Re: [MMM] Pure Pythagorean - mistake? (attn. nonoctave)

I copied this from that page:
"...tune the instrument so that each fifth is pure, all the way
around the circle of fifths until you reach the note you started
with. In doing so, you would expect a perfectly tuned scale. Ok,
let's tune one. For the sake of simplicity, we're going to start our
tuning at a frequency of 100hz and we'll call it 'c' even though a
real 'c' would be closer to 130-something. The first fifth would be
tuned by ear by adjusting the pitch until a completely clear tone is
produced with no beats. (Beats are that 'wah wah' sound that happens
when your kids elementary school band is 'tuning up'.) If you put an
electronic frequency analyzer on the string you tuned, you would find
it vibrating 'g' at exactly 150hz.

Mathematically, that's the fundamental (100hz) times 3 (300hz for the
second harmonic), divided by 2 to drop it back into the same octave
as your starting pitch. This relationship is frequently expressed in
terms of the ratio 3:2. If you had the luxury of a tuning hammer and
a professional tuner to repair the damage, you could assault your own
piano. I don't recommend it. Since we can't demonstrate this
process auditorially, let's do the math for the rest of the scale.
Tune the next fifth up - 150 * 3 = 450/2 = 225, still more than an
octave above the starting pitch, so we'll drop it another octave to
112.5 'd'. Moving on up... 112.5 * 3= 337.5 / 2 = 168.75 'a' * 3 =
506.25 / 2 = 253.125 / 2 = 126.5625 'e' * 3 = 379.6875 / 2=
189.84375 'b' * 3 = 569.53125 / 4( see footnote 3 ) =
142.3828125 'f#' * 3 = 427.1484375 / 4 = 106.787109375 'c#' * 3 =
320.361328125 / 2 = 160.1806640625 'g#' * 3 = 480.5419921875 / 4 =
120.1354980469 'd#' * 3 = 360.4064941406 / 2 = 180.2032470703 'a#'
* 3 = 540.6097412109 / 4 = 135.1524353027 'e#(f)' * 3 =
405.4573059082 / 2 = 202.7286529541 'c'.
OOPS! Do you see the problem? Earlier, we predicted (guided by well
understood and established laws of physics) the octave above c(100)
would be c(200). When we ran the practical proof, using a circle of
perfectly tuned fifths, we ended up at c(202.7286529541), wide by
nearly 3 cycles!

> on 1/3/03 10:37 PM, X. J. Scott wrote:

> Rather than go down the path he chose, I octave reduce them
according to
> their natural octave.
.
.

> The historical thought-block where it was thought necessary to
octave-reduce
> was a conceptual error which my tuning elegantly corrects.
>
.
.
> This is a good example of how releasing the restraining constraints
of the
> pure 2/1 octave can solve many problems --

This is really good stuff.
I need a drink.