TITLE: Musical and combinatorial proof
QUESTION [6 upvotes]: How many distinct rhythms can a musical measure have?

Obviously the answer is not "$\infty$", so to answer this question we set a minimum rhythm  $\frac{1}{4}$. We will consider both notes and rests and not consider irregular groups like triplets and others. So we'll consider notes and rests as $\frac{1}{4 }$, $\frac{2}{4}$, $\frac{3}{4 }$ ...
As regards musical measure we'll consider $\frac{L}{4}$ where $L$ is the length of measure and consider only $L>0$.
Then calculate the number of combinations of rhythms in a measure:




So the sequence found is: $1,2,5,13,34,...$ and this is a possible relation with Fibonacci's bisection:
$$ F(0)=1 $$
$$ F(1)=2 $$
$$F(L)= 3 \cdot F(L-1) - F(L-2); \quad L>1$$
How is it possible to proof that this recurrence relation is valid for each $L>1$?
IMPORTANT EDIT:
Sorry i have forgotten to write that we consider rhythms from the acoustic view point, so we consider for example that two rests of $\frac{1}{4}$ acoustically equal to a rest of $\frac{2}{4}$ and so we will consider only the rest of $\frac{2}{4}$.

Furthermore rhythms of arbitrary duration are allowed, for example we can have a note of $\frac{5}{4}$ with a invented symbol.

REPLY [2 votes]: Here's a different proof, which makes the connection to the Fibonacci numbers explicit as well as using some basic musical concepts. If people have trouble with unicode musical notes, let me know and I'll try to replace them with images, but they have fairly low code points, so hopefully it'll be okay as is. (Rests seem to be more poorly supported, so I'm using images for the examples at the end which involve them.)
The standard combinatorial interpretation of the Fibonacci numbers is that $f_k$ (suitably indexed) counts the number of ways of tiling a path of length $n$ with tiles of length either $1$ or $2$. In musical terms, $f_k$ counts the number of possible rhythms which are $k$ eighth-notes long, using only eighth notes and quarter notes (and no rests). If we let the quarter note have the beat and consider sequences which are $n$ beats long, it follows that there are $f_{2n}$ such sequences — the same number as there are of the types of rhythm you are trying to count.
Given this observation, it's very natural to look for a bijection between these two types of rhythm. And in fact we can find one.
Any sequence of quarter and eighth notes can be divided into:

Syncopated sections: These consist of an eighth note that falls on the quarter-note beat (i.e., is preceded by an even number of eighth notes), followed by some number of quarter notes, followed by an offbeat eighth note (e.g., ♪♩♩♩♪).
Unsyncopated sections: These consist of some number of quarter notes, all of which fall on the quarter-note beat (♩♩♩♩♩$\dots$).

Each of these sections takes up an integer number of quarter-note beats. Note that it is convenient for our purposes to regard a pair of adjacent eighth notes on the beat (♫) as a very short syncopated section, even though in musical terms it would not be considered syncopated.
Note that it is possible for one syncopated section to follow another (e.g., ♪♩♩♪♪♩♩♩♪). But if you concatenate two unsyncopated sections, you just get a single longer unsyncopated section (e.g., ♩♩ + ♩♩♩ = ♩♩♩♩♩). This is exactly analogous to the unrestricted rhythms in the question! In that case, you can play two notes in sequence, but if you "play" two rests in sequence, you just get a longer rest.
So our bijection will work as follows: given a sequence of eighth notes and quarter notes, we replace each syncopated section with a single note of the same length, and each unsyncopated section by a rest of the same length. Going in the other direction, if we have a sequence of notes and rests, all of which are some integer number of quarter-notes long, replace each note with a syncopated section of the same length, and each rest with an unsyncopated section of each length. These are clearly well-defined and inverse to each other, which completes the proof.
For concreteness, here are the $n=2$ examples of this bijection (quarter notes and eighth notes in the first measure of each line, longer notes and rests in the second measure):





and a longer example: