One benefit of my fruitless efforts to determine the meaning of the word bruma was to cause me to read bits of the Birthday book (De die natalis) by Censorinus. This work was written in 238 AD (as he tells us) as a present for a friend, and is a compilation of material, mostly linked to the theme of birthdays. I came across this portion, which quotes from a lost book of Livy, on the secular games. It shows how much uncertainty there was on these matters, even in antiquity.
7. Some think that the Roman “age” or saeculum is marked by the Secular Games. If we could believe this, it would mean that the time span of the Roman saeculum or “century” is variable, since when it comes to the intervals at which the games ought to be held, we are ignorant not only of how long they were in the past but also how long they ought to be now. 8. We have it on the authority of Antias and other historians that they were originally founded to occur every hundred years. Varro wrote the same thing in Book I of The Early Theater at Rome: “Since there were many portents, and both the wall and tower between the Colline and the Esquiline gates were touched by heaven, the Council of Fifteen consulted the Sibylline books and announced that the Tarentine Games should be held in the Field of Mars in honor of Father Jupiter and Persephone for three days, and that black animals should be sacrificed, and that the games should be held every hundred years.”
9. Livy says the same in Book 136:
“In that same year [17 BC], Caesar Augustus held the Secular Games with great pomp, which once was the custom to hold every hundred years, for these games marked the end of a saeculum.”
On the other hand, the Records of the Council of Fifteen and the edicts of the divine Augustus seem to testify that the games were repeated after 110 years. So Horace in the song which was sung at the Secular Games designated the length of time as:
A fixed circle of ten times eleven years
to bring back song and crowded games
This from Censorinus: the birthday book, tr. Holt N. Parker, Chicago (2007), p.36. And he then goes on to list instances of uncertainty on this matter. Slightly earlier he adds:
An “age” (or saeculum) is the longest period of human life, delimited by birth and death.
which is again an interesting view.