Formalisms and Interfaces to Manipulate Music Information: The Case

of Music Petri Nets

Adriano Barat

e, Goffredo Haus and Luca A. Ludovico

Laboratorio di Informatica Musicale, Dipartimento di Informatica “Giovanni Degli Antoni”,

Universit

a degli Studi di Milano, Via Celoria 18, Milan, Italy

Keywords:

Music, Petri Nets, Formalization, Manipulation, Human-Computer Interfaces.

Abstract:

Music Petri nets are a mathematical formalism that has been already adopted in the context of music infor-

mation analysis and manipulation, concerning both symbolic scores and audio content. Among the main

advantages, it is worth mentioning their possibility to clearly describe music processes and transformations

inside a music composition, their easy-to-understand graphical representation, and the availability of formal

tools to analyze the nets thus obtained. But previous research has also highlighted some limits in the usabi-

lity of music Petri nets, due to an interface far from the ones composers and performers are used to, such as

digital score editors and digital audio workstations. In this paper, we propose a solution that combines the

descriptive power of such a mathematical formalism with the ease of use of editing tools to which musicians

are accustomed. The idea is to design human-computer interfaces that, standing between the musician and the

mathematical formalism, can hide the details of the underlying Petri nets.

1 INTRODUCTION

In the ﬁeld of sound and music computing, the pro-

blem of ﬁnding suitable interfaces to represent and

manipulate music information is a matter of interest.

In 1985, Pennycook wrote an inspiring survey on

computer-music interfaces and their basic principles,

dealing speciﬁcally with three musical tasks: ma-

nuscript preparation, music language interfaces for

composition, and real-time performance interaction

(Pennycook, 1985). In more recent times, reference

(Assayag et al., 1999) mentioned examples of inter-

faces for computer-assisted composition developed at

IRCAM, (Cook, 2001) described the principles to de-

sign computer music controllers, and (Paradiso and

Omodhrain, 2003) dealt with trends in electronic mu-

sic interfaces. Finally, it is worth mentioning the

works annually presented at the International Con-

ference on New Interfaces for Musical Expression

(NIME), an initiative dedicated to scientiﬁc research

on the development of new technologies and their role

in musical expression and artistic performance.

A successful example of computer-music inter-

face is the reacTable (Jord

a et al., 2007), an electronic

musical instrument with a tabletop tangible user inter-

face conceived in 2003 and developed by a research

team at the Pompeu Fabra University in Barcelona.

In its original conﬁguration, the interface is basically

a round translucent table, used in a darkened room as

a backlit display.

By placing blocks called tangibles

on the table, and interfacing with the visual display

via the tangibles or ﬁngertips, a virtual modular synt-

hesizer is operated, thus creating music or sound ef-

fects. The reacTable is a musical instrument that can

be proﬁtably used for extemporary composition and

performance. Similarly to our proposal detailed be-

low, tangibles can be associated with the concepts of

music object and musical operator that we will intro-

duce, but such an instrument has not been conceived

to describe music processes in a formal way.

MAX/Msp (Didkovsky and Hajdu, 2008) and Pure

Data (Puckette et al., 1996) are dataﬂow program-

ming languages where functions or objects are linked

in a graphical environment which models control and

audio ﬂows. These software tools can also support

formal approaches to composition and manipulation

of music information, an aspect that brings them clo-

ser to our proposal. Since MAX/Msp and Pure Data

are commonly in use among musicians, this kind of

approach seems to be suitable also for people without

a speciﬁc training in sound and music computing.

Now, a key question is: why should we employ a

More recently, also software versions have been released.

Baratè, A., Haus, G. and Ludovico, L.

Formalisms and Interfaces to Manipulate Music Information: The Case of Music Petri Nets.

DOI: 10.5220/0007228500810090

In Proceedings of the 2nd International Conference on Computer-Human Interaction Research and Applications (CHIRA 2018), pages 81-90

ISBN: 978-989-758-328-5

formal tool to describe music objects and their trans-

formational processes, which implies a number of

theoretical and practical problems to solve, when non-

formal solutions to create and manipulate music and

sound are available and commonly in use?

The answer is that formal tools present a number

of advantages with respect to non-formal representa-

tion formats. First, for the representation, transfor-

mation, and establishment of relationships involving

music information, it is possible to rely on the theory

behind, also beneﬁting from previous research. Mo-

reover, a formal representation typically allows the

investigation of mathematical as well as topological

properties; an example is the automatic recognition of

identical structures via an isomorphism. Finally, the

adoption of a formalism encourages the unveiling of

hidden structures and implicit processes, thus foste-

ring a deeper understanding of the creative/analytical

phenomenon.

The paper is structured as follows: Section 2 will

make research questions explicit, Section 3 will pro-

vide an overview of the formalism known as Petri

nets, Section 4 will introduce a specialization called

music Petri nets, Section 5 will discuss some propo-

sals of interfaces which can answer the mentioned re-

search questions, and Section 6 will draw the conclu-

sions.

2 RESEARCH QUESTIONS

Petri nets (PNs) are a formal tool suitable to study and

describe systems that are concurrent, asynchronous,

distributed, parallel, and non-deterministic. Also mu-

sical compositions can be seen as the result of trans-

formational processes on music objects that share

many of the mentioned features. For example, parts

and voices in a composition can be interpreted as pa-

rallel, concurrent and distributed sequences of music

objects, which can be in turn decomposed into sim-

pler information entities.

In this context, we deﬁne music object any mea-

ningful aggregation of symbols, at different degrees

of abstraction. Given this very broad deﬁnition, ex-

amples of music objects may be sections, periods,

phrases, motifs, chord sequences, rhythmic patterns,

etc. Within a composition, music objects can be reu-

sed, as in the case of verse-chorus structures for po-

pular music or subject-answer-countersubject in con-

trapuntal compositional techniques. Besides, music

objects can be transformed through ad hoc music ope-

rators, such as transpositions, inversions, pitch substi-

tutions, etc. Since music is intrinsically hierarchical

(Lerdahl and Jackendoff, 1983), the process of de-

composition of music objects into simpler ones can

occur recursively, until the most atomic level (e.g., the

one of music symbols) has been reached.

Recent research in the ﬁeld of music theory, music

cognition, and computational musicology has brought

to the formulation of theories where music processes

are made explicit, so as to unveil the structure of indi-

vidual compositions (Lerdahl and Jackendoff, 1985).

The research questions of the present work do not

focus on the possibilities offered by Petri nets to mu-

sic composition, analysis, and a posteriori manipula-

tion, since these aspects have been already investiga-

ted in literature (see Section 4.2 for further details).

Rather, our goal is to bridge the gap between a Petri

net-based representation of music content, which is

a formal tool comprehensible to mathematicians and

computer scientists, and a performance-oriented tool,

namely an interface that allows easy interaction with

music content suitable for composers and musicians.

In our intention, this initiative does not explicitly

address musicians, nor is aimed at providing only

domain experts with a new music-content manipula-

tion tool. Needless to say, musicians are a privileged

category of users both to validate the theoretical ap-

proach and to test the computer-based tools that im-

plement it, but our goal is to propose a new paradigm

to manipulate music content addressing also untrai-

ned people.

The main research questions for this work are:

a) Is it possible to hide the details of the Petri net des-

cription of a music piece, so that it can be easily ma-

naged even by non-experts while formal aspects are

being preserved? b) Is the proposed approach suitable

to deﬁne musical objects and support the main trans-

formational processes occurring on them, even in a

real time context? c) Is it possible to create a software

environment implementing the mentioned theoretical

framework?

Before answering the aforementioned research

questions, it is worth introducing some basic noti-

ons on Petri nets (PNs) and music Petri nets (MPNs),

which constitute a specialization of PNs in the music

ﬁeld.

3 AN OVERVIEW OF PETRI

NETS

A formal description of the general net theory by

Carl Adam Petri would fall beyond the scope of the

present paper. For details about this subject, please

refer to works such as (Petri, 1980), (Peterson, 1981)

and (Murata, 1989). For the sake of clarity, we will

only summarize the key elements to let the reader un-

CHIRA 2018 - 2nd International Conference on Computer-Human Interaction Research and Applications

Figure 1: An example of Petri net, showing places (cir-

cles), transitions (rectangles), arcs (oriented lines) and their

weight (numbers above oriented lines). Numbers inside pla-

ces represent the current and the maximum number of to-

kens allowed, respectively. The former number provides

place marking.

derstand the theoretical approach proposed in the fol-

lowing.

A PN is an abstract and formal model to represent

the dynamic behavior of a system presenting asyn-

chronous and concurrent activities. PNs are made of

a combination of basic objects falling in one of these

categories: places, transitions, and arcs. Usually re-

presented in a graphical forms, the instances of such

categories are drawn as circles, rectangles, and orien-

ted lines respectively. Places and transitions are also

referred to as nodes. Arcs can have a number associ-

ated, called the arc weight. The graphical appearance

of a simple PN is shown in Figure 1.

PNs are not static models, rather they present an

evolution from a state to another. The current state is

indicated by place marking, namely by the number of

tokens in each place. The dynamic evolution of a PN

is determined by the following ﬁring rules:

• A transition is enabled when all the incoming pla-

ces of that transition present a number of tokens

greater or equal to the weights of the correspon-

ding incoming arcs, and, after the ﬁre of the tran-

sition, the marking of all the output places will be

less than or equal to their capacities;

• When a transition is enabled, its ﬁring subtracts

from the incoming places a number of tokens

equal to the weights of the incoming arcs, and

adds to each outgoing place a number of tokens

equal to the weights of the corresponding out-

going arc.

An example of transition ﬁring, showing markings

before and after, is provided in Figure 2.

When their application to the music ﬁeld was ﬁrst

proposed, the following properties were considered:

PNs are associated with a graphical form of notation

that requires few symbols; they support hierarchical

descriptions and the deﬁnition of macro-structures;

they are able to describe music-objects processing,

supporting timed representations and deterministic as

well as non-deterministic models (Haus and Sametti,

1991).

4 MUSIC PETRI NETS

Music Petri nets (MPNs) are a specialized extension

of PNs that supports the association of music objects

to places. A music object (MO) may be anything that

can have a musical meaning and can be thought as

an entity, either simple or complex, either abstract or

detailed. As we will explain below, MOs can be com-

bined, reused as they are, or transformed into new in-

stances.

For the sake of clarity, it is important to underline

the possibility to associate different kinds of music-

related content with places. In detail:

• In the case of symbolic content, the MO contains

score excerpts encoded in terms of Common Wes-

tern Notation (e.g., notes, rests, etc.) or other no-

tational systems. The main advantage is the pos-

sibility to manipulate objects at any degree of ab-

straction and granularity. For example, it is pos-

sible to implement algorithms such as “transpose

all notes of the ﬁrst measure one octave below”,

Figure 2: The upper image shows a net where transition T1

is enabled, i.e. it is ready to ﬁre; the lower image illustrates

the new net marking after ﬁring. Please note that tokens

have not been transferred, rather consumed in input places

and created in output places in accordance with arc weights.

Formalisms and Interfaces to Manipulate Music Information: The Case of Music Petri Nets

or “double the duration of all C-pitched notes”.

On the other side, the corresponding audio is not

directly encoded within MOs; rather, playback re-

quires the interpretation of symbolic sequences to

be sent to a sound module;

• In the case of performance content, the MO em-

beds information encoded through a computer-

driven performance language, such as MIDI or

Csound. This kind of approach stands in the

middle of symbolic and audio content;

• In the case of audio content, the MO contains

waveforms instead of symbolic information. The

possibilities of content manipulations are far more

limited than in the ﬁrst case (e.g., alteration of

the original sequence of audio fragments, volume

changes, equalization, pitch shifting, time stret-

ching, etc.); on the other side, for net playback,

only a suitable media player is required.

In the following, these cases will be managed se-

parately, and the proposed interface will remark dif-

ferences through ad hoc graphical features.

4.1 Formalization

Concerning places, in MPNs the following cases can

occur:

• A place can have no music fragment associated

and no music fragment in input. In this way, it has

only a structural function (e.g., a counter, a selec-

tor, a semaphore, etc.) in a given net topology, in

accordance with the deﬁnition of places in ordi-

nary PNs, where markings represent the state of

the system;

• A place can host a music fragment that will be

transferred to output places after the ﬁring of the

corresponding transitions. In this case, the music

fragment will be delivered to output places after

the manipulation operated by transitions;

• A place can receive a music fragment from either

a single or multiple input transitions. If multiple

fragments arrive simultaneously and/or a MO is

already present, fragments are mixed to form a

more complex MO, potentially available for out-

going transitions.

Moreover, in MPNs a place can be either enabled

to play music objects or not. When an enabled place

containing a MO receives tokens, the fragment (either

already present or transferred from other places) is

played; when a non-enabled place hosts or receives

MOs, its only function is to mix inputs, store music

fragments and send them in output when transitions

ﬁre.

Concerning transitions, in MPNs they can host

music algorithms (deﬁned as abstract transformati-

ons). This kind of transitions is used to process MOs

in input, modify them accordingly, and transfer MOs

thus obtained in output.

For example, one can create a simple net with a

place with an associated MO containing a single note

(say, a C pitch) and the Play ﬂag set to false, an output

transition with an associated algorithm that creates a

major scale in the key of the input note, and an output

place that plays the objects thus modiﬁed (in this case,

the C major scale). Then, the same net topology can

be reused by changing the MO associated with the

input place: e.g., if the original note is set to D, the

D major scale is obtained; as another example, if a

sequence of pitches is used as the input instead of a

single note, the ﬁnal result is a progression.

For a more detailed formal description of MPNs,

please refer to (Barat

e and Haus, 2013).

4.2 Related Works on MPNs

The relationship between PNs and music has been in-

vestigated in a number of scientiﬁc works. This re-

search ﬁeld has been explored since late ’80s mainly

at the Laboratory of Music Informatics (LIM – Labo-

ratorio di Informatica Musicale) of the University of

Milan.

Early works discussing the description and perfor-

mance of musical processes by means of Petri nets

are (Camurri et al., 1986) and (Haus and Rodriguez,

1988).

Reference (Haus and Sametti, 1991) introduces

ScoreSynth, an experimental software tool for score

synthesis through MPNs (see Section 4.3).

A discussion about the effectiveness of MPNs

in the formalizations of extended music pieces can

be found in (Haus and Rodriguez, 1993), focu-

sing on Ravel’s “Bolero”, and in (De Matteis and

Haus, 1996), dealing with Stravinsky’s “The Rite of

Spring”.

The possibility to rely on MPNs in order to re-

present the structure of a piece within a multi-layer

environment has been explored in (Barat

e, 2008) and

(Barat

e and Haus, 2013). The results of this investiga-

tion brought MPNs to become one of the formalisms

natively supported in the IEEE 1599 format (Baggi

and Haus, 2009) to provide structural descriptions for

music pieces.

Moreover, research has highlighted the applicabi-

lity of MPNs to music analysis (Barat

e et al., 2006)

and composition (Barat

e et al., 2007), even in real

time environments (Barat

e et al., 2014). A very re-

cent effort in this sense was the application of MPNs

CHIRA 2018 - 2nd International Conference on Computer-Human Interaction Research and Applications

to the formalization of Schoenberg’s fundamentals of

musical composition (Barat

e et al., 2018).

4.3 Available Software Tools

Currently, a number of software tools for the design

and execution of generic PNs is available. PIPE (Plat-

form Independent Petri net Editor)

is an open-source,

platform-independent tool for creating and analysing

Petri nets, including Generalised Stochastic Petri nets

(Dingle et al., 2009). The Petri Net Toolbox

is a soft-

ware tool for the simulation, analysis, and design of

discrete-event systems based on Petri Net models em-

bedded in the MATLAB environment. WoPeD (Work-

ﬂow Petri Net Designer)

is an open-source software

which aims to provide an easy-to-use interface for

modelling, simulating and analyzing processes des-

cribed by workﬂow nets.

Concerning the musical extension of PNs, there is

only one available tool supporting MPNs: ScoreSynth

(Haus and Sametti, 1991). Its goal is to draw and exe-

cute MPNs supporting MOs encoded in IEEE 1599

format. This application, written in C], runs under

Microsoft Windows, even if recent tests have demon-

strated a good compatibility with Linux-based opera-

ting systems thanks to the Mono project.

Scoresynth has many features designed to draw

net models. Its multi-window GUI permits to ar-

range all Petri nets that concur to design a single

complex model. Customizable graphical elements in-

clude: place and transition sizes; place and transi-

tion names with customizable positions; place, tran-

sition, and arc background and foreground colors; arc

smoothness; pen widths; fonts; drawing grids. The

interface of ScoreSynth is shown in Figure 3.

In ScoreSynth, MPNs can be executed and debug-

ged in various ways:

• Complete execution – The standard execution

mode, where nets are processed until no transition

can ﬁre, or the user stops the execution;

• Timed execution – Similar to the complete execu-

tion, but transition ﬁrings occurs every n seconds;

• Step-by-step execution – The user controls the

process by triggering single steps, such as transi-

tion ﬁrings, tokens transfer, performance of MOs.

To help debugging operations, an automatic text

report of net execution is provided, step by step. The

execution history is saved too, and the user can invoke

previous/next step buttons: this is also useful to test

http://pipe2.sourceforge.net/

https://www.mathworks.com/products/connections/

product detail/petri-net-toolbox.html

https://woped.dhbw-karlsruhe.de/

non-deterministic situations, stepping back and choo-

sing another execution branch of the net.

Since MOs are coded in IEEE 1599, a net exe-

cution creates a comprehensive IEEE 1599 document

that mixes all these objects, thus creating a ﬁnal result

where the logic part reﬂects the mixing process, and

the linked media maintain their synchronization.

Together with printing and export graphics facili-

ties, an interesting feature consists in the ﬁle format

used to save models: PNML (Petri Net Markup Lan-

guage), which is an XML-based standard interchange

format used to represent PNs.

Currently, a Web-based multi-platform version of

ScoreSynth is under development at the Laboratory of

Music Informatics of the University of Milan.

The features of the original ScoreSynth have been

analyzed in order to support them in the proposed in-

terface, even if the purpose of this work is to hide PN-

implementation details.

5 A COMPUTER INTERFACE

FOR MUSIC PETRI NETS

In this section we will describe a proposal of soft-

ware tool, conceived as a browser application, which

aims to bridge the gap between traditional interfaces

for music manipulation and MPNs.

The ﬁrst phase is the realization of a diagram

where MOs can be placed, modiﬁed through suita-

ble operators, and connected together as a graph. In

the following, please refer to Figure 4 for interface-

related aspects.

This phase can be decomposed into steps. First,

MOs are loaded into the Object Gallery, organized

by type, marked through a customizable name (e.g.,

“Fragment A”, “Main theme”, “Chorus”, etc.) and as-

sociated with a color. After the import step, using

drag-and-drop operations, MOs can be placed over

the canvas and connected by cords that join outlets

and inlets.

Inside the canvas, MOs are represented as rectang-

les, identiﬁed by a name, an icon, and a conventional

color that speciﬁes the MO type. Depending on the

latter aspect, the rectangle adopts different graphical

representations for content: music notation in the case

of symbolic content, piano roll for computer-driven

performances, and waveforms for audio (see Figure

5). Each rectangle embeds basic media controls to

play its content; in case of non-audio blocks, an inter-

nal synthesizer is invoked.

For each block, multiple ingoing and outgoing

connections are supported. Cords may belong to two

Formalisms and Interfaces to Manipulate Music Information: The Case of Music Petri Nets

Figure 3: The interface of ScoreSynth for Microsoft Windows.

Figure 4: Screenshot of the proposed web-based interface.

CHIRA 2018 - 2nd International Conference on Computer-Human Interaction Research and Applications

Figure 5: Graphical representations of audio, computer-driven performance, and symbolic MOs respectively.

Figure 6: Diagram of a “contrasting verse-chorus with bridge” song form, reconstructed by joining audio fragments obtained

from a segmentation process.

different types: a) concurrent arcs, represented by so-

lid lines, meaning that all concurrent outgoing paths

are traveled simultaneously, and b) exclusive arcs, re-

presented by dashed lines, implying that a path exclu-

des the others. Priority depends on numeric values

over arcs, that create a sorted sequence of arcs to be

navigated.

If no operator is invoked, the ﬁnal result will

be a graph of music excerpts without any modiﬁca-

tion. A simple example that illustrates a typical song

structure, speciﬁcally a contrasting verse-chorus with

bridge, is shown in Figure 6. In this case, MOs cor-

respond to high-level descriptions of the composition.

Please note that such an abstract representation con-

siders Verse 1 and Verse 2 as identical, whereas audio

signals (as well as score excerpts) would be typically

different, at least concerning lyrics.

The proposed framework supports another impor-

tant feature of MPNs, namely the availability of musi-

cal operators that can be applied to MOs in order to al-

gorithmically transform them into new MOs. Musical

operators can manipulate information concerning me-

lodic, harmonic and rhythmic aspects. Examples in-

volving symbolic content are transposition by a given

interval, inversion, calculation of the retrograde, aug-

mentation and diminution, and so on. Processes can

be applied in a selective way (e.g., “Transpose only

G-pitched eighth notes one octave below”) and they

can carry parameters (e.g., “Transpose all notes in the

sequence x halftones up”, with x ∈ N). Audio content

can be modiﬁed through different operators, such as

pitch shifting, time stretching, fade-in and fade-out,

ﬁlters, and other effects.

In MPNs, musical operators are typically linked to

transitions: when the transition ﬁres, the correspon-

ding musical operator is triggered, thus modifying

the MOs contained in input places and “passing” the

transformed MOs to output places. This process, for-

mally deﬁned in MPN theory, is hard to understand

for non experts. In the proposed interface, musical

operators are associated with blocks, and they can be

stacked one on top of the other. Rectangles, originally

conceived to carry only music content, now contain

tab views that allow to track step-by-step the transfor-

mations achieved by single musical operators. Figure

7 shows a clarifying example about cascading trans-

formations of audio content: a sine wave is ﬁrst trans-

posed one octave below (pitch shifting by a factor 0.5)

and then faded out after a given number of samples.

The ﬁgure illustrates the graphical aspect of the three

panels contained in the same block.

After consolidating the structure of the graph, the

second phase, called macro expansion, can occur. The

goal is to turn such a diagram into a MPN. In com-

puter science, a macroinstruction (or simply macro)

is a rule or pattern that speciﬁes how a certain input

sequence should be mapped to a replacement output

sequence according to a deﬁned procedure. Macro

expansion stands for the mapping process that instan-

tiates a macro use into a speciﬁc sequence.

Under our working hypotheses, macro expansion

is trivial. Each block of the original graph can be ea-

sily substituted by a suitable subnet. When musical

operators have to be applied, the corresponding algo-

rithms are implemented within transitions. The cords

become arcs that join subnets together, possibly weig-

hted by probabilistic values in order to support exclu-

sive paths.

The third phase, called post-processing, works on

MPNs resulting from the previous phase and seeks for

known structures in order to improve their representa-

tion. For example, if in the original graph a MO cal-

Formalisms and Interfaces to Manipulate Music Information: The Case of Music Petri Nets

Figure 7: Panels of a block containing musical operators, showing step-by-step transformations that content undergoes and

allowing to set parameters.

led A is followed by A

∗

, obtained from A by applying

some musical operators, the macro expansion would

create a place loading and playing A, followed by a

transition that enables a new place loading A again,

followed by a transition that ﬁrst applies the required

transformation on A (thus obtaining A

∗

) and then pas-

ses the modiﬁed music fragment to a new place that

performs it. Conversely, post-processing would pro-

vide a beautiﬁed MPN, more effective and compact

from a structural point of view: A would be loaded

into a sounding place only one time and played the

ﬁrst time as is, than a transition would modify it and

transfer the result back to the same place, thus obtai-

ning the performance of A

∗

In conclusion, even if the macro expansion phase

produces semantically correct PNs by applying very

simple conversion algorithms, post-processing can

signiﬁcantly improve PN structure, making some ad-

ditional music features emerge.

A complete example based on the nursery rhyme

called Fr

ere Jacques is shown in Figure 8. The tune

has been segmented into 4 fragments, called A, A

, B,

and C. Fragment A

can be obtained from A by dele-

ting the last note and transposing the remaining pit-

ches two grades up. The second voice is identical to

the ﬁrst one, with an offset of 2 measures; this voice

is triggered by the second instance of A in the ﬁrst

voice. Please note the use of both concurrent and ex-

clusive arcs, the latter with a priority value. Finally,

the MPN shown in Figure 8 is the result of the post-

processing phase, where the occurrence of two iden-

tical macro-expansions is recognized and encoded by

adding a backward arc from “Alg” to “Start”, and mo-

difying both the marking and the capacity of place

“StopMain” to 2 tokens.

6 CONCLUSIONS

In this work, we have outlined the functional and

graphical characteristics of a tool that lets users mani-

pulate music structures and content via a GUI, produ-

cing also a formal description in terms of music Petri

nets.

Recalling research question a (see Section 2),

from an early and limited experimentation performed

so far on a mock-up solution, the proposed appro-

ach seems to be effective also for people with no

speciﬁc training in Computer Science and Petri nets.

Answering research question b required to analyze

MPNs’ expressive power (e.g., the possibility to focus

on different degrees of aggregation/abstraction, orga-

nize structures thanks to the recursive use of subnets,

etc.), and consequently design a graph-oriented envi-

ronment where to place and connect blocks suppor-

ting most features of MPNs. Unfortunately, real-time

transformations of net topology and on-the-ﬂy substi-

tutions of music content are still posing practical pro-

blems to solve. Another unsupported feature, at the

moment of writing, is non-determinism. These as-

pects will be further explored in our future research.

Concerning research question c, the theoretical fra-

mework is ready to be implemented, and a release in

form of a browser application is under development.

As it regards future work, after completing the

implementation, this solution requires to be extensi-

vely tested and validated by multiple categories of

users (e.g., composers, performers, musicologists,

non-experts, etc.). Besides, its functions can be exten-

ded, for instance by supporting a language to imple-

ment user-deﬁned musical operators and improving

the post-processing phase.

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