Bert, I thought I remembered from Latin class:

Kalends = 1st
Nones = 5th
Ides = 15th

Except that your poem made sense in that "nones" should be the ninth.

So I did a google search and found the answer: Kalends, Nones, and Ides The Kalends is always the first of the month, the Ides is the 13th in 8 months and the 15th in 4, and the Nones is the 5th when the Ides is the 13th, but the 7th when the Ides is the 15th. The Romans identified days as so many before the next Kalends, Nones, or Ides.

So your poem was correct, but only if you substitute a 7th for a 9th (See? I made this into a musical thread!) The Nones was so named because it was 9 days before the Ides (including the Ides in the count).

There's also a fascinating Roman Calendar site that describes which days were for business, which for voting, which for religious festivals, etc. And it gives the original names for the months:
1. Martius
2. Aprilis
3. Maius
4. Iunius
5. Quinctilis (later named for Julius Caesar, who made Ianuarius the first month of the year)
6. Sextilis (later named for Augustus Caesar)
7. September
8. October
9. November
10. December
11. Ianuarius
12. Februarius

Aren't you glad you asked?

Aloha,
Marcus