morphing series into series

One of the things I personally gravitate towards in my own tuning
designs and work is inclusive generalizations; where given the
characteristic limits of a design the potential possibilities are
still quite limitless. That most of these designs tend towards
blurring rather than distinguishing the various distinctions between
free, linear, and logarithmic tuning methodologies only makes sense as
it is consistent with my own musical practice and aesthetic beliefs.

Here's a nice little method that allows one to either morph a given
n-over series into its n-under series or a given n-under series into
its n-over series.

OVER INTO UNDER:

Where "n" and "x" are any given numbers, an over to under series is
defined as 2n*x with a sequential numerator rule of +2 and a
sequential denominator rule of -(x-1).

Letting x=1 gives a corresponding n-over series. Increasing the value
of x by 1s works n towards ever more accurate approximations of its
under series.

Here's an example using n=4 where x=1,2,3,4,5:

1/1 5/4 3/2 7/4 2/1

1/1 6/5 10/7 22/13 2/1

1/1 13/11 7/5 5/3 2/1

1/1 34/29 18/13 38/23 2/1

1/1 7/6 11/8 23/14 2/1

UNDER INTO OVER:

Where "n" and "x" are any given numbers, an under to over series is
defined as n*x with a sequential numerator rule of +(x-2) and a
sequential denominator rule of -1.

Letting x=2 gives a corresponding n-under series. Increasing the value
of x by 1s works n towards ever more accurate approximations of its
over series.

Here's an example using n=7 where x=2,3,4,5,6:

1/1 14/13 7/6 14/11 7/5 14/9 7/4 2/1

1/1 11/10 23/19 4/3 25/17 13/8 9/5 2/1

1/1 10/9 16/13 34/25 3/2 38/23 20/11 2/1

1/1 19/17 41/33 11/8 47/31 5/3 53/29 2/1

1/1 46/41 5/4 18/13 29/19 62/37 11/6 2/1

--Dan Stearns