In Part 1 of this series, the foundation of the highland bagpipe scale was outlined in the context of staff notation. Out of a possible scale of 12 notes found in the chromatic scale: A Bb B C C# D Eb E F F# G G#, the highland bagpipe uses only 7: A B C# D E F# G, placing it square in the key of D-major. It is important to understand what the highland bagpipe scale is in relation to the more universal chromatic scale if the goal is to understand how to tune a highland bagpipe chanter using a standard chromatic, non-bagpipe tuner.
Even if you don't intend to ever use a tuner to help tune your bagpipes, this information is very useful in playing music with other melody instruments. What follows is an explanation of the equal temperament scale as programmed into universal, non-bagpipe tuners and why it can't simply be used to tune our chanter notes that are to be played against a set of drones.
As discussed in Part 1, octaves of a note are simply the previous note frequency multiplied by 2. For example, if the bass drone is tuned to 120 Hz, the tenor drones being 1 octave above that would tune at 120*2 = 240 Hz. This is also true for all other instruments. What remains is how to divide up the space between one octave and the next to create all 12 notes of the chromatic scale. One would think the easiest way to do this is to just to subtract the lower octave frequency (120 Hz) from the upper octave frequency (240 Hz), using the drones as an example, 240 - 120 = 120 Hz, and then divide that by the number of notes needed (12), 120 Hz/12 = 10 Hz. That means each of our 12 notes could be obtained by just adding 10 Hz to the fundamental frequency (120 Hz) to get the next note. The scale would then be 120 Hz, 130 Hz, 140 Hz, 150 Hz, 160 Hz, 170 Hz, 180 Hz, 190, Hz, 200 Hz, 210 Hz, 220 Hz, 230 Hz, and then finally the octave of the first note, 240 Hz, after 12 additions of 10 Hz. However, there is a problem. If we continue this series the next note is at 250 Hz, but this is not 1 octave above 130 Hz. Instead of adding 10 Hz, upon starting the second octave, we'd need to add 20 Hz each time to get the next note. This scheme of creating a scale is quickly falling apart. How would you know when to switch from adding 10 Hz to adding 20 Hz? All of it would depend on which note you defined first, in our case 120 Hz. There must be a better way, and there is.
Instead of adding frequencies to the fundamental (the reference, lowest pitch), we multiply the fundamental and each successive note by 2 raised to the 1/12 power, 2^(1/12) ~ 1.0595; this is the 12th root of 2. 12 is used because we need 12 notes and at that 12th note, one is simply multiplying by 2^(12/12) = 2 which is what is needed to get the octave! Again, starting at 120 Hz (A), the next note is 120*2^(1/12)=127.135571323 Hz (Bb). The following table fills out this octave by multiplying each successive note by 2^(1/12).
|Note name||Frequency (Hz)|
Whew, most of those frequencies look ugly. But, if you continue the trend of multiplying each new note by 2^(1/12) the octaves fall into place naturally as can be seen by the octave of A at 240 Hz. The next note in the series is 254.271142646 Hz which is exactly twice of 127.135571323 Hz because they are displaced by twelve multiplications of 2^(1/12), which is equivalent to multiplying 127.135571323 Hz by [2^(1/12)]^12 = 2^(12/12) = 2^(1) = 2! How clever.
Instead of multiplying each successive note by 2^(1/12), one could just multiply the chosen fundamental by 2^(x/12) where x is the number of notes above the fundamental that you want to calculate the note frequency of. For example, if you wanted the fifth note: 120*2^(5/12) = 160.1807825 Hz.
However, this scheme for generating the equal temperament chromatic scale doesn't work to generate the bagpipe scale. The bagpipe's internal tuner, i.e. the drones, prevent the use of the equal temperament scale. All the ugly note frequencies found in the equal temperament scale would be out of tune with the drones, some more so than others. The drones do not play just one note frequency. Starting at the fundamental (the first note in the harmonic series), the drones also play overtones whose frequencies are obtained by multiplying the fundamental by 2, 3, 4, 5, etc. The first overtone (the second note in the harmonic series), obtained by multiplying the fundamental by 2, falls on the definition of the octave having multiplied by 2. Using the bass drone as an example, if the fundamental is at 120 Hz (A), the first overtone plays at 120*2 = 240 Hz (A, this is also the fundamental pitch of the tenor drone). The second overtone of the bass drone is 120*3 = 360 Hz which is not an octave of A, but rather an octave of 360/2 = 180 Hz. Looking above at the ugly frequencies generated by the equal temperament scale, the closest note to 180 Hz is E at 179.797 Hz. Those frequencies are actually quite close, however, an E tuned to 2*179.797 Hz = 359.594 Hz would be out of tune with a bass drone tuned to 120 Hz because of the slight difference, 360 - 359.594 = 0.406 Hz, which is called the beat frequency. If we take the reciprocal of the beat frequency, 1/0.406 Hz = 2.46 seconds, we obtain the period over which a wah-wah-wah sound will be heard, indicating the two notes are out of tune. Two and a half seconds is rather slow, but things get worse as we work up the overtone scale produced by the drones. Of course, the chanter doesn't produce an E note at 360 Hz but rather at 720 Hz so we need to go up a few overtones to find frequency overlap with the chanter. The 3rd overtone is another octave of the fundamental, 120*4 = 480 Hz (A, the frequency of low A on the chanter); for all the octaves of A, the equal temperament tuning is the correct tuning because multiplying by 2 gives "clean" frequencies. The 4th overtone is 120*5 = 600 Hz (600/2 = 300 Hz, 300 Hz/2 = 150 Hz) which is an octave of C# referencing the table above. If the chanter's C# was tuned according to equal temperament tuning, 4*151.191 Hz = 604.764 Hz, the beat frequency compared to the bass drone's 4th overtone would be 604.764 - 600 = 4.764 Hz which has a period of 0.210 seconds; every .210 seconds the wah-wah sound of out-of-tune notes would be heard; 5 wahs every second is pretty fast.
Below is a table which fills out the rest of the bass drone's harmonic series as overtones of the bass drone fundamental. The amplitude of the overtones generally decreases as you you go up so the higher overtones are much quieter than the lower overtones and it's sufficient to stop at the 13th overtone for now.
|Multiplicative Factor||Frequency (Hz)||Note name|
What's interesting about the overtones is we can start to pick out the bagpipe scale. The most harmonic notes are seen in the lower overtones, A, E, C#, and then G. Starting at the 7th overtone and going to the 13th, the bagpipe scale is listed in order: A, B, C#, D, E, F#, G! The tuning of the bagpipe chanter must produce notes which align with the overtone frequencies of the drones, but we can see these frequencies are all very clean looking. Thus, the notes cannot follow the equal temperament algorithm of generating the scale based around the 12th root of 2 because that produces ugly frequencies. (Note that the bagpipe chanter note frequencies range from low G = 420 Hz to high A = 960 Hz if the bass drone is tuned to 120 Hz.)