Wednesday, January 7, 2015

Decimals of π in 10TET

The digits of π have been translated into music a number of times. Sometimes the digits are translated to the pitches of a diatonic scale, perhaps accompanied by chords. The random appearance of the sequence of digits is reflected in the aimless meandering of such melodies. But wouldn't it be more appropriate, in some sense, to represent π in base 12 and map the numbers to the chromatic scale? After all, there is nothing in π that indicates that it should be sung or played in a major or minor tonality. Of course the mapping of integers to the twelve chromatic pitches is just about as arbitrary as any other mapping, it is a decision one has to take. However, it is easier to use the usual base 10 representation and to map it to a 10-TET tuning with 10 chromatic pitch steps in one octave.

Here is an etude that does precisely that, with two voices in tempo relation 1 : π. The sounds are synthesized with algorithms that also incorporate the number π. In the fast voice, the sounds are made with FM synthesis where two modulators with ratio 1 : π modulate a carrier. The slow voice is a waveshaped mixture of three partials in ratio 1 : π : π2.

Despite the random appearance of the digits of π, it is not even known whether π is a normal number or not. Let us recall the definition: a normal number in base b has an equal proportion of all the digits 0, 1, ..., b-1 occuring in it, and equal probability of any possible sequence of two digits, three digits and so on. ("Digit" is usually reserved for the base 10 number system, so you may prefer to call them "letters" or "symbols".) A number that is normal in any base is simply called normal.

Some specific normal numbers have been constructed, but even though it is known that almost all numbers are normal, the proof that a number is normal is often elusive. Rational numbers are not normal in any base since they all end in a periodic sequence, such as 22/7 = 3.142857. However, there are irrational, non-normal numbers, some of which are quite exotic in the way they are constructed.

1 comment:

  1. If you combine Pi with the mathematical constant e you get a very tasteful irrational pastry.


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