Taking Music Seriously

Taking Music Seriously: on the Dynamics
of ‘Mathemusical’ Research with a Focus
on Hexachordal Theorems††This paper is a contribution to the Special Issue on
Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre
Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html

Moreno ANDREATTA a, Corentin GUICHAOUA b and Nicolas JUILLET c

M. Andreatta, C. Guichaoua and N. Juillet

a) IRMA-CNRS-CREAA-University of Strasbourg and IRCAM, Paris, France
<EMAIL_ADDRESS>http://repmus.ircam.fr/moreno-en/home

b) Independent Researcher, SMIR Project, France<EMAIL_ADDRESS>
c) IRIMAS, Université de Haute-Alsace, France<EMAIL_ADDRESS>https://juillet.perso.math.cnrs.fr/

Received July 01, 2023, in final form January 11, 2024; Published online
January 25, 2024

After presenting the general framework of ‘mathemusical’ dynamics, we focus on
one music-theoretical problem concerning a special case of homometry theory
applied to music composition, namely Milton Babbitt’s hexachordal theorem. We
briefly discuss some historical aspects of homometric structures and their
ramifications in crystallography, spectral analysis and music composition via
the construction of rhythmic canons tiling the integer line. We then present
the probabilistic generalization of Babbitt’s result we recently introduced in
a paper entitled “New hexachordal theorems in metric spaces with probability
measure” and illustrate the new approach with original constructions and
examples.

mathemusical research; homometric sets; distance-sets; metric measure spaces;
ball volume; scalar curvature; Patterson function

00A65; 28A75; 05C12; 60D05

##  1 An introduction to “mathemusical” research

Despite a very long historical relationship between mathematics and music,
which went through the centuries from Pythagoras to nowadays [23], the real
interest of the community of “working mathematicians” in this research field
is a relatively recent phenomenon. One may surely find the germs of a new way
of looking at the relations between music and (modern) mathematics in the
second half of the twentieth century, with some remarkable figures of
mathematically-inclined music-theorists and composers such as Iannis Xenakis
(in Europe) and Milton Babbitt (in the United States). Both played an
important role in the emergence of an algebraic approach in the formalization
of twentieth-century music theory and composition, which constitutes a crucial
moment in the development of a systematic orientation in contemporary
musicology, both in the European and in the American tradition [11]. As widely
discussed in another contribution of this special issue,111See the eleven
variations on the maths and music theme by François Nicolas, published in the
present volume. Jean-Pierre Bourguignon has undoubtedly given a major
contribution in the progressive institutionalization process of this research
domain. He was in fact one of the main catalyzers of the Diderot forum on
mathematics and music, which took place in 1999 simultaneously in Paris,
Vienna and Lisbon and that was organized under the auspices of the European
Mathematical Society. The conference, as well as the volume that was published
subsequently by Springer [15], represented a real milestone in the change of
perspective by mathematicians on music and mathematics as a truly
interdisciplinary research field. This led in 2007 to the constitution of an
international society (the “Society for Mathematics and Computation in
Music”)222See http://www.smcm-net.info. and the launch of the first
mathematical journal devoted to maths and music research (J. Math. Music,
edited by Taylor and Francis).333See
https://www.tandfonline.com/journals/tmam20. The recognition of the relevant
mathematical dimension of the research carried on in this domain enabled the
inscription in 2010 of “Mathematics and Music” as an official topic within the
mathematics subject classification under the code 00A65.

###  1.1 Around the domain of structural music information research

In contrast to statistical methods and signal-based approaches currently
employed in different domains of computational musicology and music
information retrieval,444See [43] for a survey of the different approaches in
computational music analysis. The reader may refer to [46] for a roadmap in
the domain of music information research in both symbolic and signal-based
approaches and edited by the MIReS (Music Information Research) Consortium.
what we originally suggested to call “mathemusical” research [7] shows the
interest of introducing a structural perspective into the multidisciplinary
field of music information research making use of advanced mathematics. The
research presented in this paper is carried on, in particular, within the
ongoing SMIR Project devoted to structural modern mathematics applied to the
field of music information research. This project was initially supported by
the University of Strasbourg Institute for Advanced Study (USIAS) and is
currently hosted as a permanent transversal axis at the Institut de Recherche
Mathématique Avancée (IRMA) in collaboration with the CREAA (Centre de
Recherche et d’Expérimentation sur l’Acte Artistique) and the Institut de
Recherche et Coordination Acoustique/Musique (IRCAM) in Paris.555For a
detailed description of the SMIR project, together with the list of
participants and scientific production, including academic work at Master and
PhD levels, see http://repmus.ircam.fr/moreno/smir.

“Mathemusical” research, as carried on within the SMIR Project, is based on
the interplay between several mathematical disciplines: algebra,
combinatorics, geometry, topology, category theory, statistics and probability
theory. It opens promising perspectives on important prevailing challenges in
the domain of relations between music and mathematics, such as the automatic
classification of musical styles or the solution of open mathematical
conjectures. It therefore asks for new collaborations between mathematicians,
computer scientists, musicologists, and composers.

As we suggest with the present article, music surely deserves to be taken
seriously by mathematicians since it provides a number of difficult
theoretical problems, some of which can even be the starting point for
approaching, in a new way, open mathematical conjectures. Among the music-
theoretical problems showing a remarkable link with interesting mathematical
constructions and sometimes open conjectures, one may quote the following ones
that still constitute active research axes in the maths and music domain.
According to the idea of “mathemusical” dynamics we first indicate in the list
the original musical problem followed by the mathematical theory in which such
a problem can be formalized:666Obviously, the list does not entirely cover the
domain of music-theoretical problems that can be interesting to approach from
a mathematical perspective. The reader may find some more examples in J. Math.
Music as well as in the Proceedings of the Mathematics and Computation in
Music Conferences that have been regularly edited by Springer.

  * •

Tiling rhythmic canons and their Fourier-based characterization (with a
connection to Fuglede’s spectral conjecture). See [5, 35].

  * •

Z-relation in music theory as connected to the study of homometric structures
in crystallography (and their extensions via the notion of $k$-deck). See [40,
41];

  * •

Transformational music theory and the functorial representation of poly-
Klumpenhouwer networks as categorical graph-theoretical constructions. See
[47].

  * •

Neo-Riemannian music analysis, spatial computing and lattice-based
representations of musical structures (with tools derived from formal concept
analysis and mathematical morphology). See [26, 36].

  * •

Diatonic theory, maximally even sets and the discrete Fourier transform [3].

  * •

Periodic musical sequences and finite difference calculus taking values in
generic finite groups [14].

  * •

Chord and rhythms classification in music composition expressed in terms of
combinatorial block-designs [31].

  * •

Voice-leading theory and the geometry of orbifolds [50].

As a pedagogical illustration of the “mathemusical” dynamics, we will briefly
present in the final part of this introductory section the two first research
topics from the previous list, insisting on their mutual relations.
Interestingly, these two topics, as most of the problems listed before, are in
fact deeply interrelated which shows the existence of a remarkable interplay
between algebraic formalization and geometric representations of musical
structures and processes. From a more philosophical perspective, this
interplay provides a further example of the duality between “temporal” and
“spatial” constructions which are surely two of the most fundamental
ingredients of music. To quote, for example, Alain Connes’ opinion, as
expressed in a conversation with composer Pierre Boulez devoted to creativity
in mathematics and music: “Concerning music, it takes place in time, like
algebra. In mathematics, there is this fundamental duality between, on the one
hand, geometry – which corresponds to the visual arts, an immediate intuition
– and on the other hand algebra. This is not visual, it has a temporality.
This fits in time, it is a computation, something that is very close to the
language, and which has its diabolical precision. […] And one only perceives
the development of algebra through music” [19].

The crucial point is the existence of a permanent feedback loop between
musical thought, mathematical formalisation and computational modeling, which
constitutes the heart of a new kind of dynamics between music and mathematics
via computer-science. This dynamic back-and-forth is illustrated in the
diagram of Figure 1 which clearly shows the place of the three main
disciplines (music, computer science and mathematics) as well as the
contribution of two additional fields within contemporary “mathemusical”
research, namely epistemology and cognition.777See [8] for a more
philosophical account of the research in music and mathematics also including
an epistemological discussion and some reflexions about the role of cognitive
science with respect to mathemusical dynamics.

Figure 1: A diagram showing the underlying “mathemusical” dynamics between
music and mathematics through computer-science, also including some
epistemological and cognitive aspects.

Moreover, the mathemusical dynamics (from music to mathematics to music via
computer science and the possible epistemological and cognitive implications)
constitutes a radical change of perspective with respect to the traditional
application of mathematics in the musical domain. Mathemusical problems are
characterized by the fact that settling them in an appropriate mathematical
framework not only gives rise to new musical applications, but also paves the
way to new mathematical constructions. By carefully analyzing the various
steps of this mathemusical dynamics, one observes that it can be decomposed
into the following three stages (see Figure 2):

  * •

Formalization: the initial music-theoretical problem is approached by means of
a combination of mathematical tools enabling its formalization and revealing
its computational character;

  * •

Generalization: the formalized problem is generalized by using a panoply of
mathematical constructions, ranging from abstract algebra to topology and
category theory and leading to general statements (or theorems);

  * •

Application: once a generalized result has been obtained, it can be applied to
music by focusing on one of the three main aspects, i.e., the theoretical, the
analytical and the compositional one.

This decomposition of the mathemusical dynamics, together with the triple
perspective of the possible musical applications of a general result is shown
in Figure 2.

Figure 2: A more detailed perspective on the “mathemusical” research diagram
as presented in Figure 1, with the indication of the three main ingredients of
the dynamics (namely “formalization”, “generalization” and “application”).

It is this fruitful double movement, from music to mathematics and backwards,
which is at the heart of a research activity, where computer science is
positioned in the middle of this feedback loop, as an interface for connecting
the musical and mathematical domains. We simplify the picture by considering a
homogeneous intermediate level corresponding to the place occupied by computer
science with respect to music theoretical and mathematical research. Through a
more careful analysis of the different music theoretical problems, one may
nevertheless distinguish the cases in which the computer-aided models are
directly built in the formalization process such as problems asking for a
computational exploration of the solution space, more than a search of a
general underlying mathematical theory. To this family belong, for example,
typical enumeration problems such as the classification of all possible
Hamiltonian paths and cycles in some graph-theoretical musical spaces, such as
the Tonnetz and their multi-dimensional extensions [12]. Conversely, there are
cases in which the computational models are built starting from some general
algebraic results, as in the case of the construction of tiling rhythmic
canons that we briefly recall by stressing their link with two fields of
mathematics: homometry and spectral theory.

###  1.2 Approaching spectral theory and homometry musically

Applying advanced mathematics to the field of computational musicology is
necessary, we believe, to successfully tackle difficult “mathemusical”
problems which are linked to open conjectures in mathematics. This is the case
of two major problems that have been the object of study in the last fifteen
years and which can be approached in a new way by stressing their mutual
theoretical and computational interplay: the construction of tiling rhythmic
canons and the classification of homometric musical structures.

Tiling canons are special rhythmic canons having the property of tiling the
time axis by temporal translation of a given rhythmic pattern. Algebraically,
it corresponds to the decomposition of a cyclic group of order n into a direct
sum of two subsets (which can be or not periodic). See [10] and [6] for two
journal special issues devoted to the theoretical and computational aspects of
the construction of tiling rhythmic canons.

Concerning homometry theory, it is a field in mathematical combinatorics that
originates in crystallography, where as originally shown by Arthur Lindo
Patterson [44, 45] one may find crystals having the same X-ray spectrum
without being the same. In other words, these crystal structures are
undistinguishable by spectroscopic analysis although the is no isometric group
transforming one structure into the other one. Analogously, music composition
naturally provides examples of homometric structures in different domains such
as harmony, melody or rhythm. By definition, two musical structures are
homometric if they share the same multiplicity of occurrence of distance
between their elements. Since distances in music are expressed in terms of
intervals, we can say that two musical structures are homometric if they have
the same distributions of intervals without necessarily being equivalent up to
elementary musical transformations such as transpositions (i.e., translations)
or inversions (i.e., axial symmetries) [40, 41].

The deep connection between tiling and homometry comes from the observation
that considering a cyclic group $\mathbb{Z}_{n}$ and its decomposition into
two non-periodic factors (i.e., two subsets of periodicity equal to $n$), the
tiling process constitutes a special case of an open conjecture in mathematics
dating from the 1970s, namely Fuglede’s spectral conjecture.

####  1.2.1 A digression on Fuglede’s spectral conjecture
and its application to the discrete integer line

Interestingly, by going through the wide literature devoted to group
factorization in the perspective of tiling problems in geometry, there is no
evidence that all these problems can be related to the functional analysis
domain and, more specifically, to an analytical problem concerning the
commutativity of self-adjoint operators in the space $L^{2}(\Omega)$ of
square-integrable functions defined on a domain
$\Omega\subseteq\mathbb{R}^{n}$. This problem was raised by Bent Fuglede [27]
who was interested in the link between the spectrality of a domain $\Omega$
and the fact that it can tile the n-dimensional Euclidean space by
translation. More formally, a domain $\Omega\subseteq\mathbb{R}^{n}$ is
spectral if every function $f\in L^{2}(\Omega)$ can be represented in the
following way: $f(x)=\sum f_{k}{\rm e}^{2\pi{\rm i}\lambda_{k}\cdot x}$, where
$(\lambda_{k})_{k\in\mathbb{Z}}$ is a family of vectors and the exponentials
${\rm e}^{2\pi{\rm i}\lambda_{k}\cdot x}$ are mutually orthogonal maps.

In a more informal way, a domain $\Omega$ is spectral if it admits an
orthonormal basis of complex exponentials, i.e., a Fourier decomposition.
Fuglede’s spectral conjecture can be stated in the following way.

######  Conjecture 1.1 (Fuglede).

A domain $\Omega\subset\mathbb{R}^{n}$ is spectral if and only if it tiles
$\mathbb{R}^{n}$ by translation.

Contributions by several mathematicians including Fields medalist Terence Tao
[49], as well as Mate Matolcsi and Mihalis Kolountzakis [33], have shown that
the conjecture is false in all dimension $n>2$. In contrast to what happens in
convex domains, where the conjecture has very recently been proved true in all
dimensions [37], it is therefore possible to find domains $\Omega$ which tile
$\mathbb{R}^{n}$ with $n>2$ without admitting a spectrum. Surprisingly, the
conjecture is still open in dimension 1 and 2. The case $n=1$, once restricted
to the integer $\mathbb{Z}$ line, enables to make a fruitful link between the
spectral conjecture and the tiling rhythmic canons. In order to understand
this connection, we need to show that tiling rhythmic canons can be expressed
in a natural way in spectral terms via (the set of the zeroes of) the discrete
Fourier transform associated to a given subset $A\subseteq\mathbb{Z}$. First
of all, note that, since any given finite subset $A\subseteq\mathbb{Z}$ tiling
the integer line by translation always tiles with a given period,888This
result has been shown independently by Nicolaas Govert de Bruijn [22] and
György Hajós [30] in the 1950s. we are in fact factorizing $\mathbb{Z}_{n}$
into a direct sum of two subsets, one of which is the set $A$. Formally, it
exists a set $B$ such that $\mathbb{Z}_{n}=A\oplus B$. We can therefore define
the discrete Fourier transform associated to a given subset
$A\subseteq\mathbb{Z}_{n}$ without loss of generality on the tiling process.

######  Definition 1.1.

Let $A\subseteq\mathbb{Z}_{n}$ be a subset of $\mathbb{Z}_{n}$, then the
discrete Fourier transform $F_{A}$ is the discrete Fourier transform of its
characteristic function, i.e., the map from $\mathbb{Z}_{n}$ to $\mathbb{C}$
which sends every element $t\in\mathbb{Z}_{n}$ to $\sum{\rm
e}^{\frac{-2\pi{\rm i}kt}{n}}$, where the sum is done for all elements $k\in
A$.

We will denote with $Z_{A}$ the set $Z_{A}=\\{t\in\mathbb{Z}_{n}\mid
F_{A}(t)=0\\}$ of the zeroes of $F_{A}$. We recall the following property
which expresses the factorization of a cyclic group in terms of the zeroes of
the Fourier transforms of the respective factors (see [4] for a proof):

######  Proposition 1.2.

Let $A$, $B$ be two subsets of $\mathbb{Z}_{n}$. Then $\mathbb{Z}_{n}=A\oplus
B$ if and only if $Z_{A}\cup Z_{B}=\mathbb{Z}_{n}\backslash\\{0\\}$ and
$\\#A\times\\#B=n.$

Fuglede’s conjecture (in dimension 1 and restricted to the discrete case)
turns out to be naturally linked to homometry by virtue of the following
observation: if a rhythmic pattern tiles the musical line by translation
(i.e., it generates a tiling rhythmic canon), so does any rhythmic pattern
that is homometric to the initial one. Moreover, from a spectral conjecture
perspective, one only has to consider tiling canons associated to
factorizations of a cyclic group as a direct sum of two non-periodic subsets
(i.e., Vuza canons), since, as it has been shown by Emmanuel Amiot, all the
other canons verify Fuglede’s conjecture [5]. The interested reader can refer
to Greta Lanzarotto’s Ph.D. Thesis [35] for the most recent theoretical and
computational account of Vuza Canons as potential candidates for approaching
Fuglede’s spectral conjecture as well as other open problems, such as Coven
and Meyerowitz T2 conjecture.999See [21]. For a description of tiling canons
as a key to approach open mathematical conjectures, see the survey [9] as well
as the special issue of J. Math. Music entitled “Tiling problems in music”
[10].

####  1.2.2 A special case of homometric structures: Babbitt’s theorem

Milton Babbitt’s hexachord theorem is surely one of the most celebrated
results in “mathemusical” research. It expresses a property of invariance of
the intervallic content of a chord with respect to its complement. Cyclic
groups are traditionally used to represent musical structures such as chords,
melodies or rhythms. In particular, a hexachord is a subset of $6$ notes over
the $12$ of the chromatic scale

$\mathbb{Z}/12\mathbb{Z}\equiv\\{C,C\\#,D,\ldots,B\\}.$

Babbitt realized that the same _intervals_ appear with the same _multiplicity_
in the complementary hexachord $A^{c}$ as in the hexachord $A$. The
hexachordal theorem simply states that $A$ and $A^{c}$ of measure $1/2$ are
homometric. Note that this is a special case of a property – known as
Z-relation in the musical set-theoretical tradition [24] – which can hold for
sets $A$ and $B$ of equal cardinality and having the same interval content
without being each other’s complement. More generally, given a $n$-tone equal
temperament which divides the octave into $n$ equal parts, it is possible to
find $m$-tuples of sets of equal cardinality which are homometric. In the
quarter-tone equal temperament, for example, there are three 12-tuples of sets
of cardinality equal to 12. This is at the present the highest value of
homometric sets that have been obtained by computational methods [51].
Surprisingly, although many composers made use either consciously or
unconsciously of Z-relation in their composition [29], it must be noted that
the question of the perceptual relevance of homometry still remains unanswered
even in the simple case of couples of Z-related chords in the twelve-tone
equal temperament. Nevertheless, the classification of homometric structures
of different cardinalities within a given cyclic group $\mathbb{Z}_{n}$, i.e.,
generalized Z-related sets – is an interesting combinatorial problem that
constitutes an active field of research in computational musicology and micro-
tonal composition [32, 51].

Whereas some examples of homometric structures can be found in the research
carried on in the 1940s in the field of crystallography (see [20, 45, 48]),
the American music-theorist and composer Milton Babbitt made systematic use of
Z-related musical structures within his serial technique, in which he applied
the hexachordal theorem to the twelve-tone equal temperament
$\mathbb{Z}/12\mathbb{Z}$ [16]. Since Babbitt’s original formulation [16] and
its first complete proof by Ralph Hartzler Fox [25], the hexachordal theorem
has been discussed, reproved and sometimes generalized by several authors
including David Lewin [38], Howard J. Wilcox [52], Steven K. Blau [18],
Daniele Ghisi [28], Emmanuel Amiot [2] and Brian J. McCartin [42]. One also
finds a full characterization of simple graphs exhibiting the hexachordal
property in [1] by T.A. Althuis and F. Göbel. The hexachordal property has
also been studied by David Lewin in [39] within the so-called
_transformational music theory_ and in particular in the context of
(transformation) groups $T$ acting on a musical space $S$ in a simply
transitive way (the so-called generalized interval systems). The uniquely
determined group element mapping $x$ to $y$ is called _interval_ and denoted
by $\mathrm{Int}(x,y)$. By choosing $e\in S$ as a reference, we can identify
$T$ with $S$ through

$x\in S\mapsto\mathrm{Int}(e,x)\in T.$

In this way, the group action of $\mathrm{Int}(x,y)$ is identified with the
left product by $z\mapsto\big{(}yx^{-1}\big{)}z$. The triplet
$(S,T,\mathrm{Int})$ was called _generalized interval system_ or _GIS_ by
David Lewin. This is the language for the proof of the hexachordal theorem for
locally compact groups presented in [40, 41].

In the remaining part of this article, we discuss the new general framework
that we introduced in the short companion paper [13], where the statements and
their proofs are given in the framework of metric spaces with a probability
measure and we illustrate our generalized hexachordal theorems with original
constructions and examples. It must be observed that the approach we describe
in this paper does not entirely fit within the previous diagram describing
what we called the “mathemusical” dynamics, since there is no need, in
general, to go through a computer-aided model to connect the musical original
problem and the final mathematical result. Moreover, these constructions
remain, at the present, highly speculative and the generalized results still
need to find the appropriate musical interpretation which is far beyond the
scope of the present study.

##  2 Generalizing Babbitt’s hexachordal theorem

In our recent contribution [13, Theorem 1.3], we give a full characterization
of the spaces that host the hexachordal property. There the word _interval_ is
given a metric meaning similar to [1] and the _multiplicity_ is measured
through a new probabilistic approach. The heart of the very short proof is the
same as [25]. We also generalize our result to more symbolic spaces [13,
Theorems 4.2 and 4.5]. These results are recalled in Section 2.2.

###  2.1 Naive illustration of the hexachordal phenomenon
and heuristic proof in the case of the sphere $\boldsymbol{\mathbb{S}^{2}}$

Among the closest extensions of the original hexachordal theorem on the
discrete circle $\mathbb{Z}/12\mathbb{Z}$ are the ones to the continuous
circle $\mathbb{S}^{1}$ and to the spheres $\mathbb{S}^{3}$ and
$\mathbb{S}^{7}$ [17, 40, 41]. Our recent paper [13] adds the other spheres
$\mathbb{S}^{d}$ to the list. Since it speaks a lot to the inhabitants of the
Earth, in this paragraph we would like to naively illustrate the hexachordal
property on $\mathbb{S}^{2}$ and suggest a statistical proof for it. Our
discussion should also be designed for non mathematician readers. To describe
a set $A\subset\mathbb{S}^{2}$, one can look at the mean distance between two
points picked randomly from $A$, where we assume that $\mathbb{S}^{2}$ is
equipped the surface measure $\mu$ and the chord distance $d$. By mean
distance, we mean

$M_{1}(A)=\mu(A)^{-2}\iint_{A\times A}d(x,y)\mathrm{d}\mu(x)\mathrm{d}\mu(y)$

the value corresponding to $p=1$ in the range of the power mean distances
$(M_{p}(A))_{p>0}$, where

$M_{p}(A):=\left(\frac{1}{\mu(A)^{2}}\iint_{A\times
A}d(x,y)^{p}\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)\right)^{1/p}.$

It is clear that rotating $A$ on $\mathbb{S}^{2}$ does not modify $M_{1}(A)$.
To state the obvious the other power mean distances – as for instance the
quadratic mean distance $M_{2}(A)$ – are also conserved after rotation.
Finally, the (essential) diameter of $A$ is obviously conserved – besides the
fact it is $\lim_{p\to\infty}M_{p}(A)=\sup_{p}M_{p}(A)$. Nothing surprising in
all that: the set $A$ is “the same” before and after rotation.

As we have proved in [13], if $\mu(A)=\mu\big{(}\mathbb{S}^{2}\big{)}/2$
another – this time nontrivial – operation conserves $M_{1}(A)$ and any other
power mean $M_{p}(A)$, namely the complementary map

$\displaystyle A\mapsto A^{c}:=\mathbb{S}^{2}\setminus A.$

This invariance by the complementary map is the signature of the hexachordal
theorem. Recall from Section 1.2.2 that we are looking for invariance of the
multiplicity of the intervals. The proper notion for this is the one of
_distribution_ (or _law_) of the random distance between two independent
points. Recall that the law of a random variable $X\colon\Omega\to F$ from a
probability space $(\Omega,\mathcal{A},\mathbb{P})$ to a measured space
$(F,\mathcal{B})$ is a probability measure $\mathbb{P}^{X}$ on $F$ defined by
${\mathbb{P}^{X}=\mathbb{P}\circ X^{-1}}$. Concretely $\mathbb{P}^{X}(A)$
represents the probability for $X$ to be in $A$. By definition, it is

$\mathbb{P}\big{(}X^{-1}(A)\big{)}=\mathbb{P}(\\{\omega\in\Omega\colon
X(\omega)\in A\\})$

and is usually denoted by $\mathbb{P}(X\in A)$. Our proof of this invariance
in [13] is very basic – see below for the heuristic – and is similar to the
one of Fox [25]. Note that the invariance of the distribution implies the one
of the power moments. The reason why we first mentioned the invariance of
$M_{p}$ is that for many readers mean distances are more intuitive than
probability distribution. However, notice that in the special case of
$\mathbb{S}^{2}$, where the diameter is finite both are equivalent since
according to the theory of the _Hausdorff moment problem_ the distribution of
the random distance is uniquely characterized by the power means sequence
$(M_{p}(A))_{p\in\mathbb{N}}$. To give another concrete and, we
think,illustration of what distribution invariance implies one can consider
$A$ to be the set of points with latitude between $-30^{\circ}$ and
$30^{\circ}$, as illustrated in Figure 3 (in white). We have the following
application: the probability that the random distance is smaller than
$\sqrt{2}$ (the distance between the poles and the equator) is the same for
$A$ and $A^{c}$. While with simple geometric considerations on the set $A^{c}$
one proves that this probability is $1/2$, a direct computation on $A$ may
first appear out of reach.

Figure 3: Two points randomly picked in the bright region of the sphere have
distance distributed equally as the one between points picked in the dark
region (made of two caps).

Before giving a heuristic proof of the $\mathbb{S}^{2}$ case, let us stress
the relevance of generalizing the hexachord theorem to this specific case. As
discussed in [41], the case of the sphere $\mathbb{S}^{2}$ is a challenging
one with respect to traditional music-theoretical constructions since David
Lewin’s notion of generalized interval systems (GIS) and related interval
content are meaningless. In fact, as the authors recalled, unlike the
continuous circle $\mathbb{S}^{1}$and the spheres $\mathbb{S}^{3}$ and
$\mathbb{S}^{7}$, there is no group structure and a Haar measure which can be
provided on the sphere $\mathbb{S}^{2}$ which will be compatible with its
natural topology. Our general framework enables to overcome this difficulty
but at the same time it asks for new musical interpretations that cannot be
based anymore on well-known music-theoretical constructions such the GIS.

Let $(x_{1},y_{1}),\ldots,(x_{N},y_{N})$ be a large sample of pairs of random
points from $\mathbb{S}^{2}$. Intuition (backed up by theorems such as the law
of large numbers) tells us that we can estimate $M_{p}(A)$ and $M_{p}(A^{c})$
as well as other metric quantities depending on the random distance by sorting
pairs $(x_{k},y_{k})$ such that $x_{k}\in A$ and $y_{k}\in A$ on the one side
and pairs satisfying $(x_{k},y_{k})\in(A^{c})^{2}$ on the other side. Note
that the expected size of these subsamples is $N/4$. We claimed that the
distribution of the distance is conserved by $A\mapsto A^{c}$ so our two
samples $S_{1}=\big{\\{}d(x_{k},y_{k})\colon(x_{k},y_{k})\in A^{2}\big{\\}}$
and $S_{2}=\big{\\{}d(x_{k},y_{k})\colon(x_{k},y_{k})\in(A^{c})^{2}\big{\\}}$
should have the same statistical aspect when $N$ tends to infinity. The deep
reason for this phenomenon is only revealed once a third sample is added to
the two others simultaneously, namely
$S_{3}=\\{d(x_{k},y_{k})\colon(x_{k},y_{k})\in A\times A^{c}\\}$. Doing this,
the first sample $S_{1}$ becomes $S_{1}\cup S_{3}$ that collects the distances
$d(x_{k},y_{k})$ with $x_{k}\in A$ and $y_{k}\in A\cup A^{c}=\mathbb{S}^{2}$ –
i.e., there is no restriction on $y_{k}$ – whereas $S_{2}$ becomes $S_{2}\cup
S_{3}$, where the pairs $(x_{k},y_{k})$ are in $\mathbb{S}^{2}\times A^{c}$ –
again one point, here $x_{k}$, is free. Now it appears that in both cases we
are considering the typical random distance to one given point of the sphere.
The fact that this point, $x_{k}$ (respectively $y_{k}$) is in $A$
(respectively $A^{c}$) has no incidence on the random distance. Therefore, the
two augmented samples have the same properties (up to variations due to the
sampling) and since we added the same sample to both, so do the initial
samples. This concludes our heuristic proof.

In [13], we directly implement this strategy with probability measures in
place of random samples and metric spaces more general than $\mathbb{S}^{2}$.
One can check above that the geometric property of $\mathbb{S}^{2}$ that was
important for the proof and this scheme of proof is that the random distance
to a fixed point does not depend of this point. It is stated in [13,
Definition 1.2] in purely geometric terms through the constant volume
condition (CVC).

######  Definition 2.1 (constant volume condition).

A metric measure space $(\mathfrak{X},d,\mu)$ is said to satisfy the _constant
volume condition_ if there exists a function $\rho$ on $[0,\infty)$ such that
for any center $x\in\mathfrak{X}$ and radius $r\in[0,\infty)$ the closed ball
$\mathcal{B}(x,r)=\\{y\in\mathfrak{X}\colon d(x,y)\leq r\\}$ has measure
$\rho(r)$. In an equivalent way,

$\displaystyle\forall x,y\in\mathfrak{X},\ \forall r\geq
0,\qquad\mu(\mathcal{B}(x,r))=\mu(\mathcal{B}(y,r)).$ (CVC)

In the following, we sometimes call _CVC space_ a metric probability space
which satisfies the constant volume condition.

###  2.2 Statement of three generalized hexachordal theorems

We recall the statement of the three theorems we proved in [13].

######  Theorem 2.2 (hexachordal theorem for metric probability spaces).

Let $(\mathfrak{X},d,\mu)$ be a metric probability space. Assume that it
satisfies the constant volume condition. Then for every Borel set $A$ of
$\mu$-measure $1/2$, with notation $A^{c}=\mathfrak{X}\setminus A$ one has

$\displaystyle\mu^{2}\big{\\{}(x,y)\in A^{2}\colon d(x,y)\in
E\big{\\}}=\mu^{2}\big{\\{}(x,y)\in(A^{c})^{2}\colon d(x,y)\in E\big{\\}}$
(Hex)

for every open subset $E\subset[0,\infty)$, where $\mu^{2}$ is the product
measure $\mu\times\mu$ used for the $($measurable$)$ sets of pairs
$(x,y)\in\mathfrak{X}^{2}$.

From the perspective of homometry theory, as interpreted within this new
probabilistic framework, two subsets $A$ and $B$ are homometric if for every
$E$

$\displaystyle\mu^{2}\big{(}\big{\\{}(x,y)\in A^{2}\colon d(x,y)\in
E\big{\\}}\big{)}=\mu^{2}\big{(}\big{\\{}(x,y)\in B^{2}\colon d(x,y)\in
E\big{\\}}\big{)}.$

Therefore, Theorem 2.2 stated that subsets $A$ of mass $1/2$ are homometric to
$A^{c}$. This definition generalizes to homometric probability measure spaces
as follows. The spaces $(\mathfrak{X},d_{1},\mu)$ and
$(\mathfrak{Y},d_{2},\nu)$ are homometric if

$\displaystyle\mu^{2}\big{(}\big{\\{}(x,y)\in\mathfrak{X}^{2}\colon
d_{1}(x,y)\in
E\big{\\}}\big{)}=\nu^{2}\big{(}\big{\\{}(x,y)\in\mathfrak{Y}^{2}\colon
d_{2}(x,y)\in E\big{\\}}\big{)}$

for every Borel $E\subset\mathbb{R}$.

Now we recall the statements of two more general theorems [13, Theorems 4.2
and 4.5]. We need to recall that a _balanced decomposition_
$(\mu_{0},\mu_{1})$ of $\mu$ is a pair of probability measures such that
$\mu_{0}+\mu_{1}=2\mu$ (see [13, Definition 4.1]).

######  Theorem 2.3 (characterization for metric probability spaces).

Let $(\mathfrak{X},d,\mu)$ be a metric probability space. The following
properties are equivalent:

  1. $({\rm CVC}^{\prime})$

There exists a set $\mathfrak{X}^{\prime}\subseteq\mathfrak{X}$ of full
measure for $\mu$ such that the constant volume condition is satisfied on
$(\mathfrak{X}^{\prime},d,\mu)$.

  2. $({\rm Ind})$

For any independent random variables $X$ and $Y$ of law $\mu$ and $D=d(X,Y)$,
the random variables $X$, $Y$ and $D$ are pairwise independent.

  3. $({\rm Hex}^{\prime})$

For every balanced decomposition $(\mu_{0},\mu_{1})$ of $\mu$ and two random
triples $(X_{i},Y_{i},D_{i})_{i=0,1}$, where for every $i$, $(X_{i},Y_{i})$ is
a pair of independent random variables of law $\mu_{i}$ and
$D_{i}=d(X_{i},Y_{i})$, we have the equality on distributions
$\mathbb{P}(D_{0}\in\cdot)=\mathbb{P}(D_{1}\in\cdot)$.

######  Theorem 2.4 (characterization for abstract probability spaces).

Let $(\mathfrak{X},\mathcal{F},\mu)$ be a probability space and $f$ a
measurable symmetric function into a measured space
$(\mathcal{M},\mathfrak{M})$. The following properties are equivalent:

  1. $({\rm Ind})$

For any independent random variables $X$ and $Y$ of law $\mu$ and $F=f(X,Y)$,
the random variables $X$, $Y$ and $F$ are pairwise independent.

  2. $({\rm Hex}^{\prime})$

For every balanced decomposition $(\mu_{0},\mu_{1})$, considering the triples
$(X_{0},Y_{0},F_{0})$ and $(X_{1},Y_{1},F_{1})$, where for $i=0,1$ the pair
$(X_{i},Y_{i})$ is made of independent random variables of law $\mu_{i}$ and
$F_{i}=f(X_{i},Y_{i})$, we have equality of both distributions,
$\mathbb{P}(F_{0}\in\cdot)=\mathbb{P}(F_{1}\in\cdot)$ as measures on
$\mathcal{M}$.

  3. $({\rm Hex}^{\prime\prime})$

For any balanced decompositions $(\mu_{0},\mu_{1})$ and $(\nu_{0},\nu_{1})$,
where for $i=0,1$, $X_{i}$ has law $\mu_{i}$, $Y_{i}$ has law $\nu_{i}$ and
$F_{i}=f(X_{i},Y_{i})$, we have equality of both distributions
$\mathbb{P}(F_{0}\in\cdot)=\mathbb{P}(F_{1}\in\cdot)$.

Moreover, if $f$ is no longer supposed to be symmetric $({\rm
Ind})\Leftrightarrow({\rm Hex}^{\prime\prime})$ still holds as well as $({\rm
Hex}^{\prime\prime})\Rightarrow({\rm Hex}^{\prime})$.

The next section is related with the two first generalized hexachordal
theorems since it is devoted to the CVC spaces. Section 4 is about Theorem
2.4. In this case, the “interval” function $f$ is not necessary a distance but
may for instance be $(x,y)\mapsto x^{-1}\cdot y$ in the case of a group.

##  3 Metric probability spaces satisfying (CVC)

In this section, we provide examples of spaces satisfying the constant volume
condition whose definition is recalled in Definition 2.1. Recall that it is a
sufficient and almost necessary condition for the hexachordal theorem
(Theorems 2.2 and 2.3). In Section 3.1, we report on some examples obtained
from the simple fact that transitive spaces satisfies the CVC. See also Remark
3.2 for the intriguing nontransitive spaces with CVC. In Section 3.2, we
report some easy constructions to create new CVC spaces from established CVC
spaces.

###  3.1 Discrete and continuous CVC spaces

A classical group-theoretical framework which ensures (CVC) is the one of
transitive group actions. We consider the metric measure spaces
$(\mathfrak{X},d,\mu)$ such that for every $x$ and $y$ in $\mathfrak{X}$ there
exists a map $f\colon\mathfrak{X}\to\mathfrak{X}$ that satisfies

  * •

$f(x)=y$,

  * •

$d(f(z),f(z^{\prime}))=d(z,z^{\prime})$ for every
$z,z^{\prime}\in\mathfrak{X}$, so that $f$ is an isometry,

  * •

$f_{\\#}\mu=\mu$, where $f_{\\#}\mu:=\mu\big{(}f^{-1}(\cdot)\big{)}$ is the
law of $f(X)$ if $X$ has law $\mu$.

The constant volume condition follows:

$\rho_{x}(r)=\mu(\mathcal{B}(x,r))=\mu\big{(}f^{-1}\mathcal{B}(f(x),r)\big{)}=\mu(\mathcal{B}(f(x),r))=\rho_{y}(r).$

######  Definition 3.1 (transitive space).

We call the metric measure space $(\mathfrak{X},d,\mu)$ a _transitive space_
if the group $G$ of maps $f$ as above (isometries preserving the measure) acts
transitively on $\mathfrak{X}$, i.e., such that for every $x,y\in\mathfrak{X}$
there exists $f\in G$ such that $f(x)=y$.

Let us first focus on the class of transitive examples satisfying (CVC) in the
discrete setting of finite graphs with their counting measure. Note that in
this case the condition $f_{\\#}\mu=\mu$ is automatically satisfied because
$f$ is one-to-one. Graphs satisfying the two first conditions are usually
called (vertex-)transitive graphs and $f$ a graph isomorphism. Many of these
graphs are the Cayley graph of a finite group. Recall that if a group
$\mathfrak{X}$ is generated by a finite system of generators
$\Sigma\subseteq\mathfrak{X}$, the Cayley graph attached to
$(\mathfrak{X},\Sigma)$ is the graph with vertices $\mathfrak{X}$ and edges
the pairs $(x,y)$ such that $x^{-1}y\in\Sigma$ or $y^{-1}x\in\Sigma$. As
usual, we denote this adjacency relation by $x\sim y$. Let us check that these
spaces $\mathfrak{X}$ with the counting measure and the path distance are of
transitive type and hence satisfy the constant volume condition. Given $x$ and
$y$ we choose for $f$ the translation defined by $\tau_{v}\colon z\mapsto vz$,
where $v=yx^{-1}$, so that $f(x)=y$. It is an isometry because
$\tau_{v}(z)\sim\tau_{v}(z^{\prime})$ if and only if $z\sim z^{\prime}$, which
follows from $\tau_{v}(z)^{-1}(\tau_{v}(z^{\prime}))=z^{-1}z^{\prime}$.
Finally, recall that it preserves the counting measure since it is one-to-one.
Basic examples of this type (i.e., finite Cayley graphs) are

  * •

The symmetric group $\mathfrak{S}(n)$ with, for instance, for $\Sigma$ the set
of transpositions.

  * •

The group
$(\mathbb{Z}/n_{1}\mathbb{Z})\times\cdots\times(\mathbb{Z}/n_{k}\mathbb{Z})$
with $\Sigma=\\{(0,\ldots 0,\pm 1,0,\ldots,0)\\}$. Note that for
$n_{1}=\cdots=n_{k}=2$, we find the so-called hypercube $\\{0,1\\}^{k}$. For
$k=1$ and $n_{1}=12$, we recover the chromatic scale
$\mathbb{Z}/12\mathbb{Z}$.

Note that a vertex-transitive graph may not be the Cayley graph attached to
some $(\mathfrak{X},\Sigma)$, a counterexample being the _Petersen graph_ (a
famous graph with 10 vertices and 15 edges) another one the graphs made of the
edges and vertices of the _dodecahedron_ , _icosahedron_ and the _truncated
icosahedron_

We list now some continuous transitive examples among the most basic:

  * •

The (hyper)torus
$\mathbf{T}^{d}=\mathbb{S}^{1}\times\cdots\times\mathbb{S}^{1}$ of dimension
$d$ with its normalized volume (among other tori).

  * •

Any sphere $\mathbb{S}^{d}$ or product of spheres with their normalized
volumes.

  * •

The Klein bottle. When the Euclidean space $\mathbb{R}^{2}$ is made a quotient
through the group spanned by the translation $(x,y)\mapsto(x+1,y)$ and the
glide reflexion $(x,y)\mapsto(1-x,y+1)$101010The elements of this group are
the isometries of the form $(x,y)\mapsto(k\pm x,y+l)$, where $\pm$ is $+$ if
and only if $l$ is even. the translations of $\mathbb{R}^{2}$ remains
isometries that are acting transitively. Topologically the quotient space a
Klein bottle with fundamental domain the square $[0,1)\times[0,1)$ (the lower
and upper sides are identified after inversion of the orientation). Any Klein
bottle of volume 1 obtained in a similar way will be transitive, satisfy the
constant volume condition and hence host the hexachordal property. Note that
two such Klein bottles are generally not isometric.

  * •

For more exotic examples, we can think to Albanese tori. Their topology is
different from the one of usual tori.

######  Remark 3.2 (nontransitive CVC spaces).

As shown in [1, 13], there also exists nontransitive CVC spaces that are
graphs. It is unknown whether there exists a nontransitive Riemannian manifold
of finite measure with the CVC. We proved in [13, Section 3] that it does not
exist for surfaces. Note also that considering open submanifolds with infinite
Riemannian volume and at most one singularity it has been shown in [34] that
the cone $C=\big{\\{}x\in\mathbb{R}^{4}\colon
x_{4}^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\big{\\}}$ with the (Euclidean induced
by $\mathbb{R}^{4}$) chord distance $d_{\mathbb{R}^{4}}$ and the induced
volume (of dimension 3) $\mu$ satisfies CVC. This space
$\big{(}C,d_{\mathbb{R}^{4}}\,\mu\big{)}$ is nontransitive: it possesses for
instance a singularity at 0. Moreover, the volume of the balls of given radius
is the same as for $\mathbb{R}^{3}$, so that $C$ and $\mathbb{R}^{3}$ can be
qualified homometric in a general sense. A uniqueness result for $C$ among
subsets of Euclidean spaces is also established in [34].

###  3.2 Constructions with CVC spaces

In the following, we denote by $\rho$ the _$($ constant$)$ volume function_ of
a metric space $(\mathfrak{X},d,\mu)$ that satisfies (CVC). It is defined by

$\rho(r)=\mu(\mathcal{B}(x_{0},r)),$

where $\mathcal{B}(x,r)$ is the closed ball of radius $r\geq 0$ and center
$x$. Here $x_{0}$ is some (or, due to CVC, any) point of $\mathfrak{X}$.

In the case of probability spaces, $\rho$ is the cumulative distribution of
$D=d(X,Y)$, where $X$ and $Y$ are independent of law $\mu$, i.e.,

$\displaystyle\rho(r)$ $\displaystyle=\mathbb{P}(d(X,Y)\leq
r)=\iint\mathds{1}(d(x,y)\leq r)\mu\otimes\mu({\rm d}x,{\rm d}y)$
$\displaystyle=\mu^{2}\big{(}\big{\\{}(x,y)\in\mathfrak{X}^{2}\colon
d(x,y)\leq r\big{\\}}\big{)}$

even though $(\mathfrak{X},d,\mu)$ does not satisfies the CVC. Since
cumulative distribution functions characterize the distribution of a random
variable, two metric probability spaces are homometric if and only if they
have the same volume function. Note, for instance, that Example 3.2 in [13]
and $\mathbb{Z}/7\mathbb{Z}$ with the relation $(x\sim y\Longleftrightarrow
y-x\in\\{-1,1,-3,3\\})$ have the same volume function but are not isometric:
the first is nontransitive and the second is transitive.

######  Example 3.3 (products).

Let $(\mathfrak{X}_{1},d_{1},\mu_{1})$ and $(\mathfrak{X}_{2},d_{2},\mu_{2})$
be two spaces satisfying (CVC) with volume functions $\rho_{1}$ and
$\rho_{2}$, respectively. Then the product space
$\mathfrak{X}:=\mathfrak{X}_{1}\times\mathfrak{X}_{2}$ also satisfies it, with
the product measure $\mu:=\mu_{1}\times\mu_{2}$. Several choices are possible
to combine the distances. We focus on the $\ell^{p}$ norms of
$(d_{1},d_{2})\in\mathbb{R}^{2}$, where $p\in[1,\infty]$:

  * $\ell^{\infty}$

We can set
$d((x_{1},y_{1}),(x_{2},y_{2}))=\max(d_{1}(x_{1},y_{1}),d_{2}(x_{2},y_{2}))$.
Let us comment on the example of the product of two finite graphs with their
path distances. The resulting space is
$\mathfrak{X}_{1}\times\mathfrak{X}_{2}$ with the path distance resulting of
the so-called _strong product_ of the two graphs. In fact,
$d((x_{1},y_{1}),(x_{2},y_{2}))\leq 1$ if and only if $d((x_{1},y_{1})\leq 1$
and $d((x_{2},y_{2})\leq 1$. Denoting by $(x_{1},y_{1})\simeq(x_{2},y_{2})$
the relation $\\{(x_{1},y_{1})\sim(x_{2},y_{2})$ or
$(x_{1},y_{1})=(x_{2},y_{2})\\}$, it follows that
$(x_{1},y_{1})\simeq(x_{2},y_{2})$ if and only if $x_{1}\simeq x_{2}$ and
$y_{1}\simeq y_{2}$. One can check that (CVC) is satisfied for
$\rho=\rho_{1}\times\rho_{2}$.

  * $\ell^{p}$

We can set
$d^{p}((x_{1},y_{1}),(x_{2},y_{2}))=d_{1}(x_{1},y_{1})^{p}+d_{2}(x_{2},y_{2})^{p}$
and obtain for this choice the constant volume function
$\rho(r)=\int_{[0,r]}\rho_{2}\big{(}(r^{p}-t^{p})^{1/p}\big{)}d\rho_{1}(t)$.

  * $\ell^{1}$

The product of two graphs is the so-called _Cartesian product_ for which
$(x_{1},y_{1})\sim(x_{2},y_{2})$ if and only if ($x_{1}=x_{2}$ and $y_{1}\sim
y_{2}$) or ($y_{1}=y_{2}$ and $x_{1}\sim x_{2}$).

  * $\ell^{2}$

If $\mathfrak{X}_{1}$ and $\mathfrak{X}_{2}$ are isometrically embedded in
Euclidean spaces, so is the product with the $\ell^{2}$ distance. For
instance, the hexachordal phenomenon can be observed on
$\mathbb{S}^{1}\times\\{0,1\\}\subseteq\mathbb{R}^{2}\times\mathbb{R}=\mathbb{R}^{3}$.

This example inductively extends to product spaces
$\mathfrak{X}:=\mathfrak{X}_{1}\times\cdots\times\mathfrak{X}_{n}$ and the
product measure $\mu_{1}\otimes\cdots\otimes\mu_{n}$. This corresponds to the
setting of independent random variables ${X=(X_{1},\ldots,X_{n})}$ and
$Y=(Y_{1},\ldots,Y_{n})$ that we can moreover equip with the Hamming distance
$D=d(X,Y)=\sum_{i=1}^{n}\mathds{1}(X_{i}\neq Y_{i})\times a_{i}$ with
$a_{i}>0$, $i=1,\ldots,n$. For an illustration of Theorem 2.2, one can, for
instance, consider the uniform head and tail space
$\mathfrak{X}=\\{h,t\\}^{5}$ and apply to $A$ “at least three consecutive
tosses are the same” since it has probability 1/2, as one can check.
(Moreover, one can check that $A$ and $A^{c}$ are generically not isometric.)

######  Example 3.4 (union of two spaces with the same constant volume
function).

We consider for $i=1,2$ two spaces $(\mathfrak{X},d_{i},\mu_{i})$ satisfying
the constant volume condition for the same function $\rho$. We assume moreover
that the two spaces are bounded. As noticed in Remark 3.2, they can be
different. Define $\mathfrak{X}$ the disjoint union
$\mathfrak{X}_{1}\sqcup\mathfrak{X}_{2}$ with probability measure
$\mu=(1/2)(\mu_{1}+\mu_{2})$ and distance defined by

$d(x,y)=\begin{cases}d_{i}(x,y)&\text{if }x,y\in\mathfrak{X}_{i}\text{ for
some }i,\\\ L&\text{otherwise.}\end{cases}$

The constant volume condition and $({\rm Hex})$ are satisfied for any $L\geq
0$. This makes sense even though $d$ is not a distance (see Theorem 2.4) but
in order to save the triangle inequality we have to require that for every
$i=1,2$ the distance between any two points of $\mathfrak{X}_{i}$ is less than
$2L$.

######  Example 3.5 (graphs whose points are replaced by metric spaces).

Let $(G,d_{0})$ be a finite graph with the constant volume condition for the
counting measure. We scale it so that adjacent points have distance $L$. We
replace $G=\bigcup_{i=1}^{N}\\{x_{1}\\}$ by
$\mathfrak{X}=\bigsqcup_{i=1}^{N}\mathfrak{X}_{i}$ a family of metric spaces
$(\mathfrak{X}_{i},d_{i},\mu_{i})$ with diameter smaller than $2L$ and
satisfying (CVC) with moreover the same constant volume function,
$\rho_{1}=\cdots=\rho_{N}$. On $\mathfrak{X}$ we set $d(x,y)=d_{i}(x,y)$ if
$x,y\in\mathfrak{X}_{i}$ and $d(x,y)=d_{0}(x_{i},x_{j})$ if
$x\in\mathfrak{X}_{i}$, $y\in\mathfrak{X}_{j}$, $i\neq j$. It can be checked
that the resulting space satisfies the constant volume condition. A special
case is Example 3.4.

######  Example 3.6.

We have indicated examples of metric probability spaces $(\mathfrak{X},d,\mu)$
satisfying (CVC) such that $D$ is an absolutely continuous or a discrete
random variable. With $\\{0,1\\}\times\mathbb{S}^{1}$ in Example 3.3 if one
takes $\ell_{\infty}$ and the Examples 3.4 and 3.5 we see the possibility for
$D$ to have both a nontrivial atomic and absolutely continuous part. We
introduce now the situation of a space with the (CVC) such that $D$ is diffuse
but not absolutely continuous. Precisely its law is the Cantor law and its
cumulative distribution function $\rho$ is the Devil’s staircase. For
$\mathfrak{X}$ we take the sequences $x=(x_{1},x_{2},\ldots)$ with
$x_{i}\in\\{0,1\\}$ for every $i\geq 1$. The measure is the one of head/tail
model, i.e., we weight each digit with 1/2 independently. For the distance
between $x$ and $y$, we set
$d(x,y)=\sum_{i=1}^{\infty}|x_{i}-y_{i}|(2/3^{i})$. Note that it corresponds
to a $\ell^{1}$ distance on an infinite product $\\{0,1\\}^{\mathbb{N}^{*}}$
weighted by a scaling factor $(2/3^{i})$ on the $i$-th coordinate.

##  4 Remarks on the theorems

###  4.1 Remark on weakening $\boldsymbol{\mu(A)=1/2}$ in Theorem 2.2

Generalized versions of Babbitt’s hexachordal theorem have been given, where
the subset of interest and its complementary do not have the same size [42].
For Theorem 2.2, this translates into $A$ not having $\mu$-measure $1/2$ (and
hence neither has $A^{c}$). The principle of considering $A\times A^{c}$ is
still useful, but does not simplify as well in this more general case. Namely,
for any $r\in[0,+\infty]$, we have

$\mu^{2}\big{\\{}(x,y)\in A^{2}\colon d(x,y)\leq
r\big{\\}}-\mu^{2}\big{\\{}(x,y)\in(A^{c})^{2}\colon d(x,y)\leq
r\big{\\}}=\rho(r)[\mu(A)-\mu(A^{c})].$

However, as the two sets $A^{2}$ and $(A^{c})^{2}$ have different measures,
this generalization does not carry over neatly to the probabilistic
formulations of Theorems 2.3 and 2.4. Finally, note that $\\{d(x,y)\leq r\\}$
in the left-hand side can be replaced by $\\{d(x,y)\in E\\}$ if one also
replaces $\rho(r)$ by $\int_{0}^{\infty}\mathds{1}_{E}(r){\rm d}\rho(r)$ in
the right-hand side.

###  4.2 Remarks on $\boldsymbol{({\rm Ind})}$ in Theorem 2.4

Whereas in Theorems 2.2 and 2.3 the law of $(X,D)=(X,d(X,Y))$ equals the law
of $(Y,D)=(Y,d(X,Y))$ and $X$ and $D$ are independent if and only if $Y$ and
$D$ are, the fact that in Theorem 2.4 the function $f$ does not have to be
symmetric (see the two last lines) makes that $({\rm Ind})$ “$X$, $Y$ and
$F=f(X,Y)$ are pairwise independent” is different from “$X$ and $F$ are
independent”. In this paragraph we comment on this difference.

To illustrate that the pairwise independence $({\rm Ind})$ is necessary we
consider $f(x,y)=y$ and observe Theorem 2.4 does not apply properly. If $X$
and $Y$ are independent, so are $X$ and ${F=f(X,Y)=Y}$. It is clear that $Y$
and $F=Y$ are not independent so that $({\rm Ind})$ is not satisfied.
Moreover, for $X_{0}$ and $Y_{0}$ independent of law $\mu_{0}$ the law of
$F_{0}$ is $\mu_{0}$ and for $X_{1}$ and $Y_{1}$ independent of law $\mu_{1}$
the law of $F_{1}=f(X_{1},Y_{1})$ is $\mu_{1}\neq\mu_{0}$. Therefore, $({\rm
Hex}^{\prime})$ is false and $({\rm Hex}^{\prime\prime})$ is also false for
$(\nu_{0},\nu_{1})=(\mu_{0},\mu_{1})$.

We continue with a corollary of the second case of Theorem 2.4 (not necessary
symmetric functions, see the last two lines). It applies, in particular, to
topological groups with a left-invariant probability measure through the
function $f(x,y)=x^{-1}\cdot y$, which is important with respect to the
existing literature. See [13, Corollary 4.6] for a direct proof.

######  Corollary 4.1 (weakening $({\rm Ind})$ in Theorem 2.4 for
antisymmetric functions $f$).

Let $(\mathfrak{X},\mathcal{F},\mu)$ be a probability space and $f$ a
measurable function defined on $\mathfrak{X}\times\mathfrak{X}$ with values in
a measurable space $(\mathcal{M},\mathfrak{M})$. Let us assume that $f$ is
antisymmetric in the sense there exists a measurable involution $i$ on
$\mathcal{M}$, i.e., a function $i\colon\mathcal{M}\to\mathcal{M}$ such that
$i\circ i(m)=m$, for every $m\in\mathcal{M}$ with $f(x,y)=i\circ f(y,x)$.

Let $X$ and $Y$ be independent random variables of law $\mu$. If $F=f(X,Y)$
and $Y$ are independent, then $X$ and $F$ are also independent so that $({\rm
Ind})$ is satisfied and $({\rm Ind})\Rightarrow({\rm Hex}^{\prime})$ and
$({\rm Ind})\Leftrightarrow({\rm Hex}^{\prime\prime})$ apply in Theorem 2.4.

Similarly if $X$ and $Y$ as well as $X$ and $F$ are independent so are $F$ and
$Y$ and $({\rm Ind})$ is satisfied.

###### Proof.

Let us assume that $(X,Y)$ and $(Y,F)$ are pairs of independent random
variables. Thus $Y$ and $i(F))=i\circ f(X,Y)=f(Y,X))$ are independent random
variables. But the law of $(Y,f(Y,X))$ is the law of $(X,f(X,Y))$. Therefore,
$X$ and $F$ are independent. The same argument to prove $({\rm Ind})$ also
works if $X$ and $F$ are independent. ∎

We now concretely illustrate Theorem 2.4 and Corollary 4.1 with
$f(x,y)=x^{-1}\cdot y$ in the special case of some $A$ of cardinality 6 in the
group $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ of cardinality 12.
We also give the invariant distribution of the distance as an illustration of
Theorems 2.2 and 2.3.

######  Example 4.2.

We consider $G=\mathbb{Z}/3Z\times\mathbb{Z}/4\mathbb{Z}$ with
$f\colon(x,y)\mapsto x^{-1}y$. For

$A=\\{(1,0),(1,2),(2,0),(2,1),(2,2),(2,3)\\}$

and its complementary set, the random variables $F_{i}$, $i=0,1$ are
distributed as follows:

$v$ | $(0,0)$ | (0,2) |  $(0,1)$ and $(0,3)$ |  $(1,0)$ and $(2,0)$
---|---|---|---|---
$\mathbb{P}(F_{i}=v)$ | $6/36$ | 6/36 | 4/36 | 2/36
$v$ |  $(1,1)$ and $(2,3)$ |  $(1,2)$ and $(2,2)$ |  $(1,3)$ and $(2,1)$
$\mathbb{P}(F_{i}=v)$ | 3/36 | 2/36 | 2/36

.

For the set of generators $\\{(\pm 1,0),(0,\pm 1)\\}$, this corresponds to the
following distribution of the distances $D_{i}$, $i=0,1$:

$r$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
$\mathbb{P}(D_{i}=r)$ | $6/36$ | 12/36 | 10/36 | 8/36

.

Beside $f(x,y)=x^{-1}\cdot y$ the most popular example of function for which
Theorem 2.4 applies is probably $f(x,y)=x\cdot y$ in the case of groups with a
left-invariant probability measure. This was explained in [13, Corollary 4.6].
Here we state it again for finite groups (where the uniform measure is
invariant) since it will be useful in the following subsections.

######  Example 4.3 (Caley tables).

Observe that $({\rm Ind})$ in Theorem 2.4 is satisfied for finite groups
$(\\!G,\\!\cdot)$ with their uniform measure $\mu$ and the function
$f(x,y)=x\cdot y$. In fact, for $X$, $Y$ independent of law $\mu$ and every
$(x,y)\in G^{2}$,

$\displaystyle\mathbb{P}(X=x,\,F=y)=\mathbb{P}\big{(}X=x,Y=x^{-1}\\!\cdot
y\big{)}=\mathbb{P}(X=x)\mathbb{P}\big{(}Y=x^{-1}\\!\cdot
y\big{)}=\mathbb{P}(X=x)\\!\times\\!\frac{1}{\\#G}.$

Therefore, $F$ is uniform on $G$ and independent from $X$. One proves in the
same way that $F$ is independent from $Y$.

In fact, the value of $(X,Y,F)$ corresponds to a uniform intersection among
$(\\#G)^{2}$ possibilities in the Cayley tables, i.e., the multiplication
table for the group $G$, where $X$ is the row and $Y$ the column. Therefore,
this type of tables permits one to visualize the hexachordal property. See,
e.g., [25, 52] where this fact is commented.

###  4.3 Remark on $\boldsymbol{({\rm Ind})\Rightarrow({\rm Hex}^{\prime})}$
in Theorem 2.4

Let us show that in Theorem 2.4 the implication $({\rm
Hex}^{\prime})\Rightarrow({\rm Ind})$ is false. To see this, let
$\mathfrak{X}$ be the space $\\{\star,\\#,\S,\bullet\\}$ equipped with the
uniform measure $\mu$. Let the values $f(X,Y)$ of a nonsymmetric function $f$
be given in Table 1, where $X$ is the uniform choice of a row, $Y$ of a column
and $F$ is the intersection. Our example is the right table of Table 1, the
left and the middle being parts of the explanation. Observe that the law of
$F=f(X,Y)$ conditioned on the choice of a row (the value of $X$) is not
constant. Hence, $({\rm Ind})$ is not satisfied. The left table corresponds to
a Cayley table as in Example 4.3 so that $({\rm Hex}^{\prime\prime})$ is
satisfies. In the middle table below, we have swapped the two last rows. We
can notice that the function is no longer symmetric. However, we stress that
$({\rm Ind})$, $({\rm Hex}^{\prime\prime})$ and $({\rm Hex}^{\prime})$ remain.
In $({\rm Hex}^{\prime})$, given a balanced decomposition $(\mu_{0},\mu_{1})$
we have for $\mathbb{P}((X_{i},Y_{i})=(a,b))=\mathbb{P}((X_{i},Y_{i})=(b,a))$
for any pair $(a,b)$ and $i=0,1$. Therefore, we conserve $({\rm
Hex}^{\prime})$ if we swap the values of $f(a,b)$ and $f(b,a)$. This is what
we did for $(\S,\star)$ between the middle and the right table. As commented
above $({\rm Ind})$ (and $({\rm Hex}^{\prime\prime})$) are no longer true
after this operation.

| $\star$ | # | § | $\bullet$
---|---|---|---|---
$\star$ | 0 | 1 | 2 | 3
# | 1 | 2 | 3 | 0
§ | 2 | 3 | 0 | 1
$\bullet$ | 3 | 0 | 1 | 2

| $\star$ | # | § | $\bullet$
---|---|---|---|---
$\star$ | 0 | 1 | 2 | 3
# | 1 | 2 | 3 | 0
§ | 3 | 0 | 1 | 2
$\bullet$ | 2 | 3 | 0 | 1

| $\star$ | # | § | $\bullet$
---|---|---|---|---
$\star$ | 0 | 1 | 3 | 3
# | 1 | 2 | 3 | 0
§ | 2 | 0 | 1 | 2
$\bullet$ | 2 | 3 | 0 | 1

Table 1: The left and middle functions satisfy $({\rm Hex}^{\prime})$ and
$({\rm Hex}^{\prime\prime})$. The right one satisfies $({\rm Hex}^{\prime})$
but not $({\rm Hex}^{\prime\prime})$.

###  4.4 Nontransitive example where Theorem 2.4 applies

In the metric setting, the constant volume condition is satisfied by many
transitive metric probability spaces for which simple formulas exist. However,
as recalled in Remark 3.2 (CVC) is not equivalent to being transitive.
Although it is known since at least 1964 [25], we would like to stress that
exactly the same happens in the symbolic setting of Theorem 2.4: not all
spaces $(\mathfrak{X},f,\mu)$ with (Ind) are isomorph to groups with the
function $f(x,y)=x\cdot y$ or $x^{-1}\cdot y$. To take the Cayley table of a
group as in Example 4.3 is only one way to have $X$, $Y$ and $F$ pairwise
independent. In the discrete setting, it was already mentioned, for instance,
in [25] that any Latin square gives rise to the hexachordal theorem. In Table
2, we represented such a Latin square with six symbols. ‘Latin’ stands for the
fact that each symbol appears once and only once on each row and column. A
rather simple argument in the theory of Latin squares/quasigroups tells that
if the following table were a Cayley table it would also be the Cayley table
of a group, where the neutral element would be both left and up in the upper
and left header, respectively. Here it means that $\heartsuit$ could be
considered as the neutral and the elements on the first row and column are the
ones of the two headers. Notice that they appear in the same order:
$\heartsuit$, $\Box$, $\triangle$, $\clubsuit$, $\diamondsuit$, $\spadesuit$.
Thus the elements on the diagonal should be the squares. Since on the diagonal
$\heartsuit$ only appears once, the group can neither be
$\mathbb{Z}/6\mathbb{Z}$,
$(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})$ nor $\mathfrak{S}(3)$
for which there are at least two elements of order 2. But these groups
precisely constitute the list of all groups of cardinality 6.

$\heartsuit$ | $\Box$ | $\triangle$ | $\clubsuit$ | $\diamondsuit$ | $\spadesuit$
---|---|---|---|---|---
$\Box$ | $\triangle$ | $\diamondsuit$ | $\spadesuit$ | $\heartsuit$ | $\clubsuit$
$\triangle$ | $\clubsuit$ | $\Box$ | $\diamondsuit$ | $\spadesuit$ | $\heartsuit$
$\clubsuit$ | $\spadesuit$ | $\heartsuit$ | $\Box$ | $\triangle$ | $\diamondsuit$
$\diamondsuit$ | $\heartsuit$ | $\spadesuit$ | $\triangle$ | $\clubsuit$ | $\Box$
$\spadesuit$ | $\diamondsuit$ | $\clubsuit$ | $\heartsuit$ | $\Box$ | $\triangle$
Table 2: A Latin square that is not a Cayley table.

###  4.5 Remark on the Patterson function

A popular setting for Theorem 2.4 is the one of separable groups with a left
invariant probability measure $\mu$. Recall, in particular, [13, Corollary
4.6] and Example 4.3. In this setting, some authors introduce the Patterson
function of $A\subset\mathfrak{X}$ defined by $\operatorname{Pat}_{A}\colon
g\in\mathfrak{X}\mapsto\mu(A\cap g\cdot A)$ and reformulate Babbitt’s theorem
as the functional equality $\operatorname{Pat}_{A}=\operatorname{Pat}_{A^{c}}$
that has to be satisfied for every $A$ of measure $1/2$. Let us explain that
this is an equality of densities (with respect to $\mu$) that corresponds to
our equality of measures $({\rm Hex})$ adapted to the case of general
functions $f$. Note first that our formulation with measures is justified by
the fact that in general $F=f(X,Y)$ does not possess a density with respect to
$\mu$ (or to the Lebesgue measure, see Example 3.6). Assume $F=X^{-1}\cdot Y$
and that the three components of $(X,Y,F)$ are pairwise independent of law
$\mu$. We have the following rewriting of $({\rm Hex})$ for groups:

$\displaystyle\mathbb{P}(F\in E\mid X\in A\text{ and }Y\in A)=\mathbb{P}(F\in
E\mid X\in A^{c}\text{ and }Y\in A^{c}).$ (4.1)

The left-hand side also writes $4\mathbb{P}(F\in E\text{ and }X\in A\text{ and
}X\cdot F\in A)$. Since $F$ and $X$ are independent, this is

$\displaystyle 4\int_{E}\mathbb{P}(X\in A\text{ and }X\cdot g\in A){\rm
d}\mu(g)$
$\displaystyle=\int_{E}4\operatorname{Pat}_{A}\big{(}g^{-1}\big{)}{\rm
d}\mu(g).$

Using the left invariance of $\mu$ in the definition of
$\operatorname{Pat}_{A}$, we get
$\operatorname{Pat}_{A}\big{(}g^{-1}\big{)}=\operatorname{Pat}_{A}(g)$ and
finally see that the law of $F$ conditional upon $(X\in A\text{ and }Y\in A)$
admits the density $4\operatorname{Pat}_{A}$ with respect to $\mu$. One can
proceed identically for the right-hand side of (4.1) so that for every $E$
measurable

$\mathbb{P}(F\in E\mid X\in A^{c}\text{ and }Y\in
A^{c})=\int_{E}4\operatorname{Pat}_{A^{c}}(g){\rm d}\mu(g).$

From (4.1), it follows the equality of the two Patterson functions at almost
every $g\in\mathfrak{X}$. As proved in [17, 20, 41, 52] in the adequate
setting this identity in fact holds not only for almost every but for every
$g\in\mathfrak{X}$.

### Acknowledgements

The authors wish to thank the editors of this special issue for their
invitation to celebrate Jean-Pierre Bourguignon’s 75th birthday through a
paper centered around some aspects of contemporary ‘mathemusical’ research. We
are particularly grateful to the anonymous referees for their careful reading
of the manuscript and their valuable criticisms that helped us to improve many
aspects of the present article. Many thanks to the colleagues at IRMA-
University of Strasbourg and IRCAM-Sorbonne University for all the discussions
that contributed to clarifying some of the ideas developed in this article.
This research is partially supported by the Interdisciplinary Thematic
Institute CREAA, as part of the ITI 2021-2028 program of the Université de
Strasbourg, the CNRS, and the Inserm (funded by IdEx Unistra ANR-10-IDEX-0002,
and by SFRI-STRAT’US ANR-20-SFRI-0012 under the French Investments for the
Future Program).

## References

  * [1] Althuis T.A., Göbel F., Graph theoretic aspects of music theory, University of Twente, Memorandum no. 1573, 2001, available at https://research.utwente.nl/files/5115800/1573.pdf.
  * [2] Amiot E., Une preuve élégante du théorème de Babbitt par transformée de Fourier discrète, Quadrature 61 (2006), 1–4.
  * [3] Amiot E., David Lewin and maximally even sets, J. Math. Music 1 (2007), 157–172.
  * [4] Amiot E., New perspectives on rhythmic canons and the spectral conjecture, J. Math. Music 3 (2009), 71–84.
  * [5] Amiot E., Music through Fourier space: Discrete Fourier transform in music theory, Comput. Music Sci., Springer, Cham, 2016.
  * [6] Amiot E., Rahn J. (Editors), Tiling rhythmic canons, Perspect. New Music 49 (2011), 6–273, available at https://muse.jhu.edu/issue/43618.
  * [7] Andreatta M., Méthodes algébriques en musique et musicologie du XXe siècle: aspects théoriques, analytiques et compositionnels, Ph.D. Thesis, EHESS-IRCAM, Paris, 2003.
  * [8] Andreatta M., Une introduction musicologique à la recherche mathémusicale, Circuit 24 (2014), 51–66.
  * [9] Andreatta M., Tiling canons as a key to approach open mathematical conjectures?, in Mathemusical Conversations, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 32, Wiley, 2015, 86–104.
  * [10] Andreatta M., Agon C. (Editors), Special issue: Tiling problems in music, J. Math. Music 3 (2009), 63–70.
  * [11] Andreatta M., Bardez J.-M., Rahn J. (Editors), Around set theory. A French/American musicological meeting, Delatour France, Sampzon, 2008\.
  * [12] Andreatta M., Baroin G., An introduction on formal and computational models in popular music analysis and generation, in Aesthetics and Neuroscience: Scientific and Artistic Perspectives, Springer, Cham, 2016, 257–269.
  * [13] Andreatta M., Guichaoua C., Juillet N., New hexachordal theorems in metric spaces with probability measure, Rend. Semin. Mat. Univ. Padova, to appear.
  * [14] Andreatta M., Vuza D.T., On some properties of periodic sequences in Anatol Vieru’s modal theory, Tatra Mt. Math. Publ. 23 (2001), 1–15.
  * [15] Assayag G., Feichtinger H.G., Rodrigues J.F. (Editors), Mathematics and music: A Diderot mathematical forum, Springer, Berlin, 2002.
  * [16] Babbitt M., Some aspects of twelve-tone composition, The Score 12 (1955), 53–61.
  * [17] Ballinger B., Benbernou N., Gomez F., O’Rourke J., Toussaint G., The continuous hexachordal theorem, in Mathematics and Computation in Music, Commun. Comput. Inf. Sci., Vol. 38, Editors E. Chew, A. Childs, C.H. Chuan, Springer, New Haven, 2009, 11–21.
  * [18] Blau S.K., The hexachordal theorem: a mathematical look at interval relations in twelve-tone composition, Math. Mag. 72 (1999), 310–313.
  * [19] Boulez P., Connes A., Creativity in music and mathematics, A meeting organized within the Mathematics and Computation in Music Conference at IRCAM (15 June 2011), Video available at http://agora2011.ircam.fr.
  * [20] Buerger M.J., Proofs and generalizations of Patterson’s theorems on homometric complementary sets, Z. Kristallographie 143 (1976), 79–98.
  * [21] Coven E.M., Meyerowitz A., Tiling the integers with translates of one finite set, J. Algebra 212 (1999), 161–174, arXiv:math/9802122.
  * [22] de Bruijn N.G., On bases for the set of integers, Publ. Math. Debrecen 1 (1950), 232–242.
  * [23] Fauvel J., Flood R., Wilson R. (Editors), Music and mathematics: From Pythagoras to fractals, Oxford University Press, Oxford, 2006.
  * [24] Forte A., The structure of atonal music, Yale University Press, 1973.
  * [25] Fox R.H., Solution by R.H. Fox, Canad. Math. Bull. 7 (1964), 623–626.
  * [26] Freund A., Andreatta M., Giavitto J.-L., Lattice-based and topological representations of binary relations with an application to music, Ann. Math. Artif. Intell. 73 (2015), 311–334.
  * [27] Fuglede B., Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101–121.
  * [28] Ghisi D., Vettori intervallari: Non degenerazione e Z-relation, Masters Thesis, University of Milano Bicocca, Italy, 2006.
  * [29] Goyette J., The Z-relation in theory and practice, Ph.D. Thesis, University of Rochester, 2012.
  * [30] Hajós G., Sur la factorisation des groupes abéliens, Časopis Pěst. Mat. 74 (1949), 157–162.
  * [31] Jedrzejewski F., Andreatta M., Johnson T., Musical experiences with block designs, in Mathematics and Computation in Music, Commun. Comput. Inf. Sci., Vol. 38, Springer, New Haven, 2009, 154–165.
  * [32] Jedrzejewski F., Johnson T., The structure of Z-related sets, in Mathematics and Computation in Music, Lecture Notes in Comput. Sci., Vol. 7937, Springer, Heidelberg, 2013, 128–137, arXiv:1304.6608.
  * [33] Kolountzakis M.N., Matolcsi M., Tessellations by translation, Gac. R. Soc. Mat. Esp. 13 (2010), 725–746.
  * [34] Kowalski O., Preiss D., Besicovitch-type properties of measures and submanifolds, J. Reine Angew. Math. 379 (1987), 115–151.
  * [35] Lanzarotto G., Extended vuza canons, Ph.D. Thesis, University of Padova, Université de Strasbourg 2022.
  * [36] Lascabettes P., Agon C., Andreatta M., Bloch I., Computational analysis of musical structures based on morphological filters, in Mathematics and Computation in Music, Lecture Notes in Comput. Sci., Vol. 13267, Springer, Cham, 2022, 267–278.
  * [37] Lev N., Matolcsi M., The Fuglede conjecture for convex domains is true in all dimensions, Acta Math. 228 (2022), 385–420, arXiv:1904.12262.
  * [38] Lewin D., Re: Intervalic relations between two collections of notes, J. Music Theory 3 (1959), 298–301.
  * [39] Lewin D., Generalized musical intervals and transformations, 2nd ed., Oxford University Press, Oxford, 2007.
  * [40] Mandereau J., Ghisi D., Amiot E., Andreatta M., Agon C., Discrete phase retrieval in musical structures, J. Math. Music 5 (2011), 99–116.
  * [41] Mandereau J., Ghisi D., Amiot E., Andreatta M., Agon C., $Z$-relation and homometry in musical distributions, J. Math. Music 5 (2011), 83–98.
  * [42] McCartin B.J., Geometric proofs of the complementary chords theorems, Int. Math. Forum 11 (2016), 27–39.
  * [43] Meredith D. (Editor), Computational music analysis, Springer, Cham, 2016.
  * [44] Patterson L., Homometric structures, Nature 143 (1939), 939–940.
  * [45] Patterson L., Ambiguities in the X-ray analysis of crystal structures, Phys. Rev. 65 (1944), 195–201.
  * [46] Peeters G., Roadmap for music information research, 2013, unpublished.
  * [47] Popoff A., Andreatta M., Ehresmann A., Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis, J. Math. Music 12 (2018), 35–55.
  * [48] Rosenblatt J., Seymour P.D., The structure of homometric sets, SIAM J. Algebraic Discrete Methods 3 (1982), 343–350.
  * [49] Tao T., Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), 251–258, arXiv:math.CO/0306134.
  * [50] Tymoczko D., A geometry of music, Oxf. Stud. Music Theory, Oxford University Press, Oxford, 2011.
  * [51] Wang M., Homologie persistante appliquée à l’analyse automatique des styles musicaux, Ph.D. Thesis, Université de Bordeaux et Université de Strasbourg, 2023.
  * [52] Wilcox H.J., Group tables and the generalized hexachord theorem, Perspect. New Music 21 (1983), 535–539.