Generalized Voice-Leading Spaces
Via music.princeton.edu
Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which both chords and chord sequences are represented. 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.
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Tags: analysis·system
Via technology.newscientist.com
Next time you are lost in an unfamiliar city, console yourself with the knowledge that the layout of its roads are probably much the same as in any other.
French and US physicists have shown that the road networks in cities evolve driven by a simple universal mechanism despite significant cultural and historical differences. The resulting patterns are much like the veins of a leaf.
Marc Barthélemy of the French Atomic Energy Commission in Bruyères-le-Châtel and Alessandro Flammini of Indiana University, US, analysed street pattern data from roughly 300 cities, including Brasilia, Cairo, Los Angeles, London, New Delhi, and Venice.
They found that cities’ road patterns have a lot in common mathematically, as well as looking similar to the eye.
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Tags: analysis·science·system
