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Jay,<br>
<br>
The pair <i>schwebend/gerade</i> may mean wavering/straight, but
also many other things. Have a look at these two web pages:<br>
<a class="moz-txt-link-freetext" href="http://www.linguee.com/german-english/translation/schwebend.html">http://www.linguee.com/german-english/translation/schwebend.html</a><br>
<a class="moz-txt-link-freetext" href="http://www.linguee.com/german-english/translation/gerade.html">http://www.linguee.com/german-english/translation/gerade.html</a><br>
From this, I would suggest that the most neutral translation would
be something like unstable/stable, which may or may not connote
beats.<br>
<br>
Your description of the interference, similarly, may or may not
describe what Schlick experienced. In order to perceive the beat
rate between harmonic 3 of D (~147Hz) and harmonic 2 of A (220), as
in the case you describe, a tuner would have to concentrate on
hearing what happens at the level of about 440Hz, and may indeed be
able to count beats there at a rate of about 1.4Hz if the fifth D-A
is in quarter-comma meantone. This is no easy trick, one that must
be learned. <br>
<br>
I stress this because, especially in the case of the organ (because
of its high harmonic content and its high stability), many
fluctuation rates can be perceived at the same moment and most
people would remain unable to concentrate on a specified one. In the
case of the fifth D-A just described, the interferences between the
first few harmonic partials would include not only harmonics 3
against 2, beating about 1.4Hz, but also harmonics 4 against 3
(71Hz), 5 against 3 (75Hz), 6 against 4 (2.7Hz), 7 against 5 (70
Hz), etc. All these partials remain below 1100Hz.<br>
<br>
Schlick, like anyone, may have perceived tempered intervals as
unstable, just ones as stable. To perceive them as beating, in the
technical sense of the term, is quite another matter.<br>
<br>
Nicolas Meeùs<br>
<a class="moz-txt-link-abbreviated" href="mailto:nicolas.meeus@paris-sorbonne.fr">nicolas.meeus@paris-sorbonne.fr</a><br>
<br>
<br>
<br>
<br>
<br>
<br>
Le 12/09/2010 23:20, JAY RAHN a écrit :
<blockquote cite="mid:740703.56248.qm@web88102.mail.re2.yahoo.com"
type="cite">
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style="font-style: inherit; font-variant:
inherit; font-weight: inherit; line-height:
inherit; font-size-adjust: inherit;
font-stretch: inherit; font-family: arial;
font-size: 10pt;">I don't think it is too
overreaching to regard Schlick's use of
schwebend (wavering) as referring to beats.
Schlick contrasts schweben with gerade
(straight)--actually a pretty good pair of
metaphors for what 'in-' and
'slightly-out-of-tune' perfect 5ths sound like.
<br>
<br>
At A 440, the rate of interference for a perfect
5th in quarter-comma tuning, which is what he
seems to be trying to prescribe, would be about
1.4 Hz, i.e., audible as undulation, especially
if the tones were played on an organ, which is
Schlick's immediate concern, and if the tones
were held for a long time, as he stipulates.
(The undulation at A 440 would arise between a D
(~146.7 Hz) in the middle of the bass-clef staff
and the A (220) on the top line. For D and A an
octave higher, the undulation rate would be ~2.8
Hz.)<br>
<br>
Since at least as early as the 4th to 2nd
centuries BCE in Greece and China, theorists
would have occasion to describe interference
effects between the tones of, e.g., a
Pythagorean major-3rd interval. In the 14th
century, Marchetto's tuning of sharpened degrees
would result in quite pronounced interference.
But did anyone describe such interference before
Schlick?<br>
<br>
Jay Rahn, York University (Toronto)<br>
<br>
<br>
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