Format problems with Brian's Posts

Brian's last Psychoacoustics post, 21/25, arrived badly garbled in
Tuning Digest 530, so I am posting a re-formatted copy. Apparently when
his word processor wraps a long line, the line gets truncated somewhere
in the chain of systems it passes through. Unfortunately, it usually
looks complete on my screen, so only if I pass his text through my word
processor, insert a CR at the end of every line, then save it again as
an ascii file, can I be assured that it will be legible. This takes some
time and I have been loath to do it. Anyway, I hope this version is
readable.

--John

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> but 12 multiplications will tell
> you if you're close - and 12 more will tell you if your next guess is
> closer or not.

Bingo! (Thanks Johnny.) And of course it's not 12 multiplications; it's
only 4:
1. Square the guess => guess sixth root of 2.
2. Square that => guessed cube root of 2.
3. Multiply by result of step 1 => guessed square root of 2.
4. Square that => guess of 2.


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I don't know who wrote this because they didn't sign their name. My software
does not import the authors' names only Tuning...Eartha..Whatever..

>1. Square the guess => guess sixth root of 2.
>2. Square that => guessed cube root of 2.
>3. Multiply by result of step 1 => guessed square root of 2.
>4. Square that => guess of 2.

How would you guess the sixth root of two????

My point seems to be lost. We all seem so delighted with the ease at which
we are able to do these things. So let me shed some additional light on the
subject.

Hindu-Arabic number system along with its place holder notation was assimilated
by west around 825 A.D. The decimal system appears about 1600 A.D. and shortly
thereafter we get logarithms. Next comes the 2 space coordinate system followed
by Calculus. With calculus we get a deep insight into the nature of solving
problems which previously were very demanding even though solutions of sorts
had been known for millenniums.

To guess the sixth root of two? The method thought to be employed since Greek
times for comparing the sizes of two ratios is that of continued fractions. It
ain't easy by a long shot.

As for the post from our pool player (I don't have his name either only his
Email address) which gave the formula for the twelfth with the powers of eleven,
this formula is a direct result of Newton's Method hence it requires the knowledge
of Calculus. I believe that the same could be said about Manuel Op de Coul's
formula. I think there is a slight difference between his and the Babylonian
rendition. I looked up in my Math History textbook and found that the
Babylonians were able to do cube roots. The Chinese have had a type of decimal
system for nearly as long with algorithms for these two problems as well.

I also remember reading that the man who discovered logarithms used the twelfth
root of two as the first problem solved using this new system.

I believe that the answer was not as important to them as was the method and its
basis were, or what they could learn about the deeper nature of things by
studying the problem. Such is the life of a mathematician.


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On Tue, 17 Oct 1995, Michael Wathen 556-9565 wrote:
> As for the post from our pool player (I don't have his name either only
> his Email address) which gave the formula for the twelfth with the
> powers of eleven,

I guess that would be me.

> this formula is a direct result of Newton's Method
> hence it requires the knowledge of Calculus. I believe that the same
> could be said about Manuel Op de Coul's formula.

Nnnnnooooo, I don't think so. All it really takes is the ability to
expand the nth power of a binomial (or just the first two terms,
actually)--a little algebra, no calculus required.

--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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Another source of interest is an article in the Science>, Volume 44 (1987), pp.471-488. The article by H. Floris
Cohen is entitled "Simon Stevin's Equal Division of the Octave".
It analyzes his treatise "Vande Spiegheling der Singconst". It
argues that Stevin was trying to deal with the problem of
Consonance of intervals.

Through the analysis of the source material available...
we find that Stevin's theory, which makes no sense if
interpreted as an early stage in the 'evolution'of equal
temperament, was meant as a solution---as freshly
original as it was wrongheaded--to this perennial problem
of consonance, which has continued to baffle some of the
best scientific minds from the very beginning of science
to the present day.

One of Stevin's ideas was that of considering the ratio of an
equal tempered fifth as "true" and the just ratio 3/2 as an
approximation!


Michael Wathen


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