metal mining and Pell-icans

OK Dan -- let's throw around a few things.

Silver ratios are defined by the equation 1/x = x + n.

For n=1, this is the golden ratio, phi, (sqrt(5)-1)/2.

For n=2, this is the Breed/Hahn fifth, sqrt(2)-1.

For n=3, you get (sqrt(13)-1)/2. . . .

see
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#silver
for a wonderful, calculator-oriented exploration of the silver means.

It appears that "bronze" already has _two_ different meanings. The page
http://members.tripod.com/vismath1/spinadel/ defines it as the
(sqrt(13)-1)/2 above, while the page
http://www.mi.sanu.ac.yu/vismath/kappraff/kap4.htm has another definition
that opens up a whole new dimension of possibilities -- which include the
constants that http://members.tripod.com/vismath1/spinadel/ would call the
Copper mean and the Nickel mean.

But I take it you're interested in the ratios that relate to sqrt(2)-1 in
the same way as the noble ratios relate to phi. In other words, the ratios
whose continued fraction representations end in all 2s. It appears, from the
above page, that we should call these the "Pell" means, or ratios, or
whatever. The proper scales generated by these numbers always have L/s =
sqrt(2), and obey the following recursion:

n(s(new)) = 2*n(L(old)) + n(s(old))
n(L(new)) = n(L(old)) + n(s(old))

In the Hahn/Breed case, the silver Pell, (or what
http://members.tripod.com/vismath1/spinadel/ calls "THE" Silver mean)
pseudo-Pythagorean case, the first two examples of this are -- first you
have the octave divided into just a fourth and a fifth, in the ratio L/s =
sqrt(2). Then the fifth is broken down into two major seconds and a minor
third, and the fourth is broken down into a major second and a minor third,
and you have a pentatonic scale. These two new intervals are in the ratio
L/s = sqrt(2). Then the minor third is broken down into two diatonic
semitones and one chromatic semitone, and the major second is broken down
into one diatonic semitone and one chromatic semitone, and you have a
chromatic scale. These two new intervals are in the ratio L/s (chromatic
semitone to diatonic semitone) = sqrt(2). Next you get a 29-tone scale, a
70-tone scale, and so on.

Notice that we skip the 3-, 7-, 17-, 41-, 99-, etc. scales in this process
-- the improper MOSs, with s/L = sqrt(2)-1. That's what differentiates this
Pell process from the noble process. For the Pell generators, you get two
different types of scale division, which alternate with one another. The
first type is like the noble type:

n(s(intermediate)) = n(L(old))
n(L(intermediate)) = n(L(old)) + n(S(old))

i.e., each old large step is divided into one new large and one new small
step, and each old mall step remains undivided, becoming a new large step;
but unlike the noble case, here the ratio L/s changes from sqrt(2) to
sqrt(2)+1 [=1/(sqrt(2)-1)]. The second type is the "opposite":

n(s(new)) = n(L(intermediate)) + n(s(intermediate))
n(L(new)) = n(L(intermediate))

i.e., each old large step is divided into one new large and one small step,
and each old small step remains a small step (kind of the obvious way of
"remedying" the impropriety) -- the large step, previously sqrt(2)+1 times
as large as the small step, is split into a small step (1 times as large as
the small step) and a less-large step (sqrt(2) times as large as the small
step).

And the process begins again.

So the overall sequence of MOS scales, in this case

2, 3, 5, 7, 12, 17, 29, 41, 70, 99 . . .

but it would be different for another Pell generator, obeys an alternating
recursion:

*******************************
if n is even, S(n+1)=S(n)+S(n-1)
if n is odd, S(n+1)=S(n)+S(n-2)
*******************************

_OR_, we can consider _just_ the "intermediate", improper MOSs (3, 7, 17,
41, 99, etc.), and ignore the proper ones. First we have two perfect fourths
and one major second, in the ratio L/s = sqrt(2)+1. Then the perfect fourth
is broken down into two major seconds and a minor second, and the major
second remains intact, giving a diatonic scale with L/s = sqrt(2)+1. Then
the major seconds are broken down into two minor seconds and a "komma", and
the minor second remains intact, giving a 17-tone "medieval Arabic" scale
with L/s = sqrt(2)+1. And so on.

The sequence of improper scales

3, 7, 17, 41, 99 . . .

obeys the recursion S(n+1)=2*S(n)+S(n-1), which is, you may recall, the same
recursion as the sequence of proper scales

2, 5, 12, 29, 70 . . .

Hope this helps and maybe someone can now justify Dan's amazing math.

Another Pell generator would be the one that divides the octave into the
proportion sqrt(2) : 1, 848.53 cents. Aside from an initial few with extreme
L/s values, the MOS scales generated by this generator alternate between L/s
= sqrt(2) and L/s = sqrt(2)+1 and have cardinalities

2, 15, 17, 19, 36, 53, 89 . . .

which obey the general Pell recursion

*******************************
if n is even, S(n+1)=S(n)+S(n-1)
if n is odd, S(n+1)=S(n)+S(n-2)
*******************************

For a final example, I'll choose to divide the octave into the proportion
sqrt(2)+3 : 1, or 928.15 cents. The sequence of scales alternating between
L/s = sqrt(2) and L/s = sqrt(2)+1 is:

4, 5, 9, 13, 22, 31, 53 . . .

which obey the general Pell recursion

*******************************
if n is even, S(n+1)=S(n)+S(n-1)
if n is odd, S(n+1)=S(n)+S(n-2)
*******************************

Okay, enough Pell-icans for now . . .

Paul!
If such things are gold, bronze and Silver, must we conclude that JI and ET are Lead ;-)

"Paul H. Erlich" wrote:

> OK Dan -- let's throw around a few things.
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Paul H. Erlich wrote,

> But I take it you're interested in the ratios that relate to
sqrt(2)-1 in the same way as the noble ratios relate to phi. In other
words, the ratios whose continued fraction representations end in all
2s.

Yes, at least as a sort of improper compliment to the strictly proper
L/s = Phi scales. That's what I meant by generalized Silver scales;
though I was trying to keep it hinged to K. Pepper's original "Silver
fifth", so I suggested a reversed Fibonacci Ls expansion as a way to
frame this improper Golden generator.

> It appears, from the above page, that we should call these the
"Pell" means, or ratios, or whatever.

Hmm, I don't know. Not quite as immediately appealing sounding 'ey,
but maybe it's a better term? I don't know... Anyway, here's a
generalized Silver (or whatever) generator algorithm for all scales
that have Myhill's property within a given periodicity where L/s = S:

X = P/((b+S*d))*(a+S*c)

where "P" = any given periodicity, "S" = sqrt(2), "a"/"b", "c"/"d" =
the adjacent fractions of a given Ls index, and "X" = the generalized
Silver generator

Note that the weighted generator of any Metallic constant >1.5 gives
an improper scale, and that any weighted generator of any metallic
constant <1.5 gives a strictly proper scale, and that MC = 2 (i.e., a
weighted generator of the Metallic constant 1.5) is the ET, MOS
generator (2x+y and x+2y on the Stern-Brocot tree where x and y are
generalized as L and s).

Perhaps this family of MGs (Metallic generators) can be generalized
for all scales that have Myhill's property within a given periodicity
where L/s = MC as:

X = P/((b+MC*d))*(a+MC*c)

where "P" = any given periodicity, "MC" = any continued fraction
constant >0, "a"/"b", "c"/"d" = the adjacent fractions of a given Ls
index, and "X" = the generalized Metallic generator

So continued fraction constants converging on zero represent the
"limit" of one border of a given Ls index ("a"/"b" as b/1200*a), and
continued fraction constants converging on infinity represent the
"limit" of the other border of a given Ls index ("c"/"d" as d/1200*c).

Any ideas?

I don't really care for the "Metallic" term (i.e., Metallic constants,
Metallic generators) aesthetically. Anyone have any better "sounding"
suggestions that are still theoretically "sound" (in other words ones
that don't cause any friction with another terms existing meaning
here)?

--Dan Stearns