Higher-dimensional partial generalization of MOS

I just did a monster review session of the tuning archives, such that
I now have no idea what has or hasn't been proposed yet. This was a
simple idea that I had, which at this point I assume has probably been
done. I'll suggest it anyway.

I know that there have been a few proposals to extend the MOS
structure into higher dimensions; here's one I stumbled across that
seems sensible, although I'm not sure exactly what the full picture
here is. I think that it has a number of interesting properties that
are desirable, and is worth exploring. It's a "partial generalization"
in that it doesn't yield the 3 step sizes per octave for planar
temperaments as you'd expect, rather it generalizes things in a
different way.

MOS's are defined with respect to a generator and a period. So
although Pythagorean tuning can be said to have "two generators" - 2/1
and 3/2 - the 5, 7, 12, 19, etc note MOS's come up only if 2/1 is the
period (e.g. tiled across the spectrum indefinitely) and 3/2 is the
generator (e.g. it only extends up and down the spectrum for a limited
number of iterations).

One interesting thing that happens is if you run across something like
Blackwood, where you have a 5L5s scale. The period here could be said
to be a sharp 3/2, and the generator could be said to be a sharp 5/4.
In this case, it just so happens that Blackwood makes it that five
3/2's itself run into an octave, so the entire thing turns into a nice
10-note sized 5L5s MOS with only 2 steps per octave.

So I wanted to extend this to planar temperaments. What happens if,
rather than taking 5-equal and offsetting it by a major third to get
blackwood, you take some fifths-based MOS, generated by a diprime
comma, and then offset a major third to get something else? So let's
say you're in the 7-limit and working within the very sharp 28/27
temperament, yields a 3L2s father-ish MOS. What happens if then, you
offset the entire thing by a 5/4?

So the generalization here is that the sharp 3/2 acts as a "period"
for the 5/4, which acts as a "generator," and from this perspective
the resultant major triad is a 1L1s scale. The 3/2's then act as
generators for the 2/1, which acts as the period.

This doesn't usually seem to yield scales with 3 step sizes, but it
often yields interesting results. Has anyone explored something like
this?

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> So I wanted to extend this to planar temperaments. What
> happens if, rather than taking 5-equal and offsetting it
> by a major third to get blackwood, you take some
> fifths-based MOS, generated by a diprime comma, and then
> offset a major third to get something else? So let's say
> you're in the 7-limit and working within the very sharp
> 28/27 temperament, yields a 3L2s father-ish MOS. What
> happens if then, you offset the entire thing by a 5/4?

Can be done, yes. I've thought of two chains of schismatic
temperament instead of 29-equal for something like Mystery
(29&58). You can also extend schismatic notation to
multiple chains by inventing new accidentals.

Graham

On Thu, Feb 10, 2011 at 6:04 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia <battaglia01@gmail.com> wrote:
>
> > So I wanted to extend this to planar temperaments. What
> > happens if, rather than taking 5-equal and offsetting it
> > by a major third to get blackwood, you take some
> > fifths-based MOS, generated by a diprime comma, and then
> > offset a major third to get something else? So let's say
> > you're in the 7-limit and working within the very sharp
> > 28/27 temperament, yields a 3L2s father-ish MOS. What
> > happens if then, you offset the entire thing by a 5/4?
>
> Can be done, yes. I've thought of two chains of schismatic
> temperament instead of 29-equal for something like Mystery
> (29&58). You can also extend schismatic notation to
> multiple chains by inventing new accidentals.

How does this relate to the generalization of MOS that you posted
before? Is there a simple explanation for it? I didn't quite
understand the original thread.

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:
> On Thu, Feb 10, 2011 at 6:04 AM, Graham Breed
> <gbreed@gmail.com> wrote:

> > Can be done, yes. I've thought of two chains of
> > schismatic temperament instead of 29-equal for
> > something like Mystery (29&58). You can also extend
> > schismatic notation to multiple chains by inventing new
> > accidentals.
>
> How does this relate to the generalization of MOS that
> you posted before? Is there a simple explanation for it?
> I didn't quite understand the original thread.

The two are pretty much orthogonal. Multiplying an MOS
scale the new scale has a multiple of the notes of the old
one. Or, turning it backwards, the number of notes in the
MOS has to be a divisor of the number of notes you're
looking for. That's useless if you want a prime number of
notes. The other generalization had to do that.

What I ended up using was a Fokker periodicity block. Gene
and Carl have better ways of choosing the unison vectors if
you want to read back over that. I still have Pari
sessions recorded for a message on it.

Graham

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

> How does this relate to the generalization of MOS that you posted
> before? Is there a simple explanation for it? I didn't quite
> understand the original thread.

I don't know if this is what you are looking for, but the hendecatonic temperament tempering out 6144/6125 and 10976/10935 has period 1/11 octave, and a generator which can be taken as 5/4. Edos 22, 55, 77, 99.
Also of note is the enneadecal microtemperament with period 1/19 octave and tempering out 4375/4374 and 703125/702464, with edos 152, 171, 665, 836 and 1007. There's also the icosidillic temperament with 1/22 octave period and tempering out 3388/3375, 6144/6125 and 9801/9800. Both of these can also have generators we can take to be 5/4. If you allow 6/5 instead, there's ennealimmal and hemiennealimmal, and if you want island harmony, decitonic with a 15/13 generator.