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No, Jay, no interference can be perceived (at least in normal
conditions) between the partials of a single tone with harmonic
spectrum, and there wouldn't be "various rates" in this case. The
difference between the harmonic partials of a single tone is always
the same, by definition, and equal to the fundamental frequency of
this tone. This may produce an interference, more specifically a
difference tone, which is not exactly of the same nature as beating.
It has the frequency the fundamental of the tone and is perceived as
reinforcing it. Other difference (and summation) tones may arise,
but they always correspond to the frequency of one or another of the
harmonic partials themselves. There is no way that a harmonic sound
could produce frequencies outside its harmonic series: this is an
arithmetic property of the harmonic series itself.<br>
[When I say that no interference can be perceived "in normal
conditions", I mean that it is technically possible to differentiate
the energy originating from the difference tones from that of the
partials themselves; but they blend in a normal perception.]<br>
<br>
Whenever two different tones with harmonic spectra sound
simultaneously, on the other hand, several interference rates appear
at the same time. It is not always so that the slowest rate
predominates. Some (piano) tuners tuning D3-A3 as in the case you
described would choose to listen to the 2.8Hz rate aroung A5
(880Hz), rather than the 1.4Hz one around A4 - and they would
succeed in doing that! That is to say that various beating rates can
be heard, and isolated by an attentive listening.<br>
<br>
Whether Schlick heard beats, we will never know. The only clue we
have is what he said: <i>schweben</i>. Now if one is convinced that
<i>schweben </i>meant 'to beat' in 1511 because that is what it
means today, there is not much that I can add, but that I strongly
doubt this and won't change my mind about it.<br>
<br>
Martin Braun writes:<br>
<blockquote>...the German term "schweben" already in those days
[Werckmeister's] was a technical term and had the precisely
defined meaning of "to beat", in the same way as it is used today.<br>
"herunter/unterwärts schweben" means downward deviation realtive
to the second vibration
<br>
"aufwärts schweben" means upward deviation relative to the second
vibration
<br>
The term "gleichschwebend" is not puzzling, if one remembers that
in those days there were no machines to measure the number of
beats. People just heard the difference between slow and fast
beating on the one hand, and the difference between highside
beating and lowside beating on the other hand. "Gleichschwebend"
simply meant and still means today "equally tolerable" or "equally
acceptable" beating across the tone scale. It was, and still is, a
qualitative term, not a quantitative one.
</blockquote>
<i>Schweben</i> may mean 'deviate' (downwards of upwards): I fully
agree and I think this is the best translation proposed up to now.
But it certainly is NOT the precisely defined meaning of 'to beat'
as it is used today.<br>
<br>
The beating rate for a given interval doubles from octave to octave:
even without machines to measure beats, this can in no way be
considered "equal". Also, intervals in equal temperament are not
"equally acceptable" across the scale. It is well known that
ET-thirds, for instance, are rather less acceptable in the low
range. This is because the roughness of beating is maximum within a
rather limited range within which thirds fall in the low, not in the
high.<br>
<br>
Jay Rahn argues that it is not easy to perceptually evaluate
intervals between successive sounds: this is right. He appears to
consider it improbable that Werckmeister <i>gleichschwebend </i>could
have denoted an equal deviation (say, by 1/12-comma), because that
could not easily be perceived. But Werckmeister is not speaking of
perception, he describes tunings to be performed with the help of
calculation, of a monochord and a compas. With such tools, equal
deviation can be obtained.<br>
Equal temperament is characterized neither by equal beating, nor
by equal acceptability of its intervals. It is characterized by the
equal <u>size</u> of its intervals, even although this cannot
easily be perceptually evaluated.<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>
Le 14/09/2010 2:29, JAY RAHN a écrit :
<blockquote cite="mid:848909.62264.qm@web88101.mail.re2.yahoo.com"
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<p class="MsoNormal">Nicholas Meeus claims that between
two tones with harmonic
spectra many fluctuation rates can be perceived at the
same moment but that most
people would remain unable to concentrate on a specified
one. To be sure,
various rates would be present acoustically as, in
principle, they are within a single tone. However,
hearing interference between two tones is quite a
different task than hearing out the partials within a
single tone. Nonetheless, as among the partials of a
single tone with a harmonic spectrum, the slowest rate
of interference between two tones would tend to
predominate perceptually, especially if the lowest
partials were most intense, which they would tend to be
on plucked strings and
particular organ stops. In contrast, hearing a
difference of, e.g., a quarter
comma between <b>successive</b><span
style="font-weight: normal;"> tones </span><b>as</b><span
style="font-weight: normal;"> a quarter of a comma
seems much more demanding than
Schlick’s somewhat vague prescription to listen for
the undulation produced by
a flattened A sounded simultaneously with D a 5th
below ‘as much as it can endure.’</span></p>
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<p class="MsoNormal">In any event, I find it remarkable
that nobody prior to
Schlick seems to have drawn attention to audible
interference between pairs of
tones, especially as harmonics (in the sense of
flageolet tones) and resonance
(in the sense of sympathetic vibration) appear to have
been observed at least
as early as, respectively, ancient Greece and China. In
principle,
interference, harmonics, and resonance would have been
semi-objective ways of verifying
whether particular numerical (or geometrical) entities
had been transferred more or less precisely to a
monochord
or other instrument.</p>
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<br>
Jay Rahn, York University (Toronto)<br>
<br>
<br>
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