Golden Secor

One Miracle alternative would be to use the Golden Secor of
(4 phi +3)/(41 phi + 31), of about 116.77 cents. Then there's always
(3 phi + 4)/(31 phi + 41) at about 116.55 cents if you get tired of
the first one. Something for Jacky or Joseph to consider, or anyone
who likes scales fitting into other scales endlessly.

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29851.html#29851

> One Miracle alternative would be to use the Golden Secor of
> (4 phi +3)/(41 phi + 31), of about 116.77 cents. Then there's always
> (3 phi + 4)/(31 phi + 41) at about 116.55 cents if you get tired of
> the first one. Something for Jacky or Joseph to consider, or anyone
> who likes scales fitting into other scales endlessly.

Hello Gene!

This raises some really interesting questions, since it has been made
clear to me that there are many "miracles" with generators around 116
cents. I believe it was stated that 31-tET even fits into this
pattern.

I'm not sure about all the mathematics behind this... I'm sure you
can work easily with it... but I remain fascinated that so many
scales with quasi just intervals are created from generators that are
very similar but different!

__________ _______ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> Hello Gene!
>
> This raises some really interesting questions, since it has been
made
> clear to me that there are many "miracles" with generators around
116
> cents. I believe it was stated that 31-tET even fits into this
> pattern.

(3/31) * 1200 = 116.129..., so there it is. Anything around 116.7
cents will work; the point of 3/31, 4/41 or 7/72 is that the circle
of secors closes up. The point of the Golden Secor is that while it
doesn't close up, it does something else interesting--the circle of
secors continues in a way which makes for smooth patterns of steps,
as it fits itself between the previous steps by a golden ratio
division. It can also be written (3 phi + 1)/(31 phi + 10), and after
completing the golden version of canasta, it fills in ten of the
steps to get 41-miracle-golden, and then 31 more to get 72-miracle-
golden; after which it simply keeps on going, according to the
sequence 10,31,41,72,113,185...

> I'm not sure about all the mathematics behind this... I'm sure you
> can work easily with it... but I remain fascinated that so many
> scales with quasi just intervals are created from generators that
are
> very similar but different!

There are always infinitely many numbers next to any number,
including 7/72.