“There is a natural and, to my mind, extremely beautiful mapping between concepts that have been around for hundred of years among musicians and concepts that are natural to mathematicians,” explains Dmitri Tymoczko, Assistant Professor of Music at Princeton. The mapping is so natural, he continues, that it’s not that much of a stretch to say that musicians have been doing geometry for the past two centuries without perhaps quite realizing it.
When you translate what musicians talk about into mathematical language, the geometry that results is a relatively recent one, he notes. Hence, perhaps the reason that his recent paper on Geometry and Music was accepted for publication in the journal Science, the first paper on music theory to be published in the prestigious journal in its 127 year history.
Not only are the ideas relatively new, they are also relatively simple, he adds, in that most people can appreciate them even without much background in music. So Dr. Tymoczko began his Lunch ‘n Learn talk, Geometry and Music, on March 29. During the talk, he presented the current state of music theory. He first illustrated a mapping of the linear keyboard with its discrete keys onto a circle that reflects the repeating structure of the keyboard. Like a clock, twelve points on the circle represent the repeating musical scale on the keyboard.
Collections of notes on the keyboard, chords can also be illustrated as collections of points on the circle. It was Jean-Philippe Rameau, a 18th century music theorist, who first proposed thinking about chords in this circular space. He postulated that major and minor chords could map onto the circle in consistent ways, and that all major and minor chords had relationships to one another. They formed similar triangles that are related by translation or rotation or transposition. The rotation preserves the distance between the notes. The other distance preserving transformation is a reflection or inversion, which happens to modify major chords into minor chords.
Tymoczko asserts that good counterpoint involve similar chords that divide the octave space relatively evenly and involve relatively little movement within this geometric space. Humans appear not to like counterpoint that involve dissimilar chords and which involve large leaps within the space.
Tymoczko also presented an alternative visual depiction of this same information in a two-dimensional configuration space. Rather than rely upon points on a one dimensional circle, he places ordered pairs of notes within in a two-dimensional grid. Mappings from one note to another are now formed by line segments. This alternative geometrical perspective beautifully illustrates Tymoczko’s principles. Good-sounding chords cluster in the center of the mapping, and western melodic music tends to involve short movements within the space. He is using these organizing principles to review the history of western music, viewing commonalities that might otherwise not be apparent.
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Posted by Lorene Lavora