Re: [tuning-math] Digest Number 2115

Paul Hj wrote:

> If one counts the septachords in a 31 symbol system (not neccessarily
> 31-tET) you get 32,000 different scales. This is based on having one
> C, one D, etc.
>
> Therefore,
>
> 4 X 5 X 4 X 4 X 5 X 5 X 4 = 32,000.

Paul - I would guess you'd also want to weed out scales that weren't in ascending order. For example <Cx Dbb E Fx Gbb Ax Bbb>. If this is the case, then you need to specify the ascending order of symbols before the question can be answered (in some tuning systems Cx might be lower than Dbb; in some they might be the same, etc.)

I wrote a program some time ago to answer this question for the modern harp: each of 7 strings can be in a flat, natural or sharp position. You weed out scales with e.g. E# and Fb, you decide whether you want to keep scales with fewer than 7 pitch classes (e.g. scales with both F# and Gb), then you reduce for transposition. I have the code around somewhere but it would probably be quicker to start from scratch for your problem. It was interesting to see which scales could only be found at only one transpositional level!

--Jon

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> Paul Hj wrote:
>
> > If one counts the septachords in a 31 symbol system (not
neccessarily
> > 31-tET) you get 32,000 different scales. This is based on
having one
> > C, one D, etc.
> >
> > Therefore,
> >
> > 4 X 5 X 4 X 4 X 5 X 5 X 4 = 32,000.
>
> Paul - I would guess you'd also want to weed out scales that
weren't in
> ascending order. For example <Cx Dbb E Fx Gbb Ax Bbb>. If this is
the
> case, then you need to specify the ascending order of symbols
before the
> question can be answered (in some tuning systems Cx might be lower
than
> Dbb; in some they might be the same, etc.)
>
> I wrote a program some time ago to answer this question for the
modern
> harp: each of 7 strings can be in a flat, natural or sharp
position. You
> weed out scales with e.g. E# and Fb, you decide whether you want
to keep
> scales with fewer than 7 pitch classes (e.g. scales with both F#
and Gb),
> then you reduce for transposition. I have the code around
somewhere but it
> would probably be quicker to start from scratch for your problem.
It was
> interesting to see which scales could only be found at only one
> transpositional level!
>
> --Jon

Yes, I agree. The LHS of my grid is backwards, BTW. But you only
need one side: The formula is merely ((31-max) + min) to get the
number of transpositions. Max and min can never be 7x apart (because
they'd be in the same column) and can never get closer than 6 apart.
So you never have values of 3, 10, 17 or 24. Once you find a value,
say 24, you know that it represents 24 scales. The grid is triangular
going through 21 values, then 20, then 19 etc. So now I just need to
figure out the pathing and the rest is easy. 25 is the highest.

I have the grid at work, (playing hookey, sort of), which shows
tranpose values for all the possible endpoints of the paths. Now to
weed out scales you mentioned. Say, do you know a good language
(perhaps a MATLAB knock-off) to use for this, and my Single Steiner
System program?

There are two approaches: Using Curtis Kitten modulo-11 based, or
the one Dr. Elkies' has described (which I cannot figure out how to
do). Wish I were a better programmer. What language did you use for
your Z-relations program again?
Thanks,

PGH

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> Paul Hj wrote:
>
> > If one counts the septachords in a 31 symbol system (not
neccessarily
> > 31-tET) you get 32,000 different scales. This is based on
having one
> > C, one D, etc.
> >
> > Therefore,
> >
> > 4 X 5 X 4 X 4 X 5 X 5 X 4 = 32,000.
>
> Paul - I would guess you'd also want to weed out scales that
weren't in
> ascending order.

A far more manageable set of 31-et, 7-note scales are the proper
scales; up to transposition there are 181 of these. A number of them
can be found in the files section, in the directory called "proper".