The common confusion between "a tempo" and "Tempo I."
One of the most common mistakes I see in scores is the accidental misuse of markings such as Tempo I and a tempo. The accidental interchanging of these two terms is common not only in the work of young composers, but I have also seen it in scores from world-class composers. Perhaps it comes from people thinking "a tempo" means "A Tempo," which could be confused with "Tempo I." I see it often enough that I felt the necessity of this article to help clear up the misunderstanding and hopefully guide people towards using the terms correctly in their music.
The "a tempo" marking is used after a temporary change, such as a ritardando. It signifies a return to the tempo that was in place immediately before the temporary change. Please notice that it is "a tempo," not "A tempo" or "A Tempo." Please also notice that it (along with the rit. marking) is not in italics and on each individual staff, but in bold Roman type and placed only above the top staff in the score. This is because it is a global marking which affects everyone and not just a local marking. You will find lots of old scores with these markings written in italics in each staff, but those were made before this particular practice was standardized.
Tempo I is more of a "big picture" marking, signifying the return to the original tempo of the piece after a significant change of tempo (more than just a ritardando). In order to demonstrate how it is best used, I'll need to make a larger example so you can see the overall form. Notice that the first time it is used (Measure 50) I have also included the BPM in parenthesis to confirm the tempo. In any successive occurrences, simple marking Tempo I will suffice (Gould, p.184).
If you really want to lean into it, you can also use Tempo II as an indicator for the Waltz section.
I hope this helps clarify the confusion that sometimes surrounds these two terms. If you have any questions, please feel free to leave a comment or send me an email.