Generalization of tetrachordality as pseudo-periods within periods, approach to scale construction

Hi all,

There are a lot more MOS's that a temperament defines than just the
ones that we usually look at. For example, if you're working within
meantone, one MOS is

Period: 2/1, Generator: 3/2
C D E F G A B | C D E F G A B

But, there's also

Period: 3/2, Generator: 2/1 (or 9/8)
C D E F | G A B C | D E F# G | A B C# D | E F# G# A

And also

Period: 4/3, Generator: 2/1 (or 9/8)
C D E | F G A | Bb C D | Eb F G

The reason I think that this is worth noting is that there's nothing
stopping us from defining two periods, one within the other. So let's
say we're defining

Period 1: 6/5, Period 2: 2/1, Generator: 3/2 (NOTE: 648/625 does NOT
vanish here)
|| C D E | Eb F G | Gb Ab Bb | Bbb Cb Db || C D E | Eb F G | Gb Ab Bb
| Bbb Cb Db ||

The periods overlap, so putting the pitches in correct and ascending order gives

| C Db D Eb E F Gb G Ab Bbb Bb Cb C |

In 31-tet, this is

3 2 3 2 3 3 2 3 3 2 3 2

Which is a MODMOS of meantone[12], but it isn't meantone[12]. But it
does exhibit some limited tetrachordal symmetry at the 6/5, although
I'm not quite sure how - patterns like C-D-E-F exhibit some kind of
tetrachordal symmetry, but I'm not seeing the directly adjacent steps
int he scale exhibiting it. Perhaps this approach can be further
refined.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi all,
>
> There are a lot more MOS's that a temperament defines than just the
> ones that we usually look at. For example, if you're working within
> meantone, one MOS is
>
> Period: 2/1, Generator: 3/2
> C D E F G A B | C D E F G A B
>
> But, there's also
>
> Period: 3/2, Generator: 2/1 (or 9/8)
> C D E F | G A B C | D E F# G | A B C# D | E F# G# A
>
> And also
>
> Period: 4/3, Generator: 2/1 (or 9/8)
> C D E | F G A | Bb C D | Eb F G
>
> The reason I think that this is worth noting is that there's nothing
> stopping us from defining two periods, one within the other. So let's
> say we're defining
>
> Period 1: 6/5, Period 2: 2/1, Generator: 3/2 (NOTE: 648/625 does NOT
> vanish here)
> || C D E | Eb F G | Gb Ab Bb | Bbb Cb Db || C D E | Eb F G | Gb Ab Bb
> | Bbb Cb Db ||
>
> The periods overlap, so putting the pitches in correct and ascending order gives
>
> | C Db D Eb E F Gb G Ab Bbb Bb Cb C |
>
> In 31-tet, this is
>
> 3 2 3 2 3 3 2 3 3 2 3 2
>
> Which is a MODMOS of meantone[12], but it isn't meantone[12]. But it
> does exhibit some limited tetrachordal symmetry at the 6/5, although
> I'm not quite sure how - patterns like C-D-E-F exhibit some kind of
> tetrachordal symmetry, but I'm not seeing the directly adjacent steps
> int he scale exhibiting it. Perhaps this approach can be further
> refined.
>
> -Mike
>

How about these, and would you find them useful?

Period: 6/5 Generator 5/4

C E G | Eb G B | Gb Bb D | Bbb Db F

Period: 5/4 Generator 6/5

C Eb Gb Bbb | E G Bb Db | G# B D F

(I define these, in a different manner, on musicalsettheory, among others closer to the above such as C D E | Eb F G ...etc in my
hexachord theory)

PGH

On Tue, Apr 12, 2011 at 4:36 PM, Paul <phjelmstad@msn.com> wrote:
>
> How about these, and would you find them useful?
>
> Period: 6/5 Generator 5/4
>
> C E G | Eb G B | Gb Bb D | Bbb Db F
>
> Period: 5/4 Generator 6/5
>
> C Eb Gb Bbb | E G Bb Db | G# B D F
>
> (I define these, in a different manner, on musicalsettheory, among others closer to the above such as C D E | Eb F G ...etc in my
> hexachord theory)

They are interesting, but I guess what I was really getting at was
that within a rank-2 system, it's possible to define a period, a
generator, and a "pseudo-period," and that to do so may be useful to
create tetrachordally symmetric scales.

-Mike

On Tue, Apr 12, 2011 at 4:36 PM, Paul <phjelmstad@msn.com> wrote:
>
> Period: 6/5 Generator 5/4
>
> C E G | Eb G B | Gb Bb D | Bbb Db F

So wait, let's think about this again - an altered version of the
above, which is just

Sub-period: 6/5, Period: 2/1, Generator: 5/4

|| A C# | C E | Eb G | Gb Bb ||

Doesn't make use of any meantone unison vector, but here you go. So
then, in order, you get

| C C# Eb E Gb G A Bb |

This is now omnitetrachordal at the 6/5, with the augmented second
functioning as the "disjunct" interval, analogously to how 9/8
functions as the disjunct interval with the sub-period as 4/3. Hell
yes.

I wonder if it's that these scales just have moments when they become
omnitetrachordal - a Moment of Omnitetrachordality, if you will, which
will of course be called a MOO. You should also note that the above is
kind of like a detempered diminished[8].

A MOS is a scale that tempers out all of the unison vectors in a
periodicity block except for one. I wonder if MOOs are, in a certain
way, temperings out of all of the unison vectors in a periodicity
block except for two, but are then re-tempered in a certain sense. I
think it has to do with redefining the concept of an "equivalence"
vector.

I also think that many MOOs may be MODMOS's (say that 5 times fast).

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Apr 12, 2011 at 4:36 PM, Paul <phjelmstad@...> wrote:
> >
> > Period: 6/5 Generator 5/4
> >
> > C E G | Eb G B | Gb Bb D | Bbb Db F
>
> So wait, let's think about this again - an altered version of the
> above, which is just
>
> Sub-period: 6/5, Period: 2/1, Generator: 5/4
>
> || A C# | C E | Eb G | Gb Bb ||
>
> Doesn't make use of any meantone unison vector, but here you go. So
> then, in order, you get
>
> | C C# Eb E Gb G A Bb |
>
> This is now omnitetrachordal at the 6/5, with the augmented second
> functioning as the "disjunct" interval, analogously to how 9/8
> functions as the disjunct interval with the sub-period as 4/3. Hell
> yes.
>

I see, but wasn't 9/8 also the generator there? Or just 3/2 ?

In this case, using 5/4 twice would smooth it all out...Hell, yes, I think.

> I wonder if it's that these scales just have moments when they become
> omnitetrachordal - a Moment of Omnitetrachordality, if you will, which
> will of course be called a MOO. You should also note that the above is
> kind of like a detempered diminished[8].
>
> A MOS is a scale that tempers out all of the unison vectors in a
> periodicity block except for one. I wonder if MOOs are, in a certain
> way, temperings out of all of the unison vectors in a periodicity
> block except for two, but are then re-tempered in a certain sense. I
> think it has to do with redefining the concept of an "equivalence"
> vector.

Interesting, could you give an example of the two unison vectors in a MOO not tempered out?

(We used to discuss chromatic and commatic unison vectors, the first not being
tempered out....Not to get off subject but I crunched a lot of matrices in Excel where
you would have one not tempered out to look at vals / monzos with adjutants and
determinants (I'd have to look up what i did..)

PGH

> I also think that many MOOs may be MODMOS's (say that 5 times fast).

Yes, many MODs may be MODMOS's in Tuning Tonality and Twenty Two Tone Temperament (sorry)

>
> -Mike