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Contract: Push 4b tranche 2b — the ten historical temperaments

Repo root /home/jeans/Repos/active/epiphany. The plan is spec/PLAN_PUSH4B_TUNING.md. Tranche 2 (6fa14c7) landed the in-memory tuning resolver: nine of twenty systems resolve, the ten historical temperaments return TuningCatalogEntry::Deferred. This tranche makes those ten resolve. Read this file in full before editing anything.

Like tranches 1 and 2 it stays in memory — no Codec, no wire movement, canonical bytes byte-identical. It is deliberately its own pass, separate from the resolver plumbing, for one reason: this is where the arithmetic must be re-verified. The S6 draft that produced these constructions first shipped two temperaments that were arithmetically impossible and one false ambiguity — every one properly cited to a real source — and only the closure invariant caught them. The code re-derivation gets the same scrutiny. A temperament transcribed from its cents table without re-deriving it from the construction is the same defect wearing the same clothes.

Blast radius

crates/epiphany-core/src/tuning.rs — the extension points below, plus the ten construction descriptors and their tests.
crates/epiphany-core/src/pitch.rs — only if you add a TuningFunctionId catalog-id newtype there beside the others (pitch.rs:69 onward); otherwise leave it.
crates/epiphany-core/src/lib.rs — re-export any new public type.
crates/epiphany-core/DECISIONS.md — record the tranche.

You edit no .tex file and add no requirement, so the three counts in requirement_labels.rs stay 212 / 282 / 282. You touch no other crate. In particular crates/epiphany-editor-gui/** and spec/PLAN_EDITOR_APP.md are the user's parallel work — do not go near them.

The central prohibition, unchanged: no `Codec`, no wire movement

Canonical bytes MUST be byte-identical before and after. These types and this data stay in memory. Do not write a Codec impl for any new type, do not add a field to any encoded struct. If any golden file or fuzz digest moves, stop and report — nothing here should reach the wire.

The source of truth

core_spec.tex §"Temperament Constructions" (3696–4011), landed normatively at 5e465a1, is the authority. Each \subsubsection gives one construction: which fifths are tempered, by what fraction of which comma, and — for the non-circulating ones — where the wolf falls. It also gives derived cents tables.

The fifth-tempering rules are the construction; the cents tables are derived. Build each temperament by walking the circle of fifths and applying the tempering rules — do not paste the cents tables in as data. The tables are there for you to check your walk against, exactly as tranche 2 computed the ji-static ratios from the lattice block rather than pasting them. A walk that reproduces the spec's derived cents and closes correctly is right; a pasted table proves nothing and re-derives nothing.

What to build

1. The `Function` variant and its plumbing

TuningResolution (tuning.rs:95) currently has two variants. Add the spec's third, Function { function: TuningFunctionId, parameters: TuningParameters } (core_spec.tex:3320). This is where historical temperaments live, per the variant's own doc.

TuningFunctionId — a catalog_id! string newtype like the others (pitch.rs:69+). The ten temperaments use reserved built-in ids (their catalog identifiers: "pythagorean", "werckmeister-iii", …).
TuningParameters — its shape is not specified anywhere in core_spec.tex and no built-in parameterizes (each construction is fixed by its id). Define it as a documented zero-field / marker type, exactly as tranche 2 handled SpellingParameters and the unconstructed payloads. Report it. Do not invent a parameter schema.

Then teach the two resolver functions the new variant:

coordinate_ratio(resolution, s) (tuning.rs, the match on resolution): a Function arm that, for a reserved built-in id, returns the temperament's ratio at coordinate s — degree = s.rem_euclid(12), octave = s.div_euclid(12), ratio = temperament_ratios[degree] · 2^octave. An unknown TuningFunctionId returns None (fails closed — Function is an extension point and no registry exists).
frequency_for_position's divisions match: for Function, divisions is the pitch space's chromatic cardinality (12 for cmn-12), taken from structure, not from the resolution — the Function variant carries no division count.

2. The ten construction descriptors, and the walk

Represent each temperament by its construction — the twelve fifths of the circle C–G–D–A–E–B–F♯–C♯–G♯–E♭–B♭–F–(C), each tagged with its tempering — and one walk that places the twelve notes and derives their ratios. Compute the comma sizes from their exact ratios, not from hardcoded rounded cents:

pure fifth = 3/2; Pythagorean comma = 531441/524288; syntonic comma = 81/80; schisma = 32805/32768 (= Pythagorean − syntonic — and note syntonic + schisma = Pythagorean exactly, which is why the Kirnberger closure works out; see below). Take each as 1200·log2(ratio) in f64. Frequencies are non-canonical (req:determinism:canonical-floating-point), so f64 throughout is correct — there is no exact-rational requirement here, and several fifths (meantone 1/5-, 1/6-comma) are irrational by construction.

The ten, from the spec section (verify each against core_spec.tex, do not take this summary as the source):

Non-circulating — eleven fifths specified, the twelfth is the closing wolf (the residue that brings the circle to seven octaves):

pythagorean — eleven pure fifths; conventional cut E♭–G♯, wolf G♯→E♭.
meantone-1/4-comma, -1/5-comma, -1/6-comma — eleven fifths each narrowed by 1/4, 1/5, 1/6 of the syntonic comma; same E♭–G♯ cut; the twelfth is the wide wolf.

Circulating — all twelve fifths specified, temperings summing to exactly one Pythagorean comma, no wolf:

werckmeister-iii — C–G, G–D, D–A, B–F♯ narrowed 1/4 Pythagorean; eight pure.
werckmeister-iv — C–G, D–A, E–B, F♯–C♯, B♭–F narrowed 1/3 Pythagorean; G♯–E♭ and E♭–B♭ widened 1/3 Pythagorean; five pure.
vallotti — F–C, C–G, G–D, D–A, A–E, E–B narrowed 1/6 Pythagorean; six pure.
kirnberger-ii — D–A, A–E narrowed 1/2 syntonic; F♯–D♭ (the closing fifth) narrowed 1 schisma; nine pure.
kirnberger-iii — C–G, G–D, D–A, A–E narrowed 1/4 syntonic; F♯–D♭ narrowed 1 schisma; seven pure.
young-ii — C–G, G–D, D–A, A–E, E–B, B–F♯ narrowed 1/6 Pythagorean; six pure (the same construction as vallotti, rotated to start at C).

Which comma is the load-bearing distinction, and no test outside the closure check can see it. Werckmeister/Vallotti/Young temper by the Pythagorean comma; meantone and the Kirnberger fifths by the syntonic; Kirnberger's closing fifth by the schisma. Confusing syntonic and Pythagorean is the classic error, and it was ratified (via closure) that Werckmeister uses Pythagorean. The closure check below is what catches a wrong choice.

The schisma-fifth trap. kirnberger-ii and -iii each temper their closing F♯–D♭ fifth by a schisma. It is the single most commonly omitted element of these constructions — its omission is exactly what made the S6 draft's first Kirnbergers impossible. Without it the closure lands one schisma (1.9537 c) short of the Pythagorean comma. Include it; the closure check will catch you if you don't.

3. Resolve them in the catalog

built_in_tuning_system's temperament arm (tuning.rs:245-247) currently returns Deferred(DEFERRED_TEMPERAMENT). Return Resolved(TuningSystem { … resolution: Function { function: TuningFunctionId::new(id), parameters: … } … }), pitch_space = cmn-12. ji-adaptive-5limit stays Deferred (still needs HarmonicContext).

Proof of life — the closure invariant, recomputed in code

This is the tranche's reason to exist, so its tests are the deliverable. For each of the ten, a test that recomputes the closure from the walk and asserts it — not a hardcoded constant:

Six circulating (werckmeister-iii/-iv, vallotti, kirnberger-ii/-iii, young-ii): the sum of the twelve fifths' deviations from pure equals one Pythagorean comma (1200·log2(531441/524288), ≈ 23.4600 c) within a tight tolerance, and no fifth is a wolf (every fifth within a small bound of pure).
Four non-circulating (pythagorean, the three meantone-*): the residual twelfth fifth (the wolf) equals the spec's stated value — pythagorean 678.495 c, meantone-1/4 737.637 c, -1/5 725.809 c, -1/6 717.923 c — within tolerance. "Does not close" is the positive claim here.

Then a handful of discriminating derived-value checks, each reproducing a spec-table value from the walk (so the walk is validated against the spec's independently-derived cents, not just against itself):

pythagorean E = 407.820 c, F♯ = 611.730 c.
meantone-1/4-comma's major third C→E = 386.31 c (the just 5/4, meantone's defining property).
kirnberger-ii D = 203.910 c vs kirnberger-iii D = 193.157 c — same chain skeleton, different tempering; a test that passes both proves the two are not the same construction.

And the resolver-level checks:

All ten resolve (no longer Deferred); a frequency comes out. E.g. werckmeister-iii's C♯ resolves distinctly from tet-12's C♯.
An unknown TuningFunctionId fails closed (the extension-point path returns the resolver's not-supported error, never a frequency).

Verification

Mutation-verify every new test, and let the closure tests earn their keep by mutating the construction, not just the assertion:

Drop the schisma fifth from kirnberger-ii → its closure test must fail (lands one schisma short). This is the S6 trap, reproduced as a mutation.
Change werckmeister-iii's comma from Pythagorean to syntonic → its closure test must fail (4 × ¼ syntonic = 21.506 ≠ 23.460). This proves the comma-type distinction is actually checked.
Plus the ordinary per-test mutation for the discriminators and the fail-closed path.

Assert the anchor text is present before each substitution (a no-op mutation looks like a passing test). Restore by reversing the exact substitution — never git checkout, which discards all uncommitted work in the file. Report each mutation and the test that died.

Then the full gate, reporting actual commands and output:

cargo fmt --all --check
cargo clippy --workspace --all-targets → 0 warnings
cargo test --workspace → 0 failed
RUSTDOCFLAGS="-D warnings" cargo doc --workspace --no-deps → 0
cargo run -q -p epiphany-testkit --example conformance_suite → 8/8
cargo test -p epiphany-testkit --test requirement_labels → 6 passed, counts unchanged at 212 / 282 / 282.

Zero golden churn and zero canonical-byte movement expected — in-memory only.

CI pins clippy at 1.95.0 and the MSRV at 1.85; local cargo clippy is CI's toolchain.

Citations

If you cite a req:* label in a doc comment, read the requirement first and confirm it says what your sentence claims (P13-S9: the checker proves the string resolves, not that it is relevant). Labels in play: req:tuning:builtin-tuning-catalog, req:tuning:tuning-resolution-determinism, req:determinism:canonical-floating-point.

Do not

Write a Codec impl, or let any new type or datum reach the wire.
Paste the spec's cents tables as data — build from the fifth-tempering construction and check against the tables.
Hardcode a rounded comma value — derive each comma's cents from its exact ratio.
Invent a TuningParameters schema, or a frequency for a system you cannot build.
Omit the Kirnberger schisma fifth, or use the wrong comma for any temperament.
Edit any other crate, any .tex, spec/PLAN_EDITOR_APP.md, or any DECISIONS.md but epiphany-core's.
Restore a mutation with git checkout. Reverse the substitution.
Re-bless a golden or move a requirement count to make a test pass.

Report

State: the extension points touched; how each of the ten is represented (the descriptor shape and the one walk); TuningParameters as you defined it; the closure value your code computed for each of the ten (the six Pythagorean-comma sums and the four wolves), against the ratified spec values; the two construction-level mutations (dropped schisma, wrong comma) and that they failed; the ordinary mutations; the actual gate output; whether any golden or digest moved; and anything you chose not to do and why.

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