Collapsing Monoid in a Quasi-Associative Space

We first consider the free monoid of binary strings as a model to see how the rhythm monoid collapses.  A 1 denotes an onset at a particular time  0 the lack of an onset.  Consider the strings 0, 00, 000, 0000,… .  If interpreted as rhythms, these sound identical yet they are different strings.  These different representations collapse to the same rhythm.  Similarly 1, 10, 100, 1000, … collapse to a single rhythm.  Generally, one can interlace any number of  zeros between digits of any given string without changing the rhythm identity.  This generates rhythm classes is akin to the pitch classes from octave equivalence. 

This collapse can be mapped if we “wrap”  the binary strings that represent the rhythm around the complex unit circle.  We can use the unit circle as a generalized module representing one or more beats or one or more measures depending on the context.

The complex points from string Y with length n can be found by multiplying (an nxn matrix with the nth roots of unity along the diagonal starting with 1, and zeros elsewhere) by the string itself written where each entry is an element in a column vector.  We delete any zeros and we are left with the roots of the polynomial that represents the rhythm. 

Usually we consider a root of unity to be a single point, but if we allow sets of points or diagonal matrices of those points to be eligible for root-hood then the solutions include all the non-null elements of the power sets of the roots of unity for each order.  These are described by complex polynomials that correspond to each and every onset profile.  That is to say that each rhythm maps to one hypercomplex root, but that root may map to other rhythms as well. 

We characterize the notated metric space as quasi-associative since the association matters in terms of the onsets. Consider 111 111 and 11 11 11, eighth notes in 6/8 and 3/4 respectively.  These are different rhythms that share the same hypercomplex root.  But this availability of multiple metric frames only comes into play if the modulus of the string > 4.    This aligns with the solvability of the associated hypercomplex roots.  At modulus 5 or higher, one has a responsibility to notate the association of the notes.  There is a convention in music called singleton exclusion where we never consider metric spaces (meters or subdivision groups) with only one element, so no bars of 1/4.  

We define the adjoined concatenations of characters to denote the onsets of a single measure of music.  Ordinarily each beat gets a character in the concatenation.  Superconcatenation is denoted using the space _ identity element (barline) to distinguish between measures in multi measure concatenations.  Subconcatenation uses the ( ) to allow smaller subdivisions by substitution of a string of characters for a single character in the concatenation.  This allows potentially infinite onset point coverage that acts as an adaptive grid.