The bells, the bells ....... the eighteenth century bells!

Thank you for your thoughts Yahya.

You can understand why I have had problems making any mathematical sense of it.
I was beginning to have doubts about my mathematical abilities and competence;-)

I am convinced that Harrison is aiming to sound the same "harmonic" relationships that he proposes for his tuning derived from pi, for
in other parts of the same manuscript he clearly restates all the tuning calculations (using logs), which were in his earlier book.

I didn't include that info. on the link of the transcription, as it would have made it all very long and his writing is sufficiently hard to read, without adding redundant passages.

Your logs of logs observation may be significant. I missed that.
But why might he be attempting to move into a further geometric level?
Did he find some multidimensional connection between the metal's physical dimensions and frequency?
He is certainly writing from practical experience and observations of bells.

The pi may be unnecessary in the calculations in this case, because it could be implied by the circular shape of the bells, and "cancelled out".
I assume that he was working with round plan view (conventional church bell-shaped) bells.

Could he be suggesting that integer ratios occur between pitches, when considering the physical dimensions of the bell in a further mathematical dimension. e.g. log of log?
Rather like one only readily observes the commonality of interval sizes, when observing the frequencies and their differences on a log scale e.g. as cents.

I think it may be a matter of experimenting with our perception of the bells problem which will enable us to understand how Harrison was viewing (conceptualising) it.

Judging by the designs of his very novel inventions (Grasshopper escapement, gridiron, clocks etc.), each of them required a "new and unique" way of looking at the problem;
which seems to have been one of Harrison's abilities which made him such a successful innovator.

Charles Lucy - lucy@harmonics.com
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Charles,

Getting into Harrison's mindset is very much a major part
of the problem, I think.

This is where it certainly helps to have read a few historical
maths books, rather than just those written in the last few
decades.

Speaking as a mathematician, our modes of expression have
become ever more a telegraphic shorthand, as new concepts
have taken root and flourished. Consequently, reading an
author like Harrison can seem strangely elliptical, as it takes
him forever to state a relationship that we would now describe
in a word or a short formula.

The reverse of this coin is that certain mathematical
procedures, once standard fare, have been relegated to the
dustbin. Few of my students have ever performed a long
division, or extracted a square or cube root, by hand, and
none since the late seventies would recognise log tables, let
alone know how to use them. This makes me wonder what
methods and techniques Harrison knew that I never learnt,
because my teachers or theirs deemed them unprofitable?

To me, much of the pleasure in reading the history of science
is the journey of discovery of ideas. (How DID they ever think
of that?) So, in teaching, it has been my pleasure to share
that journey with whoever is willing.

But, as GBS quipped, "You can lead a whore to culture, but you
can't make her think." I remember once trying to lead some
of my brighter students - admittedly not maths majors - to
discover the square root extraction algorithm for themselves,
and failing abysmally. Oh, well! I do hope you'll succeed better
with your exploration into how Harrison thought, than I did
with those kids.

If we are right in thinking that understanding Harrison and
his times is key to understanding his writings, the question
surely becomes - What is the best way to gain that insight?

Who, on this list, has done much reading of scientific and
mathematical literature of Harrison's era? These would
probably see what he's saying much quicker than I could.
Tho I will certainly have another look later - it's an intriguing
puzzle.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Tue, 8 Mar 2005 10:45:22 +0000
From: Charles Lucy <lucy@harmonics.com>
Subject: The bells, the bells ....... the eighteenth century bells!

Thank you for your thoughts Yahya.

You can understand why I have had problems making any mathematical
sense of it.
I was beginning to have doubts about my mathematical abilities and
competence;-)

I am convinced that Harrison is aiming to sound the same "harmonic"
relationships that he proposes for his tuning derived from pi, for
in other parts of the same manuscript he clearly restates all the
tuning calculations (using logs), which were in his earlier book.

I didn't include that info. on the link of the transcription, as it
would have made it all very long and his writing is sufficiently hard
to read, without adding redundant passages.

Your logs of logs observation may be significant. I missed that.
But why might he be attempting to move into a further geometric level?
Did he find some multidimensional connection between the metal's
physical dimensions and frequency?
He is certainly writing from practical experience and observations of
bells.
[YA] I would expect he was looking for some kind of exponential
or power law relationship, that can only be uncovered by taking
double logs.

The pi may be unnecessary in the calculations in this case, because it
could be implied by the circular shape of the bells, and "cancelled
out".
I assume that he was working with round plan view (conventional church
bell-shaped) bells.
[YA] Seems reasonable.

Could he be suggesting that integer ratios occur between pitches, when
considering the physical dimensions of the bell in a further
mathematical dimension. e.g. log of log?
[YA] If he was thinking of an exponential relationship, perhaps it
involved a rational exponent eg 3/2 or -2/3, such as occurs in some
physical formulae.

Rather like one only readily observes the commonality of interval
sizes, when observing the frequencies and their differences on a log
scale e.g. as cents.
[YA] That's exactly why an experimenter takes logs, or double logs,
so the plot of his observations finally reduces to a linear relationship.

I think it may be a matter of experimenting with our perception of the
bells problem which will enable us to understand how Harrison was
viewing (conceptualising) it.
[YA] And I think you're probably right on the money on this one!
See earlier comments.

Judging by the designs of his very novel inventions (Grasshopper
escapement, gridiron, clocks etc.), each of them required a "new and
unique" way of looking at the problem;
which seems to have been one of Harrison's abilities which made him
such a successful innovator.

Charles Lucy - lucy@harmonics.com
________________________________________________________________________

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