EDO conversion algorithm revisited

THE CONVERSION ALGORITHM...
If you let "e" be any equal division of the octave and then let "n" be
the fraction of "e" where n is any number =/> 0 and =/< e/2, then N/D
= (y+n)/(y-n) where "y" = x*e and "x" is the product of 7 and the
arithmetic mean fourth of 5 and 7e divided by 12, i.e., the "magic
number" 2.9.

"X" - MAGIC NUMBERS...
I doubt that there is any one optimal magic number derivation of x
here as different derivations do different things; x = 2.9 sets the
half octave as 17/12 for example. Dave Keenan (in a TD post way back
last winter) took a look at this algorithm and empirically determined
~2.90708338 (or 125/43) to be the best magic number due to its maximum
underlying error of only +/- ~1.5 cents. When Dave says "underlying
error" he means an error apart from those due to rounding the
numerator and denominator to the nearest integers. So in other words,
in 15e if we let x = 125/43, then we have underlying errors of +/- .6,
1.1, 1.4, 1.5, 1.2, .5, and .7 cents:

1.014 / 1.014
22.302 / 21.302
45.605 / 41.605
23.302 / 20.302
47.605 / 39.605
24.302 / 19.302
49.605 / 37.605
25.302 / 18.302

and rounded errors of +/- 0.54, 2.51, 1.96, 4.36, 4.44, 4.89, and 8.72
cents:

25/36, 25/38, 12/19, 3/5, 23/40, 23/42, 11/21, 1/1, 22/21, 23/21,
23/20, 6/5, 24/19, 25/19, 25/18

Or if we let x = 2.9 and use 12e as the example, then there would be
underlining errors of +/- .5, .8, .8, .2, .9, and 3 cents:

1.024 / 1.024
35.800 / 33.800
9.200 / 8.200
37.800 / 31.800
19.400 / 15.400
39.800 / 29.800
10.200 / 7.200

and rounded errors of +/- 1.05, 3.91, 2.49, 9.24, 1.96, and 17.49
cents:

5/7, 2/3, 19/30, 19/32, 9/16, 9/17, 1/1, 18/17, 9/8, 19/16, 19/15,
4/3, 10/7

I've also used (LOG(4)-LOG(3))*((7/1)/LOG(2)), or ~2.905 for "x" to
make the minimum deviation from n/e the nearest approximation to 4/3,
i.e., round (LOG(4)-LOG(3))*(e/LOG(2)). Letting x = ~2.905, 13e would
have underlining errors of +/- .6, 1.1, 1.3, 1.1, .5, and .9 cents:

1.021 / 1.021
19.384 / 18.384
39.768 / 35.768
20.384 / 17.384
41.768 / 33.768
21.384 / 16.384
43.768 / 31.768
22.384 / 15.384

and rounded errors of +/- 1.30, 2.21, 4.44, 3.41, 9.24, and 2.53
cents:

11/16, 21/32, 21/34, 10/17, 5/9, 19/36, 1/1, 19/18, 10/9, 20/17,
21/17, 21/16, 11/8

INVERSIONAL SYMMETRY...
As the algorithm I'm using here sets "n" as any number =/> 0 and =/<
e/2 there is always inversional symmetry, and all equal numbered EDOs
will always have two ratios representing the half octave (when x =2.9
these are always separated by an underlying 289/288). And in the
rounded examples above you can see that I've written them in a fashion
that is analogous to reading this from left to right on a tonality
diamond.

MUSICAL USEFULNESS...
In looking at this algorithm Dave Keenan also commented that he
thought it was "too accurate," in other words it gives ratios which
are too complex to be musically significant, and that the whole point
of using ET instead of JI is to make commas disappear and allow the
same equal tempered fraction of an octave to serve more than one
purpose, to function as more than one ratio. And he is of course quite
correct, and I really have nothing to offer in the way of a counter
argument (especially if punning and eradicating commas are your
primary concerns!). But I will say that I personally prefer to let
effective music make its structural components musically useful above
and beyond whatever obvious utility those components are said to
serve. That said, I really do think that most EDOs and their ratio
conversions here would just disappear into each other in a blur of
naturally occurring errors.

X = 3 AND MUSICAL USEFULNESS REVISITED...
Letting x = 3 and n be any number =/> 0 and =/< e allows one to work
EDOs into a manner of just intonation - a sort of reverse tempering if
you will. Earlier I wrote the rounded examples in a fashion that is
analogous to reading inversional symmetry from left to right on a
tonality diamond, here this would be analogous to reading the diamond
center to right, i.e., the 6/6, 7/5, 8/4, 2e in the following:

1/1
1/2 2/1
1/3 2/2 3/1
1/4 2/3 3/2 4/1
1/5 2/4 3/3 4/2 5/1
1/6 2/5 3/4 4/3 5/2 6/1
1/7 2/6 3/5 4/4 5/3 6/2 7/1
1/8 2/7 3/6 4/5 5/4 6/3 7/2 8/1
2/8 3/7 4/6 5/5 6/4 7/3 8/2
3/8 4/7 5/6 6/5 7/4 8/3
4/8 5/7 6/6 7/5 8/4
5/8 6/7 7/6 8/5
6/8 7/7 8/6
7/8 8/7
8/8

Changing the EDOs to non-whole number values -- i.e., the stretched
and compressed extremes of what the algorithm will still round to
2/1 -- offers a wealth of possibilities that often times radically
differ from the original EDOs (despite the fact that the amount of
stretching or compressing required is often trivial) or a "classic" JI
to ET (or ET to JI) interpretation.

For example if e = 12, we get 1/1, 37/35, 19/17, 13/11, 5/4, 41/31,
7/5, 43/29, 11/7, 5/3, 23/13, 47/25, and 2/1, or as it would read
center to right on a ("48-limit") diamond: 36/36 37/35 38/34 39/33
40/32 41/31 42/30 43/29 44/28 45/27 46/26 47/25 48/24. But say we
stretch the octave by ~1� by letting e = 11.99, we then get a rounded
ratio conversion of 1/1, 18/17, 19/17, 19/16, 5/4, 4/3, 7/5, 3/2,
11/7, 22/13, 9/5, 23/12, 2/1. Or using the same ~1� stretching of 11e,
let e = 10.99 and you then get a rounded conversion of 1/1, 17/16,
17/15, 6/5, 9/7, 19/14, 19/13, 20/13, 5/3, 7/4, 21/11, 2/1.

Etc., etc.

Dan