Re: [tuning] Digest Number 1048

Hi Dan,

> It's not trivalent. Notice that it has only two unique intervals at
> the 4th -- 4/3 and 3/2 -- and not three.

Yes, you're right, missed that!

> So what I was musing before was that this should pretty simply mean
> that no rational scale with and equal number of notes can be
> trivalent!

Well that is a pretty strong conjecture to make!

Don't quite see the rest yet - but I need to read through it slowly, and
see what questions I need to ask.

Thanks!

Robert

Robert Walker wrote,

<< Well that is a pretty strong conjecture to make! >>

Indeed, though as I've said before, it is largely a guess on my
part... but I've yet to see a counter example or condition that leads
me to think otherwise.

<< Don't quite see the rest yet - but I need to read through it
slowly, >>

Maybe if I use the example you gave as a starting off point it'll all
make more sense.

The scale you gave, 1/1 49/48 7/6 4/3 49/32 7/4 2/1, can be
generalized as [1,4,1] scale. This index would meet the conditions
that I posted before, where both the 4th and 6th term are even
numbers:

1, 4, 1, 6, 11, 18, ...

So a 6-out-of-18 should give a trivalent scale with an equal number of
notes. Lets try it in the SMMLMM order that you originally gave your
scale in:

0 133 333 533 800 1000 1200
0 200 400 667 867 1067 1200
0 200 467 667 867 1000 1200
0 267 467 667 800 1000 1200
0 200 400 533 733 933 1200
0 200 333 533 733 1000 1200

Hmm, not trivalent! But wait, note that no rotation of the scale has
the P/2 midpoint "condition"... so let's try abcbbb instead:

0 133 333 600 800 1000 1200
0 200 467 667 867 1067 1200
0 267 467 667 867 1000 1200
0 200 400 600 733 933 1200
0 200 400 533 733 1000 1200
0 200 333 533 800 1000 1200

Ah, much better!

This fiddling with the scales step order is one reason why I always
want a strong internal ordering rule in the various methods that I
come up with here. The need to avoid chasing one's own tail looking
for the "right" one is a strong one here!

Anyway, let me know if any of this is making any more sense now.

thanks,

--Dan Stearns