Klarinet Archive - Posting 000613.txt from 2002/03

From: "Tim Roberts" <timr@-----.com>
Subj: Re: [kl] R-13 and intonation. . .good grief
Date: Thu, 21 Mar 2002 12:58:52 -0500

On Wed, 20 Mar 2002 13:27:46 -0800 (PST), w6w@-----. Wright) wrote:
>
>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.

Bill, I think you have the right idea but the wrong details.

>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 @-----.

Here, you are contradicting the conclusion you quite correctly reached in
your previous paragraph. You have already noticed that tones are computed by
multiplication, not addition, and yet here you are trying to compute the
differences between individual notes with addition. That's not right.

As I'm sure you know, the space between 440 and 880 is not divided into seven
intervals using simple division. Rather, it is divided into twelve intervals
using exponentiation. The delta between any two half steps in equal
temperament is computed by taking the twelfth root of two, which is roughly
1.059463, and multiplying the lower frequency by this magic number. So 440
plus one half step is actually 440 x 1.059463, or 466.

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

880 x 1.059463 @-----.

>It's obvious enough. The interval of one "note" becomes progressively
>larger as we move up the scale. ...

This statement is true, and demonstrates that you already knew what I just
described: the difference between N and 1.06xN grows as N grows.

>... But it's not practical to shift the
>positions of the holes in our clarinets as we move up the scale.

Here, I think, your analysis goes awry. It is not NECESSARY to "shift the
positions of the holes" as we move up the scale. Ignoring for know their
important roles in venting and color effects, a single hole is basically
responsible for creating one harmonic series. The R2 hole, when opened,
gives us concert Ab 207. It also sends out Ab 415, F 587, Ab 830, and a host
of others, in a nice exponential series. When we open up the register key,
we collapse the lower frequency Ab 207 and Ab 415 components, and what is
left gives us F 587.

>But mathematically, your instrument can never be in 'perfect' tune.

You can never be in perfect tune throughout the entire range, but that's due
to the compromises in the register key placement, not necessarily because of
the distances between the whole. Within a single register, a clarinet
certainly can be in "perfect" tune, assuming one appropriately defines
"perfect".

--
- Tim Roberts, timr@-----.com
Providenza & Boekelheide, Inc.

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