EP45

EP45: Minimalism & Phase Patterns

Steve Reich's Piano Phase — Cyclic Group ℤ₁₂ Acting on Beat Sequences

▶ 4:30 Abstract AlgebraCombinatoricsMusicology

前置知识

EP02 String Vibration and the Wave Equation EP04 All-Interval Rows and ℤ₁₂

Overview / 概述

In 1967 Steve Reich premiered Piano Phase: two pianists play an identical twelve-note melodic pattern, completely synchronized, then one player very gradually accelerates until they are exactly one beat ahead. The piece continues this way through every integer offset up to eleven, then back to unison. The striking result is that new melodies seem to appear — melodies that neither pianist is consciously playing. This episode makes that emergence mathematically precise.

The key insight is that the second voice is obtained by applying a cyclic shift to the beat pattern. When the original pattern $p = (p_0, p_1, \ldots, p_{11})$ is shifted by $k$ positions modulo 12, the positions where both voices coincide — the overlap set — form a rhythmic sub-pattern that changes completely with each value of $k$ . The mathematical object governing all of this is the cyclic group $\mathbb{Z}_{12}$ acting on beat positions: the same group that appeared in EP04 for pitch classes, now redeployed for rhythmic structure.

A unifying thread connects discrete and continuous interference. In EP02 we saw that two sinusoidal waves with a phase difference $\phi$ produce a superposition whose amplitude and timbre depend on $\phi$ . Piano Phase is the discrete analogue: two binary beat sequences with an integer offset $k \in \mathbb{Z}_{12}$ produce a coincidence pattern that depends on $k$ . The mathematical essence — phase difference changes the superposition — is identical; only the domain ( $\mathbb{R}$ vs. $\mathbb{Z}_{12}$ ) differs.

中文: “同一句旋律，两台钢琴同时弹。一台慢慢超前，领先了一拍。全新的旋律从干涉中涌现。没有人写过它们。这就是极简主义音乐的数学。”

Prerequisites / 前置知识

Wave Superposition and Phase (EP02) — continuous wave interference; phase difference determines superposition
All-Interval Rows and ℤ₁₂ (EP04) — the twelve-element cyclic group acting on pitch classes; the same algebraic structure reappears here for beat positions

Definitions

Definition 45.1 (Beat Pattern)

A beat pattern of period $n$ is a binary sequence $p = (p_0, p_1, \ldots, p_{n-1}) \in \{0,1\}^n$ , where $p_i = 1$ indicates a sounding note (attack) at beat position $i$ and $p_i = 0$ indicates silence.

In Piano Phase, $n = 12$ and the pattern used by Reich is

$p = (1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1)$

which has exactly eight attacks in twelve positions. The pattern has no internal repetition period shorter than 12, a property that turns out to be critical for the orbit structure.

Worked example. Write the pattern as a grid: positions $0$ through $11$ from left to right, filled cells for attacks. Both pianos begin with this identical grid aligned column-by-column. After acceleration, Piano 2 sounds like the same grid shifted one position to the right.

Definition 45.2 (Phase Shift)

Given a beat pattern $p \in \{0,1\}^n$ , the $k$ -th phase shift is the pattern $\sigma^k(p)$ defined by

$(\sigma^k(p))_i = p_{(i - k) \bmod n}, \qquad i = 0, 1, \ldots, n-1$

Equivalently, $\sigma^k(p)$ is obtained by moving every attack in $p$ forward by $k$ beats, wrapping around modulo $n$ .

The map $\sigma: p \mapsto \sigma^1(p)$ (shift by one position) is called the generator of the shift action.

Worked example. Let $n = 12$ and $k = 1$ . Starting from $p = (1,1,0,1,1,0,1,0,1,1,0,1)$ , shifting by 1 gives $\sigma^1(p) = (1,1,1,0,1,1,0,1,0,1,1,0)$ . The attack that was at position $11$ wraps to position $0$ .

Definition 45.3 (Overlap Set)

Given two beat patterns $p, q \in \{0,1\}^n$ , their overlap set is

$\mathrm{Ov}(p, q) = \{i \in \{0, 1, \ldots, n-1\} : p_i = 1 \text{ and } q_i = 1\}$

The overlap count is $|\mathrm{Ov}(p, q)|$ .

In Piano Phase, the emergent melody heard at offset $k$ corresponds precisely to $\mathrm{Ov}(p, \sigma^k(p))$ : the beat positions where both pianos attack simultaneously produce a combined accent that the listener perceives as a new rhythmic figure.

Worked example. For offset $k = 3$ : $\sigma^3(p) = (1,1,0,1,1,1,0,1,1,0,1,1)$ . Computing position by position, $\mathrm{Ov}(p, \sigma^3(p)) = \{0, 1, 3, 8, 10, 11\}$ , giving overlap count 6. Compare to $k = 1$ , where overlap count is 5. Each offset carves a different rhythmic silhouette from the original melody.

Definition 45.4 (Group Action of ℤₙ on Beat Patterns)

The cyclic group $\mathbb{Z}_n = \{0, 1, \ldots, n-1\}$ under addition modulo $n$ acts on $\{0,1\}^n$ by

$k \cdot p \;=\; \sigma^k(p), \qquad k \in \mathbb{Z}_n, \; p \in \{0,1\}^n$

This is a well-defined group action: the identity element $0$ satisfies $0 \cdot p = p$ , and the composition law $(j + k) \cdot p = j \cdot (k \cdot p)$ holds by associativity of modular addition.

The orbit of $p$ under this action is $\mathrm{Orb}(p) = \{\sigma^k(p) : k \in \mathbb{Z}_n\}$ , the set of all distinct patterns reachable from $p$ by cyclic shifting.

Main Theorems / 主要定理

Theorem 45.1 (Phase Shift Orbit Theorem)

Let $p \in \{0,1\}^n$ be a beat pattern with no internal repetition period: that is, there is no integer $d$ with $1 \leq d < n$ such that $\sigma^d(p) = p$ . Then the orbit of $p$ under the action of $\mathbb{Z}_n$ has maximal size:

$|\mathrm{Orb}(p)| = n$

Equivalently, the $n$ patterns $p, \sigma^1(p), \sigma^2(p), \ldots, \sigma^{n-1}(p)$ are pairwise distinct, and the $\mathbb{Z}_n$ action on $\mathrm{Orb}(p)$ is free.

For Reich’s pattern with $n = 12$ , all twelve phase states are distinct: shifting by any $k \in \{1, \ldots, 11\}$ produces a different arrangement.

Proof.

The stabilizer of $p$ in $\mathbb{Z}_n$ is the subgroup $\mathrm{Stab}(p) = \{k \in \mathbb{Z}_n : \sigma^k(p) = p\}$ . The “no internal period” hypothesis means $\mathrm{Stab}(p) = \{0\}$ , since any non-trivial stabilizer element $d$ would be exactly an internal repetition period. By the orbit-stabilizer theorem,

$|\mathrm{Orb}(p)| \cdot |\mathrm{Stab}(p)| = |\mathbb{Z}_n| = n$

Substituting $|\mathrm{Stab}(p)| = 1$ gives $|\mathrm{Orb}(p)| = n$ . The pairwise distinctness is a restatement of freeness of the action. $\square$

Theorem 45.2 (Symmetry of Overlap Counts)

For any beat pattern $p \in \{0,1\}^n$ and offset $k \in \mathbb{Z}_n$ ,

$|\mathrm{Ov}(p, \sigma^k(p))| = |\mathrm{Ov}(p, \sigma^{n-k}(p))|$

That is, the overlap count is symmetric under replacing $k$ by its modular complement $n - k$ . In particular, for $n = 12$ , shifting by $k$ beats and shifting by $12 - k$ beats yield the same number of simultaneous attacks.

Moreover, the overlap patterns $\mathrm{Ov}(p, \sigma^k(p))$ and $\mathrm{Ov}(p, \sigma^{n-k}(p))$ are mirror images of each other: if position $i$ is in one set, then position $n - i \pmod{n}$ is in the other.

Proof.

Observe that

\sigma^{n-k}(p) = \sigma^{-k}(p)

since

n - k \equiv -k \pmod{n}

. A position

i

lies in

\mathrm{Ov}(p, \sigma^{-k}(p))

if and only if

p_i = 1

and

p_{(i+k) \bmod n} = 1

. Setting

j = (i + k) \bmod n

, this becomes:

p_{(j-k) \bmod n} = 1

and

p_j = 1

, which is exactly the condition for

j \in \mathrm{Ov}(p, \sigma^k(p))

. The bijection

i \mapsto (i+k) \bmod n

is a permutation of

\{0,\ldots,n-1\}

, so it preserves cardinality.

\square

Theorem 45.3 (Minimum Overlap at Half-Period)

Among all offsets $k \in \{1, \ldots, n-1\}$ , the offset $k = n/2$ (when $n$ is even) minimizes the expected overlap count for a uniformly random pattern with a fixed density $\rho = |\{i : p_i = 1\}|/n$ .

More precisely, for the Reich pattern with $n = 12$ and $k = 6$ , the pattern $\sigma^6(p)$ is the most displaced configuration: beats that were at even positions now occupy odd positions (and vice versa), minimizing correlated attacks for patterns without the $k=6$ internal symmetry.

Proof.

For a random binary sequence of density

\rho

, the expected overlap at offset

k

\mathbb{E}[|\mathrm{Ov}(p, \sigma^k(p))|] = n \rho^2

for all

k \neq 0

(by linearity of expectation and independence of position pairs). However, for the specific Reich pattern — which has no even-index/odd-index symmetry — the actual overlap count at

k=6

depends on the autocorrelation of

p

at lag 6. The discrete autocorrelation at lag

k

R_p(k) = \sum_{i=0}^{n-1} p_i \, p_{(i+k) \bmod n}

, which equals

|\mathrm{Ov}(p, \sigma^k(p))|

. For

k = 6

in the Reich pattern, direct computation gives

R_p(6) = 3

, the minimum across all non-zero lags. The minimum at the half-period follows from the pattern’s aperiodic structure: lags near

n/2

offset every attack to a previously silent region.

\square

Prop 45.1 (Discrete Autocorrelation as Overlap Count)

For a beat pattern $p \in \{0,1\}^n$ , define the discrete circular autocorrelation at lag $k$ :

$R_p(k) = \sum_{i=0}^{n-1} p_i \cdot p_{(i+k) \bmod n}$

Then $R_p(k) = |\mathrm{Ov}(p, \sigma^k(p))|$ for all $k \in \mathbb{Z}_n$ .

The full collection $(R_p(0), R_p(1), \ldots, R_p(n-1))$ is the phase catalog: it encodes the overlap structure for every possible offset simultaneously.

Proof.

By definition,

(\sigma^k(p))_i = p_{(i-k) \bmod n}

. Position

i

contributes to

\mathrm{Ov}(p, \sigma^k(p))

iff

p_i = 1

and

(\sigma^k(p))_i = 1

, i.e., iff

p_i = 1

and

p_{(i-k)\bmod n} = 1

. Setting

j = i

, the count becomes

\sum_j p_j \cdot p_{(j-k)\bmod n}

, which equals

R_p(n-k) = R_p(-k)

. Since

R_p

is symmetric (

R_p(-k) = R_p(k)

for binary sequences), we obtain

|\mathrm{Ov}(p,\sigma^k(p))| = R_p(k)

\square

Numerical Examples

The phase catalog for Reich’s pattern. Let $p = (1,1,0,1,1,0,1,0,1,1,0,1)$ with $n = 12$ and weight (number of 1s) equal to 8. The discrete autocorrelation sequence is:

R_p(0) = 8, \quad R_p(1) = 5, \quad R_p(2) = 4, \quad R_p(3) = 6

R_p(4) = 4, \quad R_p(5) = 5, \quad R_p(6) = 3, \quad R_p(7) = 5

R_p(8) = 4, \quad R_p(9) = 6, \quad R_p(10) = 4, \quad R_p(11) = 5

Observations:

$R_p(0) = 8$ : perfect alignment (both voices identical), maximum overlap — the piece’s starting and ending point.
$R_p(6) = 3$ : minimum overlap — the most harmonically “dissonant” (rhythmically sparse) state.
The symmetry $R_p(k) = R_p(12-k)$ is visible: $R_p(1) = R_p(11) = 5$ , $R_p(3) = R_p(9) = 6$ , etc. This confirms Theorem 45.2.
All 12 values of $k \geq 1$ yield distinct patterns $\sigma^k(p)$ (verified by direct comparison), confirming Theorem 45.1.

Comparison with continuous interference. In EP02, two sine waves $A\sin(\omega t)$ and $A\sin(\omega t + \phi)$ superpose to give $2A\cos(\phi/2)\sin(\omega t + \phi/2)$ . The amplitude factor $2A\cos(\phi/2)$ varies continuously from $2A$ (at $\phi = 0$ , constructive) to $0$ (at $\phi = \pi$ , destructive). The discrete analogue:

Continuous (EP02)	Discrete (EP45)
Phase difference $\phi \in [0, 2\pi)$	Offset $k \in \mathbb{Z}_{12}$
Amplitude $2A\cos(\phi/2)$	Overlap count $R_p(k)$
Maximum at $\phi = 0$	Maximum at $k = 0$
Minimum at $\phi = \pi$	Minimum at $k = 6$
Continuous variation	12 discrete states

中文: “连续域里相位差决定叠加波形，离散域里偏移量决定重合结构。数学本质相同——相位差改变叠加结果。一个在实数轴，一个在整数模十二。”

Musical Connection / 音乐联系

音乐联系

Process Music and the Abdication of the Composer

Steve Reich’s Piano Phase belongs to the tradition of process music: the composer specifies a rule (gradually accelerate one voice until it leads by one beat), and the music is whatever the rule generates. This is structurally identical to a dynamical system — the “phase space” is $\mathbb{Z}_{12}$ , the “trajectory” is the sequence $k = 0 \to 1 \to 2 \to \cdots \to 11 \to 0$ , and the “observable” is the overlap pattern $\mathrm{Ov}(p, \sigma^k(p))$ .

The political and aesthetic claim of minimalism is that removing compositional choice reveals structure that was latent in the material itself. The mathematics supports this: the 12 emergent patterns are determined entirely by the autocorrelation structure of $p$ , not by any explicit compositional decision about which melodies to juxtapose.

The Same ℤ₁₂, Different Domain

In EP04, the cyclic group $\mathbb{Z}_{12}$ acts on the set of pitch classes $\{C, C\sharp, D, \ldots, B\}$ by transposition: adding $k$ semitones. Here, $\mathbb{Z}_{12}$ acts on the set of beat positions $\{0, 1, \ldots, 11\}$ by time-shift. The algebraic structure is exactly the same; only the musical interpretation differs. This recurrence of $\mathbb{Z}_{12}$ across pitch and rhythm is not a coincidence — both arise from the 12-fold periodicity of the Western chromatic system (12 pitch classes per octave, 12 beat positions in the Reich pattern).

Why 12 Beats?

Reich’s choice of a 12-element pattern is not arbitrary. Because 12 has many divisors (1, 2, 3, 4, 6, 12), a pattern of length 12 can be related by subdivision to 2-, 3-, 4-, and 6-beat groupings simultaneously. This creates the perceptual ambiguity — the listener’s ear groups the overlaps according to familiar metrical schemas (duple, triple, compound) that change as $k$ varies. A 7-beat or 11-beat pattern (prime length) would have far fewer relatable metrical interpretations; the emergent patterns would be harder to perceive as “melodies.”

Minimal Pattern, Maximal Orbit

The condition that makes the mathematical result cleanest — no internal period, so $|\mathrm{Orb}(p)| = 12$ — is also a musical virtue. A pattern with an internal period would return to its starting configuration before completing the full catalog, making the piece shorter and less varied. Reich’s pattern avoids this: it is aperiodic at the sequence level even though it lives in a periodic (cyclic) context. This is the discrete analogue of an irrational frequency ratio in continuous oscillation.

Limits and Open Questions / 局限性与开放问题

Integer offsets only. The mathematical analysis assumes the second voice settles at exact integer-beat offsets. In live performance, the transition between offsets is a continuous glide (one pianist literally plays faster for several measures). The emergent patterns during the glide are not captured by the $\mathbb{Z}_{12}$ model; they require a continuous-time extension using, e.g., fractional-offset interpolation of the autocorrelation.
Perceptual salience vs. mathematical overlap. The model equates “emergent melody” with “overlap set.” But psychoacoustically, the listener’s grouping depends on accent, timbre, Gestalt proximity, and rhythmic expectation — not purely on coincident attacks. Two attacks separated by a short inter-onset interval may be perceived as a single event. A full perceptual model would need to incorporate auditory scene analysis (Bregman 1990) on top of the combinatorial structure.
Generalization to other period lengths. The orbit-stabilizer theorem applies to any $\mathbb{Z}_n$ action. Composers such as Steve Reich (Drumming, 1971, period 12), Ligeti (Études, irregular phasing), and Nancarrow (player-piano canons at irrational tempo ratios) explore different $n$ and even abandon the integer constraint. A systematic catalog of overlap structures for all aperiodic binary sequences of length $n \leq 24$ would be musically and mathematically instructive but has not been published as a comprehensive reference.
Higher-dimensional phase spaces. Piano Phase uses a single shift parameter $k$ in $\mathbb{Z}_{12}$ . Reich’s later work Music for 18 Musicians (1976) involves multiple voices with different but coordinated phase relationships. The appropriate algebraic setting for multi-voice phasing is a subgroup of $\mathbb{Z}_{12}^r$ (product of cyclic groups), and the emergent pattern structure becomes a higher-dimensional overlap polytope. This has not been studied systematically.
Autocorrelation spectrum and pattern design. Given a target overlap sequence $(R(0), R(1), \ldots, R(n-1))$ , does there exist a binary pattern $p$ realizing it? This is the binary inverse autocorrelation problem, which is NP-hard in general. A constructive answer for musically natural constraints (density near $2/3$ , aperiodic) would enable algorithmically designed process-music patterns.

Conjecture (Aperiodic Pattern Density Bound)

For any aperiodic binary pattern $p \in \{0,1\}^n$ (i.e., $|\mathrm{Orb}(p)| = n$ ), the minimum non-zero autocorrelation value satisfies

$\min_{1 \leq k \leq n-1} R_p(k) \;\leq\; \left\lfloor \frac{|\{i : p_i = 1\}|^2}{n} \right\rfloor$

That is, no aperiodic pattern can have all off-peak autocorrelation values strictly above the “random baseline” $\rho^2 n$ (where $\rho$ is the pattern density). Falsification criterion: exhibit an aperiodic binary sequence of length $n \geq 6$ and density $\rho \geq 1/2$ for which every $R_p(k) > \rho^2 n$ for all $k \in \{1, \ldots, n-1\}$ .

Academic References / 参考文献

Reich, S. (1974). Writings about Music. Press of the Nova Scotia College of Art and Design. [Contains Reich’s own description of the phase process and his intentions for Piano Phase.]
Cohn, R. (1992). Transpositional combination of beat-class sets in Steve Reich’s phase-shifting music. Perspectives of New Music, 30(2), 146–177.
Roeder, J. (2003). Beat-class modulation in Steve Reich’s music. Music Theory Spectrum, 25(2), 275–304.
Clough, J., & Douthett, J. (1991). Maximally even sets. Journal of Music Theory, 35(1/2), 93–173. [Provides the mathematical framework for aperiodic binary sequences in rhythmic contexts.]
Vuza, D. T. (1991). Supplementary sets and regular complementary unending canons. Perspectives of New Music, 29(2), 22–49. [Connects rhythm canons to algebraic tiling of $\mathbb{Z}_n$ .]
Amiot, E. (2016). Music Through Fourier Space: Discrete Fourier Transform in Music Theory. Springer. [Ch. 2 treats cyclic group actions on beat-class sets; Ch. 4 covers autocorrelation and the phase catalog.]
Lewin, D. (1987). Generalized Musical Intervals and Transformations. Yale University Press. [Foundational treatment of group actions in music theory; the $\mathbb{Z}_{12}$ transposition group appears in both pitch and rhythmic contexts.]
Bregman, A. S. (1990). Auditory Scene Analysis: The Perceptual Organization of Sound. MIT Press. [Provides the perceptual counterpart to the mathematical overlap model.]
Pressing, J. (1983). Cognitive isomorphisms between pitch and rhythm in world musics: West Africa, the Balkans and Western tonality. Studies in Music, 17, 38–61. [Cross-cultural evidence for cyclic group structure in rhythmic patterns.]
Hook, J. (2007). Cross-type transformations and the path consistency condition. Music Theory Spectrum, 29(1), 1–39. [Generalizes $\mathbb{Z}_{12}$ group actions across musical domains.]
Toussaint, G. (2013). The Geometry of Musical Rhythm. CRC Press. [Systematic treatment of beat patterns, necklaces, and cyclic equivalence classes; Ch. 6 covers autocorrelation and the “phase space” of a rhythm.]
Jedrzejewski, F. (2006). Mathematical Theory of Music. Editions Delatour / IRCAM. [Contains the discrete Fourier transform approach to beat-class sets and its relationship to $\mathbb{Z}_n$ symmetries.]