Klarinet Archive - Posting 000560.txt from 2002/03

From: w6w@-----. Wright)
Subj: Re: [kl] R-13 and intonation. . .good grief
Date: Wed, 20 Mar 2002 17:37:40 -0500

Fred, this conversation is going in many different directions. In
order to avoid a flame war, someone needs to summarize a basic
mathematical dilemma --- in order that the other conversations can focus
on their appropriate points. My apologies to those who don't want to
read this.

In Western music, the human ear and brain are deemed to respond most
strongly to multiples, not to additions. For example, "octave" means to
double the frequency. If A = 440 cycles per second, then one octave
higher is 440 x 2 @-----. But how about the next
octave? Is it 880 + 440 = 1320? Or is it 880 x 2 = 1760?

Your ear will tell you immediately that the answer is 1760. Thus it is
multiplication, not addition that we care about.

So far, so good.

But how are we going to divide the space between 440 and 880 into seven
intervals? Perhaps we should do the arithmetic of (880-440) / 7 =
62.9. In order to simplify the typing, let's round it off to "63".
Then 440 + "one note" would be 440 + 63 @-----.

Doing the same with the next octave: 880 + "one note" @-----.

OOOOPS !!!!! One octave above 503 would be 503 x 2 = 1006, but we just
calculated that "one note" higher than the next octave is 943. What's
going on here?

It's obvious enough. The interval of one "note" becomes progressively
larger as we move up the scale. But it's not practical to shift the
positions of the holes in our clarinets as we move up the scale.
(mental image of the golf cartoon where the golfer is using a hydraulic
jack to move the hole)

So somehow we need to make a compromise that is least offensive. But
mathematically, your instrument can never be in 'perfect' tune. Your
mouth and embouchure and so forth must make up the difference.

Cheers,
Bill

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