Bruce,
> While I'm not
> sure about the rate of pitch rise at shorter than 1.4 lengths, as
> you get longer than 2.1 or so the corresponding drop in pitch gets
> smaller for each .1 sun. (Maybe someone else can explain the
> physics of why this is so.)
The question is how the twelve notes of an octave are spaced. They could
have equal spacing or various other schemes. Western music has settled on
the Even Tempered Scale which employs a simple and unique spacing
method--the twelfth root of two. What number multiplied times itself three
times equals 27? It=B9s three, the cube root of 27. What number multiplied
five times itself equals 3125? Five or the fifth root of 3125. The Even
Tempered Scale spacing reduces to finding what number when multiplied twelv=
e
times itself will equal two. We want to fill the interval between one and
two (an Octave) with twelve regularly, but not necessarily equally, spaced
notes. That=B9s what the twelfth root of two does. Numerically it equals
1.059463094. Multiply that number times itself twelve time and you=B9ll end
up with two. As the twelfth root of two steps through an Octave it slowly
=8Cpicks up speed=B9. The first spaces are smaller than the last-- the first
divisions are shorter than the last, but the ratio is constant.
Further, the frequency of any note on the scale (think piano) multiplied or
divided by the twelfth root of two will be the frequency of the next or
previous note. Thus any two keys which are 12 notes apart on a piano
comprise an Octave. The Even Tempered Scale spacing is close to equal but
slowly and steadily increases as the frequency rises. It=B9s a percentage
deal, each note is about 6% bigger or smaller than the previous or next.
With the 'shaku' system each flute is 1/10 shaku bigger or smaller than the
next. We've got two systems going on here--one is mathematically linear
(shaks) and the other isn't. But we can ask: At what point do the two
systems coincide? This is the same as asking at what flute length does 1/1=
0
shaku equal 5.9463% of the whole? It's the inverse of 6% or the 1.6817
flute. Either side of that length the two systems of measurement start
getting out of step.
So the 1.7 flute is very close to the length where the two systems match.
Let's double that length to 3.4. Should play one octave (12 notes) lower,
right? But according to the shaku system it should be 17 notes lower.
There's the rub--12 notes verus 17. The two systems match up best the
closer you are to 1.7--the further away the worse.
Nelson Zink
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