[seqfan] Polyrhythm and A258024

[Veikko Pohjola]

Dear seqfans,

Concerning the origin of the polyrhythmic structure of A258024, please check first the link in A258200 to my initial discussion in SeqFans. According to my conjecture, any integer n maps to either 0 or 1 by nested application of floor(tan) on n. Thus the sequence of positive integers starting from 0 maps into the sequence A258021. From rhythmic point of view the appearance of 1’s is more interesting. The 0’s can be taken to represent the rests and the 1’s are the beats. 

A258024 starts like this 1, 4, 23, 26, 45, 48, ... the numbers being the roots of the equation floor(tan...(floor(tan(n)))=1. The lowest root >1 is 4, but it is periodic as can be expected due to the presence of trigonometric function. The period is 44 with periodic hick-ups, which have their origin in the floor function. The beats starting from 4 and having this period form the dominant rhythmic pattern, because the period 44 is by far the shortest. Polyrhythm forms when overlapping other patterns starting from other roots having their individual periods being much longer.

When playing a composition of several patterns, the beats can be assigned different pitches to distinguish the role of each pattern and to make the whole resemble music. Because of the unlimited choice of parameters in composing, anyone can reach a feeling to be a musical composer.

Best regards,
Veikko Pohjola