changes the cohomology of our $G_{2}$ orbifold. This problem has been studied
from the perspective of the Joyce orbifold $T^{7}/\mathbb{Z}_{2}^{3}$ in
joyce1996compactII  and then explained at the level of discrete torsion in
gaberdiel2004generalised . Here we take a slighly different approach, and
instead consider

$M_{\sigma}=\frac{\widetilde{\Big{(}\frac{T^{6}}{(\alpha,\beta)}\Big{)}}\times
S^{1}}{\sigma},$ (182)

where the tilde means we resolve the singularities of the $(\alpha,\beta)$
action. This will, of course, give the same result as the references above.

Here the choice of the $a_{i},b_{i}$ in (174) matters. We will work with
$a_{4}=b_{6}=1/2$ and all others vanishing, i.e.

$\begin{split}\alpha:(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})&\mapsto(+x_{1},+x_{2},-x_{3},\frac{1}{2}-x_{4},-x_{5},-x_{6},x_{7})\\\
\beta:(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})&\mapsto(-x_{1},-x_{2},+x_{3},+x_{4},-x_{5},\frac{1}{2}-x_{6},x_{7})\\\
\sigma:(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})&\mapsto(x_{1},-x_{2},x_{3},-x_{4},x_{5},-x_{6},-x_{7}),\end{split}$
(183)

First we want to consider the Calabi-Yau given by the resolution of
$T^{6}/(\alpha,\beta)$. We know from our previous discussion that the
untwisted sector gives contributions to the Hodge numbers

$(h^{0,0},h^{1,1},h^{2,1},h^{3,0})=(1,3,3,1),$ (184)

along with their matching Hodge duals, $h^{m,n}=h^{3-m,3-n}$. The contribution
from the twisted sector comes from considering the fixed points. Both $\alpha$
and $\beta$ have $16$ fixed points, however the action of the other on these
fixed points leaves $8$ in each case. We show this graphically in Figure 1.
Locally these fixed points are modelled by $T^{2}\times\mathbb{C}^{2}/\\{\pm
1\\}$, and as standard we can choose to either blow up or deform the
$\mathbb{C}^{2}/\\{\pm 1\\}$. In either case the Hodge numbers are the same,
and each fixed point contributes one to both $h^{1,1}$ and $h^{2,1}$ (and
their Hodge duals). So in total our Calabi-Yau has
$(h^{1,1},h^{2,1})=(19,19)$, and so is self-mirror. This geometry is known as
the Schoen Calabi-Yau $X_{S}$. In terms of Betti numbers, we have

$b^{2}(X_{S})=\underset{T^{2}}{3}+\underset{\alpha}{8}+\underset{\beta}{8}=19\quad\text{and}\quad
b^{3}(X_{S})=\underset{T^{2}}{8}+\underset{\alpha}{16}+\underset{\beta}{16}=40.$
(185)

We now want to consider the action of $\sigma$ in $M_{\sigma}=(X_{S}\times
S^{1})/\sigma$, i.e. we want to compute $b^{2}_{\pm}(X_{S})$ and
$b^{3}_{\pm}(X_{S})$ and use them to compute the Betti numbers for the
smoothing $M=\widetilde{M_{\sigma}}$ via

$b^{2}(M)=b^{2}_{+}(X_{S})+e_{2}\quad\text{and}\quad
b^{3}(M)=b^{3}_{+}(X_{S})+b^{2}_{-}(X_{S})+e_{3},$ (186)

where $e_{2,3}$ denotes the contributions from the fixed points of $\sigma$.

Let’s start with $b^{2}(X_{S})$. It is clear that the $3$ contribution from
$T^{2}$ are all odd. The $8$ that arises from the fixed points of $\alpha$
gives $4$ even and $4$ odd: the fixed points are identified in pairs (see
Figure 1) and so we can form one even and one odd combination. The $8$ from
the $\beta$ fixed points are more subtle: here $\sigma$ acts trivially (as its
action is equivalent to $\beta$s action, but we have modded out by $\beta$),
and so introduces discrete torsion. As detailed in gaberdiel2004generalised ,
if we resolved the singularity in $\beta$ via a blow up then $\sigma$ will
preserve the orientation of the exceptional divisor and so the ground state
$(1,1)$-form is invariant. However, if we had deformed the
$\beta$-singularity, then $\sigma$ reverses the orientation and so our
$(1,1)$-form is odd. Each of the $8$ fixed points can be blown up or deformed
independently, and so we have the above choice for each one. Therefore, if
$\ell\in\\{0,...,8\\}$ denotes the number of blow ups, then we get a
contribution of $\ell$ to $b^{2}_{+}(X_{S})$ and $(8-\ell)$ to
$b^{2}_{-}(X_{S})$.

Now let’s discuss $b^{3}_{\pm}(X_{S})$. The story is very similar to above:
the $8+16=24$ that comes from $T^{2}$ and $\alpha$ are identified in pairs and
so we have $12$ even and $12$ odd contributions. The $16$ from $\beta$ is
dependent on the discrete torsion, and contributes $(8-\ell)$ to
$b^{3}_{+}(X_{S})$ and $\ell$ to $b^{3}_{-}(X_{S})$.

Finally we need to add in the contributions from the $\sigma$ fixed points, of
which there are $4$ independent ones. This is easily seen from the fact that
in $T^{7}/\sigma$ we have $16$ fixed points, but these are reduced to $4$
under $(\alpha,\beta)$, which are modded out to define $X_{S}$. These four
fixed points are $T^{3}$s and give contributions of $4$ to
$b^{2}(\mathcal{M})$ and $12$ to $b^{3}(\mathcal{M})$. In total the smoothed
$G_{2}$ orbifold then has Betti numbers

$b^{2}(M)=8+\ell\quad\text{and}\quad b^{3}(M)=47-\ell\,,$ (187)

in agreement with joyce1996compactII ; gaberdiel2004generalised .

$\alpha$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$\frac{1}{4}$$\frac{3}{4}$$\frac{1}{2}$$\sigma$$x_{5}$$x_{6}$$\frac{1}{2}$$\frac{1}{2}$$\beta$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$
Figure 1: Fixed points in $T^{6}/(\alpha,\beta)$ and how they are mapped under
$(\alpha,\beta,\sigma)$. All red objects correspond to $\alpha$, blue to
$\beta$ and green to $\sigma$. The graphs indicate a decomposition of
$T^{6}=T^{2}\times T^{2}\times T^{2}$, and the number of fixed points
understood multiplicitivly. The 16 fixed points of $\alpha$ are reduced to 4
under the action of $(\beta,\sigma)$. However $\alpha$ and $\sigma$ act on the
fixed points of $\beta$ in the same manner and so $\beta\sigma$ acts
trivially. This leads to the introduction of discrete torsion in the $G_{2}$
manifold.

####  A.2.4 Mirror Symmetry

As detailed in gaberdiel2004generalised , here we have 4 notions of mirror
symmetry. Just as with the Calabi-Yau torodial orbifold considered previously,
these take the form of T-dualities:

$\begin{split}\mathcal{T}^{+}_{3}&=\\{(3,4,7),(2,4,5),(1,4,6)\\}\\\
\mathcal{T}^{-}_{3}&=\\{(2,3,6),(5,6,7),(1,2,7),(1,3,5)\\}\\\
\mathcal{T}^{+}_{4}&=\\{(1,2,5,6),(1,3,6,7),(2,3,5,7)\\}\\\
\mathcal{T}^{-}_{4}&=\\{(1,4,5,7),(1,2,3,4),(3,4,5,6),(2,4,6,7)\\}.\end{split}$
(188)

The combinations appearing in here are exactly the terms that appear in $\Phi$
and $X$ above, i.e. these are calibrated submanifolds. The subscripts indicate
how many T-dualities we do, and from chirality arguments we can see that
$\mathcal{T}^{\pm}_{3}$ map compactifications on Type IIA/B to those on Type
IIB/A, while $\mathcal{T}^{\pm}_{4}$ map Type IIA/B to Type IIA/B. The $\pm$
superscript indicates whether the discrete torsion signs are reversed or not,
i.e. whether we blow up or deform the fixed points of $\beta$. This changes
the topology of the resulting $G_{2}$ manifold, i.e. $\ell\mapsto(8-\ell)$ in
(187).

The key thing we want to notice is that within the $\mathcal{T}_{4}$ actions
we have $(1,3,6,7)$, $(2,3,5,7)$, $(1,4,5,7)$ and $(2,4,6,7)$ which have the
effect of mirroring the Calabi-Yau plus a T-duality in the additional $S^{1}$
direction. That is, we can take our Calabi-Yau mirror maps in (165) and add on
a T-dual along the $S^{1}$ direction and generate a $G_{2}$ mirror
map.191919We note that, just as in the Calabi-Yau case, mirror symmetry does
not simply map a $4$-form to a $3$-form, and vice versa. This we can see from
the fact that our $4$-form $X$ contains exactly the terms that appear in
$\mathcal{T}_{4}$, and so these terms will be mapped to the vacuum. For
example, if we did the $(1,3,6,7)$ map, then $X\ni\ket{1367}\mapsto\ket{0}$,
which geometrically is the $0$-form. Similarly
$\Phi\ni-\ket{245}\mapsto\ket{1234567}$. It is then easy to see that under any
of the maps we actually exchange $\Phi+\ket{0}$ and $X+\ket{1234567}$. This is
just the equivalent of the fact that a $3$-form in the Calabi-Yau is mapped to
a linear combination of the diagonal forms.

Here we want to ask how the mirror maps affect the involution action $\sigma$.
Namely we want to ask how $\sigma^{\vee}$ is related to $\sigma$. It is clear
from the above calculation that

$\bigg{(}\frac{X_{S}\times
S^{1}}{\sigma}\bigg{)}^{\vee}=\frac{X_{S}^{\vee}\times(S^{1})^{\vee}}{\sigma},$
(189)

and so we can set $\sigma^{\vee}=\sigma$. There are 9 independent $G_{2}$
manifolds we can form via the resolution of these spaces, labelled by the
number of blow ups, $\ell$. These blow ups appear in $X_{S}$, and so we can
label the 9 $G_{2}$s via

$G_{2}^{\ell}=\widetilde{\bigg{(}\frac{X_{S}^{\ell}\times
S^{1}}{\sigma}\bigg{)}}.$ (190)

Mirror symmetry maps $G_{2}^{\ell}$ either back to itself or to
$G_{2}^{8-\ell}$, depending on whether we use $\mathcal{T}_{4}^{+}$ or
$\mathcal{T}_{4}^{-}$, respectively.

The interesting thing in this case is that $X_{S}$ is self mirror, and so even
though the $X_{S}^{\ell}$ look different, they are all diffeomorphic. This
diffeomorphism alters the action of $\sigma$ in the required way, namely we
give $\sigma$ an $\ell$ index and obtain

$G_{2}^{\ell}=\widetilde{\bigg{(}\frac{X_{S}\times
S^{1}}{\sigma^{\ell}}\bigg{)}}$ (191)

We can therefore take two viewpoints on the situation: we either have a
collection of different Calabi-Yaus, or we have a collection of different
anti-holomorphic involutions.

##  Appendix B Mirror Symmetry and Toric Hypersurfaces

In this appendix we review the geometrical constructions of mirrors for
Calabi-Yau manifolds and building blocks using toric geometry.

###  B.1 Polytopes and Hypersurfaces in Toric Varieties

Given a toric variety $\mathbb{P}_{\Sigma}$ defined in terms of a fan $\Sigma$
(see, e.g., cox2011toric  for a detailed review of toric varieties), we can
define a hypersurface as the vanishing locus of a section of a line bundle
$\mathcal{L}$. Given $\Sigma$, the divisor class of a line bundle
$\mathcal{L}$ is specified as

$D=c_{1}(\mathcal{L})=\sum_{i}a_{i}D_{i},$ (192)

in terms of the toric divisors $D_{i}$ corresponding to the ray generators of
the fan $\Sigma$.

The relevance of polytopes is that they allow us to encode both a fan and a
line bundle in terms of a single object. A generic polytope is defined as
follows.

######  Definition B.1 (Polytope)

Let $M_{\mathbb{R}}$ be some real vector space of dimension $d$. Consider some
set of points $S\subset M_{\mathbb{R}}$. Then we can define a polytope by the
convex hull of the set $S$, i.e.

$\Delta=\text{Conv}(S):=\Big{\\{}\sum_{i}\lambda_{i}m_{i}\,\Big{|}\,\sum_{i}\lambda_{i}=1,\,\forall
m_{i}\in S,\text{ and }\lambda_{i}\in\mathbb{R}^{+}_{0}\Big{\\}}\subseteq
M_{\mathbb{R}}.$ (193)

The dimension of the polytope is equal to the dimensional of the smallest
affine subspace in $M_{\mathbb{R}}$ that contains $\Delta$, and we will take
this dimensional to be equal to $d$. Polytopes can always be made top-
dimensional, i.e. $\dim\Delta=\dim M_{\mathbb{R}}$, simply by appropriately
reducing the dimension of $M$. We will furthermore focus on the cases where
$M_{\mathbb{R}}=M\otimes\mathbb{R}$ for some lattice $M$. We then call a
polytope $\Delta\subseteq M_{\mathbb{R}}$ a lattice polytope if the vertices
of $\Delta$ are lattice points in $M$.

######  Definition B.2 (Polytope Face)

Let $(N_{\mathbb{R}},M_{\mathbb{R}})$ be a set of dual vector spaces and let
$\Delta\subseteq M_{\mathbb{R}}$ be a polytope. Then given a non-zero vector
$v\in N_{\mathbb{R}}$ and an $a\in\mathbb{R}$, we can define

$H_{v,a}:=\\{m\in M_{\mathbb{R}}\,|\,\langle
m,v\rangle=a\\}\quad\text{and}\quad H_{v,a}^{+}:=\\{m\in
M_{\mathbb{R}}\,|\,\langle m,v\rangle\geq a\\}.$ (194)

$H_{v,a}$ is clearly a hypersurface in $M_{\mathbb{R}}$, and $H_{v,a}^{+}$ is
the upper half plane associated to this hypersurface. We call a subset
$\Theta\subseteq\Delta$ a face of $\Delta$ if there exists a $H_{v,b}$ and
$H_{v,b}^{+}$ such that

$\Theta=H_{v,a}\cap\Delta,\quad\text{and}\quad\Delta\subseteq H_{v,a}^{+}.$
(195)

We will denote a dimension $k$ face by $\Theta^{[k]}$.

This definition is intuitively clear: consider some hypersurface in
$M_{\mathbb{R}}$ that “touches” $\Delta$, and then the intersection of this
hypersurface with $\Delta$ is a face of $\Delta$. We call a face of
codimension-$1$ a facet, a face of dimension $1$ an edge and a face of
dimension $0$ a vertex. Note we can think of a polytope as the convex hull of
its vertices. We give a pictorial example of this for a $2$D polytope
corresponding to a triangle below.

$H_{v,a}$$\Theta^{[1]}\quad$$\Delta$

Note that a polytope $\Delta$ is given precisely by the intersection of the a
finite number of half planes $H_{v_{i},a_{i}}^{+}$, i.e.

$\Delta=\bigcap_{i=1}^{\ell}H_{v_{i},a_{i}}^{+}$ (196)

is a polytope. It then follows from the definition of $H_{v_{i},a_{i}}$ that
the vectors $v_{i}$ are perpendicular to surfaces $H_{v_{i},a_{i}}$ and point
into the intersection, as this is exactly what we need to ensure that $\langle
m,v_{i}\rangle\geq a_{i}$ for all $m\in\Delta$. We give a pictorial example
for a $2$D polytope with $\ell=4$ below.

We now note that for a lattice polytope, in which case $\dim\Delta=\dim
M_{\mathbb{R}}=d$, each facet $\Theta^{[d-1]}$ has a unique supporting
hyperplane. It follows from this that there is a unique choice such that
$v_{\Theta}^{[d-1]}$ is a primitive lattice point in $N$, which implies that
$a_{\Theta}^{[d-1]}$ is an integer.

$H_{v_{\Theta}^{[d-1]},a_{\Theta}^{[d-1]}}=\\{m\in M_{\mathbb{R}}\,|\,\langle
m,v_{\Theta}^{[d-1]}\rangle=-a_{\Theta}^{[d-1]}\\}\quad\text{and}\quad
H_{\Theta^{[d-1]}}^{+}=\\{m\in M_{\mathbb{R}}\,|\,\langle
m,v_{\Theta}^{[d-1]}\rangle\geq-a_{\Theta}^{[d-1]}\\}\,.$ (197)

Using these, our polytope is given by

$\Delta=\bigcap_{\\{\Theta^{[d-1]}\\}}H_{\Theta^{[d-1]}}^{+}=\\{m\in
M_{\mathbb{R}}\,|\,\langle m,v_{\Theta}^{[d-1]}\rangle\geq-
a_{\Theta}^{[d-1]},\,\forall\text{ facets }\Theta^{[d-1]}\subset\Delta\\},$
(198)

The minus sign appearing in the above expressions is included for later
convenience.

The polytope defines a fan, which in turn can be used to define a toric
variety.

######  Definition B.3 (Normal Fan)

To any face $\Theta$ of a polytope $\Delta$, we can associate a cone

$\check{\sigma}_{\Theta}:=\bigcup_{r\geq
0,p_{\Delta}\in\Delta,p_{\Theta}\in\Theta}r\cdot(p_{\Delta}-p_{\Theta}),$
(199)

and its dual

$\langle\check{\sigma}_{\Theta},\sigma_{\Theta}\rangle\geq 0\,.$ (200)

The collection of the cones $\sigma_{\Theta}$ for all faces $\Theta$ defines a
complete fan202020Here, it is customary to consider the whole of $\Delta$ as a
face of itself with $\sigma_{\Delta}$ being the zero-dimensional cone.  called
the normal fan $\Sigma_{n}(\Delta)$ of $\Delta$.

Here, a $k$-dimensional face $\Theta^{[k]}$ of $\Delta$ is associated to a
$(d-k)$-dimensional cone in $\Sigma_{n}(\Delta)$. In particular, the facets in
$\Delta$ correspond to the rays in $\Sigma_{n}(\Delta)$. By an appropriate
translation of $\Delta$, the ray generators of this fan become equal to the
lattice points $v_{\Theta}^{[d-1]}$.

We can then use our polytope to define not only a fan $\Sigma_{n}(\Delta)$,
but also the divisor class of a line bundle $\mathcal{L}$ on the associated
toric variety $\mathbb{P}_{\Sigma_{n}(\Delta)}$. This construction relies on
the piecewise linear support function on $\Sigma_{n}(\Delta)$ that is defined
by $\Delta$, see fulton  for details, and implies that we can identify the
$a_{\Theta}^{[d-1]}$ in (198) with the $a_{i}$ in (192).

The Newton polytope describing of global holomorphic sections of $\mathcal{L}$
is then equal to $\Delta$, and the group of holomorphic sections in
$\mathcal{L}$ has monomial basis the elements of which are in one-to-one
correspondence with lattice points on $\Delta$. They can be given explicitly
as

$p(m)=\prod_{i}z_{i}^{\langle m,\nu_{i}\rangle+a_{i}}.$ (201)

where $z_{i}$ are the homogeneous coordinates associated with ray generators
$\nu_{i}$ of $\Sigma_{n}(\Delta)$. Note that holomorphicity of these sections
is guaranteed by the relations $\langle m,v_{\Theta}^{[d-1]}\rangle\geq-
a_{\Theta}^{[d-1]}$.

The normal fan $\Sigma_{n}(\Delta)$ does not define a smooth or even orbifold
toric variety $\mathbb{P}_{\Sigma_{n}(\Delta)}$ in general, and we may need to
refine it to fan

$\widetilde{\Sigma}_{n}(\Delta)\rightarrow\Sigma_{n}(\Delta)$ (202)

to resolve singularities.

###  B.2 Calabi-Yau hypersurfaces and Reflexive Polytopes

To get a Calabi-Yau hypersurface, adjunction tells us that we need to choose
$\mathcal{L}$ such that its divisor class equals the first Chern class of
$\mathbb{P}_{\Sigma}$:

$D=c_{1}(\mathcal{L})=\sum_{i}D_{i},$ (203)

i.e. $a_{i}=1$ for all $i$ in (192). For the above construction to work we
hence need the polar dual to $\Delta^{\circ}$ defined by

$\langle\Delta,\Delta^{\circ}\rangle\geq-1$ (204)

to be a lattice polytope.

######  Definition B.4 (Reflexive Polytope)

A lattice polytope $\Delta$ is called reflexive if its polar dual is also a
lattice polytope.

This definition implies that $\Delta^{\circ}$ is reflexive as well, and that
$\Delta^{\circ}$ and $\Delta$ both have the origin as their unique interior
point. Furthermore, one can show that $\Sigma_{n}(\Delta)$ is equal to the
face fan212121The face fan of a polytope $\Delta^{\circ}$ is the fan found by
taking all cones over the faces of $\Delta^{\circ}$.
$\Sigma_{f}(\Delta^{\circ})$ of $\Delta^{\circ}$ for a pair of reflexive
polytopes. The faces of $\Delta$ and $\Delta^{\circ}$ are related to each
other by

$\langle\Theta^{[k]},\Theta^{\circ[n-k-1]}\rangle=-1,$ (205)

where $n$ is the dimension of the polytopes. The important thing is that a
$k$-dimensional face in $\Delta$ is related to a $(n-k-1)$-dimensional face in
$\Delta^{\circ}$.

Refinements of the fan $\Sigma_{n}(\Delta)$ which introduce no new ray
generators or ray generators which are lattice points on $\Delta^{\circ}$
correspond to crepant (partial) resolutions of the hypersurface in the class
(203). For a maximal regular222222Regularity of a triangulation is implies
projectivity of the toric variety, see fulton ; de2010triangulations  for
details and definitions. triangulation of the polytope we hence find what is
called a maximal projective crepant partial (MPCP) desingularisation.

As we can use both $\Delta$ and $\Delta^{\circ}$ as the starting point of this
construction, we obtain a pair of Calabi-Yau hypersurfaces in toric varieties
with fans $\widetilde{\Sigma}_{n}(\Delta)$ and
$\widetilde{\Sigma}_{n}(\Delta^{\circ})$. The two hypersurface equations are
explicitly given as

$\begin{split}G(z)&=\sum_{m\in\Delta}\prod_{n\in\Delta^{\circ}}\alpha_{m}z_{n}^{\langle
m,n\rangle+1}=0\\\
G^{\circ}(z^{\circ})&=\sum_{n\in\Delta^{\circ}}\prod_{m\in\Delta}\alpha_{n}^{\circ}z_{m}^{\circ\,\,\langle
m,n\rangle+1}=0\end{split}$ (206)

Note that the ray generators $\widetilde{\Sigma}_{n}(\Delta)(1)$ of
$\widetilde{\Sigma}_{n}(\Delta)$ are equal to the lattice points on
$\Delta^{\circ}$, and the ray generators
$\widetilde{\Sigma}_{n}(\Delta^{\circ})(1)$ of
$\widetilde{\Sigma}_{n}(\Delta^{\circ})$ are equal to the lattice points on
$\Delta$ using MPCP triangulations. We have labelled the homogeneous
coordinates associated with a lattice point $n$ ($m$) by $z_{n}$
($z_{m}^{\circ}$).

In case the MPCP triangulation results in a smooth hypersurface, we will
denote a generic member of the resulting family by
$X_{\Delta,\Delta^{\circ}}$. The non-trivial Hodge numbers of
$X_{\Delta,\Delta^{\circ}}$ are Batyrev94dualpolyhedra

$\begin{split}h^{1,1}(X_{\Delta,\Delta^{\circ}})&=\ell(\Delta^{\circ})-(d+1)-\sum_{\Theta^{\circ[n-1]}}\ell^{*}\left(\Theta^{\circ[d-1]}\right)+\sum_{(\Theta^{\circ[d-2]},\Theta^{[1]})}\ell^{*}\left(\Theta^{\circ[d-2]}\right)\ell^{*}\left(\Theta^{[1]}\right)\\\
h^{d-2,1}(X_{\Delta,\Delta^{\circ}})&=\ell(\Delta)-(n+1)-\sum_{\Theta^{[d-1]}}\ell^{*}\left(\Theta^{[d-1]}\right)+\sum_{(\Theta^{\circ[d-2]},\Theta^{[1]})}\ell^{*}\left(\Theta^{[d-2]}\right)\ell^{*}\left(\Theta^{\circ[1]}\right)\\\
h^{m,1}(X_{\Delta,\Delta^{\circ}})&=\sum_{(\Theta^{\circ[d-m-1]},\Theta^{[m]})}\ell^{*}\left(\Theta^{\circ[d-m-1]}\right)\ell^{*}\left(\Theta^{[m]}\right)\qquad\text{for}\qquad
d-2>m>1.\end{split}$ (207)

Here $\ell(...)$ denotes the number of lattice points of its argument,
$\ell^{*}(...)$ only counts the lattice points in the relative interior of its
argument, and $d-1$ is the complex dimension of the Calabi-Yau hypersurface.

Exchanging $\Delta\leftrightarrow\Delta^{\circ}$ maps the Hodge numbers as
expected for mirror symmetry, i.e. $h^{1,1}\leftrightarrow h^{d-2,1}$. In our
main case of interest, which are Calabi-Yau threefolds, we have $d=4$, so that
mirror symmetry swaps $h^{1,1}\leftrightarrow h^{2,1}$. In this case, there
always exists a MPCP that gives rise to a smooth Calabi-Yau hypersurface
Batyrev94dualpolyhedra , so we do not have to explicitly check the existence
of such a triangulation.

###  B.3 K3 fibrations, Projecting Tops and Building Blocks

Whenever the $N$-lattice reflexive polytope $\Delta^{\circ}$ contains a
reflexive subpolytope, i.e. there is a sublattice $N_{F}\in N$ such that
$\Delta^{\circ}\cap N_{F}=\Delta_{F}^{\circ}$ with $\Delta_{F}^{\circ}$
reflexive, the associated Calabi-Yau hypersurface $X_{\Delta,\Delta^{\circ}}$
may admit a fibration with fibres $X_{\Delta_{F},\Delta^{\circ}_{F}}$
Klemm:1995tj ; Avram:1996pj ; Candelas:1996su . The projection of $N$ to
$N/N_{f}$ gives rise to a projection of the hypersurface if there is an
appropriate triangulation of $\Delta^{\circ}$ that turns this into a toric
morphism.

Our case of interest are K3 fibrations of Calabi-Yau threefolds, where
$\Delta^{\circ}$ is four-dimensional and $\Delta^{\circ}_{F}$ is three-
dimensional. In this case $\Delta^{\circ}_{F}$ separates $\Delta^{\circ}$ into
two halves $\Diamond_{1}^{\circ}$ and $\Diamond_{2}^{\circ}$ called tops (or
top and bottom). Letting $m_{0}$ be the primitive normal vector to $N_{F}$,
$\langle m_{0},N_{F}\rangle=0$, we can set

$\displaystyle\Diamond_{1}^{\circ}$
$\displaystyle:=\text{Conv}\left(\left\\{n\in\Delta^{\circ}|\langle
m_{0},n\rangle\geq 0\right\\}\right)$ (208)
$\displaystyle\Diamond_{2}^{\circ}$
$\displaystyle:=\text{Conv}\left(\left\\{n\in\Delta^{\circ}|\langle
m_{0},n\rangle\leq 0\right\\}\right)\,$

so that

$\Delta^{\circ}=\Diamond_{1}^{\circ}\cup\Diamond_{2}^{\circ}\hskip
28.45274pt\Delta^{\circ}_{F}=\Diamond_{1}^{\circ}\cap\Diamond_{2}^{\circ}\,.$
(209)

Motivated by the above, we will adopt the following

######  Definition B.5 (Top)

A top is a lattice polytope $\Diamond^{\circ}$ such that

$\Diamond^{\circ}=\text{Conv}\left(\left\\{n\in\Delta^{\circ}|\langle
m_{0},n\rangle\geq 0\right\\}\right)$ (210)

for a reflexive polytope $\Delta^{\circ}$ and primitive lattice point $m_{0}$,
and

$\Delta^{\circ}_{F}:=\text{Conv}\left(\left\\{n\in\Delta^{\circ}|\langle
m_{0},n\rangle=0\right\\}\right)$ (211)

is reflexive. We will make the simple choice $m_{0}=(0,0,0,1)$ by exploiting
the $SL(4,\mathbb{Z})$ acting on $N$ in the following.

######  Definition B.6 (Projecting Top)

A top $\Diamond^{\circ}$ is called projecting if the projection of
$\Diamond^{\circ}$ to $N_{F}\otimes\mathbb{R}$ is contained in
$\Delta_{F}^{\circ}$.

Projecting tops can be used to construct building blocks for TCS $G_{2}$
manifolds in analogy to Batyrev’s construction of Calabi-Yau threefolds
Braun:2016igl . Given a projecting top, we can define its dual as

$\langle\Diamond,\Diamond^{\circ}\rangle\geq-1\hskip
28.45274pt\langle\Diamond,n_{0}\rangle\geq 0\,.$ (212)

where $n_{0}=(0,0,0,-1)$. Using $\Diamond\in M_{\mathbb{R}}$, in the
construction outlined above in Section B.1 results in a compact hypersurface
in the toric variety $\mathbb{P}_{\Sigma_{n}(\Diamond)}$ which in general is
not smooth. A fan refinement

$\widetilde{\Sigma}_{n}(\Diamond)\rightarrow\Sigma_{n}(\Diamond)\,.$ (213)

for which all rays introduced have generators which are lattice points on
$\Diamond$ gives rise to a crepant partial desingularisation. The associated
MPCP (again for a regular triangulation) then defines a smooth hypersurface
which we denote by $Z_{\Diamond,\Diamond^{\circ}}$. The defining equation of
$Z_{\Diamond,\Diamond^{\circ}}$ is

$F(z)=\sum_{m\in\left(\Diamond\cup(0,0,0,1)\right)}\alpha_{m}z_{0}^{\langle
m,n_{0}\rangle}\prod_{n\in\Diamond^{\circ}}z_{n}^{\langle m,n\rangle+1}=0\,.$
(214)

Here $z_{i}$ are the homogeneous coordinates associated with the ray
generators $n_{i}\in\Diamond^{\circ}$, note that $n_{0}$ is always a ray
generator of $\Sigma_{n}(\Diamond)$.

The hypersurface $Z_{\Diamond,\Diamond^{\circ}}$ admits a K3 fibration with
base $\mathbb{P}^{1}$ such that

$c_{1}(Z_{\Diamond,\Diamond^{\circ}})=[S_{0}]$ (215)

where $[S_{0}]$ is the cohomology class dual to the divisor class of a generic
K3 fibre, i.e. $Z_{\Diamond,\Diamond^{\circ}}$ is a building block and
$X_{\Diamond,\Diamond^{\circ}}=Z_{\Diamond,\Diamond^{\circ}}\setminus S_{0}$
is an asymptotically cylindrical Calabi-Yau threefold.

The topological data of $Z_{\Diamond,\Diamond^{\circ}}$ required for our
purposes can be described by combinators Braun:2016igl . Denoting
$k$-dimensional faces of $\Diamond$ by $\Theta^{[k]}$, the Hodge numbers of
$Z_{\Diamond,\Diamond^{\circ}}$ are $h^{i,0}(Z_{\Diamond,\Diamond^{\circ}})=0$
for all $i>0$ and

$\begin{split}h^{1,1}(Z_{\Diamond,\Diamond^{\circ}})&=-4+\sum_{\Theta^{[3]}}1+\sum_{\Theta^{[2]}}\ell^{*}(\sigma_{n}(\Theta^{[2]}))+\sum_{\Theta^{[1]}}\ell^{*}(\Theta^{[1]})\ell^{*}(\sigma_{n}(\Theta^{[1]}))\\\
h^{2,1}(Z_{\Diamond,\Diamond^{\circ}})&=\ell(\Diamond)-\ell(\Delta_{F})+\sum_{\Theta^{[2]}}\ell^{*}(\Theta^{[2]})\ell^{*}(\sigma_{n}(\Theta^{[2]}))-\sum_{\Theta^{[3]}}\ell^{*}(\Theta^{[3]})\end{split}$
(216)

where $\ell^{*}(\sigma_{n}(\Theta^{[k]}))$ counts lattice points on
$\Diamond^{\circ}$ in the relative interior of the normal cone to
$\Theta^{[k]}$, and $\ell$ and $\ell^{*}$ of polytopes/faces are defined as
before.

The ranks of the lattices $N$ and $K$ defined for building blocks in (16) are

$\displaystyle|N(Z_{\Diamond,\Diamond^{\circ}})|$
$\displaystyle=\ell^{1}(\Delta_{F})-3+\sum_{ve\,\,\Theta_{F}^{\circ[1]}}\ell^{*}(\Theta_{F}^{\circ[1]})\ell^{*}(\Theta_{F}^{[1]})$
(217) $\displaystyle|K(Z_{\Diamond,\Diamond^{\circ}})|$
$\displaystyle=h^{1,1}(Z_{\Diamond,\Diamond^{\circ}})-|N|-1$

where $\ell^{1}()$ counts points on the one-skeleton, and
$ve\,\,\Theta_{F}^{[1]}$ denotes only those one-dimensional faces of $\Delta$
which are bounding a face of $\Diamond^{\circ}$ that is ‘vertical’, i.e.
parallel to $n_{0}$.

As for reflexive polyhedra, one may interchange the roles played by $\Diamond$
and $\Diamond^{\circ}$ which results in another building block
$Z_{\Diamond^{\circ},\Diamond}$ that satisfies Braun:2017ryx

$\begin{split}h^{2,1}(Z_{\Diamond,\Diamond^{\circ}})&=|K(Z_{\Diamond^{\circ},\Diamond})|\\\
h^{2,1}(Z_{\Diamond^{\circ},\Diamond})&=|K(Z_{\Diamond,\Diamond^{\circ}})|\,.\end{split}$
(218)

Furthermore, the lattice $N(Z_{\Diamond,\Diamond^{\circ}})$ and
$N(Z_{\Diamond^{\circ},\Diamond})$ admit a primitive embedding

$N(Z_{\Diamond,\Diamond^{\circ}})\oplus N(Z_{\Diamond^{\circ},\Diamond})\oplus
U\hookrightarrow\Gamma^{3,19}$ (219)

for $\Gamma^{3,19}=H^{2}(S_{0},\mathbb{Z})$ the unique even self-dual lattice
of signature $(3,19)$. This implies that the K3 fibres of
$Z_{\Diamond,\Diamond^{\circ}}$ and $Z_{\Diamond^{\circ},\Diamond}$ are from
algebraic mirror families Aspinwall:1994rg . The above relations play a
crucial role in the construction of mirror $G_{2}$ manifolds of TCS type.

Any two projecting tops $\Diamond_{1}^{\circ}$ and $\Diamond_{2}^{\circ}$ for
which $\Delta^{\circ}_{1F}=\Delta^{\circ}_{2F}$ can be joined to create a
reflexive polytope $\Delta^{\circ}_{12}$ Candelas:2012uu . A large fraction of
the polytopes in the Kreuzer-Skarke list are of this type, and as their Hodge
numbers can be understood from this decomposition as well, a number of
patterns in the plot of Hodge numbers can be explained by this. Conversely,
given a reflexive polytope $\Delta^{\circ}$ that can be decomposed into two
projecting tops, the Calabi-Yau threefold $X_{\Delta,\Delta^{\circ}}$ admits a
stable degeneration limit in which it becomes reducible into the two building
blocks $Z_{\Diamond,\Diamond^{\circ}}$ and $Z_{\Diamond^{\circ},\Diamond}$
Braun:2017ryx . This limit can be understood as stretching the base
$\mathbb{P}^{1}$ of the K3 fibration on $X_{\Delta,\Delta^{\circ}}$,
separating the singular K3 fibres to its two ends. Cutting along the stretched
base along the middle then decomposes $X_{\Delta,\Delta^{\circ}}$ into the
asymptotically cylindrical threefolds $X_{\Diamond_{1},\Diamond^{\circ}_{1}}$
and $X_{\Diamond_{2},\Diamond^{\circ}_{2}}$.

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We have $\fs\Sy_2= \{0, (\text{id}),(1,2),(\text{id})+(1,2)\}$. We have maps given by

\begin{align*}
\begin{array}{rrcl}
ε:&\fs\Sy_2 & \longrightarrow & \fs\\
& a(\text{id})+b(1,2) &\longmapsto & a+b \\
% \end{array}\\
% \begin{array}{rrcl}
D:&\fs\Sy_2 & \longrightarrow & \fs\Sy_2\\
& a(\text{id})+b(1,2) &\longmapsto & (a+b)\left((\text{id})+(1,2)\right).
\end{array}
\end{align*}

We see that $ε$ is surjective and $\ker ε = \ker D = \im D = \{0, (\text{id})+(1,2)\}$. The maps $ε$ and $D$ are $\fs\Sy_2$-linear, where $\fs$ is the $\fs\Sy_2$-module that corresponds to the trivial representation of $\Sy_2$. So we have a projective resolution of $\fs$ with augmentation $ε$ by

\begin{align*}
\pres \fs := (\cdots \xrightarrow{D} \underbrace{\fs\Sy_2}_{1} \xrightarrow{D} \underbrace{\fs\Sy_2}_{0}  \rightarrow \underbrace{0}_{-1} \rightarrow \cdots),
\end{align*}

where the degrees are written below.

We set $e_1$ to be the identity on $\fs \Sy_2$.

Let $A:=\Hom^*_{\fs\Sy_2}(\pres\fs,\pres\fs)$ and let the $\A_∞$-structure on $A$ be  $(m_n)_{n\geq 1}$ (cf. <ref>). Recall the conventions concerning $\Hom_B^k(C,C')$ for complexes $C,C'$ and $k∈\Z$.

An $\fs$-basis of $\Hm^*A$ is given by $\{\overline{ξ^j} \mid j\geq 0\}$ where

\begin{align*}
ξ := \dum_{i\geq 0} \ls e_1\rs_{i+1}^i∈A.
\end{align*}

Straightforward induction yields, for $j\geq 0$,

\begin{align*}
ξ^j = \dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\,.
\end{align*}

We have

\begin{align*}
m_1(ξ^j) \eqs& d \circ ξ^j - (-1)^j ξ^j \circ d \overset{} = d \circ ξ^j + ξ^j \circ d\\
\eqs& \left(\dum_{i\geq 0} \ls D\rs_{i+1}^i\right)\circ  \left(\dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\right) + \left(\dum_{i\geq 0} \ls e_1 \rs_{i+j}^i\right)\circ \left(\dum_{i\geq 0} \ls D\rs_{i+1}^i\right)\\
\eqs& \dum_{i\geq 0} \ls D\rs_{i+j+1}^i + \dum_{i\geq 0} \ls D\rs_{i+j+1}^i = 0,
\end{align*}

so $ξ^j$ is a cycle.
As $\Hom_{\fs \Sy_2}(\fs \Sy_2, \fs) = \{0,ε\}$ and $ε\circ D=0$, the differentials of $\Hom^*(\pres \fs, \fs)$ (cf. <ref>) are all zero. So $\{ε\}$ is an  $\fs$-basis of $\Hm^k \Hom^*(\pres \fs, \fs)$ for $k\geq 0$. Since in the notion of <ref>, $\bar{Ψ}_k(\overline{ξ^k}) = ε$, the set $\{\overline{ξ^k}\}$ is an $\fs$-basis of $\Hm^k\Hom^*(\pres\fs,\pres\fs)$ for $k\geq 0$. For $k<0$ we have $\Hm^k\Hom^*(\pres\fs,\pres\fs) \cong \Hm^k \Hom^*(\pres \fs, \fs) = 0$. So $\{\overline{ξ^j} \mid j\geq 0\}$ is an $\fs$-basis of $\Hm^* A$.

We define families of maps $(f_n:(\Hm^*A)^{\otimes n}→A)_{n\geq 1}$ and $(m'_n:(\Hm^*A)^{\otimes n} → \Hm^* A)_{n\geq 1}$ as follows. $f_1$ and $m'_2$  are given on a basis by

\begin{align*}
f_1(\overline{ξ^j}) :=\eqsp& ξ^j &&\text{for $j\geq 0$}\\
m'_2(\overline{ξ^j}\otimes \overline{ξ^k}) :=\eqsp& \overline{ξ^{j+k}} &&\text{for $j,k\geq 0$}.
\end{align*}

All other maps are set to zero.

It is straightforward to check that $(\Hm^* A, (m'_n)_{n\geq 1})$ is a pre-$\A_∞$-algebra and $(f_n)_{n\geq 1}$ is a pre-$A_∞$-morphism from $\Hm^*A$ to $A$. As $m'_2$ is associative, $(\Hm^* A, (m'_n)_{n\geq 1})$ is a dg-algebra, so in particular an $\A_∞$-algebra.
As $f_k=0$ for $k\neq 1$, (<ref>)[$n$] simplifies to

\begin{align*}
f_1\circ m'_n \eqs& m_n\circ (\underbrace{f_1\otimes \cdots \otimes f_1}_{n\text{ factors}}).
\end{align*}

As $m'_n=0$ and $m_n=0$ for $n\geq 3$, (<ref>)[$n$] is satisfied for all $n\geq 3$. For $n∈\{1,2\}$, we have

\begin{align*}
f_1\circ m'_1 \eqs& m_1\circ f_1\\
f_1\circ m'_2 \eqs& m_2(f_1\otimes f_1).
\end{align*}

The second equation follows immediately from the definition of $m'_2$ and $f_1$. The first equation holds as $m'_1=0$ and the images of $f_1$ are all cycles. So (<ref>)[$n$] holds for all $n$ and  $(f_n)_{n\geq 1}$ is an $\A_∞$-morphism from $(\Hm^* A, (m'_n)_{n\geq 1})$ to $(A, (m_n)_{n\geq 1})$. By the construction of $f_1$,  it induces the identity on homology. Thus $(\Hm^* A, (m'_n)_{n\geq 1})$ is a minimal model of $(A, (m_n)_{n\geq 1})$.

[Comparison with primes $p\geq 3$]
At a prime $p \geq 3$, we have constructed a projective resolution with period length $l=2(p-1)$ in (<ref>).

If one constructs a projective resolution of $\Z_{(2)}$ analogous to the case $p\geq 3$, we have a sequence of the form

\[\cdots \rightarrow \Z_{(2)}\Sy_2 \xrightarrow{\hat e_{2,2}^*} \Z_{(2)}\Sy_2\xrightarrow{\hat e_{2,2}}\Z_{(2)}\Sy_2 \xrightarrow{\hat e_{2,2}^*} \Z_{(2)}\Sy_2\xrightarrow{\hat e_{2,2}}\Z_{(2)}\Sy_2 \rightarrow 0 \rightarrow \cdots\]

with a period length of $2$, where

\begin{align*}
\hat e_{2,2}\colon&\,(\text{id})\,\longmapsto\, (\text{id})-(1,2)\\
\hat e_{2,2}^*\colon&\,(\text{id})\,\longmapsto\, (\text{id})+(1,2).
\end{align*}

However, modulo $2$ the differentials  $\hat e_{2,2}$ and $\hat e_{2,2}^*$ reduce to the same map $D: \fs \Sy_2\rightarrow \fs \Sy_2$, so we obtain a period length of $1$.

The maps $ι$ resp. $χ$ from <ref> may be identified with $ξ^2$ resp. $ξ$. This way, the definition of $m'_2$ at the prime $2$ is readily compatible with <ref>.

On the bar construction

§.§ Applications. Kadeishvili's algorithm and the minimality theorem.

In this subsection we will discuss the construction of minimal models of $\A_∞$-algebras. Firstly, <ref> states that certain pre-$\A_n$-structures and pre-$\A_n$-morphisms that arise in the construction of minimal models are actually $\A_n$-structures and $\A_n$-morphisms. Secondly, we give a proof of <ref>. We will review Kadeishvili's original proof of [9] as it gives a an algorithm for constructing minimal models which can be used for the direct calculation of examples.
Note that Lefèvre-Hasegawa has given a generalization of the minimality theorem, see <cit.>, which we will not cover.

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11institutetext: Osservatorio di Astrofisica e Scienza dello Spazio
INAF – National Institute for Astrophysics
22institutetext: Sound artist, sensorialist, expert of perception processes
33institutetext: LIM – Laboratory of Music Informatics
Department of Computer Science “Giovanni Degli Antoni”
University of Milan 44institutetext: electornic music performer

# A Sonification of the zCOSMOS Galaxy Dataset

Sandro Bardelli 11 0000-0002-8900-0298 Claudia Ferretti 22 Giorgio Presti 33
0000-0001-7643-9915 Maurizio Rinaldi 22

###### Abstract

This paper proposes a sonification for zCOSMOS, an astronomical dataset that
contains information about 20,000 galaxies. The goals of such an initiative
are multiple: providing a sound-based description of the dataset in order to
make hidden features emerge, hybridizing science with art in a cross-domain
framework, and treating scientific data as cultural heritage to be preserved
and enhanced, thus breaking down the barriers between scientists and the
general audience. In the paper, both technical and artistic aspects of the
sonification will be addressed. Finally, some relevant excerpts from the
resulting sonification will be presented and discussed.

###### Keywords:

Sonification Galaxies Astronomical data.

##  1 Introduction

Sonification is the transformation of data into acoustic signals, namely a way
to represent data values and relations as perceivable non-verbal sounds, with
the aim to facilitate their communication and interpretation [23]. Like data
visualization provides meaning via images, sonification conveys meaning via
sound.

As discussed in [8], non-verbal sounds can represent numerical data and
provide support for information processing activities of many different kinds.

A first scenario is the possibility to receive information while keeping other
sensory channels unoccupied, as required in medical environments, process
monitoring, driving, etc. Common experiences in everyday life range from the
sounds naturally produced by physical phenomena and automatically associated
with specific events (e.g., a whistling kettle) to sound-augmented objects
(e.g., a Geiger counter). This approach is explicitly used in sensory-
substitution systems, like orientation and navigation applications for blind
or visually impaired (BVI) people [1].

Sonification techniques prove to be useful also when the data to represent are
complex and have multiple dimensions to track [9]. In fact, music and sound
present multidimensional features (e.g., pitch, intensity, timbre,
spatialization, etc.), and these dimensions can be simultaneously employed to
provide understandable representations of complex phenomena. An effective
design of sonification can draw, e.g., on musicality, musical acoustics, sound
synthesis and human perceptual capacities [10]. Listening to data can open new
scientific frontiers, thanks to the human ability to parse sound for patterns
and meaning. This approach can make non-trivial structures emerge and help to
unveil hidden patterns [14]. Social processes [25], natural events [2], and
physical observations [17] are only a few examples of applicability fields.

Finally, sonification can be applied to those scenarios where a set of data is
only the base to build an experience with artistic goals. In this context, it
is worth mentioning also the concept of musification, namely the
representation of data through music. The resulting musical structures can
take advantage of higher-level features, such as polyphony or harmony, in
order to engage the listener. The relationships of sonification to music and
sound art have been explored in [20]. After providing these definitions, we
can affirm that data can be visualized by means of graphics, sonified by means
of sound, and musified by means of music.

Sonifications can be enjoyed as scientific inquiries, aesthetic experiences,
or both. The idea of bridging the gap between art and science in the context
of scientific dissemination and “edutainment” initiatives has been explored in
a number of works. An interesting point of view is reported in [6]: according
to the author, sonification, as opposed to visualisation, is still an under-
utilised element of the “wow” factor of science.

Proposing a sonification initiative during an exhibition or another public
event can also add value to the dataset itself. First, a sound-based
multimedia installation can be an engaging way to make a non-expert audience
enjoy scientific subjects; in this sense, the experience can be enhanced
through suitable support materials (e.g., wall-mounted panels), the
stimulation of other sensory channels (e.g., a video installation), and real-
time interaction with the audience (e.g., through motion detectors and
ambient-light sensors). Moreover, such an initiative can play a cultural role
by raising scientific data and achievements to the rank of cultural heritage
to be preserved and exploited. Examples are reported in [4], [18], and [34].

In the context of sensory substitution techniques, sonification can make
scientific data accessible to specific categories of users, e.g. BVI people,
with an important impact on their education, too [24, 32, 35].

For the sake of completeness, it is worth underlining that the legitimacy of
sonification as a scientific method of data display is being debated by
scholars and experts, as discussed in [37]. According to [31], widespread
adoption of sonification to display complex data has largely failed to
materialize, and many of the challenges to successful sonification identified
in the past are still persisting. Nevertheless, since the goal of the
initiative described below is dissemination, even if scientifically accurate,
our proposal does not fall within the scope of problematic applications.

The rest of the paper is structured as follows: in Section 2 we will review
some background work about astronomical data sonification, in Section 3 we
will present the zCOSMOS galaxy dataset, in Section 4 we will describe our
sonification strategy, in Section 5 we will discuss the achieved results, and,
finally, in Section 6 we will draw some conclusions.

##  2 Sonification of Astronomical Data

The idea of sonifying astronomical data is not original at all. A forerunner
of such an approach is the photophone, a device invented by Alexander Graham
Bell that used light modulation, caught by means of photosensors, in order to
transmit audio signals to a distant station [38]. In a letter written in 1880,
the inventor showed excitement about the possibility to “hear a ray of the sun
laugh and cough and sing”. He was intrigued by the idea of applying such a
technology to study the spectra of stars and sunspots by listening to the
sounds produced by the photophone receiver [11].

Many years later, space agencies promoted sonification as a mean to explore
astronomical data. For example, NASA created a Java-based software tool called
xSonify [13, 15] aiming to encourage investigation in the field of space
physics through sonification. Another initiative by NASA, dating back to 2020,
is a project aiming to sonify the center of the Milky Way. Users can listen to
data from this region captured by Chandra X-ray Observatory, Hubble Space
Telescope, and Spitzer Space Telescope. Such data can be enjoyed either as
solos or together, as an ensemble in which each telescope plays a different
instrument.

Lunn and Hunt [29] described case studies of astronomy-based sonification,
specifically addressing the sonification of radio-astronomy data as part of
the Search for Extra-Terrestrial Intelligence (SETI).

Hadhazy [21] mentioned and exemplified some interesting musical compositions
based on astronomical data. Among them, “Deep-Space Sonata” converted the
gamma-ray burst GRB 080916C, one of the most powerful explosions recorded in
the Universe, to audible sound. In this sonification, the number of notes
played represents the gamma rays received by the Fermi Gamma-ray Space
Telescope, while the accompanying sounds correspond to the probability of the
rays emanating from the burst itself (lowest-likelihood rays are played as a
harp, medium by a cello and highest-probability by a piano).

Another experience reported in [21] is “Sunny Anthem”, based on the data of
charged atoms within the solar wind from 1998 to 2010 recorded by a
spectrometer onboard NASA’s Advanced Composition Explorer spacecraft.

Many other sonification experiments starting from astronomical data could be
mentioned. For instance, “Jovian Notes” is a sonification captured by Voyager
1’s plasma wave instrument as the spacecraft crossed the bow shock at the edge
of Jupiter’s magnetosphere. Another initiative was based on the data gathered
by a scientific device onboard the Lunar Reconnaissance Orbiter concerning
radiation spewed by the sun as it floods the vicinity of the moon; the
radiation intensity, converted into musical sounds, can be considered as a
sort of live lunar music. Other relevant experiences are discussed in [33].

This kind of initiatives is also a way for stimulating interest when teaching
astronomy. An example of auditory model of the solar system integrated within
a planetarium show is described in [39]. Moreover, Ballesteros and Luque [5]
reported a successful application to the educational field, originally
conceived for a science-dissemination radio program called “The sounds of
science”, heard on Radio Nacional de Espana (RNE), but easily reproducible
also in a classroom environment. Finally, it is worth underlining that a
sound-based approach can be particularly effective for BVI students [19].

Concerning accessibility and inclusion for people who cannot fully enjoy
experiences where visual media is the primary communication mechanism, it is
worth mentioning also initiatives dealing with astronomic data but relying on
senses other than hearing. An example is the creation of three-dimensional
printed data sets, as in the case of tactile 3D models from NASA’s Chandra
X-ray Observatory [3]. Also the sense of smell can be used to design
astronomy-oriented programs, as reported in [41]. Finally, the multi-sensorial
project called “The Hands-On Universe Project” employs edible models and
common food to convey complex concepts in cosmology and astrophysics [40].

Going back to sonification of astronomical data, there are also initiatives
coupling dissemination purposes with artistic goals. For example, Ballora [7]
applied sonification techniques for astronomical data in the framework of a
film-making process, so as to create the soundtrack for a short movie titled
“Rhythms of the Universe”. In [7] he described how the sonifications had been
obtained from datasets describing pulsars, the planetary orbits, gravitational
waves, nodal patterns in the sun’s surface, solar winds, extragalactic
background light, and cosmic microwave background radiation.

##  3 The zCOSMOS Dataset

Figure 1: Overview of the original variables. Variables distribution
histograms are shown on the main diagonal; other positions display a
dispersion graph of each pair of variables.

Figure 2: Overview of the re-scaled and filtered variables. Galaxies
considered for sonification are shown in dark color, while discarded galaxies
are shown in light gray.

The zCOSMOS dataset [22, 26, 27] is the spectroscopic follow-up of the wider
Cosmic Evolution Survey (COSMOS) [36], a coordinated international effort to
study the galaxy evolution in various wavelength. zCOSMOS obtained spectra for
18,143 galaxies at the apparent magnitude $I_{AB}<22.5$ at the VLT-ESO
telescope (Chile). Such a dataset describes the evolution of a relatively
small portion of the Universe in the last 10 million years.

With the use of these high-quality spectra, it was possible to derive a number
of physical quantities of the galaxies. Among the variables provided in the
dataset, we selected:

  * •

Stellar Mass $M_{\odot}$, a value describing how many stars are formed in a
galaxy and, therefore, a proxy for the galaxy history. It is the sum of all
the star masses;

  * •

Star Formation Rate SFR, i.e. the total mass of stars formed per year, which
reflects how active a galaxy is at the moment of the observation;

  * •

Redshift $z$, namely the measure of the recession velocity of the galaxy as a
consequence of the expansion of the Universe. Due to the Hubble’s law, the
higher the redshift (measured as the shift of spectral lines toward the red
part of the spectrum), the higher the galaxy distance;

  * •

Age of the Universe $A$ at the galaxy position, measured in billions of years.
Objects that are close to us are observed with little time delay due to the
finiteness of the speed of light; therefore, the value of $A$ for this galaxy
is similar to the actual age of the Universe (i.e. about $13.8$ billion
years). Conversely, distant objects are seen with a remarkable delay, thus, in
our observations, $A$ at that distance is lower. In other terms, even if we
are observing objects at a given moment, depending on their distance we are
observing a younger or a later stage of the Universe. $A$ is linked to a
measure called lookback time, hereafter referred to as $t_{L}$, obtained by
subtracting $A$ from the age of the Universe: $t_{L}$ $=13.8-A$. The lookback
time of a galaxy represents how much time we are looking back to obtain its
current observation;

  * •

Absolute magnitude $M_{V}$, i.e. the absolute luminosity of the observed
galaxy (related to the intrinsic luminosity). Please note that lower values of
$M_{V}$ imply brighter galaxies;

  * •

Position, in terms of right ascension $\alpha$ and declination $\delta$. These
are the coordinates of the galaxy over the celestial sphere in the equatorial
coordinate system.

The number of galaxies under exam has been reduced to 18,098, since the ones
presenting $M_{V}<-26$ or $M_{\odot}\leq 6$ could come from measurement errors
or insufficient quality of data.

Prior to the choice of a sonification strategy, we conducted a simple
statistical inspection of the dataset. In fact, a sonification is expressive
when changes in sound significantly reflect those in data. For this to happen,
it is important that such data take values with a sufficient resolution and
within a perceptible range. Furthermore, in case of a strong correlation
between some variables, we had to understand if the reason was trivial, e.g.
the calculation of one datum from another, or rather it was an unexpected
feature of the phenomenon. In the former scenario, the sonification should
avoid redundancy, while, in the latter, it should remark these
characteristics.

The top-left to bottom-right diagonal of Figure 2 shows the statistical
distribution of variables. Its analysis allows to infer the range of variation
of each variable. The other areas of Figure 2 depict the dispersion graphs of
each pair of variables, so as to remark their mutual dependence. Three
problematic issues mainly emerge:

  1. 1.

Almost all values for SFR are condensed in the leftmost area;

  2. 2.

Variables $z$ and $A$ are, not surprisingly, one function of the other, since
$z$ is a proxy for $t_{L}$ in standard cosmological models;

  3. 3.

Due to an observational bias known as Malmquist bias [30], at large distances
the sample loses faint galaxies;

In Section 4.2 we will describe the design choices we implemented in order to
solve these problems.

##  4 Sonification Strategy

Sonification can be seen as the junction point between the artistic use of
science and the scientific use of art, thus combining the separate viewpoints
of the artist and the scientist. Coherently, the project described in this
paper has been designed and implemented by a working group made of scientists,
technicians, and artists.

The software tools used in the whole process include: MATLAB for data
inspection and preprocessing, Supercollider to parse the CSV exported by
MATLAB and perform real-time sound synthesis, Ableton Live to record the
distinct audio tracks generated by Supercollider, and, finally, Steinberg
Cubase for post-production.

As better detailed below, the proposed sonification is based upon three main
layers:

  * •

Galaxies, sonified through a dense stream of events, each modulated
independently, thus generating a synthetic sound texture (Section 4.3);

  * •

Statistics, producing a very simple, continuously modulated, synthetic drone
sound (Section 4.4);

  * •

Outliers, causing a rare occurrence of events, each modulated independently,
thus generating complex sound icons (Section 4.5).

Sonification examples of the listed items and portions of the final outcome
can be found at the following URL:
https://www.lim.di.unimi.it/demo/zcosmos.php.

###  4.1 Data Pre-Processing

In Section 3 we listed three issues emerging from an a-priori statistical
inspection of the dataset. These problems have been addressed in a pre-
processing phase.

In order to solve the first issue, namely an excessive clustering of SFR
values in the leftmost range, an early idea was to consider $\log$(SFR)
instead of SFR, but this would have involved an excessive flattening of the
values to the right. We therefore decided to transform the data in a monotonic
way by extracting the twelfth root of SFR, namely SFR1/12, which showed a more
balanced distribution (see Figure 2).

The second issue, namely the correlation between $z$ and $A$, was solved by
ignoring $z$, whose flattening to the left was higher than the one of $A$.
This is the reason why $z$ is not present in Figure 2.

The zCOSMOS survey is limited in apparent luminosity and therefore at higher
distances it observed only the brightest galaxies. In order to avoid this
observational bias and to have an homogeneous sample, we decided to limit the
time span and the brightness range: galaxies with $A$$<6$ million years and
$M_{V}>-21$ have been excluded. In this way, the number of sonified galaxies
has been drastically cut, from $18,098$ to $8,156$. Even if the sonification
of the full set of galaxies could have been desirable from multiple points of
view, we chose not to represent misleading data, such as the false correlation
between magnitude and age. The remaining galaxies are visible in dark color in
Figure 2.

The reduction in the number of represented galaxies pushed us to reflect on
the opportunity to provide an acoustic feedback to uncertainty in
observations. The mechanism employed to achieve this goal will be described in
Section 4.4.

Concerning $A$, as mentioned before, such a parameter has been converted into
$t_{L}$, since the latter will better suit the design described in Section
4.3.

Another adjustment has involved $M_{V}$, that has been inverted in order to
handle values in a more intuitive way. This parameter, indicated as $-M_{V}$
from now on, takes low values for dim objects and high values for bright ones.

Features have been normalized in the interval $[0,1]$ so as to provide a
simple interface to the modular sonification engine. In order to avoid very
small and very big outliers that would shrink most values into a middle range,
some clipping has been introduced. Clipped values have been marked, so as to
be properly treated in the sonification. Please refer to Table 1 and Figure 4
for further details.

Table 1: Lower and upper thresholds and number of outliers. Dim galaxies, presenting high $M_{V}$, have not been marked. Variable |  Lower threshold |  No. of lower outliers |  Upper threshold |  No. of upper outliers
---|---|---|---|---
$M_{\odot}$ |  9.25 |  7 |  11.58 |  1
SFR |  0.1 |  138 |  76 |  102
$M_{V}$ |  -24.19 |  9 |  - |  -

As the final step, a moving average has been calculated for normalized
$M_{\odot}$, SFR1/12, and $-M_{V}$ to provide explicit information about the
trend of these variables along the temporal axis. The results thus obtained
are visible in Figure 4.

Figure 3: Distributions of the normalized variables that are actually
sonified. The uniformly distributed variable $\alpha$ is not shown. Outliers
assigned to auditory icons are highlighted through dark histogram bars.

Figure 4: Moving average of principal features against lookback time.

###  4.2 Design Choices

The sonification is completely procedural and parametric, so it can be easily
modified to respond to different requirements and use cases. In particular,
modularity has been exploited to explore the space of sound-synthesis
parameters, overall duration, and feature associations, with the aim of
matching the sonification goals with the desired aesthetics.

In all design phases we followed the principle of ecological metaphors, thus
trying to make the sonification coherent with users’ real-world sensory and
cognitive experience [12]. This approach implies that variations of auditory
dimensions are consistent with those of physical parameters; for example,
position values may be mapped onto left/right sound panning. The use of
ecological metaphors should improve intuitiveness and learnability.

One of the goals was to keep the technical setup required to play the
sonification as simple as possible, so as to make it easily reproducible in a
wide range of contests. For this reason, the final outcome was a standard
stereo file. Sound spatialization through an array of loudspeakers or a
binaural approach would have extended the possibilities connected to
ecological metaphors, but it would have prevented the performance in a great
number of environments not adequately equipped.

###  4.3 Sonification of Galaxies

Each galaxy is sonified as a single and short sound event, occurring at a time
which is proportional to its lookback time. When the event density is very
low, single galaxies can be easily spotted and compared, while, in case of
very dense and crowded sections, the overlapping of many events generates a
complex texture which is more informative about the overall trend.

Each sound event is generated by $3$ distinct sinusoidal oscillators, called
$O_{1-3}$, presenting an exponential-decay envelope. Each oscillator can be
controlled in terms of pitch, level, decay time, stereophonic position, and
frequency modulation. $O_{1}$, $O_{2}$, and $O_{3}$ are the master, the
harmonic, the sub-harmonic oscillator respectively. After computing the
parameters for $O_{1}$, those for the other oscillators follow according to
the schema shown in Table 2. The rationale is to have a fundamental frequency
generated by $O_{1}$ which is louder and lasts longer than the harmonic and
sub-harmonic sounds generated by $O_{2,3}$.

Table 2: Relationship between the parameters of the oscillators. Oscillator |  Pitch |  Level |  Release |  Position |  FM
---|---|---|---|---|---
$O_{1}$ |  $f_{0}$ |  $a$ |  $r$ |  $p$ |  None
$O_{2}$ |  $2f_{0}$ |  $0.25a$ |  $0.5r$ |  $p$ |  None
$O_{3}$ |  $0.75f_{0}$ |  $0.6a$ |  $0.6r$ |  $p$ |  Yes

As it regards $O_{1}$, its parameters are modulated according to the following
data bindings (a detailed description of modulation ranges is presented in
Table 3).

The volume is determined by the absolute magnitude $-M_{V}$ of the galaxy
under exam by exploiting an analogy with vision: brighter galaxies are
represented with louder sounds, while dim galaxies (harder to see) are
represented with softer sounds (harder to hear). The resulting dynamic range
is about $-24$dB, which is sufficient to discriminate between bright and dim
galaxies without making the latter inaudible.

Since the sonification has been conceived to be reproduced through a
stereophonic speakers layout, it was natural to bind galaxy right ascension
with the sound position in the stereophonic space. The resulting
representation of spatial information is magnified, since original right
ascension of the galaxies is included in about $1$ degree of the sky, while
the stereophonic field can reach $180$ degrees, depending on the installation
conditions. Declination $\delta$ could have been treated in a similar way,
thanks to quadraphonic listening environments, but we decided to privilege a
simpler setup, as explained in Section 4.2.

The frequency of $O_{1}$, namely $f_{0}$, is inversely proportional to
$M_{\odot}$. This binding has been chosen since lower pitches are generally
associated with heavier and bigger sources, while high-pitched sounds easily
recall smaller sources. In order to produce well-sounding events, many
sonifications (including the ones mentioned in Section 2) usually map values
onto notes of the equal-tempered scale or consonant frequencies. Conversely,
we decided to let the frequency binding be continuous; in this way, the
presence of beatings as opposed to the perception of distinct sounds lets the
listener clearly perceive when two galaxies are similar (beatings) or
different (distinct sounds) in terms of $M_{\odot}$. As an aesthetic
consideration, the adoption of a musical scale would have produce a
sonification more pleasant in the short term, but more boring on the long run.
Another potential problem was the possibility to introduce a phenomenon of
data misinterpretation in case of peculiar musical structures (e.g., consonant
chords, cadences, etc.), which are strongly rooted in the tonal-harmony
perception of music, but have no particular meaning in the sonification.

Star formation rate SFR is linked to the parameters of the frequency
modulation of $O_{3}$. In order to give the idea of very active galaxies for
high values of SFR1/12 and more relaxed galaxies for low values of SFR1/12, we
carefully tuned the sinusoidal modulator frequency $f_{m}$. This parameter
runs below the audio rate (i.e. $f_{m}<20$ Hz) for low star formation rate,
thus producing a tremolo-like effect, while high values for SFR1/12 produce a
more distinctive and frantic modulation. For the same reason, the frequency
deviation $d$, a measure of the frequency modulation amount [16], is modified
proportionally to SFR1/12, too.

Finally, the amplitude envelope exponentially decays with a factor
proportional to SFR1/12, so that galaxies with high SFR1/12 present a longer
tail, while low values of SFR1/12 cause a quicker decay.

Table 3 provides a detailed view of the parameters.

Table 3: Sonification parameters for a single galaxy. Variable |  Parameter |  0 |  1 |  Unit |  Type
---|---|---|---|---|---
$t_{L}$ |  Time |  0 |  1500 |  s |  Linear
$-M_{V}$ |  Level |  -34 |  -10 |  dBfs |  Linear
$\alpha$ |  Position |  Left |  Right |  Pan |  Linear
$M_{\odot}$ |  $f_{0}$ |  7000 |  400 |  Hz |  Exponential
SFR1/12 |  $f_{m}$ |  2.88 |  252 |  Hz |  Exponential
SFR1/12 |  $d$ |  12 |  1050 |  Hz |  Exponential
SFR1/12 |  Release |  0.3 |  9.6 |  s |  Linear

###  4.4 Sonification of Statistics

Statistics include the average of $M_{\odot}$, SFR1/12, and $-M_{V}$ computed
within a moving window across lookback time. These are continuous signals
controlling the frequency $f_{1-3}$ of 3 distinct resonant bandpass filters,
each one filtering white noise, with different pan values. Filters frequency
are set by multiplying $M_{\odot}$, SFR1/12, and $-M_{V}$ by $200$, $1000$,
and $2000$ respectively. The result is a drone sound, a non-tempered chord
which is consonant only under favorable circumstances.

The quality $Q$ of the filters is very high at the beginning of the
sonification, thus producing well defined pitches; it linearly decreases in
time, so as to produce band-limited noise at the end of the sonification. The
idea is to suggest an increase in data variability and uncertainty of the
observations as long as more distant time and space is under exam. Please note
that the reciprocal of $Q$ is the actual modulated parameter.

Exact ranges are shown in Table 4, and the actual frequencies over time are
visible in Figure 5.

Table 4: Statistics sonification parameters. Variable |  Parameter |  Min |  Max |  Unit |  Pan
---|---|---|---|---|---
Avg. $M_{\odot}$ |  $f_{1}$ |  97 |  120 |  Hz |  Center
Avg. $-M_{V}$ |  $f_{2}$ |  327 |  562 |  Hz |  Left
Avg.SFR1/12 |  $f_{3}$ |  401 |  887 |  Hz |  Right
$t_{L}$ |  $Q_{1-3}$ |  0.0001 |  0.2 |  $Q^{-1}$ |  -
Figure 5: Actual frequencies played by statistics-controlled filters.

###  4.5 Sonification of Outliers

Outliers are galaxies whose values are out of range for at least one
variable111Actually, no galaxies were marked as outliers for more than one
variable. and are sonified by means of auditory icons, modulated (when
possible) with the same principles of single-galaxies modulations.

The icons have been carefully crafted using sound design principles coherent
with other sonification-design choices. The goal is to make outliers emerge
from the overall sonification, but linking their perceptibility to the
frequency of their occurrence: uncommon events have to stand out with respect
to more common outliers. Please refer to Table 1 for the number of outliers of
each kind.

The icon for the biggest galaxy is a low-pitched percussive sound, while
smaller galaxies are associated with high-pitched bells; both approaches rely
on the original binding, but provide more emphasis on their outlier nature.
Such sounds are modulated in position and intensity, according to $\alpha$ and
$-M_{V}$.

Similarly, high and low SFR1/12 are represented by fast and slow pulsing
rumbles, respectively. These are generated through filtered noise, and
modulated in pitch, level, and position according to the original bindings,
and release and non-linear distortion according to $M_{\odot}$ and SFR1/12
respectively.

Finally, very bright galaxies are represented through sound glitches, so as to
suggest a saturation effect for the sensors, modulated in pitch and position
only.

###  4.6 Post-Production and Final Outcome

The post-production phase consisted in fine-tuning the level balance between
galaxies, outliers, and statistics. Some reverberation and equalization was
added in order to improve the aesthetic result.

In particular, more reverberation has been added to background drones respect
to foreground sounds in order to create a sense of depth.

Since background drone sounds are modulated quite slowly, they were slightly
processed with granular synthesis based effects, so to avoid adaptation
effects and to render them more appealing, by providing a more ruffled texture
without compromising the intelligibility of the played frequencies.

The final outcome was a 25 minutes long sonification, whose musical meaning
will be discussed in the next section.

##  5 Discussion

The proposed sonification can be formally framed according to the taxonomy
outlined in [28] and called the sonification space. In that framework, any
sonification is described as a set of data bindings (i.e. the single mappings
between data and sound features) and one main feature (i.e. the overall sonic
outcome). Both data bindings and the main feature can be placed over a two-
dimensional plane characterized by the following axes: time granularity
(continuous/regular/asynchronous), and level of abstraction (direct
representation of data vs. symbolic representation). The areas that can be
found are:

  * •

Soundscapes, i.e. holistic descriptions of a system by means of continuously
modulated symbolic sounds;

  * •

Symbolic samples, i.e. asynchronous symbolic signaling such as auditory icons;

  * •

Feature modulations, i.e. continuous or regular sonic plotting;

  * •

Sound events, i.e. asynchronous and interactive sonic plotting.

Figure 6: Representation in the sonification space. Three groups of data
bindings with different levels of temporal granularity combine into one main
feature which provides a comprehensive description of the dataset.

As qualitatively depicted in Figure 6, galaxies, statistics, and outliers lay
in three different areas of the sonification space. Sounds of galaxies are the
constitutive atomic particles of the sonification, very small sounds which are
meant to be mostly heard in swarms. The sound driven by statistics represents
the scene, the space where all takes place, bent by the cumulative properties
of the galaxies. Finally, outliers events are infrequent distinctive sounds,
counterpointing the swarm movements.

The outcome, marked as zCOSMOS in Figure 6, can be perceived as both an
artificial soundscape and a contemporary music composition.

This sonification is like a journey back in time, which starts from the
present, close to the Earth, and proceeds away, revealing a more and more
distant past.

From a musical point of view, an overall crescendo can be clearly perceived,
with a number of intermediate full-orchestra moments. Such a crescendo effect
is due to both the increasing in average dynamics linked to each event (see
the dotted line in Figure 4) and to the increasing density of events (see the
shape of the gray area in Figure 4). In this sense, a 25-minute-long listening
session is a good compromise to make the general dynamic trend emerge without
having an excessive density of sound events. Moreover, the dashed line in 4
shows that also the average frequency-modulation amount increases with time,
thus producing a darker sound at the beginning and a brighter and richer sound
at the end of the piece.

The trend shown by the three lines in Figure 5, that correspond to the
frequencies of drone sounds, creates a contrary-motion effect between the bass
line and the leading voices of the background layer.

In order to make the listening experience more interesting, the role played by
the sonification of outliers is fundamental. In fact, in musical terms, the
function of outliers can be compared to sudden variations in the
orchestration, impulsive percussive events, and articulation signs. In
particular, the outlier related to the biggest galaxy, rendered as a low
frequency percussion, happens to be played only once, and very close to the
end of the sonification: this sound event recalls the musical function of a
final cadence.

##  6 Conclusions

In this work we have described the multiple stages bringing from the design to
the realization of a sonification driven by astronomical data.

The whole process is articulated and requires heterogeneous competences. The
working group embraced domain experts in different fields (physicists and
astronomers, music composers, performers, sound and music computing experts,
etc.) able to share ideas and cooperate.

The main activities conducted to realize the sonification (and the key actors
involved) have been:

  * •

the acquisition of the dataset (physicists and astronomers);

  * •

its transformation in a suitable computer format (computer scientists);

  * •

its filtering according to the scientific and aesthetic goals of the
sonification (artists and sound-and-music computing experts);

  * •

design choices concerning data bindings (the whole working group);

  * •

data pre-processing (physicists, astronomers and computer scientists);

  * •

technical production of single sounds (sound designers);

  * •

mixing and post-production (sound-and-music computing experts).

This project has been mainly conceived for scientific dissemination. The goal
of a sonification activity in general, and of the zCOSMOS initiative in
particular, is to make scientific data, originally hard to retrieve and
understand, easily accessible and enjoyable even by a non-expert audience.
Public presentations are planned to take place in cultural institutions,
together with explanatory wall-mounted panels (for museum-like installations)
or support videos (for planetariums).

Thanks to the choice to diverge from tonal-harmony rules, rather focusing on
the best possible rendering of the original data, this sonification challenges
the listener by proposing uncommon musical structures typical of contemporary
music. Such a listening activity is expected to raise users’ awareness about
their own perception, and highlight the importance of sound as a carrier of
meaning.

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