Resultant Bass, Beats and Difference Tones - the facts
by Colin Pykett
Posted: 21 July 2011
Last revised: 9 April 2012
Copyright © C E Pykett 2011-2012
Losing an illusion makes you wiser than finding a truth
Abstract. This article addresses the physics of resultant bass stops
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.
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.
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
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.
(click on the headings below to access the desired section)
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  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.
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  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 ).
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 . 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 . 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.
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.
5. Resultant and real suboctave
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.
6. The beat envelope of a resultant
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.
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"  apply more to the tuner, voicer and possibly the player than to Joe Public away down in the body of the church.
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 .
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 a commercial electronic organ with a resultant bass stop (though since this article appeared I have been made aware of some which do ), 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.
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.
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!
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 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
= (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
= 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.
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:
G A Audsley:
(c) Edward L Stauff:
(d) W L Sumner:
(e) Wikipedia (anonymous author):
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.
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:
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).