Chord Geometries

[junglelord]

ChordGeometries represents chords and voice leadings in a variety of 3D geometrical spaces. You can enter chords on a MIDI keyboard or using the Keyboard window. Voice leadings between successive chords are represented by continuous paths in the spaces. The program is meant to accompany the paper "The Geometry of Musical Chords" [Science 313 (2006): 72-74]. Further information can be found in "Generalized Voice-leading Spaces," with Clifton Callender and Ian Quinn. (Link temporarily removed pending publication.)

Three movies demonstrate the program, using the opening of Chopin's E minor prelude. The first shows Chopin's piece as it appears in circular "pitch class space." The second shows how the two pairs of voices chart a path on a Mobius strip. The third depicts Chopin's piece as it travels through a slice of the four-dimensional space containing seventh chords. You can also watch Deep Purple on the Mobius strip.
http://music.princeton.edu/~dmitri/ChordGeometries.html

That was totally rad.

Completely mind blowing for musicians who like to dabble in philosophical geometry.

Transposition = rotation.
Inversion = reflection.

Puts a whole new 'spin' on jamming.

It's amazing to actually see spatial relationships of the musical structures we feel in mindspace - and how aurally coherent music has a mathematical coherency that also 'looks nice' when expressed geometrically.

I also see parallels to human consciousness the in the way musical elements mathematically behave.

Melody behaves a lot like thought.
Harmony behaves a lot like emotion.
Rhythm behaves a lot like a heartbeat or a respiratory cycle.

Finally, everything that guy talked about seemed to conceptually resonate perfectly with the geometry of Treeincarnation and Justin's archetypal number articles.
http://www.breakfornews.com/forum/viewtopic.php?t=2253

[junglelord]

The Geometry of Musical Chords
Dmitri Tymoczko
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 chords possess different near-symmetries and suggest different musical uses.

Department of Music, Princeton University, Princeton, NJ 08544, USA, and Radcliffe Institute for Advanced Study, 34 Concord Avenue, Cambridge, MA 02138, USA.

E-mail: dmitri@princeton.edu
Read the Full Text
http://www.sciencemag.org/cgi/content/a ... siteid=sci