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Newent Orchestral
Society Bill's Musical Notes, June, 2009 Temperament - are you bovvered? I read a really interesting short book with a long title recently called How Equal Temperament Ruined Harmony And Why You Should Care (by Ross Duffin, published by W.W. Norton). It's full of information about the history of and the problem of temperament in tuning. At last, I thought I had understood this fascinating and peculiar subject. Then, a friend asked me to explain it - and I couldn't. So, I thought again, go back to the basics, check what the problem is, then write a short, easy explanation without referring to the book. In that way I'll know I know and also I'll be able to explain it in a couple of paragraphs to anyone who is bovvered. First, let me describe the "harmonic series". When a note is sounded on an instrument, it creates a whole series of different tones, called "overtones". These are related to that fundamental note in a simple way. If the fundamental vibrates at 100 Hz (cycles per second), then the overtones are at 200 Hz, 300 Hz, 400 Hz, etc., in simple mathematical ratios. What gives every instrument its characteristic sound, what differentiates an oboe from, say, a violin, is how much of each overtone is present in the instrument's harmonic series. OK. Now forget all that for a moment. Here's some different, but relevant, information: if you divide a vibrating string, or resonating tube, by two (divide the length by 2/1), the note it sounds is an octave higher. Concentrate now. If you divide it in the proportion 5/4 you create the interval of a major third. Now any musician can work out on their instrument that there are exactly three intervals of a major third adding up to form a perfect octave. If you multiply 5/4 x 5/4 x 5/4, you should get the same ratio as the octave, that is, 2/1. The answer is, unfortunately, not quite (it's 1.953). In other words if you add three major thirds together you don't get a perfect octave. It's just that bit less. What a shame, but there it is and the difference is easily audible. Keyboard tuners have been compensating for this for a long time now by making the interval of the fifth smaller (and the intervals of the fourth and third bigger). That's what temperament is all about in a nutshell. Return now to my introductory note about the harmonic series. On investigation, this is found to contain the intervals of the fifth, fourth and third, as well as the octave and this fact of nature gives a basis for harmony in western music. We didn't just invent harmony, it is inherent in the harmonic series and any discrepancy caused by the physics of vibrating strings or pipes when we play music is simply a horrible nuisance. I picked out the major third because the maths is simple, but the same principle applies to all the other intervals. They simply don't fit into the octave series. In addition it is, according to Ross Duffin, the major third that is most obviously a horrendous audible problem. As a musician, I am certainly bovvered, partly by the impenetrable complexity of the subject, but mainly because I now know why I can't tune my instrument to suit all musical circumstances. So that's about it. I've gone as far as I can with this for now without losing my cool. A little Mozart might be soothing, me thinks.
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Bill Anderton, June, 2009