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/c² (m = E/c²). 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.