MathNovatory/Applied Maths/Music Maths/The Structure of Time

Binary and Ternary
The most current forms of rhythmic subdivision and grouping are binary and ternary,
     usually considered similar siblings (as the sun and moon used to be),
          but actually quite different, as we will soon see.
All rhythmic structure is basically binary (Prime 2) at all levels,
          the smaller levels of subdivision as well as the larger levels of grouping,
     and most (at least popular) music is written with this simple, basic binary regularity.
Ternary grouping (and subdivision) is of a completely different nature -
     It is the first and simplest form of irregularity,
          the result of a transformation called ablation of the first quarter (a fusion of two levels),
               which produces Fibonacci forms when there is maximum ablation,
          making ternary rhythm Fib 3 and not Prime 3.
     There is actually far less ternary in our music than we might think.
          A waltz is considered a ternary piece, but only one of its seven levels is ternary
               (the small Level -3-2), the other six levels being binary.
This relationship of ternary evolving from the basic binary (actually "4") makes it possible to -
          either transform a piece from 4/4 to 3/4, or "restore" a piece from 3/4 to 4/4.
     It also opens a whole field of possibilities, including complete pieces in a Fibonacci structure.
          A five-minute violin piece has been written with 377 Quarter notes,
               of which you might like to hear the first 21 Quarter notes.

The Price to Pay for Binary
The incredible simplicity and regularity of all the larger rhythmic levels of grouping , from Level 0 up, do not give the listener any appreciable point of reference, such as the kind offered by the prime number incompatibilities of the Structure Of Pitch. The best the listener can feel is the number of bars, evidently preferred in binary exponentials (2, 4, 8, 16) but the qualitative dimension of alternating Beat and Off-Beat, so evident at smaller levels of subdivision, below Level 0, is intuitively less accessible at the larger levels. If a Bridge has the required 8-bar length, it does not seem to disturb anyone that it be placed a bar too soon or a bar too late.

The Rcode
There exists a precise and detailed Rhythm Code, the Rcode,
          which uses the Binary System (of Prime 2) in an unusual way.
     1) The numbering of each bar is placed before the "binary dot"
          with "0" indicating a Beat and "1" indicating an Off-beat at each level.
     2) The numbering of each note within a bar is placed after the "binary dot", indicating -
          (a) its quantitative position and length (duration) by the number of digits and
          (b) its qualitative alternation by the choice of digit.

The quality of Beat ("0") and Off-beat ("1")
is thus indicated at all levels,
grouping and subdivision.

To The Structure Of Pitch