Way to further classify MODMOS's - request for feedback

I think this is another important property that should be looked at as
fundamental in organizing the MODMOS's of a certain MOS. However, I'm
having trouble seeing the mathematical "big picture" here, so if
anyone has any insight, it would be appreciated.

The idea is that - even if you're altering notes by chroma, only
certain note alterations actually exist in the chromatic scale of the
albitonic MOS that you're working with. The way I'm used to using
these MODMOS's is against the backdrop of a larger "chromatic" scale,
and you really have to learn to think that way when you play music
that shifts often between meantone MOS's and MODMOS's. So it may be
that many of the most useful MODMOS's are those that form a subset of
the chromatic scale of the albitonic pair, which in most cases is just
the next MOS up.

So, for example, in 31-tet, C D E# F G A B C doesn't actually exist in
the chromatic MOS above meantone[7], despite that you have made a
chroma alteration and the resulting scale is proper. So as far as
modulation within a chromatic superstructure is concerned - it is best
thought of as being part of the hyperchromatic MOS two levels above
albitonic, instead of the chromatic MOS one level above albitonic.
This is something important I think that needs to be kept track of -
how many levels you have to go down the scale tree to find a superset
of the resulting MODMOS.

Obviously it's not a hard rule that 1 level down is best, otherwise
porcupine[7] would be screwed. If porcupine[7] is an albitonic scale,
then porcupine[8] is best thought of as a mega-albitonic scale. So in
this case, even though the next level up from porcupine[7] is
porcupine[8], it's useful to generate scales that go two levels down
the scale tree to porcupine[15].

Nonetheless, it would perhaps be useful to arrange the MODMOS's in
this fashion so we can see how much further order each chromatic
alteration implies.

So now we have three fundamental attributes that MODMOS's have, aside
from their propriety:
1) The number of alterations that you've made to the scale; e.g. C Db
E F G Ab B C necessitates a minimum of two alterations, C D Eb F G A B
C necessitates a minimum of one
2) How many orders of magnitude you have to go up to find a
"chromatic" MOS that encapsulates the resultant scale; e.g. C D E# F G
A B C requires going two levels up, whereas C D E F G# A B C only
requires going one
3) Which interval it is you're MODding the scale by - is it the
chroma, c = L-s? Is it the diesis, d = |s-c|?

Does anyone see the big picture here? We're getting closer...

-Mike

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

> So, for example, in 31-tet, C D E# F G A B C doesn't actually exist in
> the chromatic MOS above meantone[7], despite that you have made a
> chroma alteration and the resulting scale is proper.

I'd say the best way to phrase that is that from E# to F is a diesis, which is the chroma for the MOS size above 7. So one could require that a MODMOS of a particular kind (name?) not have any steps of such a size. Hyperchromatic steps, could we call them?

> 3) Which interval it is you're MODding the scale by - is it the
> chroma, c = L-s? Is it the diesis, d = |s-c|?

Please let's not call |s-c| the diesis, an already seriously overloaded term. Hyperchroma?

On Wed, Mar 16, 2011 at 10:42 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So, for example, in 31-tet, C D E# F G A B C doesn't actually exist in
> > the chromatic MOS above meantone[7], despite that you have made a
> > chroma alteration and the resulting scale is proper.
>
> I'd say the best way to phrase that is that from E# to F is a diesis, which is the chroma for the MOS size above 7.

OK but - it looks like the presence of a diminished second in the
scale isn't sufficient to determine this. Sometimes it works out to
there being other intervals that don't fit right. For example, check
out C D# E F G Ab B C in 31tet - All of the seconds are fine, but the
D# and Ab form an ~11/8 which requires the hyperchromatic scale to
exist.

You can also prove this by seeing that to get to D#, you'd have to go
through G# first, and if there's also an Ab in the scale, then you'd
have to go through Eb to get to it. So you're already within the realm
of enharmonic intervals.

Also, could it ever be that all of the intervals themselves in
isolation are present in the containing MOS, but the scale that they
produce when you "fit" them together isn't present? Not sure...

I think a method that gives some insight is to view these as unmapped,
rank 2 "periodicity strips" - to ensure that the scale ends up being
the right size, make sure that the size of the block (counting holes)
doesn't end up being larger than the next-highest MOS. More about this
below, and even more in an upcoming message...

> So one could require that a MODMOS of a particular kind (name?) not have any steps of such a size. Hyperchromatic steps, could we call them?

I think that the periodicity block approach laid out above is best,
but yeah - there are now three properties - the number of alterations,
the interval you're altering by, and the resultant span of the
periodicity block, counting holes. I'm not sure what to name them.

For example,

F C G D A E B - diatonic, total width 7. now if we sharp the second
note from bottom
F x G D A E B x C# - melodic minor, total width 9. if instead we sharp
the third note
F C x D A E B x x G# - harmonic minor, total width 10. If we sharp
the fourth note
F C G x A E B x x x D# - I dunno what this is called, total width
11. If we sharp the fifth note
F C G D x E B x x x x A# - I also dunno what this is called, total width 12.

All of the above will fit within a single chromatic scale, since the
resultant width is 12. But God help the poor soul who tries to sharp
E...

F C G D A x B x x x x x E# - Total width 13.

Total width 13 - now you can't represent it by a single chromatic
scale anymore. You also have a diesis within the scale, which is a
nice clue that you've done something wrong. But there are times when
this will happen with no diesis appearing, like if you make two
alterations as follows:

Eb x F C G D x x B x x x x A# - Total width 14, but no diesis.

Looking at the periodicity blocks like this helps a lot. More to come...

> > 3) Which interval it is you're MODding the scale by - is it the
> > chroma, c = L-s? Is it the diesis, d = |s-c|?
>
> Please let's not call |s-c| the diesis, an already seriously overloaded term. Hyperchroma?

I really was going to suggest the diesis, since the heirarchy chroma
-> diesis -> comma suggests itself here (when, in antiquity, folks
would refer to the "diesis," were they thinking specifically about
128/125, or about the interval that subdivides the chroma?). But I
guess that might be confusing.

Graham suggested that the level above "chromatic" be called
"enharmonic" instead of "hyperchromatic", which I can dig; maybe we
could choose a related term from "enharmonic?" Or maybe hyperchroma is
fine too. I don't have strong feelings either way.

-Mike