Re: TD 861 -- Wilson's footprints on plateau!

[Dave Keenan wrote...]
>>>Sorry Paul, this _was_ concise. I just failed to unpack from it the
>>>reason why it was useful for L/s to be close to Phi, rather than merely
>>>proper or strictly proper. i.e. the fact that this gives maximum
>>>distinction between interval classes.
>>
>>Well, if you're going to create an infinite number of intervals, then in
>>a sense this is true. But for a single MOS scale along the path, is there
>>any way you can define "distinction between interval classes" such that
>>L/s = Phi implies "maximum distinction between interval classes"?
>
>A very good question. I should have written "the claim that" rather than
>"the fact that". I believe it was Dan's claim. Do you have such a
>definition Dan?

My post _New Work on Rothenberg's Model_ may be of interest here...

http://www.egroups.com/message/tuning/11092

...I cover a few of my own attempts to measure distinction between interval
classes.

>At first glance I thought it was possible, but now it seems to me that the
>greatest distinction between interval classes occurs when all the scale
>steps are the same size. So what is it we are really trying to optimise
>here?

Don't forget scales which "cover" their ET (like the diatonic scale in 12-
tET). They ought to be as good, at least until we start looking at how the
intervals are distributed. Pure ETs might seem best then, but as we found in
our discussion of efficiency, completely symmetrical interval matrices loose
the ability to encode a great wealth of information. I think Erv would
welcome a description of MOS as a view into a kind of symmetry breaking here.
Perfect symmetry is boring, but scales with poorly distinguished interval
classes are too messy -- maybe it's the middle ground where the interesting
stuff goes on.

At first glance, MOS seems a rather arbitrary slice of even this narrow
ground. We also have "symmetrical" scales like the recently-discussed
octatonic, or Easley Blackwood's decatonic scale in 15-tET. These scales
will have two interval sizes in all but one class, about which a kind of
mirror symmetry appears in the interval matrix. On the other side of MOS,
there must be scales satisfying a sort of Myhill's property which insists
that every interval class have three varieties. This end of things is
somewhat generalized in Rothenberg's own "mean variety", which is the
average number of interval sizes per class. But who knows -- maybe further
work on this subject will reveal that MOS is special... maybe it's most
special when a series is slow to converge, as Dan suggests?

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> At first glance, MOS seems a rather arbitrary slice of even this
narrow
> ground. We also have "symmetrical" scales like the recently-
discussed
> octatonic, or Easley Blackwood's decatonic scale in 15-tET.

And don't forget the 22-tET pentachordal decatonic and hexachordal
dodecatonic and the 26-tET heptachordal tetradecatonic . . .

Carl Lumma,

> At first glance, MOS seems a rather arbitrary slice of even this
narrow ground. We also have "symmetrical" scales like the
recently-discussed octatonic, or Easley Blackwood's decatonic scale in
15-tET.

Paul Erlich,

> And don't forget the 22-tET pentachordal decatonic and hexachordal
dodecatonic and the 26-tET heptachordal tetradecatonic . . .

So Myhill's property within a given fractional (or even arbitrary)
periodicity is or is not MOS? I assumed it is. If it's not, I would
have to think it's only because it generally hasn't been used to say
as much... or am I missing something?

--Dan Stearns

>So Myhill's property within a given fractional (or even arbitrary)
>periodicity is or is not MOS? I assumed it is. If it's not, I would
>have to think it's only because it generally hasn't been used to say
>as much... or am I missing something?

Not sure of your question Dan. Myhill's property is equivalent to MOS.
What do you mean by "fractional periodicity"?

-Carl

Carl Lumma wrote,

> Myhill's property is equivalent to MOS.

OK, but both you and Paul seem to be making a clear distinction
between MOS and symmetrical scales. And I'm considering any scale
"MOS" so long as Myhill's property exists within some periodicity --
which it does with symmetrical scales. So I consider symmetrical
scales to be MOS...

Does that help frame the, "isn't Myhill's property within any given
periodicity MOS", question any better?

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Carl Lumma wrote,
>
> > Myhill's property is equivalent to MOS.
>
> OK, but both you and Paul seem to be making a clear distinction
> between MOS and symmetrical scales. And I'm considering any scale
> "MOS" so long as Myhill's property exists within some periodicity --
> which it does with symmetrical scales. So I consider symmetrical
> scales to be MOS...

Symmetrical scales are not MOS, and neither are scales with Myhill's
property. Wilson and Myhill recognized the octave as the only
interval of equivalence. Any further periodicity results in a
generic interval with only one specific size (contradicting Myhill's
property) and requires an additional generating interval besides the
generator and the octave (contradicting Wilson's MOS conception).

I meant to say, "Symmetrical scales are not MOS, and symmetrical
scales don't have Myhill's property either".

Paul Erlich wrote,

> Wilson and Myhill recognized the octave as the only interval of
equivalence.

So then a relaxed MOS (or whatever) where Myhill's property occurs in
any given "P" is a condition in need of a term?

--Dan Stearns

>>>Myhill's property is equivalent to MOS.
>
>OK, but both you and Paul seem to be making a clear distinction
>between MOS and symmetrical scales.

Right- symmetrical scales don't have Myhill's property, because
there's always one interval class with only one size.

>And I'm considering any scale "MOS" so long as Myhill's property
>exists within some periodicity

Eh? Re-read the def. of Myhill's property and get back to me.

-Carl

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> > Wilson and Myhill recognized the octave as the only interval of
> equivalence.
>
> So then a relaxed MOS (or whatever) where Myhill's property occurs
in
> any given "P" is a condition in need of a term?

MOS and Myhill's property are synonymous, so I take it a "relaxed
MOS" is automatically pseudo-Myhill in this way.

Carl Lumma wrote,

> symmetrical scales don't have Myhill's property, because there's
always one interval class with only one size.

Right, I got that. But I'm still looking at these scales as single
generator scales where Myhill's property occurs within a given "P".

One of my favorites from the 20-tET guitar is this LsLsLsLsLs 10-tone
symmetrical scale:

15.-------,7.------,19
\`. .' /|\ `. .' /|\
\ 11--/-+-\--3 / | \
\ | / 16--\-+-/--8 \
\|/.' `. \|/ ,' `. \
0'------`12'-------`4

So besides being a 5s/5L octave scales, I also look at it as a 1s/1L
single generator scale (in 20-tET the 180� generator being most like a
Silver generator).

Here's what I would call the three "archetypal" <or "alchemical"
depending on your point of view I suppose...> generators.

The 1/1, 1/2, 2/3, 3/5, 5/8, ... series generator at 148.3282�:

0 148 240 388 480 628 720 868 960 1108 1200
0 92 240 332 480 572 720 812 960 1052 1200

The 1/1.5, 1.5/3.25, 3.25/6.375, 6.375/12.8125, ... series generator
at 160�:

0 160 240 400 480 640 720 880 960 1120 1200
0 80 240 320 480 560 720 800 960 1040 1200

And the 1/2, 2/5, 5/12, 12/29, 29/70, ... series generator at
169.7056�:

0 170 240 410 480 650 720 890 960 1130 1200
0 70 240 310 480 550 720 790 960 1030 1200

I guess the rhythmic analogy with these scales would be that of
multiple subgroupings within a meter. So by way of the analogy the
5s/5L scale is a 5/4 meter as:
__ __ __ __ __
|. | |. | |. | |. | |. |

--Dan Stearns

Paul Erlich wrote,

> MOS and Myhill's property are synonymous, so I take it a "relaxed
MOS" is automatically pseudo-Myhill in this way.

I think so. For what I'm doing, it just gives these scales a context
that allows for a single generator interpretation which in turn tends
to integrate them back into the big body of MOS scales as opposed to
separating them from it.

--Dan Stearns

I wrote,

> I guess the rhythmic analogy with these scales would be that of
multiple subgroupings within a meter. So by way of the analogy the
5s/5L scale is a 5/4 meter as:
> __ __ __ __ __
> |. | |. | |. | |. | |. |

Sorry, that example should have read:
___ ___ ___ ___ ___
| -| | -| | -| | -| | -|
|. | |. | |. | |. | |. |

--Dan Stearns

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/14119

> Perfect symmetry is boring, but scales with poorly distinguished
interval classes are too messy -- maybe it's the middle ground where
the interesting stuff goes on.
>

This seems to me like a very interesting comment by Carl Lumma....
________ ___ __ _ __
Joseph Pehrson

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/14127

>
> So Myhill's property within a given fractional (or even arbitrary)
> periodicity is or is not MOS? I assumed it is. If it's not, I would
> have to think it's only because it generally hasn't been used to say
> as much... or am I missing something?
>
>
> --Dan Stearns

Could someone please briefly explain "Myhill's property" again to me
in "layman's" terms... It almost sounds like something Dan Sterns
would dream up... "My hill, rather than *yourn* hill..."

It's not in the Monzo dictionary (alors!) so I would tend to think it
doesn't exist, except that people are discussing it, which would tend
to contradict that theory...
________ ___ __ __
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:

> Could someone please briefly explain "Myhill's property" again to
me
> in "layman's" terms...

Myhill's property is equivalent to MOS. Myhill's property is that
each generic interval besides the unison (second, third,
fourth . . .) comes in exactly two specific sizes (minor/major,
augmented/perfect) . . . So the diatonic scale has Myhill's property,
while the melodic minor scale (having three sizes of fourth:
diminished, perfect, and augmented) does not. . . .the whole-tone
scale is profoundly non-Myhill, having only _one_ specific size for
each generic interval. . . .getting the drift?

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14179

Myhill's Property

. . . .getting the drift?

Thanks, Paul... that clears this up for now!

_______ ___ __ _
Joseph Pehrson

P.S. I thought that Dan Stearn's said "My hill's property goes there
up to the fence, and your hill's property is on the other side!"

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14162

> --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> > Carl Lumma wrote,
> >
> > > Myhill's property is equivalent to MOS.
> >
> > OK, but both you and Paul seem to be making a clear distinction
> > between MOS and symmetrical scales. And I'm considering any scale
> > "MOS" so long as Myhill's property exists within some periodicity
--
> > which it does with symmetrical scales. So I consider symmetrical
> > scales to be MOS...
>
> Symmetrical scales are not MOS,

Yes, John Chalmers was showing me this... MOS scales need at least
two different sized intervals, correct??
______ ___ __ __ _
Joseph Pehrson

>> Symmetrical scales are not MOS,

Joseph wrote,

>Yes, John Chalmers was showing me this... MOS scales need at least
>two different sized intervals, correct??

Well, many symmetrical scales have at least two different sized intervals --
some even have at least two different-sized steps. MOS scales, though, have
Myhill's property -- every generic interval size (except the unison) comes
in _exactly_ two specific sizes.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14218

> >> Symmetrical scales are not MOS,
>
> Joseph wrote,
>
> >Yes, John Chalmers was showing me this... MOS scales need at
least
> >two different sized intervals, correct??
>
> Well, many symmetrical scales have at least two different sized
intervals -- some even have at least two different-sized steps. MOS
scales, though, have Myhill's property -- every generic interval size
(except the unison) comes in _exactly_ two specific sizes.

Umm. Here's something I probably should know... In the case of the
diatonic collection, which is MOS, what do we do with the perfect
fifths and the tritone?? Is the *inversion* considered the 2nd
size?? That seems a little strange...
________ ___ __ _ _
Joseph Pehrson

Paul H. Erlich wrote,

> Well, many symmetrical scales have at least two different sized
intervals -- some even have at least two different-sized steps. MOS
scales, though, have Myhill's property -- every generic interval size
(except the unison) comes in _exactly_ two specific sizes.

And what I've been doing is generalizing, or "relaxing" this so that
it's applicable within any given periodicity thereby allowing a single
generator interpretation for a huge body of scales; "symmetricals"
included.

-- Dan Stearns

Joseph Pehrson wrote:

>Umm. Here's something I probably should know... In the case of the
>diatonic collection, which is MOS, what do we do with the perfect
>fifths and the tritone?? Is the *inversion* considered the 2nd
>size?? That seems a little strange...

Joseph, the diatonic scale has two sizes of fourth -- perfect and augmented;
and two sizes of fifth -- diminished and perfect. The fact that in 12-tET
the augmented fourth is the same size as the diminished fifth is of no
consequence to Myhill's property, since they are different _generic_ sizes.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14228

> Joseph Pehrson wrote:
>
> >Umm. Here's something I probably should know... In the case of
the
> >diatonic collection, which is MOS, what do we do with the perfect
> >fifths and the tritone?? Is the *inversion* considered the 2nd
> >size?? That seems a little strange...
>
> Joseph, the diatonic scale has two sizes of fourth -- perfect and
augmented;and two sizes of fifth -- diminished and perfect. The fact
that in 12-tET the augmented fourth is the same size as the
diminished
fifth is of no consequence to Myhill's property, since they are
different _generic_ sizes.

Got it! Thanks!
___________ ___ __ __
Joseph Pehrson