molad falls part way through that day, or (as you suggested) due to the possibility of witnesses appearing during the day, similar to Rosh HaShanah.

Now, R. Melamed points out (as you noted) that they would not have offered up a korban musaf on the thirtieth of the month, unless witnesses arrived and beit din declared that day to be Rosh Chodesh. Nevertheless, because we treat the day as Rosh Chodesh, he writes that we still say tefillat musaf (along with hallel and ya'aleh veyavo.)

I would suggest that the reason for doing so is due to the difficulty that would be involved in instructing people to treat a day as Rosh Chodesh, but telling them not to recite tefillat musaf.

  • The reason for the first part of the quote is to address the OP's preliminary questions: "It seems to me from the Mishna that they did not offer a Korban Mussaf on the 30th of Elul. Is that right?" and "We observe some two-day Roshei Chodoshim. Are they modelled on the observance described in the above Mishna?" – Joel K Jan 2 at 13:51
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The musaf offerings were certainly never brought before the Sanhedrin declared that that day was actually rosh chodesh based on the testimony of witnesses to the new moon. The recitation is only in places where one wouldn't yet know if that declaration had already occurred, akin (as noted by @doubleAA in a comment) to the two day yom tovs of the diaspora (see Rosh Hashana 2:2; see also Rambam Hilchoth Qiddush Hachodesh Ch. 5).

  • unless you're asking how could they recite musaf without knowing if (and/or knowing that likely not) that day would be declared; or why then there shouldn't also be 2 days after short months. I would speculate that the practice is based on places where word would generally reach by the second day if the first had been declared, so they always recited musaf on the first day just in case, so that they wouldn't lose track of the months. – Loewian Jul 13 '18 at 15:35

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