some help with MOS, CPS, subtends (was: Schoenberg and octave displacement)

> From: unidala <JGill99@imajis.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, December 29, 2001 4:46 AM
> Subject: [tuning] Re: Schoenberg and octave displacement
>
>

> While you are at it (though this may seem like the Nth
> time some moron hasn't "gotten it"), could you give a
> stab at (briefly, and in your own language) the phrases:
>
> "Moment of Symmetry" (MOS)

I have something pretty comprehensive about this, which I
thought I put online but apparently didn't. I'll look for it.

> "Constant Product Set" (CPS)

The "C" in "CPS" stands for "Combination", not "Constant".

It helps here to understand how Pascal's triangle works.

(It was rediscovered by Pascal, but also known as "Mt. Meru"
in ancient times.)

The MathResource interactive Dictionary of Mathematics
[by J.M. Borwein, C.R.Watters and E.J. Borowski] defines
the Pascal's triangle as follows:

>> Pascal's triangle, n. the triangular array of integers, with
>> 1 at the apex, in which each number is the sum of the two
>> numbers above it in the preceding row; an initial segment
>> is shown in Fig. 279 [ reproduced below ]. The nth line of
>> the triangle is the sequence of coefficients of x^{k}a^{n-k}
>> in the expansion of the binomial (x + a)^{n}.
>>
>>
>>
>> 1
>> 1 1
>> 1 2 1
>> 1 3 3 1
>> 1 4 6 4 1
>> 1 ......................... 1

The best full definition I've found is here:
http://mathworld.wolfram.com/PascalsTriangle.html

And there's a lot more interesting info about it here:
http://ptri1.tripod.com/

Here's an interface where you can enter various values
and see different types of Pascal's Triangles:
http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html

There are some printable worksheets for Pascal's Triangles here:
http://mathforum.org/workshops/usi/pascal/

http://www.cs.washington.edu/homes/jbaer/classes/blaise/blaise.html

> "sub-tends" (everyone seems to assume that the reader
> understands the intended meaning of this; I'm still scratching
> my head about it).

Yeah, when Carl Lumma and then Paul first explained MOS to
me about 3 years ago I was scratching *my* head over that word too.

From
http://www.dictionary.com/cgi-bin/dict.pl?term=subtend

The second definition in the first entry seems to give the
best meaning as intended in the definition of MOS:

"subtend: To underlie so as to enclose or surround."

And the etymology is best of all:

"[Latin _subtendere_, to extend underneath]"

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Saturday, December 29, 2001 4:46 AM
> > Subject: [tuning] Re: Schoenberg and octave displacement
> >
> >
>
> > While you are at it (though this may seem like the Nth
> > time some moron hasn't "gotten it"), could you give a
> > stab at (briefly, and in your own language) the phrases:
> >
> > "Moment of Symmetry" (MOS)
>
>
> I have something pretty comprehensive about this, which I
> thought I put online but apparently didn't. I'll look for it.
>
>
> > "Constant Product Set" (CPS)
>
> The "C" in "CPS" stands for "Combination", not "Constant".
>
> It helps here to understand how Pascal's triangle works.
>
> (It was rediscovered by Pascal, but also known as "Mt. Meru"
> in ancient times.)
>
>
> The MathResource interactive Dictionary of Mathematics
> [by J.M. Borwein, C.R.Watters and E.J. Borowski] defines
> the Pascal's triangle as follows:
>
> >> Pascal's triangle, n. the triangular array of integers, with
> >> 1 at the apex, in which each number is the sum of the two
> >> numbers above it in the preceding row; an initial segment
> >> is shown in Fig. 279 [ reproduced below ]. The nth line of
> >> the triangle is the sequence of coefficients of x^{k}a^{n-k}
> >> in the expansion of the binomial (x + a)^{n}.
> >>
> >>
> >>
> >> 1
> >> 1 1
> >> 1 2 1
> >> 1 3 3 1
> >> 1 4 6 4 1
> >> 1 ......................... 1
>
> The best full definition I've found is here:
> http://mathworld.wolfram.com/PascalsTriangle.html
>
>
> And there's a lot more interesting info about it here:
> http://ptri1.tripod.com/
>
>
> Here's an interface where you can enter various values
> and see different types of Pascal's Triangles:
> http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html
>
>
> There are some printable worksheets for Pascal's Triangles here:
> http://mathforum.org/workshops/usi/pascal/
>
>
> http://www.cs.washington.edu/homes/jbaer/classes/blaise/blaise.html
>
>
>
> > "sub-tends" (everyone seems to assume that the reader
> > understands the intended meaning of this; I'm still scratching
> > my head about it).
>
>
> Yeah, when Carl Lumma and then Paul first explained MOS to
> me about 3 years ago I was scratching *my* head over that word too.
>
> From
> http://www.dictionary.com/cgi-bin/dict.pl?term=subtend
>
> The second definition in the first entry seems to give the
> best meaning as intended in the definition of MOS:
>
> "subtend: To underlie so as to enclose or surround."
>
> And the etymology is best of all:
>
> "[Latin _subtendere_, to extend underneath]"
>
>
>
>
> -monz

JG: I guess I just won't "see" this "subtending"
without a simple, but specific, example. Could
you provide one or two?

J Gill
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com

> From: unidala <JGill99@imajis.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, December 29, 2001 10:26 AM
> Subject: [tuning] Re: some help with MOS, CPS, subtends
>
>

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: unidala <JGill99@i...>
> > > To: <tuning@y...>
> > > Sent: Saturday, December 29, 2001 4:46 AM
> > > Subject: [tuning] Re: Schoenberg and octave displacement
> > >
> > >
> >
> > > While you are at it (though this may seem like the Nth
> > > time some moron hasn't "gotten it"), could you give a
> > > stab at (briefly, and in your own language) the phrases:
> > >
> > > "Moment of Symmetry" (MOS)
> >
> >
> > I have something pretty comprehensive about this, which I
> > thought I put online but apparently didn't. I'll look for it.
> >
> > ...
> >
> > > "sub-tends" (everyone seems to assume that the reader
> > > understands the intended meaning of this; I'm still scratching
> > > my head about it).
> >
> >
> > Yeah, when Carl Lumma and then Paul first explained MOS to
> > me about 3 years ago I was scratching *my* head over that word too.
> >
> > From
> > http://www.dictionary.com/cgi-bin/dict.pl?term=subtend
> >
> > The second definition in the first entry seems to give the
> > best meaning as intended in the definition of MOS:
> >
> > "subtend: To underlie so as to enclose or surround."
> >
> > And the etymology is best of all:
> >
> > "[Latin _subtendere_, to extend underneath]"
> >
> >
> >
> >
> > -monz
>
> JG: I guess I just won't "see" this "subtending"
> without a simple, but specific, example. Could
> you provide one or two?

The MOS explanation that I wrote (lost on my hard-drive
somewhere... I'm still looking for it) will make that clear.

Patience.

-monz

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Get your free @yahoo.com address at http://mail.yahoo.com

> JG: I guess I just won't "see" this "subtending"
> without a simple, but specific, example. Could
> you provide one or two?

I think that'd be "a third (any third) subtends 2 steps on the diatonic
scale" or "a major third subtends 4 steps on the chromatic scale".
Assuming that *I* have the meaning of "subtend" right. I've never found
not knowing it to be a problem in reading texts that use it. You can work
out from the context what they're trying to say.

MOS has lots of different definitions. The basic idea is that you have a
scale built up of only two step sizes. For the diatonic scale, these are
the tone and semitone. It's also spaced evenly, so you don't have two
semitones next to each other. More specifically, it has a "generator" and
a "period". For the diatonic (or chromatic) scale, these are a fourth or
fifth and an octave respectively. It should be obvious if you think about
it, or even if you don't, that you can get from any note on the keyboard
to any other, with strict spelling, by only moving by fifths and octaves.
That's what makes it an MOS.

For CPS, try the original papers by Erv Wilson, either in Xenharmonikon or
the Wilson Archives at <http://www.anaphoria.com/wilson.html>.
D'Alessandro, Like A Hurricane covers everything, but in a concentrated
way. You may need to stare at the diagrams for a while until they make
sense.

Graham