The Ground & Its
Thirty Variations

I · The materialThe audio and the notes both already exist, free and clear

We do not need to derive audio from the score, and we do not need to license a recording.

The Open Goldberg Variations project, funded on Kickstarter and released on 28 May 2012, put a professional studio recording into the public domain along with a newly engraved score. The pianist is Kimiko Ishizaka, recorded over five days on a Bösendorfer 290 at Teldex Studio, Berlin. Both the recording and the score were dedicated under Creative Commons Zero.

Verified. The archive.org item metadata carries the license as a machine-readable field, not marketing language: licenseurl: creativecommons.org/publicdomain/zero/1.0/

This matters more than a normal licence would, because CC0 §2 waives not only copyright but Related Rights — which includes the producer’s rights in a phonogram. That is the layer that ordinarily makes classical recordings unusable. It is waived here. The project’s own stated intent is that the material be free for “schools, universities, musicians, private persons and even commercial productions.”

Two small obligations survive. CC0 §4(a) does not waive trademark, so the project’s name and MuseScore’s branding cannot be used in a way implying endorsement. And CC0 cannot waive personality rights in every jurisdiction, so a one-line credit removes the residual risk at no cost. We should credit Ishizaka regardless; it is a fine performance.

Glenn Gould is not available to us

Worth settling now, since it is the recording everyone pictures. The composition is public domain everywhere. The sound recording is a separate right with a separate term, and the answer differs by continent.

The 1955 recording’s fifty-year European term expired in 2006, and the extension to seventy years was explicitly non-retroactive, so it was never revived. But the American Music Modernization Act places recordings published between 1947 and 1956 under a 110-year term, which puts Gould’s 1955 into the United States public domain on 1 January 2067. For a site served worldwide he is unusable, and geofencing is not worth the engineering. We lose nothing.

Where the notes come from

The same project’s engraved score — 990 measures, 58 pages, publicly peer-reviewed twice against the manuscript — exports to MusicXML and MIDI, also under CC0. That is the primary source, and it is the same edition as the audio.

One wrinkle. MuseScore’s storefront has historically pushed a subscription on the download button even for public-domain scores. If that blocks us, dl-librescore extracts it — working around a paywall misapplied to CC0 content, not circumventing a licence.

The caveat that actually affects scope: the file is a single piano part on two staves, and voice numbers are scoped per staff. In the canons a contrapuntal voice migrates between staves mid-phrase. So we get the raw material for voice tracking, not finished voice tracks. Separating the canonic voices is the real engineering work — larger than the audio alignment step, and it falls on precisely the nine movements the piece most wants to show.

The PDF in this repository is not a data source

It is a MuseScore export with a family dedication on page two — a community upload rather than a scholarly edition — and its text layer is empty, because the variation headings are vector outlines. Keep it for reading.

Optical music recognition is the wrong path, and not by a narrow margin. On piano music Audiveris shows symbol error rates around 94 per cent; even current state-of-the-art on cleanly engraved keyboard writing sits near 14 per cent, roughly one error every seven notation events. Audiveris’ own documentation concedes it cannot represent a notehead shared between two voices at all — and shared noteheads are everywhere here, alongside hand-crossing in nine variations and Bach’s ornament tables.

The decisive point: OMR fails hardest at voice assignment, which is not incidental to this project but the entire thing we want to show.

Making the visuals land on the performance

The score gives idealized time; Ishizaka plays with rubato. Alignment closes that gap. parangonar, from Gerhard Widmer’s group at JKU Linz, reports 21 ms median error, with 84 per cent of notes inside 50 ms — below the threshold at which people perceive audiovisual desynchronization. Not what was built: the pipeline uses librosa chroma DTW at a ~93 ms hop, which is cursor-grade and has no ground truth. These figures describe a tool the project evaluated and did not adopt; do not read them as ours. That is the number that makes a piece like this feel exactly right rather than approximately right.

Dynamic time warping does not degrade gracefully when the repeat structure is wrong — it threads a monotone path through unrelated material and reports no error at all. So the first task is the least technical one in the project: listen through once, and note for each of the thirty-two movements whether each half is repeated.

There is also a CC0 harpsichord rendering of the whole work, synthesized from MIDI and therefore perfectly aligned by construction. Nobody would ship it as the headline recording, but it is an excellent scaffold: build every visual system against zero alignment error first, then swap in Ishizaka. It means never debugging a rendering fault and an alignment fault at the same time — and it could survive into the finished piece as a harpsichord toggle, which is arguably the correct instrument anyway.

II · Prior artThe obvious idea has been had, repeatedly, since 1983

Anyone who sits with this piece arrives at “one artwork per variation” within ten minutes. So has everyone else, for forty years.

The thirty-two-part mapping is thoroughly occupied, and by good work. Elena Presser’s mixed-media series of 1983–84 is not merely one-per-variation but explicitly structural — a folded script for the canon at the unison in Variation 3, symmetrical rectangles for the canon’s three levels in Variation 9. Farhad Ostovani’s thirty-panel work is hanging at the Nelson-Atkins as this is written. Nancy Huston wrote thirty chapters, each the interior monologue of one listener during one variation. Gabriel Josipovici wrote thirty interlocking stories. Alice Fogel wrote thirty-two poems of thirty-two lines, the line count borrowed from Bach’s bar count. Richard Powers built The Gold Bug Variations on thirty chapters plus two arias. François Girard made thirty-two short films about Glenn Gould.

Live prior art. Ostovani at the Nelson-Atkins, 7 March 2026 – 28 February 2027. And Ólafsson with Olafur Eliasson premiered a Goldberg light installation at Lucerne on 8–10 May 2026, using a church’s translucent marble walls as the screen.

In visualization proper, Stephen Malinowski’s Music Animation Machine owns the scrolling coloured bar — pitch as height, a fixed playline, notes lighting as they cross it. His Art of Fugue is far more inventive and worth studying as the counter-example. Nicholas Rougeux rendered the whole work as a radial disc. Ceal Floyer played thirty recordings of the Aria simultaneously and let them decay into a ghost chorus within seconds — conceptually the sharpest Goldberg artwork found, and it has no visual component at all.

What nobody has done

These are verified absences rather than guesses, and the first one is the opening.

• 01Nobody visualizes the ground bass as the invariant. Every existing project draws the surface melody. The subject of the piece is a harmonic scheme that never changes — carried by a bass that is re-composed every time.
• 02No arc diagram of the full cycle. Wattenberg’s Shape of Song never covered it.
• 03No keyscape. Craig Sapp’s gallery covers the Well-Tempered Clavier exhaustively and skips this.
• 04BWV 1087 — the fourteen canons in Bach’s own copy — is almost untouched outside music. Since qualified: Rachel Wells Hall treated it mathematically at Bridges 2021, and Haakenson had done so in Music Theory Spectrum in 2017.
• 05No published entropy, fractal or Markov study specific to BWV 988.
• 06Canon has never been formalized as a braid. Unverified absence
• 07Interactive, game and immersive treatments are essentially empty.

III · The mathematicsWhat is exact, what is contested, and what has never been drawn

Measurements marked Computed were taken from the score during this research, not quoted from the literature.

The Aria is thirty-two bars of 3/4 carrying one structural bass note per bar. The work has thirty-two movements. The distinction usually glossed over is that the notated bass is a decorated line of hundreds of notes; the thirty-two-note ground is the fundamental bass, one per measure. It opens with the descending tetrachord G–F♯–E–D, the standard European passacaglia ground, and closes with a cadential turn back to G.

That correspondence — thirty-two notes, thirty-two movements — is not documented as Bach’s stated intention, but it is arithmetically exact, and it is the best organizing idea available to us.

The canon ladder, with the delays normalized

Every third variation is a canon, climbing one interval each time from the unison to the ninth. The delay column is the under-reported part: everyone reports it in bars, where the meters differ. Normalize it to ground-notes and an anomaly surfaces that appears in no prose account of the piece.

Variation 18 is the unique half-unit canon. Every other delay is a whole number of ground-notes. And Variation 27 is the only canon with no free bass — two voices, pure canon, the ground implied and never sounded. At the very top of the ladder, the invariant goes invisible.

Ignoring octave displacement, the nine canons use only five distinct pitch transformations, so they pair off: 3↔24, 6↔27, 9↔18, 12↔15. Dan Tepfer calls them sister canons.

Variation 30 is a quodlibet rather than a canon at the tenth, and the reason is arithmetic: a tenth is a compound third, already spent at Variation 9, so it would have been the first redundant rung on the ladder. In its place, a movement about stacking borrowed melodies replaces one about stacking a melody with itself.

The bass is a promise kept unevenly

Received wisdom says all thirty-two movements share the Aria’s bass. True harmonically; much looser as sounding pitch — and the looseness is patterned. Nobody has ever plotted this.

The shape of the whole

Thirty-two movements in three repeating roles. Every third is a canon; the movement before each canon is a virtuosic hand-crossing duet; the one before that is a character or dance piece. Only three movements leave G major, and the tonic never modulates across eighty minutes.

Proportion — and where to stop believing it

The credible fact is that the work totals 960 bars, split exactly 480 : 480 at Variation 16. The French overture, the movement that opens the second half, begins at the precise bar midpoint — and is simultaneously the only movement of irregular length. The thing that makes the halves equal is the thing that breaks the length pattern.

We should state the caveat rather than hide it. Reaching five sixteen-bar movements requires counting Variation 22 alla breve; counted as notated, the total is 976 and the split is 480 : 496. And a Monte Carlo test finds 156,306,160 distinct one-to-one partitions of the thirty-two movements — a probability of 7.3 per cent Attributed to Shepherd; the passage is paywalled and was never reached, so treat the figure as unconfirmed at source., high precisely because nearly every movement is the same length. The split is musically meaningful because it falls at the structural double bar. It is not statistically remarkable on its own.

Drop the golden ratio entirely. Ruth Tatlow finds no evidence that any composer of the period had aesthetic interest in the golden section. The claim is anachronistic.

One numerical claim does hold. In 1974, on the inside back cover of Bach’s own copy of the print, fourteen unrealized puzzle canons were found — Verschiedene Canones über die ersteren acht Fundamental-Noten vorheriger Arie, various canons on the first eight fundamental notes of the preceding Aria. That there are exactly fourteen, and that Bach sums to fourteen in the natural-order alphabet, is the one gematria claim with real support — because the count is a hard fact rather than a derived number.

IV · DirectionInk absorbed into paper, not light emitted in darkness

Being wholly different from the Reich work is not a matter of changing the palette. It means inverting every axis.

What the comparison is to. Simulated Monolithes is the researcher’s earlier piece: three films setting real-time light simulation against Steve Reich’s Piano Phase, Music for Pieces of Wood and New York Counterpoint, in Jean-Louis Marchand’s recordings. Every image is generated by a single fragment shader rather than filmed or drawn. It is the immediate predecessor to this project, which is why the visual direction here is defined against it.

Monolithes is a fragment shader raymarching signed-distance fields: volumetric light gathered along rays, mirrored floors, exponential fog, near-black grounds with colour only in emissive elements, glacial scripted camera arcs, filmic grain. It is cinematic and atmospheric, and it is seen from a moving eye.

Four traditions, in a deliberate order

The spine is change-ringing notation. English bell-ringers solved our exact problem during Bach’s own lifetime: how do you draw a permutation unfolding in time so a human can learn it by eye and then perform it from memory? Their answer is a static field of numerals with one or two voices traced through it as continuous coloured lines — the blue line. That gives us the ground as the invariant grid and the canonic voices as traced paths, and it supplies what most music visualization lacks: a reason for the eye to move, rather than things moving past a passive eye.

The colour logic is Oliver Byrne’s Euclid of 1847. Byrne replaced the labels A, B and C in geometric proofs with the coloured figures themselves, so a proposition reads: the and the together equal the . Flat spot colour as pure identity, never as decoration or heat. Setting diagram fragments inline in running prose — as in this paragraph — does more to establish that a thing is a proof rather than a music player than any amount of layout.

This document is built in the language it argues for. The palette, the plate rule, the marginal notes and the computed diagrams are the first prototype.

The material surface is the 1741 engraving. Balthasar Schmid cut the first edition into copper by hand, so the noteheads, beams and title lettering are literally an engraver’s handwriting in reverse. Warm laid paper rather than white, a plate-mark frame, staff lines of variable weight. This is what makes the piece specific to this work rather than to canon in general.

The generative logic is the weaving draft. In a draft, the threading, tie-up and treadling are given, and the drawdown — the cloth itself — is computed, not authored. That is exactly the epistemology of canon: specify a subject and an interval, and the second voice is a consequence, not a choice. Rendering the derived voice differently from the given one makes the deduction visible without a word of explanation. Sol LeWitt supplies the level above it: every case in an identically framed cell, none privileged, the rule stated in plain language.

Held in reserve: the braid. A canon genuinely is a braid word — strands are voices, crossings are voice-crossings with a real over and under determined by register. Since no formal treatment appears to exist, this is the project’s best shot at an original contribution. But braids are visually thin, so they belong as a layer one toggles into, not as the primary surface.

What to avoid

Scrolling piano-roll bars, which Malinowski owns and a thousand tutorial channels have flattened into wallpaper. Circle-of-fifths hue wheels. Amplitude-reactive particles. Spectrograms and frequency ribbons — actively wrong for solo keyboard, the material with the least timbral variety in the repertoire. Radial note mandalas. Score-following highlight. Synesthesia claims.

V · ConceptsThree directions, and what they cost

Each answers a different verified gap. They are not mutually exclusive.

VI · OpenFour questions that need answering before anything is built

• 01Interactive piece, rendered film, or both? The CC0 audio means the interactive work can be public, which Monolithes could not be. That argues for making the website the work rather than a trailer for it.
• 02How much does the reader do? Passive reading, or can they set an interval and watch a canon be generated? The latter is where the untouched territory is.
• 03The whole work, or a chosen subset? Eighty-five minutes is long. The nine canons are twenty.
• 04How honest about contested mathematics? Recommendation: completely. Showing the 480:480 split and the 7.3 per cent is more interesting than asserting the first alone — and it is the difference between a work about Bach and a work about pattern-finding.