The Tonnetz Environment: A Web Platform for Computer-aided

“Mathemusical” Learning and Research

Corentin Guichaoua

1,2

, José L. Besada

, Erica Bisesi

and Moreno Andreatta

2,1

Music Representations Team, IRCAM, 1 Igor Stravinsky Square, Paris, France

Institute for Advanced Mathematical Research, CNRS, University of Strasbourg, Strasbourg, France

Department of Musicology, Complutense University of Madrid, Madrid, Spain

Music Faculty, University of Montreal, Montreal, Canada

Keywords: Tonnetz, Computational Musicology, Computer-Supported Education, Geometry, Topology, Cognition.

Abstract: We describe the Tonnetz web environment and some of the possible applications we have developed within

a pedagogical workshop on mathematics and music that has been conceived for high-school students. This

web environment makes use of two geometrical representations that constitute intuitive ways of accessing

some theoretical concepts underlying the equal tempered system and their possible mathematical

formalizations. The environment is aimed at enhancing “mathemusical” learning processes by enabling the

user to interactively manipulate these representations. Finally, we show how Tonnetz is currently being

adapted in order to lead computer-based experiences in music perception and cognition that will be mainly

carried at universities. These experiences will explore the way in which geometrical models could be

implicitly encoded during the listening process. Their outcome may reinforce educational strategies for

learning mathematics through music.

1 INTRODUCTION

Within the panoply of computer-based approaches on

music education, there is a common agreement that

web environments constitute an excellent device to

support music learning processes (Meredith, 2016;

Conway, 2020). The Tonnetz environment

is a

web application based on previous research on

computer-aided topological music analysis (Bigo et

al., 2013; Bigo et al., 2014; Bigo and Andreatta,

2016) and fostering mathematical and computational

thinking through music.

In this paper, we first provide a very short

overview of the most active research axes of the

Structural Music Information Research Project

(SMIR project in short) within which the web

environment has been developed. After discussing the

“mathemusical” dynamics underlying the SMIR

project, we focus in particular on some computational

models of harmonic spaces that have been

implemented in Tonnetz. This environment enables

an interactive exploration of a great variety of

We henceforth use Tonnetz for our open-source web

environment and Tonnetz for the neo-Riemannian

representation. The web environment is available at:

geometric spaces for music-theoretical, analytical and

compositional purposes. We will also present a

current research project, supported by the CNRS,

which constitutes the cognitive component of the

SMIR project. This project, entitled Processes and

Learning Techniques of Mathemusical Knowledge

(ProAppMaMu project in short), aims to explore the

link between geometric representations and cognitive

sciences by following some previous work carried out

by researchers in neurosciences and experimental

psychology and by focusing on the interplay between

structural mathematics and computational models.

After some considerations on music cognition, we

show how Tonnetz is currently being adapted to

lead computer-based educational experiences for

music perception and cognition. These experiences

are planned to be systematically conducted at

universities and might shed some new light on the

way in which geometrical models are implicitly

encoded in the listening process, and eventually

suggest how to reinforce the learning techniques in

mathematics through music.

https://guichaoua.gitlab.io/web-hexachord/. We recom-

mend to open it with Chrome.

680

Guichaoua, C., Besada, J., Bisesi, E. and Andreatta, M.

The Tonnetz Environment: A Web Platform for Computer-aided “Mathemusical” Learning and Research.

DOI: 10.5220/0010532606800689

In Proceedings of the 13th International Conference on Computer Supported Education (CSEDU 2021) - Volume 1, pages 680-689

ISBN: 978-989-758-502-9

2 THE SMIR PROJECT AND ITS

“MATHEMUSICAL”

DYNAMICS

The SMIR project, hosted by the Institut de

Recherche Mathématique Avancée (IRMA) of the

University of Strasbourg, is currently carried out in

collaboration with musicologists from the Groupe de

Recherche Expérimentale sur l'Acte Musical

(GREAM) and computer science researchers from the

Music Representations Team at IRCAM in Paris.

Ongoing research includes axes such as mathematical

morphology, generalized Tonnetze, formal concept

analysis and computational music analysis (Agon et

al., 2018), persistent homology and automatic

classification of musical styles (Bigo and Andreatta,

2019), category theory and transformational music

analysis (Popoff, Andreatta and Ehresmann, 2018),

homometry and phase retrieval in music analysis

(Mandereau et al., 2011), or tiling musical problems

and Fuglede spectral conjecture (Andreatta, 2015).

All these topics share a common trait, which

positions music at the core of mathematical

formalization and computational modelling. Their

dynamics is captured by the diagram in Figure 1,

which shows how to navigate between music and

mathematics by using computer science as an

interface in the formalization ascending process

(from music to mathematics) and the application

descending process (from mathematics back to

music). This dynamical “mathemusical” process is

detailed deeper in Figure 2. Notice that some musical

problems make use of computational models in an

ascending formalization process, before necessarily

reaching the state of general theorems in the

mathematical realm. This is typically the case of NP-

complete music-theoretical problems for which,

unfortunately, it is not possible to exhibit a

constructive algorithm. Some well-known

compositional problems belong to this class. It is the

case, for instance, of the enumeration of all-interval

twelve-tone series, the estimation of Z-related chords,

or the enumeration of Hamiltonian paths and cycles

in music-theoretical graphs, such as the Tonnetz or its

possible extensions (Cannas and Andreatta, 2018).

The most interesting case for “mathemusical”

research occurs however when a musical problem

reaches the state of a mathematical theorem after the

formalization or generalization process and becomes

Available at: https://guichaoua.gitlab.io/web-

hexachord/hexachordTheorem. We recommend to

visualize it with Chrome.

integrated within a computer-aided environment to be

used by composers or music theorists in different

situations within the field of music. One of the most

celebrated theorems that one may find in the literature

is Milton Babbitt’s “hexachordal theorem”, stating

that a hexachord and its complement always have the

same intervallic content (Blau, 1999). Tonnetz, and

in particular its circular representation associated to

the equal-tempered system, can be easily adapted to

this special musical problem. It becomes an

interactive platform for visualising the intervallic

content of two generic collections of pitches related

by complementation. Figure 3 shows the environment

as it has been integrated in an online article retracing

the history of this famous combinatorial problem

(Bayette, 2019). The reader may try to discover the

possible relations between the intervallic content of a

given collection of pitches and its complement by

exploring this special adaptation

Figure 1: The “mathemusical” dynamics at the core of the

SMIR project.

Figure 2: A more detailed perspective of Figure 1 showing

the different steps of the “mathemusical” dynamics.

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681

Figure 3: Adaptation of Tonnetz for interactively

visualizing Babbitt’s hexachordal theorem.

3 THE TONNETZ GEOMETRY-

BASED WEB ENVIRONMENT

Tonnetz makes use of two kinds of geometric

representations: a circular representation and a tiling

of the plane by triangles (alternatively by hexagons).

We go now into some details of these geometric

representations by emphasizing their relevance in

supporting music learning and teaching via computer

models.

3.1 The Circular Representation

As in the case of the HexaChord computer

environment

, which was the inspirational source for

this web application, two circular representations are

available to the user: the chromatic circle and the

cycle of fifths representation. These representations

are extensively used in both American and European

research around the pitch-class set theory (Andreatta,

Bardez, and Rahn, 2008). Although both circles are

algebraically equivalent (since they are related by an

affine transformation), they highlight different

properties of chord collections and their associated

intervallic content with respect of symmetry, as

shown in Figure 4.

An interesting case, that can be taken as an

exercise for advanced maths students, occurs when

the two representations are exactly the same for a

specific pitch collection. The reason of this

equivalence goes deep into algebraic properties of the

cyclic groups underlying the previous geometric

representations. Apparently simple musical problems

may give rise to deep algebraic investigations that can

be easily modelled within an interactive environment

such as Tonnetz.

Available at: https://louisbigo.com/hexachord

Figure 4: Two possible representations of the C7 chord in

Tonnetz. On the left, within the chromatic circle; on the

right, within the cycle of fifths.

3.2 Two Ways of Tiling the Space

Beyond the circles, Tonnetz incorporates further

geometric representations: the tiling of the plane by

triangles and by hexagons. These representations are

widely spread within the neo-Riemannian music

analysis (Gollin and Rehding, 2010). The

triangulation of the 2-dimensional Euclidean space

offers a first geometrical perspective on musical

transformations between two given chords.

According to the simplicial approach previously

introduced by Louis Bigo and Moreno Andreatta

(2015), triangles are 2-simplexes and they are glued

together, as shown in Figure 5, by a self-assembly

process when they share a common edge.

Figure 5: On the left, the first three n-simplexes; on the

right, the self-assembly process. Adapted from Bigo and

Andreatta (2015).

In the standard Tonnetz space, triangles represent

major and minor chords, as Figure 6 puts in evidence,

but one may use any 3-note chord as a 2-simplex

generating the triangular tessellation of the plane via

the self-assembly process. The structural properties of

the resulting non-standard Tonnetz space will be

different depending on the properties of the

generating 2-simplexes (Catanzaro 2011). This fact

leads to a panoply of topological cases, from the torus

(as in the case of the standard Tonnetz) to non-

connected spaces (for instance, when the space is

generated by only augmented triads).

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Figure 6: Multiple visualizations of pentachord {C#, D, E, G, A} within the Tonnetz. On the left, its representation within

the standard Tonnetz. On the right, the circular representations associated to this chord.

Figure 7: Multiple visualizations of pentachord {C#, D, E, G, A,} within the Tonnetz, but represented this time in the dual

space of the standard Tonnetz.

Any triangulation of the plane is associated to its

dual one: the graph obtained by replacing every 2-

simplex with a 0-simplex (i.e., a point) and by

connecting two points when they corresponding 2-

simplexes share a common 1-simplex (i.e., an edge).

Figure 7 shows how the pentachord of Figure 6 is

now represented in the dual space, also called

“chicken-wire” torus in the neo-Riemannian music

theory (Douthett and Steinbach, 1998).

In this new visual representation, the vertices are

now major triads (normal font with the name of the

chord root) or minor ones (italic font with the name

of the chord root). Now, for any given collection of

pitches, if the node corresponding to A major is

illuminated, it means that the collection contains the

A major triad as a subset. One may notice that each

hexagon of the dual space represents a single note,

more precisely the note which is common to all the

triads which are associated to the nodes of the

The Tonnetz Environment: A Web Platform for Computer-aided “Mathemusical” Learning and Research

683

hexagonal surface. For instance, pitch class F#

belongs to the F# major, F# minor, D major, D#

minor, B major, and B minor triads.

4 USING TONNETZ IN

OUTREACH AND

EDUCATIONAL CONTEXTS

Tonnetz has taken part of several large public

events, such as the LaLaLab exhibition “Mathematics

and Music” created by IMAGINARY and premiered

at Heidelberg in May 2019. This exhibition was

curated with the advice of several “mathemusical”

researchers around the world and with the

contribution of more than twenty artists and scientists

working in the field. As shown in Figure 8, Tonnetz

was presented by means of tactile screens and

keyboards enabling the public to interactively explore

the different geometric spaces available through the

environment.

Of course, Tonnetz can be used in quite more

controlled scenarios with educational purposes. It has

been recently integrated into a pedagogical workshop

which was conceived by following the new high-

school programs of the French National Education.

These programs introduce the topic of mathematics

and music at the high-school level (typically for 15

years-old students). They put a special emphasis on

the way in which several harmonic structures have

been formalized, from the Pythagorean scales to the

equal-tempered system, and on the use of computer

science to provide computational models of musical

knowledge. In this workshop, designed within the

ProAppMaMu project, we have tried to reinforce the

interplay between standard pedagogical tools (flyers,

slide

presentations, audio and video extracts) and an

interactive manipulation of Tonnetz. Figure 9

shows the typical set-up of such a workshop in a high-

school class, where the students can individually

explore further contents of the presentation through

our web environment, which is available in their

computer screen.

The conception of a traditional paper support

accompanying the “mathemusical” workshop has

been done in synergy with Marie Marty, who is a

designer from the Ecole Estienne of Design in Paris,

specialist in scientific drawings. She has also created

a series of animations for enriching the brochure

support. Some topics within the official programs of

the French National Education were selected for their

potential to be also depicted in a traditional flyer, as

shown in Figure 10. The aim was to describe, in an

visually appealing way, the main features which are

available in Tonnetz. Figure 11 highlights a portion

of the paper flyer that can be “augmented” during a

workshop.

Figure 8: A visitor interacting with Tonnetz and a

keyboard at the LaLaLab exhibition in Heidelberg

(reproduced with the kind permission of IMAGINARY).

Figure 9: A typical set-up of a “mathemusical” workshop

of the ProAppMaMu project.

Figure 10: The flyer for enriching the workshop.

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Figure 11: On the left, the topic “the Tonnetz system” as contained in the brochure (third inner panel); on the right, a still of

its matching gif animation for slide presentations.

Both the animations and the computer-aided

visualisations have been integrated into a pedagogical

film (entitled “Musique et mathématiques: histoire

d’une rencontre”) that aims to artistically summarize

the theoretical content of the high-school workshop.

The two main characters of this film, which appear in

Figure 12, are playing on a stage (which is a

keyboard) surrounded by several animations

generated with Tonnetz. The dialogue between

these characters provides at the same time a journey

through the history of the relations between maths

and music and a condensed presentation of the main

concepts introduced in the workshop via the web

environment

Figure 12: Screenshot of the pedagogical film “Musique et

mathématiques: histoire d’une rencontre”.

The film is hosted at the CNRS AudiDiMath outreach

platform: http://video.math.cnrs.fr/musique-et-

mathematiques/

5 PERCEPTUAL AND

COGNITIVE IMPLICATIONS

So far, the main outcomes of the SMIR project have

been the online uploading of the Tonnetz

environment, its adaptation within the contexts of art

and popular music, and a detailed dissemination plan

trough “mathemusical” workshops and exhibitions,

suitable for pedagogical applications. Besides all

these achievements, the SMIR project disclosed new

interesting research questions belonging to the area of

psychology and cognition. These questions concern

the way in which individuals process the information

carried by the geometrical representations discussed

in the previous sections:

 Do the structures implicitly encoded by

Tonnetz have some perceptual foundation,

somehow innate in human cognition?

 How could we grasp plausible connections

between its geometrical representations and a

multimodal (in our particular context, auditory

and visual) musical perception?

 Which strategies would be the most suitable and

efficient for helping the listeners become more

The Tonnetz Environment: A Web Platform for Computer-aided “Mathemusical” Learning and Research

685

aware and familiar with such structures under

the effect of such stimuli?

ProAppMaMu aims at answering these questions

by means of perceptual tests. Moreover, by focusing

on the interplay between structural mathematics and

computational models, the project will explore the

way in which geometrical models of music theory

might be used to reinforce the learning techniques

within the context of mathematical education.

5.1 Transformational Music Theory

and Cognition

One of the main goals of systematic musicology

during the last decades has been to explore the links

between human cognitive capabilities to process

musical signals and the theoretical foundations of

music (Leman, 1995). As music develops quite

differently within each human culture, the theoretical

representations conceived by Western music theorists

are far from standing as universals. This fact does not

mean, however, that Western representations are

completely arbitrary. Consequently, a search for

some objectification in this sense is highly desirable.

Moreover, a better comprehension of the cognitive

foundations and implications of such representations

would help to improve the pedagogical strategies in

musical and “mathemusical” educational contexts.

Many music representations can be traced back to

two main schemes, which are embedded in Western

music notation: the horizontal flow of time and the

vertical ordering of pitches, as outlined by the image

schema theory (Johnson, 1987). Although such

schemas may seem to be almost trivial within our

culture, the way individuals interact with them is,

cognitively speaking, quite complex. For instance,

there is no one-to-one relationship between the

vertical distribution of pitches and the plethora of

cognitive strategies individuals adopt for pitch

recognition (Letailleur, Bisesi, and Legrain, 2020).

Concerning time, some composers have twisted the

straight left-to-right musical timeline while drafting

their oeuvre (Besada and Pagán Cánovas, 2020;

Besada, Barthel-Calvet, and Pagán Cánovas, 2021).

All these examples rely on relatively simple music

features (pitch and time), which are unfolded

throughout a unidimensional image schema. The

standard Tonnetz is conversely much more complex,

as pitches are placed in the crossroads of three

different linear directions, giving rise to geometrical

shapes matching with particular chords. Previous

research in the field of musical psychology has

encompassed these kinds of problems from diverse

approaches. A key related topic is the study of

perceived distance when hearing pitch collections in

tonal and post-tonal music (Rogers and Callender,

2006; Bisesi, 2017). Concerning our particular

research, Carol L. Krumhansl (1998) compared three

neo-Riemannian models of triadic distance with

psychological data. She surmised that the perceived

distance between chord roots is akin to the dual

standard Tonnetz. Andrew J. Milne and Simon

Holland (2016) have shown through experimental

data that spectral pitch-class distances and the

Tonnetz have high correlations. This evidence led

them to provide a psychoacoustical explanation for

perceived triadic distance. By contrast, very few is

known about how individuals actually apprehend

such abstract structures.

The standard Tonnetz probably stands as the most

exploited model of transformational music theory.

This branch of music theory was formally stablished

by David Lewin (1987; 1993) and privileges the

analysis of the way musical objects transform over

time instead of focusing on their intrinsic features. As

previously defined, the aforementioned gluing self-

assembly process of the Tonnetz unfolds along three

different directions which are respectively orthogonal

to the three linear directions of individual pitches.

Three different transformations (those that generate

the space) relate each triangle (i.e., each triad) with

each contiguous neighbour. As the standard Tonnetz

is a cyclic and connected space (the aforementioned

torus), any pair of triangles can be related by means

of a finite series of combinations of these only three

transformations generating the space.

As highlighted above, how music evolves over

time is a key issue in transformational music theory.

Consequently, these kinds of transformations have

been sometimes described as metaphors of motion by

music theorists (Attas, 2009). Musical software like

Tonnetz allows to perceptually reify this metaphor,

and therefore to observe and potentially describe the

way a user conceives them.

5.2 Adapting Tonnetz for Empirical

Psychology Experiences

We are currently developing a variant of the standard

Tonnetz, in order to collect quantitative and

qualitative data for discussing the cognitive features

of human interaction with its environment. Several

chord sequences will be submitted to listeners of

different expertise, in the double format of sounds and

coloured patterns. All these sequences are

parsimonious (i.e. any pair of contiguous triads share

two of their three respective pitches), but the last triad

does not necessarily follow a parsimonious

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movement, as it happens for instance in the sequence

of Figure 13. There, pitch names on the vertices will

be erased to avoid potential bias, and the first six

chords are associated with red (for major chords) and

blue triangles (for minor chords). When the last chord

sounds, six new triangles are illuminated in a third

colour (yellow). Participants are asked to choose,

according to their subjective intuition, the yellow

triangle that fits best with the sound stimulus. The

task will be presented twice: for the first time, the

participant is unaware of what the Tonnetz is, whereas

the second attempt is performed after a short video

tutorial. The device stores, for each sequence, the

user’s response time and the selected triangle

matching with its triad for a quantitative comparison

with the correct one. In addition, a follow-on

questionnaire will be aimed at qualitatively grasping

the mental strategies brought into play when

elucidating the answer.

Figure 13: Top: A musical sequence among those planned

for the cognitive experiences with our adaptation of

Tonnetz. Bottom: Partial screenshot when executing the

sequence. We provide the pitch names for a better

understanding of this figure, although they will not be

shown during the tests.

We currently test the viability of the protocol with

a few individuals. Our goal is to run it with different

populations. First, we are going to compare subjects

whose cognitive styles are supposed to be very

contrasting from each other. Following the E-S theory

(Baron-Cohen, Knickmeyer, and Belmonte, 2005),

we may distinguish between “the capacity to respond

to feeling states of other individuals” (empathizing

cognitive style) and the “capacity to respond to

regularities of objects and events” (systemizing

cognitive style). The E-S theory has been applied to

the musical domain, (Kreutz, Schubert, and Mitchel,

2008) leading to the development of music

empathizing (ME) and music systemizing (MS)

scales as emerging from principal component

analysis procedures. Questions aimed at distinguish

between the two groups concerned thoughts about the

emotional state of the composer or the performers

when listening to the music, the importance ascribed

to physics and acoustics of musical instruments,

music structure, or the different layers of instruments

and voices.

In our study, we are going adapt the ME-MS

scales to the participants’ mother languages (Schubert

et al., 2014) for a population of trained musicians and

scientific professionals to explore some potential

specificities in achieving a “mathemusical” goal.

While no significant difference between these groups

is awaited at a first stage without training, we expect

to disclose a number of correlations between the

preferred cognitive style and the amount and

specificity of the improvements, after some training

at a second stage. In a follow-on experiment, a group

of university students will be involved, adhering to a

very similar protocol. In the future, we plan to

consider the dual representation of the Tonnetz, as

well as musical sequences related to non-standard

Tonnetze or even more general related structures

(Tymoczko, 2012).

6 CONCLUSIONS

In this paper we have shown how geometry-based and

interactive web environments such as Tonnetz can

be useful for supporting “mathemusical” learning and

research activities. Within the larger SMIR project,

the main goal was to develop a dissemination plan

through several “mathemusical” workshops and

exhibitions, suitable for pedagogical applications.

Our activities primarily targeted high-school and

university students, but were also conceived for larger

audiences.

Besides this outreach achievement, the SMIR

project disclosed new interesting research questions

belonging to the areas of psychology and cognition.

These questions are currently addressed by the

ProAppMaMu project, focusing on how the more

complex geometrical representations implemented in

Tonnetz are processed during an active listening. In

order to tackle this cognitive component of the SMIR

project, we are adapting this web environment by

integrating a series of perceptual tests which are

aimed at better understanding how these geometrical

The Tonnetz Environment: A Web Platform for Computer-aided “Mathemusical” Learning and Research

687

representations are processed by the listeners. These

tests will start by observing small populations with

different musical and scientific skills, but will soon be

generalized to larger groups. Our experimental

approach may hopefully shed some light on the way

in which some geometrical models of music theory

could be used to reinforce educational strategies, even

those beyond music education. They will surely

enhance existing learning techniques within the

exciting context of a musically driven mathematical

education.

ACKNOWLEDGEMENTS

This research has been partially carried on within the

ProAppMaMu project that receives financial support

from the CNRS through the MITI interdisciplinary

programs. In addition José L. Besada is currently

supported by the Atracción del talento investigador

grant (ref. 2019-T2/HUM-14477) from the Madrid

region, and Corentin Guichaoua by the European

Union’s Horizon 2020 research and innovation

program (ref. 788960) under the ERC ADG project

COSMOS.

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