To begin with, try that software:
http://www.klavierstimmen.at/InstrumenteTuner.htm
install the English 30 days version.
See the label "stretch".
Stretch is used to adapt to the physiological effect that
low notes are perceived higher and high notes are perceived
lower by many listeners.
A side effect is, to straighten out some unwanted beat frequencies
between the higher harmonics of the strings.
Also the 3 strings of one note are tuned slightly different
for that reason and also for the side effect of giving
a beautiful "attack".
On a piano the lower octaves are tuned to lower,
and the higher octaves are tuned higher than mathematically just,
and only 2 octaves in the center of the keyboard are just.
On digital pianos one can adjust the stretch curve.
I always wonder, why the Japanese know all about tuning,
and the Westerners don't.
more on this later, need some rainy days.
w.
Peter Larsen - 22 Oct 2005 08:16 GMT
> To begin with, try that software:
> http://www.klavierstimmen.at/InstrumenteTuner.htm
> install the English 30 days version.
> See the label "stretch".
Will do, most interesting topic, thanks.
> Stretch is used to adapt to the physiological effect that
> low notes are perceived higher and high notes are perceived
[quoted text clipped - 4 lines]
> for that reason and also for the side effect of giving
> a beautiful "attack".
According to the article in Scientific American I have previously
mentioned it is required to so do to make them vibrate at exactly the
same frequency because the mechanical coupling effects at each end cause
a distuning when all three strings vibrate. If this is not done, then
the _sustain_ of the notes will suffer and beats will arise. I'm sorry
that I can not remember volume and number, I tried their website, but
the concept of a searchable magazine index does at last not go that far
back in time if exists.
> w.
Kind regards
Peter Larsen

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Angelo Campanella - 25 Oct 2005 18:23 GMT
> Stretch is used to adapt to the physiological effect that
> low notes are perceived higher and high notes are perceived
[quoted text clipped - 8 lines]
> and only 2 octaves in the center of the keyboard are just.
> On digital pianos one can adjust the stretch curve.
You are describing multipole filter phenomena well known by mechanical
and electical filters designers.
That is, two attributes can be adjusted by several resonators whose
independednt frequencies are very near one another.
1- when it is desired to make a narrow bandpass filter that only passes
a asmall selected range of frequencies, several such resonators must be
connected in parallel. The result is a pass band that can be made
constant in purveyance from, say 100 Hz to 110 Hz, etc.
2- The transient response, the time it takes oscillations to rise to a
high and constant level, is quicker the wider this passband and vice
versa. That time is approximately the reciprocal of the bandwidth. Thus,
excitation to full sound level (the "attack") of a 105 Hz tone through
such a filter of 10 Hz bandwidth will ccur within 1/10th of a second, etc.
> I always wonder, why the Japanese know all about tuning,
> and the Westerners don't.
The Japanese may have merely thought all of this out, combining
electrical & mechancial engineering with music inctrumenttion technology .
Angelo Campanella