Distribution variant

Farey series or related series are particular Stern-Brocot subtrees which
differ by a contingent way of limiting values. Mediant property has nothing
to do with restriction modalities but only with fact of being Stern-Brocot
subtrees. Besides, with calculation depending of distance between ratios,
jump to subsequent series results in few abrupt change in places.

For the sake of coherence it could be useful for particular calculation to
use series of N-first [strata/layers/levels??] of Stern-Brocot tree. What
is expecting ?

- well-distributed changes from a series to the next
- possibility to rely on coherent distribution (new variant).

Here are N-first complete layers series in Stern-Brocot tree for N=1 to N=5 :

1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5
5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 4 3 5 2 5 3 4 1 3 2 3 1 2 1 1

1 1 2 1 3 2 3 1 4 3 5 2 5 3 4
4 3 5 2 5 3 4 1 3 2 3 1 2 1 1

1 1 2 1 3 2 3
3 2 3 1 2 1 1

1 1 2
2 1 1

1
1

And here are N-first layers series limited to first octave for N=5 and N=6 :

1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2
1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1

1 5 4 7 3 8 5 7 2
1 4 3 5 2 5 3 4 1

Now, what I mean by coherent distribution ?

As you know probably, by citation of Alex Bogomolny, I have soon
demonstrated that Arithmetic Sum of simplicities on each layer is equal to
1. This is equivalent to Harmonic Sum of complexities defined as 2^sonance
or ab, in the case of reduced a/b. I recall what is Harmonic Sum :

HS (a, b, c ...) = 1/(1/a + 1/b + 1/c ...)

If we attribute to each reduced ratio a/b of N-first layers series a weight
equal to Nab then Harmonic Sum of those weights is equal to 1. I don't need
to demonstrate that since it's almost an evidence.

New fitted weights are defined at limits and vary only by factor N/N-1 when
jumping to N.

I did'nt read sufficiently on entropy theory to know if relevant as
starting series. It seems only, at first glance, that it would be
independant of subsequent physiological (and other) hypothesis by which
smoothing and upper limitation of distribution could be obtained.

Here are weights for N=1 to N=4 of complete series :

16 12 40 8 60 24 48 4 48 24 60 8 40 12 16
9 6 18 3 18 6 9
4 2 4
1

And here are weights for N=1 to N=6 for series limited to first octave :

6 80 120 378 72 528 210 420 36 462 240 624 80 504 168 270 12
5 100 60 175 30 200 75 140 10
4 48 24 60 8
3 18 6
2 4
1

I wouldn't encroach on Paul's theory concerning smoothing and upper
limitation of dissonance curve, but I would like to express general
comments about all possible theories where perception could be implied.

I would say that perception don't create phenomenon but only sensation that
tends to reveal part of reality. Art may freely focus on impression and
texture but science may not. How generally (in mind) presence at musical
work is built implies understanding of how intelligibility is preserved in
complex sound perception. By analogy, I ask : "Has mind only PhotoShop
working on bitmaps or also FreeHand working on vectorialy built images ?"

My remark don't concern perceptive theories as such but possible
unrestricted interpretation out of validity domain. Besides, in the same
spirit, I recall that "lot of mathematics" is not synonymous with "most
scientific".

I don't remind who has written something like that : "Computers aren't very
useful, they give only anwsers".

Pierre Lamothe