[Smt-talk] Early account of beats

Nicolas Meeùs nicolas.meeus at paris-sorbonne.fr
Tue Sep 14 09:17:21 PDT 2010

  No, Jay, no interference can be perceived (at least in normal 
conditions) between the partials of a single tone with harmonic 
spectrum, and there wouldn't be "various rates" in this case. The 
difference between the harmonic partials of a single tone is always the 
same, by definition, and equal to the fundamental frequency of this 
tone. This may produce an interference, more specifically a difference 
tone, which is not exactly of the same nature as beating. It has the 
frequency the fundamental of the tone and is perceived as reinforcing 
it. Other difference (and summation) tones may arise, but they always 
correspond to the frequency of one or another of the harmonic partials 
themselves. There is no way that a harmonic sound could produce 
frequencies outside its harmonic series: this is an arithmetic property 
of the harmonic series itself.
     [When I say that no interference can be perceived "in normal 
conditions", I mean that it is technically possible to differentiate the 
energy originating from the difference tones from that of the partials 
themselves; but they blend in a normal perception.]

Whenever two different tones with harmonic spectra sound simultaneously, 
on the other hand, several interference rates appear at the same time. 
It is not always so that the slowest rate predominates. Some (piano) 
tuners tuning D3-A3 as in the case you described would choose to listen 
to the 2.8Hz rate aroung A5 (880Hz), rather than the 1.4Hz one around A4 
- and they would succeed in doing that! That is to say that various 
beating rates can be heard, and isolated by an attentive listening.

Whether Schlick heard beats, we will never know. The only clue we have 
is what he said: /schweben/. Now if one is convinced that /schweben 
/meant 'to beat' in 1511 because that is what it means today, there is 
not much that I can add, but that I strongly doubt this and won't change 
my mind about it.

Martin Braun writes:

    ...the German term "schweben" already in those days [Werckmeister's]
    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
    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. 

/Schweben/ may mean 'deviate' (downwards of upwards): I fully agree and 
I think this is the best translation proposed up to now. But it 
certainly is NOT the precisely defined meaning of 'to beat' as it is 
used today.

The beating rate for a given interval doubles from octave to octave: 
even without machines to measure beats, this can in no way be considered 
"equal". Also, intervals in equal temperament are not "equally 
acceptable" across the scale. It is well known that ET-thirds, for 
instance, are rather less acceptable in the low range. This is because 
the roughness of beating is maximum within a rather limited range within 
which thirds fall in the low, not in the high.

Jay Rahn argues that it is not easy to perceptually evaluate intervals 
between successive sounds: this is right. He appears to consider it 
improbable that Werckmeister /gleichschwebend /could have denoted an 
equal deviation (say, by 1/12-comma), because that could not easily be 
perceived. But Werckmeister is not speaking of perception, he describes 
tunings to be performed with the help of calculation, of a monochord and 
a compas. With such tools, equal deviation can be obtained.
     Equal temperament is characterized neither by equal beating, nor by 
equal acceptability of its intervals. It is characterized by the equal 
_size_ of its intervals, even although this cannot easily be 
perceptually evaluated.

Nicolas Meeùs
nicolas.meeus at paris-sorbonne.fr

Le 14/09/2010 2:29, JAY RAHN a écrit :
> Nicholas Meeus claims that between two tones with harmonic spectra 
> many fluctuation rates can be perceived at the same moment but that 
> most people would remain unable to concentrate on a specified one. To 
> be sure, various rates would be present acoustically as, in principle, 
> they are within a single tone. However, hearing interference between 
> two tones is quite a different task than hearing out the partials 
> within a single tone. Nonetheless, as among the partials of a single 
> tone with a harmonic spectrum, the slowest rate of interference 
> between two tones would tend to predominate perceptually, especially 
> if the lowest partials were most intense, which they would tend to be 
> on plucked strings and particular organ stops. In contrast, hearing a 
> difference of, e.g., a quarter comma between *successive* tones *as* a 
> quarter of a comma seems much more demanding than Schlick’s somewhat 
> vague prescription to listen for the undulation produced by a 
> flattened A sounded simultaneously with D a 5th below ‘as much as it 
> can endure.’
> In any event, I find it remarkable that nobody prior to Schlick seems 
> to have drawn attention to audible interference between pairs of 
> tones, especially as harmonics (in the sense of flageolet tones) and 
> resonance (in the sense of sympathetic vibration) appear to have been 
> observed at least as early as, respectively, ancient Greece and China. 
> In principle, interference, harmonics, and resonance would have been 
> semi-objective ways of verifying whether particular numerical (or 
> geometrical) entities had been transferred more or less precisely to a 
> monochord or other instrument.
> Jay Rahn, York University (Toronto)
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