From 986b90aef3c772e63559237eb42852b49962e950 Mon Sep 17 00:00:00 2001 From: Levi Neuwirth Date: Wed, 22 Jul 2026 11:50:06 -0400 Subject: [PATCH] P13-S6: the tuning constructions, drafted -- and why sourcing was not enough `req:tuning:builtin-tuning-catalog` MUST-resolves 20 tuning identifiers while specifying only the six `tet-*`. Ruling B keeps all 20 and pins them by **construction** rather than by cents table, because published sources agree on constructions and differ mainly in rounding. This lands the draft of the other 14 as a **non-normative** artifact for review; nothing enters `core_spec.tex` until the four open ratifications are decided. Drafting rather than editing was deliberate, and it earned its keep. An agent writing Werckmeister III from memory into a normative document is the `NOTEHEAD_ANCHORS` failure exactly: hand-written, authoritative-looking, unverifiable in-tree, and load-bearing. **The first draft was well-sourced and wrong.** Ten entries, ten live citations, zero recalled, zero unknown -- and three of them could not exist. A twelve-fifth circle must absorb exactly one Pythagorean comma (23.4600 c). Checked: kirnberger-ii 2 x 1/2 syntonic, "remaining ten pure" -> 21.5063 c kirnberger-iii 4 x 1/4 syntonic, "remaining eight pure" -> 21.5063 c Both short by 1.9537 c -- exactly one schisma, the signature of a dropped schisma-tempered fifth. The draft said of the first, in terms, "This closes the circle." It does not. And `werckmeister-iii` carried the syntonic-vs-Pythagorean comma forward as a "genuinely ambiguous" hedge into its summary table, when closure settles it: only the Pythagorean reading closes, and a well temperament that does not close is not one. The supporting reasoning had been *correct* -- the derivation that C-E, G-B and D-F# come out pure is right. The error was inferring from it that every remaining fifth is pure. Making those thirds just does not discharge the full comma. That is precisely the failure a citation count cannot detect and an invariant catches in one line. **Corrected and re-verified by recomputation, not by re-reading.** All six circulating temperaments now close to exactly 23.4600 c; all four non-circulating residues reproduce independently (`pythagorean` 678.495 c; meantone 1/4, 1/5, 1/6 wolves 737.637 / 725.809 / 717.923 c). The schisma fifth is sourced at F#-Db, and Kirnberger III's four fifths are now a direct quotation rather than a reconstruction. **The best result is a disagreement.** Recomputing Kirnberger II's thirds from its own chain, the agent found its new source's prose claims four pure thirds where the construction yields three -- F-A comes out at 397.067 c, not 386.314. It reported the conflict instead of deferring to the source or dropping it. Independently confirmed to the cent. The count is still 10 verified / 0 recalled / 0 unknown, but `verified` now means sourced **and** invariant-checked. The number did not move; its meaning did. A stable metric hiding a changed reality is this session's recurring lesson in another costume. `CONTRACT_P13S6_TEMPERAMENTS.md` gains a permanent section, "Check the arithmetic, not just the source", stating the general principle first so it transfers: a citation proves the source said it, not that it is right, that you read it correctly, or that you transcribed it completely. Five obligations -- closure; closure as a decision procedure for vague sources; non-circulating temperaments must *not* close, with the wolf computed; twelve fifths each exactly once; recompute every derived claim and treat disagreement as a finding. Plus the guard that matters most: never adjust a construction to make the arithmetic work and present it as sourced -- that manufactures a temperament nobody published. An unresolved entry is the correct output. Still open, surfaced not decided: the three `ji-static-5limit-*` scales and `ji-adaptive-5limit`'s algorithm, the latter recommended to follow `req:pitch:spelling-algorithm`'s versioned-identifier pattern. Co-Authored-By: Claude Opus 4.8 (1M context) --- spec/CONTRACT_P13S6_TEMPERAMENTS.md | 138 +++++ spec/DRAFT_P13S6_TEMPERAMENTS.md | 907 ++++++++++++++++++++++++++++ 2 files changed, 1045 insertions(+) create mode 100644 spec/CONTRACT_P13S6_TEMPERAMENTS.md create mode 100644 spec/DRAFT_P13S6_TEMPERAMENTS.md 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`.