Posts Tagged ‘temperament’

Aristoxenus

Tue 5th Jul 11

I read this in the Smog Blog: “But the fact is, if you go to blogs like WattsUpWithThat or Climate Audit, you certainly don’t find scientific and mathematical illiterates doubting climate change. Rather, you find scientific and mathematical sophisticates itching to blow holes in each new study.”

A bell rang in my head. This is the same treatment that Aristoxenus got.

Aristoxenus was the earliest proponent of the theory of equal temperament in music. He was a pupil of Aristotle and the son of a professional musician. He was a competent musician himself, and his musical theory was endorsed by J. S. Bach and has been the standard ever since. All electronic tuners give equal temperament by default, and 99% (say, probably more) of bands, orchestras and solo performers in the western world use it.

The theory divides the musical octave into twelve pitches by equal steps: the seven white and five black notes on the piano keyboard, that repeat the same pattern from bass to treble. Indian, Turkish, and other traditions use more, smaller steps. We needn’t go into the small details here, but if you are intrigued then look at my article on temperament in h2g2.

The odd thing is that between Aristoxenus proposing the theory and Bach accepting it was a gap of two thousand years, during which the system was rejected by musical theorists on mathematical grounds. That’s a long time in the wilderness.

The trouble is that the maths doesn’t quite add up: equal temperament is an approximation. Pythagoras had famously established the physical relation between sound-producing bodies (strings, pipes, bells) and their size, weight, and tension. To sound an octave lower, one string must be twice as long as another of the same material, thickness and tension. To sound a perfect fifth lower, it must be 1.5 times as long, and to sound a major third lower it must be 1.25 times as long.

A compromise must be reached, or else a discord must be tolerated, when you tune a harp to the notes A, B, C, D, E, F, G. If your A is a major third above your F, and your F is a perfect fifth below your C, and your C and G and D are also tuned to perfect fifths, then your D will not be a perfect fifth from your A. You have enough figures above to work out by how much, if you are handy with a calculator (though a pen and paper is all you need). To keep the figures manageable you will have to multiply or divide by two here and there: this keeps the notes unaltered, since twice C or half C gives an octave, still C.

This compromise is simply achieved: spread out the discrepancy throughout all the intervals, so that none but the octaves are pure fractions, but all are bearably close. The brain decoding the vibrations in the air categorises something close to a major third so that we ‘hear it as’ a major third. This ‘hearing-as’ is the miracle of perception; not a simple thing, as brain-circuit diagrams show us, but a capacity we inherit at birth and without which we could not live.

Aristoxenus was the first to challenge the mathematical theory of Pythagoras: while Pythagoras claimed that the perfection of music was its mathematical purity, Aristoxenus claimed that the ear, not the measuring tape, is the proper judge of musical excellence.

What’s wrong with that?

He gave a lecture, which has been transcribed, demonstrating his theory. Having gone through a number of transpositions, he arrived at an outlandish interval, what we might call in modern notation E-flat to G-sharp. Then he said: “Is this a perfect interval? We should let our ear decide.”

To let the ear decide, he must have been tuning things—strings, probably, as they are easiest—as he went along. And to arrive at a perfect interval (tuning his G-sharp the same as A-flat) he must have been tempering the intervals all along. The point of the demonstration was that his listeners couldn’t tell that each perfect fifth he tuned was a tiny bit defective. They ‘heard them as’ perfect fifths.

For two millennia Aristoxenus was disrespected and his theory rejected. A century before Bach, the organist and composer Frescobaldi recommended equal temperament, but, apart from the theoreticians’ distaste for compromise, it was found awkward to tune keyboards in equal steps; they hadn’t hit on the system that organ and piano tuners now use, counting the beats. In the meantime (it seems) lutes and other fretted instruments had been tuned to equal temperament for centuries, since it is actually quite easy to set frets proportionately by eye.

The ancient and medieval theorists rejected equal temperament because their calculations showed (rightly) that it was not a hundred per cent accurate in placing the intervals. It took two thousand years for theorists (led by Rameau) to admit that the discrepancies in the maths were not really so significant in perception; human judgement is satisfied with ninety-nine per cent, and often very much less.

Cavillers. Hair splitters. Look at the big picture.


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