Re: a tuning-math question

--- In tuning-math@yahoogroups.com, "daniel_anthony_stearns
<daniel_anthony_stearns@y...>" <daniel_anthony_stearns@y...> wrote:
> I have a tuning-math question that I'm hoping someone can answer in
a
> purely mathematical fashion.
>
> If you were to divide a line segment into a given number of parts
> of a given number of sizes, what strategies could you use to order
> those parts?
>
> (I'm hoping the strategies would be general enough so that the
number
> of parts and separate sizes would be irrelevant--thanks.)

hi dan -- great to have you back!

how about ordering them from left to right?

(i'm probably not understanding your question . . .)

-paul

--- In tuning-math@yahoogroups.com, "daniel_anthony_stearns
<daniel_anthony_stearns@y...>" <daniel_anthony_stearns@y...> wrote:
> Hi Paul,
>
> Unfortunately I really can't be back, but I have had enough free
time
> lately so that I can at least visit a bit.
>
> This problem seems so simple, but it is one that has vexed me for
> years. What I'm looking for is another way to approach the aspect
of
> scale generalization that has to do with how one orders scale
steps.
> So by "line segment" I really mean period,

circle?

> and what I'm trying to
> answer is something like this:
>
> If you have a line segment and you want to divide it into 9 pieces
of
> 3 different sizes where 5 are smallest, 3 are largest, and 1 is
> larger than the smallest and smaller than the largest, what are
some
> general methods for organizing the 9 pieces within the segment?

general methods? so you'd like a method that produces every possible
ordering? counts them? produces them more cleverly than brute force?

> Obviously what I'm looking for is a mathematical generalization
that
> agrees well enough with existing theory

existing combinatorics? i suspect it all exists already, and gene
knows it . . .

> so that it might be a useful
> blueprint when applied to scales that we know nothing about. Surely
> the math is very simple and it's simply a matter of finding a
method
> that works. So that's what I'm looking for some help with... and if
I
> get a flood of suggestions I'm sure I'll find it (and then I can
> sleep at night too).

in the past you've asked something similar, but it sure looks like
this is a distinct question. so i guess you just need to clarify it.
can you go further with the example? maybe a much simpler example, so
we can lay out all the possibilities, if need be?

sorry for being such a dunce,
paul

--- In tuning-math@yahoogroups.com, "daniel_anthony_stearns
<daniel_anthony_stearns@y...>" <daniel_anthony_stearns@y...> wrote:
> Yes, the "line segment" that I'm using as an example could just as
> easily be a circle or a square or a space (or whatever) so long as
it
> can be reduced to a linear sequence in the end.
>
> The obvious method that comes to mind is something like ME, with
its
> tendency to adhere to palindromic symmetry.

ME is defined on a circle, and if you break the circle into a line,
you'll only get a palindrome occasionally -- usually you won't.

> The problem is that while
> a:
>
> *---*---*---1/1---*---*---*
>
> and its LsLLLsL agrees with existing theory well enough, a:
>
> (*)--*
> / \ / \
> *---*--1/1--*---*
> \ / \ /
> *--(*)
>
> and its LsMLMsL does not.

is this a 5-limit lattice? what are the notes in parentheses? what do
you mean by "agrees with existing theory"??

> It's certainly possible that no math model will ever fully satisfy
> the rather opposing demands of this sort of a generalization--one
> that's attempting to make an end-run around just intonation--

look, dan, if you can pin down for us just what your generalization
entails, i'm sure it can be "mathematized" -- but, as usual, i have
extreme difficulty comprehending your ideas. that won't stop me from
continuing to try with all my might!

--- In tuning-math@yahoogroups.com, "daniel_anthony_stearns
<daniel_anthony_stearns@y...>" <daniel_anthony_stearns@y...> wrote:
> Yes it's supposed to be a symmetric (1/1, 9/8, 6/5, 4/3, 3/2, 5/3,
> 16/9, 2/1) 5-limit lattice.
>
> We understand how to generalize the order the step sizes for scales
> with two step sizes--the generator does the work for us.
> What I want
> to be able to do is generalize the ordering of the step sizes for
any
> scale with any amount of step sizes.

note that the scale above is epimorphic (i.e., a periodicity block),
and so is what gene refers to as a "scale along a line". maybe
that'll help somehow.

> We understand that a 2:3 gives a 7-tone scale with 2 small steps
and
> 5 large steps ordered LLsLLLs.

ok . . .

> We also understand that a 4:5:6 gives
> a 7-tone scale with 2 small steps, 2 medium steps and 3 large steps
> ordered LMsLMLs.

a 4:5:6 "gives" this? if i treat 4:5:6 as a generator in the same
sense as 2:3 is a generator, i get a 6-tone scale, a 10-tone scale,
etc . . . but 7?

> What I seek to understand is a generalized ordering principle or
> scheme (how step sizes could be ordered in random scales) that
could
> tie these two scales to say an 8-tone scale with 3 small steps, 1
> larger step, 2 even larger steps, and 2 largest steps.

i really don't know what you have in mind. can you give an example of
such an 8-tone scale, and how it relates to the others?

> This is the sort of generalization I want. I already understand how
> to generalize the size of these steps, but not their order.

why don't you flesh out the part you understand for us, so that we
might get a better grip on what you don't?