P13-S6: all twenty tuning systems now say what they mean

Ten temperaments and the static 5-limit construction were bare names in a
catalog whose own requirement calls the surrounding semantics normative.
They now carry generative rules: which fifths are tempered, by what fraction
of which comma, where the wolf sits when the construction does not force it,
exact ratios, and the closure sum that lets a reader check the whole thing
without leaving the page.

The comma distinction is the load-bearing part and no test in this repo can
see it. Pythagorean for pythagorean, werckmeister-iii and -iv, vallotti and
young-ii; syntonic for the three meantones and both Kirnberger sets, each of
which also carries the schisma-tempered F-sharp--D-flat closing fifth whose
absence made two of these temperaments arithmetically impossible in the first
draft. Verified by recomputation rather than by re-reading: 2 x 10.753 +
1.9537 and 4 x 5.377 + 1.9537 both land on 23.4600 cents exactly.

ji-adaptive-5limit gets version 1 as the key-anchored static scale, identity
"default-v1" with the version inside the machine-visible string, hard error on
anything unregistered. The anchor derivation is pinned to (7 * fifths) mod 12
off the prevailing key signature -- and pinned twice over, because
key_sequence is per-staff and time-anchored, so the staff and the moment both
had to be named or two conforming implementations would disagree on a
modulating score.

The Forward References block stops claiming KeyContext is partially defined
somewhere it is not. It is defined nowhere, stays out of scope, and now
carries the one obligation that matters: whatever completes it must expose a
tonal-centre pitch class the anchor rule can use.

Three new requirements, 209 -> 212. Verified independently of the agent that
did the work: the count by grep, the count constant by mutation, the ten
comma types against the draft, the twelve lattice ratios against a generator,
and the ten deleted lines against the four sites they were supposed to come
from.

Closes P13-S6.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

crates/epiphany-testkit/tests/requirement_labels.rs

@@ -9,9 +9,9 @@ use std::collections::{BTreeMap, BTreeSet};
 use std::fs;
 use std::path::{Path, PathBuf};
 
-const CORE_REQUIREMENT_COUNT: usize = 209;
+const CORE_REQUIREMENT_COUNT: usize = 212;
-const SUITE_REQUIREMENT_COUNT: usize = 279;
+const SUITE_REQUIREMENT_COUNT: usize = 282;
-const SUITE_LABEL_COUNT: usize = 279;
+const SUITE_LABEL_COUNT: usize = 282;
 
 /// The normative chapter-to-area assignment. Keeping this as data makes adding a
 /// requirement under the wrong chapter fail without encoding chapter names in

spec/core_spec.tex

@@ -3362,10 +3362,15 @@ pub enum TuningResolution {
 \subsection{Adaptive Tuning}
 
 Adaptive tuning systems compute frequency from position \emph{plus
-harmonic context}. In a 5-limit adaptive JI system, for example, the
+harmonic context}. A registered adaptive tuning function \MAY{}
-frequency of a given E depends on whether it is the major third of a C
+resolve position per sonority --- for example, tuning a given E
-chord, the perfect fifth of an A chord, or a passing tone between
+differently depending on whether it sounds as the major third of a C
-other harmonic interpretations.
+chord, the perfect fifth of an A chord, or a passing tone --- subject
+only to the purity constraint below
+(Requirement~\ref{req:tuning:adaptive-tuning-purity}). The built-in
+\texttt{ji-adaptive-5limit}, at its pinned version~1
+(Requirement~\ref{req:tuning:adaptive-default-version}), does not: it
+consumes only the harmonic context's tonal centre.
 
 \begin{lstlisting}[language=Rust]
 pub struct HarmonicContext {
@@ -3395,6 +3400,64 @@ systems ignore it; adaptive tuning systems consume it.
   changes.
 \end{requirement}
 
+\begin{requirement}
+  \label{req:tuning:adaptive-default-version}
+  \textbf{The built-in adaptive function, version 1.} The catalog
+  identifier \texttt{ji-adaptive-5limit} resolves to
+  \texttt{TuningResolution::Adaptive} with function identity
+  \texttt{"default-v1"} --- the version is part of the
+  machine-visible identifier string, not prose beside it. Version~1
+  is a pure function of (position, anchor pitch class):
+  \begin{itemize}
+    \item the position resolves through the construction of
+      Requirement~\ref{req:tuning:ji-static-construction}, transposed
+      so the anchor takes the role of $1/1$;
+    \item the anchor is the tonal centre supplied by the harmonic
+      context (Requirement~\ref{req:tuning:adaptive-anchor-derivation}
+      where that context is score-graph-derived); C (chromatic
+      position~0) when no tonal centre is supplied;
+    \item \texttt{concurrent}, \texttt{recent}, \texttt{hints},
+      \texttt{parameters}, and mode are \emph{ignored} by version~1,
+      exactly as \texttt{"default"} version~1 of
+      Requirement~\ref{req:pitch:spelling-algorithm} ignores the
+      context inputs it does not consult: consuming any of them is a
+      new version, not a silent behaviour change under the same
+      identity;
+    \item every result derives from the anchor and the reference
+      pitch alone. No adjustment is ever carried forward from a
+      previous resolution, so the construction is comma-drift-free by
+      shape, not by a correction step.
+  \end{itemize}
+  An unregistered or unknown \texttt{AdaptiveTuningFunctionId}
+  \MUST{} be a hard error; there is no silent fallback.
+  \texttt{AdaptiveTuningFunctionId} remains an extension point and
+  other functions \MAY{} be registered, but the built-in
+  \texttt{ji-adaptive-5limit} is bound to \texttt{"default-v1"} and
+  that binding \MUSTNOT{} be overridden.
+\end{requirement}
+
+\begin{requirement}
+  \label{req:tuning:adaptive-anchor-derivation}
+  Where a \texttt{HarmonicContext}'s tonal centre is derived from a
+  score graph, the anchor pitch class \MUST{} be computed from the
+  prevailing key signature as
+  \[
+    \texttt{anchor\_pc} = (7 \times \texttt{fifths}) \bmod 12,
+  \]
+  reading \texttt{fifths} (\texttt{KeySignature}, Chapter~\ref{ch:graph})
+  as the major tonic; mode is not consulted, so a signature of 0
+  anchors at C whether the prevailing key is C major or A minor. The
+  \emph{prevailing} key signature is the one on the staff containing
+  the pitch being resolved, taken from that staff's
+  \texttt{key\_sequence} (\texttt{StaffInstance::key\_sequence},
+  Chapter~\ref{ch:graph}) at the latest \texttt{KeySignatureChange}
+  whose anchor is at or before the pitch's onset. Both selections ---
+  staff and time --- \MUST{} be made this way: without them, a
+  polytonal or modulating score would resolve differently in two
+  conforming implementations, which
+  Requirement~\ref{req:tuning:tuning-resolution-determinism} forbids.
+\end{requirement}
+
 \section{Reference Pitch}
 \label{sec:tuning:reference}
 
@@ -3615,8 +3678,8 @@ transposition algebra; automatic 24-chromatic spelling inference is deferred.
   \texttt{kirnberger-iii} & Kirnberger III well temperament. \\
   \texttt{young-ii} & Thomas Young's second temperament. \\
   \texttt{ji-static-5limit-C} & Static 5-limit JI anchored to C tonic. \\
-  \texttt{ji-static-5limit-G} & Static 5-limit JI anchored to G. \\
+  \texttt{ji-static-5limit-G} & Static 5-limit JI anchored to G tonic. \\
-  \texttt{ji-static-5limit-D} & Static 5-limit JI anchored to D. \\
+  \texttt{ji-static-5limit-D} & Static 5-limit JI anchored to D tonic. \\
   \texttt{ji-adaptive-5limit} & Adaptive 5-limit JI; resolves based on
   harmonic context. \\
   \bottomrule
@@ -3630,6 +3693,368 @@ transposition algebra; automatic 24-chromatic spelling inference is deferred.
   redefine the semantics of those listed.
 \end{requirement}
 
+\subsection{Temperament Constructions}
+\label{sec:tuning:temperament-constructions}
+
+This subsection gives the normative construction for each of the ten
+non-equal-tempered, non-just-intonation identifiers in the catalog
+above. Per the pitch-space/tuning-system independence stated in
+Section~\ref{sec:tuning:principles}, all ten are constructions over
+\texttt{cmn-12}'s twelve chromatic positions per octave (C,
+C$\sharp$/D$\flat$, D, \ldots, B) --- not over the open-ended
+\texttt{ji-5limit} lattice pitch space, which is a separate built-in
+with its own structure. Cents figures throughout are derived from the
+stated ratio or fraction-of-comma, never primary.
+
+\subsubsection{\texttt{pythagorean}}
+
+A chain of eleven pure $3/2$ fifths (twelve notes); every other interval
+is stacked fifths reduced by octaves. No comma is tempered anywhere in
+the chain --- the entire Pythagorean comma ($531441/524288 \approx
+23.460$ cents) is concentrated in the single interval where the chain
+does not close.
+
+The twelve-note selection is a choice of \emph{which} eleven consecutive
+fifths to keep, not a consequence of ``pure fifths'' alone. The
+conventional cut runs E$\flat$--B$\flat$--F--C--G--D--A--E--B--F$\sharp$--C$\sharp$--G$\sharp$
+(C at the center), leaving the wolf on the diminished sixth
+G$\sharp$--E$\flat$. An equally valid alternative cuts the spiral one
+step the other way (chain D$\flat$--\ldots--F$\sharp$, wolf at
+F$\sharp$--D$\flat$); either is a legitimate Pythagorean tuning. The
+E$\flat$--G$\sharp$ cut is the construction this identifier denotes.
+
+\begin{longtable}{p{1.8cm} p{4.5cm} p{4.5cm}}
+  \toprule
+  \textbf{Note} & \textbf{Ratio} & \textbf{Cents (derived)} \\
+  \midrule
+  \endhead
+  C & $1/1$ & 0.000 \\
+  C$\sharp$ & $2187/2048$ & 113.685 \\
+  D & $9/8$ & 203.910 \\
+  E$\flat$ & $32/27$ & 294.135 \\
+  E & $81/64$ & 407.820 \\
+  F & $4/3$ & 498.045 \\
+  F$\sharp$ & $729/512$ & 611.730 \\
+  G & $3/2$ & 701.955 \\
+  G$\sharp$ & $6561/4096$ & 815.640 \\
+  A & $27/16$ & 905.865 \\
+  B$\flat$ & $16/9$ & 996.090 \\
+  B & $243/128$ & 1109.775 \\
+  \bottomrule
+\end{longtable}
+
+The closing wolf fifth G$\sharp\to$E$\flat$ (octave-reduced) is
+$262144/177147 \approx 678.495$ cents.
+
+\textbf{Closure (non-circulating, by design).} Eleven pure fifths at
+701.955~c each, plus the closing wolf at 678.495~c:
+$11 \times 701.955 + 678.495 = 8400.000$~c exactly (seven octaves, as
+any assignment of twelve distinct pitch classes must sum to by
+construction). Relative to twelve
+\emph{pure} fifths ($12 \times 701.955 = 8423.460$~c), this construction
+is short by exactly $8423.460 - 8400.000 = 23.460$~c --- one Pythagorean
+comma, concentrated entirely on the single G$\sharp$--E$\flat$ interval
+rather than distributed. That concentration is what makes the tuning
+non-circulating: no pair of its twelve notes can be treated as
+interchangeable fifths the way a well temperament's can.
+
+\subsubsection{\texttt{meantone-1/4-comma}, \texttt{meantone-1/5-comma}, \texttt{meantone-1/6-comma}}
+
+Meantone is a \emph{regular} temperament: every one of the twelve fifths
+in the chain is tempered by the same fraction of the \emph{syntonic}
+comma ($81/80 \approx 21.506$ cents) --- not the Pythagorean comma used
+by the well temperaments below. All twelve fifths are tempered
+uniformly, so there is no ``which fifths'' selection the way there is
+for a well temperament; there is still a twelve-note chain-closing
+choice, structurally identical to \texttt{pythagorean}'s, and the same
+conventional cut places the wolf between G$\sharp$ and E$\flat$.
+
+The tempered fifth is $(3/2) \cdot (80/81)^{1/n}$ for $1/n$-comma. For
+$n=4$ this collapses to the exact form $5^{1/4}$; $n=5$ and $n=6$ are
+irrational and are given only as derived cents.
+
+\begin{longtable}{p{2.6cm} p{4.2cm} p{2.8cm} p{3.6cm}}
+  \toprule
+  \textbf{Variant} & \textbf{Fifth ratio} & \textbf{Cents (derived)} &
+  \textbf{Narrowing vs.\ pure $3/2$} \\
+  \midrule
+  \endhead
+  1/4-comma & $(3/2)(80/81)^{1/4} = 5^{1/4}$ exactly & 696.578 &
+    5.377~c $= \tfrac{1}{4} \cdot 21.506$~c \\
+  1/5-comma & $(3/2)(80/81)^{1/5}$ & 697.654 &
+    4.301~c $= \tfrac{1}{5} \cdot 21.506$~c \\
+  1/6-comma & $(3/2)(80/81)^{1/6}$ & 698.371 &
+    3.584~c $= \tfrac{1}{6} \cdot 21.506$~c \\
+  \bottomrule
+\end{longtable}
+
+Four ascending 1/4-comma fifths, lowered by two octaves, give the just
+major third: $4 \times 696.578 - 2 \times 1200 = 386.31$~c, the cents of
+$5/4$ --- the defining property of quarter-comma meantone.
+
+\textbf{Closure (non-circulating, by design; all three).} Because all
+twelve fifths are tempered equally, the residual wolf is the twelfth,
+closing interval forced to bring the total to 8400~c (seven octaves):
+
+\begin{longtable}{p{2.6cm} p{3.4cm} p{3.6cm} p{3.4cm}}
+  \toprule
+  \textbf{Variant} & \textbf{11 $\times$ tempered fifth} &
+  \textbf{Wolf (12th, closing)} & \textbf{Wolf vs.\ pure $3/2$} \\
+  \midrule
+  \endhead
+  1/4-comma & 7662.363~c & 737.637~c & $+35.682$~c (wide) \\
+  1/5-comma & 7674.191~c & 725.809~c & $+23.854$~c (wide) \\
+  1/6-comma & 7682.077~c & 717.923~c & $+15.968$~c (wide) \\
+  \bottomrule
+\end{longtable}
+
+All three wolves land on the wide side of a pure fifth --- the opposite
+sense from \texttt{pythagorean}'s narrow wolf.
+
+\subsubsection{\texttt{werckmeister-iii}}
+
+Fifths C--G, G--D, D--A, and B--F$\sharp$ are each narrowed by 1/4 of
+the \emph{Pythagorean} comma; the remaining eight fifths are pure.
+Because only four of twelve fifths are tempered and the pure fifths
+absorb none of the comma, all twelve notes remain usable as a tonic:
+there is no wolf.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  C--G, G--D, D--A, B--F$\sharp$ & narrow, 1/4 Pythagorean comma (4) \\
+  A--E, E--B, F$\sharp$--C$\sharp$, C$\sharp$--G$\sharp$,
+    G$\sharp$--E$\flat$, E$\flat$--B$\flat$, B$\flat$--F, F--C &
+    pure (8) \\
+  \bottomrule
+\end{longtable}
+
+Tempered-fifth cents: $(3/2)/(3^{12}/2^{19})^{1/4} \approx 696.090$
+(narrowing 5.865~c $= \tfrac{1}{4} \cdot 23.460$~c).
+
+\textbf{Closure.} $4 \times 5.865 = 23.460$~c exactly --- the chain
+closes.
+
+\subsubsection{\texttt{werckmeister-iv}}
+
+Fifths C--G, D--A, E--B, F$\sharp$--C$\sharp$, and B$\flat$--F are each
+narrowed by 1/3 of the Pythagorean comma; fifths G$\sharp$--D$\sharp$
+and E$\flat$--B$\flat$ are each \emph{widened} by 1/3 of the Pythagorean
+comma; the remaining five fifths are pure.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  C--G, D--A, E--B, F$\sharp$--C$\sharp$, B$\flat$--F &
+    narrow, 1/3 Pythagorean comma (5) \\
+  G$\sharp$--D$\sharp$, E$\flat$--B$\flat$ &
+    wide, 1/3 Pythagorean comma (2) \\
+  G--D, A--E, B--F$\sharp$, C$\sharp$--G$\sharp$, F--C & pure (5) \\
+  \bottomrule
+\end{longtable}
+
+Narrow-fifth cents: $(3/2)/(3^{12}/2^{19})^{1/3} \approx 694.135$
+(narrowing 7.820~c $= \tfrac{1}{3} \cdot 23.460$~c). Wide-fifth cents:
+$(3/2) \cdot (3^{12}/2^{19})^{1/3} \approx 709.775$ (widening 7.820~c).
+No wolf: all twelve notes remain usable as a tonic.
+
+\textbf{Closure.} $5 \times 7.820 - 2 \times 7.820 = 3 \times 7.820 =
+23.460$~c exactly --- the chain closes.
+
+\subsubsection{\texttt{vallotti}}
+
+Fifths F--C, C--G, G--D, D--A, A--E, and E--B (six consecutive) are each
+narrowed by 1/6 of the Pythagorean comma; the other six fifths are pure.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  F--C, C--G, G--D, D--A, A--E, E--B & narrow, 1/6 Pythagorean comma (6) \\
+  B--F$\sharp$, F$\sharp$--C$\sharp$, C$\sharp$--G$\sharp$,
+    G$\sharp$--E$\flat$, E$\flat$--B$\flat$, B$\flat$--F & pure (6) \\
+  \bottomrule
+\end{longtable}
+
+Tempered-fifth cents: $(3/2)/(3^{12}/2^{19})^{1/6} \approx 698.045$
+(narrowing 3.910~c $= \tfrac{1}{6} \cdot 23.460$~c). No wolf.
+
+\textbf{Closure.} $6 \times 3.910 = 23.460$~c exactly --- the chain
+closes.
+
+\subsubsection{\texttt{kirnberger-ii}}
+
+Fifths D--A and A--E are each narrowed by 1/2 the syntonic comma;
+fifth F$\sharp$--D$\flat$ (i.e.\ F$\sharp$--C$\sharp$, named with the
+flat spelling because the chain below is built outward from
+D$\flat$) is narrowed by a schisma (the difference between the
+Pythagorean and syntonic commas, $32805/32768 \approx 1.9537$~c); the
+remaining nine fifths are pure.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  D$\flat$--A$\flat$, A$\flat$--E$\flat$, E$\flat$--B$\flat$,
+    B$\flat$--F, F--C, C--G, G--D, E--B, B--F$\sharp$ & pure (9) \\
+  D--A, A--E & narrow, 1/2 syntonic comma (2) \\
+  F$\sharp$--D$\flat$ (closing) & narrow, 1 schisma (1) \\
+  \bottomrule
+\end{longtable}
+
+Cents by note, built by stacking the chain
+D$\flat$--A$\flat$--E$\flat$--B$\flat$--F--C--G--D--A--E--B--F$\sharp$
+from C~$= 1/1$:
+
+\begin{longtable}{p{2.2cm} p{3.5cm}}
+  \toprule
+  \textbf{Note} & \textbf{Cents (derived)} \\
+  \midrule
+  \endhead
+  C & 0.000 \\
+  D$\flat$ & 90.225 \\
+  D & 203.910 \\
+  E$\flat$ & 294.135 \\
+  E & 386.314 \\
+  F & 498.045 \\
+  F$\sharp$ & 590.224 \\
+  G & 701.955 \\
+  A$\flat$ & 792.180 \\
+  A & 895.112 \\
+  B$\flat$ & 996.090 \\
+  B & 1088.269 \\
+  \bottomrule
+\end{longtable}
+
+\textbf{Closure.} $2 \times 10.753$ (half-syntonic-comma fifths) $+ 1
+\times 1.9537$ (schisma fifth) $= 21.5063 + 1.9537 = 23.4600$~c exactly
+--- the chain closes.
+
+\subsubsection{\texttt{kirnberger-iii}}
+
+Fifths C--G, G--D, D--A, and A--E (four consecutive) are each narrowed
+by 1/4 the syntonic comma; fifth F$\sharp$--D$\flat$ is narrowed by a
+schisma (same position and same chain skeleton as
+\texttt{kirnberger-ii}); the remaining seven fifths are pure.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  D$\flat$--A$\flat$, A$\flat$--E$\flat$, E$\flat$--B$\flat$,
+    B$\flat$--F, F--C, E--B, B--F$\sharp$ & pure (7) \\
+  C--G, G--D, D--A, A--E & narrow, 1/4 syntonic comma (4) \\
+  F$\sharp$--D$\flat$ (closing) & narrow, 1 schisma (1) \\
+  \bottomrule
+\end{longtable}
+
+Cents by note, same chain skeleton as \texttt{kirnberger-ii}, from
+C~$= 1/1$:
+
+\begin{longtable}{p{2.2cm} p{3.5cm}}
+  \toprule
+  \textbf{Note} & \textbf{Cents (derived)} \\
+  \midrule
+  \endhead
+  C & 0.000 \\
+  D$\flat$ & 90.225 \\
+  D & 193.157 \\
+  E$\flat$ & 294.135 \\
+  E & 386.314 \\
+  F & 498.045 \\
+  F$\sharp$ & 590.224 \\
+  G & 696.578 \\
+  A$\flat$ & 792.180 \\
+  A & 889.735 \\
+  B$\flat$ & 996.090 \\
+  B & 1088.269 \\
+  \bottomrule
+\end{longtable}
+
+\textbf{Closure.} $4 \times 5.377$ (quarter-syntonic-comma fifths) $+ 1
+\times 1.9537$ (schisma fifth) $= 21.5063 + 1.9537 = 23.4600$~c exactly
+--- the chain closes.
+
+\subsubsection{\texttt{young-ii}}
+
+Thomas Young's \emph{second} temperament. Fifths C--G, G--D, D--A, A--E,
+E--B, and B--F$\sharp$ (six consecutive) are each narrowed by 1/6 of the
+Pythagorean comma; the remaining six fifths are pure.
+
+\begin{longtable}{p{9.5cm} p{2.5cm}}
+  \toprule
+  \textbf{Fifths} & \textbf{Tempering} \\
+  \midrule
+  \endhead
+  C--G, G--D, D--A, A--E, E--B, B--F$\sharp$ &
+    narrow, 1/6 Pythagorean comma (6) \\
+  F$\sharp$--C$\sharp$, C$\sharp$--G$\sharp$, G$\sharp$--E$\flat$,
+    E$\flat$--B$\flat$, B$\flat$--F, F--C & pure (6) \\
+  \bottomrule
+\end{longtable}
+
+This is the same six-tempered/six-pure, 1/6-Pythagorean-comma
+construction as \texttt{vallotti}, rotated: \texttt{young-ii}'s
+tempered run starts at C; \texttt{vallotti}'s starts at F. Tempered-fifth
+cents are therefore identical to \texttt{vallotti}'s: $\approx 698.045$
+(narrowing 3.910~c $= \tfrac{1}{6} \cdot 23.460$~c). No wolf.
+
+\textbf{Closure.} $6 \times 3.910 = 23.460$~c exactly --- the chain
+closes.
+
+\subsection{The Static 5-Limit Construction}
+\label{sec:tuning:ji-static}
+
+\begin{requirement}
+  \label{req:tuning:ji-static-construction}
+  The static 5-limit just-intonation systems are constructed from the
+  contiguous lattice block $\{3^a 5^b \mid a \in [-1,2],\ b \in
+  [-1,1]\}$ --- twelve cells, octave-reduced, assigned in ascending
+  order to the twelve chromatic positions of \texttt{cmn-12} starting
+  from the anchor, which takes the role of $1/1$. The block is
+  generated by its bounds; nothing is selected and nothing is
+  discarded.
+\end{requirement}
+
+The twelve, anchor-relative (position 0 is the anchor):
+
+\begin{longtable}{p{1.4cm} p{2.6cm} p{2.2cm} p{3.4cm}}
+  \toprule
+  \textbf{Step} & \textbf{Cell} & \textbf{Ratio} &
+  \textbf{Cents (derived)} \\
+  \midrule
+  \endhead
+  0 & $3^{0}5^{0}$ & $1/1$ & 0.000 \\
+  1 & $3^{-1}5^{-1}$ & $16/15$ & 111.731 \\
+  2 & $3^{2}5^{0}$ & $9/8$ & 203.910 \\
+  3 & $3^{1}5^{-1}$ & $6/5$ & 315.641 \\
+  4 & $3^{0}5^{1}$ & $5/4$ & 386.314 \\
+  5 & $3^{-1}5^{0}$ & $4/3$ & 498.045 \\
+  6 & $3^{2}5^{1}$ & $45/32$ & 590.224 \\
+  7 & $3^{1}5^{0}$ & $3/2$ & 701.955 \\
+  8 & $3^{0}5^{-1}$ & $8/5$ & 813.686 \\
+  9 & $3^{-1}5^{1}$ & $5/3$ & 884.359 \\
+  10 & $3^{2}5^{-1}$ & $9/5$ & 1017.596 \\
+  11 & $3^{1}5^{1}$ & $15/8$ & 1088.269 \\
+  \bottomrule
+\end{longtable}
+
+\texttt{ji-static-5limit-C}, \texttt{ji-static-5limit-G}, and
+\texttt{ji-static-5limit-D} are this one construction at three anchors,
+the named note taking the role of $1/1$. The catalog keeps exactly
+these three rows; additional anchors are an implementation extension
+and \MUSTNOT{} be read as a conformance obligation.
+
+No 12-note set built this way is inversionally symmetric: $45/32$ is
+present and its inverse $64/45$ is not.
+
 \subsection{Default Score Configuration}
 
 A newly-created score with no explicit tuning context \MUST{} default to:
@@ -3648,10 +4073,18 @@ A newly-created score with no explicit tuning context \MUST{} default to:
   \item \texttt{VoiceId}, \texttt{StaffId}, \texttt{RegionId},
     \texttt{VoiceSelector}, and the score graph's hierarchical
     organization are defined in Chapter~\ref{ch:graph}.
-  \item \texttt{KeyContext} and the harmonic-context construction
+  \item \texttt{KeyContext} is not defined anywhere in this
-    machinery are partially defined in Chapter~\ref{ch:graph} and
+    specification; completing it is out of scope for this document.
-    completed in the audio engine specification (out of scope for this
+    Whatever completes it \MUST{} expose a tonal-centre chromatic
-    document).
+    pitch class consistent with
+    Requirement~\ref{req:tuning:adaptive-anchor-derivation}.
+    Chapter~\ref{ch:graph} defines \texttt{KeySignature}, not
+    \texttt{KeyContext}. \texttt{ContextHint} and
+    \texttt{AdaptiveTuningParameters} are likewise undefined; the
+    built-in \texttt{ji-adaptive-5limit}, version~1
+    (Requirement~\ref{req:tuning:adaptive-default-version}), ignores
+    both, so defining them now would freeze a type surface on a
+    chapter with no consumer.
   \item The specific spelling algorithms referenced by
     \texttt{SpellingAlgorithmId} remain an open question; see the
     related open question in Section~\ref{sec:pitch:prepass}.