Meantones and ETs - correction

Although I said that I sent the "Pi;meantones and ETs" message to the
list accidentally, I do feel I ought to warn anyone who bothered to note
down the formula that there was an error, namely a missing bracket. My
thanks to Jonathan Wild for noticing that there must have been a typo in
Friday's version.

What I had previously sent was:

> |((nlog_2(3^((4-m)/m)5^(-1/m))-4/m))/[nlog_2(3)])+1| < 0.00011

This should have read:

|((n(log_2(3^((4-m)/m)5^(-1/m))-4/m))/[nlog_2(3)])+1| < 0.00011

So there is an extra bracket between the first "n" and "log_2". For
greater legibility, I'll provide the following version.


| (4-m)/m -1/m |
| n(log_2(3 * 5 )) - 4/m) |
| ____________________________________ + 1 | < 0.00011
| |
| [nlog_2(3)] |


Just to remind anyone who might be interested, m is the root of the
syntonic comma in the meantone system for which an ET approximation is
being sought, while n is the number of steps per octave in the ET being
tested; [x] is the integer value of x. Here are the margin-of-error
figures I've obtained for the six historic meantones and their standard
ET near-equivalents, in the form (1/m,n):

(1/3,19) 0.0000260699
(2/7,50) 0.0001000280
(1/4,31) 0.0001032090
(2/9,74) 0.0000640236
(1/5,43) 0.0000108953
(1/6,55) 0.0000994642

The margin of error which I have given (<0.00011) can, of course, be
altered in either dirction, depending on the level of accuracy you want;
I've set it so that it easily catches all of the above standard
near-equivalents.
--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/


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Neil said:
> When the term "organizing fields" came up the other day, in relation to
> the ability of humans to pick up "cosmic" types of wavelengths...Gary, I
> think the spirit is the organizing field that holds the fabric of the
> body and mind together

I think it's very important for us to realize that we've been discussing
three fundamentally different types of topics here:
1. Things that we know about.
2. Speculations about unknown relationships between things that we know
about to
other things.
3. Things that are irrelevant to science in the first place.

Electromagnetic waves are an example of the first. Electromagnetic
waves are no more mysterious than E-mail. As with sound waves, everything
about them - their frequency, amplitude, and waveform - can be precisely
quantified and measured. The are a quite simple physical phenomenon as
natural as a rock falling when you drop it.

An example of the second is the possibility of as-yet-unknown medical or
psychological effects of electromagnetic waves. Since one of these two
things, the electromagnetic waves, are scientifically well-understood,
possible correlations can be located scientifically. It makes no sense,
however, to speculate about such correlations mystically. If I were to say
that on my way to work this morning I had an overwhelming deja vu
experience, what value could possibly come from also suggesting that this
was caused by an electromagnetic phenomenon? That's irrelevant to the
topic at hand, which is the freaky sensation of that experience, and could
mislead people about the nature of Electromagnetics.

The third general sort of topic we've discussed are mystical, religious,
and metaphysical. Whether God has set me on a personal quest to bring new
musical resources to the public and other composers is irrelevant of the
precise reason why xylophone bars vibrate with nonharmonic partials.
Divine influence has no precisely quantifiable basis (and if you'll pardon
the pun I'll add "Thank God!"), so electromagnetic waves also have nothing
to do with it.

By no means am I suggesting that imply that mystical ideas are
unimportant, or meaningless. In fact, when we're on our death beds, many
of us will conclude that they are the ONLY meaningful ideas. They're just
not of much concern to science, nor is science of much concern to
mysticism. They contribute very little to each other.

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