Grasping the Jaki Liebezeit Set

Working with the Belgian composer Simon Halsberghe to gain a better understanding of the Jaki Liebezeit Set. The Jaki Liebezeit set $J$ is the union of its subsets $J_n$, where each $J_n$ is the set of all possible rhythmical patterns consisting of 1s and 2s (represented as . and - further down) of total length $n$, up to cyclic permutation:

$$ J := \bigcup_{n \in \mathbb{N}} J_n $$

$$ J_n := \left\{ a_0a_1\dots a_{k} \mid a_j \in \{ 1, 2 \}, k \in \mathbb{N} \text{ and } \sum_{i=0}^k a_i = n \right\}_{/ \mathord{\sim}}$$

where

$$ a_0a_1\dots a_k \sim b_0b_1 \dots b_k $$ $$ \iff $$ $$ \exists l \in \mathbb{N} \quad \forall j \in \{0,1, \dots, k\}: \qquad a_{(j + l)\mod (k+1)} = b_j$$

Empirically, we have found the following progression and are still looking for a direct way of computing the cardinality of each subset $J_n$. Please contact us if you recognize the set and know more about it!

$n=1$, number of patterns: 1

.

$n=2$, number of patterns: 2

-
..

$n=3$, number of patterns: 2

.-
...

$n=4$, number of patterns: 3

--
..-
....

$n=5$, number of patterns: 3

.--
...-
.....

$n=6$, number of patterns: 5

---
..--
.-.-
....-
......

$n=7$, number of patterns: 5

.---
...--
..-.-
.....-
.......

$n=8$, number of patterns: 8

----
..---
.-.--
....--
...-.-
..-..-
......-
........

$n=9$, number of patterns: 10

.----
...---
..-.--
.-..--
.....--
.-.-.-
....-.-
...-..-
.......-
.........

$n=10$, number of patterns: 15

-----
..----
.-.---
....---
.--.--
...-.--
..-..--
.-...--
......--
..-.-.-
.....-.-
....-..-
...-...-
........-
..........

$n=11$, number of patterns: 19

.-----
...----
..-.---
.-..---
.....---
..--.--
.-.-.--
....-.--
...-..--
..-...--
.-....--
.......--
...-.-.-
..-..-.-
......-.-
.....-..-
....-...-
.........-
...........

$n=12$, number of patterns: 31

------
..-----
.-.----
....----
.--.---
...-.---
..-..---
.-...---
......---
...--.--
..-.-.--
.-..-.--
.....-.--
..--..--
.-.-..--
....-..--
...-...--
..-....--
.-.....--
........--
.-.-.-.-
....-.-.-
...-..-.-
..-...-.-
.......-.-
..-..-..-
......-..-
.....-...-
....-....-
..........-
............

$n=13$, number of patterns: 41

.------
...-----
..-.----
.-..----
.....----
..--.---
.-.-.---
....-.---
.--..---
...-..---
..-...---
.-....---
.......---
.-.--.--
....--.--
...-.-.--
..-..-.--
.-...-.--
......-.--
...--..--
..-.-..--
.-..-..--
.....-..--
.-.-...--
....-...--
...-....--
..-.....--
.-......--
.........--
..-.-.-.-
.....-.-.-
....-..-.-
...-...-.-
..-....-.-
........-.-
...-..-..-
.......-..-
......-...-
.....-....-
...........-
.............

$n=14$, number of patterns: 64

$n=15$, number of patterns: 94

$n=16$, number of patterns: 143

$n=17$, number of patterns: 211

$n=18$, number of patterns: 329

$n=19$, number of patterns: 493

$n=20$, number of patterns: 766

$n=21$, number of patterns: 1170

$n=22$, number of patterns: 1811

$n=23$, number of patterns: 2787

$n=24$, number of patterns: 4341

$n=25$, number of patterns: 6713