## A New Geometry of Musical Chords in Interval Representation: Dissonance, Enrichment, Degeneracy and ComplementationThis monograph covers a fresh and original look at musical chords. The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition. This condition itself leads to a convenient geometric representation of the chordal space as a simplicial grid. Chords appear as points in this grid and musical inversions of the chord would generate beautiful polyhedra inscribed in concentric spheres centered at the barycenter. The radii of these spheres would effectively quantify the evenness and thus the consonance of the chord. Internal symmetries would collapse these chordal structures into polar or equatorial displays, creating a platform for a thorough degeneracy study. Appropiate morphisms would allow us to navigate through different chordal cardinalities and ultimately to characterise complementary chords. |

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### Contents

1 | |

Interval Space | 11 |

Enrichments and Reductions | 29 |

Degenerated Chords | 41 |

Chord Complementation | 57 |

Appendix | 77 |

115 | |

### Other editions - View all

A New Geometry of Musical Chords in Interval Representation: Dissonance ... Miguel Gutierrez,Makoto Taniguchi No preview available - 2010 |

### Common terms and phrases

Assume barycentric chord barycentric coordinate Cayley-Menger determinant Chord Complementation chord types chords collapse circulant matrices co-I-symmetric consonance converting x2 coordinate system cyclic permutation matrices DDDD deﬁned deﬁnition degenerated chords é-radius eigenvalues eigenvectors Enrichment and reduction enrichment f epimorphism equatorial chords Euclidian space R3 Figure ﬁnd ﬁom ﬁrst form Forte Name Fourier matrix Given a chord Hankel matrix hexachord I-sym I-symmetric I-symmetry index I(PC idempotent Index f Index of dissonance inscribed interval notation interval space intervallic distances inversion argument inversion exponent major chord metric space minor triad musical chords n-tuple note chord Notice nth root o3 e o Orbit of chord orbital meshes original chord permutations and inversions pitch-class notation polar chord Prime form Forte properties Quotient space representation representative of PC respect to Q root of unity S1 equatorial satisﬁes simplex simplicial sphere symmetry tetrahedron vertices volume x2 x3 xn