Klarinet Archive - Posting 000588.txt from 2002/03

From: Eric Mumpower <nocturne@-----.com>
Subj: Re: [kl] R-13 and intonation. . .good grief
Date: Thu, 21 Mar 2002 00:44:19 -0500

>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)

I apologize that I'm not sure I have the time to explain clearly why I think
you misunderstand the underlying principles.

But I get the impression you're approaching this all wrong. It's all about
relative proportions and ratios of frequency and length, not additive
differences.

In even temperment, the frequency of any "d" is in precise proportion to the
frequency of the "c" which is a step beneath it. (For this interval, this
factor is the 2^(2/12), or the sixth root of two.) It's the same interval
for the frequency of an "e" to a "d", and f#/e, g#/f#, a#/g#, and c/a# --
these are all whole steps. Thus, when you go up six whole steps, you've
multiplied your frequency by 2^(1/6) six times, so your factor is
(2^(1/6))^6, or simply 2. Hence the frequency-doubling property of the
octave.

Similarly, if you're plucking an ideal string (or playing an ideal clarinet
within any particular register of the instrument), for each increase in
frequency of a whole step, you shorten the string (or bore) by a precise
fraction of the current length. What matters isn't the number of millimeters
difference between holes, but the ratio of the effective lengths of the
bore, between the tip of the reed and those holes.

> So somehow we need to make a compromise that is least offensive. But
> mathematically, your instrument can never be in 'perfect' tune.

Yes, but not for the reasons you describe. Part of the problem is that even
temperment doesn't sound quite perfect; it's just a compromise that lets us
play in all keys with equally bad facility, rather than having some really
good keys and some really awful-sounding ones.(*)

Another part of the problem has to do with the fact that the harmonic series
inherent in musical instruments are themselves not in even temperment --
they're just close. Once you get to the third harmonic, which is roughly an
octave and a fifth above the fundamental (i.e. the second register of the
clarinet), you're no longer in even temperment. For higher harmonics, it
just moves farther from the mark.

And, yes, some of this can be compensated for by skillful instrument design,
but the rest is up to your lips and fingers.

(* Find, if you can, a digital piano which lets you change its temperment.
For a grossly apparent example, pick Pythagorean. For something with
historical basis, pick "just" temperment. Play, say, C,F, and G major
chords. Hear how much sweeter they sound than even temperment. Then play,
say, C#, F#, and G# major chords. Try not to wince too much.)

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