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

The numbers begin

$$ 0,1,-1,2,1,0,-2,3,-1,2,0,1,2,-1,-3,4. $$

They are the opening of Per Nørgård’s infinity series, a rule for generating signed integers that can be mapped to pitches. Take every second entry, starting at zero, and the same sequence reappears upside down. Take every fourth entry and it reappears unchanged.

This is also a compositional resource. In his comments on Symphony No. 2, Nørgård described wanting to expand his use of the series from roughly a thousand notes to roughly four thousand, while intervening to create melodies and climaxes. The accompanying account identifies the symphony’s main strand with the first 4,096 entries. [[1]]

The rule explains how related material can exist at several speeds. It does not specify an orchestra or compose a symphony by itself. Understanding the distinction requires only signed numbers, even and odd indices, and a careful separation of pitch from note number.

Two Operations Generate the Whole Sequence

Write \(a_n\) for entry number \(n\), counting from zero. The defining recurrence is

$$ a_0=0,\qquad a_{2n}=-a_n\quad(n\geq1),\qquad a_{2n+1}=a_n+1\quad(n\geq0). $$

An even index tells us to halve the index and reverse the sign. An odd index tells us to subtract one, halve the index, and add one to the resulting value. Each step reduces a positive index, so the rule eventually reaches the specified starting value. This is the integer version analysed by Au, Drexler-Lemire and Shallit. [[2]]

For example,

$$ a_{13}=a_6+1=-a_3+1=-(a_1+1)+1=-1. $$

The same rule can generate blocks. Replace every entry \(x\) with the pair

$$ x\longmapsto(-x,x+1). $$

Starting from \(0\), successive replacements give

$$ (0),\quad(0,1),\quad(0,1,-1,2),\quad (0,1,-1,2,1,0,-2,3). $$

Each block extends the previous one and doubles its length. The replacement for zero begins with zero, so these finite blocks fit together into one infinite sequence. The historical musical name is “series”; mathematically, we are considering an ordered sequence rather than adding an infinite sum.

A Number Becomes a Pitch Only After a Choice

For a simple model, let zero denote middle C and let each integer step mean one equal-tempered semitone. Negative values lie below C. The first eight entries then become C, C♯, B, D, C♯, C, B♭ and E♭, with B and B♭ below the starting C.

If the frequency assigned to zero is \(f_0\), the model assigns

$$ f_n=f_0\,2^{a_n/12}. $$

Here the subscript \(n\) labels the event, while \(a_n\) specifies its pitch displacement. With \(f_0\approx261.63\,\mathrm{Hz}\), the second entry gives about \(277.19\,\mathrm{Hz}\), and the third about \(246.94\,\mathrm{Hz}\).

This chromatic mapping is an explanatory choice, not a transcription of Symphony No. 2. A composer can instead interpret the integers as degrees of a chosen scale. In a diatonic scale the degree steps need not all span equal semitone distances, so a reflection of the integers need not be an exact reflection in logarithmic frequency.

Duration is another choice. Equal numbers can receive different rhythms, articulations and instruments. The recurrence says nothing about any of them. Those decisions determine how the arithmetic becomes music.

Thinning the Sequence Preserves Its Family Resemblance

The recurrence immediately identifies the two subsequences obtained by retaining alternate entries:

$$ (a_{2n})_{n\geq0}=(-a_n)_{n\geq0},\qquad (a_{2n+1})_{n\geq0}=(a_n+1)_{n\geq0}. $$

The even subsequence is the original reflected about zero. The odd subsequence is the original shifted upwards by one unit. The even formula also holds at \(n=0\), since negating zero changes nothing.

In the chromatic mapping, these are pitch inversion and semitone transposition. If the original entries occupy equally spaced time slots, a voice playing the even subsequence at its original event times progresses through its related melody at half the rate. That describes an opportunity the sequence offers, not a claim that Nørgård’s music uses that literal arrangement throughout.

Apply the recurrence again:

$$ \begin{aligned} a_{4n}&=a_n,\\ a_{4n+1}&=1-a_n,\\ a_{4n+2}&=-a_n-1,\\ a_{4n+3}&=a_n+2. \end{aligned} $$

Every fourth entry therefore yields either the original or its reflection, with a fixed shift. For instance, the third line follows from \(a_{4n+2}=-a_{2n+1}=-(a_n+1)\).

The general statement is

$$ a_{2^r n+j}=\varepsilon a_n+c, \qquad 0\leq j<2^r, $$

where \(r\) is a nonnegative integer, and \(\varepsilon\in\{-1,1\}\) and the integer \(c\) depend on \(r\) and \(j\), not on \(n\). Applying either operation \(x\mapsto-x\) or \(x\mapsto x+1\) to an expression of the form \(\varepsilon x+c\) preserves that form. Repeating the operations proves the claim. This precise self-similarity is also established in the mathematical analysis. [[2]]

The useful property is stronger than an occasional repeated motif: the prescribed subsequences remain related through the entire infinite construction.

A Finite Register Contains Surprisingly Many Entries

The sequence is not periodic. At the indices made by repeatedly doubling and adding one,

$$ a_{2^r-1}=r. $$

The proof is induction: the odd-index rule adds one at each step, starting from \(a_0=0\). Consequently the values are unbounded. A periodic integer sequence would repeat a finite list and remain bounded; even an eventually periodic tail cannot accommodate this growth.

But the growth is slow. Among the first \(2^r\) entries, for \(r\geq1\), the minimum and maximum are

$$ 1-r\quad\text{and}\quad r. $$

To verify this, the first two entries span zero to one. Replacing a block spanning \(L\) to \(U\) produces values from the reflected range \(-U\) to \(-L\) and the shifted range \(L+1\) to \(U+1\). Substituting the claimed endpoints gives the next pair of endpoints.

For 4,096 entries, \(r=12\), so the range is \(-11\) to \(12\): a span of 23 units. In our semitone model, thousands of events fit inside slightly less than two octaves. That is a derived property of this mapping; it is not an assertion about the full orchestral register of the symphony.

Folding the values modulo twelve would discard register altogether. The unboundedness proof would then cease to establish nonperiodicity, because the folded values are bounded. A changed representation needs its own argument.

The Parity Is Thue–Morse, Not a Second Melody

Now retain only whether each value \(a_n\) is even or odd. Define \(t_n=a_n\bmod2\), using remainders zero or one, including for negative integers. The recurrence reduces to

$$ t_{2n}=t_n,\qquad t_{2n+1}=1-t_n. $$

These rules generate the Thue–Morse sequence:

$$ 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0. $$

Equivalently, \(t_n\) is the parity of the number of ones in the binary representation of \(n\). Doubling appends a zero and leaves that count unchanged; doubling and adding one appends a one and reverses its parity. For \(n=13\), binary \(1101\) contains three ones, agreeing with the odd value \(a_{13}=-1\). The parity is read directly from the pitch values, not from the intervals between successive notes. [[2]]

There is a precise physics connection if these bits become scattering weights at integer positions. Assign balanced weights \(w_n=(-1)^{t_n}\), taking values \(+1\) and \(-1\). The associated infinite Thue–Morse chain has purely singular continuous diffraction: its intensity measure has neither isolated Bragg peaks—sharp peaks at definite wavevectors—nor an ordinary integrable density over wavevector. Baake and Grimm give a direct analysis. [[3]]

Changing the weights to zero and one adds a nonzero mean and hence a lattice Bragg component, alongside the singular continuous part. This is a different scattering model. Neither construction is the acoustic spectrum of the melody, and calling the entire infinity series a “quasicrystal” would blur the distinction that makes the connection interesting.

The Rule Leaves Room for a Composer

Nørgård’s own comments on Symphony No. 2 emphasise intervention: melodic shaping, climaxes and orchestration while preserving the continuing flow. The booklet’s account describes sixteen phases of 256 entries within the main 4,096-entry strand. These are compositional decisions made around the sequence, rather than consequences of its three defining equations. [[1]]

The arithmetic supplies material whose relationships survive a systematic change of scale. It also supplies a relatively compact register without forcing the material into a repeating period. Neither result decides when a phrase should arrive, which instrument should carry it, or whether it deserves to be heard twice.

The sequence continues automatically. The symphony requires choices.

Sources, pitch conversions and sequence identities checked through 2026-10-12.

References

[1] Cornelius, J. (2016). A Delta of Other Worlds, section “Symphony No. 2 — In One Movement”. Liner notes originally issued with Dacapo 6.220645; reprinted in Per Nørgård: 8 Symphonies, Dacapo Records 8.204002, booklet, pp. 46–47. English translation by James Manley. Includes Nørgård’s own account of his use of the infinity series. Publisher’s booklet.

[2] Au, Y. H., Drexler-Lemire, C., and Shallit, J. (2017). Notes and note pairs in Nørgård’s infinity series. Journal of Mathematics and Music, 11(1), 1–19. DOI: 10.1080/17459737.2017.1299807. Authors’ manuscript, dated 18 February 2017, §§1–5.

[3] Baake, M., and Grimm, U. (2008). The singular continuous diffraction measure of the Thue–Morse chain. Journal of Physics A: Mathematical and Theoretical, 41, 422001. DOI: 10.1088/1751-8113/41/42/422001. Full-text preprint.

Changelog

  • 2026-10-12: Research and calculation review. Derived and checked the recurrence, affine subsequences, register bounds and parity relation; distinguished explanatory pitch mappings and weighted diffraction from the composer’s musical choices.