$V\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}\times{\mathbb{C}}$
is similar. If
$X_{0}\xleftarrow{s_{i}}X_{1}\xleftarrow{s_{i}}\dots\xleftarrow{s_{i_{m}}}X_{m}$
represents a point in $V$, then $X_{m}$ is not weakly $w_{0}$-related to
$X_{1}$. Because $X_{0}\stackrel{{\scriptstyle
w_{0}}}{{\Longleftarrow}}X_{m}$, we have in fact $X_{1}\stackrel{{\scriptstyle
w_{0}s_{i}}}{{\Longleftarrow}}X_{m}$. By Lemma 6.5, this means
$X_{2}\stackrel{{\scriptstyle w_{0}}}{{\Longleftarrow}}X_{m}$. So we may
define a map
$V\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}\times{\mathbb{C}}$
by

$(g\dot{s}_{i}z_{i}(t)U_{+}=X_{0}\xleftarrow{s_{i}}g\dot{s}_{i}U_{+}=X_{1}\xleftarrow{s_{i}}gU_{+}=X_{2}\leftarrow\dots\xleftarrow{s_{i_{m}}}X_{m})\mapsto\left((X_{2}\xleftarrow{s_{i_{2}}}\dots\xleftarrow{s_{i_{m}}}X_{m}),t\right).$

(The fact that all points in $V$ are represented by a tuple of the form
written above follows from Lemma 6.5 as well.) The inverse of this map is
clear. ∎

###  9.3. Triangularity

The grid minors $\Delta_{c,j}$ on $\accentset{\circ}{R}_{\beta}$ restrict to
the corresponding grid minors on $\accentset{\circ}{R}_{i\beta^{\prime}}$
(resp., $\accentset{\circ}{R}_{\beta^{\prime}}$), except for grid minors for
the leftmost (resp., the leftmost two) crossings.

Functions on $\accentset{\circ}{R}_{i\beta^{\prime}}$ (resp.,
$\accentset{\circ}{R}_{\beta^{\prime}}$) can be pulled back to $W$ (resp.,
$V$) under the projection
$\pi_{i\beta^{\prime}}:\accentset{\circ}{R}_{i\beta^{\prime}}\times{\mathbb{C}}^{\times}\to\accentset{\circ}{R}_{i\beta^{\prime}}$
(resp.,
$\pi_{\beta^{\prime}}:\accentset{\circ}{R}_{\beta^{\prime}}\times{\mathbb{C}}\to\accentset{\circ}{R}_{\beta^{\prime}}$).

By induction, we have shown that
${\mathbb{C}}[\accentset{\circ}{R}_{i\beta^{\prime}}]={\mathcal{A}}_{i\beta^{\prime}}$
and
${\mathbb{C}}[\accentset{\circ}{R}_{\beta^{\prime}}]={\mathcal{A}}_{\beta^{\prime}}$.
These equalities are compatible with the one that we want to prove, as the
following result demonstrates.

######  Proposition 9.3.

Let $x_{c}$ be a cluster variable of ${\mathbf{x}}_{\beta}$, not equal to
$x_{1}$ (resp., $x_{1}$ or $x_{2}$). Then $x_{c}$ restricts to the pullback
under $\pi_{i\beta^{\prime}}$ (resp., $\pi_{\beta^{\prime}}$) of similarly
denoted cluster variable on $\accentset{\circ}{R}_{i\beta^{\prime}}$ (resp.,
$\accentset{\circ}{R}_{\beta^{\prime}}$). That is,
$x_{c}\in{\mathcal{A}}_{\beta}$ and
$x_{c}^{\prime}\in{\mathcal{A}}_{i\beta^{\prime}}$ (resp.,
$x_{c}^{\prime}\in{\mathcal{A}}_{\beta^{\prime}}$) have the same image in the
following diagrams:

${\mathcal{A}}_{\beta}\hookrightarrow{\rm
Frac}({\mathbb{C}}[\accentset{\circ}{R}_{\beta}])\xrightarrow{\text{restriction}}{\rm
Frac}({\mathbb{C}}[W])\xleftarrow{\text{pullback}}{\mathbb{C}}[\accentset{\circ}{R}_{i\beta^{\prime}}]={\mathcal{A}}_{i\beta^{\prime}},$

${\mathcal{A}}_{\beta}\hookrightarrow{\rm
Frac}({\mathbb{C}}[\accentset{\circ}{R}_{\beta}])\xrightarrow{\text{restriction}}{\rm
Frac}({\mathbb{C}}[V])\xleftarrow{\text{pullback}}{\mathbb{C}}[\accentset{\circ}{R}_{\beta^{\prime}}]={\mathcal{A}}_{\beta^{\prime}}.$

###### Proof.

The claimed compatibility is true for grid minors, and the Laurent monomial
expressions for $x_{c}$ and $x_{c}^{\prime}$ in terms of grid minors are the
same. ∎

###  9.4. Regularity of mutated cluster variables

The results of Section 8 show that the mutation of $x_{2}$ is a regular
function on $\accentset{\circ}{R}_{\beta}$. Indeed, applying (B5) rescales a
single frozen variable by a Laurent monomial in frozens and applying (B1)
mutates at $x_{2}$. The resulting seed is ${\mathbf{x}}_{i(-i)\beta^{\prime}}$
and it consists of regular functions on $\accentset{\circ}{R}_{\beta}$. The
cluster variable $\tilde{x}_{2}$ in this seed differs from the mutation of
$x_{2}$ by a unit, hence the mutation of $x_{2}$ is also a regular function.

For all other mutated cluster variables $\tilde{x}_{c}$, we note that the
formula for the mutation is the same in
$\Sigma_{\beta},\Sigma_{i\beta^{\prime}}$ and $\Sigma_{\beta^{\prime}}$. Now,
either (a) $x_{c}$ is mutable in both $\Sigma_{i\beta^{\prime}}$ and
$\Sigma_{\beta^{\prime}}$ or (b) $x_{c}$ is mutable in
$\Sigma_{i\beta^{\prime}}$ and frozen in $\Sigma_{\beta^{\prime}}$. If (a),
then by induction $\tilde{x}_{c}$ is regular on both $W$ and $V$ and thus on
$\accentset{\circ}{R}_{\beta}$. If (b), then by induction $\tilde{x}_{c}$ is
regular on $W$. Since $x_{c}$ is a unit on $V$ and
$\tilde{x}_{c}=(M+M^{\prime})/x_{c}$ where $M,M^{\prime}$ are monomials in the
other cluster variables, $\tilde{x}_{c}$ is also regular on $V$.

###  9.5. Tori in the braid variety

For $c\in J^{\operatorname{mut}}_{\beta}$, let $\tilde{{\mathbf{x}}}_{c}$ be
the cluster of the mutated seed $\mu_{c}(\Sigma_{\beta})$. Since we have shown
all the cluster variables and their mutations are regular, we have maps

$\varphi:\accentset{\circ}{R}_{\beta}\to{\mathbb{C}}^{d},\qquad
g\mapsto(x_{1}(g),x_{2}(g),\ldots,x_{d}(g))={\mathbf{x}}(g);$
$\varphi_{c}:\accentset{\circ}{R}_{\beta}\to{\mathbb{C}}^{d},\qquad
g\mapsto(x_{1}(g),x_{2}(g),\ldots,x_{d}(g))=\tilde{{\mathbf{x}}}_{c}(g).$

Let $T_{c}$ be the Deodhar torus associated to the cluster
$\tilde{{\mathbf{x}}}_{c}$.

######  Proposition 9.4.

There are inclusions of cluster tori
$T\hookrightarrow\accentset{\circ}{R}_{\beta}$ and
$T_{c}\hookrightarrow\accentset{\circ}{R}_{\beta}$ which are partial inverses
to $\varphi,\varphi_{c}$. In other words, we have inclusions
${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]\subset{\mathbb{C}}[{\mathbf{x}}^{\pm
1}]$ and
${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]\subset{\mathbb{C}}[\tilde{{\mathbf{x}}}_{c}^{\pm
1}]$.

###### Proof.

For $T=T_{\beta}$, this statement is Proposition 7.4. For $T_{2}$, this
follows from Proposition 7.4 applied to the isomorphic braid variety
$\accentset{\circ}{R}_{i(-i)\beta}$, whose Deodhar torus pulls back to
$T_{2}$. For $c\neq 2$, by the inductive hypothesis $T_{c}$ includes into $W$,
so also includes into $\accentset{\circ}{R}_{\beta}$. ∎

###  9.6. Finishing

We proceed by following the argument in [BFZ05, proof of Theorem 2.10].

######  Lemma 9.5.

A Laurent monomial $M={\mathbf{x}}^{\mathbf{a}}$ is regular on
$\accentset{\circ}{R}_{\beta}$ if and only if the exponents of mutable cluster
variables are nonnegative.

###### Proof.

The “if” part is trivial. For the “only if” part, consider the restriction of
$M$ to the torus $T_{c}$. Then $x_{c}^{a_{c}}$ is a regular function on
$T_{c}$ and must be a Laurent polynomial in $\tilde{{\mathbf{x}}}_{c}$. It
follows from the exchange relation that $a_{c}\geq 0$. ∎

######  Lemma 9.6.

For $c\in J^{\operatorname{mut}}_{\beta}$, $x_{c}$ is irreducible in
${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$.

###### Proof.

Every regular function on $\accentset{\circ}{R}_{\beta}$ is a Laurent
polynomial in ${\mathbf{x}}$. If $x_{c}=PQ$ is the product of two regular
functions $P$ and $Q$ then both $P$ and $Q$ must be Laurent monomials in
${\mathbf{x}}$. By Lemma 9.5, one of the factors must be a Laurent monomial in
only the frozen variables, and thus invertible. Therefore $x_{c}$ is
irreducible. ∎

######  Lemma 9.7.

Every mutation $\tilde{x}_{c}$ of a cluster variable $x_{c}$ is the product of
an irreducible element $f_{c}$ in ${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$
with a Laurent monomial in $x_{d},d\neq c$.

###### Proof.

Let $P\in{\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$ be an irreducible factor
of $\tilde{x}_{c}$. Restricting to $T_{c}$, the same argument as in the proof
of Lemma 9.6 implies that $P$ is a Laurent monomial in
$\tilde{{\mathbf{x}}}_{c}$. Restricting to $T$, we see that $P$ is a Laurent
polynomial in ${\mathbf{x}}$. Substituting the exchange relation, we deduce
that the exponent of $\tilde{x}_{c}$ in $P$, as a Laurent polynomial in
$\tilde{{\mathbf{x}}}_{c}$, must be nonnegative. We see that one of the
irreducible factors of $\tilde{x}_{c}$ is equal to $\tilde{x}_{c}$ times a
Laurent monomial in $x_{d},d\neq c$ and the remaining irreducible factors are
Laurent monomials in $x_{d},d\neq c$. ∎

######  Lemma 9.8.

The complement of

$T\cup\bigcup_{c\in J^{\operatorname{mut}}_{\beta}}T_{c}$

has codimension at least $2$ in $\accentset{\circ}{R}_{\beta}$.

###### Proof.

Let $g$ be a point in the complement. Then either $x_{c}(g)=x_{d}(g)=0$ for
distinct mutable indices $c,d$, or $f_{c}(g)=x_{d}(g)=0$ for distinct mutable
indices $c,d$. The codimension two statement follows from Lemmas 9.6 and 9.7.
∎

###  9.7. Proof of Theorem 7.14

Since $\accentset{\circ}{R}_{\beta}$ contains $T\cup\bigcup_{c}T_{c}$, the
coordinate ring ${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$ is contained in
the upper bound of $\Sigma_{\beta}$ given by

${\mathbb{C}}[{\mathbf{x}}^{\pm 1}]\cap\bigcap_{c\in
J^{\operatorname{mut}}_{\beta}}{\mathbb{C}}[\tilde{{\mathbf{x}}}_{c}^{\pm
1}].$

Since $\accentset{\circ}{R}_{\beta}$ is smooth and thus normal, Lemma 9.8
implies that this inclusion is an equality. By [BFZ05, Corollary 1.7] the
upper bound is equal to the upper cluster algebra. By [Mul13] and Corollary
5.8, the upper cluster algebra is equal to the cluster algebra. ∎

##  10\. Applications

###  10.1. Curious Lefschetz

Let $X$ be a smooth, affine, complex algebraic variety of dimension $d$. Then
the cohomology $H^{*}(X)=H^{*}(X,{\mathbb{C}})$ has a mixed Hodge structure,
which gives a Deligne (weak) splitting
$H^{k}(X)=\bigoplus_{p,q}H^{k,(p,q)}(X)$. We say that $X$ is of _mixed Tate
type_ if $H^{k}(X)=\bigoplus_{p}H^{k,(p,p)}(X)$. We say that $X$ satisfies the
_curious Lefschetz property_ if $X$ is of mixed Tate type and there is a class
$[\gamma]\in H^{2,(2,2)}(X)$ such that cup product induces isomorphisms

$[\gamma]^{d-p}:H^{p+s,(p,p)}(X)\xrightarrow{\sim}H^{2d-p+s,(2d-p,2d-p)}(X)$

for all $p$ and $s$. Curious Lefschetz implies the _curious Poincaré symmetry_

$\dim H^{p+s,(p,p)}(X)=\dim H^{2d-p+s,(2d-p,2d-p)}(X).$

In [LS16] it is shown that a large class of cluster varieties satisfy curious
Lefschetz, and the following result is a consequence.

######  Theorem 10.1.

Suppose that $\widetilde{Q}$ is an ice quiver that is sink-recurrent, full
rank, and has an even number of vertices. Then the cluster variety
$X(\widetilde{Q})$ satisfies curious Lefschetz.

The even-dimensional hypothesis is not a significant restriction since we can
always add an extra isolated frozen vertex. See also [LS16, Theorem 8.3].

###### Proof.

Let $X$ be a smooth, affine complex algebraic variety, and $X=U\cup V$ be an
open covering of $X$, and $[\gamma]\in H^{2,(2,2)}(X)$. By [LS16, Theorem
3.5], if all three $(U,[\gamma]),(V,[\gamma]),(U\cap V,[\gamma])$ satisfy
curious Lefschetz then so does $(X,[\gamma])$.

By [LS16, Proposition 8.2], full rank isolated cluster varieties satisfy the
curious Lefschetz property with respect to a (nearly canonical) _GSV form_
$[\gamma]$; cf. [GSV10].

Let $X=\operatorname{Spec}{\mathcal{A}}(\widetilde{Q})$ be a sink-recurrent,
full rank cluster variety and let $X_{1}=\operatorname{Spec}{\mathcal{A}}_{1}$
and $X_{2}=\operatorname{Spec}{\mathcal{A}}_{2}$ give an open cover of $X$, as
in the proof of Proposition 5.5. Both $X_{1}$ and $X_{2}$ are sink-recurrent,
and $X_{1}\cap X_{2}$ is also a cluster localization with mutable quiver
$Q-(N^{\operatorname{in}}_{s}(Q)\cup\\{s\\})$, which is sink recurrent. Thus,
by induction we may assume that $X_{1},X_{2},X_{1}\cap X_{2}$ all satisfy
curious Lefschetz with respect to (the respective restriction of) the GSV form
$[\gamma]$ of $X$. Thus $(X,[\gamma])$ satisfies curious Lefschetz as well. ∎

We do not know whether the Louise condition [LS16] is satisfied for the
quivers in this work. The cohomology of open Richardson varieties are
particularly interesting because of the relations to both Category
$\mathscr{O}$ and to knot homology [GL20]; see also [GHM21].

######  Corollary 10.2.

Even-dimensional braid Richardson varieties $\accentset{\circ}{R}_{{u,\beta}}$
satisfy the curious Lefschetz property and thus the curious Poincaré symmetry.
Odd-dimensional braid varieties satisfy the curious Poincaré symmetry.

###  10.2. Braid Richardson links

By applying Remark 3.3 and the moves (B1)–(B4), we reduce to considering pairs
$u\leq\beta$ such that $\beta$ has only positive indices. Then we can consider
a _braid Richardson link_ $L_{u,\beta}$ obtained as the braid closure of
$\beta\cdot\beta(u)^{-1}$, where $\beta$ is viewed as a positive braid, and
$\beta(u)$ is the positive braid lift of $u$. This construction generalizes
the _Richardson links_ of [GL20]. At the same time, a link $L_{G}$ inside
${\mathbb{R}}^{3}$ can be constructed from a plabic graph $G$; see [STWZ19,
FPST22, GL22a]. The construction can be extended to give a link
$L_{G_{u,\beta}}$ for our 3D plabic graphs $G_{u,\beta}$, using the following
convention for bridges and for hollow crossings:

The construction in [GL22a, Section 4.1] can be extended to our setting to
show that the braid Richardson link $L_{u,\beta}$ agrees with the plabic graph
link $L_{G_{u,\beta}}$.

If we treat the strands of $L_{G_{u,\beta}}$ as the boundaries of ribbons, as
suggested in the figure above, the convention for hollow crossings forces the
ribbons to intersect in ${\mathbb{R}}^{3}$. There is a natural way to alter
the surface $\mathbf{S}_{u,\beta}$ locally at hollow crossings to obtain a
Seifert surface $S(L_{u,\beta})$ for $L_{u,\beta}$. However,
$\mathbf{S}_{u,\beta}$ and $S(L_{u,\beta})$ are different as abstract
surfaces. Nevertheless, one can draw versions of the relative cycles
$(C_{c})_{c\in J_{u,\beta}}$ on $S(L_{u,\beta})$ rather than on
$\mathbf{S}_{u,\beta}$, preserving their intersection numbers.

The surface $S(L_{u,\beta})$ has the “wrong” homology, so an alternative is to
consider a different embedding $\mathbf{S}_{u,\beta}^{\prime}$ of the surface
$\mathbf{S}_{u,\beta}$ in ${\mathbb{R}}^{4}$, defined so that there are no
intersections of ribbons at hollow crossings. The surface
$\mathbf{S}_{u,\beta}^{\prime}$ is diffeomorphic to $\mathbf{S}_{u,\beta}$,
but it will now have the “correct” link $L_{{u,\beta}}$ as its boundary in
${\mathbb{R}}^{3}$.

###  10.3. Point count

Let $Q$ be a mutable quiver. Following [GL22a], define the _point count
rational function_ $R(Q;q)$ to be the function
$q\mapsto\\#X(\widetilde{Q})({\mathbb{F}}_{q})/(q-1)^{a}$ where
${\mathbb{F}}_{q}$ denotes a finite field with $q$ elements and
$\widetilde{Q}$ is a really full rank ice quiver with mutable part $Q$ and $a$
frozen vertices.

Suppose $Q$ is sink-recurrent. Then the covering considered in the proof of
Proposition 5.5 shows that $R(Q;q)$ is a rational function not depending on
the choice of $\widetilde{Q}$; see [GL22a, Section 5.3]. We obtain the
following variant of [GL22a, Conjecture 2.8].

######  Theorem 10.3.

Let $u\leq\beta$, and suppose that the 3D plabic graph $G_{u,\beta}$ has
$\operatorname{c}(G_{u,\beta})$ connected components. Then we have

$R(Q_{u,\beta};q)=(q-1)^{\operatorname{c}(G_{u,\beta})-1}P^{\operatorname{top}}(L_{u,\beta};q),$

where $P^{\operatorname{top}}(L_{u,\beta};q)$ is obtained from the top
$a$-degree term of the _HOMFLY polynomial $P(L_{u,\beta};a,z)$_ by
substituting $a:=q^{-\frac{1}{2}}$ and $z:=q^{\frac{1}{2}}-q^{-\frac{1}{2}}$.

###### Proof.

By Theorems 5.6 and 5.9 and our main result Theorem 7.14, it follows that
$R(Q_{u,\beta};q)$ is equal to
$(q-1)^{-a}\\#\accentset{\circ}{R}_{u,\beta}({\mathbb{F}}_{q})$, where $a$ is
the number of frozen variables in $\widetilde{Q}_{u,\beta}$. A recursion for
the point count of $\accentset{\circ}{R}_{{u,\beta}}$ is given in [GLTW22,
Section 5]. The comparison with the top $a$-degree term of the HOMFLY
polynomial proceeds in the same way as the proof of [GL20, Theorem 2.1]. ∎

The special case of Theorem 10.3 when $\beta$ is a reduced word for some
permutation $w\in S_{n}$ follows from our main results combined with [GL20,
Theorem 1.11]. See [GL20, GL21, GL22a] for further details and examples.

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# Smells like Teen Spirit:
An Exploration of Sensorial Style in Literary Genres

Osama Khalid, Padmini Srinivasan
University of Iowa
{osama-khalid<EMAIL_ADDRESS>

###### Abstract

It is well recognized that sensory perceptions and language have
interconnections through numerous studies in psychology, neuroscience, and
sensorial linguistics. Set in this rich context we ask whether the use of
sensorial language in writings is part of linguistic style? This question is
important from the view of stylometrics research where a rich set of language
features have been explored, but with insufficient attention given to features
related to sensorial language. Taking this as the goal we explore several
angles about sensorial language and style in collections of lyrics, novels,
and poetry. We find, for example, that individual use of sensorial language is
not a random phenomenon; choice is likely involved. Also, sensorial style is
generally stable over time - the shifts are extremely small. Moreover, style
can be extracted from just a few hundred sentences that have sensorial terms.
We also identify representative and distinctive features within each genre.
For example, we observe that 4 of the top 6 representative features in novels
collection involved individuals using olfactory language where we expected
them to use non-olfactory language.

##  1 Introduction

Sensory perceptions shape how we use language and communicate Paradis (2003).
When we use sensorial words (i.e. words with meanings connected to our senses)
like fuzzy or stinky, besides communicating sensorial experiences these also
stimulate perceptual systems in the recipient’s mind Speed and Majid (2020).

The space of senses – sometimes called the “Aristotelian” senses Sorabji
(1971), include the five modalities: visual, auditory, haptic, gustatory, and
olfactory. Relatively recently, linguistics and psychologists have added a
sixth sense – interoception Craig (2002). This refers to the perception of
sensations from inside the body, both physical such as hunger and pain, and
emotional, such as joy. This sensory space has been the basis of much prior
research.

Sensorial Linguistics is about studying how language relates to the senses. A
key focus has been to study how different sensorial experiences and
perceptions are packaged into linguistic units Winter (2019). Researchers have
looked at how some senses dominate in language Winter et al. , how sensorial
language varies across lexical categories Lievers and Winter (2018) and how
sensory experiences influence sensorial language Croijmans et al. (2019);
Murphy (2019). However, the domain of sensorial linguistics is still nascent
with many unexplored questions.

Stylometrics: As individuals grow, besides consciously learning a vocabulary
they also develop a linguistic style. Some stylistic elements may be acquired
subconsciously and others by choice. Stylistic choices can reflect the
individual’s social reality or affective state Savoy (2020). Several
categories of stylistic features have been identified, such as the use of
function words and language complexity Holmes (1998). Stylometrics is
important for goals such as author attribution and affect classification.

Style in sensorial language use: A key limitation in stylometrics is that
linguistic style around sensorial language has not been studied
systematically. A person who wants to express being depressed has several word
choices. She could use “sad” or the less frequent “downcast”. Her propensity
to choose one or the other may be considered as part of her linguistic style.

Consider a cloudy scene. A person may use visual language focusing on color,
and say “the clouds are white”. Another may use haptic language focusing on
texture, “the clouds are fluffy”. While sensorial language is clearly
important for communication, we do not yet know if there are distinguishable
patterns in sensory language use at the level of individuals, texts, etc. This
gap in stylometrics motivates us to ask the following about sensorial language
style:

  * •

RQ 1: Is the notion of sensorial style meaningful or is it a product of random
chance?

  * •

RQ 2: How much data do we need to get a stable representation of sensorial
style?

  * •

RQ 3: Does sensorial style vary with time?

  * •

RQ 4: Which features are representative and distinctive of the individuals
within each genre?

##  2 Representing Sensorial Language Style

Sensorial style may be represented at different levels of abstractions. At the
lowest level, we can represent the proportion of an individual’s language that
is sensorial and also examine the frequencies of different sensorial words. At
a higher level of abstraction, we can ask how frequent are different sense
modalities (visual, auditory, etc.) in an individual’s language.
Alternatively, we can represent style by the extent to which a person’s use of
sensory modalities aligns with general expectations. This is related to
synaesthesia, where one sense modality is used when another is expected - a
well studied phenomenon in sensorial linguistics Lievers (2015); De Ullmann
(1945). As an example in their work, Lievers and Huang (2016a) developed a
lexicon of perception that they used to automatically identify perception
related synaesthetic metaphors. Similarly we also approach the problem of
sensorial style through the lens of their synaesthetic usage.

In the 2010 animated film ‘Despicable Me’, the character of Agnes hugs a
unicorn and says “It is so fluffy”. One reason why this quote acquired
somewhat of a meme status is because it subverted audience expectations of a
more visual word like “pretty” or “white” to describe the unicorn. Instead she
opts for the more unexpected haptic word “fluffy”. This substitution of visual
language for haptic, a synaesthesia, might indicate that Agnes’ perceives the
world in a more tactile manner rather than in a visual way. In order to assess
if this is a stylistic tendency, we can examine all of Agnes’ language use and
ask the general question: to what extent does she use haptic language in
contexts where we generally expect visual language? We can ask similar
questions related to each combination of expected versus observed sensory
modalities. Observations for all combinations, including the homogeneous non-
synaesthesia ones, are accumulated to form Agnes’ (or any other individual’s,
group’s or genre’s) sensory style representation.

Terminology and notation: More formally, we consider a sentence to be a
“sensorial sentence” if it has at least one word or phrase that appears in a
sensorial lexicon. Further, we define a “sense-focused sentence” to be a
sensorial sentence with a single sensorial term selected as focus term. Thus,
if a sensorial sentence has $n$ sensorial terms then we derive from it $n$
sense-focused sentences.

Assume $S_{i}=\\{S_{i1},S_{i2},...,S_{in}\\}$ is the set of $n$ sense-focused
sentences identified from the writings of individual $i\in I$. Let
$C=\\{H,V,I,O,G,A\\}$ represent the modalities: Haptic, Visual, Interoceptive,
Olfactory, Gustatory, Auditory respectively. Using $\bar{N}$ to represent
concepts that are “not-sensorial” we define $\widetilde{C}$ as:
$\widetilde{C}=C\cup\\{\bar{N}\\}$.

We also define two functions. $F(\hat{s}_{ij})$ is a sensory lexicon lookup
function that returns the sensorial category $c\in C$ for the focused
sensorial term $\hat{s}_{ij}$ in the sense focused sentence $S_{ij}$. E.g.,
given the sensorial sentence “The unicorn is white and fluffy”, we have two
sense-focused sentences $S_{ij}$ and $S_{ik}$ corresponding to the two
sensorial terms, $\hat{s}_{ij}$ and $\hat{s}_{ik}$. For $S_{ij}$ with the
focus term “white” $F(\hat{s}_{ij})$ will return $V$. For $S_{ik}$ with focus
term “fluffy” it will return $H$.

The second function we define is $M(S_{ij})$ which returns the “expected”
modality $c\in\widetilde{C}$ for the same focus term in $S_{ij}$

We describe this function in Section 2.1.

Calculating observed to expected ratios: We represent an individual’s sense-
focused sentences as a list of length $|S_{i}|$. Each entry is a pair of
expected and observed modalities of the form $[(M(S_{ij}),(F(\hat{s}_{ij})]$.
For observed modality $y\in C$ and expected modality $x\in\widetilde{C}$, the
observed to expected ratio $\alpha_{i}^{xy}$ for individual $i$ is:

$\alpha_{i}^{xy}=\frac{\left|\\{S_{it}:F(\hat{s}_{it})=y\text{ and
}M(S_{it})=x\\}\right|}{\left|\\{S_{it}:M(S_{it})=x\\}\right|}$ (1)

Note that these ratios are the informative units of sensorial style. For
example, if the ratio is $1$ when $x$ and $y$ are the same modality, a
homogeneous combination, then the individual’s use of that modality is highly
aligned with general expectation. On the other hand, if it is close to $0$
then she deviates considerably from the expected use of modality $x$, i.e.
there is greater synaesthesia.

Style vectors: For each $x\in\widetilde{C}$, we then concatenate its $6$
ratios into a vector of the form:

$s_{i}^{x}=\underset{y\in C}{\sqcup}\alpha_{i}^{xy}$ (2)

It follows that

$\sum_{y\in C}\alpha_{i}^{xy}=\begin{cases}1,\quad&\text{if
}|\\{S_{it}:M(S_{it})=x\\}|\geq 1\\\ 0,\quad&\text{otherwise}\end{cases}$

We can now define the sensorial style vector $u_{i}$ of $i\in I$ as a
concatenation of the seven vectors, one for each expected modality. The size
of $u_{i}$ is $x\times y=42$.

$u_{i}=\underset{x\in\widetilde{C}}{\sqcup}s_{i}^{x}$ (3)

###  2.1 Implementing function $M(S_{ij})$

Given a sense-focused sentence, function $M$ returns the expected modality
$c\in\widetilde{C}$ of the sentence’s focus sensorial term as per general
expectation in English. We leverage RoBERTa-MLM111We experimented with BERT as
well, however, RoBERTa gave us more accurate results. as a stand-in for
general English language usage. RoBERTa is a transformer based language model
pre-trained on 160 GB of data Liu et al. (2019). Prior works like Mosbach et
al. (2020) and Sinha et al. (2020) have shown that language models learn the
norms of the language on which they are trained.

This makes them ideal for our task. We mask the focus sensorial words in our
sentences and input them to the model. RoBERTa returns the probabilities for
all the words in its vocabulary at each masked location. Probabilities
represent likelihood of appearance of the words at that location. We use these
probabilities to identify the expected sense modality at each masked location
as follows.

Let $W=\\{(w_{1},p_{1}),(w_{2},p_{2})\ldots(w_{N},p_{N})\\}$ be the ranked set
of words returned by RoBERTa for a masked location (location of focus
sensorial term) in $S_{ij}$; top ranked has highest probability.

Using $F(w_{k})$, we lookup the sense for each word in the top 100. We combine
this information to get an aggregate probability score $\Pi(c,S_{ij})$ for
each modality $c$ as follows:

$\Pi(c,S_{ij})=\underset{\begin{subarray}{c}k\leq 100\\\
F(w_{k})=c\end{subarray}}{\sum}p_{k}$ (4)

For greater confidence we only include $S_{ij}$ in our analysis if its
majority modality has $\Pi(c,S_{ij})$ $>0.5$. We then define

$M(S_{ij})=\operatorname*{argmax}_{c\in\widetilde{C}}\Pi(c,S_{ij})$ (5)

In essence, $M(S_{ij})$ returns the expected modality with the highest
aggregate probability for the focused sense word in $S_{ij}$ as determined
using RoBERTa.

###  2.2 Sensorial Lexicon

| Lexicon
---|---
| Original | Modified
Modality | $N$ | $\%$ | $N$ | $\%$
Visual | 29552 | 75.0 | 9419 | 50.2
Interoceptive | 3546 | 9.0 | 3449 | 18.4
Auditory | 4528 | 11.5 | 3803 | 20.3
Haptic | 675 | 1.7 | 972 | 5.2
Gustatory | 890 | 2.3 | 890 | 4.7
Olfactory | 216 | 0.5 | 216 | 1.2
Total | 39407 | 100 | 18749 | 100

Table 1: Distribution of modalities in original Lynott et al. (2020) lexicon
and our modified subset lexicon.

We use the sensorimotor norms lexicon published recently Lynott et al. (2020)
which has 39,954 concepts from the English Language. Brysbaert et al.
estimates that the average adult lexicon is composed of approximately $42,000$
words. Therefore this lexicon approximates a significant majority of everyday
English.

Each concept was rated by annotators along a $0$-$5$ scale for the six
modalities (Auditory, Gustatory, Haptic, Olfactory, Visual, Interoceptive) For
example “fluffy” is rated $4.41$ for Haptic, $0.29$ for Gustatory, $3.77$ for
Visual, $0.35$ for Interoceptive and $0$ for Auditory and Olfactory.

The dimension with the highest rating is the dominant modality.

Dominance alone is not enough to ensure that a concept belongs to a particular
sensorial modality since almost half of the concepts score less than $2.55$ on
any sense modality. Therefore, we filter the lexicon by ranking all concepts
in a given modality by their rating and selecting only those in the top
quartile. This ensures strong alignment to dominant modalities. Table 1
describes the lexicons222There are other sensory lexicons like Lievers and
Huang (2016b) and Lynott and Connell (2013). Besides being the largest and
most recent the Lynott et al. (2020) sensorimotor lexicon is the only one to
include ratings for interoception.

##  3 Methods

###  3.1 RQ 1: Is Sensorial Style a Product of Random Chance?

In order to be meaningful our representation of sensorial style should not be
a product of random chance.

If an individual chooses sensorial modalities randomly and not deliberately
then we expect her observed and expected modality distributions to be
independent of each other.

For example in “the clouds are white” the expected modality may be visual but
the individual randomly chooses from one of the six senses. We use this random
model in our analysis.

As a first step, for any $i\in I$ with a set of $n$ sense-focused sentences
$S_{i}$, we define, $\Gamma(i)$, the distribution of the sense modalities in
$C$ observed in $S_{i}$. For each sense-focused term $\hat{s_{ij}}$ in
$S_{i}$, we use a function $\bar{F}(\hat{s_{ij}},\Gamma(i))$ that returns a
random modality $c\in C$ with distribution $\Gamma(i)$.

For each $i\in I$, we create $m$ random pseudo-documents
$\mathcal{R}_{i}=\\{\mathcal{R}_{i}^{1},\mathcal{R}_{i}^{2}\ldots\mathcal{R}_{i}^{m}\\}$.
Each random pseudo-document has the same set of sense-focused sentences
$S_{i}$. However, instead of using $F(\hat{s}_{ij})$ to look up the modality,
we use $\bar{F}(\hat{s}_{ij},\Gamma(i))$ to get a random modality. Equation 6,
a modification of equation 1, gives us $\bar{\alpha}_{ik}^{xy}$ which is used
to calculate the style vector $u_{i}^{k}$ for random pseudo-document
$\mathcal{R}_{i}^{k}$.

$\bar{\alpha}_{ik}^{xy}=\frac{\left|\\{S_{ij}:\bar{F}(\hat{s}_{ij},\Gamma(i))=y\text{
and }M(S_{ij})=x\\}\right|}{\left|\\{S_{ij}:M(S_{ij})=x\\}\right|}$ (6)

Thus, for each $i\in I$ with sensorial style vector $u_{i}$, we have $m$
random style vectors $\\{u_{i}^{1},u_{i}^{2}\ldots u_{i}^{m}\\}$ generated
from the random pseudo-documents in $\mathcal{R}_{i}$.

Let $U_{i}=\\{u_{i}\\}\cup\\{u_{i}^{1},u_{i}^{2}\ldots u_{i}^{m}\\}$. For each
vector $v\in U_{i}$, we calculate its average cosine similarity with all other
elements in $U_{i}$. Ranking the elements of $U_{i}$ by decreasing order of
average similarity we check whether the style vector $u_{i}\in U_{i}$ has
lower average similarity than at least $95\%$ of the vectors in $U_{i}$ (i.e.
$p$-value $<0.05$ ). If so, we infer with $95\%$ confidence that $i$’s style
vector, $u_{i}$, is not random and therefore likely a product of an individual
stylistic choice.

###  3.2 RQ 2: How much data is needed to describe sensorial style?

Given the set of sense focused sentences $S_{i}$, where $|S_{i}|=n$, we
randomly sample subsets from $S_{i}$ of size $k$ and compute the style vector
from each sample. We explore how increasing the values for $k$ affect style
convergence.

For a given sentence set size $k$, we identify $m$ random samples (with
replacement) of $S_{i}$, each of size $k$. Thus, we create a set of sentence
sets,
$T_{i}^{k}=\\{\hat{T}_{i}^{k1},\hat{T}_{i}^{k2},\ldots,\hat{T}_{ki}^{m}\\}$.
For each sentence set $\hat{T}_{i}^{kj}\in T_{i}^{k}$, we use the method
discussed in Section 2 to generate the corresponding style vector
$\hat{u}^{kj}_{i}$. This gives us a set of $m$ sensorial style vectors,
$\bm{\hat{u}^{k}_{i}}=\\{\hat{u}^{k1}_{i},\hat{u}^{k2}_{i},\ldots,\hat{u}^{km}_{i}\\}$.
We then use cosine similarity to calculate the average pairwise similarities
between all elements in $\bm{\hat{u}^{k}_{i}}$. We recompute this average
self-similarity $\overline{\text{sim}(\bm{\hat{u}^{k}_{i}})}$ for different
values of $k$ in increasing steps of $r$. We say that the style of the
individual has converged for a minimum of $k$ sensorial sentences if
$\overline{\text{sim}(\bm{\hat{u}^{k}_{i}})}\approx\overline{\text{sim}(\bm{\hat{u}^{k+r}_{i}})}$,
where $k+r$ is the next sentence set size tested.

###  3.3 RQ 3: Does sensorial style vary over time?

Here we investigate whether style vectors evolve over time spans. We segment
the writings by time and consider how the average similarity in style varies
with temporal distance. We first identify all pairs of time points $t_{a}$ and
$t_{b}$ that are $\gamma$ duration apart. We then build style vectors for each
author with text anchored at $t_{a}$ and for each author with text anchored at
$t_{b}$. We then compute the average pairwise similarity between $t_{a}$ and
$t_{b}$ style vectors. We repeat this for all values of $\gamma$ that are of
interest.

We use a notion of windowing around the time points ($t_{a}$ and $t_{b}$) to
reduce noise. Each window is of size $\delta$ and distributed equally around
each time point. For example, for $t_{a}$ we create an individual’s style
vector from all the sense-focused sentences that were published in the range
$t_{a}-\frac{\delta}{2}<\tau_{a}<t_{a}+\frac{\delta}{2}$.

###  3.4 RQ 4: Which features are representative and distinctive of the
individuals within each genre?

A genre can be represented by the set of sensorial style vectors of its
members. Each style vector is composed of 42 features that explore
synaesthesia. We are interested in exploring which features are representative
of the members of a genre, and also features that make the members distinct.

We consider a sensorial style feature to be representative if the variation in
its usage is low. This would indicate that the members use the feature in a
consistent manner. Formally, a stylistic feature $\alpha^{p}$ is more
representative for the members of a genre than another feature $\alpha^{q}$ if
its standard deviation $\sigma(\alpha^{p})$, across all the members is lower
than the standard deviation $\sigma(\alpha^{q})$. At the other end, a high
variation would indicate that the feature is distinctive amongst the members.

##  4 Datasets

Genre | # Authors | # Works | #Sentences |  # Sensorial Sentences |  # Sensorial Expressions
---|---|---|---|---|---
Novels | 130 | 317 | 1,525,894 | 156,570 (10%) | 474,299
Lyrics | 5,321 | 20,785 | 1,007,090 | 754,572 (75%) | 1,501,501
Poetry | 1,246 | 3,315 | 85,236 | 4,979(6%) | 8,209

Table 2: Dataset details for each genre. The percentage of total sentences
that are sensorial is in parentheses.

We analyze 3 literary genres — novels, poems, music lyrics. Compared to poems
and lyrics, novels can span tens of thousands of sentences. Additionally,
novels and poetry are generally associated with a single author. Lyrics are
sometimes collaborations, however, we assume an artist would not perform a
song that is in a style they do not like. Thus, we assume music lyrics to be a
reflection of the artist’s style.

Novels:  We collected English language novels from the Domestic fiction genre
of Project Gutenberg333https://www.gutenberg.org/. There were 317 works
written by 130 authors, with the earliest by Henry Fielding from the early
18${}^{\text{th}}$ century and the latest by Rebecca West from the
mid-20${}^{\text{th}}$ century.

Lyrics:  We collected songs that were listed on the Billboard Hot 100 charts,
1963 to 2021 (inclusive). This weekly chart ranking of song popularity is
considered the industry standard Whitburn (2010). We assume the first time a
song is listed to be its year of production. We obtained song lyrics using the
Genius API444Some lyrics were not available on http://www.genius.com, because
they were instrumentals like the “Star Wars Theme” which hit No. 1 in 1977, or
were not in the Genius database.. There are $20,785$ song lyrics.

Poetry: Following works like Lou et al. (2015), we used the corpus of poems
available on the Poetry Foundation’s555The Poetry Foundation was established
in 2003, and one of its goals is to make “the best poetry” accessible
(https://www.poetryfoundation.org/foundation/about). website. To make this
dataset more comparable to the lyrics dataset, we only included works
published after 1963.

##  5 Results

Genre | $>95^{th}$ | N
---|---|---
Novels | 112 | 123
Lyrics | 701 | 735
Poetry | 20 | 85
Table 3: Number of individuals with lower average similarity than $95\%$ of
random vectors.

We present our results in two parts. First, we make our general observations.
Second, we present results related to our specific research questions.

###  5.1 General Observations

| Novels | Lyrics | Poetry
---|---|---|---
| $I-O$ | $I-O$ | $A-G$
| $\bar{N}-O$ | $\bar{N}-O$ | $A-O$
| $A-O$ | $V-O$ | $G-I$
| $I-G$ | $A-O$ | $H-O$
| $V-O$ | $I-G$ | $O-A$
| $A-G$ | $V-G$ | $O-H$
Range | (0.00,0.01) | (0.00,0.02) | (0.00,0.00)
Table 4: The top representative features for each genre.

| Novels | Lyrics | Poetry
---|---|---|---
| $G-G$ | $H-H$ | $A-A$
| $O-O$ | $G-G$ | $I-I$
| $H-H$ | $A-A$ | $H-H$
| $G-V$ | $H-V$ | $\bar{N}-V$
| $H-V$ | $G-V$ | $G-G$
| $A-A$ | $A-I$ | $\bar{N}-I$
Range | (0.43,0.15) | (0.44,0.19) | (0.47,0.27)
Table 5: The top distinctive features for each genre.

Each feature is an (expected,observed) pair e.g., $I-O$ in means that we
observe Olfactory language when we expected Interoceptive language. This is
the most representative feature for Novels and Lyrics as it has the lowest
standard deviation. We include the range of variances of the top most
distinctive and representative features.

Domination over lower senses:  The five Aristotelian sensorial modalities have
classically been thought of as part of a hierarchy with vision and audition
dominating over the three so-called “lower senses”—Touch, Taste, Smell Howes
(2010)666Note interoception is generally not considered in discussions of this
hierarchy, possibly because of its relatively recent inclusion Connell et al.
(2018).. This hierarchy manifests in the frequency of language use with the
visual and auditory modalities being used more often than the lower senses
Majid et al. (2018). We have consistent results. Figure 2 shows that for all
three genres visual and auditory dominate over the “lower senses”. Concepts
associated with haptic, gustatory and olfactory modalities — combined, form
less than $10\%$ of the total sensorial language.

While auditory dominates the “lower senses” in all three genres, it occurs
less than half as often as visual. Going beyond the classical five senses, in
all cases interoception dominates the three “lower senses” surpassing audition
in this regard. Additionally, in lyrics, interoception is as common as visual.
Clearly interoception with its emphasis on sensations within the body, both
physical and emotional, is important in language.

Sensorial style across genres: We investigate how sensorial style varies
across genres. Using the method described in Section 2 we calculate genre-
level sensorial style vectors by combining sentence sets at the genre level.
We show the distribution over 42 sensorial combinations in Figure 1 for just
Lyrics. Each cell represents an expected-observed modality combination.

Figure 1: Distribution of expected-observed modalities in Lyrics. Note that we
calculate proportions using equation 1, however, for illustrative purposes we
show sensorial distribution as percentages.

Observed modalities are largely as expected with some exceptions:  The
diagonal values which are in the range 56 to 88% indicate that the observed
modalities are generally consistent with expected modalities. That is, the
individuals in our datasets select from the 6 sensorial modalities in a manner
that is consistent with the general norms of language use. The highest
consistency is for interoceptive and the lowest is for olfactory. We observe
this trend across all genres (see appendix).

Looking at off diagonal values, we observe visual language used in $25\%$ of
the cases where we expected olfactory language, $19\%$ and $11\%$ of the cases
where we expected gustatory and haptic language, respectively. This usage of
visual language as a replacement for lower senses was observed across all 3
genres. This replacement or _cross-modal compensation_ might be because the
lower senses do not have a strong relation with the perceptual system and
consequently individuals might be relying on visual language as a semantic
scaffold to compensate for the weaker perceptual system of the lower senses
Speed and Majid (2020).

We also observe (in all three genres) that in more than $90\%$ of instances
where we expected to see non-sensorial language we instead observed
interoception. This might also be because interoception dominates in our data
and is consistent with observations about higher senses in the literature
Majid et al. (2018).

Figure 2: Distribution of observed modalities.

###  5.2 Results for research questions

RQ1: Is sensorial style a product of random chance? We investigate whether
sensorial style is motivated by the individual choices or whether it is a
product of randomness.

Table 3 provides the results. If sensorial style is a non random phenomenon
and a product of individual choice and intent, we expect the sensorial style
vectors to be distinct from random vectors. That is, we expect them to have a
lower average similarity as compared to random vectors (generated by our
random model).

Methodological details are in Section 3.1).

Considering all individuals with more than 10 sense-focused
sentences777Because of the volume of lyrics, we limit the analysis to
individuals with $>500$ sense-focused sentences., more than $90\%$ of the
individual sensorial style vectors in the novels and lyrics datasets are non-
random. However, in the poetry dataset only around $23\%$ of the individual
vectors in the poetry dataset were non-random.

A possible reason for the difference in poetry is likely data sparsity — fewer
sensorial expressions/ author (see Table 2). Exploring this intuition further
we find that the non-random vectors in poetry have on average 159 sense-
focused sentences while the remaining vectors that looked random had on
average 24 sense-focused sentences. Similarly, 8 out of the 10 most prolific
individuals had non-random vectors. However, none of the 10 least prolific
individuals had non-random vectors. These support our intuition regarding data
sparsity being the cause of the difference in poetry.

RQ2: How much data do we need to get a stable representation of sensorial
style? We evaluate the average similarity for each individual with
progressively larger samples sizes, $k$, of their sense-focused sentences. We
chose a range of values of $k$ from $k=1$ to $k=10$ with a granularity of $1$,
from $k=10$ to $k=100$, and $k=100$ to $k=1,000$ with granularities of $10$
and $100$. In Figure 3 we summarize these results with the median of average
similarity across all individuals in each genre. We say the sensorial style
vector has converged at a $k$ value when the graph becomes more or less
horizontal from that point onward.

Figure 3: Convergence of style vectors as a function of $k$, the number of
sense-focused sentences sampled.

From the figure we see that as $k$ increases similarity increases (within the
$m$ samples of size $k$). We note that lyrics reaches a median average
similarity of $0.90$ with a sample of less than $100$ sense focused sentences.
In contrast, we need between $200$ and $300$ sentences to get the same 0.9
median average similarity for novels. Compared to novels and lyrics the plot
for poetry has some fluctuations at $k\geq 400$, perhaps because there are
only $7$ poets with more than $100$ sentences.

RQ 3: Does sensorial style vary over time?

Figure 4: Average similarity between music lyrics as a function of their
temporal distance. The blue line represents the linear approximation of the
relationship. As temporal distance increases similarity decreases slightly.

We now explore whether sensorial style has changed over time in lyrics. As
observed earlier, the poetry dataset is sparse so we do not include it. And
with novels, it is sometimes challenging to pinpoint a single publication
year, since some texts are written over multiple years888As an example, Louisa
May Alcott’s Little Women was published in two volumes spanning two years,
$1868$ and $1869$..

Using the method described in Section 3.3, we measure average similarity in
style of lyrics between pairs of individual sensorial style vectors that are a
temporal distance $\gamma$ away from each other. We use window length
$\delta=1.5$ years. Figure 4 illustrates our findings.

We observe that as temporal distance between works increases the similarity of
sensorial style decreases. The augmented Dickey-Fuller test had a $p-$value
$=0.96$ meaning we cannot reject $H_{0}$, that the distribution is not-
stationary Cheung and Lai (1995). However, the decrease is very slight; the 50
year drop in similarity is only $0.01$.

Approximating the relationship between the average similarity and temporal
distance as a linear process, we note that the average similarity decreases
very slowly at a rate of $3.37\times 10^{-4}$ per year.

RQ 4: Which features are representative and distinctive of the individuals
within each genre? Table 4, shows the top-6 features that are representative
of the members of each genre. We observe that these are all synaesthetic.
Additionally the use of olfactory language in non-olfactory contexts is a
representative feature in a majority of table cells (10/18).  The standard
deviations of top representative features is relatively low (between 0.00 to
0.02). This would indicate that the level of consistency for these top
features is generally consistent for the three genres. One takeaway from these
observations can be that in synaesthetic contexts, individuals are more prone
to using lower senses (like olfaction) in a more consistent manner.

Table 5, shows the top-6 features that had the highest standard deviation
between the members of each genre and were the most distinctive. The
distinctive features were predominantly non-syneasthetic. In the cases where
the distinctive feature was synaesthetic, the observed modality was either
interoceptive or visual.  This would indicate that there is greater diversity
in expressions that rely on higher senses as a semantic scaffold.

For each genre, we rank all the sensorial style features by the standard
deviation and compare them using Pearson’s correlation. We observe that, the
features are highly correlated. Lyrics had a high correlation with both novels
(0.75) and poetry (0.81), while poetry and novels had a slightly lower
correlation of 0.48.

Can sensorial style be used for prediction tasks?

We investigate whether sensorial style features can be used to identify genre.
We compare against other standard style representations: LIWC Tausczik and
Pennebaker (2010) and content-free words (CFW) vectors Hughes et al. (2012).
We use standard 5 fold cross validation for each experiment to train and test
a random forest classifier. We consider the most prolific $50$ authors/genre.

Method | Baseline | Sensorial Style | LIWC | CFW
---|---|---|---|---
Features | — | 42 | 73 | 307
Accuracy | 0.33 | 0.91 | 0.99 | 0.99

Table 6: Prediction accuracy of the different features.

Table 6 shows the results. We observe that sensorial style predicts genre with
a high level of accuracy ($>90\%$).

While the other representations achieve close to perfect accuracies, key to
note is that our goal is less about beating baselines and more about
understanding the kinds of signals conveyed by sensorial style.

##  6 Case Study: Sensorial style in Lyrics

As a small illustration, we explore how sensorial style varies across
different songs composed by the same artist. We consider all $962$ artists who
had at least $5$ songs in the Hot 100 and extract a sensorial vector for each
song. We then measure the average pair-wise cosine similarity (self
similarity) amongst the songs of each artist. Almost $80\%$ of the artists had
an average self similarity $\geq 0.70$. Only two artists had a self similarity
$<0.50$.

The rapper NF was the most consistent artist with an average self similarity
of $0.93$. Conversely, the least consistent artist was the rock musician Tommy
James with an average similarity of $0.42$. For example, in the song “When I
Grow Up”, NF used auditory language non-synaesthetically in $85.7\%$ of the
cases and for the song “NO NAME” this happened in $76.9\%$ of the cases. The
similarity between these two songs was over $95\%$.

In contrast, “Nothing to Hide” and “Ball and Chain” by Tommy James’ had a
similarity of $0.32$. In “Ball and Chain”, the artist uses visual language in
all the instances where it was expected. However in “Nothing to Hide”, he uses
interoceptive language synaesthetically instead of visual in about $57\%$ of
the cases. This case study demonstrates a method for exploring sensorial style
and their variations across writings at the individual level.

##  7 Related Work

There are no directly comparable studies examining sensorial style for large
numbers of individuals that consider interoception. Instead we briefly review
closely allied topics.

Sensorial language: Sensorial language is not uniformly distributed across the
six sensory modalities as reflected in sensorial lexicons, such as the one we
use Lynott et al. (2020). This is also observed in large text collections. In
their analysis of 8 million words from around 7,000 English texts Koblet and
Purves (2020) found over 28,000 visual descriptions and only 78 referring to
the olfactory modality. Similar findings for multiple corpora are observed in
Winter et al. (2018). Our results are consistent with these prior works.

Sensorial language, the brain & emotion: The salience of sensorial words is
known to be highly correlated with the volumes of cortical activation in the
brain Reilly et al. (2020). Lievers (2015) show that there is directionality
to how senses are substituted for each other.

Winter (2016) found that gustatory and olfactory words (e.g., ‘stinky’,
‘delicious’) are on average more emotionally valenced than visual and auditory
words and these also appear in more emotionally valenced sentences.

Bubl et al. (2010), show that alterations in mental states have a direct
effect on perception; specifically, that depression directly impacts how the
color blue was perceived. Kernot et al. (2016), found a decrease in the
novelist Iris Murdoch’s use of olfactory language following her diagnosis of
depression and Alzheimer’s. We credit this study for providing us with the
hint that sensorial language may lead to a sensorial style. We take their
sensorial style analysis forward with larger collections of authors, several
genres and a more informative representation of sensorial style.

##  8 Limitations and Conclusions

We have shown that individuals have sensorial language style and that this
sensorial style is a non random phenomenon for novelists and musicians and
therefore is likely developed intentionally. Interestingly, we also found that
it takes just a few hundred sentences to extract stable sensorial style
representations. We also show that sensorial style in lyrics largely stable
over time; the average similarity decreases at a rate of $3.37\times 10^{-4}$
per year.

Additionally, we show that sensorial style vectors seem to perform well at
genre identification. The performance was high ($>0.90$), however, it was not
close to perfect as with other style representations. The question about how
sensorial style representations can be improved to increase performance
requires further investigation.

Our study is limited in that our method relies heavily on the underlying
Lynott et al. (2020) lexicon, and as with similar studies, is only as good as
the lexicon. Additionally, we assume that each term is associated with a
single sensorial modality. However, as research in psychology and neurology
has shown, sensorial language is cross-modal. We leave this analysis to future
work. In summary, we take a first step towards showing that sensorial style
has a legitimate role in stylometrics research.

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##  Appendix A Distribution of Modalities

Figure 5: Distribution of expected-observed modalities in the Novels Dataset.

Figure 6: Distribution of expected-observed modalities in the Poetry Dataset.


$\displaystyle\lesssim\epsilon^{2-d}\frac{1}{N^{5}}\mathbb{E}\int_{0}^{T}\sum_{i\neq
j;k,\ell\neq
i,j}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))|v^{\epsilon,z}(x_{i}^{N}(t)-x_{k}^{N}(t))|\phi(x_{i}^{N}(t))^{2}|\psi(x_{k}^{N}(t))\psi(x_{\ell}^{N}(t))|dt,$

where we have used (62). Using the auxiliary free system, its exchangeability,
and that $(y^{\epsilon}_{1}(t),...,y^{\epsilon}_{4}(t))$ has a uniformly
bounded density, see Proposition 13, we further bound the above by

$\displaystyle\leq\frac{1}{N^{4}}\int_{0}^{T}\sum_{i\neq j;k,\ell\neq
i,j}\theta^{\epsilon}(y_{i}^{\epsilon}(t)-y_{j}^{\epsilon}(t))|v^{\epsilon,z}(y_{i}^{\epsilon}(t)-y_{k}^{\epsilon}(t))|\phi(y_{i}^{\epsilon}(t))^{2}|\psi(y_{k}^{\epsilon}(t))\psi(y_{\ell}^{\epsilon}(t))|dt$
$\displaystyle\leq\int_{0}^{T}\theta^{\epsilon}(y_{1}^{\epsilon}(t)-y_{2}^{\epsilon}(t))|v^{\epsilon,z}(y_{1}^{\epsilon}(t)-y_{3}^{\epsilon}(t))|\phi(y_{1}^{\epsilon}(t))^{2}|\psi(y_{3}^{\epsilon}(t))\psi(y_{4}^{\epsilon}(t))|dt$
$\displaystyle\leq\int_{0}^{T}\iiint_{\mathbb{R}^{4d}}\theta^{\epsilon}(y_{1}-y_{2})|v^{\epsilon,z}(y_{1}-y_{3})\phi(y_{1})^{2}|\psi(y_{3})\psi(y_{4})|f_{t}^{1234,\epsilon}(y_{1},y_{2},y_{3},y_{4})dt$
$\displaystyle\lesssim\int_{\mathbb{K}}|v^{\epsilon,z}(y_{3})|dy_{3},$

where $\mathbb{K}$ is a compact set determined by the supports of $\phi,\psi$,
and we used that $\int_{\mathbb{R}^{d}}\theta^{\epsilon}(y_{1}-y_{2})dy_{2}=1$
for any $y_{1}$. We already analyzed in (5) that
$\int_{\mathbb{K}}|v^{\epsilon,z}(y_{3})|dy_{3}\lesssim\epsilon^{2}+(3|z|+2\epsilon)^{2}+|z|$.
This yields (80).

This completes the verification that all the terms (38), (40), (41), (42),
(44), (45), (47), (50), (51), (4), (53) and (54) are negligible in the sense
that $\lim_{|z|\to 0}\limsup_{\epsilon\to 0}\mathbb{E}|(*)|=0$, where $(*)$
stands for each of these terms.

Acknowledgments. We thank Andrea Papini, Alessandra Lanotte and Jeremie Bec
for important discussions, Francesco Russo and the organizers of the Winter
School “Stochastic and deterministic analysis of irregular models”, Luminy
2024 which motivated the development of the result presented here. This
research is funded by the European Union (ERC, NoisyFluid, No. 101053472).
Views and opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them.

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