Part of The Beauty of Physics, Mathematics, and Music (and Maybe Cats).

Two regular streams of piano chords begin together. One proceeds faster than the other. Later attacks almost align, then separate again. In Conlon Nancarrow’s Study No. 33, the tempo proportion is $2:\sqrt2$, equivalent to a faster-to-slower ratio of $\sqrt2:1$. [[1]]

A player piano makes this an unusually concrete mathematical proposal. Its paper roll places events in space; the mechanism converts that spacing into time. The ratio can organise a composition even though no physical roll, mechanism or measurement has infinite precision. Martin Schlumpf’s account describes both Nancarrow’s roll-punching practice and the practical differences between manuscripts and realised tempos. [[2]]

The interesting question is what survives the journey from an irrational number to a finite performance.

Rational Beats Eventually Share a Grid

First remove melody, note lengths and articulation. Imagine two ideal pulse trains that begin together at $t=0$. Let the slower train produce one pulse each second and the faster train $r$ pulses per second. Their attack times are

$$ t_m=m\ \mathrm{s}, \qquad u_n=\frac{n}{r}\ \mathrm{s}, \qquad m,n=0,1,2,\ldots $$

The integers $m$ and $n$ count elapsed intervals in the two trains. An exact coincidence after the opening requires $m=n/r$, hence $r=n/m$.

If $r=p/q$ is a rational number in lowest terms, the first positive coincidence occurs at $q$ seconds: the slower train has completed $q$ intervals and the faster $p$. For $r=3/2$, three faster intervals occupy the same time as two slower ones. The joint arrangement repeats every two seconds.

Rescale the tempos and the coincidence time rescales with them. The ratio determines the counts; the absolute tempo determines how much clock time those counts occupy.

This is the familiar basis of a regular polyrhythm. It need not be easy to perform, but there is a common exact grid on which both pulse streams fit.

The Square Root Removes Exact Recurrence

Now set $r=\sqrt2$. A later exact coincidence would require $\sqrt2=n/m$. That is impossible for positive integers $n$ and $m$.

The proof fits in a few lines. Suppose $n/m$ were a fraction in lowest terms whose square was two. Then $n^2=2m^2$, so $n$ would be even. Writing $n=2j$ gives $m^2=2j^2$, so $m$ would also be even. Both would share a factor of two, contradicting the assumption that the fraction was reduced.

The two ideal pulse trains therefore share their opening attack and no later one. Their combined pattern has no positive repetition period: a common period would have to contain an integer number of intervals from both trains, again expressing the ratio as a fraction.

That is a theorem about perfectly regular pulses with a common start. It is not a theorem that all the voices of Nancarrow’s composition can never meet. Entrance offsets and changes of tempo are compositional choices, and they change the problem.

Seventeen Intervals Almost Equal Twelve Seconds

Exact equality is one question. Temporal separation is another.

With the illustrative one-pulse-per-second slower train, the seventh faster pulse occurs at

$$ \frac7{\sqrt2}\ \mathrm{s}\approx4.949747\ \mathrm{s}, $$

about $50.25$ milliseconds before the fifth slower pulse. The seventeenth occurs at

$$ \frac{17}{\sqrt2}\ \mathrm{s}\approx12.020815\ \mathrm{s}, $$

about $20.82$ milliseconds after the twelfth slower pulse. Later, $41/\sqrt2$ is approximately $28.991378$: the forty-first faster pulse precedes the twenty-ninth slower one by only $8.62$ milliseconds.

These are calculations for chosen pulse rates, not measurements from a Nancarrow recording and not experimentally established hearing thresholds. Changing the slower rate changes all the separations proportionally.

The pairs arise because $7/5$, $17/12$ and $41/29$ approximate $\sqrt2$. Their errors lie on alternating sides of it. The faster pulse can arrive just before or just after the slower one while missing exact alignment every time.

Callender identifies related near-alignments in the study’s opening, with staccato articulation for the earlier chord. [[1]] The arithmetic describes the gap. Articulation helps determine what a listener can make of it.

Number Theory Supplies Arbitrarily Close Encounters

The approximation sequence has a compact origin. If $y=\sqrt2-1$, then $y=1/(2+y)$. Substituting this expression into itself repeatedly gives

$$ \sqrt2=1+\cfrac1{2+\cfrac1{2+\cfrac1{2+\cdots}}}. $$

Stopping this continued fraction at successive depths produces $1$, $3/2$, $7/5$, $17/12$, $41/29$, $99/70$ and so on. These fractions converge to $\sqrt2$.

Each numerator–denominator pair $(n,m)$ gives the next as $(n+2m,n+m)$. This operation reverses the sign of $n^2-2m^2$. Starting from $(1,1)$ therefore gives $n^2-2m^2=\pm1$ throughout. Rationalising the difference between the corresponding pulse times gives the exact identity

$$ \Delta t=\left(\frac n{\sqrt2}-m\right)\mathrm{s} =\frac{n^2-2m^2}{\sqrt2(n+m\sqrt2)}\ \mathrm{s}. $$

The numerator remains $+1$ or $-1$ while the denominator grows. The gaps therefore approach zero. For example, $(99,70)$ gives about $3.57$ milliseconds; $(239,169)$ gives about $-1.48$ milliseconds.

For any positive tolerance, however small, the ideal pulse trains eventually contain a pair of attacks closer than that. A finite composition may end before such a pair occurs. The theorem guarantees increasingly close encounters over unbounded time. This says nothing by itself about perceived simultaneity.

Irrationality excludes exact recurrence. It does not impose a minimum audible distance between attacks.

A Better Approximation Can Create Another Problem

Replacing $\sqrt2$ by $17/12$ gives a small tempo error, but restores an exact joint period. In the illustrative model, the two pulse streams now repeat together after twelve seconds. The replacement buys a common grid at the price of the periodicity that the irrational ratio excludes.

It also accumulates timing error. After $N$ intervals of the faster train, the discrepancy between the rational and irrational schedules is

$$ \delta t_N=N\left(\frac{12}{17}-\frac1{\sqrt2}\right)\mathrm{s}. $$

For $N=100$, this is approximately $-0.1224\ \mathrm{s}$. An approximation that misses one interval by roughly a millisecond has drifted by over a tenth of a second after a hundred of them.

One way to control drift is to alternate approximations that err in opposite directions. Another is to place individual events close to their target times. These strategies optimise different things: maintaining a steady local pulse, preserving the alignment of particular events, and making a part playable are not identical requirements.

Paul Usher arranged the study for the Arditti Quartet using rational subdivisions and compensation for accumulated error. [[1]] The arrangement is a solution for human players, with the ideal ratio as the target being approximated.

A Canon Can Converge Despite Its Irrational Ratio

A canon repeats musical material between voices. A tempo canon gives those voices different speeds; a convergence point is where they reach the same position in the material together. Callender distinguishes these structural intersections from the opening’s pulse relations. [[1]]

A small model makes the distinction exact. Let a phrase have length $L$ in abstract score units. The slower voice proceeds at speed $v_s$, the faster at $v_f$, and their entrance times are $t_s$ and $t_f$. Their finishing times match when

$$ t_f-t_s=L\left(\frac1{v_s}-\frac1{v_f}\right). $$

Take $L=12$, $v_s=1$ unit per second and $v_f=\sqrt2$ units per second. The slower voice starts at zero; the faster begins approximately $3.514719$ seconds later. Both finish at twelve seconds.

The entrance offset contains an irrational value, but there is no contradiction. The earlier impossibility proof required both pulse trains to start together. Here their start times were chosen to produce an intersection. A score can also exchange the tempos between voices and alter their subsequent relationship.

“The voices never meet” is therefore an inadequate description of a tempo canon. It confuses a property of one pulse model with the architecture of a composition.

A Performance Cannot Prove Irrationality

Any finite set of ideal attack times can be approximated by rational schedules within a specified positive error. Measurement also reports values with finite precision. Neither a roll nor a recording can empirically certify that a tempo ratio has infinitely many non-repeating digits.

That does not make the ratio decorative. It specifies the ideal relationship from which timings and approximations are derived. The practical question is which features survive: independent pulse streams, the absence of a readily apparent short common cycle, near-alignments, and intended intersections of musical material.

The beauty here is a specific tension between an exact relation and a physical realisation. The number excludes a repeating grid. The composer still chooses where the music arrives.

Sources and calculations checked through 2026-10-12.

References

[1] Callender, C. (2014). Performing the irrational: Paul Usher’s arrangement of Nancarrow’s Study No. 33, Canon $2:\sqrt2$. Music Theory Online, 20(1). Full article. Especially §§2 and 4.

[2] Schlumpf, M. (2004). Conlon Nancarrow—Martin Schlumpf: The Art of Tempo Canon. Programme/booklet essay, Schott Music & Media, Mainz. Publisher-hosted PDF. Especially pp. 2–3 on the player piano and pp. 8–10 on punching scores and tempo grids.

Changelog

  • 2026-10-12: Research and calculation review. Distinguished exact pulse recurrence, rational approximation, finite timing tolerance and composed convergence points, using Study No. 33 as the musical anchor.