MOS's that shouldn't be MOS's, but somehow are

Something that I found interesting: meantone forms an MOS at 11 notes
if its fifth is exactly 700 cents, and an MOS at 6 notes if its fifth
is exactly 4\7 of an octave. And 19-equal's fifth will form an MOS at
18 notes, etc.

On the surface, this is trivial, because in general, any generator
from n-EDO that is coprime with n will form an MOS at n-1 notes. Let's
call these the trivial cases. Does anyone know of any non-trivial
cases for this?

To define this precisely, the "anomalous MOS's" that I'm talking about are
1) those formed for only certain precise generator values, but are not
formed if the generator is altered at all in either direction. For
example, meantone[11] is only an MOS if the fifth is 700 cents
exactly, but not 700.001 or 699.999 cents. On the other hand, meantone
forms an MOS at 12 notes if the fifth is anywhere from 4\7 to 7\12 (or
to 3\5 if you want to be creative), so this isn't anomalous.
2) If the generator is taken from some n-note equal temperament, a
nontrivial anomalous MOS would does not correspond to an MOS of any
number of notes except for n-1.

Does anyone know about this?

-Mike

Yeah. Funny thing, this. There are other examples like this. In 12-TET, you can get an "MOS" out of 700 cents at 8, 9, and 10 notes, too, though they're pathological in terms of not being DE. But...they have two step-sizes! And L:s is 2:1, so they should be proper...but they're not! It is wacky. But it's also not what you're talking about.

-Igs

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Something that I found interesting: meantone forms an MOS at 11 notes
> if its fifth is exactly 700 cents, and an MOS at 6 notes if its fifth
> is exactly 4\7 of an octave. And 19-equal's fifth will form an MOS at
> 18 notes, etc.
>
> On the surface, this is trivial, because in general, any generator
> from n-EDO that is coprime with n will form an MOS at n-1 notes. Let's
> call these the trivial cases. Does anyone know of any non-trivial
> cases for this?
>
> To define this precisely, the "anomalous MOS's" that I'm talking about are
> 1) those formed for only certain precise generator values, but are not
> formed if the generator is altered at all in either direction. For
> example, meantone[11] is only an MOS if the fifth is 700 cents
> exactly, but not 700.001 or 699.999 cents. On the other hand, meantone
> forms an MOS at 12 notes if the fifth is anywhere from 4\7 to 7\12 (or
> to 3\5 if you want to be creative), so this isn't anomalous.
> 2) If the generator is taken from some n-note equal temperament, a
> nontrivial anomalous MOS would does not correspond to an MOS of any
> number of notes except for n-1.
>
> Does anyone know about this?
>
> -Mike
>

On Sat, Feb 19, 2011 at 4:40 PM, cityoftheasleep
<igliashon@sbcglobal.net> wrote:
>
> Yeah. Funny thing, this. There are other examples like this. In 12-TET, you can get an "MOS" out of 700 cents at 8, 9, and 10 notes, too, though they're pathological in terms of not being DE. But...they have two step-sizes! And L:s is 2:1, so they should be proper...but they're not! It is wacky. But it's also not what you're talking about.

The 10-note one happens to also be Paul's standard pentachordal major
scale, in 12-tet. These aren't actually MOS's, though, because some of
the interval classes come in three specific sizes. Those would be
near-MOS's or "MODMOS's" or whatever we're calling them today.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Does anyone know about this?

All I know is that when I was writing up the definition of a MOS on the Xenwiki I thought about discussing what happens if it is in a rank one tuning and not rank two, and decided not to get into it. Maybe I should have?

On Sat, Feb 19, 2011 at 8:24 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Does anyone know about this?
>
> All I know is that when I was writing up the definition of a MOS on the Xenwiki I thought about discussing what happens if it is in a rank one tuning and not rank two, and decided not to get into it. Maybe I should have?

All I can say is that I would certainly find it interesting.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> All I can say is that I would certainly find it interesting.

You've come across one of the points already. Another is that a MOS need no longer either proper or improper. Another is that the MOS can degenerate into having only one step size. Another is that any such MOS has another, complementary MOS, in the way that diatonic and pentatonic are complements in 12edo. Another is that the MOS need not be picked out by semiconvergents of the generator.

On Tue, Feb 22, 2011 at 4:12 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> You've come across one of the points already. Another is that a MOS need no longer either proper or improper.

Do you mean it doesn't have to be either strictly proper or improper,
and can now be proper?

> Another is that any such MOS has another, complementary MOS, in the way that diatonic and pentatonic are complements in 12edo.

Sometimes it's multiples of a complementary MOS though. In 22-equal,
the fifth produces a 5L7s chromatic MOS, but 22-12 = 10 and there's no
fifth-based 10-note MOS that I can see here. 5 is an MOS though.

-Mike