meta meantone bis

In tuning@yahoogroups.com, Kraig Grady wrote:

> A basic criteria of meantone is that 4 fifths will be equal to a third.
> In order for this relationship to be a proportional triad, the
> difference tones between the upper and lower tones in relation to the
> middle tone must be the same. If we look at 4-5-6 we notice that the
> difference tones are one. Now if we continue to feed a recurrent
> sequence with the formula of meta-meantone (expressed in simplest terms
> (A+B)x2=E we converge upon a certain number. This formula says that A
> which we could call the root plus B otherwise known as the "fifth" is
> equal to E the third. Why E? Because in the chain, E is four fifths
> above. Now it is possible to seed this formula how one wishes. In Erv's
> example on page 4 he starts 1 2.5 3 5 7 11 but it might be
> clearer if we start with 16 24 36 54 80 as A, B, C, D, E. we see that
> (16+ 24)x2=80 so next we treat 24 as A and then 36 as B which generates
> 120 and so on. Going back to Page 1 we see the series carried out beyond
> that on page 4 . Already with the last few numbers we can see that it
> closing in on converging on the stated ratio. Which comes out BTW to be
> 695.6304373 cents and the triads will be proportional and equal beating.

At first I thought I knew this series, as years ago I found one starting strictly by the same 7
frequencies.
But mine was only very close (1.493 358 556 560 19 or 694,27 c,
instead of Erv Wilson's 1.494 530 180 48 ... or 695.63 c)
the fifth in my algorithm fulfills the equation 4x - x^4 = 1 while his does x^4 - 2x = 2
taking your example we can express it 4B - E = A and the series (simplified by 2) goes
8 12 18 27 40 60 90 133 200 300 442 667 1000 1468 2226 3333... instead of
8 12 18 27 40 60 90 134 200 300 448 668 1000 1496 2232 3336...
my musical interest here was to have the difference tones generated by the minor thirds (4B - E)
to belong to the scale, being equal to A. The first difference comes only with the eigth term, 133
(instead of 134 in the other serie) and indeed 160 - 133 = 27 makes the minor third coherent,
while in Erv's it's the major sixths which have their first-order difference tone in the scale,
ex. 134 - 80 = 54 and so on.
It is rare to find fractal series that close, and this co�ncidence should help to find specially
interesting sounds for the meta-meantone.

If you like that type of series which provide, I think, fantastic alernatives to current
temperaments,
this is another one : take the five first terms of the above metas, 8 12 18 27 40 and turn them

upside down, which create a cycle of fourths, 20 27 36 48 64 - and note there is a another
type of minor third here, 64 - 54 (octave of 27) = 10, which is the octave below 20. If we repeat
that sequence recurrently, we generate a series where all minor thirds have also their difference
tone
in tune with the fifth interval, resumed in the algorithm 2x^4 - 4x = 1 (or 2E - 4B = A)
Now the ratios between consecutive frequencies are converging towards 1.334 157 447 130 13...
or 499.11 c, which is not far from 4/3. Because of a need for even numbers for A, the series need
constant octave-refreshing, such as 80 108 144 192 256 342 456 608 812 1083 1444 1928
...and so on (quite lovely for medieval music)