The Tunings That Ring

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

Erv Wilson spent his lifetime searching, by hand, for scales that resolve one of music's oldest tensions. A scale that serves melody wants to be stepwise and singable and built from just a few step sizes, and a scale that serves harmony wants chords whose combination tones reinforce each other rather than beat against each other, and Erv wanted structures that do both at the same time. I worked at his side from 1995 to 2005, turning his diagrams into software while he drew the next one. Erv passed away in 2016, and I have kept going because the search he started was never finished. In this post I describe how I have tried to make that search falsifiable, with a scoring function anyone can read, a queue of experiments whose predictions are registered before they run, and an open invitation to tear both apart.

I want to say two things up front. First, I run the entire computational program in collaboration with an AI agent (Claude), and I will be precise below about what that means and why I believe it works. Second, everything here is public, including the scorer, the receipts, and the failures. If you think the scoring function is wrong, and in at least one measured way I know it is incomplete, then I want to hear from you. Skip to the invitation if that is 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 of the math happens in frequency space rather than in cents:

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

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

Several conventions make the numbers mean something, so I will spell them out:

Exact arithmetic for JI. The rational path runs on fractions.Fraction from end to end, so there are no floats and no epsilon anywhere in it. 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). I chose this over the naive two-octave window because it is exactly self-dual, meaning that inverting the scale swaps the proportional and subcontrary counts (P ↔ S), and it is exactly transposition-invariant. The window convention fails both tests, so 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 it is not the chord this criterion is about. I validated this rule by ear, and 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. I explain below why this guard exists, because it has an instructive origin story.

Known limitations

I have measured these rather than hedged about them, because I believe 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. The consequence is that the aggregate min(P, S), which was once primary, cannot discriminate within these families at all, so I have demoted it to a statistic that is recorded but no longer optimized.
The aggregator is unsettled, and my ear is the reason. My listening check validated the classifier, because the triads it finds really do lock, but it rejected min(P, S) as a ranking. I found that I want proportional-heavy and subcontrary-heavy and geometric-heavy scales as distinct musical characters rather than as a scalarized compromise. Current reporting uses balance buckets instead, and an aggregator I would be willing to freeze 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, and 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, which makes it twice as strict at the top of the octave, and nothing in the theory motivates that. I measured this against the real compiled C++, and I am tracking it as a plugin issue because I did not want it 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 one 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.

The inner loop is plain Python with no AI in it: exhaustive and quality-diversity batch search over scale families (Combination Product Sets and MOS generators), with every candidate generated, scored, and archived, headless and reproducible.
The outer loop is an AI agent (Claude) that proposes a hypothesis, writes the pre-registration entry before the run happens, including the prediction, the tolerances, and the expected failure modes, then implements, runs, and files the verdict as kept-or-reverted. The surprises become the next hypotheses.
I hold three things the loop cannot hold. I decide what is worth asking, my ear supplies the ground truth that the metric only approximates, and I decide when a measurement standard is frozen.

Why does this work for scale aesthetics in particular? Because 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 it takes a human to test one. And Wilson's own practice was already shaped like a loop, because he would draw, listen, and redraw. The machine runs the draw-and-score part of that loop at scale, and it keeps receipts.

The obvious objection is Goodhart's law, and it bit us immediately, in a way I found both alarming and instructive. 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, because if you cram all the tones together then every triple is within ε of both mean conditions simultaneously. The scale is musical nonsense, but the score was perfect, and seeing that happen so quickly convinced me that the scorer had to be protected from the thing optimizing against it. That episode is why the scorer is frozen, pinned by SHA-256, enforced by CI, and changeable only with my explicit approval, and it is why the degeneracy guard exists, because an optimizer that can edit its own verifier is not really being verified. I accept that the metric may be wrong, but I will not let it drift silently while something optimizes against it.

The second firewall is my ear. Twice now the listening check has overruled the machine, once by validating the triad classifier while rejecting the min(P, S) aggregator, and once by 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. We reproduced 70 hexanies with zero mismatches and 15 MOS scales within 1 ulp against the real compiled C++, so I can trust that an offline score describes the scale the plugin would actually produce.
All 29 eikosanies from odd seeds ≤ 15 sit exactly on P = S. This is an instance of the inversional-symmetry theorem above, and I enjoyed the order in which it arrived, because the loop discovered it empirically first and we proved it structural afterward.
MOS generator sweeps show triad-count hot spots that are stable across cardinality, near 571.6¢ at odd cardinalities, at the pure fourth at N = 12, and at 416¢ at N = 20. Mapping why these spots are hot is an open experiment, which I describe below.
The melodic axis exists as of this week. It consists of three scorers, gap-size entropy (which lives in three-gap-theorem territory), Wilson's constant-structure criterion, and Rothenberg propriety, and each one is 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, and this genuinely surprised me. The 12-EDO diatonic is not a constant structure, because the 600¢ tritone subtends both 3 and 4 steps. And the 12-tone Pythagorean chromatic is strictly proper, because the classic improper scale is actually the Pythagorean diatonic, whose 611.7¢ augmented fourth exceeds its 588.3¢ diminished fifth. I was glad to see the harness catch errors in its own spec's assumptions before running a single experiment, because that is exactly what I built it to do.

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). The predictions were registered before the run, and I quote them 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 claim has long been attributed to Wilson, and I would rather verify it than keep assuming it, and either outcome will be a finding.
H-L2: CPS scales are systematically improper and multi-gap relative to MOS at matched cardinality, which would quantify the folk claim that "CPS lacks melodic continuity," but melodic rank varies within the CPS family, so it discriminates exactly where the harmonic axis, stuck on the P = S diagonal, cannot.
H-L4 is Wilson's 1965 rank–gap conjecture, and it is the one I most want to test. In a letter to John Chalmers dated 21 Aug 1965, Wilson conjectured that the minimum number of melodic step sizes grows with the odd-limit: 2 at the 3-limit, 3 at the 5-limit, 4 at the 7-limit, 5 at the 9-limit, and 6 at the 11-limit. 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, which is the prime-rank reading and predicts a minimum of 4 gaps, or like 4 independent identities, which is Wilson's literal claim and predicts a minimum of 5? The gap counter can discriminate directly. If Wilson's literal version wins, then composite identities are load-bearing melodic dimensions even though they sit in the interior of the harmonic lattice, and that would connect to the comma experiments below.

Future: the queue, and why each experiment is there

Where do we go from here? Each experiment in the queue is there because it tests something I cannot currently answer, and I want to explain the reason alongside the design.

SHADOW-001 — comma perturbation of CPS factors. This takes a practice of mine and makes it falsifiable. I replace a CPS factor n with 2^k·n ± 1, which is always odd and therefore co-prime with the octave. The pitch displacement halves with each step of k while the prime content changes completely, so for example 255 = 3·5·17 sits 6.8¢ under the octave and 4095 = 3²·5·7·13 sits 0.42¢ under it. I predict that exact-coincidence triads drop for small k and recover discontinuously when the displacement falls inside the scorer's ε, that the dedup behavior snaps at a threshold k* which shifts by exactly one per doubling of the dedup tolerance, and, this is the thesis I care most about, that perturbations which share factors with the base set (255 shares 3 and 5) will outperform prime perturbations of comparable size such as 257, because staying connected to the existing factors matters more than the novelty of a new prime.

MOS-LAT-001 — the hidden lattice of a noble generator. Every quadratic-irrational generator is a loop in the scale tree, and its continued fraction is a walk in GL(2,ℤ), so the MOS at each level should be exactly a cut-and-project set, meaning lattice points whose conjugate coordinate falls in a window, which is the Fibonacci-chain picture. Step 1 verifies that construction against the plugin's own Brun recursion at every level. Step 2 asks whether the triad-count hot spots across generators correlate with geometric descriptors of the conjugate embedding, such as window width and spectral gap, rather than with the generator value itself. I insist on a permutation test here, because showing that hot spots exist is easy and showing that something explains them is much harder. If this works it 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). He applied a 31-tone template, which is precisely the 31-EDO patent val, homomorphically to 38 just tones, producing exactly 7 duplicated degrees that realize three kernel commas (385/384, 2079/2048, 121/120), and he resolved them with keyboard geometry rather than by tempering a single cent. The inverted version, which Kraig Grady built as a marimba, keeps the val but lifts 11 differently, and the two versions turn out to be the two standard 11-limit meantone extensions, hiding in a pair of 50-year-old diagrams. The experiment is to encode the val, reproduce all 38 placements and all 7 collisions against the primary-source scans, and then run the frozen scorer on D'Alessandro, which would be its first quantitative harmonic-fidelity measurement. I want every machine-proposed bridge that follows to have to beat it.

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

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

Further out, and speculative on purpose, are 3-term recurrences seeded with CPS values (Meru and Rauzy-fractal windows, rank-3 cut-and-project), and treating temperament as a feature, so that specific comma collisions are heard as reinforcement instead of error.

An invitation: help me improve the scoring function

This is the part I actually want responses to. The scorer measures one aesthetic, the mean-coincidence triad, and the one new axis measures melodic viability, and together that is nowhere near a theory of what makes a tuning beautiful. Here are the 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), and the balance buckets I use now are a placeholder. Is the honest answer that no scalar exists and the Pareto view is the finished product?
Balancing melody and harmony. Once PARETO-001 exists, is distance-to-frontier a musically meaningful penalty? Should the axes be weighted by use, so that a performance scale and a drone scale are judged differently?
Missing dimensions. Difference-tone audibility weighting, because register and amplitude matter and my counts ignore both. Critical-band roughness. Voice-leading economy between subset chords. Tetrads and beyond, because Wilson's own CPS structures are built from higher-order chords, so a triad-only scorer under-values them almost by construction.
Counterexamples, most of all. I am looking for scales that sound extraordinary and score badly, or that score brilliantly and sound dead. One good counterexample would teach me more than any amount of agreement, because the whole apparatus exists to be falsified.

You can engage via GitHub Discussions for ideas and argument, or Issues for concrete scorer proposals and bug reports. The scales the program surfaces are exported as Scala .scl files in the results directories, so you can listen before you argue, and I hope you do listen first.

Production note, in the spirit of transparency: the experiments, the code, and the 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, I review every finding against the primary sources and my own ear, and no measurement standard changes without my explicit approval. Whatever mistakes survive that process are my responsibility.