A three-year-old cannot put her shoes on before her socks. Not because she lacks motor skills — because the operations do not commute.

The same structural constraint, dressed in the language of operators on a Hilbert space, is why Heisenberg’s uncertainty principle holds. This post is about that connection: the accidental algebra lesson built into getting dressed, and why the physicists of 1925 had to abandon one of arithmetic’s most taken-for-granted assumptions.

Getting Dressed Is a Non-Abelian Problem

Start with the mundane. Your morning routine imposes a strict partial order on operations: underwear before trousers, socks before shoes, cap before chin-strap if you cycle. Try reversing any pair and the sequence fails — physically, not just socially. You cannot pull a sock over a shoe.

The operation “put on socks” followed by “put on shoes” produces a wearable human; the reverse produces neither, and no amount of wishing commutativity into existence will help.

In the language of abstract algebra, two operations \(A\) and \(B\) commute if \(AB = BA\) — if doing them in either order yields the same result. Everyday life is full of operations that do not commute: rotate a book 90° around its vertical axis then 90° around its horizontal axis; now reverse the order. The final orientations differ. Turn right then turn left while driving; left then right. Different positions.

The intuition is not hard to build. What is surprising is how rarely we note it, and what it costs us when we finally hit a domain — quantum mechanics — where non-commutativity is not an inconvenient edge case but the central fact.

Piaget Said Seven; Toddlers Disagreed

Jean Piaget argued that children do not acquire operational thinking — the ability to mentally perform and reverse sequences of actions — until the concrete operational stage, roughly ages seven to eleven (Inhelder & Piaget, 1958). Before that, he claimed, children lack the understanding that an operation can be undone or reordered.

Post-Piagetian research pushed back hard. Patricia Bauer and Jean Mandler tested infants aged sixteen and twenty months on novel, multi-step action sequences (Bauer & Mandler, 1989). For causally structured sequences — where step A physically enables step B — infants reproduced the correct order after a two-week delay. They were not told the order was important. They had no language to encode it. They just knew, implicitly, that the operations had a necessary direction.

A 2020 study by Klemfuss and colleagues tested 100 children aged roughly two-and-a-half to five on temporal ordering questions (Klemfuss et al., 2020). Children answered “what happened first?” questions correctly 82% of the time. The errors that did appear followed an encoding-order bias — children defaulted to reporting the next event in the sequence as originally experienced, regardless of what was asked. The ordering knowledge was intact. What children lack, for Piaget’s full seven years, is the formal recursive conception of reversibility. The procedural knowledge — that some sequences must be done in the right order and cannot be freely rearranged — is there from the second year of life.

Which means: learning that \(AB \neq BA\) is not learning something exotic. It is articulating something the nervous system already knows.

The Mathematician’s Commutator

Abstract algebra formalized this intuition in the nineteenth century. A group is abelian (commutative) if every pair of elements satisfies \(ab = ba\). Integers under addition: abelian. Rotations in three dimensions: not.

Arthur Cayley’s 1858 memoir established matrix algebra as a formal theory (Cayley, 1858). Multiply two \(2 \times 2\) matrices:

\(AB \neq BA\). Non-commutativity is not a curiosity; it is the generic condition for matrix products. Commutativity is the special case — and requiring justification.

William Rowan Hamilton had already gone further. On 16 October 1843, walking along the Royal Canal in Dublin, he discovered the quaternions and carved their multiplication rule into the stone of Broom Bridge:

From this it follows immediately that \(ij = k\) but \(ji = -k\). Hamilton’s four-dimensional number system — the first algebraic structure beyond the complex numbers — was non-commutative by construction. He did not apologize for it. He celebrated it.

The Lie algebra structure underlying these commutator relations is the same skeleton that governs Messiaen’s modes of limited transposition, which I traced in a previous post on group theory and music — a very different physical domain, but identical algebraic machinery.

Born, Jordan, and the Physicist’s Shock

Classical mechanics treats position \(x\) and momentum \(p\) as ordinary real numbers. Real numbers commute: \(xp = px\). The Poisson bracket \(\{x, p\} = 1\) encodes a classical relationship, but the underlying quantities are scalars, and scalars commute.

In July 1925, Werner Heisenberg published a paper that could not quite bring itself to say what it was doing (Heisenberg, 1925). He replaced classical dynamical variables with arrays of numbers — what we would now call matrices — and found, uncomfortably, that the resulting quantum condition required order to matter.

While Heisenberg was on vacation, Max Born and Pascual Jordan finished the translation into matrix language (Born & Jordan, 1925). They wrote the commutation relation explicitly, recognized it as the fundamental law, and showed that it reproduced the known quantum results:

Non-commutativity of position and momentum was not a mathematical accident. It was the theory.

The uncertainty principle followed four years later as a theorem, not an additional postulate. Howard Robertson proved in 1929 that for any two observables \(\hat{A}\) and \(\hat{B}\), the Cauchy–Schwarz inequality on Hilbert space yields (Robertson, 1929):

Substituting \(\hat{A} = \hat{x}\), \(\hat{B} = \hat{p}\), \([\hat{x}, \hat{p}] = i\hbar\):

This is the uncertainty principle. It does not say nature is fuzzy or that measurement disturbs systems in some vague intuitive sense. It says: position and momentum are operators that do not commute, and the Robertson inequality then constrains their joint variance. Non-commutativity is the uncertainty principle. Put the shoes on before the socks and the state is not defined.

The same logic applies to angular momentum. The three components satisfy:

This is the Lie algebra \(\mathfrak{su}(2)\). You cannot simultaneously determine two components of angular momentum to arbitrary precision — not because the measurement apparatus is noisy, but because the operations of measuring them do not commute.

The fiber bundle language that underlies these rotation groups also appears, in different physical dress, in the problem of the falling cat and geometric phases — another case where the order of rotations has non-trivial physical consequences (see that post).

Connes and Non-Commutative Space

Alain Connes asked what happens if we allow the coordinates of space itself to be non-commutative. In ordinary geometry, the algebra of coordinate functions on a manifold is commutative: \(f(x) \cdot g(x) = g(x) \cdot f(x)\). Connes’ non-commutative geometry replaces this with a spectral triple \((\mathcal{A}, \mathcal{H}, D)\): an algebra \(\mathcal{A}\) of operators (possibly non-commutative) acting on a Hilbert space \(\mathcal{H}\), with a generalized Dirac operator \(D\) encoding the geometry (Connes, 1994).

The payoff was remarkable. With Ali Chamseddine, Connes showed that if \(\mathcal{A}\) is chosen as a specific non-commutative product of the real numbers, complex numbers, quaternions, and matrix algebras, the spectral action principle reproduces the full Lagrangian of the Standard Model coupled to general relativity from a single geometric principle (Chamseddine & Connes, 1996). The Higgs field, the gauge bosons, the graviton: all from the geometry of a non-commutative space.

Classical geometry is the special case where the coordinate algebra is commutative. Drop that assumption and you open up a vastly richer landscape. Quantum mechanics lives in that landscape. Possibly, so does the structure of spacetime at the Planck scale.

The Lesson Pre-Schoolers Already Know

There is an irony here that I cannot quite leave alone. Students learning linear algebra for the first time consistently make the same mistake. Anna Sierpinska documented it carefully: they assume \(AB = BA\) for matrices because they have spent years in arithmetic and scalar algebra where multiplication commutes (Sierpinska, 2000). The commutativity of ordinary multiplication is so deeply internalized that abandoning it feels like breaking a rule.

But the pre-schooler in the sock-and-shoe scenario never had that problem. Her procedural memory, documented in infants as young as sixteen months by Bauer and Mandler, encoded the correct asymmetry directly. The order of operations is the first thing a developing mind learns about actions in the world, before the arithmetic of school teaches it the convenient fiction that order is irrelevant.

Arithmetic is the outlier. \(3 + 5 = 5 + 3\) because counting does not depend on where you start. But putting on clothes, multiplying matrices, rotating rigid bodies, measuring quantum observables: these operations carry memory of order, and they repay the attention a child already brings to them before she can name a number.

The universe is non-abelian. We are born knowing it. School briefly convinces us otherwise. Physics eventually agrees with the pre-schooler.

References

• Inhelder, B., & Piaget, J. (1958). The Growth of Logical Thinking from Childhood to Adolescence. Basic Books.
• Bauer, P. J., & Mandler, J. M. (1989). One thing follows another: Effects of temporal structure on 1- to 2-year-olds’ recall of events. Developmental Psychology, 25, 197–206.
• Klemfuss, J. Z., McWilliams, K., Henderson, H. M., Olaguez, A. P., & Lyon, T. D. (2020). Order of encoding predicts young children’s responses to sequencing questions. Cognitive Development, 55, 100927. DOI: 10.1016/j.cogdev.2020.100927
• Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17–37. DOI: 10.1098/rstl.1858.0002
• Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, 879–893.
• Born, M., & Jordan, P. (1925). Zur Quantenmechanik. Zeitschrift für Physik, 34, 858–888.
• Robertson, H. P. (1929). The uncertainty principle. Physical Review, 34, 163–164. DOI: 10.1103/PhysRev.34.163
• Connes, A. (1994). Noncommutative Geometry. Academic Press. ISBN 0-12-185860-X.
• Chamseddine, A. H., & Connes, A. (1996). Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Physical Review Letters, 77, 4868–4871. DOI: 10.1103/PhysRevLett.77.4868
• Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 209–246). Kluwer Academic Publishers. DOI: 10.1007/0-306-47224-4_8

Changelog

• 2026-02-03: Corrected the age range for the Klemfuss et al. (2020) study from “two to four” to “roughly two-and-a-half to five” — the actual participants were aged 30–61 months.
• 2026-02-03: Updated the characterisation of Klemfuss et al. (2020) findings to reflect the paper’s central result: errors follow an encoding-order bias (children default to the next event in encoding sequence). The paper’s title — “Order of encoding predicts young children’s responses” — names the mechanism.

This post is a companion to Fibonacci, the Golden Ratio, and Tool’s Lateralus, which found number theory in a prog-rock song. Here we find abstract algebra in twentieth-century sacred music.

Pitch Classes and the Chromatic Clock

Western music divides the octave into twelve equal semitones. For purposes of harmony and counterpoint, the absolute pitch is often less important than the pitch class — the equivalence class of all pitches related by octave transposition. Middle C and the C two octaves above belong to the same pitch class.

We label the twelve pitch classes $0, 1, 2, \ldots, 11$, with $0 = \mathrm{C}$, $1 = \mathrm{C}\sharp/\mathrm{D}\flat$, $2 = \mathrm{D}$, and so on up to $11 = \mathrm{B}$. Addition is taken modulo 12 — the integers wrap around like a clock face, with $11 + 2 = 1$ (one semitone above B is C$\sharp$).

The set of pitch classes with this operation is a group:

This is the cyclic group of order 12. It has an identity element ($0$, “no transposition”), every element has an inverse ($-n \bmod 12$), and the operation is associative. If you are used to thinking about the chromatic scale as a linear sequence ending at the octave, $\mathbb{Z}_{12}$ is the insistence that it is actually a circle.

Musical Operations as Group Elements

Two operations are fundamental in tonal and post-tonal music theory.

Transposition by $n$ semitones maps every pitch class up by $n$:

The twelve transpositions $T_0, T_1, \ldots, T_{11}$ are exactly the elements of $\mathbb{Z}_{12}$, with $T_n$ corresponding to the integer $n$. Composing two transpositions gives a transposition: $T_m \circ T_n = T_{m+n}$.

Inversion reflects the pitch-class circle:

Inversion maps C to C, D to B$\flat$, E to A$\flat$, and so on — it is the mirror symmetry of the chromatic circle about the C/F$\sharp$ axis. Combining inversion with transposition gives the inversional transpositions:

The transpositions and inversional transpositions together generate a group of order 24:

This is the dihedral group $D_{12}$ — the same abstract group that describes the symmetries of a regular 12-gon (twelve rotations and twelve reflections). The identification is not coincidental: the twelve pitch classes arranged in a circle are the vertices of a regular 12-gon, and the musical operations are geometrically the symmetries of that polygon.

Twelve-tone composition — Schoenberg’s method — is almost entirely a working-out of the consequences of $D_{12}$ acting on ordered sequences of the twelve pitch classes. The four canonical row forms (prime, inversion, retrograde, retrograde-inversion) correspond to cosets of $\mathbb{Z}_{12}$ (the transposition subgroup).

Orbits and Stabilisers

Let $S \subseteq \mathbb{Z}_{12}$ be a pitch-class set — a chord, a scale, a collection of any size.

The orbit of $S$ under $\mathbb{Z}_{12}$ is the collection of all distinct transpositions of $S$:

For most sets, all twelve transpositions produce a different set, so $|\mathrm{Orb}(S)| = 12$. The C major scale, for example, has twelve distinct transpositions, one for each key.

But some sets are symmetric under certain transpositions: there exists $n \neq 0$ such that $T_n(S) = S$. The collection of all symmetry transpositions of $S$ is the stabiliser:

Because composing two symmetry transpositions yields another, $\mathrm{Stab}(S)$ is a subgroup of $\mathbb{Z}_{12}$.

The orbit–stabiliser theorem gives the fundamental count:

The number of distinct transpositions of $S$ equals $12$ divided by the number of transpositions that leave $S$ unchanged. The more internally symmetric $S$ is, the fewer new versions you can produce by transposing it.

A set with $|\mathrm{Stab}(S)| > 1$ — one that is invariant under some non-trivial transposition — is a mode of limited transposition.

Mode 1: The Whole-Tone Scale

The whole-tone scale contains the six pitch classes at even intervals:

Transposing by $T_2$:

The set is unchanged. The same holds for $T_4, T_6, T_8, T_{10}$. The stabiliser is the full subgroup of even transpositions:

By the orbit–stabiliser theorem:

There are exactly two distinct whole-tone scales. Every pianist learns this: the one on C and the one on C$\sharp$. Composing with whole-tone harmony means working from a stock of only two harmonic pools with no way to modulate into a genuinely new version of the scale. This is Messiaen’s first charm of impossibility.

Mode 2: The Octatonic Scale

The octatonic (diminished) scale alternates half-step and whole-step intervals. Starting on C:

Does $T_3$ leave this set invariant?

Also $T_6$ and $T_9$. The stabiliser is the subgroup generated by transposition by a minor third:

The orbit size:

There are exactly three distinct octatonic scales. Composers from Rimsky-Korsakov and Bartók to Coltrane have exploited this closed system. The three scales correspond to the three cosets of the subgroup $\langle T_3 \rangle$ in $\mathbb{Z}_{12}$: the cosets $\{0, 3, 6, 9\}$, $\{1, 4, 7, 10\}$, and $\{2, 5, 8, 11\}$ are the “starting-point classes” that generate each scale. Note that the scales themselves are not pairwise disjoint — each has eight pitch classes, so any two share four — but the coset structure determines which transpositions produce the same scale and which produce a different one.

The Subgroup Lattice and All Seven Modes

The orbit–stabiliser theorem constrains which stabiliser sizes are algebraically possible. Since $\mathrm{Stab}(S)$ is a subgroup of $\mathbb{Z}_{12}$, its order must divide 12. The proper non-trivial subgroups of $\mathbb{Z}_{12}$ — those with order strictly between 1 and 12 — are precisely:

These four subgroups exist because the proper divisors of 12 that are greater than 1 are exactly $\{2, 3, 4, 6\}$. The subgroups of $\mathbb{Z}_n$ are in bijection with the divisors of $n$ — a consequence of the fundamental theorem of cyclic groups. Since $12 = 2^2 \times 3$, the proper divisors are $1, 2, 3, 4, 6$.

Each row of the table maps onto a level in Messiaen’s system:

• Mode 1 (whole-tone scale): stabiliser $\langle T_2 \rangle$, 2 transpositions
• Mode 2 (octatonic scale): stabiliser $\langle T_3 \rangle$, 3 transpositions
• Mode 3: stabiliser $\langle T_4 \rangle$, 4 transpositions
• Modes 4 – 7: stabiliser $\langle T_6 \rangle$, 6 transpositions each

The subgroup lattice of $\mathbb{Z}_{12}$ — its Hasse diagram of containment relationships — maps directly onto the hierarchy of Messiaen’s modes. The more symmetric the stabiliser subgroup, the fewer distinct transpositions the mode admits.

The containment relations are: $\langle T_2 \rangle \supset \langle T_4 \rangle$ and $\langle T_2 \rangle \supset \langle T_6 \rangle$ and $\langle T_3 \rangle \supset \langle T_6 \rangle$. Correspondingly, Mode 1 (stabiliser $\langle T_2 \rangle$, order 6) is “more limited” than Mode 3 (stabiliser $\langle T_4 \rangle$, order 3), in the sense that $\langle T_4 \rangle \subset \langle T_2 \rangle$: every symmetry of Mode 3 is also a symmetry of Mode 1’s stabiliser.

Why Exactly Seven Modes?

Messiaen was not enumerating all pitch-class sets with non-trivial stabilisers — there are many more than seven. At the level of the stabiliser $\langle T_6 \rangle$, for example, there are numerous pitch-class sets invariant under the tritone transposition $T_6$: any set $S$ such that $S = S + 6$ qualifies. Some of these sets are large (ten pitch classes), some are small (two pitch classes), some are musically coherent and some are not.

Messiaen selected seven that he found aesthetically and compositionally viable: scales of moderate cardinality, with a balance of interval types, that he could use as raw material for his harmonic language. The group theory explains the constraint (modes are possible only at the four stabiliser types listed above), not the selection (which specific sets Messiaen chose among the many that satisfy the constraint).

The question “why seven?” is therefore partly combinatorial and partly compositional. What is group-theoretically determined is the number of levels (four: orbit sizes 2, 3, 4, 6) and the impossibility of any mode with, say, five distinct transpositions (since 5 does not divide 12).

What Messiaen Knew — and Did Not Know

Messiaen described his modes in Technique de mon langage musical (1944). His account is entirely musical and phenomenological. He lists each mode by its interval sequence, notes how many transpositions it admits, and names the limitation a “charm.” The impossibility is for him a spiritual property, a form of harmonic stasis that he associated — as a devout Catholic — with divine eternity. A mode that cannot depart is, in his compositional theology, a glimpse of the unchanging.

He was not doing group theory. The orbit–stabiliser theorem (in its abstract form) postdates Lagrange (1771), Cauchy (early 19th century), and Galois (1832). But the concepts were not part of music-theoretic discourse until Milton Babbitt’s work in the 1950s, and they were not formalised in the pitch-class set framework I have used here until Allen Forte’s The Structure of Atonal Music (1973) and David Lewin’s Generalized Musical Intervals and Transformations (1987).

What Messiaen had was a musician’s ear for symmetry. He could hear that the modes were closed, without having the algebraic vocabulary to explain why. The group theory shows that he was correct, and why he was correct with a precision that no amount of phenomenological description could provide.

From Messiaen to Lewin

Lewin’s transformational theory (1987) generalises the $\mathbb{Z}_{12}$ framework to arbitrary musical spaces. A Generalized Interval System is a triple $(S, G, \mathrm{int})$ where $S$ is a set of musical objects, $G$ is a group, and $\mathrm{int} : S \times S \to G$ assigns an interval to each ordered pair of objects in a way that is consistent with the group structure.

This framework treats musical transformations — not just pitch-class transpositions but rhythmic augmentations, timbral shifts, any structurally defined operation — as elements of a group. The mathematics does not privilege any particular musical parameter; it applies wherever a transformation group acts on a set of musical objects.

Neo-Riemannian theory, which emerged from Lewin’s work in the 1980s and 1990s and was systematised by Cohn (1998), applies this framework to triadic transformations (the operations L, P, and R that map major and minor triads to their relatives, parallels, and leading-tone exchanges). The group generated by L, P, and R on the set of 24 major and minor triads is isomorphic to $D_{12}$ — the same dihedral group that governs Messiaen’s modes, but acting on a different musical space.

Emmanuel Amiot’s more recent work (2016) applies the discrete Fourier transform to pitch-class sets, using the DFT coefficients on $\mathbb{Z}_{12}$ as a continuous measure of a set’s similarity to the modes of limited transposition. The Fourier coefficients detect the algebraic symmetries that stabilisers measure discretely: a set with large coefficient at frequency $k$ (in the DFT over $\mathbb{Z}_{12}$) is close, in a precise sense, to having the stabiliser $\langle T_{12/k} \rangle$.

The group-theoretic perspective has moved, over seventy years, from a marginal curiosity to the dominant mathematical framework in music theory. Messiaen’s modes — which once seemed like personal compositional idiosyncrasies — are revealed as structurally constrained: the possible stabiliser orders are fixed by the divisors of 12, and the orbit sizes that Messiaen’s ear discovered are exactly those that Lagrange’s theorem permits. Many pitch-class sets have non-trivial stabilisers; Messiaen found the seven that are musically viable. Their limitation is not a personal choice but an algebraic fact.

The charm of impossibilities is a theorem of group theory. And it is exactly as beautiful as Messiaen heard it to be.

References

• Amiot, E. (2016). Music Through Fourier Space: Discrete Fourier Transform in Music Theory. Springer (Computational Music Science).

• Babbitt, M. (1960). Twelve-tone invariants as compositional determinants. The Musical Quarterly, 46(2), 246–259. https://doi.org/10.1093/mq/XLVI.2.246

• Cohn, R. (1998). Introduction to neo-Riemannian theory: A survey and a historical perspective. Journal of Music Theory, 42(2), 167–180. https://doi.org/10.2307/843871

• Forte, A. (1973). The Structure of Atonal Music. Yale University Press.

• Lewin, D. (1987). Generalized Musical Intervals and Transformations. Yale University Press. (Reissued Oxford University Press, 2007.)

• Messiaen, O. (1944). Technique de mon langage musical. Alphonse Leduc. (English translation: Satterfield, J., 1956.)

• Tymoczko, D. (2006). The geometry of musical chords. Science, 313(5783), 72–74. https://doi.org/10.1126/science.1126287

• Tymoczko, D. (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press.

Changelog

• 2026-01-14: Changed “cosets of $D_{12}$” to “cosets of $\mathbb{Z}_{12}$ (the transposition subgroup)” in the twelve-tone composition paragraph. $D_{12}$ (order 24) already includes both transpositions and inversions, yielding only 2 cosets in the full serial group. The four row forms {P, I, R, RI} correspond to 4 cosets of the transposition-only subgroup $\mathbb{Z}_{12}$ (order 12) in the full group of order 48.