Metrical Combinatorics and the Real Half of the Fibonacci Sequence

Languages with stress group syllables into metrical feet (Halle and Idsardi 1995, Hayes 1995)—non-exhaustive groups of contiguous syllables. The size of feet in natural languages ranges from unary (a single syllable) to unbounded (as many syllables as possible); in addition syllables can also remain unfooted. Under these conditions, the number of possible metrical footings for a string of n syllables is known to be Fib(2n) (Idsardi 2008), where Fib(n) is the Fibonacci sequence, as in (1) where the n element is the sum of the previous two elements (for example, 13 = 8 + 5).


Metrical Combinatorics and the Real Half of the Fibonacci Sequence
William J. Idsardi & Juan Uriagereka Languages with stress group syllables into metrical feet (Halle andIdsardi 1995, Hayes 1995)-non-exhaustive groups of contiguous syllables.The size of feet in natural languages ranges from unary (a single syllable) to unbounded (as many syllables as possible); in addition syllables can also remain unfooted.Under these conditions, the number of possible metrical footings for a string of n syllables is known to be Fib(2n) (Idsardi 2008), where Fib(n) is the Fibonacci sequence, as in (1) 1 where the n th element is the sum of the previous two elements (for example, 13 = 8 + 5).
(1) n: 0 1 2 3 4 5 6 7 … Fib(n): 1 1 2 3 5 8 13 21 … For example, a string of two syllables (here notated with 'x's) can be nonexhaustively footed in five ways (= Fib(4)): (xx), (x)(x), (x)x, x(x), and xx.In contrast, if footing were required to be exhaustive (that is, if every syllable had to belong to some foot) then a string of two syllables could only be footed in two ways: (xx) and (x)(x).It is easy to see from the bracketed grid representations that the number of possible exhaustive footings of a string of n syllables must be 2 n-1 as every exhaustive footing must begin and end with foot-boundaries and between each pair of x's we have a binary choice between having a foot juncture and not having one.Since there are two choices for each space between x's and there are n-1 spaces between n x's, it follows directly that there are 2 n-1 distinct exhaustive footings.
As a consequence, only half of the Fibonacci numbers (those underlined in (1): 1, 2, 5, 13, …) are solutions to the task of creating non-exhaustive footings; the other half (3, 8, 21, …) are not.An intriguing question is: Why is it the one half of the sequence and not the other?We venture some speculations about potential answers.
Only the even-numbered Fibonacci numbers (here, Fib(4)) show up in the real-valued roots, and this is the same Fibonacci subset that characterizes the number of valid metrical groupings of strings of n syllables.
In conclusion, the 'metrical' half of the Fibonacci sequence is also the 'realvalued' half of the sequence (in the sense of ( 5)).Evidently, the Fibonacci character of footing arises just when we allow for non-exhaustive footing, as exhaustive footings can be counted as a simple set of independent binary choices.Generally, the Fibonacci sequence is associated with a number of 'edge of chaos' effects, especially systems which illustrate dynamical frustration (Binder 2008); systems in which opposing forces cannot reach an equilibrium solution.We speculate that the 'forces' operative here in defining non-exhaustive footings could be the local coherence of syllables into feet clashing with word-level properties of footing.Another potential view of the emergent complexity observed here would be that sequences of footed syllables can be metrically distinct -for example, (x)(x) !(xx) -whereas all sequences of unfooted syllables are the same; thus we have asymmetric growth patterns in the footed and unfooted portions of syllabic strings resulting in Fibonacci complexity.