 
Summary: Harmonic Analysis in Mathematics
James Arthur
Harmonic analysis in musicis the studyof chords,and of how
they are used in combination to create musical effects.Harmonic
analysis in mathematics takes on a somewhat different meaning. It
too has roots in music,or at least in the mathematical analysis of
sound.However,the term can also mean a kind of universal duality
that runs throughout mathematics,as well as other sciences.
Differentmusical instruments make different kinds of sound,
even when they play the samenote.The analysis of this phenomenon
can be very complicated,but to a first approximation,it is the shape
of the instrument that determines the sound it creates.Shape is an
obviousgeometric property.Sound is an example of what is often
called aspectralphenomenon.The term spectralhas a precise mean
ing in mathematics,but we can also think of it in the everyday
sense of the wordghostlike, hard to pin downthe very opposite
of geometric.We thus have an example of somethinggeometric
the shapeof a musical instrumentwhich correspondsto something
spectralthe sound produced by the instrument. Harmonicanalysis .
in the broadest sense refers to a general principlein mathematics
