[Smt-talk] Early account of beats

reinifrosch at bluewin.ch reinifrosch at bluewin.ch
Fri Sep 17 03:00:26 PDT 2010

Dear Jay,

On my Yamaha DX11 synthesizer, I have just played the (quarter-comma) meantone fifth A4-E5, using the voice C25, Harmonica. There are very noticeable beats, about 4 pulses per second. That (comparatively high) beat rate can be understood as follows:

f(A4) = 440 Hz; f(E5) = 5^(1/4) * 440 = 1.49535 * 440 = 657.95 Hz.

3 * f(A4) = 1320 Hz; 2 * f(E5) = 1315.9 Hz; beat rate = 1320 -1315.9 = 4.1 per second.

Some form of the meantone temperament was used before 1500 already, I think, and organs generating overtone-rich, approximately  harmonic complex tones were probably available at that time, too. So it appears likely that such beats were noticed (and used for tuning?) by many before Schlick.


----Ursprüngliche Nachricht----
Von: jayrahn at rogers.com
Datum: 16.09.2010 19:46
An: <smt-talk at societymusictheory.org>
Betreff: Re: [Smt-talk] Early account of beats

Returning to my original query, namely, whether Schlick (1511) is the earliest account of interference (i.e., slow undulation or faster, phenomenally fused beating), I would emphasize that my (and others’) understanding was premised on the contrast between schwebend and gerade as between wavering and straight.
On the basis of an entry in a contemporary online German-English dictionary, Nicolas Meeus originally claimed that schweben means “to float.” However, the dictionary he cited is a general dictionary, not a dictionary of music, and it lists the following additional translations for schwebend: floating, hovering, pending, soaring, suspended, wavering, breezing, poising, impending, unadjusted, undetermined. All the same, he insisted that to claim that Schlick was referring to beats (or, even more, to an interference between harmonic partials) would be farfetched.
Subsequently, Nicolas admitted that schwebend/gerade may correspond to wavering/straight but suggested that the most neutral translation would be something like unstable/stable, which, he felt, may or may not connote beats and went on to say that interference may or may not describe what Schlick experienced.
Thereupon, Reinhart Frosch pointed out that in a contemporary German-English music dictionary the first translation of Schwebung is “beat, beating” and Martin Braun emphasized that in a context of musical acoustics Schwebung and schweben exclusively mean “beat” and “to beat.” Contrary to Nicolas, Martin went on to say that ‘if one wants to understand why the Germans called this oscillation “Schwebung,” one has to look into the history of the word. The meaning “deviation” [which Nicolas suggested as a substitute] is not part of this history, but, for example, “flying” and “hovering” are.’
Since Nicolas regards Werckmeister’s 1691 account of tuning as relevant to Schlick, I think it is worthwhile to point out the following, to which a colleague has drawn my attention: 

According to MGG 13, 216-17, Mersenne (1636-37) said a tempered 5th should schweben once per second. His terms in the Harmonie universelle are trembler (to tremble, to vibrate) and battre (to beat). Leaving aside the question of just how quickly tempered 5ths would tremble, vibrate or beat throughout the entire gamut, Mersenne’s specification of a rate would seem to indicate a rate of interference.
Subsequent to Werckmeister, but earlier than various 19th- and 20th-century general and technical music dictionaries, Johann Christoph Adelung’s Grammatisch-kritisches Wörterbuch (1793-1801) 3, 1617-18, gives the following for Schocken:
1) Stoßen, Franz. chocquer,Engl. to shock; in welcher Bedeutung es noch in einigen gemeinen Sprecharten, besonders am Nieder-Rheine, gangbar ist. 2) Hin und her bewegen, schweben; schaukeln, welches das Intensivum oder Diminutivum davon ist. Ein Schiff schocket, sagt man in Nieder-Deutschland, wenn es von einer Seite zur andern wankt, wovon man im Hochdeutschen schaukeln oder schwanken gebrauchen würde. Jemanden schocken oder schockeln, ihn schaukeln, daher in einigen Gegenden die Schaukel auch Schockel genannt wird. 
Adelung’s understanding of schwebend seems more like wavering or hovering than like floating, suspended, etc. To be sure, in contemporary English, hovering can mean hanging suspended in the air or remaining in an uncertain or irresolute state, but it can also mean wavering, fluctuating, or remaining in one place in the air by beating the wings.
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In any event, deviating does not seem to have occupied a place in the history of musical schwebend. Granted, instability overlaps some of its uses, but this instability is consistent with motion in general, and in particular with repeated change of position, albeit within a larger region, as in Mersenne’s trembling/vibrating and beating.
Jay Rahn, York University (Toronto)
Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch at bluewin.ch .

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