# singularities

this post continues the idea from my previous post about massive irrationally related sets. one impressive thing about these things is that they extend infinitely and never repeat, despite that they initially are completely aligned. another way of saying this is that for each set there is only one singularity. we can assume that any set of irrationally related periodic signals will have, in the fullness of time and if we extend them in both directions of time, one and only one singularity. but how can we predict when that singularity will occur? if our set of periodic signals happens to be an equal temperament, we can use the following formula to delay the singularity by one cycle of the lowest signal in the set:

for every signal x, xdelay = 1/f0 - 1/x

where f0 is the frequency of the lowest signal in the set, and x is the frequency of the signal to which we apply the delay.

to delay the singularity by more than one cycle, simply replace those 1's with the number of cycles.

we can now generate sequences of mostly periodic signals whose phase we occasionally manipulate to get singularities whenever we want them.