Origins in PRESTO, and early computer application developed by Guerino Mazzola.
RUBATO is a universal music software
environment developed since 1992 under the direction of Guerino Mazzola.
RUBATO COMPOSER system developed in Gérard Milmeister’s doctoral dissertation (2006) where he implemented the Functorial Concept Architecture, based on the data format of Forms and Denotators. http://www.rubato.org/
This software works with components called rubettes ( perform basic tasks for musical representation and whose interface with other rubettes is based on the universal data format of denotators).
The data format of denotators uses set-valued presheaves over the category of modules and diaffine morphisms http://www.rubato.org/
In addition to what RUBATO COMPOSER is designed to be for the composer and music theorist, it is also an excellent tool for learning sophisticated mathematical concepts.
The mathematics involved are sophisticated, and could be accessible in a formal way to the average mathematics student in their senior year, after having some experience with courses such as Linear Algebra, Modern Algebra, Analysis or Topology, but would usually be taught at the graduate level.
Possibilities of knowledge expansion and new applications
Danger of superficiality, contamination and surrender to fashion.
Possibilities that Mathematical Music Theory, and its applications, have to offer to the knowledge base of mathematics, music, and computer science students, without excluding those from other fields.
It has been generally acknowledged that there is a gap between the formality of modern mathematics as conceived and taught by trained mathematicians, and the mathematics that are seen by non mathematicians as relevant.
When the mathematics are embedded in different practical contexts, it is often easier to get students to think mathematically in a natural manner
Even mathematics students themselves often have difficulty in making meaning out of the formal presentation of their subject (MacLane, 2005).
The creation of interdisciplinary curriculum materials and courses, using RUBATO COMPOSER as a common ground, opens a realm of possibilities for Mathematical Music Theory, and for the development of research and researchers in the field. It also can be justified, in and of itself, where RUBATO COMPOSER is conceived of as a learning tool.
The International Journal of Computers for Mathematical Learning “… publishes contributions that explore the unique potential of new technologies for deepening our understanding of the field of mathematics learning and teaching.”
A revision of articles from 2006-8, illustrates that the notion of using specific software to enhance the learning of mathematics has a respectable recent history, and has been analyzed using well documented research paradigms.
Using formalism to construct meaning is a very difficult method for students to learn, but this is the only route to learning large portions of mathematics
The writing of computer programs to express mathematical concepts can be an effective way of achieving this goal of advanced mathematical learning.(Dubinsky, 2000)
The RUBATO COMPOSER software opens up the possibility of creating meaning behind the formalisms of advanced mathematical areas, and accelerating processes of learning and understanding.
These mathematical areas (Abstract Algebra beyond Group Theory, Category and Topos theory) are not really addressed in the literature on computer-based learning, or on collegiate mathematics education in general.
Computer-based learning in music is usually related to training in aural skills, sight reading, and other subjects essential to the music student.
Musical representation languages such as Common Music, OpenMusic, and Humdrum, for composing and analyzing music, that do require programming skills.
However, RUBATO COMPOSER offers the opportunity of introducing the music student to the higher mathematics involved in modern Mathematical Music Theory.
This can be done in a relatively (not completely) “painless” manner, as compared to what it would require to learn this material in the traditional way.
RUBATO COMPOSER is based on the data format of Forms and Denotators.
Forms are mathematical spaces with a precise structure, and Denotators are objects in the Form spaces.
Category theory is the mathematical foundation on which this particular conceptual basis of Mathematical Music Theory is built.
In the RUBATO COMPOSER architecture,
modules are a basic element, much like primitive
types in programming languages.
The recursive structure of a Form, if not circular,
will eventually “stop” at a Simple form which, for
all practical purposes, is a module.
Morphisms between modules (changes of
address), are built into the software.
In the development of the data base management systems, the objects must be named and defined in a recursive way and they must admit types that, in this context, are such as limit, colimit, and power.
It is necessary to work with the algebraic structure of modules, yet form constructions whose prototypes are found in the category of sets.
This is the reason why, in the context of RUBATO COMPOSER, the approach is to work in the functor category of presheaves over modules (whose objects are the functors ).
Through the creation of denotators, and the recursive structure of types when working with forms, the mathematics student accustomed to the formalism of abstract mathematics has the opportunity to participate in a concrete implementation of these concepts.
The mathematics student who still struggles to find meaning in the abstract formalism, may find a vehicle through which this process can be accelerated.
The majority of rubettes available at this time are of low level nature.
One of the objectives of the developers of RUBATO COMPOSER is to create more high level rubettes that present ‘friendlier’ interfaces and language for the non-mathematical user, in particular the composer or musicologist.
However, musicians interested in using technology in an innovative manner, cannot isolate themselves from the mathematics used to create their tools.
The musical analysis itself, and much of the musical ontology, is intricately related to the mathematical framework.
The music student does not have to deal with the mathematical objects in the same way as the mathematics student (nor does the computer science student). However, if he wants to follow the developments of research in Mathematical Music Theory, he needs an understanding of the language and concepts behind the tools.
This is especially true in the case of RUBATO COMPOSER, which has been designed as the result of a precisely defined, and perhaps revolutionary, approach to musical analysis.
Even with the high level rubettes that are, and will be, available, it is possible to retrace the steps and uncover the mathematics behind their construction.
When the terminology changes from ‘transposition’ to ‘translation’ and, in general, from the musical ‘inversion’, ‘retrograde’, ‘augmentation’, to the language of mathematical transformations, or morphisms, the music student is presented with an opportunity to develop an understanding of the meaning behind the formalism.
In RUBATO COMPOSER not only translations, but general affine morphisms as well, can be used to generate musical ornamentation.
The Wallpaper rubette, developed by Florian Thalmann, also opens the possibility of generating morphisms in any n-dimensional space (for example, using the five simple forms of the Note denotator - Onset, Pitch, Duration, Loudness and Voice- an affine transformation in 5D can be defined).
When working with affine transformations in 2D space, the command can be given by just ‘dragging’, instead of defining the morphisms.
A unit introducing the basic concepts of linear algebra, group theory, and geometry needed to understand mathematical music theory, as it has been developed over the last 40 years, can be created.
Most ‘extreme’ example, up until now, the BigBang rubette, developed by Florian Thalmann, in the context of Mathematical Gesture Theory and Computer Semiotics.
Based on a general framework for geometric composition techniques.
Given a set of notes, their image is calculated through affine invertible maps in n-dimensional space.
There is a theorem that states that the affine invertible map in n dimensions can be written as a composition of transformations, each one acting on only one or two of the n dimensions.
The component functions (act on only one or two of the n dimensions) represented geometrically as five standard mathematical operations that have their musical representation:
Translation (transposition in music)
Reflection (inversion, retrograde in music)
Rotation (retrograde inversion in music)
Dilations (augmentations in music)
Shearings (arpeggios).
Sample of a Unit and its Focus on different Majors
A sample unit has been created to show how the
analysis and creation of a musical object can give students from different disciplines, in particular mathematics, computer science and music, a deeper understanding of abstract mathematics while satisfying aesthetic interests as well.
Description of the Module: The development of a melodic phrase, recursively transformed by transformations in the plane as ornamentation, using the Wallpaper rubette in RUBATO COMPOSER
Objectives and Activities:
All students will be able to:
Identify rigid transformations in the plane and give them musical meaning. For example:
mathematical translations – musical transpositions;
mathematical reflection – musical inversion, retrograde;
mathematical rotation – musical inversion-retrograde;
mathematical dilatation – musical augmentation in time;
mathematical shearing – musical arpeggios in time.
All students will be able to:
Use the software RUBATO COMPOSER and, in particular, the Wallpaper rubette, to generate musical ornamentation by means of diagrams of morphisms (functions).
Create and interpret transformations, and compositions of transformations, like the following, in which is a rotation of 180 followed by a translation, and is a translation.
1f
2f
Select any of the coordinates of a Note denotator (which is 5-dimensional) and combine them.
When two coordinates are chosen, say Onset and Pitch, students will relate them to the rigid transformations in the Euclidean plane.
Mathematics students (and those from computer science and other related areas) will formally construct the morphisms, while Music students can use the succession of primitive transformations by dragging with the mouse
Mathematics students will be able to: Construct the composition of module morphisms from the Form
Note to the Form Note. For example, using the coordinates Onset and Pitch, they can construct the following composition of embeddings, projections and affine transformations.
The creation of a melodic phrase where and are the injections and is the embedding . The transformation (a
musical embellishment, as seen in the previous slide) is then applied, and to return the coordinates to the module morphisms
and the projections and are applied where c represents quantized to
1i 2i
2
2e
nonp
1p2p
= ○ f ○ (( ○ o) + ( ○ ○ p)): A →
= c ○ ○ f ○ ((i1 ○ o) + ( ○ ○ p)): A →
no1p
1i 2i 2e
2p 2i 2enp
General Topic Mathematics Highlights Music Highlights
Transformational Theory Group Theory, Set Theory,
Function Theory, Geometry,
Topology
Analysis of Musical Works,
from all genres. In the case of
Classical music, analysis of
modern atonal music that is
not approachable with
traditional tools from music
theory.
Subdivision of the Octave;
Maximally Even Sets
Group Theory, Number
Theory, Differential Equations,
Continuous Fractions
Exploration of different and
exotic scales; Composition
using unusual scales.
Forms and Denotators; The
Software RUBATO
COMPOSER
Category and Functor Theory,
Topos Theory, Sets and
Modules, Linear, Affine and
Diaffine Transformations.
Linear Algebra, Geometry,
Mathematical Gesture Theory
Music Composition;
Embellishment of existing
music; Algorithms for
Composition; Counterpoint
Theory
Mazzola, G, Milmeister, G, Morsy, K., Thalmann, F. Functors for Music: The Rubato Composer System. In Adams, R., Gibson, S., Müller Arisona, S. (eds.), Transdisciplinary Digital Art. Sound, Vision and the New Screen, Springer (2008).
Milmeister, Gérard. The Rubato Composer Music Software Component-Based Implmentation of a Functorial Concept Architecture. Springer-Verlag (2009).
Thalmann, Florian and Mazzola, Guerino. The BigBang Rubette: Gestural Music Composition with Rubato Composer ICMC 2008 http://classes.berklee.edu/mbierylo/ICMC08/defevent/papers/cr1316.pdf
Thalmann, Florian. Musical Composition with Grid Diagrams of Transformations, Masters Thesis, Bern (2007)
International Journal of Computers for Mathematical Learning, http://www.springer.com/education/mathematics+education/journal/10758
Dubinsky, E.: Meaning and Formalism in Mathematics, Int J Comput Math Learning, 5, 211-240 (2000).
MacLane, Saunders. Despite Physicists, Proof is Essential in Mathematics. Synthese 111, 2, 147-154 (May, 1997).
Creating and Implementing a Form Space and Denotator for Bass Using
the Category-Theoretic Concept Framework
The Dilemma
• The dilemma resides in how to maintain the algebraic structure of the category of modules(over any ring, with diaffine morphisms) and, at the same time, construct such objects as limits, colimits, and power, and classify truth.
The Dilemma
• The functorial approach leads to the resolution of this dilemma by working in the category of presheaves over modules (whose objects are the functors F: Mod → Sets, and whose morphisms are natural transformations of functors) and which will be denoted as Mod@.
• This category is a topos, which means that it allows all limits, colimits, and subobject classifiers Ω, while retaining the algebraic structure that is needed from the category Mod.
My Research
• My research consists of creating a form broad enough for the majority of simple electric bass scores, and a denotator which represents the jazz song “All of Me”. The recursivity of the mathematical definitions of form and denotator are made evident in this application
Bass Score Form
Name Form Bass Score
GeneralNotes
SimpleNote
Denotator “SimpleNote”
• For an example of how to build a denotator I will take a small part of my denotator named “SimpleNote”
Creating the Denotator of “All of Me”
• A single denotator N1 of the form: SimpleNote is created from the coordinates of the denotator which themselves are forms of type simple: Onset, Pitch, Duration, Loudness, and Voice.
Creating the Denotator of “All of Me”
• As we don't have time to see how all of the module morphisms are constructed we will build the pitch module morphism “mp”.
• All the others are built analogously.
Bassline for “All of Me”
Creating the Denotator in Rubato Composer
• To create the denotator for the Jazz song “All of Me” we must first create the Module Morphisms
Creating the Denotator in Rubato Composer
• Once we've opened our module morphism builder in Rubato Composer we will start creating a module morphism for pitch.
• First we must create “mp2” and “mp1”
and then they will be used to make “mp”, which is a composition of the two.
Creating the Denotator in Rubato Composer
• For mp2 the domain is determined from the number of musical notes in the bass line.
• For instance, the bass line I wrote for “All of Me” contains 64 notes.
• We establish the first note as the anchor note, so for our domain we use Z63.
Creating the Denotator in Rubato Composer
• mp2 is an embedding of the canonical vectors plus the zero vector, that goes from Z63 → Q63.
mp1
• mp1 will take us from Q63→ Q.
• We will set up the module morphism in the same way as mp2, except we will select affine instead of canonical.
mp1
mp
• To create mp, we bring up the module morphism builder, and create a module morphism with the domain of mp2 and a codomain of mp1.
• Which results in mp1○mp2 = mp.
mp
mp
• mp: Z63 → Q63 → Q: x → (4, 7, 9, 7, 4, 0, 2, 4, 8, 11, 8, 4, 2, -1, -4, -3, -1, 1, 4, 7, 6, 5, 4, 2, 5, 9, 12, 11, 9, 5, 2, 4, -8, 3, 4, 8, 4, 0, -1, -3, -1, 0, 4, 9, 7, 5, 4, 2, 6, 12, 11, 9, 6, 0, 2, 5, 9, 2, 6, 7, -5, -1, 2)●x + 48
• Example, when x=(0,...,0)
(4,..., 2)●(0,...0) + 48 → 0+48 → 48
Which is the first note in our bassline.
Module Morphisms
• All of the Module Morphisms are made in the same way.
• Once all of the Module Morphisms are built we can arrange them using the denotator builder.
Denotator Builder
Creating Denotator
Using Rubettes in Rubato Composer
• To play our bass line in Rubato Composer, we must create a network using rubettes.
• We will need to set up three rubettes; the Source rubette, the @AddressEval rubette, and the Scoreplay rubette.
Source
Source
Source
@AddressEval Rubette
• Next open the @AddressEval Rubette.
• This rubette will be directly connected to the Source rubette.
Scoreplay Rubette
• To play our bassline we need to open the scoreplay rubette and connect it to the @AddressEval rubette.
• This is done the same way as the Source and @AddressEval rubettes.
Finished Network
Pianola
Rubato Composer and its
Functorial Approach:
From Morphisms to Gestures through
Rubettes
Jonathan Cantrell
Junior Mathematics
Georgia State University
Perspectives on Music
Where did we start?
Melodies and harmonies applied across time
Often written in sequential fashion
Where are we now?
Asynchronous editing tools allows a composer to
work in non-linear fashion
Digital Audio Workstations e.g Pro Tools
MIDI Sequencers e.g Logic, Cubase
Still music is written in distinct phrases and compiled
together
Architecture of Rubato Composer
Defined recursively
Utilizes the Form and Denotator concept
Allows for heterogeneous types
Simple, Limit, Colimit, or Power
Implemented as free modules over rings
Working from the category of presheaves over
modules
Diaffine Transformations
Working in the category Mod of modules over
any ring whose morphisms are diaffine
transformations which gives us the ability to
perform operations from one module to another
Diaffine transformations are module morphisms
plus a translation
A dilinear homomorphism from an R-module M to an
S-module N plus a translation in N
Why Topoi?
Rigidly defined categories which are a
generalization of the category of Sets
Allows the composer to perform set-valued
operations where the elements in the sets are
module morphisms over any ring
Sets are required within the framework of the
Form and Denotator concept as the evaluation
of the colimit of Score:Note
Geometric Representation
Module morphisms in n-dimensional space
embedded into a Form of type Simple
Any diaffine transformation h in n-dimensional
space can be written as a composition of
transformations hi which involve only one or two
dimensions of the n dimensions and leave the
others unchanged
In our particular example, we are in ℝ5 we want
to operate exclusively on pitch and onset,
therefore we apply this construct to work in ℝ2
Why Mod?
Why does the algebraic structure need to be
retained?
It allows us to map individual Simple Forms to
the plane and perform affine transformations in
a different space, as exemplified in the
Wallpaper rubette
We apply the mathematical tools of translation
as musical transpositions, and reflections as
retrograde
My Research
I have developed an example of a 12-note
melodic phrase recursively transformed using
the Wallpaper rubette. I further generalize this
series of transformations using the high-level
tool of the BigBang rubette available in Rubato
Composer
Compound Transformations
An example of two translations applied
recursively.
For a specific Note denotator, we operate
exclusively on the module morphisms Onset
and Pitch
Compositions
As stated, the module morphisms contained in
the Simple forms Onset, o: A → ℝ and Pitch,
p: A → ℚ are extracted from the Note denotator
We now need to compose these module
morphisms as follows
Compostitions
This is described by the following compositions:
i₁ ○ o : A → ℝ2,
i₂ ○ e₂ ○ p : A → ℝ2,
Where i₁ and i₂ are the injections ℝ → ℝ2, and e₂ is the
embedding ℚ → ℝ.
In order to combine these two morphisms into a
single instance of ℝ2 we must sum them such
that onset and pitch become respective axes in
ℝ2
Compositions
The transformation f is then applied, and finally,
to return the coordinates to the module
morphisms on and p
n, we apply the projections
p1 and p
2 as follows where c represents ℝ
quantized to ℚ
on = p
1 ○ f ○ ((i
1 ○ o) + (i
2 ○ e
2 ○ p)): A → ℝ
pn = c ○ p
2 ○ f ○ ((i
1 ○ o) + (i
2 ○ e
2 ○ p)): A → ℚ
Tracing the modules on which these
compositions take place we have
onset: ℤ11 → ℝ11 → ℝ → ℝ2 → ℝ2 → ℝ
pitch: ℤ11 → ℚ11 → ℚ → ℝ → ℝ2 → ℝ2 → ℝ → ℚ
Morphisms to Gestures
The Wallpaper rubette is an example of a low-
level process
We are working in a very mathematical context
For the composer, this will not always be
appropriate, as mathematics may be a means
rather than an end
For this reason Gesture Theory is being
developed by Dr. Guerino Mazzola and Florian
Thalmann as implemented in the BigBang
rubette
Gestures as curves in topological space
Future Applications
The Rubato Framework gives the composer an
alternate view of composition, working from a
functorial perspective
Also the musician can gain insight into a branch
of mathematics using intuition as a guide
opening up exciting educational avenues
The highly characterizable nature of the
category theoretic framework opens up the
opportunity for any system to modeled
effectively
Bibliography
– Milmeister, Gérard. The Rubato Composer Music Software: Component-Based Implementation of a Functorial Concpet Architecture Zürich: 2006
– Thalmann, Florian and Mazzola, Guerino. The BigBang Rubette: Gestural Music Composition with Rubato Composer
– Thalmann, Florian. Musical Composition with Grid Diagrams of Transformations Bern: 2007
– “Pro Tools.” http://www.digidesign.com/
– “Logic.” http://www.apple.com/logicstudio/
– “Cubase.” http://www.steinberg.net