the tooth fairy's algorithm

Pentagonal numbers in the form of

n(3n-1)/2

and

n(3n+1)/2

can be seen as a unique way to tie equal temperaments to overtone and
undertone series.

Say n = 4. If you take an overtone sequence as

n*2-1, ..., n/2

and an undertone sequence as

n/2+1, ..., n

you would then have

7,6,5,4

and

5,6,7,8

These sequences can be seen as superparticular ratios in the guise of
a consecutive integer sequence where their sum is an equal
temperament. Interestingly the superparticular ratios and the uniquely
articulated fractions of an octave both increase by consecutive
increments of 1.

So if n = 5, n(3n-1)/2 = 35. And this pentagonal number corresponds to
a 5-10 overtone series in 35-tET.

0 309 583 823 1029 1200

9 = 6/5
8 = 7/6
7 = 8/7
6 = 9/8
5 = 10/9

And if n = 5, n(3n+1)/2 = 40. And this secondary pentagonal number
corresponds to a 10-5 undertone series in 40-tET.

0 180 390 630 900 1200

6 = 10/9
7 = 9/8
8 = 8/7
9 = 7/6
10 = 6/5

Hopefully this will help make the other figurative number post -- the
square number "mirrored symmetry and a centralized tonic" one from the
other day -- a bit clearer.

--Dan Stearns