Idsardi, William J.

Uriagereka, Juan

English

Biolinguistics (E-Journal)

 BRIEFS  
 
 
 
 
 
Biolinguistics 3.4: 404–406, 2009 
ISSN 1450–3417       http://www.biolinguistics.eu  
Metrical Combinatorics and the Real Half  
of the Fibonacci Sequence 
 
William J. Idsardi  &  Juan Uriagereka 
 
 
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)1 where the nth 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 non-
exhaustively 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 2n-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 2n-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. 
 In 1680, Cassini (1733) discovered a relation among successive members of 
the Fibonacci sequence, expressed in (2):2 
 
(2)  Fib(n)2 – Fib(n–1) · Fib(n+1) = (–1)n 
                                                
    1 The Fibonacci sequence can also be defined to start with 0: 0, 1, 1, 2, 3, … 
    2 This relation was independently discovered by Simson (1753). 
Biolinguistics    Briefs   
 
 
405 
That is, the square of any Fibonacci number is equal to the product of the two 
flanking Fibonacci numbers, give or take one. For example, Fib(4)2 – Fib(3) · 
Fib(5) = 25 – 3 · 8 = 1 = (–1)4 and Fib(5)2 – Fib(4) · Fib(6) = 82 – 5 · 13 = –1 = (–1)5. 
Rearranging (2) gives (3): 
 
(3)  Fib(n)2 – (–1)n = Fib(n–1) · Fib(n+1)  
 
The left-hand side of (3) has two possible expansions, depending on whether n is 
odd or even, as in (4):  
 
(4) a. n is even: Fib(n)2 – 1 
 b. n is odd:  Fib(n)2 + 1 
 
We can now see that (4a) has the form (x2 – 1) and (4b) has the form (x2 + 1). Ele-
mentary algebraic polynomial factorization (Herstein 1977) shows that (4a) has 
real-valued roots, (5a), whereas (4b) only has complex-valued roots, (5b): 
 
(5) a. Fib(n)2 – 1 =   [Fib(n) – 1][Fib(n) + 1]  
 b. Fib(n)2 + 1 =   [Fib(n) – i][Fib(n) + i] (where i2 = –1) 
 
Thus, for example, Fib(3) · Fib(5) = 3 · 8 = 24 = 4 · 6 = [Fib(4) – 1][Fib(4) + 1]. 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 ‘real-
valued’ half of the sequence (in the sense of (5)). Evidently, the Fibonacci charac-
ter 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.  
 
 
References 
 
Binder, Philippe M. 2008. Frustration in complexity. Science 320, 322–323. 
Cassini, Jean-Dominique. 1733. Une nouvelle progression de nombres. Histoire de 
l’Acae ́mie Royale des Sciences, volume 1. Paris.  
Biolinguistics    Briefs   
 
 
406 
Halle, Morris & William J. Idsardi. 1995. General properties of stress and metrical 
structure. In Eric S. Ristad (ed.), Language Computations, 37–69. Providence, 
RI: American Mathematical Society. 
Hayes, Bruce. 1995. Metrical Stress Theory: Principles and Case Studies. Chicago: 
University of Chicago Press. 
Herstein, Israel N. 1977. Topics in Algebra. New York: John Wiley and Sons.  
Idsardi, William J. 2008. Combinatorics for Metrical Feet. Biolinguistics 2, 233–236. 
Simson, Robert. 1753. An explication of an obscure passage in Albert Girard’s 
commentary upon Simon Stevin’s works. Philosophical Transactions of the 
Royal Society of London 48, 368–376. [doi:10.1098/rstl.1753.0056] 
 
 
 
 
William J. Idsardi / Juan Uriagereka 
University of Maryland 
Department of Linguistics 
1401 Marie Mount Hall 
College Park, MD 20742 
USA 
idsardi@umd.edu / juan@umd.edu 


Metrical Combinatorics and the Real Half of the Fibonacci Sequence

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Metrical Combinatorics and the Real Half of the Fibonacci Sequence

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