over-equal and under-equal examples

Here's some examples to go with the O/U-equal morphing algorithms I've
recently posted.

Using the under-equal example, where x=2+(sqrt(2)), here's
n=7,8,9,10,11,12 in cents. Note that the maximum underlying errors for
any given "n" are only ~2�. This means that O/U-equal series are
virtually indistinguishable from their corresponding n-tETs:

0 174 345 515 685 855 1026 1200
0 152 302 451 600 749 898 1048 1200
0 135 269 402 534 666 798 931 1065 1200
0 122 242 362 481 600 719 838 958 1078 1200
0 111 220 329 438 546 654 762 871 980 1089 1200
0 102 202 302 402 501 600 699 798 898 998 1098 1200

I use three rational under-equal variations; x=3, x=3.5, and x=4.
These reset the 1:2^(1/2) as 5:7, 12:17, and 7:10 respectively.

Here's an example of each where n=7. Note that while the x=3.5
variation is essentially a rational "well-temperament" of the
=2+(sqrt(2)), itself a sort of well-temperament of its corresponding
n-tET, the x=3 and x=4 variations are more not. They are much more
like some very colorful rational recasting of the corresponding n-tET,
and could be said to represent a sort of O/U interpretational extreme:

1/1 11/10 23/19 4/3 25/17 13/8 9/5 2/1
1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1
1/1 10/9 16/13 34/25 3/2 38/23 20/11 2/1

Here's the same where n=12:

1/1 37/35 19/17 13/11 5/4 41/31 7/5 43/29 11/7 5/3 23/13 47/25 2/1

1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

1/1 50/47 26/23 6/5 14/11 58/43 10/7 62/41 8/5 22/13 34/19 70/37 2/1

The over-equal would be the same as the under-equal in every case
except for rational x=2.5 "well-temperament" of the x=1+(sqrt(2))
which sets the 1:2^(1/2) as 17:24. Here's the n=12 example which is of
course just a series inversion of the x=3.5 under-equal which sets the
half-octave as 12:17:

1/1 124/117 64/57 44/37 34/27 4/3 24/17 148/99 19/12 52/31 16/9 164/87
2/1

For summery sake, here's the two rules again...

O-series:

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.

Letting x=1+(sqrt(2)) gives an "over-equal series" with perfect
symmetry at the half-octave.

U-series:

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.

Letting x=2+(sqrt(2)) gives an "under-equal series" with perfect
symmetry at the half-octave.

--Dan Stearns