$$ John Baez wrote: > > (Dik T. Winter) wrote: >
In article <C0B8F249.155F3%rbj@audioimagination.com> robert
bristow-johnson <rbj@audioimagination.com> writes: >
this is why we in the "west" ended up with the 12 note/octave
equal tempered scale.
-=-
In the currently popular "equal-tempered" tuning system, each note
on the piano vibrates 2^{1/12} times as fast as the one below it...
at least if we ignore the slight "stretching" of this system that
people use on actual pianos.
There were a number of "well-tempered" systems in Bach's day,
none of which was the "equal-tempered" system: >
http://en.wikipedia.org/wiki/Well_temperament
Nobody really knows which system Bach used. This page: >
http://www.kirnberger.fsnet.co.uk/Temps3.htm
argues that it was the Werckmeister III system. They also explain
how this system works.
The advantage of non-equal-tempered systems, of course, is
that different keys have different personalities, making for
more interesting music.
$$ GUESS iSS STANDARD GOLDEN Music Scale frequency per iDEAL Second:
$$
$$ Middle C @ 261.8034 cycles/(GUESS iSS STANDARD second); 261.6
$$ C# @ 277.3710 " /( " " " " ) 277.2
$$ D @ 293.8644 (Given as 293.7 Equally Tempered).. 293.6
$$ D# @ 311.3385 311.1
$$ E @ 329.8516 ( .. .. .. ).. 329.6
$$ F @ 349.4656 349.2
$$ F# @ 370.2459 ( .. .. .. ).. 370.0
$$ G @ 392.2619 392.0
$$ G# @ 415.5870 ( .. .. .. ).. 415.3
$$ A @ 440.2991 440.0
$$ A# @ 446.4806 ( ..given as 466.2 " " ).. 466.1
$$ B @ 494.2190 ( .. " " 493.9 " " ).. 493.8
$$ C. @ 523.6068 ( .. .. .. ).. 523.2
$$
$$ Each is multiplied by 1.059463, for a series of twelve intervals.
$$ Upper C is Middle C @ 261.8034 x 12th rt2 x 12th rt2 ..etc etc.,.
$$ All from 100*(GOLDEN Ratio)^2 to 200*(GOLDEN Ratio)^2 cycles/Sec.
$$
$$ 100*(GOLDEN Ratio)^2 1
$$ 100*(GOLDEN Ratio)^2 + -------------------- = 1 + -- = 1.0594631.
$$ 12 12
$$
$$ One STANDARD Meter-long PENDuLUM beats ONE Second each way @ STP.
$$ One STANDARD Meter beats ONE HALF cycle / Second ..here on EARth.
$$ One STANDARD Meter beats ONE cycle per TWO Second, here on EARth.
$$ One STANDARD Meter beats ONE Second each way, on EARth's surface.
$$ One STANDARD Meter beats ONE Second each way, at EARth radius r3.
$$ Any STANDARD Meter beating ONE Second each way, is NOT ARBiTRARY.
$$ Especially, where the gravitational acceleration is (pi)^2*M/Sec.
$$ You may call that "co-incidental" if you will, but not ARBiTRARY.
Re: Go-go NETSCAPE: sci.physics.electromag,alt.sci.nanotech,bit.org.
Re: The physics or mathematics of musical scales.
Re: The geometric Golden Scale of music. End of POST.
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