# The idea of “sonification” in mathematical terms

In order to achieve a “sonification” of the standard model or based on the standard model, we took advantage of a happy coincidence. The fundamental particles (fermions) of the standard model are 12 (6 quarks: up, charm, top, down, strange, bottom and 6 leptons: electron, muon, tau, electron neutrino, muon neutrino, tau neutrino) while the notes of the chromatic scale are 12 (C, C#, D, D#, E, F, F#, G, G#, A, A#, B). Thus, we had to create a transformation that matches the 12 particles (Image 1) to the 12 notes of the musical scale (Image 2).

*Image 1. Particles of the Standard Model.*

*Image 2. Chromatic Musical Scale.*

In mathematical terms, we define two sets, the set of the particles and the set of the musical notes. We call P the set whose elements are the particles:

P = {u, d, c, s, t, b, e, μ, τ, ν_{e}, ν_{μ}, ν_{τ}}

We call N the set whose elements are the musical notes of the chromatic scale:

N = {C, C#, D, D#, E, F, F#, G, G#, A, A#, B}

So, we have two sets of the same cardinality (twelve elements each) (Image 3) and we are looking for an order.

*Image 3. Particles and notes.*

To carry out the transformation that will match the particles to the musical notes, we have to define a function f from P to N that will match every element of set P to an element of set N. Set P will be the domain and set N will be the range of the function f. Function f should be an injection (one-to-one), so that each element of the range is mapped to by at most one element of the domain as we don’t want two different particles correspond to the same note. Function f should be a surjection (onto), so that each element of the range is mapped to by at least one element of the domain as we don’t want any notes remain unmatched. Therefore, function f should be a bijection, i.e. a one-to-one and onto mapping between the two sets P and N. What remains is to define the rule under which each element of set P will match to a unique element of set N, i.e. each particle will match to a note. The characteristic of the particles that was selected is their mass measured in eV/c^{2} (m = E/c^{2}). We take an idea for the mass distribution among the particles from the following table (Image 4):

*Image 4. Approximate mass distribution.*

The rule was selected to be a correspondence between the mass and the pitch. More mass will correspond to heavier pitch, while less mass will correspond to higher pitch (Image 5).

*Image 5. Matching table.*

Therefore, the output values of function f are defined as follows:

f(t) = C

f(b) = C#

f(τ) = D

f(c) = D#

f(m) = E

f(s) = F

f(ν_{τ}) = F#

f(d) = G

f(u) = G#

f(e) = A

f(ν_{μ}) = A#

f(ν_{e}) = B

and the following table with the input and output values for function f is derived (Image 6):

*Image 6. Input and output values table. *

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