Mathematical and Computational Modeling of Tonality: Theory and Applications

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Reljin N. Roig C. Rosner A. Shepard R.

Shmulevich I. Sigtia S. Temperley D.

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Toiviainen P. Tverdokhleb E. Zhou X. Article Tools Print the abstract. Indexing metadata. How to cite item. Supplementary files. Email this article Login required. Email the author Login required. Notifications View Subscribe. Font Size. Archives of Acoustics, 44 , 3, pp. The paper presents the key-finding algorithm based on the music signature concept.

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The proposed music signature is a set of 2-D vectors which can be treated as a compressed form of representation of a musical content in the 2-D space. Each vector represents different pitch class. Its direction is determined by the position of the corresponding major key in the circle of fifths.

Harmonic Structure Predicts the Enjoyment of Uplifting Trance Music

The length of each vector reflects the multiplicity i. The paper presents the theoretical background, examples explaining the essence of the idea and the results of the conducted tests which confirm the effectiveness of the proposed algorithm for finding the key based on the analysis of the music signature.

The developed method was compared with the key-finding algorithms using Krumhansl-Kessler, Temperley and Albrecht-Shanahan profiles. The experiments were performed on the set of Bach preludes, Bach fugues and Chopin preludes. Keywords : music information retrieval; computational music cognition; music data mining; music visualisation. Unfortunately, this technique does not scale well and would be impossible to embed in a quantitative algorithm as it would cause huge delay.

This problem is often referred to as the human fitness bottleneck [2]. Next, we could turn to music theory to obtain a set of rules that defines what is permissible in a given style.

Mathematical and Computational Modeling of Tonality

Music generation algorithms that rely on rules focus primarily on counterpoint, a style of music popular in the Renaissance period that has multiple harmonically interdependent voices [10, 23]. Counterpoint is one of the most formally defined musical styles; for most other genres of music, no such well-behaved set of rules exist.

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.

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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.

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.

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Figures reproduced from Chew , p. Figure 3. In order to work with music in a quantitative way, we need a mathematical representation. 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. 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. A study by Margulis [19] revealed that listeners rated unfamiliar music that had repetitions as more enjoyable, interesting and artistic than the original non-repeated version. With repetition, patterns begin to form; Plato alludes to the importance of patterns in music:. Thus, in MorpheuS, we aim to generate music with patterns having the same length and that recur at the same places as the original template piece.

In order to generate music with recurring patterns, we need to first find them. Using the elements discussed above, we can define music generation as an optimization problem. MorpheuS takes as an input a template piano piece. From the exemplar piece, the rhythmic structure is extracted and retained as a template.

The algorithm aims to find new pitches for each note in the original piece, and thus morph the piece into a new one. These patterns form hard constraints during the optimization so as to enforce perceptible musical motifs and to ensure long-term coherence.

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The objective of the algorithm is to generate music with a specific tension profile; the user can choose to have the algorithm match the tension profile of the template piece, or follow a user-given tension profile. To solve the multi-objective tension profile matching problem, we have implemented an efficient variable neighborhood search VNS metaheuristic based on the algorithm developed by Herremans [11] for generating counterpoint music.

This original VNS has been extended — with a move that changes multiple notes in a time slice — in order to handle more complex polyphonic music.