Yahoo Tuning Groups Ultimate Backup tuning-math Question about omnitetrachordality

Let's define the a/b-omnitetrachordality of a scale to be its
omnitetrachordality with respect to some sub-period a/b. So for
example, meantone[7] is 4/3-omnitetrachordal.

Is it that a scale can be a/b-omnitetrachordal if and only if it one
of its generating elements is a/b? For example, the pajara decatonic
scales have a generator of 4/3 and a period of 1/2 oct. Meantone has a
generator of 4/3 and a period of an octave. Magic[7] is
5/4-omnitetrachordal - for every 5/4 trichord, there's either a
conjunct or disjunct trichord with the same patterns on either side of
it.

The one thing I don't get is Blackwood[10], which is omnitetrachordal
- it has a generator of 5/4, but it's period is a half of a 4/3.
Having the period be a fraction of 4/3 doesn't seem to matter too
much, but having the generator be a fraction of a 4/3 does - I don't
think there exists an omnitetrachordal variant of semaphore or
porcupine anywhere, ever anywhere. I keep looking, but I can't find
one.

Halp?

-Mike

🔗Carl Lumma <carl@lumma.org>

Mike wrote:

>Let's define the a/b-omnitetrachordality of a scale to be its
>omnitetrachordality with respect to some sub-period a/b. So for
>example, meantone[7] is 4/3-omnitetrachordal.
>Is it that a scale can be a/b-omnitetrachordal if and only if it one
>of its generating elements is a/b? For example, the pajara decatonic
>scales have a generator of 4/3 and a period of 1/2 oct. Meantone has
>a generator of 4/3 and a period of an octave. Magic[7] is
>5/4-omnitetrachordal - for every 5/4 trichord, there's either a
>conjunct or disjunct trichord with the same patterns on either side
>of it.

I think there's a misunderstanding of the term here...
"omnitetrachordal" refers only to symmetry at the 3:2. The
"omni" just means the 4:3 may be divided into some number of
parts other than four.

So I'd suggest a different name for your concept...
a/b-periodicity maybe? a/b-tetrachordality?

>The one thing I don't get is Blackwood[10], which is omnitetrachordal
>- it has a generator of 5/4, but it's period is a half of a 4/3.

Er, generators are always smaller than periods. It's
period + generator = 5:4
period * 2 = 4:3

>Having the period be a fraction of 4/3 doesn't seem to matter too
>much, but having the generator be a fraction of a 4/3 does - I
>don't think there exists an omnitetrachordal variant of semaphore
>or porcupine anywhere, ever anywhere. I keep looking, but I can't
>find one.

Maybe you're using these terms differently than I'm used to,
but plain ol' porcupine[7] is tetrachordal in the conventional
sense

1/1.....4/3 3/2.....2/1
...s s s...L...s s s

-Carl

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Apr 14, 2011 at 3:42 AM, Carl Lumma <carl@lumma.org> wrote:
>
> I think there's a misunderstanding of the term here...
> "omnitetrachordal" refers only to symmetry at the 3:2. The
> "omni" just means the 4:3 may be divided into some number of
> parts other than four.
>
> So I'd suggest a different name for your concept...
> a/b-periodicity maybe? a/b-tetrachordality?

OK, well Paul was telling me that omnitetrachordal means that, under
any type of transposition, the pattern of steps within any 4/3 is
repeated within another 4/3, either conjunct or disjunct, and either
going up or down from the root. But I thought it was a silly name
anyway, and the whole "tetra" thing needs to go if we're generalizing
it, so maybe we should call it a/b-pseudoperiodicity, since that's
what it is. Or a/b-subperiodicity, or a/b subsymmetry, or something
like that.

> >The one thing I don't get is Blackwood[10], which is omnitetrachordal
> >- it has a generator of 5/4, but it's period is a half of a 4/3.
>
> Er, generators are always smaller than periods. It's
> period + generator = 5:4
> period * 2 = 4:3

That was how I used to use the term, but then I'd see Gene saying that
you can think of the generator for diminished[8] as 3/2 instead of
just as 15/8, so I went with that instead.

> >Having the period be a fraction of 4/3 doesn't seem to matter too
> >much, but having the generator be a fraction of a 4/3 does - I
> >don't think there exists an omnitetrachordal variant of semaphore
> >or porcupine anywhere, ever anywhere. I keep looking, but I can't
> >find one.
>
> Maybe you're using these terms differently than I'm used to,
> but plain ol' porcupine[7] is tetrachordal in the conventional
> sense
>
> 1/1.....4/3 3/2.....2/1
> ...s s s...L...s s s

I meant it in the sense Paul was using, which is that it remains
conjunctly or disjunctly tetrachordal under every modal transposition.
Porcupine doesn't have this feature, but the diatonic scale does. I'm
not sure if the point of it is to ease modulation, or if it's just a
neat way of "chunking" to reduce the effective amount of information
in a scale that someone has to remember, but it seems like an
interesting property to have.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Apr 14, 2011 at 8:24 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> OK, well Paul was telling me that omnitetrachordal means that, under
> any type of transposition, the pattern of steps within any 4/3 is
> repeated within another 4/3, either conjunct or disjunct, and either
> going up or down from the root. But I thought it was a silly name
> anyway, and the whole "tetra" thing needs to go if we're generalizing
> it, so maybe we should call it a/b-pseudoperiodicity, since that's
> what it is. Or a/b-subperiodicity, or a/b subsymmetry, or something
> like that.

"Quasi-periodic" or "locally quasi-periodic" seems to be the
mathematically correct term for this behavior, since within some
bounded interval there exists a periodic structure, but not outside of
that.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> "Quasi-periodic" or "locally quasi-periodic" seems to be the
> mathematically correct term for this behavior, since within some
> bounded interval there exists a periodic structure, but not outside of
> that.

Sadly, this is not the case. The mathematically precise definition of "quasiperiodic" is here:

http://en.wikipedia.org/wiki/Quasiperiodic_function

The vague, mushy concept is here:

http://en.wikipedia.org/wiki/Quasiperiodicity

These are not the same and don't have much to do with each other, a fact which should probably be pointed out in these articles. I really hope you abandon the idea of using "quasiperiodic" in the second sense, as I'm already using it in the first sense on the Xenwiki:

http://xenharmonic.wikispaces.com/Periodic+scale

The usual treatment in scale theory is made more complex and confusing by the fact that a scale is not defined as a quasiperiodic function, so I'd like to to keep it.

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Apr 14, 2011 at 2:27 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> Sadly, this is not the case. The mathematically precise definition of "quasiperiodic" is here:
>
> http://en.wikipedia.org/wiki/Quasiperiodic_function
>
> The vague, mushy concept is here:
>
> http://en.wikipedia.org/wiki/Quasiperiodicity
>
> These are not the same and don't have much to do with each other, a fact which should probably be pointed out in these articles. I really hope you abandon the idea of using "quasiperiodic" in the second sense, as I'm already using it in the first sense on the Xenwiki:
>
> http://xenharmonic.wikispaces.com/Periodic+scale
>
> The usual treatment in scale theory is made more complex and confusing by the fact that a scale is not defined as a quasiperiodic function, so I'd like to to keep it.

So what's the actual mathematical term for this? That it's "locally symmetric?"

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Fri, Apr 15, 2011 at 6:51 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So what's the actual mathematical term for this? That it's "locally symmetric?"
>
> It's tough to have a mathematical term for vague mush.

OK, no more vague mush. The following is a graphic description of
what's going on:

As Paul described it, an "omnitetrachordal" chord is one in which
every single mode has the "tetrachordal" property. The "tetrachordal"
property is that the pattern of steps within any 4/3 is repeated
within another 4/3 in the scale. The tetrachords can either be
conjunct or disjunct from one another. For example (view fixed width):

Disjunct tetrachordal symmetry:
| || |
|C D E F||G A B C|
| || |

Conjunct tetrachordal symmetry:
| | ||
|B C D E F G A||B
| | ||

As you can see, "disjunct" tetrachordal symmetry is just conjunct
tetrachordal symmetry, but with one of the tetrachords going down
instead of up. For example

Disjunct tetrachordal symmetry:
| || | || | || |
|C D E F||G A B C D E F||G A B C D E F||G A B C
| || | || | || |

So Paul defined the "omnitetrachordal" property as being one in which
every mode of the scale has this property. The tetrachords can either
both be going down, or going up. So in the case of the diatonic scale:

LLs tetrachords:
Lydian:
|| | |
F||G A B C D E F|
|| | |

Ionian:
| || |
|C D E F||G A B C|
| || |

Mixolydian:
| | ||
|G A B C D E F||G
| | ||

LsL tetrachords:
Mixolydian (Mixolydian is doubly tetrachordal):
|| | |
G||A B C D E F G|
|| | |

Dorian:
| || |
|D E F G||A B C D|
| || |

Aeolian:
| | ||
|A B C D E F G||A
| | ||

sLL tetrachords:
Aeolian (Aeolian is doubly tetrachordal):
|| | |
A||B C D E F G A|
|| | |

Phrygian:
| || |
|E F G A||B C D E|
| || |

Locrian:
| | ||
|B C D E F G A||B
| | ||

Maybe this diagram explains it better:

|| | || | || | ||
||G A B C D E F||G A B C D E F||G A B C D E F||G A
|| | || | || | ||
|| | || | || | ||
G||A B C D E F G||A B C D E F G||A B C D E F G||A
|| | || | || | ||
|| | || | || | ||
G A||B C D E F G A||B C D E F G A||B C D E F G A||
|| | || | || | ||

So you can see what's going on here. So I want to generalize this to
any interval, not just 4/3. And it seems that "tetrachordality" is
really just something like "local periodicity," e.g. within some
bounded interval the scale is periodic at the 4/3, but not beyond
that. For example, this chunk of the diatonic scale is periodic with
respect to the 4/3:

G A B C D E F G A

But, this is then tiled as a "sub-period" with respect to a larger
period, namely the 2/1. That's the point.

Does that all make sense now?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Sun, Apr 17, 2011 at 10:27 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> > Does that all make sense now?
>
> It looks like you are in a good position to make a definition, so I'd suggest you do. Defining it as a property of quasiperiodic scales is what I's suggest.

I've come up with some theorems about it since I made this, so I'll
make a larger post shortly. But like I said above -

> So Paul defined the "omnitetrachordal" property as being one in which
> every mode of the scale has [the tetrachordal] property. The tetrachords can either
> both be going down, or going up.

That's the actual definition, which implies a lot of things about
local periodicity, not all of which I have figured out yet.

-Mike