Formula for constructing omnitetrachordal scales

Paul is telling me that it really is called omnitetrachordality. I'm
going to call it that for now, until someone comes up with a better
name to describe the effect for scales that don't repeat at the 4/3.

Let's stick with the 4/3 for now. Start with two fourths (let's say
we're in 24-et for convenience):

C ----- F ----- Bb

Now throw some generator over each one - could be tempered or not.
Let's say the generator is 150 cents (notated in 24-et for
convenience):

C - Dv - F - Gv - Bb - Cv - (C)

This is an omnitetrachordal scale. This gets a bit tricky if the
generator thrown on top of the Bb on top ever exceeds the octave,
however. Now let's try it with semaphore:

C - D^ - F - G^ - Bb - (C) - C^

That (C) will screw things up, because the D^ - F - G^ trichord is no
longer repeated at G^ (the G^ -> C is now in the way). To fix this, we
can put a G in there, so that the D^ F G^ trichord now becomes D^ F G
G^ to mimic it:

C - D^ - F - G - G^ - Bb - (C) - C^

This screws up the symmetry between C D^ F and F G^ Bb, however, but
the addition of a D fixes that:

C - D - D^ - F - G - G^ - Bb - (C) - C^

Not done yet though - see that C^ on top? Now it has to get transposed
down to the bottom, where it screws up the symmetry between C--F and
F--Bb:

C - C^ - D - D^ - F - G - G^ - Bb - (C) - C^

By throwing in F^ and Bb^, we can fix it:

C - C^ - D - D^ - F - F^ - G - G^ - Bb - Bb^ - (C)

And we now have an omnitetrachordal scale. This is 1 3 1 5 1 3 1 5 1
3, which also happens to be two pentatonic scales that are offset by
1\24. Likewise, Paul's pentachordal scales are two pentatonic scales
that are offset by 2\22. Can anyone see the bigger pattern here? It
seems to have something to do with "quasi"-periodicity, as outlined
above, but I can't quite nail it down.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Paul is telling me that it really is called omnitetrachordality.

By whom? Google supplies me with no reason to think it is called that by anyone outside of the tuning list gang.

I'm
> going to call it that for now, until someone comes up with a better
> name to describe the effect for scales that don't repeat at the 4/3.

It's an amazingly idiotic name for that purpose, and I think a bad one for any purpose. I can feel myself tuning out just reading it.

> C - Dv - F - Gv - Bb - Cv - (C)
>
> This is an omnitetrachordal scale.

A term which has not been given a definition as yet.

On Sun, Apr 17, 2011 at 10:32 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Paul is telling me that it really is called omnitetrachordality.
>
> By whom? Google supplies me with no reason to think it is called that by anyone outside of the tuning list gang.

By Paul. But we can come up with a better name for it. Maybe some
insight into what's really going on under the hood would suggest some
things/

> I'm
> > going to call it that for now, until someone comes up with a better
> > name to describe the effect for scales that don't repeat at the 4/3.
>
> It's an amazingly idiotic name for that purpose, and I think a bad one for any purpose. I can feel myself tuning out just reading it.

So you're not a fan of Moments of Omnitetrachordality (MOOs)?

> > C - Dv - F - Gv - Bb - Cv - (C)
> >
> > This is an omnitetrachordal scale.
>
> A term which has not been given a definition as yet.

Hopefully the other thread has now cleared that up.

For some ideas, here's two omnitetrachordal variants of porcupine in
22-tet, subsymmetric about the 4/3:
3 6 3 6 3 1 - 6 notes
2 1 1 2 3 2 1 1 2 3 2 1 1 - 13 notes

Here's some omnitetrachordal negri in 19-tet:
2 6 2 6 2 1 - 6 notes
1 1 1 1 4 1 1 1 1 4 1 1 1 - 13 notes

-Mike

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

> Hopefully the other thread has now cleared that up.

"Tetrachordal" "in all modes"? I'm not sure what you are saying.

> For some ideas, here's two omnitetrachordal variants of porcupine in
> 22-tet, subsymmetric about the 4/3:
> 3 6 3 6 3 1 - 6 notes
> 2 1 1 2 3 2 1 1 2 3 2 1 1 - 13 notes

Why are these "tetrachordal" "in all modes"? What is an example of something which is tetrachordal, but not in all modes?

On Mon, Apr 18, 2011 at 1:15 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> > For some ideas, here's two omnitetrachordal variants of porcupine in
> > 22-tet, subsymmetric about the 4/3:
> > 3 6 3 6 3 1 - 6 notes
> > 2 1 1 2 3 2 1 1 2 3 2 1 1 - 13 notes
>
> Why are these "tetrachordal" "in all modes"? What is an example of something which is tetrachordal, but not in all modes?

A mode is tetrachordally symmetric if

1) The pattern of steps in a 4/3 that is connected to the root is
repeated at another 4/3 adjacent to it
2) The tetrachords can both be going up from the root, both be going
down from the root, or have one going up and one going down

An example of a tetrachordally symmetric mode with both tetrachords
going up from the root is C Db Eb F Gb Ab Bb C, since G---C and C---F
are both sLL. An example with them both going down is C D E F# G A B
C, where D-G and G-C are both LLs. An example where one goes down and
one goes up is C D Eb F G A Bb C, where C-G is LsL, and G-C is also
LsL.

Porcupine[7] is an example of a scale that displays tetrachordal
symmetry in some, but not all modes. It has identical sss tetrachords
both going up at ssssssL, both going down at Lssssss, and one going up
and one going down at sssLsss.

I notice that the omnitetrachordal scales tend to be various 3-limit
MOS's transposed by a certain generator; i.e. Pajara's
omnitetrachordal mode is two interlocking superpyth[5] pentatonic
MOS's offset by a Pajara generator. Semaphore's omnitetrachordal mode
is the same thing with the semaphore generator. Meantone[10] is
omnitetrachordal and consists of two meantone[5]'s offset by 25/24.
Meantone[5] itself is two copies of meantone[3] that are shifted by a
whole step.

-Mike

On Mon, Apr 18, 2011 at 12:13 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I notice that the omnitetrachordal scales tend to be various 3-limit
> MOS's transposed by a certain generator; i.e. Pajara's
> omnitetrachordal mode is two interlocking superpyth[5] pentatonic
> MOS's offset by a Pajara generator. Semaphore's omnitetrachordal mode
> is the same thing with the semaphore generator. Meantone[10] is
> omnitetrachordal and consists of two meantone[5]'s offset by 25/24.
> Meantone[5] itself is two copies of meantone[3] that are shifted by a
> whole step.

A last note - it may make more sense to think about these as tempered
versions of rank-3 scales, consisting of the octave as one period, the
fourth (or whatever interval of repetition you want) as a second
sub-period, and then a generator. If you do that, the following
5-limit JI major scale is omnitetrachordal:

Take a 3-limit pentatonic scale, C D F G A (1/1 9/8 4/3 3/2 27/16),
and transpose it upwards by 10/9:

D- E- G- A- B-
C D F G A

Smushing this within an octave yields

C D- D E- F G- G A- A B- C

This scale is a 10-note omnitetrachordal scale. Now, if you temper out
81/80, three pairs of notes in this scale end up converging on one
another - D- and D, G- and G, A- and A -- and so this 10 note scale
gets turned into a 7-note scale, which is the meantone diatonic scale
- and is also omnitetrachordal. On the other hand, if you temper out
256/243, then this ends up turning into two 5-equal chains offset by
10/9 instead of two pythagorean[5] chains - and you now have
Blackwood[10], which is also omnitetrachordal. The same applies if you
start with two chains of pythagorean[3]:

D- E- A-
C D G C

Putting this altogether in an octave yields

C D- D E- G A- C

This 6-note hexatonic scale is omnitetrachordal. If you eliminate
81/80, D- and D converge on one another, and you're left with
meantone[5], which is also omnitetrachordal. So omnitetrachordal
scales are, in a certain sense, higher-rank analogues of MOS, but
generalize them in a different way than do hobbit scales.

What I don't understand is why no omnitetrachordal variant of
semaphore exists until 10 notes up.

-Mike

On Mon, Apr 18, 2011 at 12:25 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> By Paul. But we can come up with a better name for it. Maybe some
> insight into what's really going on under the hood would suggest some
> things/

Geez, alright, last last note - Paul's now telling me that these
scales aren't actually omnitetrachordal unless the "disjunction" only
contains step sizes used in the tetrachord. So for porcupine, 2 1 1 2
3 2 1 1 2 3 2 1 1 is still omnitetrachordal, but under Paul's
definition 3 6 3 6 3 1 is not, because the "1" in the disjunction
doesn't occur in any of the tetrachords.

We need a new name for all of this. These scales are kind of like -
take an MOS and then use that as the period for a third generator.

So while blackwood[10] is an MOS, if you detemper 256/243, you end up
with two pythagorean[5] chains offset by 10/9, and that's a ______
(insert new name here that is no longer omnitetrachordal).

-Mike