[Smt-talk] Early account of beats

Nicolas Meeùs nicolas.meeus at paris-sorbonne.fr
Mon Sep 13 06:34:14 PDT 2010

  This is exactly the kind of translation that begs the answer. So far 
as I can tell, the German /schweben /has no connotation of fluctuation, 
wavering, or beating, outside the context of music. Schlick's usage of 
the term to describe a tempered fifth may well have been its first usage 
in a musical context (there are not that many texts about music in 
German before 1511). To say today that when Schlick said that tempered 
fifths produce a 'floating' effect, he meant a 'beating' effect, because 
we know today that the effect results from beating, merely reverses the 

The expression /gleichsschwebend/, as applied to equal temperament, is 
puzzling because the beating rate for two identical intervals is never 
the same. As Reinhart Frosh rightly describes, the beat rate is 
proportional the the frequencies involved. This also is what 
Werckmeister already stated (albeit implicitly) in 1707 (/Musikalische 
Paradoxal Discourse/): "equal temperament is when all the fifths 'float' 
(/schweben/) by 1/12 of a comma" : it is the size of the tempering 
interval that remains constant, not the beating rate.

When Werckmeister speaks of /schweben/, he usually refers to the 
tempering or the tempered interval. In /Musikalische Temperatur/ (1691), 
for instance, he writes:

    /Einige bringen es vor/es mussen alle Quinten ein Viertel eines
    commatis herunter schweben/ (p. 1)
    Some claim that all the fifths must 'float' a quarter of a comma lower
    /Etliche lassen alle Quinten theils auffwarts / theils unterwarts
    schweben /(p. 2)
    Some let all fifths 'float' partly upwards, partly downwards
    .../und das D soll wieder 1/4 Commat. schweben mit dem G. /(p. 53)
    ...and D shall again 'float' by 1/4 comma with respect to G

In other words, for Werckmeister, /schweben /cannot have meant 'to 
beat'; it may have meant 'to temper'.

Nicolas Meeùs
nicolas.meeus at paris-sorbonne.fr

Le 13/09/2010 14:02, reinifrosch at bluewin.ch a écrit :
> 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.
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