Pete Stewart explores the mysteries of drone harmonics in search of an elusive concordance

Some of my favourite pipe tunes, particularly those written in the last fifty years, are in D, rather than in A. Playing these tunes with drones in A, however, has always left me less than satisfied. Recently, in response to a different question, I decided to figure out why. What follows is described in terms of smallpipes; conical bore chanters with drones one and two octaves below the chanter’s root have a difference of balance of harmonics, both with the drones and in themselves.
We all know that for a piper, the great joy of the instrument is the wondrous effect of the chanter being in accord with the harmonics of the drones. I was therefore both consoled and surprised to finally convince myself that, travel as far up the stack of harmonics that a drone can produce as you may, you will never find a ‘just’ fourth above the root; that is, an A drone just doesn’t produce a good ‘D’ anywhere in its harmonic series.
Perhaps it’s worth stating exactly what is meant by a ‘just’ interval, although to do so means dealing with numbers, which I know are not everyone’s cup of tea, so I’ll try to keep it as straightforward as the topic allows.
The bagpipe drone that is said to sound an ‘A’, you are probably aware, in fact sounds a whole series of notes, thanks to which we are able to get the instrument in tune with itself (the poem quoted in the article about the Dumfries pig piper in this issue includes a description of the drone as ‘a stick that holds the notes’) For reasons of simplicity I am going to assume that the frequency [the rate of vibrations per second] of the lowest of these notes, the one we use to identify the pitch of the drone, is 110 cycles per second. Now, the series of sounds produced by such a drone consists of a mathematical series known, not surprisingly, as the ‘harmonic series’ (although it is only indirectly linked to the idea of ‘harmony’); it consists of frequencies related to each other in a simple mathematical way. The second in the series (taking the ‘root’ A as the first) is 2 times the frequency of the original [220cps, the pitch of the ‘tenor’ drone and the tonic of the smallpipe chanter], the third is 3 times that of the first, 330cps, and so on. You may be aware that these notes, starting from the bottom A are, nominally, A3, A4, E4, A5, C#5, E5, A6 etc. (middle C being C4)

You can perhaps see these notes are getting closer together, the intervals between them getting smaller.¹ We have the octave, the fifth, the fourth, the ‘major’ third and the ‘minor’ third. After this come a couple of unacknowledged intervals: 7/6 of the ‘minor’ third generates a note which Table 1 calls F## and 8/7 generates an A above that note.(these two intervals don’t appear in conventional European music). Following on, we have two ‘tones’, one ‘major’ and one ‘minor’, notes which are respectively 9/8 and 10/9 times the frequency of their preceding neighbour.2
Now, we can establish the frequencies of the notes within a single octave by reducing each by a suitable fraction so that they all lie between 220 and 440;

But what about our ‘D’, the tonic ‘home’ note of so many wonderful tunes? We know that it is a fourth above our tonic A so its frequency should be 220*4/3 [the same proportion that relates the E’ and the A’’ in our drone harmonics], that is 293.33cps.
Now there is probably some erudite proof somewhere, but I’ve convinced myself by doing brute calculations for long enough to satisfy me that nowhere in the harmonic series of a drone pitched at 110cps will you hear a note that is 293.33 multiplied by any power of 2.³ That is to say, you cannot tune your ‘D’ to a consonant tone in an A drone, and your ‘pure D’ tonic is therefore unsupported by it. The nearest harmonic is the 11th, which, by our nominal values, would require a frequency of 302.5cps for the D, very sharp of pure. This is the principal explanation for the ‘dry’ sound that a tune in D produces, compared to one in A.⁴
Is there a solution to this problem? Well, no, strictly speaking there isn’t. But there might be a compromise to be reached which will enable us to continue to play all those wonderful D tunes with satisfaction. Let us leave aside the matter of the drone for the moment and consider how to tune the chanter so that we get the best possible intervals in the key of D.
Let’s consider the intervals of a ‘major’ scale of D. The crucial ones are D/F# and D/A [the third and fifth], but also of interest is that of A/D [the fourth] the determinate of our V/I cadence, the fundamental cadence of most of the ‘D’ tunes we play.
Looking at the table above, it is clear that, in addition to D, we have not yet established from our drone harmonics, an appropriate frequency for either F# or G. Now if we want a ’just’ value [that is, one that is mathematically determined by the harmonic series] for the important intervals of fourth and third, then our D, as we have seen, must be 293.33cps. Our third, F# must then be 293.33*5/4=366.66. Note that the interval between this F# and the E below is now 10/9, what we can call a ‘minor tone’, equivalent to that between the 9th and 10th harmonics. Also, the interval between our D and E is a ‘major tone’, 9/8 [frequencies 293.33 and 330]. Together, these two intervals, a major tone and a minor tone, make up a ‘perfect’ third. [9/8*10/9=5/4].
This is all well and good and to it we can add the one missing note, the G, which for playing in D should ideally be a perfect fourth above the D, 586.66*4/3=782.213.⁵

But, and the but is inevitable in a time-bound world, what about the intervals between the other notes in this D scale? Ideally for playing in D we would want just intervals for the triads of G too; especially we would like a perfect major third G-B. What does the system outlined so far give for this interval? G=195.5; B=247.5; now a perfect third above 195.5 is 244.44 [195.5*5/4]; our third is thus a little sharp. In fact, it is a major tone above the A; since our G is a major tone below the A, what we need for G-B to be a perfect third is the minor tone. So let’s put a little bit of tape over the six-finger hole, to lower the B. However, the result now gives us a B which can be heard to conflict with the 3rd harmonic E of the bass drone, being no longer a perfect a 5th above it. So, perhaps we should leave the B in its whole tone tuning above the A and raise the low G instead, to 198cps. This retains the true third between the low G and the B, but what has now happened to the ‘perfect’ 5th that we had established between the low G and the D? It is now short of a true 5th (198*3/2 =297).
Ok, so suppose we raise our D to this new pitch of 297cps. Sounds unlikely? It was at this stage in my musings on this topic that I was reminded of an article Barnaby Brown had written in Piping Today some while ago, where he demonstrated that up until around 1950, the recordings of some of the finest Highland pipers revealed just such a ‘sharp’ D, and that this appeared to be the same with the late 17th century Iain Dall chanter that he and Julian Goodacre had been working on reproducing.⁶ Using such a pitch it was possible for the two triads ACE and GBD, which form the basis of so much early pipe music, to be pure intervals, sacrificing the ‘pure’ fourth of A-D. (Using this tuning, the D is now a minor tone below the ‘pure’ E, rather than the usual major tone). The interval between this new D and the F# is now two minor tones rather than the usual combination of a major and a minor. The true third interval from this D would need to be 371.25cps; (I find that my chanter, which is tuned to play covered-fingering, actually has this tuning for the F# - in order to play a ‘pure’ 6th with A as the tonic requires the 3rd finger of the top hand to be raised; it is only during the compiling of this article that I have uncovered this subtlety.)

All we need to do now in order to play happily in D is to do something about the drones. Firstly we can plug off the bass, which is sounding E and C# harmonics in conflict with our chanter’s D. Assuming we can re-tune our baritone drone to D3 (148.5cps), it so happens that nature has devised a means whereby a bass D is (to a certain extent) available. This comes in the form of what are called ‘difference tones’. By the magic of acoustics, when two sounds of different frequencies are sounded together they generate an ethereal extra sound which has the frequency of the difference between the two played frequencies.⁷

Now if we sound two tones a fourth apart, a D at 146.67cps and an A at 220cps (our tenor drone), they will produce a low D at 73.3cps.

However, we have just re-tuned our chanter D to be 297cps. If we tune the baritone drone to an octave below this D (148.5cps) then the ‘difference tone’ against the A is 71.5cps and thus the bottom D is now slightly more than two octaves below our chanter D. You are, I hope, beginning to understand the problem.⁸

These frequency differences are tiny, it’s true, but in training themselves to tune their pipes, pipers learn to attune their ear to such things - in an imperfect world compromise is the only solution; it’s a question of which compromise is the best in any situation.
For smallpipe players a compromise set-up for playing in D can be achieved by slightly raising the pitch of the chanter’s G and D, shutting off the bass drone and re-tuning the baritone to D. Unfortunately, this last device cannot be deployed on the conical bore chanters with all drones in A and no baritone drone. But many of those D tunes are so good that conical-bore pipers are gonna play ‘em regardless …

 NOTES

1. You will probably also notice that the difference in the actual cycles per second remains constant; this is because the increase in ‘pitch’ is logarithmically related to the increase in frequency. Isn’t mathematics wonderful?

2. The existence of these two slightly different intervals of a ‘tone’ is something of a musical secret, but it is the root cause of all the problems we have in tuning the chanter to play just intervals in different ‘keys’.

3. The frequencies of the D’s rise by powers of 2 every octave, whereas those of the harmonics rise by 300cps each step. The mathematical proof probably relies on the fact that these two series can never coincide, that is, that no power of 2 is divisible by 3. It might be succinctly stated in the form ‘there is no whole number value of n such that 3n =2^x where x is a whole number. It should also be noted that how the harmonics that are present in any drone actually sound is a product of its material, its geometry and its manufacture.

4. It is true that when playing a D on the smallpipe chanter, it itself generates a harmonic series of which the third is an A which will be in tune with the fourth harmonic of the tenor drone; this may not be easy to hear.

5. With allowance made for recurring decimal 3’s or 6’s, the result of dividing even numbers by 3.

6. In the extended version of this article, to be included in The Highland Bagpipe, Music, History, Tradition, Barnaby Brown describes this wide fourth as ‘colourful’ and points to examples of its use in fiddle traditions, particularly in Norway..

7. There are also ‘sum tones’, which are what we might expect. I have not taken them into account here. For a striking demonstration of the magic of difference tones, I recommend the slow air played on two whistles on the first Chieftains’ LP. So distinct is this ‘third’ voice that it is possible to transcribe it and demonstrate that the pitches are the mathematical consequence of the voicing of the two ‘real’ parts.

8. I should add, perhaps, that this tuning is ideal for the earlier Lowland repertoire which is sometimes described as ‘double-tonic’ in character. In this article I have been addressing the more modern problem of tunes in D major.