http://denisevellachemla.eu/mai8-en.pdf
The spectrum of the sphere, as I said, is practically the integers, these are the numbers of the square root form of J(J + 1). Here if you look at the natural frequencies of the sphere, that gives you that, the frequencies, proper frequencies of the sphere ?
a point is given by correlations between different frequencies. So, you can think these correlations, these are complex numbers. But you can exactly think of these correlations between different frequencies like a chord, we take a chord between these notes and that’s it, a point, it’s that, okay. So think in your head, and keep a geometric object, a geometric
shape is given by its music, by its scale, it is given by its range, and the set of points
is given by the set of possible chords and a point is given by a chord.
it is the arithmetic fact that exists, which means
that if we take the number 2 to the power of a twelfth, if you take the twelfth root
of 2, that’s very, very close to the nineteenth root of 3.
See, I gave those numbers. You see that the twelfth root of 2 is 1.059..., etc. The
nineteenth root of 3 is 1.059... Where does 12 come from ?
The 12 comes from the fact that there are 12 notes when you make the chromatic
range. And the 19 comes from the fact that 19 is 12 + 7 and that the seventh note
in the chromatic scale, this is the scale that allows you to transpose. So what does
it mean ? It means that going to the range above is multiplying by 2 and the ear is
very sensitive to that. And transpose is multiplication by 3 [Perfect Fifth], except that it returns to the range before, i.e. so it is to multiply by 3 / 2, that agrees.
the object in question must be of dimension 0. ..there was an object, well known to mathematicians, that study non-commutative geometry or quantum things, which works for that thing and which gives you the right range.
It is none other than what is called the quantum sphere
S2 index q. So this object is therefore a more delicate object. It was considered in
particular by these three names (on the transparency are noted the names Poddles,
Brain and Landi). It has a spectrum, it has a spectrum. And this spectrum ? If you
choose the number q carefully, it will correspond exactly to the musical spectrum.
we will find the non-commutative sphere Sq 2 in nature and one will be able to use it as a musical instrument,
this definition was then replaced in 1983
by the current definition which using the speed of light as a conversion factor is
expressed in terms of inverse-frequencies rather than wavelength, and is based on a
hyperfine transition in the caesium atom. The advantages of the new standard are
obvious. No comparison to a localized “m`etre des archives” is necessary,
A VIEW OF MATHEMATICS Alain CONNES
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