Brian A M
Sat, Jun-24-06, 17:16
$$ John Baez wrote: > > Dik.Winter@cwi.nl (Dik T.
Winter) wrote: >
> > In article <C0B8F249.155F3%rbj@audioimagination.com>
> > robert bristow-johnson <rbj@audioimagination.com>
> > writes: >
> > > this is why we in the "west" ended up with the 12
> > > note/octave equal tempered scale.
-=-
> In the currently popular "equal-tempered" tuning
> system, each note on the piano vibrates 2^{1/12} times
> as fast as the one below it... at least if we ignore
> the slight "stretching" of this system that people use
> on actual pianos.
>
> There were a number of "well-tempered" systems in Bach's
> day, none of which was the "equal-tempered" system: >
> http://en.wikipedia.org/wiki/Well_temperament
>
> Nobody really knows which system Bach used. This page: >
> http://www.kirnberger.fsnet.co.uk/Temps3.htm
>
> argues that it was the Werckmeister III system. They also
> explain how this system works.
>
> The advantage of non-equal-tempered systems, of course, is
> that different keys have different personalities, making for
> more interesting music.
$$ 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. $$ You
may call that "co-incidental" if you will, but not ARBiTRARY.
Re: Go-go NETSCAPE:
sci.physics.electromag,alt.sci.nanotech,bit.org. Re: The
physics or mathematics of musical scales. Re: The geometric
Golden Scale of music. End of POST.
Winter) wrote: >
> > In article <C0B8F249.155F3%rbj@audioimagination.com>
> > robert bristow-johnson <rbj@audioimagination.com>
> > writes: >
> > > this is why we in the "west" ended up with the 12
> > > note/octave equal tempered scale.
-=-
> In the currently popular "equal-tempered" tuning
> system, each note on the piano vibrates 2^{1/12} times
> as fast as the one below it... at least if we ignore
> the slight "stretching" of this system that people use
> on actual pianos.
>
> There were a number of "well-tempered" systems in Bach's
> day, none of which was the "equal-tempered" system: >
> http://en.wikipedia.org/wiki/Well_temperament
>
> Nobody really knows which system Bach used. This page: >
> http://www.kirnberger.fsnet.co.uk/Temps3.htm
>
> argues that it was the Werckmeister III system. They also
> explain how this system works.
>
> The advantage of non-equal-tempered systems, of course, is
> that different keys have different personalities, making for
> more interesting music.
$$ 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.