The Tunings That Ring

Marcus Hobbs · first posted 2026-07-25 · status: living document — this will be revised as experiments land. Receipts for every claim live in the repository.

Erv Wilson spent a lifetime searching, by hand, for scales that solve music's oldest tension: structures that serve melody (stepwise, singable, few step sizes) and harmony (chords whose combination tones reinforce rather than beat) at the same time. I worked at his side from 1995 to 2005, turning his diagrams into software while he drew the next one. He's gone; the search isn't. This post describes how I've made that search falsifiable — a scoring function anyone can read, a queue of pre-registered experiments, and an open invitation to tear both apart.

Two things up front. First, the entire computational program is run in collaboration with an AI agent (Claude), and I'll be precise below about what that means and why it works. Second, everything here is public: the scorer, the receipts, the failures. If you think the scoring function is wrong — and in at least one measured way it is incomplete — I want to hear from you. Skip to the invitation if that's why you came.

The scoring function, fully disclosed

The harmonic axis is a frozen Python module, experiments/triads/scorer.py (v1.1.0). For frequencies a < b < c, all math in frequency space, never cents:

proportional (major prototype 4:5:6): 2b = a + c — middle tone at the arithmetic mean of the outer tones
subcontrary (minor prototype 10:12:15): b(a + c) = 2ac — middle tone at the harmonic mean
geometric (neutral): b² = ac — counted, never scored

Why these three: when the middle tone sits at the arithmetic or harmonic mean, first-order sum and difference tones land on other scale degrees. The triad audibly locks. This is Wilson's proportional-triad criterion, and it is the sonority the plugin's own analyzer looks for.

The conventions that make the numbers mean something:

Exact arithmetic for JI. The rational path runs on fractions.Fraction end to end — zero floats, zero epsilon. The tempered path reconstructs f = 2^(cents/1200) and admits a single hyperparameter ε (default 2¢), which appears only in the comparison layer.
Middle-anchored sampling. For each scale degree b, the outer tones range over the unique octave representatives in (b/2, b) and (b, 2b). Chosen over the naive two-octave window because it is exactly self-dual (inverting the scale swaps proportional and subcontrary counts, P ↔ S) and exactly transposition-invariant. The window convention fails both; we keep it only for comparison.
Triads fit within an octave (outer ratio ≤ 2/1, v1.1.0). 1:2:3 is arithmetically proportional but is not the chord this criterion is about. Ear-validated; it re-ranked results enough that v1.0.0 and v1.1.0 numbers are never mixed.
Octave reduction is half-open [1, 2), matching the shipping plugin bit for bit.
A degeneracy guard on the tempered path: a triple whose arithmetic and harmonic means are closer than ε cannot be shown to be proportional rather than subcontrary, so it counts as neither. More on why below — this guard has a good origin story.

Known limitations — measured, not hedged

Transparency means the defects list is part of the disclosure:

P = S exactly for every MOS and every CPS(2k, k). Both families are inversionally symmetric as pitch-class sets, and the anchored scorer commutes exactly with inversion. Consequence: the once-primary aggregate min(P, S) cannot discriminate within these families at all. It has been demoted to a recorded-but-not-optimized statistic.
The aggregator is unsettled — by ear. My listening check validated the classifier (the triads it finds do lock) but rejected min(P, S) as a ranking: I want proportional-heavy and subcontrary-heavy and geometric-heavy scales as distinct characters, not a scalarized compromise. Current reporting uses balance buckets instead. An aggregator worthy of freezing does not exist yet. This is the single best entry point for outside critique.
Counts are not weighted by audibility. A triad at the edge of ε counts the same as an exact rational coincidence; register, timbre, and critical band are ignored entirely.
The plugin's own analyzer has a register-dependent tolerance. Its 0.0005 linear-frequency threshold is ≈0.865¢ for triads rooted near 1/1 but ≈0.433¢ near the octave — twice as strict at the top of the octave, which nothing in the theory motivates. Measured against the real compiled C++; tracked as a plugin issue, deliberately not silently "fixed."
The melodic axis (below) is brand new and has not yet had its own ear check.

Why let an AI optimize against a scoring function?

The working pattern is what Andrej Karpathy has described for AI-driven research: put the machine in the outer loop, operating on experiments, and keep deterministic computation in the inner loop, operating on parameters.

Inner loop — plain Python, no AI: exhaustive and quality-diversity batch search over scale families (Combination Product Sets, MOS generators), every candidate generated, scored, and archived. Headless and reproducible.
Outer loop — an AI agent (Claude) proposes a hypothesis, writes the pre-registration entry before the run (prediction, tolerances, expected failure modes), implements, runs, and files the verdict as kept-or-reverted. Surprises become the next hypotheses.
The human — I hold three things the loop cannot: the spec (what is worth asking), the ear (ground truth the metric approximates), and the freeze (what counts as a measurement).

Why this works for scale aesthetics in particular: the search spaces are enormous but cheap to evaluate — a scorer verdict costs microseconds, so an agent can test a thousand parameterizations of a hypothesis in the time a human tests one. And Wilson's own practice was already loop-shaped: draw, listen, redraw. The machine just runs the draw-score part of the loop at scale, with receipts.

The obvious objection is Goodhart's law, and it bit us immediately — concretely, and instructively. Early on, the tempered-path optimizer discovered that a 1-cent generator was the global optimum of min(P, S) at every cardinality from 5 to 10 and at every ε: cram all the tones together and every triple is within ε of both mean conditions simultaneously. The scale is musical nonsense; the score was perfect. That episode is why the scorer is frozen — SHA-256-pinned, CI-enforced, changeable only by explicit human approval — and why the degeneracy guard exists: an optimizer that can edit its own verifier is not being verified. The metric is allowed to be wrong; it is not allowed to drift silently while something optimizes against it.

The second firewall is the ear. Twice now the listening check has overruled the machine — validating the triad classifier but rejecting the min(P, S) aggregator, and forcing the within-an-octave span rule. Optimization pressure is applied only to metrics that have survived a human ear, and every archived number records which scorer version produced it.

Past: what the loop has already found

Everything below is receipt-backed in the repo (FINDINGS.md, results/):

The offline mirror is bit-exact against the shipping plugin. 70 hexanies reproduced with zero mismatches and 15 MOS scales within 1 ulp against the real compiled C++ — an offline score can be trusted as the scale the plugin would actually produce.
All 29 eikosanies from odd seeds ≤ 15 sit exactly on P = S, an instance of the inversional-symmetry theorem above — discovered empirically by the loop, then proven structural.
MOS generator sweeps show triad-count hot spots that are stable across cardinality — near 571.6¢ at odd cardinalities, the pure fourth at N = 12, 416¢ at N = 20. Mapping why these are hot is an open experiment (below).
The melodic axis exists as of this week: three scorers — gap-size entropy (three-gap-theorem territory), Wilson's constant-structure criterion, and Rothenberg propriety — each a pure function of a pitch-class set with every tolerance logged.
The melodic unit tests immediately corrected two pieces of folklore that had been written into our own spec: the 12-EDO diatonic is not a constant structure (the 600¢ tritone subtends both 3 and 4 steps), and the 12-tone Pythagorean chromatic is strictly proper — the classic improper scale is the Pythagorean diatonic, whose 611.7¢ augmented fourth exceeds its 588.3¢ diminished fifth. A harness that catches errors in its own spec's assumptions before running a single experiment is the pattern working as intended.

Present: scoring Wilson's harmonic objects for melody

LAT-MEL-001 runs the melodic scorers over the full harmonic corpus — all 70 hexanies, all 29 eikosanies, MOS controls (which should saturate the melodic scores), and random controls (which should floor them). Pre-registered predictions, quoted from the spec:

H-L1: the eikosany on {1,3,5,7,9,11} is a constant structure by machine check at 0.5¢. This verifies a claim long attributed to Wilson rather than assuming it — and either outcome is a finding.
H-L2: CPS scales are systematically improper and multi-gap relative to MOS at matched cardinality — quantifying the folk claim that "CPS lacks melodic continuity" — but melodic rank varies within the CPS family, discriminating exactly where the harmonic axis (stuck on the P = S diagonal) cannot.
H-L4 — Wilson's 1965 rank–gap conjecture, the one I most want to test. In a letter to John Chalmers (21 Aug 1965), Wilson conjectured the minimum number of melodic step sizes grows with the odd-limit: 3-limit 2, 5-limit 3, 7-limit 4, 9-limit 5, 11-limit 6. The 9-limit case forks the conjecture, because 9 = 3² adds no new prime: does a {1,3,5,7,9} region behave like rank 3 (prime-rank reading → minimum 4 gaps) or like 4 independent identities (Wilson's literal claim → minimum 5)? The gap counter can discriminate directly. If Wilson's literal version wins, composite identities are load-bearing melodic dimensions even though they are lattice-interior harmonically — which would connect to the comma experiments below.

Future: the queue, and why each experiment is there

SHADOW-001 — comma perturbation of CPS factors. A practice of mine turned falsifiable: replace a CPS factor n with 2^k·n ± 1 (always odd, so co-prime with the octave). Pitch displacement halves per step of k, while the prime content changes completely — 255 = 3·5·17 sits 6.8¢ under the octave; 4095 = 3²·5·7·13 sits 0.42¢. Predictions: exact-coincidence triads drop for small k and recover discontinuously when displacement falls inside scorer ε; dedup behavior snaps at a threshold k* that shifts by exactly one per doubling of the dedup tolerance; and — the thesis I care about — perturbations that share factors with the base set (255 shares 3 and 5) beat prime perturbations of comparable size (257): connectivity beats pure novelty.

MOS-LAT-001 — the hidden lattice of a noble generator. Every quadratic-irrational generator is a loop in the scale tree; its continued fraction is a walk in GL(2,ℤ), and the MOS at each level should be exactly a cut-and-project set — lattice points whose conjugate coordinate falls in a window (the Fibonacci-chain picture). Step 1 verifies that construction against the plugin's own Brun recursion at every level. Step 2 asks whether triad-count hot spots across generators correlate with geometric descriptors of the conjugate embedding (window width, spectral gap) rather than with the generator value itself — with a permutation test, because "hot spots exist" is cheap and "hot spots are explained" is not. This would turn the MOS sweep's empirical bumps into arithmetic.

BRIDGE-000 — D'Alessandro as calibration standard. Wilson executed the melody⇄harmony bridge by hand in D'Alessandro (1975/1989): a 31-tone template — precisely the 31-EDO patent val — applied homomorphically to 38 just tones, producing exactly 7 duplicated degrees that realize three kernel commas (385/384, 2079/2048, 121/120), resolved by keyboard geometry rather than by tempering a single cent. The inverted version (Kraig Grady's marimba) keeps the val but lifts 11 differently — the two standard 11-limit meantone extensions, hiding in a pair of 50-year-old diagrams. The experiment: encode the val, reproduce all 38 placements and all 7 collisions against the primary-source scans, then run the frozen scorer on D'Alessandro — its first-ever quantitative harmonic-fidelity measurement. Every machine-proposed bridge that follows must beat it.

BRIDGE-001 — CPS structure inside an MOS. The endgame question: given an MOS whose hidden lattice approximates a Combination Product Set, does the CPS subset structure survive — can you still navigate hexanies inside it, with stepwise voice leading between subset chords? Rank accounting makes the search finite: embedding the {1,3,5,7} genus (rank 3) into an MOS (rank 2) means choosing exactly one comma to temper; enumerate kernels, reject any whose val order contradicts pitch order, measure triad survival on what remains. D'Alessandro proves a third regime exists — keep every pitch exactly just and temper only the addressing — where the cost is paid in duplicate keyboard positions instead of cents.

PARETO-001 — the melody × harmony frontier. With both axes scored, plot every scale the program has ever generated on (melodic score, harmonic score). The conjecture worth testing: Wilson's published designs sit on or near the Pareto frontier of that plot — that his lifetime of hand search was, in effect, multi-objective optimization, and the frontier's shape (smooth trade-off? sharp knee?) is itself a finding about what scale design is.

Further out, and speculative on purpose: 3-term recurrences seeded with CPS values (Meru/Rauzy-fractal windows, rank-3 cut-and-project), and temperament-as-feature — treating specific comma collisions not as error but as reinforcement.

An invitation: help me improve the scoring function

This is the part I actually want responses to. The scorer measures one aesthetic — mean-coincidence triads — and one new axis measures melodic viability. That is nowhere near a theory of what makes a tuning beautiful. Things I would genuinely like to argue about:

Aggregation. Given (P, S, G) and triad quality, what is the right harmonic objective? My ear rejected min(P, S). Balance buckets are a placeholder. Is the honest answer that no scalar exists and the Pareto view is the finished product?
Melody + harmony balance. Once PARETO-001 exists: is distance-to-frontier a musically meaningful penalty? Should the axes be weighted by use — a performance scale vs. a drone scale?
Missing dimensions. Difference-tone audibility weighting (register and amplitude matter; my counts ignore both). Critical-band roughness. Voice-leading economy between subset chords. Tetrads and beyond — Wilson's own CPS structures are built from higher-order chords, and a triad-only scorer under-values them almost by construction.
Counterexamples, most of all. Scales that sound extraordinary and score badly, or score brilliantly and sound dead. One good counterexample is worth more than any amount of agreement — the whole apparatus exists to be falsified.

Engage via GitHub Discussions for ideas and argument, or Issues for concrete scorer proposals and bug reports. Scales the program surfaces are exported as Scala .scl files in the results directories if you want to listen before you argue — and I hope you do listen first.

Production note, in the spirit of transparency: the experiments, code, and first drafts of documents like this one are produced by an AI agent (Claude) operating inside the research loop described above; I set the questions, review every finding against the primary sources and my own ear, and no measurement standard changes without my explicit approval. The mistakes that survive that process are mine.