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 f 1 followed by a translation, and is a translation. f 2
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. i The creation of a melodic phrase where and are the injections i 1 2 2 e and is the embedding . The transformation (a 2 musical embellishment, as seen in the previous slide) is then applied, and to return the coordinates to the module morphisms and o p n n the projections and are applied where c represents p p 1 2 quantized to
o = ○ f ○ (( ○ o ) + ( ○ ○ p )): A → p i e i n 1 2 2 1 p = c ○ ○ f ○ (( i 1 ○ o ) + ( ○ ○ p )): A → i e p n 2 2 2
General Topic Mathematics Highlights Music Highlights Transformational Theory Group Theory, Set Theory, Analysis of Musical Works, Function Theory, Geometry, from all genres. In the case of Topology Classical music, analysis of modern atonal music that is not approachable with traditional tools from music theory. Subdivision of the Octave; Group Theory, Number Exploration of different and Maximally Even Sets Theory, Differential Equations, exotic scales; Composition Continuous Fractions using unusual scales. Forms and Denotators; The Category and Functor Theory, Music Composition; Software RUBATO Topos Theory, Sets and Embellishment of existing COMPOSER Modules, Linear, Affine and music; Algorithms for Diaffine Transformations. Composition; Counterpoint Linear Algebra, Geometry, Theory Mathematical Gesture 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.p df 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+educatio n/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 N 1 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 “ mp 2 ” and “ mp 1 ” and then they will be used to make “ mp” , which is a composition of the two.
Creating the Denotator in Rubato Composer • For mp 2 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 Z 63 .
Creating the Denotator in Rubato Composer • mp 2 is an embedding of the canonical vectors plus the zero vector, that goes from Z 63 → Q 63 .
mp 1 • mp 1 will take us from Q 63 → Q. • We will set up the module morphism in the same way as mp 2 , except we will select affine instead of canonical.
mp 1
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 mp 1 . • Which results in mp 1 ○ mp 2 = mp .
mp
mp • mp : Z 63 → Q 63 → 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 h i which involve only one or two dimensions of the n dimensions and leave the others unchanged In our particular example, we are in ℝ 5 we want to operate exclusively on pitch and onset, therefore we apply this construct to work in ℝ 2
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: i ₁ ○ o : A → ℝ 2 , i ₂ ○ e ₂ ○ p : A → ℝ 2 , Where i ₁ and i ₂ are the injections ℝ → ℝ 2 , and e ₂ is the embedding ℚ → ℝ . In order to combine these two morphisms into a single instance of ℝ 2 we must sum them such that onset and pitch become respective axes in ℝ 2
Compositions The transformation f is then applied, and finally, to return the coordinates to the module morphisms o n and p n , we apply the projections p 1 and p 2 as follows where c represents ℝ quantized to ℚ o n = p 1 ○ f ○ (( i 1 ○ o ) + ( i 2 ○ e 2 ○ p )): A → ℝ p n = 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: ℤ 11 → ℝ 11 → ℝ → ℝ 2 → ℝ 2 → ℝ pitch: ℤ 11 → ℚ 11 → ℚ → ℝ → ℝ 2 → ℝ 2 → 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
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