For a fixed LameJuis configuration and lens, harmonic analysis starts with a relation between time slices and available pitch labels. This page defines its Galois connection and expresses it using Boolean functions on a finite Stone space. Harmonic concepts and persistence develops the associated complexes, filtrations, and barcodes.
These pages specify a mathematical analysis model. They do not assert that the engine currently computes concepts or persistent homology. See LameJuis for the pitch engine and its implementation terminology.
Fix an accepted configuration $M$ and a lens $U$. Let $I=\{0,1\}$ and
X=I^d,\qquad T=X/{\sim_U},\qquad \pi_U:X\longrightarrow T.
Two inputs are equivalent when they agree on the coordinates read by $U$.
Thus $T\cong I^{|U|}$, and each element of $T$ represents an entire fiber of
co-muted inputs. The current engine has $d=6$. The symbol $d$ denotes this input
dimension; $k$ will denote a filtration threshold and $q$ a homological degree.
Choose a finite set $\Sigma$ of pitch or section identities and write
$p_M:X\to\Sigma$ for the resulting output map. This is the map called $M$ in
the expression $F_x^M(U)$ in the engine documentation. Here the subscript
distinguishes the configuration from the function it determines.
The identity convention is part of the analysis: accumulator sections, exact
evaluated pitches, and pitches modulo octave can give different sets $\Sigma$.
Choose one convention before constructing the relation. Take $\Sigma$ to be
the image of $p_M$, so every label occurs somewhere.
| Symbol | Meaning |
|---|---|
$x,y\in X$ |
Full Boolean input states |
$t\in T$ |
A time slice modulo the fixed lens |
$S\subseteq T$, $s\in S$ |
A collection of slices and one of its slices |
$\sigma,\tau\in\Sigma$ |
Pitch labels |
$C\subseteq\Sigma$ |
A collection of pitch labels |
$\mathcal P(E)$ |
The power set of a set $E$ |
For $t=\pi_U(x)$, define
F_t^M(U)=p_M(\pi_U^{-1}(t))=F_x^M(U).
This is independent of the representative $x$. The incidence relation is
R=R^{M,U}\subseteq T\times\Sigma,
\qquad tR\sigma\iff\sigma\in F_t^M(U).
This relation describes availability across all slices. A chronological path through those slices is additional data from the Theory of Time.
Define the two order-reversing operations
\begin{aligned}
S^\uparrow
&=\{\sigma\in\Sigma:\forall s\in S,\ sR\sigma\},\\
C^\downarrow
&=\{t\in T:\forall\sigma\in C,\ tR\sigma\}.
\end{aligned}
The intent $S^\uparrow$ is the set of pitches common to every slice in $S$.
The extent $C^\downarrow$ is the set of slices admitting every pitch in $C$.
The up/down notation is an established convention for the derivation operations
of formal concept analysis; see, for example,
this formulation of the operators.
Their defining Galois equivalence is
C\subseteq S^\uparrow\quad\Longleftrightarrow\quad S\subseteq C^\downarrow.
Universal quantification gives the empty-set conventions
$\varnothing^\uparrow=\Sigma$ and $\varnothing^\downarrow=T$. Composing the
operations gives two closures:
\mathrm{cl}_T(S)=S^{\uparrow\downarrow},
\qquad
\mathrm{cl}_\Sigma(C)=C^{\downarrow\uparrow}.
Each closure is extensive, order-preserving, and idempotent. In particular,
S^{\uparrow\downarrow\uparrow}=S^\uparrow,
\qquad C^{\downarrow\uparrow\downarrow}=C^\downarrow.
Closing a set of slices therefore adds every slice satisfying its common pitch constraints, without changing those common pitches. Closing a pitch collection adds every pitch that is always available wherever the original collection is available.
For illustration, restrict to the following three-slice relation. The operations in this example use only the displayed rows:
| Slice | $\alpha$ |
$\beta$ |
|---|---|---|
$t_1$ |
available | |
$t_2$ |
available | available |
$t_3$ |
available |
Here $\{t_1\}^\uparrow=\{\alpha\}$, so
$\{t_1\}^{\uparrow\downarrow}=\{t_1,t_2\}$.
Also $\{\alpha,\beta\}^\downarrow=\{t_2\}$. The singleton $\{t_1\}$
is not closed under the incidence closure even though it is a closed subset of
the finite Stone space.
Let
A=\mathrm{Map}(T,\mathbb F_2).
This is both a Boolean ring and a Boolean algebra: ring addition is XOR,
multiplication is AND, and complement is $f\mapsto1+f$. Define
e:\Sigma\longrightarrow A,
\qquad e_\sigma(t)=1+\mathbf1[tR\sigma].
The convention is that zero means available. Thus $e_\sigma$ is the unique
Boolean function with zero locus equal to the availability locus of $\sigma$.
The principal ideal it generates is exactly
(e_\sigma)
=\{f\in A:f(t)=0\text{ whenever }tR\sigma\}.
Indeed, every multiple of $e_\sigma$ vanishes there; conversely any such $f$
satisfies $f=e_\sigma f$.
The map $e$ need not be injective. Different pitches can have identical
availability patterns and hence equal Boolean functions and equal principal
ideals. They remain different members of $\Sigma$ and count separately in
$|S^\uparrow|$. Replacing the labelled family $(e_\sigma)_{\sigma\in\Sigma}$
by the unlabelled set $e(\Sigma)$ would lose this count.
This label multiplicity is also distinct from the number of full input states producing the same pitch. The latter is recorded by
W_{t,\sigma}=\lvert\{y\in\pi_U^{-1}(t):p_M(y)=\sigma\}\rvert.
The binary relation remembers only whether $W_{t,\sigma}>0$. The Dowker
thresholds in the companion page count pitch labels, not these input witnesses.
The multiset viewpoint remains relevant to the engine’s percentile chooser;
see co-muting.
Use $Z$ for the common-zero operation. For pitch labels this means
Z(C)=\{t\in T:\forall\sigma\in C,\ e_\sigma(t)=0\}=C^\downarrow.
For a family of functions or an ideal in $A$, $Z$ has the same common-zero
meaning. In particular,
Z(C)=Z\bigl(\langle e_\sigma:\sigma\in C\rangle\bigr).
To discuss the full algebraic vanishing ideal, use a different typeface:
\mathfrak I(S)=\{f\in A:f|_S=0\}.
The relationship to the up operation is especially direct:
S^\uparrow=e^{-1}(\mathfrak I(S)).
Thus the up operation returns labels whose designated functions vanish on
$S$. The ideal $\mathfrak I(S)$ contains all functions vanishing on $S$.
The earlier notation $I(S)$ for common pitches is superseded by $S^\uparrow$;
$\mathfrak I$ is reserved for the actual ideal.
The selected equations generate
J(S)=\langle e_\sigma:\sigma\in S^\uparrow\rangle
\subseteq\mathfrak I(S),
and the two zero-locus identities are
Z(J(S))=S^{\uparrow\downarrow},\qquad Z(\mathfrak I(S))=S.
For the availability table above, $e_\alpha=(0,0,1)$ and
$e_\beta=(1,0,0)$, listing values at $t_1,t_2,t_3$. If $S=\{t_1\}$,
then $J(S)=(e_\alpha)$ cuts out $\{t_1,t_2\}$. The full ideal
$\mathfrak I(S)$ additionally contains $(0,1,0)$, which excludes $t_2$.
The incidence closure is therefore closure relative to a labelled family of pitch equations. In the finite Boolean coordinate ring, every subset already has its own full vanishing ideal and is closed in the Zariski topology.
The Boolean cube has coordinate ring
A_X=\mathbb F_2[z_1,\ldots,z_d]/(z_i^2-z_i)_{i=1}^{d}
\cong\mathrm{Map}(X,\mathbb F_2)
\cong\prod_{x\in X}\mathbb F_2.
Consequently $X\cong\mathrm{Spec}\,A_X$. Likewise,
$T\cong\mathrm{Spec}\,A$; pullback along $\pi_U$ identifies $A$ with the
subring of functions on $X$ constant on each lens fiber. The cube has $2^d$
points, so its function ring has $2^d$ field factors. This differs from the
ring $\mathbb F_2^d$, whose spectrum has $d$ points.
Finite Stone spaces are discrete. General Stone spaces have a basis of clopen
sets and need not be discrete. In a Boolean ring the principal zero locus of
$f$ equals the principal open locus of $1+f$. This is the precise relationship
between the affine and clopen descriptions; see
Tressl on Stone duality and prime spectra.
Boolean constraints $g_1=\cdots=g_r=1$ define the solution scheme
\mathrm{Spec}\bigl(A_X/(1+g_1,\ldots,1+g_r)\bigr).
The constraints are unsatisfiable exactly when this ideal contains $1$.
The harmonic equations $e_\sigma=0$ use this same algebra of constraints.
Separate the Boolean output map from its pitch evaluation:
X\xrightarrow{\beta_M}\mathbb F_2^m
\xrightarrow{\phi}P,\qquad p_M=\phi\circ\beta_M.
Here $m$ counts Boolean outputs, and $P$ is the chosen pitch or section space;
$\Sigma$ is the finite image of the composite. The Boolean output is a global
section
\beta_M\in\Gamma(X,\mathcal O_X^{\oplus m})=A_X^m.
The free module sheaf is $\mathcal O_X^{\oplus m}$; $\beta_M$ is one of its
sections. The pitch evaluator $\phi$ is generally a map of sets rather than an
$\mathbb F_2$-linear map. The composite is a global section of the sheaf of
sets $V\mapsto\mathrm{Map}(V,P)$ on the discrete space $X$.
The lens construction instead takes the image of that chosen section on a
fiber, $F_x^M(U)=p_M(\pi_U^{-1}(\pi_U(x)))$. For fixed $x$, if $U\subseteq V$
as sets of read coordinates, there is an inclusion
F_x^M(V)\hookrightarrow F_x^M(U).
This defines a presheaf on the poset of lenses. It can be realized as a sheaf
on that poset with the Alexandrov topology of downward-closed subsets, where
the minimal open neighborhood of $U$ is the set of lenses contained in $U$.
This topology is different from treating lenses themselves as open subsets of
the coordinate set and imposing gluing for their unions.
The Boolean module sheaf on $X$, the presheaf over lenses, and the fixed-lens
incidence relation have different bases and roles. The nontrivial simplicial
topology studied next comes from the incidence relation between points of $T$
and labels in $\Sigma$.