Maximally Even Rhythms

• Maximally even rhythms are evenly distributed musical patterns characterized by two distinct interval values that ensure uniformity and deep structure.
• They are constructed using methods like Bjorklund’s Euclidean algorithm, Clough–Douthett construction, and iterative averaging techniques to achieve fixed or periodic patterns.
• These rhythms underpin diverse musical traditions and inform algorithmic approaches in rhythm analysis and scale construction across global music genres.

Maximally even rhythms are distinguished patterns in the geometry of musical rhythms and scales whose onsets (active pulses) are distributed as evenly as possible around a discrete cyclic time grid. These rhythms maximize specific notions of evenness, including uniqueness (up to rotation), uniformity of interval distribution, and deepness properties. Maximally even rhythms—often realized as Euclidean rhythms—underpin both traditional timelines in world music and mathematical characterizations of scale construction. Their structural properties are central in both rhythmical analysis and algorithmic generation of patterns (1711.01734, 0705.4085).

1. Formal Definition and Representations

Consider integers $N \geq 3$ and $0 < n < N$. A rhythm of length $N$ with $n$ onsets can be described equivalently in three primary ways:

• As a binary sequence $$r \in CR_N^{(n)} = \{ b \in \{0,1\}^N \mid \sum_{i=0}^{N-1} b_i = n \}$$, where $1$ indicates an onset.
• As a set $P = \{P_0,\ldots,P_{n-1}\} \subset \mu_N$, where $\mu_N$ is the set of $N$th roots of unity on the complex unit circle, identifying each onset with $e^{2\pi i k/N}$ for some $k$.
• As an onset index list $a = (a_0, a_1, \ldots, a_{n-1}) \in \mathbb{Z}_N^n$ in increasing cyclic order. The onset list satisfies the cyclic ascent property: exactly one "jump" $k_0$ where $a_{k_0} > a_{k_0+1}$ (indices mod $n$), and $a_k < a_{k+1}$ otherwise.

A rhythm (or its difference-sequence) is called maximally even if its $n$ inter-onset spacings, when measured modulo $N$, take only the two integer values $\lfloor N/n \rfloor$ and $\lceil N/n \rceil$. Concretely, defining $\operatorname{diff}(a) = (d_0, \ldots, d_{n-1}),$ with $d_k = a_{k+1} -_N a_k$, maximal evenness holds if $\max(d_k) - \min(d_k) \leq 1$ (1711.01734).

2. Evenness Measures and Optimization

Evenness can be rigorously quantified using chordal distances. Embedding onsets on the unit circle, map each $\theta$ to $e^{2\pi i\theta/n}$. The chordal distance between pairs is $$d(\theta_i, \theta_j) = |e^{2\pi i\theta_i/n} - e^{2\pi i\theta_j/n}|$$, and total evenness is

$$E(\theta_1,\dots,\theta_k) = \sum_{1 \le i < j \le k} d(\theta_i, \theta_j).$$

Maximally even (Euclidean) rhythms uniquely (up to rotation) maximize the total evenness $E$ for fixed $N$ and $n$. Any deviation from the maximally even pattern decreases the sum of chord lengths (0705.4085).

These patterns also satisfy an interval-balancing property:

$$(r_i, r_{i+\ell}) \in \left\{ \left\lfloor \frac{\ell N}{n} \right\rfloor, \left\lceil \frac{\ell N}{n} \right\rceil \right\}, \quad \text{for all } i,\, \ell=1,\dots,n.$$

3. Algorithmic Constructions

Multiple equivalent constructions lead to the maximally even rhythm for given $N$ and $n$:

• Bjorklund’s Euclidean Algorithm: Recursively partitions $N$ slots between $n$ onsets via an algorithm mirroring Euclid’s gcd computation. When $n \mid N$, all inter-onset distances are equal; otherwise, recursive distribution leads to only two distinct gap lengths (0705.4085).
• Clough–Douthett Construction: The onset indices are given by $\{\lfloor i N/n\rfloor: i=0, \ldots, n-1 \}$.
• Snap Heuristic: Position $n$ ideal (fractional) onsets evenly spaced and map them to the nearest lattice points mod $N$.

A practical iterative averaging algorithm exists: given a rhythm $a = (a_0, \ldots, a_{n-1})$, define the discrete average $$\operatorname{av}_{\mathbb{Z}_N}(a,b) = a +_N \lfloor (-a +_N b)/2 \rfloor$$, and update each $$a'_k = \operatorname{av}_{\mathbb{Z}_N}(a_k, a_{k+1})$$. Iterate until the difference sequence width $\max(d_k) - \min(d_k) \leq 1$. This process converges for any initial rhythm, producing a maximally even pattern after finitely many steps (1711.01734).

4. Classification, Uniqueness, and Periodicity

The fixed points of the averaging map are precisely the maximally even rhythms. If $\operatorname{width}(\operatorname{diff}(a)) = 0$, then the rhythm is the regular polygon. If $\operatorname{width}(\operatorname{diff}(a)) = 1$, the resulting orbit is periodic with period $n$. The limit of the iterative process is unique up to cyclic rotation (for periodic width-1 cases) or trivial in the regular case.

Correspondence with Toussaint’s geometric perspective asserts that maximally even rhythms maximize the minimum inter-onset distance in a uniform way, connecting physical spacing and combinatorial criteria (1711.01734, 0705.4085).

5. Deep Rhythms and Distance Multiplicities

Maximally even (Euclidean) rhythms exhibit the property of deepness. For $R = \{r_0, r_1, \ldots, r_{n-1}\}_N$, consider all pairwise nonzero circular distances (shorter arcs).

Erdős-deep: Distances occur with multiplicities $1,2,\ldots,n-1$. That is, among possible distances, exactly one occurs once, one occurs twice, and so forth. Winograd-deep: Each arc length in $\{1,\ldots,\lfloor N/2\rfloor\}$ occurs with distinct multiplicity.

All Winograd-deep rhythms are Erdős-deep but not conversely. Generated rhythms of the form $D_{n,N,m}=\{0,m,2m,\ldots,(n-1)m\}_N$ with $\gcd(m,N)=1$ and $n \le \lfloor N/2 \rfloor+1$ exhibit these deepness properties. Maximally even rhythms correspond exactly to the Erdős-deep generated class, plus the exceptional hexatonic set $F=\{0,1,2,4\}_6$ (0705.4085).

These patterns admit the shelling property: there exists an ordering of onsets so that at every removal, the subset remains deep.

6. Examples and Applications in World Music and Scales

Maximally even and deep rhythms are prevalent across world music and cyclic scales:

• $E(3,8)$ (Tresillo/Habanera): $[x\ .\ .\ x\ .\ .\ x\ .]$; distances 2 and 3.
• $E(5,8)$ (Cinquillo): $[x\ .\ x\ x\ .\ x\ x\ .]$.
• $E(5,16)$ (Bossa-Nova bell): $[x\ .\ .\ x\ .\ .\ x\ .\ .\ x\ .\ .\ .\ .\ .]$.
• $E(7,12)$ (Diatonic major scale): $[0,2,4,5,7,9,11] \subset \mathbb{Z}_{12}$.

There are over forty documented traditional timelines spanning Africa, the Balkans, India, the Middle East, and the Americas that instantiate the Euclidean (maximally even) construction (0705.4085).

These constructions apply verbatim to pitch-class sets, leading to maximally even scales and deep scales characterized by generated sets or the exceptional hexatonic subset.

7. Summary of Theoretical and Algorithmic Results

Maximally even rhythms—uniquely determined (up to rotation) by the maximization of chordal evenness and characterized by the two-value distance criterion—can be constructed via Euclidean algorithms or iterative averaging dynamics. Their deepness properties connect interval multiplicities to structural uniformity. All maximally even rhythms, and deep rhythms more broadly, can be classified as generated sets (plus the small exceptional case), and all maximally even rhythms arise as fixed or periodic orbits of the iterative averaging procedure. The geometric, combinatorial, and algorithmic approaches present a unified theory for the generation and analysis of these patterns in both rhythm and scale theory (1711.01734, 0705.4085).