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Published online 17 April 2008 | Nature | doi:10.1038/news.2008.758
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Journeys in musical space
Researchers map out the geometric structure of music.
To most of us, a Mozart piano sonata is an elegant succession of notes. To composer and music theorist Dmitri Tymoczko of Princeton University in New Jersey and his colleagues, it is a journey in multidimensional space that can be described in the language of geometry and symmetry.
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The article states: "... playing ‘Somewhere Over the Rainbow’ in the key of G rather than, as originally written, the key of E flat, involves a different sequence of notes — but no one would say that it is a different song on that account...." This is true when modulating using equal temperament. However, from a practical standpoint modulation is more limited when using other temperaments, e.g. well-temperaments, meantone temperaments, various just intonations. Thus when performing Bach's Well-Tempered Clavier on an instrument tuned to well temperament, some keys produce wolf notes; or when playing in meantone temperament some keys are unplayable. I assume that Callender et al. considered the impact of different temperaments in their paper, and I wonder how temperament affects their analysis of the geometry of music.
Concerning the material about inversions under "Transforming sounds": "The researchers say there are five common kinds of transformation that are used in judging equivalence in music, including octave shifts, reordering of notes (for example, in inversions of chords, such as C-E-G and E-G-C), and duplications (adding a higher E to those chords, say). These equivalences can be applied individually or in combination, giving 32 ways in which, say, two chords can be considered ‘the same’." The sequence A-C-E-G is called an "A minor seventh", and is often heard as a variation of an A Minor chord in both classical and popular music. However, the sequence C-E-G-A, called a "C Major sixth" is often heard as a variation of a C Major chord, primarily in popular music, often with the C doubled. I'm sure the authors would agree that inversions have distinct differences in musical effect. But I'm claiming that the example I gave creates two essentially *different* chords...
This is all very interesting to theorists but my gut feeling is that this new analysis points to another "great grey mass" of compositions like the academic enthusiasm for twelve-tone compositions in the 20th century. There seem to always be these two currents in the western "classical" tradition - one heading toward more and more esoteric examination of the standard European tonality (as in twelve-tone compositions) and one toward renewing and deepening the experience of music by the audience, performers and composers, whatever the theory behind it (which I associate with composers such as Terry Riley and Erik Satie).
I wonder how this newborn science will exert influence on MUSIC, composing, listening, evaluating , tasting, enjoying..... and all other variances that this word and world could involve. Welcome to the potent auspicious baby.