[Smt-talk] Early account of beats

Nicolas Meeùs nicolas.meeus at paris-sorbonne.fr
Mon Sep 13 02:42:48 PDT 2010


The pair /schwebend/gerade/ may mean wavering/straight, but also many 
other things. Have a look at these two web pages:
 From this, I would suggest that the most neutral translation would be 
something like unstable/stable, which may or may not connote beats.

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.

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.

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.

Nicolas Meeùs
nicolas.meeus at paris-sorbonne.fr

Le 12/09/2010 23:20, JAY RAHN a écrit :
> 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.
> 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.)
> 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?
> Jay Rahn, York University (Toronto)
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