The Geometry of Musical Chords

@article{Tymoczko2006TheGO,
  title={The Geometry of Musical Chords},
  author={Dmitri Tymoczko},
  journal={Science},
  year={2006},
  volume={313},
  pages={72 - 74}
}
A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant… 
View on AAAS
dmitri.mycpanel.princeton.edu
237 Citations
Highly Influential Citations
18
Background Citations
72
Methods Citations
25
Results Citations
1

237 Citations

Generalized Voice-Leading Spaces
TLDR
This model reveals connections between music-theoretical concepts, yields new analytical tools, unifies existing geometrical representations, and suggests a way to understand similarity between chord types.
Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes
TLDR
The notion of compliance is introduced, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly, and some examples illustrate the use of this notion to characterize musical pieces and styles.
EIGHT SHORT MATHEMATICAL COMPOSITIONS CONSTRUCTED BY SIMILARITY
  • William Turner
  • Mathematics
    Far East Journal of Mathematical Sciences (FJMS)
  • 2018
Similar sounds are a formal feature of many musical compositions, for example in pairs of consonant notes, in translated passages, and in continuous musical lines. The aim of this article is to…
Building Topological Spaces for Musical Objects
TLDR
It is shown how a topological representation for n-note chords associated with the degrees of the diatonic scale and for the All-Interval Series can be automatically built using MGS, a rule-based spatial programming language.
Contemporary methods in mathematical music theory: a comparative case study
This article offers a direct comparison of several mathematical approaches to the study of musical structure that have come to prominence in recent years. The methods include chord-based and…
Comment on "The Geometry of Musical Chords"
Tymoczko (Reports, 7 July 2006, p. 72) proposed that the familiar sonorities of Western tonal music cluster around the center of a multidimensional orbifold. However, this is not true for all tonal…
Generating Music Using Concepts from Schenkerian Analysis and Chord Spaces
TLDR
A generative grammar for classical Western chord progressions using ideas from Schenkerian analysis is described, which gives the general structure of a chord progression over time, but is still an abstract representation that requires additional information to become a complete score.
Topology of musical data
TLDR
Several kinds of metrics on musical data are described, and the implications of these metrics in two ways are explored: via techniques of classical topology where the metric space of all-possible musical data can be described explicitly, and via modern data-driven ideas of persistent homology which calculates the Betti-number barcodes of individual musical works.
Three Conceptions of Musical Distance
This paper considers three conceptions of musical distance (or inverse “similarity”) that produce three different musico-geometrical spaces: the first, based on voice leading, yields a collection of…
Modes in modern music from a topological viewpoint
AbstractThe aim of this paper is twofold: on one side we review the classical concept of musical modefrom the viewpoint of modern music, reading it as a superimposition of a base-chord (seventhchord)…
...
...

References

SHOWING 1-10 OF 48 REFERENCES
Geometrical approximations to the structure of musical pitch.
TLDR
A two-dimensional "harmonic map," obtained by an affine transformation of the melodic map, provides for optimally compact representations of chords and harmonic relations; it is isomorphic to the toroidal structure that Krumhansl and Kessler (1982) show to represent the psychological relations among musical keys.
Harmonic Practice in Tonal Music
Taking a linear functional approach, Professor Gauldin uses clear explanations and outstanding musical examples to show students how individual chords function in the overall structure of a piece,…
Tuning, Timbre, Spectrum, Scale
The Octave Is Dead ... Long Live the Octave.- The Science of Sound.- Sound on Sound.- Musical Scales.- Consonance and Dissonance of Harmonic Sounds.- Related Spectra and Scales.- A Bell, A Rock, A…
Pitch as a medium: a new approach to psychophysical scaling.
TLDR
The results of two experiments designed to scale pitch by transposition methods lead to the conclusion that the musical scale, not the mel scale, describes the morphophoric function of pitch.
Music perception.
  • D. Deutsch
  • Art
    Frontiers in bioscience : a journal and virtual library
  • 2007
TLDR
It is shown that, for certain types of configuration, the music as it is perceived can differ substantially from the music that is notated in the score, or as might be imagined from reading the score.
Voice-Leading Spaces
This paper studies networks of pitch-classes that model voiceleading-the progression of pitches to form horizontal "voices" or "lines." While tonal voice-leading depends on distinctions between step…
Tone and Voice: A Derivation of the Rules of Voice-Leading from Perceptual Principles
TLDR
The traditional rules of voice-leading in Western music are explicated using experimentally established perceptual principles that distinguish different types of part writing, such as polyphony, homophony, and close harmony.
Exploring Musical Space
New mathematical approaches can elucidate abstract musical spaces and help our understanding of harmonic processes at work in musical compositions.
Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition
Connections between parsimonious structures and modes of limited transposition from three set classes are explored. A graph-theoretic approach proves useful in illustrating the symmetries inherent in…
As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert
been grateful to a dull description that faithfully guides me to the places where great artistic experiences await."' So rhapsodized Donald Francis Tovey at the end of his 1928 essay entitled…
...
...