Re: Metastable intervals that are not simple ratios or their approxi

The entire Scale tree of Erv Wilson was a plotting of the noble tones following the leads of Kornerup and Yasser and Temes and O'Connell later. It predates the work of Erlich and Setheres along these lines.
That it is formed by the Stern Brocot tree leaves out that the converging noble numbers are not included or its musical application.
--
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:
>
> The entire Scale tree of Erv Wilson was a plotting of the noble
tones
> following the leads of Kornerup and Yasser and Temes and O'Connell
> later. It predates the work of Erlich and Setheres along these lines.
> That it is formed by the Stern Brocot tree leaves out that the
> converging noble numbers are not included or its musical application.
> --
> 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
>

Hi Kraig,

You may be confusing two different uses of phi and other noble numbers
in tuning.

One relates to fractions of an octave (noble mediants between EDOs).
That's Kornerup, Yasser, golden meantone and Erv's scale tree.

The other relates to frequency ratios (noble mediants between JI
ratios). That's what Temes and Margo and I and now apparently Cameron
Bobro are on about.

I'm sorry I'm not familiar with O'Connell's work.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The entire Scale tree of Erv Wilson was a plotting of the noble tones
> following the leads of Kornerup and Yasser and Temes and O'Connell
> later. It predates the work of Erlich and Setheres along these lines.
> That it is formed by the Stern Brocot tree leaves out that the
> converging noble numbers are not included or its musical application.
> --
> 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 pondering the Wilson archive residing at Kraig's pad and
it looks like the Scale Tree is based on ratios? Have to poke
at it some (and print it all out, the computer screen sucks).

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> Hi Kraig,
>
> You may be confusing two different uses of phi and other noble
numbers
> in tuning.
>
> One relates to fractions of an octave (noble mediants between
EDOs).
> That's Kornerup, Yasser, golden meantone and Erv's scale tree.
>
> The other relates to frequency ratios (noble mediants between JI
> ratios). That's what Temes and Margo and I and now apparently
Cameron
> Bobro are on about.
>
> I'm sorry I'm not familiar with O'Connell's work.
>
> -- Dave Keenan
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
> >
> > The entire Scale tree of Erv Wilson was a plotting of the noble
tones
> > following the leads of Kornerup and Yasser and Temes and
O'Connell
> > later. It predates the work of Erlich and Setheres along these
lines.
> > That it is formed by the Stern Brocot tree leaves out that the
> > converging noble numbers are not included or its musical
application.
> > --
> > 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
> >
>

Cameron Bobro wrote:
> I'm pondering the Wilson archive residing at Kraig's pad and
> it looks like the Scale Tree is based on ratios? Have to poke
> at it some (and print it all out, the computer screen sucks). Fractions of a period (typically an octave), in logarithmic pitch space, not frequency ratios. E.g. a 5/12 generator is 5 steps of 12-ET. The ratios on the golden horograms, e.g. (2 phi + 1) / (5 phi + 2) for Kornerup's meantone, are also fractions of an octave.

Kraig,

I can no longer find Lorne Temes letter on your website. I would like
to add a reference to it to Margo's and my paper. We should have done
that when you pointed it out to us after we posted in 2000. I thought
it used to be at the end of the "treasures", but I can't find it there
now.

I did however find Erv Wilson's 2002 note in which he refers to both
Temes (misspelled as "Themes"?) and O'Connell's work. Lorne Temes in
1970 described only a single "most dissonant" interval at a frequency
ratio of phi, (not at noble mediants in general) and iterated it
within the octave to generate a scale.

Search on "Temes" within this group's archive to see our discussion on
this in 2000.

According to Erv, Walter O'Connell's 1993 contribution was to show
that this phi interval was closely approximated by 25 degrees of 36-EDO.

Erv, in his 2002 note, points out that 72-EDO therefore contains very
good approximations to both the most just and the most "unjust"
intervals. He does not mention any other "unjust" intervals apart from
phi itself (and by reference to Temes, its integer powers and their
octave reductions). This note is the second last page of
http://www.anaphoria.com/tres.PDF

All Erv's earlier applications of noble numbers on your website
clearly involve noble logarithmic fractions of an octave (or other
interval of repetition), used to generate scales with desirable
melodic properties. Witness the Golden Horograms.

But hey, maybe Erv (or Lorne Temes) did think of the linear or
harmonic application of more general noble numbers first. No matter
what we come up with, someone has probably thought of it before. If we
find out about it we'll acknowledge it.

Erv is a genius, and much of what we do is built on foundations he has
provided us. Do you really think Erv cares so much about priority that
he want's you to risk exaggerating his contributions? (rhetorical
question). You're a good man, Kraig, and I know you only do it because
you care about him.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The entire Scale tree of Erv Wilson was a plotting of the noble tones
> following the leads of Kornerup and Yasser and Temes and O'Connell
> later. It predates the work of Erlich and Setheres along these lines.
> That it is formed by the Stern Brocot tree leaves out that the
> converging noble numbers are not included or its musical application.
> --
> 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
>

Herman Miller wrote:
> Cameron Bobro wrote:
> >>I'm pondering the Wilson archive residing at Kraig's pad and
>>it looks like the Scale Tree is based on ratios? Have to poke
>>at it some (and print it all out, the computer screen sucks). > > > Fractions of a period (typically an octave), in logarithmic pitch space, > not frequency ratios. E.g. a 5/12 generator is 5 steps of 12-ET. The > ratios on the golden horograms, e.g. (2 phi + 1) / (5 phi + 2) for > Kornerup's meantone, are also fractions of an octave.

You can interpret the scale tree as showing frequency ratios, and the word is Erv thinks of it either way. It isn't really a "scale tree" unless it shows scales. Hammond organ tuners use the Stern-Brocot tree of frequency ratios to choose the number of teeth to approximate intervals from 12-equal. That's the original application to music.

Here's a link for the mathematics:

http://www.cut-the-knot.org/blue/Stern.shtml

Graham

The scale tree can be used in both fashions. Erv i would say uses it more often between JI intervals and always has. EDO s seem to have little interest for him. This has also been the way i most often have used the scale tree.
O'connells work is in Xenharmonikon
--
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

Actually Wilson does use the scale tree in an EDO fashion in working in keyboard layouts as being a typical case. But rarely for it being more than just one instance. It is interesting to note that the ratio approach is what lead him to the scales of Mt. Meru but as one quickly finds, this leads to a whole set of different noble numbers
--
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

yes there is a link missing Temes article is here.
http://anaphoria.com/temes.PDF
I point it out because it is a feature that not only Wilson has used but most of his students.
Actually the use of mediants goes back to the Greeks as Andrew Barker points outs. Ptolemy would also use a triple division between two intervals which has i have pointed out in the past makes the JI concept of limits originating with them mistaken.
--
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

If you notice on page 4 of http://anaphoria.com/tres.PDF
You can see an application of Ratios in the manner of a scale tree applied to the tetrachord.

Posted by: "Dave Keenan" <http://profiles.yahoo.com/dkeenanuqnetau>
Tue Oct 9, 2007 7:36 pm (PST)
Kraig,

I can no longer find Lorne Temes letter on your website. I would like
to add a reference to it to Margo's and my paper. We should have done
that when you pointed it out to us after we posted in 2000. I thought
it used to be at the end of the "treasures", but I can't find it there
now.
--
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

Thanks Kraig,

I love Erv's "and yet somehow with divinity imbued" regarding the 833
cent phi neutral sixth.

I have added the following to the end of the Noble Mediant article.

----------------------------------------------------------------------
After initial publication of this article, Kraig Grady kindly referred
us to some earlier work on this topic by Lorne Temes. [9] Also the
diagram by Erv Wilson showing the noble numbers in relation to the
Stern-Brocot tree [10] and more recently, some comments by Erv Wilson
on Lorne Temes phi neutral sixth (833.1 cents). "... it is the
worstest of the worst -- and yet somehow with divinity imbued, Lord
have mercy!" [11]

9. Lorne Temes, "Golden Tones?" in a letter to Erv Wilson, 4 January
1970.
http://anaphoria.com/temes.PDF

10. Erv Wilson, "Scale Tree", 1994.
http://anaphoria.com/sctree.PDF

11. Erv Wilson, "Radical Just-Unjust property of 72-ET", 28 June 2002.
http://anaphoria.com/tres.PDF (second last page)
---------------------------------------------------------------------

-- Dave Keenan

Dave Keenan wrote:

> According to Erv, Walter O'Connell's 1993 contribution was to show
> that this phi interval was closely approximated by 25 degrees of 36-EDO.

O'Connell's paper was written before (60s?) but only published in 1993. It's about a timbre and associated tuning based on phi. The reason he chose phi is that the difference tones are in tune with the timbre. It doesn't ahve anything to do with noble mediants other than the number phi used as a frequency ratio.

Graham