Possibly dumb generator / mode question

I'm interested in exploring Negri temperament. The wiki tells me that 2/19
is a good generator, so I go to Scala, create a new linear temperament, and
ask for a maximally even regular temperament with 9 or more notes using a
generator of 126.326 cents.

My question has to do with "number of fifths down".

I note that if I say "8", I get
0: 1/1 0.000 unison, perfect prime
1: 126.326 cents 126.326
2: 252.652 cents 252.652
3: 378.978 cents 378.978
4: 505.304 cents 505.304
5: 631.630 cents 631.630
6: 821.022 cents 821.022
7: 947.348 cents 947.348
8: 1073.674 cents 1073.674
9: 2/1 1200.000 octave

...while if I say "4" (which implies 4 up as well), I get:
0: 1/1 0.000 unison, perfect prime
1: 126.326 cents 126.326
2: 252.652 cents 252.652
3: 378.978 cents 378.978
4: 505.304 cents 505.304
5: 694.696 cents 694.696
6: 821.022 cents 821.022
7: 947.348 cents 947.348
8: 1073.674 cents 1073.674
9: 2/1 1200.000 octave

...and these are modes of each other. Similarly if I use 8 generators up
and zero down.

That leads to my question: Will this always be the case? It's not obvious
to me that it will be, but half of the math around here isn't obvious to
me. :) If the number of generators up and down "doesn't matter" in the
sense that I can just hit Shift-Alt-A to see all of the modes of a given
scale instead of trying different generators, that's a big time savings.

Thanks in advance,
Jake

On Fri, Dec 30, 2011 at 12:08 AM, Jake Freivald <jdfreivald@...> wrote:
>
> That leads to my question: Will this always be the case? It's not obvious to me that it will be, but half of the math around here isn't obvious to me. :) If the number of generators up and down "doesn't matter" in the sense that I can just hit Shift-Alt-A to see all of the modes of a given scale instead of trying different generators, that's a big time savings.

Yup. That's the rule. Consider the chain of fifths in F lydian vs C
ionian, for example.

-Mike

> Yup. That's the rule. Consider the chain of fifths in F lydian vs C
ionian, for example.

I appreciate the quick answer, Mike. I could see it for the specific case
of L = 2s, as you show here, but didn't know if it were true generally.
Thanks!

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> That leads to my question: Will this always be the case? It's not obvious
> to me that it will be, but half of the math around here isn't obvious to
> me. :) If the number of generators up and down "doesn't matter" in the
> sense that I can just hit Shift-Alt-A to see all of the modes of a given
> scale instead of trying different generators, that's a big time savings.

Yes, this is always guaranteed to work. At least in the rank-2 case, everything is wrapped up in a neat and tidy package - shifting the generator chain does the same thing as shifting the "root" of the scale.

But this is an interesting question because it really points out something we can't get to work in rank-3 (and higher). In a rank-3 temperament there are two independent kinds of generator, so in the analog of a MOS / DE scale there's not only "number of generators up" and "number of generators down", there's also "number of generators left" and "number of generators right", where I picked "left" and "right" just because they're a different pair of directions than "up" and "down". If you change these independently you usually don't get a mode of the original scale in the sense of being a cyclic permutation. You can get a scale with a different pattern of steps, which Mike Battaglia has termed a "dome" (because it's like a mixed-up "mode").