Resultant Bass, Beats and Difference Tones - the facts
by
Colin Pykett
Posted:
21 July 2011
Last
revised: 9 April 2012
Copyright
© C E Pykett
Losing
an illusion makes you wiser than finding a truth
Ludwig
Börne
Abstract. This article addresses the physics of resultant bass stops on
the organ. Such stops are often
erroneously regarded as a low frequency case of the so-called difference tones
which are widely believed to exist in music.
It is shown that difference tones are never generated in the air when two
or more organ pipes speak simultaneously, thus they cannot arrive at the ears.
Occasionally they are perceived at higher frequencies, but only if the
generating tones are especially loud. Therefore
difference tones, if heard, are purely an artefact of the auditory system.
It
is a mistake to confuse difference tones with beats, which are used in resultant
bass stops. Beats are always
produced in the air when two or more pipes of different frequencies speak, but
they possess no acoustic energy at the beat frequency.
This paradox is because a beat is merely a periodic volume variation of a
sound wave at higher frequencies which does possess acoustic energy.
Only if significant nonlinearities exist in the ear will energy be
transferred from the generating tones into the beat frequency, and then we might
occasionally hear beats as difference tones at medium and high pitches. However the nonlinearities of the ear are too small to result
in perceptible difference tones in everyday musical experience.
Otherwise we would always hear difference tones between each frequency
and all others simultaneously present. Granted
that such cacophonies do not occur, it is curious that difference tones are
widely believed to exist in music.
Another
aspect of beats is important for resultant bass stops - the ear’s temporal
resolution capability. The
beat frequency between two pipes speaking a fifth apart lies at the desired
resultant or suboctave frequency of the longer pipe.
However the beats exist as a periodic sequence of discrete bunches of
acoustic energy, each bunch having a much higher frequency.
Below about EEEE (20 Hz) the bunches begin to be separately perceived in
time by the ear and it is this which gives rise to the illusion of a resultant
bass for the lowest few notes. However
there is no acoustic power at the ‘bunch frequency’ itself, even though it
lies at the desired suboctave frequency. Above
EEEE the ear begins to hear only the two generating tones, with progressively
less perception of a resultant frequency because we no longer resolve the
bunches temporally.
Notwithstanding
the above, it is possible subjective aspects of aural perception might vary
between individuals, thus too much dogmatism might be unwarranted.
Nevertheless, this does not affect the fact that difference tones are not
generated as sound waves in the air, and that they are not the same thing as the
beats which are generated. These
issues are important if one is to understand the matter properly.
Contents
(click
on the headings below to access the desired section)
Introduction
A
quick experiment
A
bit of arithmetic
What
do the waveforms look like?
Can
difference tones sometimes be heard?
Does
the ear insert the missing fundamental?
Can
a resultant bass ever work?
Conclusions
Appendix
1. Beat and Difference Tone Formation
Notes
and References
Introduction
Organ
builders have used low-pitched resultant bass (also called acoustic or harmonic
bass) pedal stops for centuries past and they continue to do so.
The general idea is that if two pipes speaking the interval of a fifth
are sounded simultaneously, they are widely supposed generate a ‘difference
tone’ with a frequency an octave below that of the lower-pitched pipe.
The two pipes speak the second and third harmonics (the octave and
twelfth respectively) of the desired suboctave tone. Sometimes additional harmonics were included as well, as in
Compton’s ‘Harmonics of 32 foot’ stops which typically sounded nine pipes
simultaneously. The mere fact that
such bizarre stops could ever have been made by a professional organ builder
shows how pervasive is the difference tone concept.
Let
us note immediately that this almost universally-held belief is completely
untrue. No such difference tones
are generated as sound waves in the air. It
simply does not happen, and it is the object of this article to explain why not.
Therefore the article, of necessity, also has to investigate why this
fond axiom of organ building has persisted for so long when it flies in the face
of the facts. One reason is simply
that the view continues to be propounded by some in the organ world and beyond
(a few examples are quoted in reference [1] below). So, because of this gulf between popular belief and what has
just been said, it should be helpful to try a quick ‘debunking’ experiment
which anyone can do.
A
quick experiment
If
difference tones were to be generated, it is reasonable to suppose that they
should exist at any frequency, that is at any point within the compass of the
organ rather than just for the lowest notes on the pedals.
Indeed, some of the quotations in [1] agree that this
is the case. This being so, why not
try an experiment? Play any two notes a fifth apart, such as middle C and the G
above it on a quiet or mezzo-forte 8 foot stop, and see if you can hear any
other note besides these two. It
does not have to be a stop on a pipe organ, and if it is more convenient you can
use an electronic organ or even a piano. If
a difference tone was generated you would hear a tone at tenor C, the octave
below middle C, but normally you will not.
In fact, if such difference tones existed, they would be created between
the fundamental frequencies of each note and every other in all the notes
sounding simultaneously in any musical composition (and between all of their
harmonics). The resulting cacophony
can scarcely be imagined but fortunately, as we know, it does not occur.
So why do people think it happens in general, and in particular that it
happens for the lowest pitches of the pedal organ?
And does it? To answer these
questions we need to look at the issue in a bit more detail, and the story turns
out to be surprisingly complicated with several unexpected twists and turns.
The article has avoided all but the simplest arithmetic, but the basic
mathematics of beats and difference tones is given in Appendix
1.
However
it is necessary first of all to add a rider at this point.
I do not perceive difference tones myself under normal conditions, those
which I encounter when listening to music at the usual volume levels encountered
as a member of an audience at a concert, when listening to recordings on high
quality audio equipment (this aspect is important), when playing an organ or
when doing experiments of the sort just described.
However it is conceivable that other people do perceive them and that
they accept it, perhaps without thinking about it, as part of their personal
aural life. One has to allow for
this possibility. In my case I only
hear very faint difference tones occasionally at certain frequencies, and even
then only when the generating tones are very loud, far louder than I encounter
in everyday experience. This aspect
of the matter is discussed and explained further below (see the section entitled
Can difference tones sometimes be heard?).
It would be insufferably arrogant not to allow for the fact that the
subjective qualia which characterise an individual’s aural perception could
conceivably vary, in much the same way as some people are colour-blind whereas
others are not. Nevertheless, these
are subjective issues and they do not affect the plain fact that difference
tones are not generated as sound waves in the air before the waves reach the
ears. The distinction is important
to bear in mind if one is to understand the matter properly.
(It
is both fortunate and interesting that I included the foregoing paragraph
because since this article appeared it has become clear, from the correspondence
it has generated, that some people do seem to hear difference tones more
strongly than I do myself. An example of the Internet chatter which has
arisen on the matter can be seen at reference [6]).
A
bit of arithmetic
Assume
we have a pipe whose fundamental frequency (first harmonic) is f
Hz or cycles per second. For
present purposes this defines the musical pitch of the pipe [2].
For example, ‘concert pitch’ today is often defined by tuning the A
above middle C to a fundamental frequency of 440 Hz.
A
pipe tuned to the interval of a perfect fifth above our first one will then have
a fundamental frequency of 1.5f, and to avoid a lot more complication it
is best to take this as read if you are unsure why this is so [3].
If we now sound these two pipes together, their frequency difference is
1.5f - f which equals
0.5f. This is an octave
below the pipe sounding f. But
this bit of simple arithmetic does not mean that we actually hear this suboctave
note. This difference frequency does not actually exist as a sound
wave, as our simple experiment above showed.
Those
who demand further proof might take a high quality microphone (one which will
work at the very low frequencies involved in 16 and 32 foot stops) into the
building containing the organ in question and connect a spectrum analyser to it.
This will produce a graph showing which frequencies are present in the
microphone signal. Any PC with the
appropriate wave-editing software can also perform spectrum analysis if you do
not have a dedicated piece of laboratory kit to do it.
Either way, you will find no trace of any sound energy at the difference
frequency, provided you are using a microphone and electronics of high enough
quality. The ‘high quality’
attribute is necessary because both sum and difference frequencies are often
generated electronically in the form of significant harmonic and intermodulation
distortion in poor equipment. I
sometimes get the impression that investigators looking for difference tones
might be misled by the shortcomings of the electronic equipment they are using.
Unfortunately it is not an issue they often discuss.
Besides
the frequency spectra, what would the various sound waveforms look like?
These would be observed if you connected the microphone to an
oscilloscope rather than to a spectrum analyser, or the aforesaid PC could also
throw them up on its screen if required. This
is an interesting and, as it turns out, vitally important aspect of the matter.
What
do the waveforms look like?
I
decided to perform exactly the experiment suggested above because of its
simplicity. Why should I ask you to
do something I am not prepared to do myself?
So, using my trusty oscilloscope, I photographed various low frequency
sound waveforms emitted by an organ. Please
forgive the less than perfect images, which resulted from camera shake.
The
first example was the waveform emitted by a 16 foot Sub Bass stop sounding DDD
(the reason for not choosing the bottom note, CCC, was because the oscilloscope
would not trigger reliably for some reason, presumably because its designers
assumed that nobody in their right mind would want to display waveforms at such
low frequencies). About two cycles
of this waveform are shown below in Figure 1.
Figure
1. Waveform of a Sub Bass stop
sounding DDD (36.8 Hz)
Next
up in Figure 2 is the waveform of the note sounding a fifth above the previous
one - AAA.
Figure
2. Waveform of a Sub Bass stop
sounding AAA (55.0 Hz)
Note
there are about three cycles visible in this trace, compared to about two in the
previous one. This confirms that
the two frequencies indeed had a ratio of 3 : 2 or 1.5 : 1.
Then
I played both notes together (well, I didn’t actually play them with my feet
because I had to take the photo as well. So
I wedged the two pedals down with a couple of pencils).
This might have been supposed to generate a 32 foot resultant bass at
DDDD whose waveform is shown in Figure 3. Just
over a single cycle of this wave can be seen in the picture and you can see that
its shape is quite complicated, perhaps unexpectedly so, and quite unlike the
simpler ones above that you might have anticipated.
Figure
3. Waveform of a resultant bass
stop notionally sounding DDDD (18.4 Hz). This
is Figure 1 added to Figure 2 by sounding both notes at once.
Finally
I photographed the waveform from a real 32 foot stop, called Contra Bass on this
organ, also sounding DDDD (Figure 4). Again,
only a single cycle or so is visible but this time the waveshape is easier to
understand. It looks pretty much like a sine wave, similar to that from
the 16 foot pipe shown in Figure 1, but because there is only a single cycle
visible it is at half the frequency as would be expected.
Figure
4. Waveform of a real 32 foot
Contra Bass stop sounding DDDD (18.4 Hz)
You
can see several things from these pictures, but the two most important are:
(a)
The frequencies of the resultant bass note (Figure 3) and the real one
(Figure 4) are the same. The peaks
of the signal in both cases, delineating one cycle of this very low frequency
sound, are separated by just over 6 horizontal divisions on the oscilloscope
screen.
(b)
The waveshapes of the two signals are very different however, as just
remarked.
The
main question we now have to answer is this: given that the resultant bass
(Figure 3) shows some evidence, on the face of it, of a suboctave frequency at
32 foot pitch, how can it be said that this frequency does not exist as an
independent sound wave? To try and
clarify this let us look at a few more waveforms, but this time generated
synthetically by a computer (Figure 5) because these are easier to interpret.
Figure
5. Resultant and real suboctave
waveforms (synthetic)
The
upper graph was derived by adding two sine waves (pure tones) whose frequencies
spanned the interval of a perfect fifth (a frequency ratio of 1.5 : 1).
The plot has some qualitative similarities to the actual waveform of a
resultant bass shown in Figure 3, although several more cycles are shown here
than the single cycle seen on the oscilloscope.
The most important similarity is the presence in both graphs of a
periodic (cyclically repeating) main peak with some lower-level structure lying
between. The positive maxima of
these peaks repeat at the resultant suboctave frequency, as can be seen by
lining them up by eye against the real suboctave waveform in the lower graph.
(One reason why the two resultant bass plots, Figure 3 and the upper in
Figure 5, do not look the same is that Figure 3 was derived using two actual
organ notes which each possessed a few harmonics in addition to their
fundamental frequencies. However
Figure 5 was derived using two pure tones only - there were no harmonics beyond
the fundamental in both cases. Another reason is that the two real organ notes were of
slightly different amplitudes, whereas the synthetically-generated sine waves
used for Figure 5 were of identical amplitudes.
Yet another reason is that there was an arbitrary phase difference
between the two real notes whereas the two synthetic waves both started at zero
phase).
What
we hear as a resultant bass is the beat between the two generating frequencies,
and the beat frequency is at the desired suboctave frequency.
However there is no acoustic power in the air for a beat for the reason
now to be described. This is the
nub of the matter, and it is always true regardless of the values of the two
generating frequencies. Let us
explore this further.
A
beat exists when the amplitude of a waveform varies periodically with time as
shown in Figure 6.
Figure
6. The beat envelope of a resultant
bass waveform
The
blue line in this graph is the same as that plotted in Figure 5 (upper), but its
envelope has now been emphasised by adding the pink and yellow lines which make
it easier for the eye to identify the beats.
It can be seen that they consist of a repeating series of wave packets or
‘bunches’ of sound energy. The
repetition frequency (beat frequency) of the bunches is at the desired suboctave
frequency, whereas the frequency of the waves making up each bunch (the blue
line) is at a higher frequency. Perhaps
surprisingly, this frequency is not the same as either of the two generating
tones; it is their average value. In
this case where the generating tone frequencies have the ratio 1.5 : 1, their
average value is 1.25 times the frequency of the lower note.
There
are two important points to grasp. The
first is that the blue line delineates the sound wave that we actually hear, and
it contains no vestige of a suboctave tone because it is at a much higher
frequency. The second point is
that, because the bunches (the beats) are symmetrical about the horizontal zero
axis, the two halves effectively cancel each other as the yellow and pink lines
show. Therefore, although the beat
frequency is indeed at the desired suboctave frequency, nothing can be heard of
it because of this cancellation effect.
Finally,
to complete the story look at the real (not resultant) bass waveforms.
Figure 4 shows the real example captured by the oscilloscope, and Figure
5 (lower) shows a synthetic one. The
latter is merely a sine wave at the desired suboctave frequency.
Note that the two plots are virtually identical, the slight lack of
purity evident in Figure 4 indicating the presence of a few harmonics in the
tone. No cancellation effect takes
place in this case, and so one can hear it as a genuine 32 foot sound.
Can
difference tones sometimes be heard?
It
has been shown that difference tones do not exist as sound waves in the air, and
a simple experiment at the keyboard confirmed that we do not (fortunately!) hear
them as part of everyday musical experience - at least, I do not and I assume
that many if not most people with normal hearing do not either.
Therefore the difference tones involved in low frequency resultant bass
pedal stops do not actually exist. It
is therefore curious that musicians in particular seem to have accepted their
existence for a very long time. For
instance Tartini, the famous eighteenth century violinist, is popularly credited
with their discovery as “Tartini’s tones”.
The fact is that difference tones can occasionally be heard in certain
circumstances, and I now need to explain how this happens.
Hitherto
I have been careful to point out that difference tones are not generated and
cannot exist as sound waves in the air.
However they can occasionally be produced and perceived as an artefact within
the ear itself under particular conditions.
Most of the time this emphatically does not happen, as witness the truism
that we do not usually hear them when listening to music.
The simple keyboard experiment above confirmed this.
Nevertheless, at some risk of oversimplification because the human
auditory mechanism is extremely complex, they can sometimes be heard when the
generating tones are very loud. This
is because the ear processes acoustic signals in a nonlinear fashion.
Nonlinearity
is a complicated subject which is difficult to explain without diving into
mathematics, but one way it can occur is as follows.
If the ear drum moves one way slightly more readily than the other, then
this will introduce a nonlinearity as the drum vibrates in both directions -
back and forth - in response to incident sound waves.
The degree of nonlinearity introduced by this means varies from one
person to another, but it increases in some types of disease.
(I recall an occasion during a heavy cold when it became impossible to
listen to music because all the difference tones became perceptible!
This was due to the motion of my ear drums and ossicles becoming so
impeded by the products of an infection that they became grossly nonlinear for a
time. It was a most distressing
experience, especially as the doctor casually mentioned that it might never go
away again). Other sources of
nonlinearity also exist within the ear and brain which will not be discussed
here. If the ear drum’s motion is
nonlinear it will perturb the symmetry of the diagram in Figure 6, which is
symmetrical about the horizontal zero axis as drawn.
In effect the curves will slightly expand or contract above the zero axis
relative to the parts lying below it, and then the pink and yellow lines
(delineating the beat envelope) will no longer always be equal and opposite
about zero. This means they will no
longer quite cancel each other out, resulting in a faint difference tone
appearing at the beat frequency. The perceived loudness of this tone will depend on the degree
of nonlinearity of the ear drum.
When
the two generating tones are very loud, the slight nonlinearities present even
in healthy ears sometimes result in faint difference tones being perceived.
It is interesting that it is musicians and musical instrument technicians
who usually speak as though these spurious frequencies are more common than they
actually are. If you play an instrument for your living, as Tartini did,
you live close to the sound generating mechanism much of the time, and its sheer
loudness is such that you will tend to hear difference tones more frequently
than your audience who are further away. The same applies to an organ
tuner or voicer, who might hear beats as separate tonal entities (difference
frequencies) between, say, the shrieking ranks of a mixture or the
near-deafening pipes of a Tuba stop when working close to them.
This usually happens to me when I go inside an organ and stand on the
passage boards. This factor might be responsible for such people in the
trade assuming that their audiences will also hear them, though generally they
do not because they are much further away from the sound sources and therefore
the difference tones tend to vanish well into inaudibility. It also means
that statements such as Bicknell’s belief
(he came from an organ building background) that "many difference
and addition tones is the thing that makes a fine organ chorus more than equal
to the sum of its parts" [1] apply more to the tuner,
voicer and possibly the player than to Joe Public away down in the body of the
church.
Does
the ear insert the missing fundamental?
A
popular belief is that the ear can insert the fundamental frequency in a
sound when in fact it is not present, and this is sometimes used to 'explain' the
phenomenon of a resultant bass. However, like difference tones, this is a
psycho-acoustic effect which only applies in certain special cases. In
brief, it can occur with low pitched pedal reeds, but it does not occur with flue
stops because the effect is only obtained when an extensive
harmonic series is present, as it always is with reeds.
But if only a few harmonics exist, as with large flue basses which frequently
have less than ten, the ear does not insert the missing fundamental in these cases.
Using
digital processing techniques, one can remove the fundamental from the waveform of a
pedal trombone pipe, for example, and demonstrate by listening tests that it can
make not the slightest difference to the perceived sound. Although this
does not happen in every case, it happens sufficiently often to be accepted as a
relatively well known subjective phenomenon. However the same experiment
carried out on the waveform of a large flue pipe leaves an entirely different sound in which the deep bass vanishes, to leave
only the second and third harmonics (plus others if present) prominent and easily identifiable.
These lie at the octave and twelfth respectively of the missing
fundamental as discussed already. Therefore the ear has not re-inserted
the missing fundamental in this case.
A
detailed discussion on the matter, which includes some audio examples, is
available elsewhere on this website [4].
Can
a resultant bass ever work?
Regardless
of the psycho-acoustic effects mentioned above, yet another factor comes into play
in the special case of a resultant bass at very low audio frequencies.
This concerns the temporal or time resolution capability of the ear.
Looking again at Figure 6, we saw earlier that a beat consists of
identical bunches of sound energy following one upon the other.
At medium and high audio frequencies these bunches repeat so rapidly that
we do not perceive them as separate entities.
Instead our ears interpret the blue curve, representing the air pressure
waveform, as a continuous sound. In
the case of the simple experiment described earlier the sound was simply that of
the two the notes middle C and the G above it.
So nothing remarkable in that.
But
at very low frequencies the bunches repeat at such a slow rate that our ears
begin to perceive them as just that - successive bunches of sound energy.
In the case of the bottom note of a 64 foot resultant bass (such as
Hope-Jones’s Gravissima at Worcester Cathedral in 1896) they would arrive so
slowly (8 per second) that one could almost count them!
This separation into perceptually discrete packets of sound starts to
occur below about 20 Hz as one goes lower in frequency, at which point the time
separation of the bunches is 50 milliseconds.
This corresponds to bottom E (EEEE) on a 32 foot resultant bass stop, and
it explains the well known phenomenon that it is only the bottom few notes of a
resultant bass which have a chance of being effective.
Above that the ear begins to hear only the two generating tones, with
little or no perception of a resultant frequency at all because the bunches
begin to merge into each other subjectively.
And, of course, the ‘join’ between BBBB (the top note of the 32 foot
resultant octave) and CCC (the bottom note of the 16 foot generating rank) is
invariably horrible! This why a
resultant bass can sometimes be more effective if it uses a separate quint rank
at 10 2/3 foot pitch rather than borrowing two pipes from a single 16 foot rank. The quint rank can be optimised in tone quality and power by
the voicer to produce the best possible resultant effect. This approach also has the considerable advantage that there
is no sudden discontinuity - no awkward ‘join’ between the lowest two
octaves - as there is with a resultant derived from a single 16 foot rank.
So,
given the above, it is true that a resultant bass can provide some sort of
underpinning gravitas to full combinations on the organ in favourable
conditions, especially for the bottom few notes on the pedals.
And, as with life generally, half a loaf is usually better than no bread
at all. But even so, one should not
delude oneself into believing one is hearing difference tones.
Finally,
a remark about electronic organs. I
have never seen an electronic organ with a resultant bass stop (though
since this article appeared I have been made aware of some which do [5]), and this might
be because it is easy to provide full-range pedal stops which go down to the
lowest frequencies. However it is certainly not easy to radiate these sounds
effectively using small and cheap loudspeaker systems.
So the majority of electronic organs which have 32 foot pedal stops in
effect possess a resultant bass capability automatically.
This occurs because, as you descend the pedal board while using a 32 foot
simulated flue stop, the loudspeakers radiate progressively less acoustic power
at the fundamental frequency (the first harmonic), and at some point in the
compass it will often fade into inaudibility.
But at the same time the second and third harmonics (sounding the octave
and twelfth above the desired tone) do not fade out so rapidly.
Therefore these continue to recombine in the listening room to provide an
‘automatic’ resultant bass. Is
this the best of both worlds? I
leave you to decide that for yourself.
Conclusions
The
physics and psycho-acoustics underlying resultant bass stops on the organ are
surprisingly complex, many-facetted and frequently misunderstood.
Such stops are a special low frequency case of the so-called difference
tones which are widely but erroneously believed to exist in music more
generally. It has been shown that
difference tones are not generated when two or more organ pipes speak
simultaneously, thus they cannot propagate as sound waves in the air to arrive
at the ears. Only occasionally are
they perceived at medium and high audio frequencies, and then only exceptionally
such as when the generating tones are especially loud.
Therefore difference tones, in the unusual circumstances when one does
hear them, are purely an artefact of the auditory system.
It
is a mistake to confuse difference tones with beats.
Unlike difference tones, beats are always produced in the air when two or
more pipes of different frequencies speak, but they possess no acoustic energy
at the beat frequency even though they are readily perceived when the beat
frequency is low. This paradox is
because a beat is merely a periodic volume variation of a sound wave at higher
frequencies which does possess acoustic energy. Only if significant nonlinearities exist in the ear will
energy be transferred from the generating tones into the beat frequency.
It is only then that we might occasionally hear beats as identifiable
difference tones. Although our
hearing mechanism is nonlinear, the nonlinearities are of too low an order in
the normal ear for them to result in perceptible difference tones in everyday
musical experience. If this was not
so, then we would hear difference tones between each frequency and all others
simultaneously present, but we do not. Granted
that such cacophonies do not occur, it is therefore curious that difference
tones are widely believed to exist and to play an apparently important role in
music.
Another
aspect of beats is important for the special case of low frequency resultant
bass stops, and this concerns the temporal
resolution or time-resolving capability of the ear.
The beat frequency between two pipes sounding the interval of a fifth
apart lies at the desired resultant or suboctave frequency of the longer pipe.
However the beats exist as a periodic sequence of discrete packets or
bunches of acoustic energy, each bunch having a much higher frequency equal to
1.25 times that of the longer pipe Below
about 20 Hz, corresponding to EEEE on a 32 foot stop, the bunches begin to be
separately perceived in time by the ear and it is this which gives rise to the
illusion of a resultant bass for the lowest few notes.
However there is no acoustic power at the beat frequency itself, even
though it lies at the desired suboctave frequency.
Above EEEE the ear begins to hear only the two generating tones, with
progressively less perception of a resultant frequency because the bunches begin
to merge into each other as far as our aural perception is concerned - we can no
longer resolve them temporally.
Notwithstanding
all of the above, it is possible that the subjective qualia which characterise
an individual’s aural perception could conceivably vary, in much the same way
as some people are colour-blind whereas others are not.
Nevertheless, these are subjective issues and they do not affect the
plain fact that difference tones are not generated as sound waves in the air
before the waves reach the ears. The
distinction is important to bear in mind if one is to understand the matter
properly.
Appendix
1. Beat and Difference Tone Formation
The
formation of beats and difference tones are explained here using mathematics,
which has been kept as simple as possible. Nevertheless it is still
necessary to keep a clear head!
Beats
Let
two pure sine wave signals having frequencies
f 1 and f
2 be denoted by sin A and sin B, where
A
= 2π f 1 t and B = 2π f 2 t
(t is time)
Then
the combined signal when they are sounded together in a linear acoustic
environment such as the air is given by R where
R
= sin A + sin B
Applying
the trigonometrical function-sum relation, this can be written as
R
= 2 sin 1/2 (A + B) cos 1/2
(A - B)
This
equation might conceivably be the source of the erroneous belief that adding two
sine waves as sound waves in the air results in their sum and difference
frequencies being generated because of the terms A + B and A - B.
But the actuality is, of course, that HALF their sum and HALF their
difference are the terms which actually appear as the arguments of the trig
functions. This affects the results profoundly as we shall now see.
For
the interval of a perfect fifth the frequencies are in the ratio of 1.5 : 1,
thus we can write f 1 = 3/2 f 2
where f 2 is therefore the lower
frequency (that of the longer pipe). Then the equation reduces to
R
= 2 sin (1.25 x 2π f 2
t ) cos (0.25 x 2π f 2
t)
This
equation is that plotted as the blue curve both in Figure 5 (upper) and Figure
6. The sine term has a frequency equal to 1.25 times the frequency of the
longer (lower frequency) pipe, i.e. it is the average frequency of the two
pipes. The cosine term has a frequency equal to 0.25 times the frequency
of the longer (lower frequency) pipe. It imposes a slow modulation on the
sine term to generate the beat envelope, and it is essentially the equation of
the pink and yellow curves in Figure 6 (though there is a 180 degree phase shift
between them) . Note that the frequency of the cosine term is at half
the beat frequency we perceive in a resultant bass stop.
Difference
Tones
Difference
tones are only generated when the two sine waves combine in a nonlinear
environment. The mathematics of nonlinear systems is generally
intractable, but it is simplified here by considering an environment which has a
square law response. This does not represent the human ear whose
nonlinearities are more complex, but it demonstrates the essential issue (the
formation of a difference tone) for present purposes.
The
composite signal R defined above having passed through a square law
environment becomes R2 . Thus
R2
= (sin A + sin B)2 = sin 2 A + sin 2 B
+ 2 sin A sin B
Using
standard trigonometrical identity formulas, this can be written as
R2
= ½ (1 – cos 2A) + ½ (1 – cos 2B) + cos (A – B) – cos (A + B)
The
first and second terms show the presence of frequency components (2A and 2B)
at twice the original frequencies, i.e. at the second harmonics of the input
frequencies. This would constitute
harmonic distortion if it occurred in the ear or a hi-fi system.
The third and fourth terms show the presence of the difference and sum
frequencies respectively, which constitute intermodulation distortion.
Therefore, in contrast to the previous case of linear combination, separately
identifiable sum and difference frequencies are obtained this time.
Notes
and References
1.
A few quotations from some who believe (or did believe) in the
‘resultant’ idea - the generation of difference tones as sound waves in the
air - are:
(a)
Stephen Bicknell:
"many
difference and addition tones is the thing that makes a fine organ chorus more
than equal to the sum of its parts".
in
"The Mixture - to be taken as before?", number 3 of 6 articles
published in Choir and Organ in 1998-9 under the heading 'Spit and
Polish'.
(b)
G A Audsley:
“
... there is no doubt that the differential tones of all the stops of
different pitch exert an influence on the general tonality of the Organ, more
or less appreciable to the cultured ear ... ”
in
The Art of Organ-Building, vol 1, New York, 1905
Audsley
devoted several pages to the topic, going on to tabulate in detail the
difference tones allegedly generated by an entire Principal chorus.
One wonders whether Bicknell was influenced by him nearly a century
later, given the qualitative similarity of their remarks.
(c)
Edward L Stauff:
“When
pure tones (sine waves) are sounded together, they combine to produce two
additional tones whose frequencies are the sum and difference of the two
original tones”.
in
http://www.organstops.org/r/Resultant.html
(Accessed 28 June 2011)
(d)
W L Sumner:
“
... a resultant 32-ft effect is obtained by playing a 16-ft open pedal stop
with a corresponding fifth rank ... ”
in
The Organ, McDonald, London, 1962
(e)
Wikipedia (anonymous author):
“If
the beating frequency rises to the point that the envelope becomes audible
(usually, much more than 20 Hz), it is called a difference tone [citation
needed] .”
“Playing
pure harmonies [sic] (i.e., a frequency pair of a simple proportional
relation, like 4/5 or 5/6, as in just intonation major and minor third
respectively) on the two upper strings, such as the C above middle C against
an open E-string, will produce a clearly audible C two octaves lower.”
“An
interesting listening experiment is to start from a perfect unison and then
very slowly and regularly increase the pitch of one tone. When one tone starts
to split out from the former twin-note, a slow rumbling can be heard,
gradually increasing into an audible tone.”
in
http://en.wikipedia.org/wiki/Beat_%28acoustics%29
(Accessed 20 July 2011)
These
quotations demonstrate several things. Firstly
that it is not only musicians who sometimes get it wrong, assuming this
moderately mathematical Wikipedia article was written by a scientist or
mathematician. Secondly, it
provides additional confirmation of the well-known fact that the rigour of a
Wikipedia article cannot always be guaranteed.
Thirdly, yes, at least one citation is certainly needed!
Fourthly, it is, of course, possible that the article was written by
someone with an uncommon hearing defect such that he/she did indeed perceive
the difference tones described so confidently.
Fifthly, the author’s command of English was inadequate to meet the
considerable demands of the subject.
2.
Absolute frequency (an objective measure) and musical pitch (subjective)
are not always the same, because they increasingly diverge at higher
frequencies. This does not affect
the arguments in this article which are based purely on frequencies.
3.
The ratio of 1.5 between the fundamental frequencies of two pipes
sounding the interval of a fifth will actually vary slightly depending on the
temperament in use. For equal
temperament the frequency ratio of all fifths is the same and slightly less than
1.5, for instance. However these
differences can be neglected here.
4.
See the article The
Tonal Structure of Organ Reed Stops, C E Pykett, January 2011.
5.
I am grateful to Robert Sproule for drawing to my attention that some Allen
digital organs have incorporated resultant bass stops at both 32 and 64 foot
pitches.
6.
The following chatter appeared on the Internet prompted by this article, and it
is typical of other examples which arose elsewhere as well as some material I
have received personally:
Correspondent 1:
I can assure you that I can hear them [difference tones], and so can those organ builders who provide pedal stops that depend on the existence of this hearing phenomenon.
Correspondent 2:
You are also correct regarding the mixing of multiple frequencies in the ear, brain, soul or whatever since any two
frequencies have resultants in summation and subtraction not to mention the harmonics. Too many resultants I suppose
could result in pink noise.
Correspondent 3:
... are you [correspondent 1] REALLY saying you can hear tenor C if you play middle C and the G above? This is the experiment he
[Pykett] suggests in his article. Surely not? Because if you can, your auditory system must be grossly non-linear. That is not
intended to be rude because (again, as Pykett says) it is possible that some people do "hear" in this manner. It
would be no different to listening to a very poor hi-fi system with high IM distortion.
Correspondent 1:
To answer your question, I most certainly CAN hear Tenor C if Middle C and the G above it are sounded. However, it must
be admitted that it is at a very low level. And I don't think there is anything particularly wrong with my hearing.
Let me suggest my own experiment: sound a fairly pure note well up in the organ manual compass, and WHISTLE a note
within a few tones of it. If your whistled pitch is BELOW the sounded note, you should hear the resultant "sound"
lower in pitch as your whistle approaches the pitch of the sounded note. Depending on the relativities, you can can
some pretty unpleasant effects!
Correspondent 3:
Many thanks for your very interesting reply ... It confirms (as Pykett said in his article) that different
individuals probably do have differing types of audio perception. He likened it to vision, where some people are color
blind whereas others are not.
I do not hear any vestige of difference tones at all. I have just tried your whistling experiment with my home organ
and do not hear anything like what you described. However please do not think I conclude that anything is wrong with
your hearing. It is simply different to mine.
Source:
http://www.magle.dk/music-forums/13491-single-wave-file-voice.html (accessed 30 August
2011)
(The
experiment suggested by Correspondent 1 is interesting. I tried it myself
and, like Correspondent 3, I hear no resultant sounds at all, as I knew I would
not. My hearing does not work like this).
