An initial idea on how to index near-MOS's

Does anyone have any ideas on how we can filter through all of this
crap to index the near-MOS's of a scale? The goal, again, is to be
able to generate a program that not only lists the MOS's for a various
temperament, but its near-MOS's as well, which may very well be more
musically important (see Paul's SPM scales for an example). MOS's are
useful in that they every triad class comes in three specific sizes
(assuming an octave period), which is important for setting up a
major, minor, diminished structure. near-MOS's would probably make it
so that SOME triad classes come in 3 specific sizes, but that others
come in more. This might not be a bad thing if it means we get greater
tetrachordal symmetry.

A long time ago I independently discovered the same thing that Charles
Lucy calls "Scalecoding." The pattern here is that the near-MOS's of
meantone generally have "holes" in the chain of generators - see here:

Diatonic - F-C-G-D-A-E-B
Melodic Minor - Eb-xx-F-C-G-D-A-xx-B (two symmetrically spaced holes)
Harmonic Minor - Ab-Eb-xx-F-C-G-D-xx-xx-B (two holes at the upper end,
one hole at the bottom end)
Harmonic Major - Ab-xx-xx-F-C-G-D-xx-E-B (two holes at the lower end,
one hole at the upper end)
Locrian Major - Gb-xx-Ab-xx-Bb-F-C-xx-D-xx-E (two holes placed alternatingly)

Does anyone see a way that we could use this to index the near-MOS's?

-Mike

On 14 March 2011 03:28, Mike Battaglia <battaglia01@gmail.com> wrote:

> A long time ago I independently discovered the same thing that Charles
> Lucy calls "Scalecoding." The pattern here is that the near-MOS's of
> meantone generally have "holes" in the chain of generators - see here:
>
> Diatonic - F-C-G-D-A-E-B
> Melodic Minor - Eb-xx-F-C-G-D-A-xx-B (two symmetrically spaced holes)
> Harmonic Minor - Ab-Eb-xx-F-C-G-D-xx-xx-B (two holes at the upper end,
> one hole at the bottom end)
> Harmonic Major - Ab-xx-xx-F-C-G-D-xx-E-B (two holes at the lower end,
> one hole at the upper end)
> Locrian Major - Gb-xx-Ab-xx-Bb-F-C-xx-D-xx-E (two holes placed alternatingly)
>
> Does anyone see a way that we could use this to index the near-MOS's?

It looks like it. It may or may not be helpful.

I'll also suggest you add combinatorics to your reading list. It
covers choosing subsets like this. I don't know much about it but I
believe a web search will turn up accessible material.

Graham

On Mon, Mar 14, 2011 at 4:50 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> > Does anyone see a way that we could use this to index the near-MOS's?
>
> It looks like it. It may or may not be helpful.

I ditched it - see my more recent thread about directly indexing them
via chromatic alterations. I guess in one sense, this is useful
because it's effectively a 3-limit periodicity block, and I'm going to
guess that we're going to have to get into periodicity blocks in order
to develop theorems about which alterations will yield permutations of
the same scale (e.g. you don't want to waste time testing both Lydian
#5 and Lydian b7, since they both yield the melodic minor parent
scale).

The other method I've proposed has the advantage of being consistent
with the terminology that's already being used in jazz circles (and
probably with shredders too, since although I'm not into that style of
music these people know everything about different near-diatonic
modes). I also personally like it because it has the advantage of
taking the paradigm that I'm are already used to and generalizing it
in a pretty simple way to handle other MOS's - you just figuring out
what the chromatic vector is for some MOS, give it an accidental sign
(like "#" or "b") and go from there. You can even generalize
half-sharp and half-flats by going two MOS levels down the tree and
finding |(L-s) - s| = |L-2s| and so on.

Since all of this jazz/impressionistic/etc theory that involves modes
of melodic and harmonic minor is, in a sense, already an organization
of the near-MOS's of the diatonic scale into a useful structure, I
find it useful to set it up this way. Herman Miller appears to have
basically used this approach in his porcupine naming convention for
37-tet (check it out in Scala) so I guess it isn't anything new.

The one drawback is, again, that something like Ionian b3 and Ionian
#1 both yield modes of the same scale. And Ionian #4 and Ionian b7
yield just another mode of the diatonic scale. So it would be useful
to be able to predict when this will happen, and I think to do so
you'd have to regress the whole thing into a periodicity block view
and develop theorems to figure out when alterations will actually a
new, unique shape, rather than just a translation of another
alteration.

> I'll also suggest you add combinatorics to your reading list. It
> covers choosing subsets like this. I don't know much about it but I
> believe a web search will turn up accessible material.

Thanks for the reference, I'll check it out. Damn, lots to read these days...

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
I also personally like it because it has the advantage of
> taking the paradigm that I'm are already used to and generalizing it
> in a pretty simple way to handle other MOS's - you just figuring out
> what the chromatic vector is for some MOS, give it an accidental sign
> (like "#" or "b") and go from there.

I don't want to quote Carl to Michael Sheiman, but would you *please* stop calling L-s a "vector"?

On Mon, Mar 14, 2011 at 11:35 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I also personally like it because it has the advantage of
> > taking the paradigm that I'm are already used to and generalizing it
> > in a pretty simple way to handle other MOS's - you just figuring out
> > what the chromatic vector is for some MOS, give it an accidental sign
> > (like "#" or "b") and go from there.
>
> I don't want to quote Carl to Michael Sheiman, but would you *please* stop calling L-s a "vector"?

OK. But can you respond to my post in the other thread where I ask you
the difference between near-MOS and MODMOS? I see you've cleaned up
the xenwiki page on it though, looks good.

-Mike

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 14 March 2011 03:28, Mike Battaglia <battaglia01@...> wrote:
>
> > A long time ago I independently discovered the same thing that Charles
> > Lucy calls "Scalecoding." The pattern here is that the near-MOS's of
> > meantone generally have "holes" in the chain of generators - see here:
> >
> > Diatonic - F-C-G-D-A-E-B
> > Melodic Minor - Eb-xx-F-C-G-D-A-xx-B (two symmetrically spaced holes)
> > Harmonic Minor - Ab-Eb-xx-F-C-G-D-xx-xx-B (two holes at the upper end,
> > one hole at the bottom end)
> > Harmonic Major - Ab-xx-xx-F-C-G-D-xx-E-B (two holes at the lower end,
> > one hole at the upper end)
> > Locrian Major - Gb-xx-Ab-xx-Bb-F-C-xx-D-xx-E (two holes placed alternatingly)
> >
> > Does anyone see a way that we could use this to index the near-MOS's?
>
> It looks like it. It may or may not be helpful.
>
> I'll also suggest you add combinatorics to your reading list. It
> covers choosing subsets like this. I don't know much about it but I
> believe a web search will turn up accessible material.
>
>
> Graham

Yes, Enumerative Combinatorics and Polya's counting methods are really
worth the time and trouble. For example, there are 80 hexachord types, 66 pentachord types, 43 tetrachord types, 19 trichord types,
6 dichord types, 1 unichord, and 1 no-chord types based on the Polya
polynomial...you can also study mirror inverse, and complementation
(for hexachords only in 12-tET) using Polya's methods..I'd love to
tie all of this into MOS's, (see my post on MOS's in 19-tET) where
one multiplies by the group of units in a ring (I think that's right)
in 12-tET its boring, you only has M1, M5, M7, M11 so you have mirror inverse, or swapping chromatic and diatonic...or both. But even THAT
can be found by means of Polya! (D4 X S3 Group to be exact)

In 19-tET (which has 18 units) a MOS scale mapped to MOS's usually or near-MOS's sometimes. I have no idea how the behaviour of MOS's and near MOS's could benefit from Polya in 12-tET though at this point.

PGH

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

> OK. But can you respond to my post in the other thread where I ask you
> the difference between near-MOS and MODMOS? I see you've cleaned up
> the xenwiki page on it though, looks good.

I read through what you wrote and concluded there isn't any. If you think otherwise, be sure to tell us about it. I moved it to MODMOS since that name has been around for years and included some examples in the Scala scale archive.

On Mon, Mar 14, 2011 at 6:38 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK. But can you respond to my post in the other thread where I ask you
> > the difference between near-MOS and MODMOS? I see you've cleaned up
> > the xenwiki page on it though, looks good.
>
> I read through what you wrote and concluded there isn't any. If you think otherwise, be sure to tell us about it. I moved it to MODMOS since that name has been around for years and included some examples in the Scala scale archive.

For the record, the reason I was calling it a chromatic "vector" is
that I originally worked all of this up with Fokker periodicity
blocks, where an actual vector is involved. I then realized that you
could redo the whole thing in terms of diatonic set theory
(MOD-diatonic set theory?) in which no recourse to the ratios is
necessary. So I should probably just call it a "chroma" for the sake
of terminological rigor.

What name do you propose for the |c-s| interval, which generalizes the
half-sharp? Should it just be called a "diesis?" Sometimes |c-s| will
end up being larger than c, although I'm not sure that's worth
worrying about.

One last thing though - what would be really useful are some theorems
on when two separate alterations of the scale will yield permuted
results for one another. So, in other words, Ionian b3 and Ionian #1
both yield modes of the melodic minor scale. It would be very useful
to know when that's going to happen. A periodicity block approach
would show that these two just yield translated versions of the same
periodicity pseudo-jigsaw block. Since MOS's can be viewed as rank 2
periodicity "blocks," I assume that this is how all of this ties into
the scalecoding stuff I mentioned before. Can you see some "big
picture" for how to uniquely index MODMOS's based off of something
like that such that Ionian b3 and Ionian #1 get the same index?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Mar 14, 2011 at 6:38 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > OK. But can you respond to my post in the other thread where I ask you
> > > the difference between near-MOS and MODMOS? I see you've cleaned up
> > > the xenwiki page on it though, looks good.
> >
> > I read through what you wrote and concluded there isn't any. If you think otherwise, be sure to tell us about it. I moved it to MODMOS since that name has been around for years and included some examples in the Scala scale archive.
>
> For the record, the reason I was calling it a chromatic "vector" is
> that I originally worked all of this up with Fokker periodicity
> blocks, where an actual vector is involved. I then realized that you
> could redo the whole thing in terms of diatonic set theory
> (MOD-diatonic set theory?) in which no recourse to the ratios is
> necessary. So I should probably just call it a "chroma" for the sake
> of terminological rigor.
>
> What name do you propose for the |c-s| interval, which generalizes the
> half-sharp? Should it just be called a "diesis?" Sometimes |c-s| will
> end up being larger than c, although I'm not sure that's worth
> worrying about.
>
> One last thing though - what would be really useful are some theorems
> on when two separate alterations of the scale will yield permuted
> results for one another. So, in other words, Ionian b3 and Ionian #1
> both yield modes of the melodic minor scale. It would be very useful
> to know when that's going to happen. A periodicity block approach
> would show that these two just yield translated versions of the same
> periodicity pseudo-jigsaw block. Since MOS's can be viewed as rank 2
> periodicity "blocks," I assume that this is how all of this ties into
> the scalecoding stuff I mentioned before. Can you see some "big
> picture" for how to uniquely index MODMOS's based off of something
> like that such that Ionian b3 and Ionian #1 get the same index?
>
> -Mike

Well, in this case, of course, #1 and b3 are just mirror images (centering the scale on 2). Both Ionian and Melodic Minor are symmetrical scales so you are just shifting from (024),(579.11) blocks
to (02),(357911), or (24),(57911.1) So Ionian #1 and Ionian b3 are literal mirror inverses of each other and are in fact related by T2. (Tranpose(2)). I would think that would give them the same index...
As a matter of fact, every scale can be found from a simple alteration
of some basic chain (such as 036947.10.1) I would think these chains
could be expressed as periodicity blocks. (generator, and offset) PGH