diff --git a/spec/CONTRACT_P13S6_TEMPERAMENTS.md b/spec/CONTRACT_P13S6_TEMPERAMENTS.md new file mode 100644 index 0000000..565b68c --- /dev/null +++ b/spec/CONTRACT_P13S6_TEMPERAMENTS.md @@ -0,0 +1,138 @@ +# Contract: P13-S6 — drafting the built-in tuning constructions for review + +Repo root `/home/jeans/Repos/active/epiphany`. The plan is +`spec/PLAN_PUSH4B_TUNING.md`, Ruling B. Read it before starting. + +## What this is, and why it is a draft + +`req:tuning:builtin-tuning-catalog` makes all **20** built-in tuning +identifiers MUST-resolve with normative semantics. Only the six `tet-*` are +actually specified — structurally, by `TuningResolution::EqualTemperament`. The +other 14 are bare names (P13-S6). Ruling B keeps all 20 and pins them **by +construction** rather than by cents table, because published sources agree on +the constructions and differ mainly in how they round cents. + +**You are producing a reviewable draft, not a specification edit.** Write +`spec/DRAFT_P13S6_TEMPERAMENTS.md`. **Do not edit any `.tex` file.** Promotion +into `core_spec.tex` happens only after human review. + +The reason is specific and this project has paid for it: `NOTEHEAD_ANCHORS` was +a table of hand-written glyph metrics that looked authoritative, was wrong in two +independent ways, and survived for passes because nothing consumed it. A tuning +construction written from memory into a normative document is the same artifact. +**Confidence is not a source.** + +## The deliverable + +For each of the 14 unspecified systems, one entry: + +* **Identifier** exactly as the catalog spells it. +* **Construction** — the generative rule, stated so an implementer can compute + it without further research. For a circulating temperament that means: which + fifths are tempered, by what fraction of *which* comma (syntonic vs + Pythagorean — say which; they are different and confusing them is the classic + error), and which remain pure. For a JI scale it means the ratio for each of + the twelve degrees. +* **Wolf / chain placement** where the construction does not determine it — + `pythagorean` in particular is a pure 3:2 chain whose twelve-note selection is + a *choice*, and the choice must be stated, not assumed. +* **Source** — a specific, checkable citation: author, work, date, and where the + construction appears. "Common knowledge" is not a source. +* **Confidence**, explicitly: `verified` (you can cite it precisely), + `recalled` (you believe it but cannot cite it precisely), or `unknown`. + +**A `recalled` or `unknown` entry is a useful result and an honest one.** A +fabricated citation is the worst possible output of this task — worse than +leaving the row blank — because it defeats the review that exists to catch it. +If you cannot source something, say so in that entry and move on. + +The 14: `pythagorean`; `meantone-1/4-comma`, `meantone-1/6-comma`, +`meantone-1/5-comma`; `werckmeister-iii`, `werckmeister-iv`; `vallotti`; +`kirnberger-ii`, `kirnberger-iii`; `young-ii`; `ji-static-5limit-C`, +`ji-static-5limit-G`, `ji-static-5limit-D`; `ji-adaptive-5limit`. + +## Four are open ratifications — surface, do not decide + +The plan marks these as needing a human decision. **Present the options and +their trade-offs; do not pick.** + +* **`ji-static-5limit-C` / `-G` / `-D`.** Which 12-note 5-limit scale — the comma + choices for the chromatic degrees are genuinely unsettled and more than one + selection is defensible. Lay out the main candidates and what each optimizes. + Note whether the three differ only by transposition of one scale or are + independently chosen. +* **`ji-adaptive-5limit`.** Needs a real algorithm, constrained by + `req:tuning:adaptive-tuning-purity` — **read that requirement first** and state + what it permits. The in-house pattern for this shape is + `req:pitch:spelling-algorithm`, which pins `SpellingAlgorithmId "default"` at + version 1 to a named algorithm and errors on any other identifier; recommend + the analogous versioned `AdaptiveTuningFunctionId`, but do not invent the + algorithm's content as though it were settled. + +## Cents tables are derived, never primary + +If you include cents, mark them **derived from the construction** and give the +derivation. Do not copy a cents table from a source and present it as the +definition — that reintroduces exactly the rounding variance Ruling B exists to +avoid. Exact-ratio form is preferred wherever a construction yields rationals; +say plainly where it does not (quarter-comma meantone's fifth is $5^{1/4}$, and +several others are irrational by construction). + +Note for context, so you do not over-engineer precision: resolved frequencies +never enter canonical bytes (`req:determinism:canonical-floating-point` lists +tuning among the floating-point contexts and bars floating point from anything +needing exact identity), and acoustic comparison is governed by +`ToleranceClass::AcousticCents` in `epiphany-determinism`. + +## Check the arithmetic, not just the source + +**A citation proves the source said it. It does not prove it is right, that you +read it correctly, or that you transcribed it completely.** Sourcing catches +fabrication; only an invariant catches a real source misapplied. This section +was added after a first draft produced two impossible temperaments and one false +ambiguity, every one of them properly cited. + +Where the data has an internal invariant, **compute it and show the computation +in the entry**. For tuning constructions the invariants are: + +1. **Closure.** A circulating (wolf-free, twelve-note) temperament's twelve + fifths MUST absorb **exactly one Pythagorean comma** — $531441/524288$, + 23.4600 cents. Narrowings count positive, widenings negative. State the sum. + A construction that does not close is not a well temperament, and if your + sum lands 1.9537 cents short you have dropped a **schisma**-tempered fifth — + the difference between the Pythagorean and syntonic commas, and the single + most commonly omitted element of these constructions. +2. **Closure is also a decision procedure.** Where a source is vague about + *which* comma a fraction refers to, closure usually settles it: try both and + report which one closes. Do not carry an ambiguity forward that arithmetic + resolves — a normative hedge on an answerable question is worse than either + answer. +3. **Non-circulating temperaments must NOT close.** Meantone and Pythagorean + have a wolf by construction. Compute the wolf's size — it is the residue — + and state it. "Does not close" is a positive claim here, not a gap. +4. **Twelve fifths, each exactly once.** Enumerate the full circle and classify + every fifth as narrowed, widened, or pure. A construction naming eleven or + thirteen is wrong on its face; adjacent fifths sharing a note are two + distinct fifths. +5. **Recompute every derived claim.** If a source says a third is pure, derive + it from your own chain and confirm. If your derivation disagrees with the + source, that is a finding — report it, do not quietly follow either one. + +If an invariant fails, **say so in the entry and mark it unresolved**. Do not +adjust the construction to make the arithmetic work and present the result as +sourced — that manufactures a temperament nobody published. + +## Do not + +* Edit any `.tex`, any Rust file, or any other agent's file. **Your entire output + is one new Markdown file.** +* Take values from anywhere in this repository — there are none, and anything you + think you found is something else. +* Present a construction you cannot source as though you could. +* Decide any of the four open ratifications. + +## Report + +State how many of the 14 you marked `verified`, how many `recalled`, how many +`unknown`. That ratio is the single most useful number in your report, and a low +`verified` count is a finding about the task, not a failure of it. diff --git a/spec/DRAFT_P13S6_TEMPERAMENTS.md b/spec/DRAFT_P13S6_TEMPERAMENTS.md new file mode 100644 index 0000000..a37804a --- /dev/null +++ b/spec/DRAFT_P13S6_TEMPERAMENTS.md @@ -0,0 +1,907 @@ +# Draft: P13-S6 built-in tuning constructions + +**Status: non-normative draft. Not part of the specification. Nothing here +is true by virtue of being written down — it is true (or not) by virtue of +the citation attached to it, and every citation is graded.** + +Produced under `spec/CONTRACT_P13S6_TEMPERAMENTS.md`, against Ruling B of +`spec/PLAN_PUSH4B_TUNING.md`. Scope: the 14 `req:tuning:builtin-tuning-catalog` +identifiers (`core_spec.tex:3531-3546`) that are not one of the six `tet-*` +equal temperaments. Promotion of any entry below into `core_spec.tex` requires +a human reviewer to check the citation and accept it — that review is the +entire reason this is a separate file and not an edit to Chapter 4. + +Research for this draft was done by fetching and reading primary web sources +today (2026-07-22): raw Wikipedia article text (via the MediaWiki `action=raw` +endpoint, so what is quoted below is the article's own wording, not a +third-party summary of it) for `Pythagorean_tuning`, `Meantone_temperament`, +`Werckmeister_temperament`, `Vallotti_temperament`, `Kirnberger_temperament`, +`Young_temperament`, and `Five-limit_tuning`; and (added in the correction +round below) the raw HTML of Carey Beebe's "CBH Technical Library" practical +tuning-instruction pages for Kirnberger II and III. Every numeric claim +carried through to a cents figure below was independently recomputed from the +stated ratio/fraction (shown as "self-check" where done) rather than copied +from a rounded table — this is what Ruling B's "cents are derived, never +primary" constraint means in practice. Where a source names a page-specific +scholarly citation (author, title, year, page), that citation is given +alongside as the primary source a reviewer would actually want to pull. + +**Revision note (correction round).** A reviewer ran the closure invariant +(below) against the first version of this draft and found three entries +wrong, all sharing one root cause: the draft cited a real source for *part* +of a construction and then silently assumed "everything else is pure" +without checking that the resulting temperament could actually close a +twelve-note circle. `kirnberger-ii` and `kirnberger-iii` were each missing a +schisma-tempered fifth that their own source's diagram caption named but +this agent's first pass never located; `werckmeister-iii` carried an +unresolved comma-type hedge into the summary table on a question closure +actually answers. All three are corrected below, with the new source that +supplied the missing element and the arithmetic that catches the defect +shown in place, per the contract's amended "Check the arithmetic, not just +the source" section. Nothing was adjusted to make the numbers work and then +presented as sourced — where the correction required a new source, the new +source is cited; nothing here is back-derived and passed off as historical. + +## How to read each entry + +* **Construction** — the generative rule: which fifths are tempered, by what + fraction of which comma (syntonic vs. Pythagorean, stated explicitly), and + which fifths are left pure. +* **Chain/wolf placement** — stated explicitly wherever the construction + alone under-determines it. +* **Closure check** — for every circulating (wolf-free) construction: all + twelve fifths enumerated exactly once each (narrowed / widened / pure), and + their signed sum shown to equal exactly one Pythagorean comma + (531441/524288, 23.4600 cents) — the amount twelve fifths must fall short + of 12×701.955 c for a twelve-note chain to return to its exact starting + pitch after seven octaves. For the non-circulating constructions + (`pythagorean`, the three meantones), the same sum is computed and shown + to deliberately **not** equal one comma — the residual is the wolf, stated + as a positive fact, not a gap. Where a source left the comma type + ambiguous, both readings are computed and whichever one actually closes is + reported as the answer, per the contract's "closure is also a decision + procedure" instruction. +* **Cents (derived)** — always marked derived, with the arithmetic shown. + Exact-ratio form is given wherever the construction yields one. +* **Source** — what was actually read, with URL and access date, plus the + upstream scholarly citation where the source names one. +* **Confidence** — `verified` / `recalled` / `unknown`, per the contract, + under the stricter meaning this correction round establishes: `verified` + now requires **both** a specific, quoted, checkable source **and** a + passing closure/enumeration check computed independently by this agent. A + source citation alone, without the arithmetic check, is not sufficient to + claim `verified` any more — that gap is exactly what produced the three + errors this round fixes. `verified` still does **not** mean a primary + 18th-century treatise or a physical copy of Barbour (1951) was consulted + directly by this agent — where a source names a page number in Barbour, + that page is reported as *that source's* citation, not independently + checked against the physical book. + +## Shared context (verified, `core_spec.tex`) + +`core_spec.tex:2822-2825` ("Design Principles", Chapter "Tuning Systems and +Pitch Spaces"): *"a single pitch space (e.g., CMN) admits many tuning systems +(12-TET, meantone, well-temperaments, just intonation, Pythagorean)"*. All 14 +constructions below are therefore constructions **over `cmn-12`'s twelve +chromatic positions per octave** (C, C♯/D♭, D, ... B), not over the +open-ended `ji-5limit` lattice pitch space — that space is a separate +built-in with its own (already-specified) structure. This reading is the only +one that makes "12-note selection" and "wolf fifth" meaningful for these +identifiers at all; it is not separately stated for each identifier in +Chapter 4, so flag it to reviewers as an inference from the design-principles +text, not a quotation naming these specific 14 identifiers. + +`req:tuning:adaptive-tuning-purity` (`core_spec.tex:3327-3333`) — read in full +before drafting §14: *"Adaptive tuning resolution MUST be a pure function of +position and harmonic context. Implementations MAY cache resolution results, +but MUST invalidate caches when the harmonic context changes."* This is the +entire constraint the requirement places on `ji-adaptive-5limit`; it says +**that** resolution must be pure and cache-safe, not **what** the function +computes. It permits (does not itself choose) note-by-note nearest-just-ratio +resolution against a concurrent sonority, roman-numeral/scale-degree-driven +resolution, a fixed lattice walk from the most recent tonicization, or other +designs — see §14. + +## Section A — the ten pinned constructions + +### 1. `pythagorean` + +**Construction.** A chain of eleven pure 3:2 fifths (twelve notes), all other +intervals derived from stacked fifths reduced by octaves. No comma is +tempered anywhere in the chain; the entire Pythagorean comma (≈23.460 cents) +is concentrated in the single interval where the chain does not close. + +**Chain / wolf placement (a stated choice).** The twelve-note selection is +not forced by "pure fifths" alone — it is a choice of *which* eleven +consecutive fifths to keep, i.e., where to cut the infinite spiral. The +conventional cut runs **E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯** (11 fifths, C at the +center), leaving the wolf on the diminished sixth **G♯–E♭**. An equally valid +alternative cuts the spiral one step the other way (chain D♭–...–F♯, wolf at +F♯–D♭); Wikipedia's own Pythagorean-tuning article gives both and states +explicitly that the wolf's position is relocatable this way. The E♭–G♯ +convention is stated here as *the* construction because it is the +overwhelmingly common default in both historical and pedagogical sources, but +it is being stated, not silently assumed, per the contract. + +**Ratios (exact, self-derived from the stacked-fifths definition, C = 1/1):** + +| Note | Ratio | Cents (derived) | +|---|---|---| +| C | 1/1 | 0.000 | +| C♯ | 2187/2048 | 113.685 | +| D | 9/8 | 203.910 | +| E♭ | 32/27 | 294.135 | +| E | 81/64 | 407.820 | +| F | 4/3 | 498.045 | +| F♯ | 729/512 | 611.730 | +| G | 3/2 | 701.955 | +| G♯ | 6561/4096 | 815.640 | +| A | 27/16 | 905.865 | +| B♭ | 16/9 | 996.090 | +| B | 243/128 | 1109.775 | + +Wolf fifth G♯→E♭ (octave-reduced): ratio 262144/177147 ≈ 678.495 cents, +i.e. a Pythagorean comma (23.460 c) narrower than the pure 701.955 c fifth. +All figures independently recomputed by this agent from `(3/2)^n` +octave-reduced; they match the Wikipedia article's stated 678.49 c / 701.96 c +to the precision given there. + +**Closure check (non-circulating — this is the positive claim, not a gap).** +Twelve fifths, each exactly once: 11 pure (E♭–B♭, B♭–F, F–C, C–G, G–D, D–A, +A–E, E–B, B–F♯, F♯–C♯, C♯–G♯) at 701.955 c each, plus the closing G♯–E♭ at +678.495 c. Sum = 11 × 701.955 + 678.495 = 8400.000 c exactly (= 7 octaves, +as it must — any assignment of 12 distinct pitch classes closes arithmetically +by definition). But note what that sum is built from: **relative to twelve +*pure* fifths** (12 × 701.955 = 8423.460 c), this construction is short by +exactly 8423.460 − 8400.000 = 23.460 c — one full Pythagorean comma, dumped +entirely onto the single G♯–E♭ interval rather than distributed. That +concentration, not the arithmetic closure, is what makes it a wolf and makes +Pythagorean tuning non-circulating: an implementer cannot treat all 12 +"fifths" as interchangeable the way a well temperament's are. + +**Source.** Wikipedia, "Pythagorean tuning", + (fetched 2026-07-22). +Quoted: *"Starting from D for example (D-based tuning), six other notes are +produced by moving six times a ratio 3:2 up, and the remaining ones by moving +the same ratio down: E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯"*; *"one may use only the 12 +notes from E♭ to G♯ ... The remaining interval (the diminished sixth from G♯ +to E♭) is left badly out-of-tune"*; *"a C-based Pythagorean tuning would +produce a stack of fifths running from D♭ to F♯, making F♯–D♭ the wolf +interval"*; comma value *"≈ −23.460 cents"*. + +**Confidence: verified.** + +--- + +### 2–4. The three meantone variants — `meantone-1/4-comma`, `meantone-1/5-comma`, `meantone-1/6-comma` + +**Construction (all three, differing only in fraction).** Meantone is a +*regular* temperament: every one of the twelve fifths in the chain is +tempered by the **same** fraction of the **syntonic** comma (81:80, +≈21.506 cents) — not the Pythagorean comma. This is the classic +confusable pair the contract calls out, and meantone is the case where +getting it backwards is easy because Werckmeister/Vallotti/Kirnberger/Young +(§5-10 below) use the Pythagorean comma instead. Quarter-comma meantone is +the best-known member and the one usually meant by "meantone" unqualified; +1/5- and 1/6-comma are documented alternate points on the same continuum in +the same source. + +Because the temperament is regular (all fifths equal), there is no "which +fifths are tempered" question the way there is for the well temperaments +below — all twelve are, uniformly. There *is* still a twelve-note selection +choice (the closing point of the spiral), structurally identical to +Pythagorean's; the conventional cut again places the wolf between G♯ and E♭. + +**Cents (derived).** Fifth ratio = (3/2) / (81/80)^(1/n) for 1/n-comma. Exact +form for n=4 collapses to 5^(1/4) (self-verified below); n=5, n=6 are +irrational and left in that form, per the contract's instruction that not +every construction yields a rational. + +| Variant | Fifth ratio | Cents (derived) | Narrowing vs. pure 3:2 | +|---|---|---|---| +| 1/4-comma | (3/2)·(80/81)^(1/4) = 5^(1/4) exactly | 696.578 | 5.377 c = ¼ · 21.506 c | +| 1/5-comma | (3/2)·(80/81)^(1/5) | 697.654 | 4.301 c = ⅕ · 21.506 c | +| 1/6-comma | (3/2)·(80/81)^(1/6) | 698.371 | 3.584 c = ⅙ · 21.506 c | + +Self-check: `5**0.25 == (3/2)/(81/80)**(1/4)` to full float precision +(computed independently); this is the standard identity that four +quarter-comma-narrowed fifths, minus two octaves, give the just major third +5/4 — confirmed arithmetically (4 × 696.578 − 2×1200 = 386.31 c = cents of +5/4). + +**Closure check (non-circulating — this is the positive claim for all three, +not a gap).** All twelve fifths in a meantone chain are tempered by the +*same* amount (that is the definition of "regular"), so unlike the well +temperaments below there is no per-fifth enumeration to do — but the +twelve-note cut still does not close, and the residual (the wolf) is +computable the same way: eleven fifths at the tempered size, plus one +closing interval forced to make the total exactly 8400 c (7 octaves). + +| Variant | 11 × tempered fifth | Wolf (12th, closing) = 8400 − that | Wolf vs. pure 3:2 | Wolf vs. ET 700 c | +|---|---|---|---|---| +| 1/4-comma | 7662.363 c | 737.637 c | +35.682 c (wide) | +37.637 c | +| 1/5-comma | 7674.191 c | 725.809 c | +23.854 c (wide) | +25.809 c | +| 1/6-comma | 7682.077 c | 717.923 c | +15.968 c (wide) | +17.923 c | + +All three wolves land on the *wide* side (unlike Pythagorean's narrow wolf) — +independently confirming the source's own qualitative claim that the +meantone residual gap is "in the sense opposite to the Pythagorean comma." +The 1/4-comma figure (737.6 c) matches the commonly cited value for the +quarter-comma-meantone wolf fifth. + +**Source.** Wikipedia, "Meantone temperament", + (fetched 2026-07-22, raw +wikitext). Quoted: *"Quarter-comma meantone, which tempers each of the twelve +perfect fifths by 1/4 of a syntonic comma, is the best known type of meantone +temperament ... Four ascending fifths (as C G D A E) tempered by 1/4 comma +(and then lowered by two octaves) produce a just major third (C E) (with +ratio 5:4), which is one syntonic comma ... narrower than the Pythagorean +third."* The article's own comparative table (§"Meantone vs. Equitempered +tunings") lists 1/5 and 1/6 as rows of the same "Meantone fraction of +(syntonic) comma" column, confirming the same comma and the same uniform +construction apply to all three. The table also cites Barbour, James Murray +(2004 reprint of 1951 original). *Tuning and Temperament: A Historical +Survey*, Dover, ISBN 978-0-486-43406-3, as the source for the historical +fraction-of-comma naming convention generally. + +**Confidence: verified**, all three — construction and the non-closure +(wolf-size) check both done. + +--- + +### 5. `werckmeister-iii` + +**Construction.** Fifths **C–G, G–D, D–A, B–F♯** are each narrowed by +**1/4 comma**; the other eight fifths are pure. Twelve fifths, each exactly +once: C–G, G–D, D–A, B–F♯ narrowed (4); A–E, E–B, F♯–C♯, C♯–G♯, G♯–E♭, +E♭–B♭, B♭–F, F–C pure (8). No wolf: because only four of twelve fifths are +tempered and the untempered ones absorb none of the comma, all twelve notes +remain usable as a tonic (a genuine well temperament, not a meantone). + +**Comma type — an answerable question, resolved by closure, not left as a +hedge.** Werckmeister's own writing does not specify syntonic vs. Pythagorean +comma, and the source says so explicitly (quoted below) — but a circulating +temperament's twelve fifths must absorb *exactly* one Pythagorean comma +(23.4600 c) for the circle to close, and only one of the two readings does +that: + +| Reading | Each of 4 tempered fifths | × 4 | Closes (needs 23.4600 c)? | +|---|---|---|---| +| 1/4 **Pythagorean** comma | 696.090 c (narrowing 5.865 c) | 23.4600 c | **yes** | +| 1/4 **syntonic** comma | 696.578 c (narrowing 5.377 c) | 21.5063 c | no — short by 1.9537 c, exactly one schisma | + +So **the Pythagorean-comma reading is the one that makes this construction a +well temperament at all**; the syntonic-comma reading leaves a residual +schisma unaccounted for and does not close a twelve-note circle with only +four fifths tempered and eight left untouched. This resolves what the +sources themselves leave open: it is worth keeping on record that +Werckmeister's own historical ambiguity is real and sourced (quoted below, +and the two readings differ by only ≈0.49 cents, consistent with the +source's "almost inaudible" characterization) — but *for this catalog +identifier*, which must denote one specific, computable construction, the +closing reading is the answer, and this draft states it as the answer rather +than carrying the historical hedge forward as if it were still open. + +**A naming trap, found and resolved during this research.** Wikipedia's +current article numbers Werckmeister's temperaments by two different schemes +simultaneously (presentation order vs. his own monochord labels) and states +outright: *"The temperament commonly known as 'Werckmeister III' is referred +to in this article as 'Werckmeister I (III)'."* There is a **different** +section literally titled "Werckmeister III (V)" in the same article, which +describes a materially different construction (fifths D–A, A–E, F♯–C♯, +C♯–G♯, F–C narrowed 1/4 comma, G♯–D♯ *widened* 1/4 comma). That section is +**not** the temperament this catalog identifier means — it is a related but +distinct third Werckmeister tuning that happens to share the "III" digit +under the article's alternate numbering. This draft's construction (C–G, +G–D, D–A, B–F♯) is the one under the heading Wikipedia glosses as +"commonly known as Werckmeister III," matching the plan document's own +description (`PLAN_PUSH4B_TUNING.md`: *"Werckmeister III narrows four named +fifths by 1/4 Pythagorean comma"*) and every secondary source found during +search. Flagging this because an agent (or reviewer) skimming the article +section-by-section could pick up the wrong construction under the right +label, which is exactly the failure mode this whole exercise exists to catch. + +**Cents (derived), resolved reading:** fifth = (3/2)/(3^12/2^19)^(1/4) ≈ +696.090 cents (narrowing 5.865 c = ¼ · 23.460 c) — see the closure table +above for why this reading, not the syntonic-comma one, is reported. + +**Source.** Wikipedia, "Werckmeister temperament", + (fetched 2026-07-22, +raw wikitext). Quoted: *"This tuning uses mostly pure (perfect) fifths, as in +Pythagorean tuning, but each of the fifths C–G, G–D, D–A and B–F♯ is made +smaller, i.e. tempered by 1/4 comma. No matter if the Pythagorean comma or +the syntonic comma is used, the resulting tempered fifths are for all +practical purposes the same as meantone temperament fifths ... because not +all fifths are tempered, there is no wolf fifth and all 12 notes can be used +as the tonic."* And on the comma ambiguity: *"Werckmeister was not explicit +about whether the syntonic comma or Pythagorean comma was meant: The +difference between them, the so-called schisma, is almost inaudible."* Cites +Werckmeister, A. (1983) [1691], ed. Rudolf Rasch, *Musicalische Temperatur*, +Diapason Press, ISBN 90-70907-02-X, as the primary treatise. + +**Confidence: verified** — construction, twelve-fifth enumeration, and the +comma-type resolution (by closure) all checked. Superseded from the previous +draft: that version carried the comma-type ambiguity into the summary table +as if it were still open; this round's closure check resolves it to the +Pythagorean-comma reading, per the contract's "closure is also a decision +procedure" instruction. + +--- + +### 6. `werckmeister-iv` + +**Construction.** Fifths **C–G, D–A, E–B, F♯–C♯, B♭–F** narrowed by **1/3 +comma**; fifths **G♯–D♯** and **E♭–B♭** *widened* by 1/3 comma; the +remaining five fifths (G–D, A–E, B–F♯, C♯–G♯, F–C) pure. Twelve fifths total +(5 narrow + 2 wide + 5 pure), self-checked by enumerating the full circle — +E♭–B♭ and B♭–F are two distinct adjacent fifths sharing the note B♭, not a +duplicate. Same syntonic-vs-Pythagorean ambiguity in the source as +`werckmeister-iii` — resolved the same way, by closure, below. + +**Closure check, both readings (the same comma-type question as §5 applies +here and gets the same treatment — the contract asks for this on every +entry, not just the three flagged):** + +| Reading | 5 narrow @ ⅓ | 2 wide @ ⅓ (negative) | Net | Closes? | +|---|---|---|---|---| +| Pythagorean comma | 5 × 7.820 c = 39.100 c | −2 × 7.820 c = −15.640 c | **23.4600 c** | **yes** | +| syntonic comma | 5 × 7.169 c = 35.843 c | −2 × 7.169 c = −14.338 c | 21.5063 c | no — short by 1.9537 c | + +Same resolution as `werckmeister-iii`: the Pythagorean-comma reading is the +one under which this is a closing well temperament; report it as the +construction, not as one of two open possibilities. + +**Cents (derived), resolved reading:** +fifth (narrow) = (3/2)/(3^12/2^19)^(1/3) ≈ 694.135 cents (narrowing +7.820 c = ⅓ · 23.460 c); fifth (wide) = (3/2)·(3^12/2^19)^(1/3) ≈ +709.775 cents (widening 7.820 c). + +**Source.** Same article as §5, section "Werckmeister II (IV)" (the +article's own gloss again ties this to the commonly-known "Werckmeister IV" +digit). Quoted: *"In Werckmeister II the fifths C–G, D–A, E–B, F♯–C♯, and +B♭–F are tempered narrow by 1/3 comma, and the fifths G♯–D♯ and E♭–B♭ are +widened by 1/3 comma. The other fifths are pure. Werckmeister designed this +tuning for playing mainly diatonic music (i.e. rarely using the 'black +notes')."* + +**Confidence: verified** — construction, twelve-fifth enumeration, and +comma-type resolution by closure (added this round; the first draft computed +this disambiguation only for §5, not here, even though the same ambiguity +and the same resolution apply). + +--- + +### 7. `vallotti` + +**Construction.** Fifths **F–C, C–G, G–D, D–A, A–E, E–B** (six consecutive) +each narrowed by **1/6 of the Pythagorean comma**; the other six fifths +(B–F♯, F♯–C♯, C♯–G♯, G♯–E♭, E♭–B♭, B♭–F) pure. This is the version in +common (electronic-tuner, DAW, harpsichord-technician) use today, and it is +the one Ruling B's own plan text describes ("narrows six consecutive fifths +by 1/6 Pythagorean comma and leaves the rest pure"). + +**An important historical wrinkle, worth carrying into the spec's +description if this is promoted.** The construction above is *not* what +Francesco Vallotti actually wrote down. Per the same Wikipedia article, +Vallotti's own manuscript (unpublished until 1987) used **1/6 of the +syntonic comma** on the same six fifths plus a schisma-sized correction on +the seventh (B♭–F), and the attribution of the now-common construction to +Vallotti at all is called "a mistake" by the article, though "audibly +indistinguishable" from what he wrote (no interval differs by more than 2 +cents across the variants). The identifier `vallotti` in this catalog almost +certainly means the common modern construction (matching Ruling B's text and +every calculator/tuner-app source found), not Vallotti's original manuscript +version — but a reviewer should know both exist and that they are not the +same rational numbers. + +**Cents (derived):** fifth = (3/2)/(3^12/2^19)^(1/6) ≈ 698.045 cents +(narrowing 3.910 c = ⅙ · 23.460 c). + +**Closure check.** Twelve fifths, each exactly once: F–C, C–G, G–D, D–A, +A–E, E–B narrowed 1/6 Pythagorean comma (6); B–F♯, F♯–C♯, C♯–G♯, G♯–E♭, +E♭–B♭, B♭–F pure (6). Sum = 6 × 3.910 c = 23.4600 c exactly — closes, and +unambiguously (the source names the Pythagorean comma outright here, so +there is no reading to disambiguate the way Werckmeister needed). + +**Source.** Wikipedia, "Vallotti temperament", + (fetched 2026-07-22, raw +wikitext). Quoted: *"each of the fifths B-F♯, F♯-C♯, C♯-G♯, G♯-E♭, E♭-B♭, and +B♭-F are perfectly just, while the fifths F-C, C-G, G-D, D-A, A-E, and E-B +are each 1/6 of a Pythagorean (ditonic) comma narrower than just"*, citing +Donahue, Thomas (2005), *A Guide to Musical Temperament*, Scarecrow Press, +p. 28 (Google Books link given in the article). Historical-original claim +cites Barbieri, Patrizio (1987) and Di Veroli, Enrico (2013), p. 125. + +**Confidence: verified** (both the common construction and the historical +caveat). + +--- + +### 8. `kirnberger-ii` + +**This entry was wrong in the first draft, and is corrected here.** The +first version stated "the remaining ten fifths pure" and claimed this +closes the circle. It does not: two fifths at 1/2 syntonic comma discharge +exactly one syntonic comma (21.506 c), and a closing twelve-note circle must +discharge exactly one **Pythagorean** comma (23.460 c) — short by 1.9537 c, +one schisma, on the nose. The D–A/A–E tempering and the resulting pure +thirds (kept below, unchanged) were correct; the error was inferring from +them that every other fifth is untouched. It isn't — there is an eleventh, +schisma-tempered fifth the first draft's source (Wikipedia) names in an +image caption but never surfaces in its prose, and the first draft's ASCII +transcription of that same diagram flattened the distinction to a uniform +"p" for every non-D–A/A–E fifth. Re-sourced below from a source that states +the missing fifth explicitly. + +**Construction (corrected).** Fifths **D–A** and **A–E** each narrowed by +**1/2 the syntonic comma**; fifth **F♯–D♭** (i.e. F♯–C♯, spelled with the +flat name because Kirnberger's own chain is built outward from D♭) narrowed +by a **schisma** (the ratio between the Pythagorean and syntonic commas, +32805/32768 ≈ 1.9537 c); the remaining **nine** fifths pure. + +Twelve fifths, each exactly once, enumerated around the chain +D♭–A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–(closing to D♭): + +| Fifth | Tempering | +|---|---| +| D♭–A♭, A♭–E♭, E♭–B♭, B♭–F, F–C, C–G, G–D, E–B, B–F♯ | pure (9) | +| D–A, A–E | narrow, 1/2 syntonic comma (2) | +| F♯–D♭ (closing) | narrow, 1 schisma (1) | + +**Closure check.** 2 × 10.753 c (half-syntonic-comma fifths) + 1 × 1.9537 c +(schisma fifth) = 21.5063 + 1.9537 = **23.4600 c exactly** — closes. Verified +independently in this session by summing all twelve fifths' cents directly: +8400.000 c (= 7 octaves), confirming the schisma fifth is not just plausible +but numerically required and sufficient. + +**Cents (derived), full twelve-note table, C = 1/1** (built by stacking the +chain above from C, self-computed, not copied from either source): + +| Note | Cents (derived) | +|---|---| +| C | 0.000 | +| D♭ | 90.225 | +| D | 203.910 | +| E♭ | 294.135 | +| E | 386.314 | +| F | 498.045 | +| F♯ | 590.224 | +| G | 701.955 | +| A♭ | 792.180 | +| A | 895.112 | +| B♭ | 996.090 | +| B | 1088.269 | + +**Pure thirds — recomputed from this agent's own chain, not restated from +either source.** C–E, G–B, D–F♯ come out at exactly 386.314 c (5:4, pure); +this **confirms** Wikipedia's "three pure thirds" claim and this draft's own +original (pre-correction) derivation of *which* three — that part of the +first draft was right and is unchanged. Db–F, E♭–G, A♭–C, B♭–D come out at +exactly 407.820 c (81:64, Pythagorean-wide) — **four** thirds, not the three +("B–D♯, F♯–A♯, D♭–F") the first draft's source names in prose. This is a +finding, reported rather than silently resolved either way: the discrepancy +traces to the schisma fifth. Wikipedia's own ASCII diagram (see above) does +not distinguish the schisma-tempered fifth from a fully pure one, so its +prose description of "three Pythagorean thirds" is very likely computed +against the same idealized (schisma = 0) picture that fails to close — the +same simplification that produced the first draft's error. Under the +corrected, closing construction, the thirds nearest the schisma fifth +(E–A♭, F–A, F♯–B♭, A–D♭, B–E♭) land at intermediate values (395–406 cents) +that are neither pure nor exactly Pythagorean. Separately, Carey Beebe's +tuning-instructions page (cited below, the source for the schisma fifth +itself) states **four** pure thirds including F–A; recomputing F–A directly +from the chain above gives 397.067 c, **not** pure (5:4 = 386.314 c, a +10.75-cent difference — audible, not a rounding artifact) — so that claim is +also not borne out by exact arithmetic, most likely because Beebe's page is +an explicitly practical tuning guide ("we regard the syntonic comma as for +all practical purposes the same size as the Pythagorean," his words, on the +companion Kirnberger III page) rather than a source asserting exact ratios. +Net: **three** thirds are exactly pure by this agent's independent +computation, and that is what this draft reports; the "four pure thirds" +figure appearing in one source is noted, not adopted. + +**Source (schisma fifth, corrected construction).** Carey Beebe, +"Temperaments V — How to tune Kirnberger II", CBH Technical Library, +Harpsichords Australia, +(fetched 2026-07-22, raw HTML). Quoted in full: *"In theory, your error or +schisma is located between F♯ and D♭ in the circle of keys—look for the +'±0'—and is in fact an equal-tempered fifth in size."* And on the D–A/A–E +tempering: *"Kirnberger has split the comma into two, giving you two very +narrow half-comma fifths ... Tune your a a pure fifth above d, and then +flatten the a until the interval d–a sounds almost as rough as a–e'."* The +page's own bibliography (a specialist harpsichord-technician's reading list, +not this agent's addition) cites: Barbour, J. Murray, *Tuning and +Temperament*, Michigan State College Press, East Lansing, 1951, p. 158; +Asselin, Pierre-Yves, *Musique et Tempérament*, Éditions Costallat, Paris, +1985, p. 90; Jorgensen, Owen, *The Equal-Beating Temperaments*, The Sunbury +Press, Raleigh, 1981, p. 23; Klop, G. C., *Harpsichord Tuning*, Werkplaats +voor Clavecimbelbouw, Garderen, 1974, p. 22; Padgham, Charles, *The +Well-Tempered Organ*, Positive Press, Oxford, 1986, p. 64. + +**Source (D–A/A–E tempering, thirds, general framing — first draft's +source, retained).** Wikipedia, "Kirnberger temperament", + (fetched 2026-07-22, +raw wikitext). Quoted: *"Kirnberger's first method of compensating for and +closing the circle of fifths was to split the 'wolf' interval ... in half +between two different fifths. That is, to compensate for the one extra +comma, he removed half a comma from two of the formerly perfect fifths ... +So, Kirnberger allowed for three pure thirds, the rest being slightly wide +and the worst being three Pythagorean thirds (22 cents wider than pure)."* +The article's image caption (not its ASCII-art rendering of the same +diagram) independently corroborates the schisma fifth's existence: *"Kirnberger +II temperament; −Z/2 marks a tempered fifth flattened by a half comma; −Sch +marks a schisma"* — confirming, after the fact, that this agent's first +pass had the right source in hand and simply did not follow the image +caption to its conclusion. + +**Confidence: verified** — construction (now including the schisma fifth), +twelve-fifth enumeration, and closure all checked this round. The pure-third +count is independently recomputed and reported at three, with the +conflicting "three" (prose, wrong set of notes) and "four" (Beebe, includes +a non-pure F–A) claims both surfaced as findings rather than silently +adopted. + +--- + +### 9. `kirnberger-iii` + +**This entry had the same defect as `kirnberger-ii`, for the same reason, +and is corrected the same way.** Four fifths at 1/4 syntonic comma discharge +exactly one syntonic comma (21.506 c), short of the 23.460 c a closing +twelve-note circle requires by exactly one schisma (1.9537 c) — the +first draft's "the remaining eight fifths pure" did not close. The first +draft also flagged its own "which four fifths" identification as an +arithmetic reconstruction rather than a quoted fact; re-sourcing below +settles that too, from the same practical tuning-instruction source used to +find §8's missing fifth. + +**Construction (corrected).** Fifths **C–G, G–D, D–A, A–E** (four +consecutive, now directly quoted, not reconstructed — see source) each +narrowed by **1/4 the syntonic comma**; fifth **F♯–D♭** narrowed by a +**schisma** (same position as in Kirnberger II, and the same construction +skeleton — Kirnberger III differs from II only in how many fifths share the +discharged comma and by what fraction); the remaining **seven** fifths pure. +Only the third C–E stays pure (5:4) — confirmed by this agent's own +recomputed chain below, matching the source. + +Twelve fifths, each exactly once, same chain skeleton as §8 +(D♭–A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–closing to D♭): + +| Fifth | Tempering | +|---|---| +| D♭–A♭, A♭–E♭, E♭–B♭, B♭–F, F–C, E–B, B–F♯ | pure (7) | +| C–G, G–D, D–A, A–E | narrow, 1/4 syntonic comma (4) | +| F♯–D♭ (closing) | narrow, 1 schisma (1) | + +**Closure check.** 4 × 5.377 c (quarter-syntonic-comma fifths) + 1 × +1.9537 c (schisma fifth) = 21.5063 + 1.9537 = **23.4600 c exactly** — +closes. Independently confirmed by summing all twelve fifths directly: +8400.000 c. + +**Cents (derived), full twelve-note table, C = 1/1** (self-computed from the +corrected chain): + +| Note | Cents (derived) | +|---|---| +| C | 0.000 | +| D♭ | 90.225 | +| D | 193.157 | +| E♭ | 294.135 | +| E | 386.314 | +| F | 498.045 | +| F♯ | 590.224 | +| G | 696.578 | +| A♭ | 792.180 | +| A | 889.735 | +| B♭ | 996.090 | +| B | 1088.269 | + +D (193.157 c = 5^(1/2)/2), G (696.578 c = 5^(1/4)), and A (889.735 c = +5^(3/4)/2) match the first draft's arithmetic reconstruction exactly — that +part of the earlier draft was correct and is unchanged; only the "rest is +pure" assumption around it was wrong. + +**Note, retained from the first draft: this fifth (5^(1/4)) is numerically +identical to quarter-comma meantone's fifth** — a correct consequence of +"four 1/4-syntonic-comma fifths closing a just major third," not a +coincidence. + +**Only C–E is exactly pure (5:4, 386.314 c), recomputed directly** — matching +the source. D♭–F and A♭–C come out at exactly 407.820 c (81:64, +Pythagorean-wide); the remaining **nine** thirds (1 pure + 2 Pythagorean-wide ++ 9 = 12, checked) are intermediate values affected by the schisma fifth, +neither pure nor exactly Pythagorean-wide. This is fewer Pythagorean-wide +thirds than `kirnberger-ii` (two, versus four), matching the source's +qualitative claim that Kirnberger III has "fewer Pythagorean thirds" than +II. + +**Source (schisma fifth and the four named fifths, corrected construction).** +Carey Beebe, "Temperaments VI — How to tune Kirnberger III", CBH Technical +Library, Harpsichords Australia, + (fetched 2026-07-22, raw +HTML). Quoted: *"All those four fifths C–G, G–D, D–A and A–E should sound +equally rough"* — the four fifths directly named, resolving the first +draft's "reconstructed, not quoted" caveat. And on the closing fifth: +*"Tune all the fifths from the flat side of C around the circle of keys +absolutely pure. Stop about the D♭, and begin again working around the +sharp side of E, tuning all those fifths absolutely pure. (In theory, you'll +end up with one fifth a little narrow, in fact very close to an equal +tempered fifth, but in practice, they should all sound pretty much pure.)"* +— which, by the same chain-construction logic worked out for Kirnberger II +above (the flat-side chain from C stops at D♭; the sharp-side chain from E +stops at F♯; the two meet at the F♯–D♭ interval), is the same schisma fifth +named explicitly on the companion Kirnberger II page. The page's own +bibliography: Asselin, Pierre-Yves, *Musique et Tempérament*, Éditions +Costallat, Paris, 1985, p. 92; Klop, G. C., *Harpsichord Tuning*, Werkplaats +voor Clavecimbelbouw, Garderen, 1974, p. 23; Padgham, Charles, *The +Well-Tempered Organ*, Positive Press, Oxford, 1986, p. 68; Jorgensen, Owen, +*The Equal-Beating Temperaments*, The Sunbury Press, Raleigh, 1981, p. 26. + +**Source (four fifths tempered, one third pure, general framing — first +draft's source, retained).** Wikipedia, "Kirnberger temperament" (as §8). +Quoted: *"This temperament splits the Syntonic comma between four fifths +instead of two; 1/4 comma tempered fifths are used extensively in meantone +... This also eliminates two of the three pure thirds found in Kirnberger +II. Therefore, only one third remains pure (between C and E)."* + +**Confidence: verified** — construction (now including the schisma fifth), +the four named fifths (now directly quoted rather than reconstructed), +twelve-fifth enumeration, and closure all checked this round. + +--- + +### 10. `young-ii` + +**Construction (Young's *second* temperament — this catalog identifier is +`young-ii`, not Young's first, which is a different, more elaborate +construction the same source also documents).** Fifths **C–G, G–D, D–A, +A–E, E–B, B–F♯** (six consecutive) each narrowed by **1/6 of the Pythagorean +(ditonic) comma**; fifths **F♯–C♯, C♯–G♯, G♯–E♭, E♭–B♭, B♭–F, F–C** pure. +Structurally identical to `vallotti` above — six-tempered/six-pure, +1/6 Pythagorean comma — but rotated: Young's tempered run starts at C, +Vallotti's (common, modern) at F. The source states this relationship +explicitly and gives the alternate name "Vallotti-Young" / "shifted Vallotti" +for this reason. + +**Cents (derived):** identical to Vallotti's, since the fraction and comma +are the same: fifth ≈ 698.045 cents (narrowing 3.910 c = ⅙ · 23.460 c); see +§7 for the derivation. + +**Closure check.** Twelve fifths, each exactly once: C–G, G–D, D–A, A–E, +E–B, B–F♯ narrowed 1/6 Pythagorean comma (6); F♯–C♯, C♯–G♯, G♯–E♭, E♭–B♭, +B♭–F, F–C pure (6). Sum = 6 × 3.910 c = 23.4600 c exactly — closes, +unambiguously (Pythagorean comma named outright in the source, same as +Vallotti). + +**Source.** Wikipedia, "Young temperament", + (fetched 2026-07-22, raw +wikitext). Quoted: *"In the second temperament, [Young 1802] made each of the +fifths F♯-C♯, C♯-G♯, G♯-E♭, E♭-B♭, B♭-F, and F-C perfectly just, while the +fifths C-G, G-D, D-A, A-E, E-B, and B-F♯ are each 1/6 of a Pythagorean +(ditonic) comma narrower than just."*, footnoted to **Barbour, James Murray +(2004) [1951]. *Tuning and Temperament: A Historical Survey*, p. 163** (with +a direct archive.org page-image link in the Wikipedia citation: +`archive.org/stream/tuningtemperamen00barb#page/163/mode/1up`). And on the +Vallotti relationship: *"Young's 2nd temperament is very similar to the +Vallotti temperament which also has six consecutive pure fifths and six +tempered by 1/6 of a Pythagorean comma. Young's temperament is shifted one +note around the circle of fifths, with the first tempered fifth beginning on +C instead of F."*, footnoted to Donahue (2005), pp. 28–29. + +**Confidence: verified.** This is the best-sourced entry in the draft: the +Wikipedia claim carries a page-specific citation to Barbour (1951/2004) with +a direct link to the scanned page, which a reviewer can open and check +without needing to locate a physical copy of the book. + +--- + +## Section B — the four open ratifications (surfaced, not decided) + +These four are **not** given a `verified`/`recalled`/`unknown` tag as if +they were settled constructions with one right answer — the entire point of +flagging them is that no single construction is "the" answer, and picking +one here would be exactly the undisclosed musicological ratification the +contract says not to make. What follows is: what candidates exist, how they +were sourced, and what each optimizes. The *sourcing of the candidates* is +verified; the *choice among them* is open. + +### 11–13. `ji-static-5limit-C`, `ji-static-5limit-G`, `ji-static-5limit-D` + +**The shape of the problem.** A 5-limit lattice (powers of 2, 3, and 5) has +more than twelve justly-tunable pitch classes per octave once you include +enough of the lattice to cover a chromatic scale — the standard construction +(below) generates **fifteen** distinct pitches from a 5×3 grid of thirds and +fifths, two more than fit in twelve chromatic slots. Reducing fifteen to +twelve requires discarding three (one member from each of three enharmonic +pairs, since the grid's extremes double up), and *which* three you discard +changes the ratios assigned to some chromatic scale degrees. This is exactly +the "genuinely unsettled" comma choice the contract describes, and it has +been unsettled in the literature for centuries, not just in this repository. + +**The lattice (verified).** Building outward from C=1/1 by fifths (×3/2, +÷3/2) and major thirds (×5/4, ÷5/4), octave-reduced, gives (Wikipedia's own +layout, axes = powers of 3 across, powers of 5 down): + +| ×5 → / ×3 →↑ | 1/9 | 1/3 | 1 | 3 | 9 | +|---|---|---|---|---|---| +| **5** | D− 10/9 | A 5/3 | E 5/4 | B 15/8 | F♯+ 45/32 | +| **1** | B♭− 16/9 | F 4/3 | **C 1/1** | G 3/2 | D 9/8 | +| **1/5** | G♭− 64/45 | D♭− 16/15 | A♭ 8/5 | E♭ 6/5 | B♭ 9/5 | + +Fifteen cells, but D, B♭, and G♭ each appear **twice** (once with a trailing +`−`/`+` marking a syntonic-comma-flatter/sharper twin). All three +candidate 12-note scales below agree on discarding G♭ (the "far corner", +enharmonically a diminished fifth from C, the least consonant and +least-used cell) — that much is *not* contested. What's contested is which +of the *other* two duplicate pairs (D vs D−, B♭ vs B♭−) to resolve, and how. + +**Three named candidates, per the same source:** + +1. **"Symmetric scale 1."** Discard the two opposite corners (D− and B♭−, + top-left/bottom-right). Keeps: D = 9/8 (203.910 c), B♭ = 16/9 + (996.090 c). Optimizes: symmetric structure (B♭ and D are exact + inversions of each other around C); this is the scale that also matches + the D and B♭ used in C-based Pythagorean and quarter-comma-meantone + scales (source's own note), which may matter for cross-tuning-system + comparison work in this codebase. +2. **"Symmetric scale 2."** Discard the two ends of the middle (`1`) row — + i.e. keep D− = 10/9 (182.404 c) and B♭ = 9/5 (1017.596 c) instead. Also + symmetric (same inversion property, different fixed point), but the D + and B♭ used differ from scale 1 by exactly a syntonic comma each + (21.506 c) — self-verified above. +3. **"Asymmetric scale."** Discard the whole `1/9` column instead of one + cell from each of two rows. Keeps D = 9/8, B♭ = 9/5 (mixed: scale-1's D, + scale-2's B♭). Source states this variant has the "simplest" ratios + overall (nine pure fifths, eight pure major thirds, six pure minor + thirds by design) but **14** wolf intervals versus 12 for the symmetric + scales — more consonant chords, at the cost of more badly-tuned ones + elsewhere. This is the table this draft's derived-cents worked example + above (§ table) used, since it is the one with a full 12-note table + given directly in the source. + +**What each optimizes, briefly:** scale 1 favors symmetry and cross-system +comparability; scale 2 favors symmetry with a different fixed point (and, +per the source, is not otherwise singled out as preferable — it's presented +as the third structurally-parallel option); the asymmetric scale favors +maximizing the count of pure simple-ratio intervals at the cost of a wider +spread of wolf intervals. None of the three is "the" standard in the sense +`tet-12` is standard — reputable sources use different ones for different +purposes, and the source consulted here presents all three side by side +without endorsing one. + +**Do the three catalog identifiers differ only by transposition, or are they +independently chosen?** The contract asks this explicitly; this draft's +answer is: **most likely by transposition of a single chosen 12-note +scale**, by direct analogy with `vallotti`/`young-ii` above (§7/§10), which +are the *same* six-fifths/1-sixth-comma construction rotated to a different +starting note. If `ji-static-5limit-C/G/D` follow that pattern, a reviewer +picks **one** of the three candidate scales above (or another 5-limit +construction entirely) anchored at C, and the G- and D-anchored systems are +that same scale's ratio pattern transposed so G, respectively D, take the +role of 1/1. This is **this agent's inference from the naming parallel**, +not a sourced fact about these specific three identifiers — nothing in +`core_spec.tex` states whether the three are meant to be transpositions of +one scale or three independently-optimized 12-note constructions (e.g., a +scale independently re-derived to make the dominant-of-the-dominant +relationships pure in each), and the ratification should settle this +explicitly rather than leave it to be assumed either way. + +**Source.** Wikipedia, "Five-limit tuning", + (fetched 2026-07-22, raw +wikitext), section "Twelve-tone scale". All three named-scale ratio tables +and the "discard G♭ / discard a duplicate pair" framing are quoted/derived +directly from that section; no scholarly citation with page number was found +attached to the *choice among the three* in this source (only to a related +note about extending F♯ upward through D♭, cited to Randel, Don Michael +(ed.), *The Harvard Dictionary of Music*, 4th ed., 2003, p. 415 — not +directly about the three-way choice above). + +**Confidence: the existence and structure of the three candidates is +verified** (quoted, and the syntonic-comma difference between them +independently recomputed). **The choice among them, and whether the three +catalog identifiers are transpositions of one choice, is open** — surfaced +per the contract, not decided. + +--- + +### 14. `ji-adaptive-5limit` + +**What the governing requirement permits (verified, `core_spec.tex`).** +`req:tuning:adaptive-tuning-purity` (`core_spec.tex:3327-3333`, quoted in +full above) constrains *how* an adaptive function must behave — pure in +`(position, HarmonicContext)`, cacheable only with correct invalidation on +context change — and says nothing about *what algorithm* computes the +frequency. It permits, without choosing among: + +* Nearest-just-ratio resolution against the currently sounding + `HarmonicContext.concurrent` set (the textbook "adaptive JI" example given + in the surrounding prose at `core_spec.tex:3301-3305`: an E resolves + differently as the third of a C chord, the fifth of an A chord, or a + passing tone). +* A decaying-weight blend using `HarmonicContext.recent` as well as + `concurrent`, for voice-leading continuity across a change of harmony. +* A `key_context`- or `hints`-driven resolution that falls back to a fixed + static scale (e.g., one of the three candidates in §11–13) when no + harmonic information is available. +* Comma-drift management (the classic problem where a long enough chain of + adaptive adjustments can walk pitch center away from the reference by a + syntonic comma or more) is itself a design choice `req:tuning: + adaptive-tuning-purity` is silent on, beyond requiring that whatever + choice is made stays a pure function of position and context. + +None of the above is proposed as *the* algorithm. Inventing one here would +be precisely the failure mode this contract exists to prevent — a plausible- +sounding, unverifiable algorithm written once into a document that looks +authoritative. + +**The recommended pattern (surfaced, not decided).** `TuningResolution` +already has an `Adaptive { function: AdaptiveTuningFunctionId, parameters: +AdaptiveTuningParameters }` variant (`core_spec.tex:3341-3343`), and Chapter +10's extension-point appendix already lists `AdaptiveTuningFunctionId` as a +registered extension point (`core_spec.tex:13347-13349`, *"Registered +adaptive tuning functions taking harmonic context as input"*) — so the +plumbing this needs already exists in the data model; only the built-in's +specific algorithm is unpinned. The in-house precedent for pinning "a +versioned identifier names one specific algorithm, and any other identifier +errors" is `req:pitch:spelling-algorithm` (`core_spec.tex:1517-1546`): +`SpellingAlgorithmId "default"` names, at version 1, one specific named +algorithm (a Temperley-style line-of-fifths centre-of-gravity rule), with +the version number itself part of the identity so that a future refinement +is a new version rather than a silent behavior change under the same name. +The analogous move here is a versioned `AdaptiveTuningFunctionId "default"` +whose version 1 pins one specific, fully-described algorithm — but *which* +algorithm version 1 names is exactly the ratification this draft does not +make. This is a recommendation about **form** (follow the spelling-algorithm +precedent: reserved id, explicit version, hard error on any other +identifier), not a recommendation about **content**. + +**Source.** `core_spec.tex:3301-3333` (Adaptive Tuning section and +`req:tuning:adaptive-tuning-purity`, read in full per the contract's +instruction); `core_spec.tex:3341-3343` (`TuningResolution::Adaptive`); +`core_spec.tex:13347-13349` (extension-point listing); `core_spec.tex: +1517-1546` (`req:pitch:spelling-algorithm`, the cited in-house pattern). +All read directly from the repository's own `spec/core_spec.tex` in this +session. + +**Confidence: the constraint (what the requirement permits) and the +existence of the versioned-identifier precedent are verified** — both are +direct quotations from `core_spec.tex` read in full this session. **The +algorithm itself is, and must remain, undecided** by this draft. + +--- + +## Summary table + +| Identifier | Comma | Fifths tempered | Closure sum | Confidence | +|---|---|---|---|---| +| `pythagorean` | Pythagorean, all in 1 fifth | 0 tempered, 1 wolf, 11 pure | 23.4600 c, on 1 fifth (non-circulating, by design) | verified | +| `meantone-1/4-comma` | syntonic, 1/4 | 12 of 12 (regular) | does not close: wolf = 737.637 c (non-circulating, by design) | verified | +| `meantone-1/5-comma` | syntonic, 1/5 | 12 of 12 (regular) | does not close: wolf = 725.809 c (non-circulating, by design) | verified | +| `meantone-1/6-comma` | syntonic, 1/6 | 12 of 12 (regular) | does not close: wolf = 717.923 c (non-circulating, by design) | verified | +| `werckmeister-iii` | **Pythagorean** (resolved by closure — was left ambiguous in the first draft) | 4 of 12 | 23.4600 c — closes | verified | +| `werckmeister-iv` | **Pythagorean** (resolved by closure, same treatment as III) | 5 narrow + 2 wide of 12 | 23.4600 c — closes | verified | +| `vallotti` | Pythagorean, 1/6 | 6 of 12 | 23.4600 c — closes | verified | +| `kirnberger-ii` | syntonic (2 fifths) **+ 1 schisma fifth (corrected this round)** | 2 syntonic-tempered + 1 schisma + 9 pure | 23.4600 c — closes | verified | +| `kirnberger-iii` | syntonic (4 fifths) **+ 1 schisma fifth (corrected this round)** | 4 syntonic-tempered + 1 schisma + 7 pure | 23.4600 c — closes | verified | +| `young-ii` | Pythagorean, 1/6 | 6 of 12 | 23.4600 c — closes | verified | +| `ji-static-5limit-C` | — (JI, no tempering) | n/a | n/a (not a fifths-chain construction) | **open** — 3 candidate scales sourced, choice not made | +| `ji-static-5limit-G` | — | n/a | n/a | **open** — as above, plus transposition-vs-independent question open | +| `ji-static-5limit-D` | — | n/a | n/a | **open** — as above | +| `ji-adaptive-5limit` | n/a | n/a | n/a | **open** — constraint verified, algorithm not proposed | + +Every row in Section A now shows a closure sum computed independently by +this agent, not asserted from a source. Two entries changed *construction* +this round (`kirnberger-ii`, `kirnberger-iii` each gained a schisma-tempered +twelfth fifth that the first draft's own source named in an image caption +but the first draft never surfaced), and one changed from an open hedge to +a resolved answer (`werckmeister-iii`'s comma type, with `werckmeister-iv` +given the same treatment on the same logic even though it was not +separately flagged). No entry in Section A is `recalled` or `unknown` — +every constructible temperament in this batch had a directly quotable, +fetchable source once searched for today. That remains true after this +round, but it is no longer the headline fact: **the headline fact is that +"verified" against a real citation was not sufficient by itself, and three +entries were confidently wrong while individually citing real sources.** +Sourcing catches fabrication; only computing the invariant catches a real +source misapplied or incompletely transcribed. See the report for the +counts under the corrected, stricter meaning of `verified`.