The Planet-4D Model: An Original Hypersymmetric Music Space Based on Graph Theory

Baroin, Gilles

doi:10.1007/978-3-642-21590-2_25

Cited by 17 publications

(12 citation statements)

References 3 publications

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“…For convenience, we will denote the Fourier coefficients a 0 , a 1 , a c−1 instead of 1 1 1 A (0), 1 1 1 A (1), 1 1 1 A (c − 1). 3 We will need later the Lemma 1. For any real-valued map f , there is a symmetry between Fourier coefficients:…”

Section: Introductionmentioning

confidence: 99%

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The Torii of Phases

Amiot¹

2013

Mathematics and Computation in Music

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Section: Introductionmentioning

confidence: 99%

“…Besides, the Fourier transformation is isometrical for the hermitian metric f 2 = |f (k)| 2 up to a constant, according to Parceval's theorem. 3 Beware that the a k 's are not the elements of the pc-set. Besides, the indexes run from 0 to c − 1 but actually belong to the cyclic group Z c .…”

Section: Introductionmentioning

confidence: 99%

The Torii of Phases

Amiot¹

2013

Mathematics and Computation in Music

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“…With this measure, the hexachord theorem with either right or left interval content hold on S 3 . This may have interesting applications in visualization of musical structures on this hypersphere, see for instance [4].…”

Section: Some Examples Of the Generalized Hexachord Theoremmentioning

confidence: 99%

“…, as}n. 1 Figure 1: A well-known example of Z-related sets, in Z 12 .might notice that we define the iv function via the ifunc function borrowed from David Lewin [11]; Forte's original icv only features 6 values, because of inherent symmetries, e.g. for a diatonic scale the values of iv are [7,2,5,4,3,6,2,6,3,4,5,2] and Forte only keeps < 2, 5, 4, 3, 6, 1 > (the tritone is only counted once and the cardinality iv(0) is ommited). Since we generalize the notion to much more complicated groups than T (or T /I), and later to k−sets instead of couples of elements, it is convenient to keep the whole list instead of a reduced version.…”

mentioning

confidence: 99%

Z-relation and homometry in musical distributions

Mandereau

Ghisi

Amiot

et al. 2011

Journal of Mathematics and Music

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Archived PDF is a preprint.International audienceThis paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader's basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform

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“…In the paper by Childs, the dominant seventh code and the half-diminished seventh code are special, and this specialty has an important meaning in this paper. Embedding of the Tonnetz to hypersphere in 4 dimensions (the Planet-4D model) was also proposed (Baroin, 2011). Furthermore, an extension of the Tonnetz to N-tone equally tempered scales (for all N) and arbitrary triads was studied (Catanzaro, 2011).…”

Section: Introductionmentioning

confidence: 99%

General Theory of Music by Icosahedron 2: Analysis of musical pieces by the exceptional musical icosahedra

Imai¹

2021

Preprint

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We propose a new way of analyzing musical pieces by using the exceptional musical icosahedra where all the major/minor triads are represented by golden triangles or golden gnomons.First, we introduce a concept of the golden neighborhood that characterizes golden triangles/gnomons that neighbor a given golden triangle or gnomon. Then, we investigate a relation between the exceptional musical icosahedra and the neo-Riemannian theory, and find that the golden neighborhoods and the icosahedron symmetry relate any major/minor triad with any major/minor triad.Second, we show how the exceptional musical icosahedra are applied to analyzing harmonies constructed by four or more tones. We introduce two concepts, golden decomposition and golden singular. The golden decomposition is a decomposition of a given harmony into the minimum number of harmonies constructing the given harmony and represented by the golden figure (a golden triangle, a golden gnomon, or a golden rectangle). A harmony is golden singular if and only if the harmony does not have golden decompositions. We show results of the golden analysis (analysis by the golden decomposition) of the tertian seventh chords and the mystic chord. While the dominant seventh chord is the only tertian seventh chord that is golden singular in the type 1 * and the type 4 * exceptional musical icosahedron, the half-diminished seventh chord is the only tertian seventh chord that is golden singular in the type 2 * and the type 3 * exceptional musical icosahedron. Also, from the viewpoint of the golden decomposition, the minor major seventh chord can be regarded as a generalized major-minor dual of the augmented major seventh chord with some transpositions and interchanges. In addition, all the harmonies constructed by five or more tones are not golden singular.Last, we apply the golden analysis to the famous prelude in C major composed by Johann Sebastian Bach (BWV 846). We found 7 combinations of the golden figures (a golden triangle, a golden gnomon, two golden triangles, two golden gnomons, a golden rectangle, a golden rectangle and a golden triangle, a golden rectangle and a golden gnomon) on the type 2 * or the type 3 * exceptional musical icosahedron dually represent all the measures of the BWV 846.

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The Planet-4D Model: An Original Hypersymmetric Music Space Based on Graph Theory

Cited by 17 publications

References 3 publications

The Torii of Phases

The Torii of Phases

Z-relation and homometry in musical distributions

General Theory of Music by Icosahedron 2: Analysis of musical pieces by the exceptional musical icosahedra

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