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Listeners fascinated by the theory of music may like to take their knowledge to the next level of detail but -- be warned -- this sort of thing can easily spiral down into number mysticism.

The ancient Greeks discovered that sounds whose frequencies had a simple numerical relationship sound sweeter together, more consonant, than those which don't. The purest consonances come from frequency ratios of 2:1, 3:2, 4:3 and so on, and these intervals (named as the octave, fifth, fourth, etc) became the basis of Western musical practice. These simple numerical ratios worked well for the intervals of a single scale but each interval in the scale was a little bigger or smaller than its neighbour, leading to endless trouble when someone innocently said, 'Can we move that up one step, please?'

Two thousand years and dozens of tuning systems later, Western European art music settled on a compromise called Equal Temperament, which made all intervals equal at the expense of making all intervals (except the octave) slightly out of tune. It was a very pragmatic solution to the problems of making fixed-pitch instruments playable in any key, but no-one except the pianists (who had no choice) ever really liked it. Violinists famously complain about having to adapt their intonation to 'out of tune' (perfectly tuned, but in equal temperament) pianos, but they are not alone. Listen to a good string quartet, vocal ensemble or woodwind quartet, then listen to the same piece on piano: the harmonies are subtly different.

Warren Burt wanted to be able to play pure consonances, so he made his nineteen note set of tuning forks; but nineteen notes per octave means more and bitterer dissonances are possible too. For The Archytan Transpositions, he went two steps further and used his computer to transpose layers of The Animation of Lists both up and down by a very small interval and then overlaid the untransposed layer with its two transposed partners to create a piece employing a 53-note scale. The pitches are far from random but their relationships are far from usual.

The same transposition process time-shifted the layers so that one was also somewhat slower and the other somewhat faster, introducing an element of arbitrariness to the rhythms.

Hence the greater tension of the Transpositions. Listen to the two extracts again for a direct comparison of the original [listen] and its variant [listen].

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