Updated scale archive

The scale archive with the collections of John Chalmers and myself
has been updated and contains now over 1250 scales.
The scales.doc file contains the complete listing. It's found here:

http://www.cs.ruu.nl/pub/MIDI/DOC/scales.doc (readme file, ASCII)
http://www.cs.ruu.nl/pub/MIDI/DOC/scales.zip (ZIP file, binary)

Use the "-a" option while unzipping. The size of the ZIP file is about
284 Kb. FTP is possible also from ftp.cs.ruu.nl.
The file format of the scales is that of my tuning program Scala,
which I hope will be available soon. The files are text files so also
usable without it.

Manuel Op de Coul coul@ezh.nl

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>The algorithm for calculating a root is actually not so difficult.
>This calculates a square root:
>
> Answer := Of_Number;
> loop
> Delta := (Of_Number / Answer) - Answer;
> if abs Delta <= Answer * Epsilon then
> return Answer;
> end if;
> Answer := Answer + 0.5 * Delta;
> end loop;

>Epsilon indicates the required precision and is the smallest number that
>makes a difference.
>I suppose that modifying this to do the twelfth root of 2 is easy.
>It converges very rapidly so it's doable by hand. It's very elementary
>so it must have been known for a very long time

>Manuel Op de Coul coul@ezh.nl


This algorithm was known by the Babylonians. I remember this from a Math
History Course I took several years ago. I'm not quite sure but I think
that dates it to 1000 B.C.


Is this adaptable to the 12th root of two? My immediate hunch is no. It is
probably only good for roots that can be expressed as powers of 2. I'll
have to think about it. Maybe Mr. Canright would have something to say about
this.

Michael Wathen


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On Fri, 13 Oct 1995, Michael Wathen 556-9565 wrote:
> >This calculates a square root:
> >
> > Answer := Of_Number;
> > loop
> > Delta := (Of_Number / Answer) - Answer;
> > if abs Delta <= Answer * Epsilon then
> > return Answer;
> > end if;
> > Answer := Answer + 0.5 * Delta;
> > end loop;
>
> Is this adaptable to the 12th root of two? My immediate hunch is no. It is
> probably only good for roots that can be expressed as powers of 2. I'll
> have to think about it. Maybe Mr. Canright would have something to say about
> this.

It's easily adaptable. Powers of two reduce the amount of calculation
required, which is nice, but not necessary. Here's the new algorithm:

Answer := Of_Number;
loop
Delta := (Of_Number / (Answer ^ 11)) - Answer;
if abs Delta <= Answer * Epsilon then
return Answer;
end if;
Answer := Answer + Delta / 12;
end loop;

The annoying part when doing it by hand is all the multiplication
involved in taking Answer to the 11th power with each iteration.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "A three-cushion player doesn't need to be married.
-\-\-- o He already has enough aggravation."

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Message-Id: <9510131643.AA12181@danhicks.math.nps.navy.mil>
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Paul Hahn's adaptation of the square root algorithm to the Nth root certainly works:

Answer := Of_Number;
loop
Delta := (Of_Number / (Answer ^ (N-1))) - Answer;
if abs Delta <= Answer * Epsilon then
return Answer;
end if;
Answer := Answer + Delta / N;
end loop;

But as he points out:

> Powers of two reduce the amount of calculation
> required, which is nice, but not necessary.

For doing by hand, the simplest way is find the square root of two, then the
square root of that, then the cube root of that. This converges to 10
significant digits in a total of 12 iterations (4 for each root). The direct
12th root approach also converges in 12 iterations, but the latter requires
calculating 11th powers 12 times, while the former only requires finding 2nd
powers 4 times.

Either way, the algorithm is not inherently difficult.

David C.
--
David Canright (408) 656-2782 (or -2206)
Math. Dept., Code MA/Ca (408) 656-2355 (FAX)
Naval Postgraduate School DCanright@NPS.Navy.mil
Monterey, CA 93943 USA http://math.nps.navy.mil/~dcanrig/

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The large cache of 12-bell/chimes per octave running several octaves and
found laying on the ground out of order are an interesting historical
find. A Chinese friend translated a Chinese article about them and Fritz
Kuttner played me a tape of these rare percussion.

It seems that the majority of them have 3 koo...different points with
which one can strike, giveing off different pitch. Non-Chinese have been
forbidden to directly play these percussion and I don't know if that has
ever changed. Surely, they have been hung in size order, but it is wrong
to deduce the tuning implications by the mere 12 bell/chimes per octave.
Even solid metal loses it exact tuning after 2 thousand years.


Chin music made use of tuning modulations from Just
(harmonic-representing heaven), Pythagorean (open strings-representing
earth), and Stopped (fingers on the strings now heard in equal
temperament-representing people).

My surmise is that before the Mongol invasions outlawed Han Chinese
social theater and music, in effect raping it over hundreds of years,
there may have been microtonal activity in the form of multiple notes per
octave - past 12. As a result of the Mongol invasions, pentatonicism as
the simplest means would dominate. There is a Chinese word for a foreign
note outside the system. (Pinyon? sp.)

Yin Fah Loo, guest musicologist from Shanghai, spent a year as guest
lecturer in Chou Wen Chung's graduate Chinese music classes at Columbia
University shared this material with us when I was a grad student in
ethnomusicology in 1982.

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@ios.com


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