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Additional skills in data science, optimization, statistics, scientific visualization, machine learning, cardiovascular signal processing or computational music perception and cognition will be appreciated. Fluency in spoken and written English is a requirement; ability to operate in French is a plus. A willingness to learn signal processing software and other dedicated languages is also expected. The candidate should be flexible and adaptable, and willing to learn and develop new skills. Requirements include a high level of competencies in the areas of formulating problems and arguing a case, performing scientific analyses, strong powers of reasoning, and an ability to summarize ideas and provide critical evaluation.

Candidates must have a positive attitude to working collaboratively in and managing a multi-disciplinary team. Hence, strong communication and organizational skills will be extremely useful.

Mathematical and Computational Modeling of Tonality - Theory and Applications - Semantic Scholar

IRCAM is a world renowned center for contemporary music production; it is organizationally linked to the Pompidou Center. The data resources will include MIDI recordings of legendary performances and anonymized electrocardiographic data. Short periods of travel within France and abroad, such as to conferences to present accepted peer-reviewed papers and posters, should be expected.


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Applications must include at the minimum a detailed CV; two references people who may be contacted ; a one-page cover letter; the abstract of the doctoral thesis; and, up to three samples of writing such as published articles or thesis reports. The closing date for sending applications is 1 September Interviews will take place in the latter half of September The expected start date is 15 October or later, depending on whether the candidate is French, European, or from a non-EU country.

Here are archives of the talk and concert on that day; audio recorded by James Weaver. A third, more robust technique is to use machine learning to construct the objective function.

Since the s, Markov models have been used to capture statistical properties of musical pieces and genres [24, 3]. While these models can capture many aspects and features of music, their traditional sampling methods e. More recently there has been interest in using deep neural networks for composition [16,17], yet the problem of constraining long-term structure remains. Herremans et al. The approach allows for the resulting music to be evaluated based on a machine-learned model, yet supports the use of hard constraints to fix a larger structure.

In our MorpheuS system [13, 14], this approach is further expanded to tackle complex polyphonic music and automatically detected recurring patterns. We also developed a way to combine machine learning and music theory to construct a new type of objective function based on musical tension.


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Figure 2: Spiral array pitch class, chord, and key representations: chords are weighted combination of their component pitches, and keys of their defining tonic rooted on the first scale degree , subdominant rooted on the fourth scale degree , and dominant rooted on the fifth scale degree. Figures reproduced from Chew , p.

Mathematical Modelling and Computation (MSc), DTU

Figure 3. In order to work with music in a quantitative way, we need a mathematical representation.

Computer Models of Music

Typically, music consists of a number of notes, each with properties such as pitch often expressed as a MIDI pitch , duration, start time and beat number. In the MorpheuS project, the music generation problem is re-cast as one of finding new pitches that satisfy certain structural constraints given a rhythm template extracted from an existing piece. Our goal is thus to morph a pre-existing piece into a new composition.

Almost all the music that we hear is tonal, i. The tension profile of a piece is one of the main defining characteristics that imbues upon the piece long-term structural, which we use as a structural constraint in the MorpheuS project. This tension profile is calculated based on the spiral array model for tonality [4]. The spiral array consists of multiple embedded helices see Figure 1 , representing tonal entities at three hierarchical levels: pitch classes, chords and keys.

The pitch class helix is effectively the line of fifths most simply explained as pitches having a simple frequency ratio wrapped around a cylinder so that pitch classes align vertically every four quarter turns. Chords and keys are each generated as convex combinations of their defining elements as shown in Figure 2. A real-time implementation of the spiral array and its tonal analysis algorithms has been used in live performances [5] to visualize tonal relationships in a music piece see Figure 3.

Expectation suspended elicits tension, and expectation fulfilled leads to resolution of that tension. The MorpheuS system aims to recreate the palpable tension that music generates; to this end, we have developed a model that captures tonal tension based on the spiral array [13].

The spiral array tension model first segments a piece into equal length subdivisions and maps the notes to clouds of points in the array. Three aspects of tonal tension are captured using these clouds: the cloud diameter measures the dispersion an indicator of dissonance of note clusters in tonal space; the cloud momentum measures the movement of pitch sets change in tonal context in the spiral array; and, tensile strain measures the distance between the local and global tonal context.

Each attribute can be visualized as tension ribbons on the score see Figure 5. Figure 4, shows a well-known tense chord in the spiral array: the German Sixth. It is noticeably spread out in the tonal space.

The Music Engagement Research Initiative

When this chord occurs, there is a noticeable increase in the tension ribbons representing cloud diameter and cloud momentum in the spiral array model. The tensile strain starts to increase slightly earlier, adding to the buildup of tonal suspense leading up to the German Sixth chord. Repetition forms another important aspect of music structure.

Perfect Balance: A novel principle for the construction of musical scales and meters, Mathematics and Computation in Music. Scratching the scale labyrinth, Mathematics and Computation in Music. Modelling the similarity of pitch collections with expectation tensors, Journal of Mathematics and Music. Tonal music theory: A psychoacoustic explanation? Tuning continua and keyboard layouts, Journal of Mathematics and Music.

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