Planning, Research & Reflection

Multiple untuned instruments: asymptotic prime number ratios

Slightly sideways idea for a composition for ‘untuned percussion’ that would break all records for the metronome mark, number of instruments and length.  Conceptual composition really.  May be possible to implement this as custom software…

Instrument 1 enters with a pulse defining interval 1 (essentially a click track) at a fast but discernible beat e.g. MM=240.

Instrument 2 at interval 2, 3 at interval 3, 4 at interval 5,  5 at interval 7

Instrument m at the interval of n the m’th prime number

Subtlety: the novelty would get spaced wider and wider in time, so the tempo needs to be progressively increased in a log ratio, i.e. constant multiplicative rate.  This will lead to pulses eventually being interpreted as a pitch e.g. a click track at MM=6000 is a tone of 100 Hz. 

To the extent that prime numbers become less frequent with size then the increase in tempo should balance the scarcity of prime numbers.   I think that this will lead to a pseudo random sequence of chords – the Wiki link at foot of this post is interesting as a source.

Clearly the progression is infinite.  With a continuously rising pitch, the original click track and then, soon after, multiples of this would become inaudible above around 15kHz so the calculation can be truncated, but the speed of processing to test primality and need for special large number formats will lead to slower processing so the implementation of this in practice needs a cached list of prime numbers. A further challenge is that with the exponential increase in tempo the number of numbers to be tested increases exponentially also.  So the piece needs to be ‘precalculated’ up to some duration, or parts of the piece between set starting and ending ‘beats’ could be calculated on demand.

A further refinement likely to be more tolerable to hear for a sustained period would be to disguise the pitch change (falling as primes become less frequent) by doubling at multiple octaves and as the frequency of finding primes falls, transfer amplitudes upwards through the octaves and so use Shepard’s illusion – ‘the auditory equivalent of Escher’s continuous descending staircase’: 

http://psylux.psych.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/auditory.html

Andrew

http://en.wikipedia.org/wiki/Von_Mangoldt_function

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