Tuning and Scales
There is not really the space to go into a lot of detail here but I feel it is appropriate to give at least the main principles a brief mention. For those who wish to look into the subject a list of useful publications is given at the end.
There are a number of different scales the most commonly met being Equal Temperament, Just Intonation, Mean Tone and Pythagorean. These are further complicated by the way Bagpipes work and the way the scales apply to different regional types. These modes such as Mixolydian, Lydian, Dorian will also be covered.
The bagpipe (discounting keyboards and free reed) is a unique instrument in that several notes can be sounding at once - the constant drone which can be based on the keynote and/or most commonly a fifth or fourth and the chanter which can sound a number of notes and on a few types regulators or subsidiary chanters. Because of this drone the notes of the chanter have to correspond with the harmonics produced by the drones otherwise beating will occur and the effect will not be pleasant. The scale of the chanter will therefore approximate to what is known as "just intonation" whereby the intervals are based on simple ratios that correspond to these harmonics. These are the fundamental at 1f, major tone 9/8f, major third 5/4f, perfect fourth 4/3f, perfect fifth 3/2f, major sixth 5/3f, major seventh 15/8f, octave 2f.
With multiple key instruments such as keyboards the scale is divided into equal values to even out the differences but this does not suit bagpipes as certain notes will clash due to the difference between the pure and tempered tones. For instance applying the just intonation to the scale of of say A major gives the following results starting at A = 440hz, we get B = 495hz, C# = 550hz, D = 587hz, E = 660hz, F# = 733hz, G# = 825hz and a = 880hz. Now if we go up a fifth to E we get E = 660hz, F# = 742.5hz, G# = 825hz, a = 880hz, b = 990hz, c# = 1100hz, d# = 1238hz and e = 1320hz. These intervals are sometimes expressed in "cents", where 1 cent is equal to the 12th root of 2. There are 1200 of them to an octave and with equal temperament they divide up conveniently into 100 per semitone. When applied to just intonation they give slightly differing values of (figures in brackets equal temperament) Major tone = 204 (200), Major third = 384 (400), Perfect fourth = 498 (500), perfect fifth = 702 (700), major sixth 884 (900), major seventh 1,088 (1100) and octave 1200 (1200). From this it can readily be seen that their is an immediate problem with the F# (major third) as to play accurately in the key of E it needs to be sharper in pitch than in the key of A, the home key of the chanter. In practice the notes have to be tempered, i.e. the difference evened out to not overly notice in either key. The more keys that a chanter has to play in then the more notes that require tempering to give an acceptable result with the trade off being how they sound against the fixed pitch of the drone.
Generally the most acceptable keys are either the fourth and fifth above the home key plus the associated minor keys. Once you start trying to go beyond these then the problems start to compound each other. I hope this is understandable as it is not something that is easy to explain. I have had trouble before trying to get the message across to musicians who are used to equal temperament instruments such as guitars and keyboards who can play in any key signature and also have this idea that "g sharp" and "a flat" are the same note, which of course they are not. From the above it can also be seen why a "natural" scale does not work on a keyboard instrument or anything with a compass of more than a couple of octaves as if you go round in a circle of any interval such as fifths where the difference is only slight (+2 cents) with the corresponding note for equal temperament you will end up with a sizable error by the time you get back to the original note several octaves higher. Having said all this and to add further confusion I would point out that the piper has a few tricks to cover up intervals where these problems occur, these being vibrato, sliding and gracing.
A problem that often occurs when using measurements of old instruments is that of the pitch they play at. Concert pitch of a = 440hz only became standardised in 1939 and prior to this each maker tended to design his instruments to suit the circumstances under which it would be used. Small adjustments can be made by varying the size of the reeds but for greater differences some redesigning may be required. One of the simplest methods is to use the frequency ratio of required frequency to existing frequency and multiply this by the measurements to arrive at the desired pitch. The bore however retains it's existing diameters although the points at which they are found will change by the same proportion in order to retain the existing characteristics of the instrument in question.
For example the measurements obtained for a Bagpipe with a conical bore chanter measured in a museum give an instrument that plays at A = 425hz. - slightly flat of modern concert pitch. Whilst it possible to reed the drones to this latter it is difficult to get the chanter to the same so a redesign is required.
The measurements of the tone holes and changes in bore are all multiplied by the ratio of the frequencies - 425/440 - giving a shortened chanter with the tone holes now positioned slightly closer together. The diameter of the bore remains as the original i.e. the bell is the same as is the throat. If it has a bi-conal bore then the point of change will also have to be multiplied by this ratio as well so as to keep the bore to tonehole configuration the same.
It should be noted that this method works best where only a small difference exists between actual and desired pitches. Where it is greater then care has to be taken that toneholes do not get either too close together or uncomfortably too far apart. Whilst it is possible on the thicker walled instruments to drill tone holes at an angle to the bore some further modification of their relevant positions will also be necessary. There are a number of books available that discuss this in all much greater detail.