Book contents
- Frontmatter
- Contents
- Mathematics: user's manual
- Appetizers
- 1 Space and geometry
- 2 Motions on the plane
- 3 The many symmetries of planar objects
- 4 The many objects with planar symmetries
- 5 Reflections on the mirror
- 6 A raw material
- 7 Stretching the plane
- 8 Aural wallpaper
- 9 The dawn of perspective
- 10 A repertoire of drawing systems
- 11 The vicissitudes of perspective
- 12 The vicissitudes of geometry
- 13 Symmetries in non-Euclidean geometries
- 14 The shape of the universe
- Appendix: Rule-driven creation
- References
- Acknowledgements
- Index of symbols
- Index of names
- Index of concepts
8 - Aural wallpaper
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Mathematics: user's manual
- Appetizers
- 1 Space and geometry
- 2 Motions on the plane
- 3 The many symmetries of planar objects
- 4 The many objects with planar symmetries
- 5 Reflections on the mirror
- 6 A raw material
- 7 Stretching the plane
- 8 Aural wallpaper
- 9 The dawn of perspective
- 10 A repertoire of drawing systems
- 11 The vicissitudes of perspective
- 12 The vicissitudes of geometry
- 13 Symmetries in non-Euclidean geometries
- 14 The shape of the universe
- Appendix: Rule-driven creation
- References
- Acknowledgements
- Index of symbols
- Index of names
- Index of concepts
Summary
All art constantly aspires to the condition of music.
W. Pater (1986: 86)[…] why do so many of us constantly try to explain the beauty of music, thus apparently depriving it of its mystery? The fact is that music is not only a mysterious and metaphorical art; it is also born of science. It is made of mathematically measurable elements […] And so any explication of music must combine mathematics with aesthetics […]
L. Bernstein (1976: 9) Reprinted by permission of the publisher. Copyright © 1976 by Leonard Bernstein.In 1893 Erik Satie composed Vexations, a musical piece consisting of a short theme – of about a minute and twenty seconds – which had to be played “840 times in succession” with the player(s) prepared beforehand “in the deepest silence, by serious immobilities”. A possible goal of this piece was, apparently, to bore his critics. In the trade-off between boredom and confusion Satie was structuring his piece to achieve the former.
Yet repetition is, in many forms and in smaller doses, an essential constituent of music, allowing themes to be re-expressed and patterns to be displayed. What is more, its many forms correspond to different geoemetric transformations whose use is neatly exemplified in the various kinds of canon, a form of imitative counterpoint common in the Baroque. They are also the building bricks which, via the notion of invariance, are used to obtain the different symmetries occurring in music.
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- Information
- Manifold MirrorsThe Crossing Paths of the Arts and Mathematics, pp. 188 - 224Publisher: Cambridge University PressPrint publication year: 2013