--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
Joseph,

As a complement to Monz's explanation, see what I just added to
http://uq.net.au/~zzdkeena/Music/MiraclePitchChart.gif

It shows the octave as a circle divided into 72 parts. Start at D> and
follow the (new) straight line segments clockwise. You'll see each one
jumps 7/72 of an octave. When you've done that 20 times and wound up
at D<, you've generated Blackjack.

> Dave Keenan found the MIRACLE generator (~116.7 cents) by use
> of the "brute force" approach: he had his computer perform
> thousands (millions?.. billions?) of calculations and analyze
> the resulting scales.
>
> The ~116.7-cent generator came out on top as implying the
> largest number of 11-limit consonances. ...

Er, no. That all came _after_ the discovery, and merely confirmed its
"miraculous" nature (as a 7-limit or 11-limit generator, but not necc.
9-limit). I was afraid there might have been some holes in my search
strategy, but since then Graham Breed has performed a search using a
completely different method to mine, and (I think?) further confirmed
it.

Strictly speaking, the MIRACLE generator was discovered by Paul
Erlich, who extracted it from a scale that I posted, the 31-noter that
we now call Canasta. Paul then recognised that there was a more
manageable (although improper) MOS with 21 notes using the same
generator. So historically it went: Canasta - MIRACLE generator -
Blackjack - Decimal scale (although, apart from "Blackjack" we didn't
call them that immediately). But logically it goes: MIRACLE generator
- Decimal scale - Blackjack - Canasta.

So Graham,

By what figure-of-demerit and at what odd-limits can we claim that the
MIRACLE generator is the best?

Does cardinality_of_smallest_MOS_containing_a_complete_otonality
divided by exp(-(minimax_error/17c)^2) do it at 7 and 11 limits?
What's the best 9-limit generator by this FoD?

I'm sure some folks would be interested in the 13-limit result too.

Regards,
-- Dave Keenan