The shul you were attending seemed to have been following the custom of the Vilna Gaon. Rabbi A Grossman has an article entitled The Vilna Gaon’s Psalms for Special Days. Some extracts: ...the Vilna Gaon was faced with a conflict. Like Maimonides, he believed that the public prayers officially ended with the reader’s full qaddish, what we call ...


Maaseh Rav 157 (a work documenting the practices of the Vilna Gaon) records the answer according to the tradition you ask about: אין אומרים שיר של יום של יומו ואומרים במקומו בב' ימים דר"ח ברכי נפשי (מזמור ק"ד) שהוא שיר של ר"ח ודוחה אפי' של שבת ופשיטא של חנוכה [On Rosh Chodesh] the Song of the Day for that day [of the week] is not said, and in its place ...


The popular sidur "תפלת כל פה" has a list of all the red-letter days' daily psalms. It says that 104 is said on rosh chodesh Teves (and also on Shabas rosh chodesh, by the way).


(Note that even with the list in Maseches Sofrim, there are a few days with arguments or multiple psalms which indicate that these choices were not absolute.) Not perfect, but the best I could do... In Chashukei Chemed to Sukkah 55a, it says the following: ז. השיר זכר למעשה בראשית - כתב בשו"ת בנין שלמה (ח"ב או"ח סימן א' הערות על 'מעשה רב - סימן קנח', ...


Here is a link to a chart of the Gra's order. At the top of the chart three regulations are listed: Two psalms are not recited on any given day. The psalm for the New Moon supersedes any other psalm. The psalm for the Sabbath supersedes the psalms of festivals and Hanukkah.


In Maseches Sofrim (which was written by the Amoraim of Eretz Yisroel) 18,2-3 there is a list of Mizmorim for every Yom tov בחנוכה (שם ל) ארוממך ה' בפורים (שם ז) שגיון לדוד ביום הראשון של פסח (שם קלה) הללו את שם ה' (וי"א (שם פג) אלהים אל דמי לך וכן חולו של מועד אלא בימים הראשונים של פסח הוא צריך לומר יהי כבוד ה' וכל העם בעמידה עד וברוך שם כבודו ...


Echad, Shtayim, Shalosh (Cardinal numbers) Rishon, Sheni, Shlishi (Ordinal numbers) Ordinal makes more sense to me: You are ranking (enumerating) the days, not counting how many there are. But Bereishis mixes the two types - which is explained here: http://www.ou.org/torah/tt/5769/bereishit69/aliya.htm "The day is called YOM ECHAD (cardinal number) ...


Rabbi Dovid Cohen (in his Sefer משאת כפי pg 104)suggests, that it is to indicate that the main part of shira was the part that was done orally, and not the musical instruments.

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