Jak's top 10 fractals (was Re: Special types of noble numbers

Of course Michael, useful property indeed of that family :

x^2 = x + 1 Phidiane
x^2 = 2x + 1 Aksaka
x^2 = 3x + 1 Béata
x^2 = 4x + 1 Chromph (= Phi^3)
x^2 = 5x + 1 Panchaï
x^2 = 6x + 1 Mixys
x^2 = 7x + 1 Perchak
x^2 = 8x + 1 Upticom
x^2 = 9x + 1 Barak
etc.,
where x = n + 1/x

Note that all those have another important property : their powers converge
towards whole numbers. Among other applications, this helped me to resolve
some fractal waveforms for each of them.

Just one thing, I don't use myself the term "noble numbers". For what we are
talking here I have been simply using instead the term "fractals" (or in
french, "fractale", since "onde" is feminine) for 25 years now and I should
not change, unless someone convinces me with very good reasons.
But "noble number" is not a good choice from my point of view because in
mathematics it defines something very specific and different.
I also realised with your help and the usual internet tools that there is'nt
such a thing like a list of "noble numbers" existing... And that apart from
the tuning list, this term is not much in use in the way we think of it.

Thus I am talking of fractal ratios, fractal algorithms, fractal series
(recurrent sequences), fractal rhythmns, fractal waveforms, fractal
temperaments, fractal sounds etc, by analogy with the absence of "scale
factor" in the fractal images that behave in 2D the same way that these
harmonies behave in the frequency domain, leaving you without a definite
idea of the fundamental, nor a beginning or an end.

You asked me some days ago what where my "top ten" fractals - I don't have
such a list but according to the number of times I played fractal disks in
concert, these should be representative of what pleased me at times :

1) Phidiane / Iph 1.61803398875 / 1,2360679774998
2) Zinith / Zira'at 2.732050808 / 3.732050808
3) Aksaka / Persi 2.41421356237 / 3,41421356237
4) Narayana 1.4655712318768
5) Fong 2.2055694304
6) Ishku 4.44948974278
7) Beata 3.30277563773
8) Tritonde 5.64575131106
9) Trimohabbi 3.66868509048
10) Natté 1.32471795725
11) Amaz 1.3802775691
12) Pollux 6.54138126515
13) Buzurg 1.23205463143
14) Ulang-Tiha 2.87938524157
15) Dhaivati 1.67169988166
16) Xring 4.64575131106
17) Bohr 4.30277563773
18) Mona 3.37228132327
19) Onyx 2.3829757679062
20) Hex 3.56155281281
21) Niris 2.35930408597
22) Semq 2.3202325356
23) Huib 2.5943130163
24) Chandrak 4.79128784748
25) Panchaï 5.19258240356 Awj 1.83928675521

"Mohajira" (1,2232849566), "Cylf" (1,14549722437), and others favorites are
absent of this list, even if I like them as much, but waveforms I don't
consider geometrically resolved at this day are ommited here.

Enjoy,
- - - - - - -
Jacques Dudon

Jacques> "For what we are talking here I have been simply using instead the term 'fractals'"
    Understood...my apologies...I guess I had just heard of such numbers described in far too many places as simply "noble numbers"...I'll call them fractals from here on in. :-)

Jacques> "I also realised with your help and the usual internet tools that there is'nt

such a thing like a list of 'noble numbers' existing"

     Good point...and I wish there was such a list.  On line I have basically seen the any numbers that converge toward whole numbers are called noble numbers. 

Jacques>
"x^2 = x + 1 Phidiane

x^2 = 2x + 1 Aksaka 

x^2 = 3x + 1 Béata   

x^2 = 4x + 1 Chromph (= Phi^3)

x^2 = 5x + 1 Panchaï

x^2 = 6x + 1 Mixys

x^2 = 7x + 1 Perchak

x^2 = 8x + 1 Upticom

x^2 = 9x + 1 Barak"
   This is an excellent list...just the type of thing I was looking for (including how the numbers are derived), thank you. :-)  I actually made a program that solved where x = 1/x + (any whole number) for all values of x and came up with the exact values your list above did.

Jacques>
"

1) Phidiane / Iph 1.61803398875 / 1,2360679774998

2) Zinith / Zira'at 2.732050808 / 3.732050808

3) Aksaka / Persi 2.41421356237 / 3,41421356237

4) Narayana 1.4655712318768

5) Fong 2.2055694304

6) Ishku 4.44948974278

7) Beata 3.30277563773

8) Tritonde 5.64575131106

9) Trimohabbi 3.66868509048

10) Natté 1.32471795725

11) Amaz 1.3802775691

12) Pollux 6.54138126515

13) Buzurg 1.23205463143

14) Ulang-Tiha 2.87938524157

15) Dhaivati 1.67169988166

16) Xring 4.64575131106

17) Bohr 4.30277563773

18) Mona 3.37228132327

19) Onyx 2.3829757679062

20) Hex 3.56155281281

21) Niris 2.35930408597

22) Semq 2.3202325356

23) Huib 2.5943130163

24) Chandrak 4.79128784748

25) Panchaï 5.19258240356 Awj 1.83928675521
...."Mohajira" (1,2232849566) , "Cylf" (1,14549722437)...
"
   Also a great list...though I wonder
A) what geometric properties explain these
B) If it would still be a good idea to use (1/(any of the above numbers)) to get a generator for a scale.  For example, for 2.414 "the silver ratio" or "Aksaka"....I simply took (1/2.414) to get 0.414 and took 1+(0.414^x) to get the decimal ratios used to make the scale:
1.414
1.1715
1.07107
and then added the notes 1.414 - 1.07107 and 1.414 - 1.1715 to get
1.414
   1.34314
1.24257

1.1715

1.07107

   Such a method seems to at least give good results for those "fractals" that satisfy x = 1/x + 1, x = 1/x + 2, etc. as shown in
   http://www.geocities.com/djtrancendance/PHI/silverrain.mp3
...which uses the above scale.

  
   As a side question, Jacques, what methods do you generally use to extract scales from fractals?

Thank you, Michael