creating and implementing a form space and denotator for...

Creating and Implementing a Form Space and Denotator for Bass Using

the Category-Theoretic Concept Framework

Introduction

• Both modern mathematical music theory and computer science are strongly influenced by the theory of categories and functors.

• The theory of categories and functors provides the framework for the design of the software system RUBATO COMPOSER.

Introduction

• In musical research software, the objects have to be named and defined in a recursive way.

• They also have to admit heterogeneous types, such as limits, colimits(of which products and coproducts are particular cases), power and list.

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

Definition of a Form

• A Form is a quadruple, (NF, TF, CF, IF) where

– NF is a string of unicode characters, called the Name of the Form.

– TF is one of the following:Simple, Power, Limit, and Colimit is called the Type of the Form.

Definition of a Form

– CF is one of the following, called the Coordinator of the Form

• For type Simple, CF is a module,

• For type Power, CF is a Form

• For types Limit and Colimit, CF is a diagram of Forms

• IF is a monomorphism of functors,

• IF : Fun(F) → X, IF in Mod@

Bass Score Form

Name Form Bass Score

GeneralNotes

SimpleNote

Definition of Denotator

• Let M be in Mod. An M addressed Denotator is a triple (ND, FD, CD) where:

– ND is a string of unicode characters, called the Name of the Denotator;

– FD is a Form, the Form of the Denotator;

– CD is an element of M@Fun(FD) and is called the Coordinates of the Denotator.

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


• 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 “mp

1”

and then they will be used to make “mp”, which is a composition of the two.


• 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.


• 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 mp

2,

except we will select affine instead of canonical.

mp

• To create mp, we bring up the module morphism builder, and create a module morphism with the domain of mp

2 and a

codomain of mp1.

• Which results in mp1○mp

2 = 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

@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

Run the program

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