Towards hyper-MOS

For MOS scales we've seen the noble generators allow for a uniquely
regular expansion of resources, in that each MOS will have L = s*phi
and the steps in the old MOS change into the steps of the new MOS as
follows:

L(old) -> L(new) + s(new)
s(old) -> L(new)

so that the sizes (number of notes) of any three consecutive scales
in the "evolution" obey a generalized Fibonacci recursion:

siz(n-1) + siz(n) = siz(n+1);

and the ratios of adjacent sizes approaches phi. The most famous
example is the Kornerup system, which has the same scale sizes as
Yasser's proposed evolution

2, 5, 7, 12, 19, 31, 50, 81 . . .

but maintains the same (noble) generator throughout.

What if we allow three step sizes, and posit the following evolution
rules:

siz(n-2) + siz(n-1) + siz(n) = siz(n+1)? Then the ratio of adjacent
sizes approaches the solution of

x^3 - x^2 - x - 1 = 0

= 1.839286755...

One example is the so-called "Tribonacci Sequence":

1, 1, 2, 4, 7, 13, 24, 44, 81,...

while another may be more musically relevant and may be familiar to
followers of Kraehenbuehl & Schmidt:

2, 2, 3, 7, 12, 22, 41, . . .

although the next term would be 75 instead of K&S's 78 -- since K&S
started with 3-limit JI and forced "inflections" reflecting
successively higher prime limits to "deform" the scale at each stage.

Like the Fibonacci sequence, the Tribonacci sequence and its
relatives can be derived from a continued fraction representation,
but this time using generalized, "third-order" continued fractions
(see http://www.mathsoft.com/asolve/constant/pythag/dgm.html).

Can anyone demonstrate a rule for how the steps of one scale
transform into the steps of the next scale, and find the set of
scales which are to Kraehenbuehl & Schmidt as Kornerup's set of
scales is to Yasser? Bonus question: What is the object here
analogous to the generator in MOS scales, and in what way is this
generator built upon itself to create these "hyper-MOS" scales? Is
there more than one solution?

Paul Erlich wrote,

> Can anyone demonstrate a rule for how the steps of one scale
transform into the steps of the next scale, and find the set of scales
which are to Kraehenbuehl & Schmidt as Kornerup's set of scales is to
Yasser? Bonus question: What is the object here analogous to the
generator in MOS scales, and in what way is this generator built upon
itself to create these "hyper-MOS" scales? Is there more than one
solution?

Hey Paul, this all sounds very interesting (and I've worked a bit with
three term sequences and other things looking for a three-stepsize
analog to MOS scales in the past), but if the above are, as I'm
assuming they are(?), questions to which you already have the answers,
could you just go ahead and post a bit more on them!

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> > Can anyone demonstrate a rule for how the steps of one scale
> transform into the steps of the next scale, and find the set of
scales
> which are to Kraehenbuehl & Schmidt as Kornerup's set of scales is
to
> Yasser? Bonus question: What is the object here analogous to the
> generator in MOS scales, and in what way is this generator built
upon
> itself to create these "hyper-MOS" scales? Is there more than one
> solution?
>
> Hey Paul, this all sounds very interesting (and I've worked a bit
with
> three term sequences and other things looking for a three-stepsize
> analog to MOS scales in the past), but if the above are, as I'm
> assuming they are(?), questions to which you already have the
answers,
> could you just go ahead and post a bit more on them!

No I don't already have the answers, but I was thinking of you
specifically as someone who might be interested in this idea, and
thought you might enjoy working this out. You, and also Carl Lumma
who once asked what a 2-d generalization of MOS might be like. I came
up with the idea while waking up Friday morning, when I discovered
that the pattern of very important scale sizes 3, 7, 12, 22, 41 is a
Tribonacci-like sequence, and one thought led to another . . . but I
haven't had a chance to get to my office and get on Matlab or Excel
yet.