Phi series

Hi Kraig,
Good to hear from you !

Yes, I know many people, mostly mathematicians, call it the Lucas
Series and I mentionned that before.
Some mathematicians also call that way ANY recurrent sequence, which
is a bit too much.
Actually you should not write it with two "1", this only starts the
Fibonacci series since 1 + 1 = 2.
You could write it 2 1 3 4 7 11... except that for us musicians is
has no real acoustic meaning,
and so would be -1 2 1 3 4 7 11... unless a "negative frequency"
concept would make some sense !

But before I heard of Lucas I saw it called the "P series" ("P" =
powers, puissances, potenza, etc...), and I choosed myself to keepcalling it that way, for Phi and for any other fractal ratio as well,
to design the series formed by the successive powers of a ratio that
would converge towards whole numbers. It well may be Edouard Lucas
who discovered those, but they are enough universal, I think, to not
be attached to one person.
However I still keep the letter F generally speaking, for what I
found to be the best precise series in any recurrent system. Usually
it is evident, but sometimes it is impossible to say what series it is .

That YOU only know this one and the F series to have a name only
comforts my thoughts : these are mostly documented by individual
searchers.
Actually a third one, very famous and musical, that you know of
course is
(3) 2 5 7 12 19 31 50...
and I have a tendancy to call it "G" ... like "Golden meantone" or in
french, "Gamme dorée"...

I have been working, years ago, on some classification of the Phi
series and I think I found at least a correct way to specify any of
these series with an exclusive quality, simply from "the first whole
number non-previously covered by more pertinent series..."
Our mathematicians I am sure will find more correct ways to express
that, but here is how it goes :
Just take the whole numbers one by one :
1, 2, 3 (then 5, 8, etc.) are already covered by the F series.
This leaves then 4, whose series has to be defined, and is disputed
by 4>6 and 4>7 :
simply multiply 4 by Phi = 6.472...: 4>6 wins : that should be
rounded to 6, which starts the "2F" series
2 2 4 6 10 16 ...
Next to be defined then is 7, that multiplied by Phi = 11.3256...,
that should be rounded to 11, which starts the "P" series (2) 1 3 4 7
11 18 ...
and so on.
Thus any new series, wether multiples of previous series or "prime"series, can be specified by only one "pertinent" whole number, and
with a basic calculator you may be able then to recover the whole
series.
Therefore any series can be named Phi(n) and this shows a simple
hierarchy between Phi-series :

Phi(1) = F
Phi(4) = 2F
Phi(7) = P
Phi(9) = 3F
Phi(12) = or the Golden meantone series (2 5 7 12 19 31 50 ...)
Phi(14) = new prime series > 23 37 60 ... (which I provisionally
called "J")
Phi(17) = another new prime> 28 45 73 ...
Phi(20) = 4F
Phi(22) = 2P
Phi(25) = 5F
Phi(27) = new prime > 44
Phi(30) = new prime > 49
Phi(33) = new prime > 53
Phi(35) = new prime > 57
Phi(38) = new prime > 61
Phi(41) = new prime > 66
Phi(43) = new prime > 70
Phi(46) = 2J, etc.

Now take the difference between successive (N) and look what hides
behind :
(3 3 2), 3 2 3, 3 2, 3 2 3, 3 2 3 3 2...
... PHI infinite word... but surprisingly not starting from 1, but
from 9 !!!,
which makes it easy anyway to predict at once all these series, that
present here a simple hierarchy.
They can be given letters according to the alphabet some way, but
this matter will certainly be debated, when will be held the next
international convention on recurrent sequences.

I totally feel like you about numerical series, by force this is only
what I deal with when making and playing photosonic disks, which are
just palettes of frequencies : not knowing what will be the perceived
1/1, and that, in best cases, can well stay as much undefined as we can.
Same thing with "becoming series" too - as I was recently
mentionning, I found one of my favorite scales by taking one every
two degrees in the F series :
3 8 21 55 144 and octaves (a Fibonacci version of Lou Harrison
Aptos School Gamelan slendro) -
Had I choosen instead
5 13 34 89 233 (one step further towards closer approximations of
Phi^2), its music would have been completely different.
- - - - - - - - - - - - - - - - - - - - -
Jacques

Jacques~ the series where you start
1 1 3 4 7 11 18......... is called the Lucas series. This is the only
other one i know that has a name. That recurrent sequences have no 1/1
is an important point and also one can seed these formulas with any
ratios one wants ( actually one could feed them with anything and they
will converge.
I have always found the numerical series more to my liking because on
limited acoustic instruments it give subtle but meaningful differences
between the same scale over different tones. What and where one chooses
becomes an 'artistic choice'. I tend to like to have the high end of my
scale very converged with working back down into the range in which it
is still 'becoming'. With Wilson's Mt. Meru scales i never start with
the seeds of 1's and go up to where one is at least in 'a ballpark' of
where the interval is heading.
--
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_