Re: Tribonacci scales

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

I was suggesting 0-8-15-19-27-35-42-46 in 46-equal, with step sizes
8748874 and Paul pointed out that this is the Indian Diatonic scale,
which in JI is 9/8 10/9 16/15 9/8 9/8 10/9 16/15 in terms of step
sizes. Of course, this has the same sizes of intervals as the Western
diatonic scale. I got this from what I've been calling a "muddle",
and they could be used to produce other 3-interval scales.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> I'm still not sure... Having historical precedents like the syntonic
> diatonic to use for comparison sake helps, and indeed the 2, 2,
3, ...
> checks out favorably... but what are some other
traditional/historical
> three-stepsize scales besides the 2, 2, 3, ... that I could check
> against?
>
> thanks,
>
> --Dan Stearns

Why not just create some others from other series -- e.g.,

1, 1, 1, . . .
1, 1, 2, . . .
1, 2, 2, . . .
2, 3, 3, . . .

and just examine and enjoy them for their own sake? Not sure what you
want "historical precedent" for . . . especially when this process
clearly ignores any frequency ratios more complex than 2/1, while the
syntonic diatonic clearly doesn't!

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
>
> You've misunderstood me. What I'm interested in checking against,
just
> for the sake of my own curiosity, is the ordering of the scale steps
> (a vexing problem to generalize if you don't have something like a
1D
> generator doing it for you), and not the properties relative JI (or
> whatnot) of the scales themselves.

I think I understood that, but as Gene inadvertantly pointed out by
bringing up the Indian diatonic, and which you can immediately see by
comparing it with the syntonic diatonic, is that the ordering of
scale steps can be contradictory from one historical example to
another . . . so I don't see what kind of "checking" you're expecting
to acheive here. Rather, it seems that your L-out-of-M-out-of-N
method for determining the ordering, and perhaps variations where you
rotate these ETs relative to one another by continuous amounts before
doing the "out-of" operation, forms a paradigm unto itself, that
perhaps a powerful mathematician (Gene?) can find some governing
principles behind.

>
> I had to check out dozens of different Tribonacci combinations just
in
> the process of making sure that things worked and were generalized
(in
> doing this I found that fractional periodicity doesn't work quite
the
> way I might've expected it would).

Can you expound on what you mean bt "fractional peridocity"? Also,
can you verify that the generating rules, and other relationships,
that Gene put forth in his latest post here are correct?

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> <<Can you expound on what you mean bt "fractional peridocity"?>>
>
> I mean dividing the period by the GCD of A, B and C--but this does
> work the way you'd expect it would (I was thinking of the 1D to 2D
> conversions).
>
Still mystified.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> I'll have to take a better look at it.

You could start with this--if we replace the steps of a scale with
steps A,B,C by a scale with steps a,b,c, such that A=c, B=a+c and
C=b+c, we plainly will get the old scale as a subscale of the new
scale. Is this what you've been doing?

--- In tuning-math@y..., genewardsmith@j... wrote:

> You could start with this--if we replace the steps of a scale with
> steps A,B,C by a scale with steps a,b,c, such that A=c, B=a+c and
> C=b+c, we plainly will get the old scale as a subscale of the new
> scale.

Well, I'm glad I read that right!

Now, in the 2-step-size case, consistently putting the new small step
_below_ the new large step, or consistently putting the new small
step _above_ the new large step, when replacing the old large step,
gets you MOS scales at every stage in the construction.

In the 3-step-size case, do some or all such consistent ordering
rules lead to scales that match the step-size-patterns of Dan's L-out-
of-M-out-of-N scales, and/or your muddles?

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
>
> Sorry. I'm sure you know what I mean, but I must be saying it in too
> personal a way--fractional periodicity, scale index, etc.
>
> By fractional periodicity I just mean the period divided by the GCD
of
> the scale index (and in a Fibonacci scale I'd call the scale index
A,
> B, ... and in a Tribonacci scale it'd be A, B, C, ...),

The same A, B, and C that satisfy C/A = t, (A+B)/C = t? Or rather a
different A, B, and C (those that "seed" the sequence?)