55 KiB
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.
RATIFIED 2026-07-22 — read this before §11–13, which it supersedes
The four open ratifications this draft surfaced have been decided. Section A (the ten pinned constructions) is accepted as written and is the source of truth for promotion. Section B is superseded by the rulings below; where the rulings and §11–13 disagree, the rulings win.
- The 12-note 5-limit scale is the contiguous lattice block
\{3^a 5^b \mid a \in [-1,2],\; b \in [-1,1]\}, octave-reduced and assigned in ascending order to the twelve chromatic positions from the anchor. This is the construction §11–13 calls "asymmetric" — that name must not be used normatively, nor must "symmetric scale 1" or "symmetric scale 2". The block is generated by its bounds; it discards nothing, so the "5×3 grid minus three cells" framing that produced those names does not survive into the spec. ji-static-5limit-C/-G/-Dare one construction at three anchors, the named note taking the role of1/1. The catalog keeps exactly three rows; further anchors are an implementation extension, not a conformance obligation.ji-adaptive-5limitversion 1 is the key-anchored static scale — anchor from the harmonic context's tonal centre, C when absent;concurrent,recent,hints,parameters, and mode are ignored. The per-sonority algorithm §14 sketches is deferred to a later version, to land with its bounds, objective, tiebreaks, implementation, and vectors together.- The identity is
"default-v1"— version machine-visible in the string, hard error on any unregistered identifier, no silent fallback.
Two corrections to §11–13, computed
- The quarter-comma-meantone argument is false and is struck. §11–13 offers, as a point in favour of "symmetric scale 1", that its D and B♭ match those of C-based Pythagorean and quarter-comma meantone. In quarter-comma meantone D = 193.157 ¢ and B♭ = 1006.843 ¢; the just values are 203.910 ¢ and 996.090 ¢. Each differs by half a syntonic comma. The claim can only hold of circle-of-fifths position, never of ratio, and even granting that reading it covers 2 notes of 12 while the other ten shift by a comma between systems — so it is not a cross-system invariant and carried no weight in the ruling.
- No 12-note set here is inversionally symmetric. All three candidates fail
at the tritone:
45/32is present and its inverse64/45is not. §11–13's inversion claim is correct only of the D/B♭ pair. The word "symmetric" in the source names the shape of the discarded cells, not a property of the resulting scale, and must not be repeated as if it named one.
Verified interval census (recomputed, all 12 roots)
| construction | pure P5 | pure M3 | pure m3 | total of 36 | I / IV / V |
|---|---|---|---|---|---|
| "symmetric 1" | 9 | 7 | 6 | 22 | ✓ ✓ ✓ |
| "symmetric 2" | 9 | 6 | 6 | 21 | ✓ ✓ ✗ (G–D = 40/27) |
| the block (ratified) | 9 | 8 | 6 | 23 | ✓ ✓ ✓ |
The block also gains pure E♭ major and G minor against "symmetric 1", losing only B♭ minor — the parallel-minor region in exchange for a remote key.
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:verifiednow 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 claimverifiedany more — that gap is exactly what produced the three errors this round fixes.verifiedstill 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", https://en.wikipedia.org/wiki/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", https://en.wikipedia.org/wiki/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", https://en.wikipedia.org/wiki/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", https://en.wikipedia.org/wiki/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, https://www.hpschd.nu/tech/tmp/kirnberger-2.html (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", https://en.wikipedia.org/wiki/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, https://www.hpschd.nu/tech/tmp/kirnberger.html (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",
https://en.wikipedia.org/wiki/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:
- "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.STRUCK — false; see the ratification block at the head of this file. Quarter-comma meantone's D and B♭ are 193.157 ¢ and 1006.843 ¢, each half a syntonic comma from the just values. Do not carry this claim into the specification. - "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. - "Asymmetric scale." Discard the whole
1/9column 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", https://en.wikipedia.org/wiki/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.concurrentset (the textbook "adaptive JI" example given in the surrounding prose atcore_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.recentas well asconcurrent, for voice-leading continuity across a change of harmony. - A
key_context- orhints-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-purityis 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.