Sporadic Simple Groups

It's kind of interesting that the prime factors of the sporadic simple
groups use primes that are important in tuning.

For example, the order of M24 is 2^10 * 3^3 * 5 * 7 * 11 *23.

Each prime is within one of some octave of 2 or 3:

2
3
6-1 or 4+1
6+1 or 8-1
12-1
24-1

The Monster is worse, it is

2^46 · 3^20 · 5^9 · 7^6 · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 · 41 ·
47 · 59 · 71

If you go one above each prime, and divide out by 2 you obtain
3, 1, 3, 1, 3, 7, 3, 5, 3, 15, 1, 21, 3, 15, 3, which are all simple
products of 1, 3, 5 and 7. The only one missing is 35, which actually
is right under 71 if you go the other direction:

1, 2, 1, 3, 5, 3, 1, 3, 11, 7, 15, 5, 23, 29, 35. Not as pretty.

So, maybe this could open up the field of "Monster Tuning?"

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:

> The Monster is worse, it is
>
> 2^46 · 3^20 · 5^9 · 7^6 · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 · 41
·
> 47 · 59 · 71

The question of why the primes dividing the Monster are what they are
is an old and fascinating one, and relates to how "Moonshine Theory"
got called that. The primes dividing the Monster are only and exactly
the supersingular primes, and when Andy Ogg noticed that, he offered
a fifth of Jack Daniels to anyone who could explain it. It turned out
not to be a coincidence, which is what most people probably expected
it was.

http://mathworld.wolfram.com/SupersingularPrime.html
http://en.wikipedia.org/wiki/Supersingular_prime

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
>
> > The Monster is worse, it is
> >
> > 2^46 · 3^20 · 5^9 · 7^6 · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 ·
41
> ·
> > 47 · 59 · 71
>
> The question of why the primes dividing the Monster are what they
are
> is an old and fascinating one, and relates to how "Moonshine
Theory"
> got called that. The primes dividing the Monster are only and
exactly
> the supersingular primes, and when Andy Ogg noticed that, he
offered
> a fifth of Jack Daniels to anyone who could explain it. It turned
out
> not to be a coincidence, which is what most people probably
expected
> it was.
>
> http://mathworld.wolfram.com/SupersingularPrime.html
> http://en.wikipedia.org/wiki/Supersingular_prime
>

Question:

Since E has to be 1 mod p, would my list have any bearing
with respect to the j-function (etc etc). Anyway, you get
this scale

CDEFGA#B The diatonic scale with a sharp 6th

and these intervals

21/20
16/15
15/14
10/9
9/8
8/7
7/6
6/5
5/4
9/7
4/3
21/16
7/5

A few used more than once
9/7