Took a bit of time but with today's clue, I think I've got a solution.
The carols are spoilered separately at first for anyone who wants to check without seeing my other identifications.

First, we start by converting the Solfege into the carols:

>! 1. Good king Wenceslas  

>! 2. Love came down at Christmas  

>! 3. We three kings of orient are  

>! 4. Our day of joy is here again  

>! 5. Hark! The herald angels sing  

>! 6. Silent night, Holy night  

>! 7. Redeemer of the nations, come  

Next, sort them into
>! Alphabetical order:  
>!  
>! - Good king Wenceslas  
>! - Hark! The herald angels sing  
>! - Love came down at Christmas  
>! - Our day of joy is here again  
>! - Redeemer of the nations, come  
>! - Silent night, Holy night  
>! - We three kings of orient are  

and extract letters by
>! Removing punctuation and spacing, and reading diagonally downwards, in keeping with the Professor's name and preferred temperature:  
>!  
>! ```Goodkingwenceslas```  
>! ```hArktheheraldangelssing```  
>! ```loVecamedownatchristmas```  
>! ```ourDayofjoyishereagain```  
>! ```redeEmerofthenationscome```  
>! ```silenTnightholynight```  
>! ```WethreEkingsoforientare```  

to give us
>! GAVDETE in [Classical Latin](https://en.wikipedia.org/wiki/Latin_alphabet#Classical_Latin_alphabet)

So the 8th carol must be

>! _Gaudete, gaudete! Christus est natus_  
>! which is only appropriate for [Gaudete Sunday](https://en.wikipedia.org/wiki/Gaudete_Sunday) - credit to @Stiv for the connection