The MOS "Hypothesis"

Let's say that we're working in the octave-equivalent space created by
the 3-limit, and let's say that our single chromatic unison vector is
81/64. This would appear to be an epimorphic block such that, if you
temper out all of the unison vectors except for the chromatic one (aka
you temper out nothing), is not an MOS.

Yes?

-Mike

And then while we're at it, what about the 5-limit block delimited by
81/80 and 81/64, where you temper out everything but 81/80? That's not
MOS either, not unless the generator ends up being sharper than
2\3-equal.

And what about generator tunings? The periodicity block denoted by
138/125 as a commatic unison vector and 648/625 as a chromatic unison
vector, which should be mavila[16], isn't MOS unless the generator is
flatter than 4\7. How does that work out when we're dealing with an
unmapped lattice where you could pick any hypothetical tuning for
anything?

-Mike

On Sat, Aug 20, 2011 at 12:38 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Let's say that we're working in the octave-equivalent space created by
> the 3-limit, and let's say that our single chromatic unison vector is
> 81/64. This would appear to be an epimorphic block such that, if you
> temper out all of the unison vectors except for the chromatic one (aka
> you temper out nothing), is not an MOS.
>
> Yes?
>
> -Mike
>

Mike wrote:
>Let's say that we're working in the octave-equivalent space created by
>the 3-limit, and let's say that our single chromatic unison vector is
>81/64. This would appear to be an epimorphic block such that, if you
>temper out all of the unison vectors except for the chromatic one (aka
>you temper out nothing), is not an MOS.

It can't be epimorphic, because 81/64 > 9/8. -Carl

On Sat, Aug 20, 2011 at 1:27 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
> >Let's say that we're working in the octave-equivalent space created by
> >the 3-limit, and let's say that our single chromatic unison vector is
> >81/64. This would appear to be an epimorphic block such that, if you
> >temper out all of the unison vectors except for the chromatic one (aka
> >you temper out nothing), is not an MOS.
>
> It can't be epimorphic, because 81/64 > 9/8. -Carl

OK, but what about the 3-limit block where 2187/2048 vanishes? That
one is MOS, despite that 2187/2048 > 256/243.

-Mike

Blehh, this sucks. Of course every periodicity block isn't MOS if all
but one of the vectors vanish. How could it be true? What tuning are
we going to use for the generator? If we're in dominant temperament,
and my commatic unison vectors are 64/63 and 81/80, with my commatic
vector set at 3125/3072, that's 19-equal, specifically the 19d val.
But that's not going to be an MOS unless the generator is less than
7\12, and the POTE generator for 19d is 701.5 cents. So it all boils
down to which weighting matrix we use at the end of the day. I'm sure
it's possible to construct absurdly large periodicity blocks that are
MOS iff the generator is within a 1 cent margin or something like
that.

And if your response is, well, every periodicity block COULD be made
MOS with the proper generator - yeah, great, we'll just use a 0.0001
cent generator for everything.

Blegh.

-Mike

Graham's now telling me that the proper interpretation is that

1) Any periodicity block in which every comma vanishes except for one
defines a rank-2 temperament, which is obvious
2) It is possible, for that rank-2 temperament, to pick a generator
that creates an MOS the same size as the original block, and one in
which the intervals stay in the same order as the chromatic vector
shrinks to zero

I guess that although this seems tautological to me coming in a decade
after the fact, this was a big deal back when the concept of a "val"
was brand new. So alright then. All is right with the world.

-Mike

On Sat, Aug 20, 2011 at 4:11 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Blehh, this sucks. Of course every periodicity block isn't MOS if all
> but one of the vectors vanish. How could it be true?

Mike wrote:
>Blehh, this sucks. Of course every periodicity block isn't MOS if all
>but one of the vectors vanish. How could it be true? What tuning are
>we going to use for the generator? If we're in dominant temperament,
>and my commatic unison vectors are 64/63 and 81/80, with my commatic
>vector set at 3125/3072, that's 19-equal, specifically the 19d val.
>But that's not going to be an MOS unless the generator is less than
>7\12, and the POTE generator for 19d is 701.5 cents. So it all boils
>down to which weighting matrix we use at the end of the day. I'm sure
>it's possible to construct absurdly large periodicity blocks that are
>MOS iff the generator is within a 1 cent margin or something like
>that.
>And if your response is, well, every periodicity block COULD be made
>MOS with the proper generator - yeah, great, we'll just use a 0.0001
>cent generator for everything.

Yes, for any MOS there's a range of allowed generator sizes.
19d can't be MOS (nor can any single val), as it is rank 1.

-Carl

Mike wrote:
>Graham's now telling me that the proper interpretation is that

Why isn't he telling it onlist, I wonder?

>1) Any periodicity block in which every comma vanishes except for one
>defines a rank-2 temperament, which is obvious
>2) It is possible, for that rank-2 temperament, to pick a generator
>that creates an MOS the same size as the original block, and one in
>which the intervals stay in the same order as the chromatic vector
>shrinks to zero
>I guess that although this seems tautological to me coming in a decade
>after the fact, this was a big deal back when the concept of a "val"
>was brand new. So alright then. All is right with the world.

The hypothesis predates the concept of vals by a year at least.
Well, Graham had the concept of vals before that, but the
significance wasn't apparent, at least to me.

-Carl

On Sat, Aug 20, 2011 at 6:25 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >Graham's now telling me that the proper interpretation is that
>
> Why isn't he telling it onlist, I wonder?

Another one of life's great mysteries! Where is your response to what
I wrote about 2187/2048?

> The hypothesis predates the concept of vals by a year at least.
> Well, Graham had the concept of vals before that, but the
> significance wasn't apparent, at least to me.

I don't understand the significance of the hypothesis then. As I wrote
above, it seems to be saying that any periodicity block in which all
of the commas vanish except for one leads you to rank 2. But even
before Gene joined and group theory was introduced, Fokker had already
figured out that tempering reduces the dimensionality of the lattice.
So surely this wasn't what was holding everybody up...?

-Mike

Carl Lumma <carl@lumma.org> wrote:
> Mike wrote:
> >Graham's now telling me that the proper interpretation
> >is that
>
> Why isn't he telling it onlist, I wonder?

Because Mike started chatting to me before I read my email
that morning.

> >1) Any periodicity block in which every comma vanishes
> >except for one defines a rank-2 temperament, which is
> >obvious 2) It is possible, for that rank-2 temperament,
> >to pick a generator that creates an MOS the same size as
> >the original block, and one in which the intervals stay
> >in the same order as the chromatic vector shrinks to zero
> >I guess that although this seems tautological to me
> >coming in a decade after the fact, this was a big deal
> >back when the concept of a "val" was brand new. So
> >alright then. All is right with the world.
>
> The hypothesis predates the concept of vals by a year at
> least. Well, Graham had the concept of vals before that,
> but the significance wasn't apparent, at least to me.

Is this the first statement of the hypothesis: May 5th 2001?

/tuning/topicId_22135.html#22135

By that time, if the concept I had was vals, I'd shown that
you could always associate one with a periodicity block. I
was also working on code to find rank 2 temperaments from
periodicity blocks. So the significance should have been
apparent. (Gene also had the concept, but he hadn't
published it at the time.)

If you understand how periodicity blocks relate to rank 2
temperaments, it should be obvious that something like the
hypothesis will be correct. There are details involved in
exactly how you state it to make sure that it's correct
and how you define the terms you use to state it. In so
far as it wasn't obvious in a broader sense it's because
these things weren't well known back then.

Graham

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

> > The hypothesis predates the concept of vals by a year at least.
> > Well, Graham had the concept of vals before that, but the
> > significance wasn't apparent, at least to me.

I entered the group burbling constantly about homomorphisms, come to that.

> So surely this wasn't what was holding everybody up...?

What held me up at first was trying to get an exact statement of what "the hypothesis" actually was, while learning about MOS and their relationship to rank two temperaments.