[Smt-talk] Early account of beats

Martin Braun nombraun at telia.com
Tue Sep 14 04:02:33 PDT 2010

Dear Nicolas and others,

thank you very much for quoting Andreas Werckmeister. These lines definitely 
confirm that in the context of tuning musical instruments the German term 
"schweben" already in those days was a technical term and had the precisely 
defined meaning of "to beat", in the same way as it is used today.

"herunter/unterwärts schweben" means downward deviation realtive to the 
second vibration
"aufwärts schweben" means upward deviation relative to the second vibration

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.


Martin Braun
Neuroscience of Music
S-671 95 Klässbol
email: nombraun at telia.com
web site: http://www.neuroscience-of-music.se/index.htm

----- Original Message ----- 
From: "Nicolas Meeùs" <nicolas.meeus at paris-sorbonne.fr>
To: <reinifrosch at bluewin.ch>
Cc: <smt-talk at societymusictheory.org>
Sent: Monday, September 13, 2010 3:34 PM
Subject: Re: [Smt-talk] Early account of beats


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 

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