Alan Coady’s Musical Blog

Imaginary Numbers

I was never much good at maths and so would believe anything to be true if it felt vaguely correct. This led to an unusual situation where, by means of a stepping stone, so incorrect* as to be unreal, a conceptual stream was crossed.

I must have dreamt that imaginary numbers could be entered into an equation when no alternative means of reaching its end were in sight. The fact that they cancelled one another out overcame the fact that they don’t really exist.

This led to the following thought process and procedure, which I have seen help many pupils cope with sizeable changes of position (horizontal location of the left hand) on the guitar.

• starting point - the shortest distance between two points is a straight line
• therefore, travelling along the string will always be faster than describing a curve in the air (stay on the x-axis, forget about the y-axis, and don’t even dream about using the z-axis)
• however, the finger on the string before the move may not be immediately involved in the new position
• now the imaginary numbers enter
• decide on which note the remaining finger could land if it were going to be involved – based on the first real note
• insert that imaginary note into the music (at this point, Sibelius comes in handy so pupils can actually see the note – there is enough imagining going on already
• practise – departure note – imaginary note – destination note(s), several times in a slow, relaxed manner
• then acknowledge the imaginary note by leaving by continuing to aim for it but not playing it – simply leaving a gap where it would have been
• finally, eliminate the gap - the shortest journey possible having been taken, the join should be as smooth as it is possible to be

Cynics might wonder at the point of going through such a long process only to undo it by eliminating the note. Without this process, most pupils would simply dive at the next real note, causing tension and an unnecessary break in the flow of the music.

If you want to see this in action, go to Guitar Group Support – Additional Parts and open up Concerto in D, Part 1 Page 3.

• find the bar before section 13
• the 6th note is the departure note
• the journey takes place during the 7th note (an open string)
• the imaginary note (coincidentally identical to the 10th note) is inserted here**
• the 8th note is the destination note

* in the interest of defending the reputation of maths teachers at Knox (my alma mater) I should stress that I had dropped out of the world of maths long before anyone would have had the chance to be associated with this misunderstanding. Despite a sizeable mathematical blind spot, I do not find it difficult to believe those who feel that numbers lie at the heart of our universe.

** there is no necessity for the imaginary note to be one which also appears in the passage. It need not even sound particularly musical, as it is going to retreat once its job is done.