Tribonacci

Well, some sources call 1,1,1,3,5,9,17,... "the" Tribonacci sequence
instead . . .

Hi Daniel,

Thanks for sharing this great moment it seems you had together, between such
talentuous specialists.
This discussion was rather entirely on fractal divisions of the octave, that
is of course one of these recurrent sequences important applications.
I am surprised that at no moment the solution ratio is mentionned !
It is 1.83928675521 and I made several disks with its waveform which is one
of the 3rd degree waveforms I consider as perfectly resolved.
Its sound is totally soft and united and the form itself is beautiful (I
liked the description Paul made of this waveform, like if he had it in front
his eyes !)
Actually thanks, I did not know it was known as "Tribonacci", I call it
"Awj", because of its strongly audible and haunting maqam-like neutral 7th.

I could write pages and pages about this fractal, that is fascinating and
connected with several others, but I don't know if it would interest people
on this list and also it's not one of the simplest to start with.
Just this, since we talked about series F and P lately and because these
speak better than pages of equations :

F is 1 1 2 4 7 13 24 44 81...

P is 1 1 4 6 11 21 39 71 131...

But most important is another one , the S series :

= 1 2 3 6 11 20 37 68 125...
because then appears one major differencial coherence : 44 - 37 = 7, etc.,
of the (sqrt of a) / (a + 1) interval which I call Mi-majinn, some kind of a
slightly strong 19/16, itself a fractal, that applies here to F and S
together in association... (and S will have his own S' and so on...)

Another cool recurrent property that applies to each single series is :
(2 times 44) - 81 = 7 also, etc. - this is the coherence of the neutral
second complementary interval which I call Scylla, itself fractal.

More complex than Phi. But great sound !

Anything to say about the octave divisions applications of Tribonacci ? were
they used by any of you, or others ?

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Jacques

It seems you've started delving into polynacci ratios.
You all should know the 2-nacci or Fibonacci ratio is PHI.
The ratio is actually the limit of n of the 2-nacci sequence: Xn/Xn-1.
Phi is ((5)^(1/2)+1)/2 = 1.6180339887498948482045868343656

Now, the traditional way to look at it is the first two numbers are INTEGERS and are 1,1. In actuality, I've created proofs showing that the first two numbers can be ANY number including ZERO and imaginary numbers and they limit to +/- infinity is always PHI.

So, you've actually just taken it one step further to 3-nacci or Tribonacci.
The ratio is again Xn/Xn-1, but the starting numbers are traditionally 1,1,1. Your "F series" is actually 0,1,1. Your "P series" is actually 2,1,1. Your "S series" is actually 0,1,0.

The limit of the Tribonacci sequence is 1.839286755214~.
Now, I've found limits of 2-nacci to 10-nacci.
The limit of n-nacci as n approaches +/- infinity is 2!

If you have any questions about musical applications please ask me!

I recently mentioned this in another thread if you search "PHITAVE" but you can create a scale based on the limit of TRIB.

A tribtave scale consists of the ratio TRIB:1, 1055~ cents.
With the regular twelve steps the equal-tempered intervals would be 87.9~ cents.