[Smt-talk] Early account of beats

reinifrosch at bluewin.ch reinifrosch at bluewin.ch
Mon Sep 13 05:02:18 PDT 2010

Dear Jay, dear Nicolas,


According to the "Dictionary of Terms in Music, English-German, German-English" by Horst Leuchtmann (Metzler, Stuttgart, 1998), the first translation of "Schwebung" is "beat, beating", and "equal temperament" translates into "gleichschwebende Temperatur", because the beat rates of ET harmonic-complex-tone fifths divided by the frequency of the lower of the two tones of the fifth are equal in all cases, namely (3 - 2^(19/12)) = 0.003386. For instance, the ET-fifth A3(220 Hz)-E4 generates 0.003386 x 220 = 0.745 beats per second.


The (quarter-comma) meantone frequency of D3 given in Jay's posting (~146.7 Hz) is not correct, I think. I get 147.123 Hz. Beat rate of the meantone harmonic-complex-tone fifth D3-A3: (3 x 147.123 - 2 x 220.0) = 1.369 beats per second, i.e., about 1.4 Hz, as stated by Jay.  



Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
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CH-5200 Brugg.
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E-mail: reinifrosch at bluewin.ch .

----Ursprüngliche Nachricht----
Von: nicolas.meeus at paris-sorbonne.fr
Datum: 12.09.2010 22:02
An: "JAY RAHN"<jayrahn at rogers.com>
Kopie: <smt-talk at societymusictheory.org>
Betreff: Re: [Smt-talk] Early account of beats

Is there any account of beats in Schlick? The German word he uses is schweben, often taken to mean "to beat" because we know better, but that is not the German sense: schweben means "to float".

What Schlich says, really, is this:
... mach [die quint] dazu nitt hoch genug, oder ganz gerade in. sonder etwas in die niederer schweben, so vil das das gehor leyden mag...
... make its fifth not high enough nor exactly, but somewhat floating in the low, as much as the ear may suffer...

We know that this probably refers to a sensation (not a consciousness!) of beating. But to deduce that Schlick accounted for beats (or, even more, for an interference between harmonic partials) would be farfetched, to say the least.

To answer the question otherwise: there is no mention of temperament, that I know (or that is known, I'd dare say), before Schlick. The tuning of Pythagorean diminished fourths (e.g. d-gb) as approximations of perfect major thirds (d-f#), as described in several 14th- and 15th-century texts (and earlier in Safi al-Din) cannot be considered temperaments, as they involve pure intervals (pure fifths, in this case) exclusively. The difference between a Pythagorean diminished fourth and a major third is so little (2 cents) that it hardly could have been considered schwebend.

Nicolas Meeùs [...]
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