In the psalm for Thursday, Psalm 81 there is a command to blow the shofar at the New Moon. We no longer follow this practice, but presumably they did when the psalm was written, during the era of either the First or Second Temple. Why did this custom fall out of practice?
Not exactly sure about this, but wasn't it part of the temple process? "Uvayom simchaschem ovamo'adaychem ovaroshei chodshaychem" - therefore presumably it would have ceased when the use of horns in the temple stopped i.e.: at the Churban. More interesting is why they wouldn't institute monthly blowing zecher limikdash... One could speculate that it would be too close to the tekiyos on Rosh Hashanah and could lead to people treating them identically...– Isaac KotlickyMar 10, 2016 at 16:07
@IsaacKotlicky Exactly. So much of our ritual is based on remembrance of the Temple.– MikeMar 10, 2016 at 16:30
Seforno translates the verse תקעו בחדש שופר בכסה ליום חגנו as follows. Blow Shofar on the Rosh Chodesh that is that is Rosh Hashana.
See also Metzudas David who translates it similarly.
1As does the Gemara Rosh Hashana 8a-b– YishaiMar 11, 2016 at 16:44