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</head><body><div class='bwmail'><P>Dear Jay, dear Nicolas,</P>
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<P>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.</P>
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<P>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. </P>
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<P>Reinhart.<BR><BR>Reinhart Frosch,<BR>Dr. phil. nat.,<BR>r. PSI and ETH Zurich,<BR>Sommerhaldenstr. 5B,<BR>CH-5200 Brugg.<BR>Phone: 0041 56 441 77 72.<BR>Mobile: 0041 79 754 30 32.<BR>E-mail: reinifrosch@bluewin.ch .<BR><BR>----Ursprüngliche Nachricht----<BR>Von: nicolas.meeus@paris-sorbonne.fr<BR>Datum: 12.09.2010 22:02<BR>An: "JAY RAHN"<jayrahn@rogers.com><BR>Kopie: <smt-talk@societymusictheory.org><BR>Betreff: Re: [Smt-talk] Early account of beats<BR><BR><!--
-->Is there any account of beats in Schlick? The German word he uses is <I>schweben</I>, often taken to mean "to beat" because we know better, but that is not the German sense: <I>schweben</I> means "to float".<BR><BR>What Schlich says, really, is this:<BR>... <I>mach </I>[<I>die quint</I>] <I>dazu nitt hoch genug, oder ganz gerade in. sonder etwas in die niederer schweben, so vil das das gehor leyden mag...</I><BR>... make its fifth not high enough nor exactly, but somewhat floating in the low, as much as the ear may suffer...<BR><BR>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.<BR><BR>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. <I>d-g</I>b) as approximations of perfect major thirds (<I>d-f</I>#), 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 <I>schwebend</I>.<BR><BR>Nicolas Meeùs [...]<BR></P></div></body></html>