Re: Introduction to MOS

Erv's technique is mathematically equivalent to Viggo Brun's subtractive
version of the Euclidean Algorithm and to a simple continued fraction.
The number of times one zigs or zags corresponds to the terms of the
continued fraction approximation of the number and to the number of
times a given subtractor is used in the Brun algorithm. Both Erv's and
the Brun algorithm reveal intermediate approximations (semiconvergents?)
which are skipped over in the continued fraction algorithm. That is to
say, the CF converges faster and skips many musically useful solutions.

What I find amazing is that Erv discovered his method and the
Stern-Brocot tree independently of previous work in number theory and
"higher" arithmetic. He is an astonishingly intuitive mathematician.

--John

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There is great advantages to working with both MOS and Viggo Brun's Algorithm.
Personally the Bi-level MOS structures are highly musical and heard as scales and are unique to other like systems

The Viggo Brun Algorithm if applied to generate constant structures (as seen here
http://anaphoria.com/viggo3.PDF using 3 intervals instead of two), we find scales extremely hard to get by Modifying MOS scales.
There is a great freedom in Brun.
In the above paper he shows examples of two patterns, a-b and a-c as well as reseeding between the two. This alteration can be done more freely. One can also use 4 of more intervals (based on limits the most common way) and these can be brought in at will. I have spent hours playing with this, working a scale down and then as an example deciding it might be nice to add 7s now, or seeing what a reseeding will give me. All in all, I find it an extremely "artistic" tool where ones taste can (and is) used at each and every step. The disadvantage of Brun is that it is not clear how which side to insert the new intervals . Here a knowledge of MOS gives one good background in knowing how to best possibly do this.

Brun is just one of the gems we find in Mandlebaum's book

John Chalmers wrote:
>
> Erv's technique is mathematically equivalent to Viggo Brun's subtractive
> version of the Euclidean Algorithm and to a simple continued fraction.
> The number of times one zigs or zags corresponds to the terms of the
> continued fraction approximation of the number and to the number of
> times a given subtractor is used in the Brun algorithm. Both Erv's and
> the Brun algorithm reveal intermediate approximations (semiconvergents?)
> which are skipped over in the continued fraction algorithm. That is to
> say, the CF converges faster and skips many musically useful solutions.
>
> What I find amazing is that Erv discovered his method and the
> Stern-Brocot tree independently of previous work in number theory and
> "higher" arithmetic. He is an astonishingly intuitive mathematician.
>
> --John
>
> -- > No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.5.516 / Virus Database: 269.20.0/1268 - Release Date: > 2/9/08 11:54 AM
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

> The Viggo Brun Algorithm if applied to generate constant structures (as
> seen here
> http://anaphoria.com/viggo3.PDF using 3 intervals instead of two), we
> find scales extremely hard to get by Modifying MOS scales.
> There is a great freedom in Brun.

What would those scales be if given in regular pitch ratios, rather
than scale steps?

These are all ratios. the numbers in the charts illustrates how many times the interval occurs. One does scale steps by substituting the number of units in stead of the ratios.

Mats �ljare wrote:
>
> > The Viggo Brun Algorithm if applied to generate constant structures (as
> > seen here
> > http://anaphoria.com/viggo3.PDF <http://anaphoria.com/viggo3.PDF> > using 3 intervals instead of two), we
> > find scales extremely hard to get by Modifying MOS scales.
> > There is a great freedom in Brun.
>
> What would those scales be if given in regular pitch ratios, rather
> than scale steps?
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

I realize now you might have been looking at the earlier articles up which indeed are scale steps. the later ones, like the one above shows ratios

Mats �ljare wrote:
>
> > The Viggo Brun Algorithm if applied to generate constant structures (as
> > seen here
> > http://anaphoria.com/viggo3.PDF <http://anaphoria.com/viggo3.PDF> > using 3 intervals instead of two), we
> > find scales extremely hard to get by Modifying MOS scales.
> > There is a great freedom in Brun.
>
> What would those scales be if given in regular pitch ratios, rather
> than scale steps?
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

I'm confused about what you mean, there's nothing "above" what you
just wrote, and none of the PDF files on the site seem to contain
anything resembling regular pitch ratios.

> I realize now you might have been looking at the earlier articles up
> which indeed are scale steps. the later ones, like the one above shows
> ratios

http://anaphoria.com/viggo3,PDF
i am confused cause page 1 of this is nothing but ratios

Mats �ljare wrote:
>
> I'm confused about what you mean, there's nothing "above" what you
> just wrote, and none of the PDF files on the site seem to contain
> anything resembling regular pitch ratios.
>
> > I realize now you might have been looking at the earlier articles up
> > which indeed are scale steps. the later ones, like the one above shows
> > ratios
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> http://anaphoria.com/viggo3,PDF
> i am confused cause page 1 of this is nothing but ratios

Still only ratios of scale steps, not relative to the tonic.

that is easy math

Mats �ljare wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > http://anaphoria.com/viggo3,PDF <http://anaphoria.com/viggo3,PDF>
> > i am confused cause page 1 of this is nothing but ratios
>
> Still only ratios of scale steps, not relative to the tonic.
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles